1

THEEUMAEUSGUIDETOEQUITYRELEASEVALUATIONRestatingtheCaseforaMarketConsistentApproach

DeanBucknerandKevinDowd[Thisversion:15July2019]

ExecutiveSummaryAcknowledgementsDisclosureStatementPARTONE:NNEGBASICSChapter1:IntroductionChapter2:OriginandHistoryChapter3:TheBasicsofNNEGandERMValuationChapter3Appendix:TheChoiceofOptionPricingModelPARTTWO:KEYINPUTSChapter4:Loan-to-ValueRatioChapter5:Risk-FreeRateChapter6:LoanRateChapter7:NetRentalandDefermentRates:TheoryChapter8:NetRentalandDefermentRates:CalibrationChapter9:DilapidationChapter10:VolatilityChapter 10 Appendix 1: A Hurst Exponent Approach to Autocorrelation in PropertyPricesChapter10Appendix2:ProofofEquation(10.5)Chapter10Appendix3:ProofofApproximation(10.13)Chapter10Appendix4:ProofofEquation(10.14)Chapter11:Mortality

2

Chapter12:Long-TermCareChapter13:DelayedPossessionChapter14:CreditSpreadsChapter15:DrawdownChapter16:PrepaymentChapter17:FeesandChargesPARTTHREE:APPLICATIONSChapter18:ScenarioAnalysisandStressTestingChapter19:ThePRA’sGoodPracticePrinciplesPARTFOUR:COMPETINGAPPROACHESTOEQUITYRELEASEVALUATIONChapter20:TheMarketConsistentApproachChapter20Appendix:AWorkedHedgeExampleChapter21:MisconceptionsAbouttheMarketConsistentApproachChapter22:The‘DiscountedProjection’ApproachChapter23:TheTunaruReportChapter24:Just’sNNEGValuationModelPARTFIVE:PROFESSIONALSTANDARDSANDRECOMMENDATIONSChapter25:TechnicalActuarialStandardsChapter26:AccountingStandardsPARTSIX:RECOMMENDATIONSChapter27:RecommendationsforGoodValuationPracticeChapter28:RecommendationsforGovernance

3

ExecutiveSummary

• TheUKequityreleasesectorispermeatedbypoorvaluationpractice:asfarasweareaware,notasingleequityreleasefirmisvaluingitsNo-NegativeEquityGuarantees(NNEGs)inascientificallyvalidmanner.

• This NNEG under-valuation problem is on a large scale and impliescorrespondinglylargeover-valuationsofEquityReleaseMortgages(ERMs).

• TheDiscountedProjectionor ‘RealWorld’approachusedbytheequityreleaseindustryisinherentlyflawedandproducesvaluationsthatviolateboundsthatareknowntobeinviolable.

• TheonlyscientificallyvalidvaluationapproachistheMarketConsistentapproach,

which is also the only approach compatible with accounting principles andtechnicalactuarialstandards.

• Market consistent valuations cast doubt on the profitability of ERM loans

especiallytoyoungerborrowers.

4

Acknowledgements

Wethankthemanypeoplewhohavekindlycontributedtothisreportthroughhelpfuldiscussions on the topics coveredhere or through comments on earlier versions.Wethank inparticular:DavidBlake, ChrisCundy, JonathanFord, CharlesGoodhart, TonyJeffery,HowardMustoe,GordonKerr,CavinO’Driscoll,AndrewSmith,JohnSkar,CraigTurnbull,membersofthePRA’sCP7.19team,seminarparticipantsattheLondonSchoolofEconomicsandQueenMaryUniversityofLondon,andvariouscorrespondentswhohaveofferedcommentsinprivate.Theusualcaveatapplies:wealoneareresponsibleforanyviewsexpressedhere,andforanyremainingerrors.DisclosureStatementThisreporthasnotbeenfundedbyanyresearchorganisationorcommercialbody,andwehavenotsoughtanysuchfunding.Dowd is currently involved as a co-investigator in the Actuarial Research Council’s“Modelling,MeasurementandManagementofLongevityandMorbidityRisk”researchprogramme.Formoredetails,see:https://www.actuaries.org.uk/learn-and-develop/research-and-knowledge/actuarial-research-centre-arc/research-programmes/modelling-measurement-and-management-longevity-and-morbidity-risk/research-team

5

ChapterOne:Introduction

“Idourgeyoualltokeepanopenmind…becausewe[actuaries]don'talways- sometimeswe've got a very narrowway of thinking, but I do think thatthere'smoretogo[onNo-NegativeEquityGuaranteevaluation],andthismaygoonforlongerthantheBrexitdiscussions.”

GinaCraske1

IntheEquityReleaseCouncil’sSpring2018MarketReport,itschairmanDavidBurrowesstruckareassuringtone:

Annuallendingactivitybyourmembershassurpassed£3billionforthefirsttime and customer numbers reached 67,000 in 2017. Property wealth isincreasingly recognised by people as a safe and sought-after source ofretirementfinance,withthemarketattractingtwiceasmanynewcustomersasitwasfiveyearsago.…Therangeofproductoptionsavailabletoequityreleasecustomershasgrown25% year-on-year, providing more choice to underpin a robust andcompetitivemarket.Lookingforward,weexpecttheneedfornewsourcesofincomeinretirementwill continue to grow asmany peoplewill be unable to rely on pressuredpensionpots.(EquityReleaseCouncil,2018,p.2)

Mr.BurrowesomittedhowevertomentionanissuethathadbeencausingsomeworriesinEquityReleaseMortgage(ERM)circlesforalittlewhilenow.Theproblemisthatfirmsare under-valuing the No Negative Equity Guarantees (NNEGs) that are a standardfeatureofmostERMproducts.Thisunder-estimationseemstobeonalargescaletoo.These concerns received somepublicitywith thepublicationon7August last yearofreportsbyBBCbusinessjournalistHowardMustoe2andtheAdamSmithInstitute3ontheissue,andwiththeairingthateveningofaBBCRadio4programme,“TheEquityReleaseTrap.”4Sincethen,therehasbeenconsiderablepublicdiscussionoftheNNEGvaluationissue.Weprovideacommentaryonourblog,TheEumaeusProject(eumaeus.org/),madea presentation on the subject to the London School of Economics in October andpublished a second analysis in our Johns Hopkins University discussion paper ofNovember2018.5AnotherreportonNNEGvaluationbyRaduTunaruwaspublishedby

1StapleInneventtranscript28Feb2019.2H.Mustoe,withadditionalreportingbyM.Keyworth(2018)“HomeEquityReleaseMayCostPensionFirmsBillions.”BBCnewswebsite(7August).3K.Dowd(2018)AsleepattheWheel:ThePrudentialRegulationAuthorityandtheEquityReleaseSector.London:AdamSmithInstitute(7August)(a)4https://www.bbc.co.uk/programmes/b0bd8h785 D. Buckner and K. Dowd (2018) “Equity Release: Another Equitable in the Making,” Johns HopkinsInstitute forAppliedEconomics,GlobalHealth,andtheStudyofBusinessEnterprise,Studies inAppliedEconomicsNo.129,November2018.

6

theInstituteandFacultyofActuariesinFebruarythisyearandgivenextensivecoverageatitsStapleInnlaunch.6AfurtherreportonNNEGvaluationwaspresentedtotheSocietyofActuariesinIrelandbyTonyJefferyandAndrewSmithinMarch,andwegaveanotherseminaronthesubjectatQMULinJune.7Itwouldbefairtosaythatthereisawiderangeofviewsandtheissueisnowmorecontroversialthanever. Consider the following 2017 quotes from UK equity release firms discussing themethodologiestheyusetovaluetheirNo-NegativeEquityGuarantees(NNEGs).Asyoudoso,askyourselfwhattheyallhaveincommon:

“When calculating the value of theno-negative equityguarantee on thelifetimemortgages, certain economic assumptions are required withinthevariantof theBlack-Scholes formula. […] In theabsenceof a reliable long-termforwardcurveforUKresidentialpropertypriceinflation,the[firm]hasmadeanassumptionaboutfutureresidentialpropertypriceinflation.…Thisresultsinasinglerateoffuturehousepricegrowthof4.25%.”8

“[Thevalueof theNNEG] iscalculatedusingavariantof theBlackScholesoptionpricingmodel.Thekeyassumptionsusedtoderivethevalueoftheno-negative equity guarantee include current property price,property growthandpropertyvolatility.”

“Stochastic modelling is used to capture the expected cost of [the NNEG],whichwilldependon theexpected rateandvolatilityof futurehousepricegrowth…“Equityreleaseandsecuritisedmortgageloans…arevaluedusinganinternalmodel.Inputstothemodelincludeprimarilypropertygrowthrates,mortalityandmorbidityassumptions,….”

“Thefairvalueoftheguaranteeisdeterminedusingastochasticmodel.Thefair value of the loans is determined using assumptions for interest rates,futurehousepriceinflationanditsvolatility…”

The answer is that they are all using incorrect valuation approaches. They all usepropertygrowthassumptionsintheirNNEGmodels,butnocorrectoptionpricingmodelsincludepropertygrowthvariables.TheiruseofanirrelevantvariablethenindicatesthattheyarenotvaluingtheirNNEGsproperly.Totheircredit,thePRAhavebeenawareofthisproblemforsometime.Referringtotheresultsofanearliersurvey,CP48/16states(p.25):

6 R. Tunaru and E. Quaye (2019) “UK Equity Release Mortgages: a review of the No Negative EquityGuarantee,”ActuarialResearchCentre/InstituteandFacultyofActuaries(19February).7 T.JefferyandA.D.Smith“EquityReleaseMortgages:Irish&UKExperience,”presentationtotheSocietyofActuariesinIreland,28March2019 8SeeF.BlackandM.Scholes(1973)“ThePricingofOptionsandCorporateLiabilities,”JournalofPoliticalEconomy81:637–654.

7

ManyrespondentsmentionedaversionoftheBlack-Scholesformulaknownas‘Black76’,wheretheunderlyingpriceisthe‘forwardprice’oftheproperty.Thisversionusesthecurrentpriceofaforwardcontract.Somerespondentsappearedtoconflate thiswiththeforecastfuturepriceoftheproperty,butprovidedno justification forwhy house price inflationwas relevant to thecurrentpriceofaforwardcontract.(Ouritalics)

The key word is “conflate”. The reason why these correspondents provided nojustificationforusingprojectionsoffuturehousepriceinflationtovaluetheseguaranteesisbecausenosuchjustificationexists.Tospellitout:somefirmssaythattheyareusingassumptionsaboutfuturehousepricegrowth,butthePRAcorrectlysaysthatthisisobviouslywrong.Fromwhichitfollows(1)thatsomefirmsareusingamethodwhollyatoddswiththeoneendorsedbythePRAand(2)thatthePRAwouldnotbebotheringtostatethispointatall,particularlythroughaprotractedconsultationperiodifithadnotexperiencedsubstantialpushbackfromfirms.Wecantheninfer(3)thatfirmswithequityreleaseexposurehavebeenundervaluingtheirnonegativeequityguarantees.WecanmakethisinferencebecausethePRAwouldnotbepublishingon the subjector seeking industry consultation if they thought thatthese guarantees were correctly valued. Consequently, some firms are presumablyundervaluing them. Also (4) by a similar logic, if firms are dedicating substantialresourcestopushingback,theymustthinkthatthevaluationofguaranteesisamaterialissue.Infact,wearenotawareofasinglefirmthathasdemonstratedthatitisvaluingitsNNEGsusingadefensiblemethodology.Ourimpressionisthattheyareallgettingitwrong.EquityReleaseandtheGhostofEquitableLifeWehaveseenthismoviebefore.Acoupleofdecadesago,therewasascandalsurroundingEquitableLife.Theworld’soldestmutualinsurer,EquitableLifewasfoundedin1762andpioneeredage-basedpremiumsbasedonmortalityassumptions.Inthemiddleofthe20thcentury, it also pioneered Guaranteed Annuity Rate (GAR) options that offeredguaranteedfixedreturns.Atitspeakinthe1990s,ithad1.5millionpolicyholderswithfundsworth£26billionundermanagement.However, it failed tovalue theseoptionsproperly,andinsomecases,itdidn’tvaluethematall.Asaresult,itfailedtoprovideforthemproperly.Equitablecametogrief in2000whenitwasnolongerabletokeepitspromises. There was then a big outcry and the insurance regulatory system wasoverhauled tomake sure that an Equitable-style fiasco never happened again. So theproblem was that the company had been under-valuing opaque and apparentlyinnocuouslong-termguaranteesandtheundervaluationoftheseguaranteeseventuallybroughtitdown.Inbothcases,therewasatoxiccombinationofintellectualerrorandshort-termthinking.In the Equitable case, there was an underlying presumption that the guarantees inquestion,Equitable’sGARoptions,didn’treallymatterandthatanyproblemsthattheymightentailwerewellintothefutureanyway.Intheequityreleasecase,theintellectual

8

error involves a profoundmisunderstanding of option pricing theory by professionalactuaries,combinedwithamindsetonthepartofindustryleadersthatputsshort-termprofitabilityand‘competitiveness’aheadofnotionsoflong-termsustainability.WhenitcomestoNNEGvaluation,thismindsetprioritiseslowNNEGvaluationsoversoundNNEGvaluationsandtherestisobvious.The intellectual error centres around the underlying variable in the option pricingformula. ANNEG involvesaportfolioofputoptionsandwearedealingwithputsonforwardcontracts.Forexample,ifacustomertakesoutanERMattheageof70,thereisaNNEGput for thepossibility that theERMloanmightendwhenthecustomer is71,anotherNNEGputforthepossibilitythattheERMloanmightendwhenthecustomeris72andsoforth.Eachoftheseputoptionsisissuednow,buthasahorizon(ordecrement)ofone, two,etc.years inthe future.Thepricethatenters intoeachputoptionpricingequationistheforwardpriceoftheunderlying,andthedeliverableisahouse.Sofortheputoptionthatendsinfutureyear𝑡,theunderlyingistheforwardhousepriceforyear𝑡,thepriceagreednowforthehousetobedeliveredandpaidforinyear𝑡.Thisapproachis based on standard option-pricing theory and is exemplified by Black (1976).9 Inactuarialcircles,thisapproachisoftenreferredtoasthe‘MarketConsistent’approachtoNNEGvaluation.TheproblemisthatanumberofpractisingactuariesintheUKequityreleasesectorhaveconvincedthemselvesthattheunderlyingpricethatisrelevantforputoptionpricingisnottheforwardhousepriceforyear𝑡butthefuturehousepriceorexpectedfuturepriceforyear𝑡.However,forwardandfuturepricesareverydifferentandtoconfusethetwoistocommitamajorerror.Thiserrorisabigdealbecauseinputtingtheexpectedfuturehousepriceintotheoption-pricingequationgivesverylowNNEGvaluations,whereasinputtingforwardhousepricesintoitgivesmuchlargerNNEGvaluations.Thissecond,incorrect,approach iscommonlyreferredto inactuarialcirclesas the ‘RealWorld’or‘DiscountedProjection’approach.AdifferencehoweverbetweentheEquitableLifeandequityreleasecasesisthatwhenEquitable started issuing GARs in the 1950s, the valuation of options was not well-understood.Theoptionpricingbreakthroughonlyoccurredin1973withthepublicationofthefamousarticlesbyBlack,ScholesandMerton(BlackandScholes,1973;Merton,1973),followedshortlyafterwardsbyBlack(1976).Boththeprinciplesandthenuancesofoptionvaluationhavebeenwellknownfordecadesandaretaughtinuniversitiesallovertheworld.Itiscurious,too,thattheUKactuarialprofessionalassociation,theInstituteandFacultyofActuaries(IFoA),hasyettospeakoutagainsttheseunsoundNNEGvaluationpractices.Onthecontrary,itisontherecordasendorsinganumberofmisconceptionsaboutNNEGvaluation.Despite copious protestations to the contrary, theUK actuarial profession appears tohavelearnednothingfromthelessonsofEquitableLife,andsowelcometoEquitable2.Thosewhofailtolearnthelessonsofthepastwillhavetotaketheclassagain.

9SeeF.Black(1976)“ThePricingofCommodityContracts,”JournalofFinancialEconomics3:167-179.

9

PurposeofthisReportThepurposeofthisreportistosetouttheissuesinvolvedinthevaluationofNNEGsandEquityReleaseMortgages(ERMs),i.e.,itprovidesahow-tohandbookforpractitionersworking onNNEG andERMvaluation issues. In doing so, it examines both valid (i.e.,Market Consistent) and invalid (e.g., Discount Projection or Tunaru) approaches, andexplainswhytheformerarevalidbutthelatterarenot.WeemphasisethatthisreportfocusesonlyonthevaluationissuesanddoesnotaddressthebroaderissuesraisedbyNNEGunder-valuation,suchastheimplicationsforpublicpolicy.Weintendtoaddressthoseissuesinafollow-onreport.OrganisationofthisReportThisarticleisorganisedasfollows.Chapter2providesanintroductiontoequityrelease:itexplainstheproductsandthesignificanceoftheNNEG,andgivesanoverviewofrecentdevelopmentsinthesector.Chapter3explainsthebasicsofNNEGandERMvaluation.Thenext14chaptersexaminekeyinputsandotherfactorsrelevanttoNNEGandERMvaluation:theloan-to-valueratio(Chapter4),therisk-freerate(Chapter5),theloanrate(Chapter6),thetheoryandcalibrationofnetrentalanddefermentrates(Chapters7and8),dilapidation (Chapter9),volatility (Chapter10),mortality (Chapter11), long-termcare (Chapter 12), delayed possession (Chapter 13), credit spreads (Chapter 14), theimpact of drawdown facilities (Chapter 15), prepayment (Chapter 16) and fees andcharges (Chapter17). Chapter18examinesERM-related scenario analyses and stresstestsandChapter19discussesthePRA’sERMGoodPracticePrinciples.Chapter20setsout the Market-Consistent approach and Chapter 21 refutes some common actuarialmisconceptions about it, Chapter 22 debunks the ‘Discounted Projection’ approach,Chapter23examines therecentTunarureportandChapter24discusses JustGroup’sNNEGvaluationmodel.Chapters25and26examinetechnicalactuarialandaccountingstandards, and Chapters 27 and 28 offer some recommendations for good valuationpracticeanditsgovernance.

10

ChapterTwo:IntroductiontoEquityReleaseTheHomeEconomicsofEquityReleaseAnERMatypeofloancollateralisedbyaproperty(‘house’),andthetypeofERMweareinterestedingoesasfollows.10Theloanistakenoutbyacustomerlateinlifewhoownsthepropertytheylivein.Thecustomerusestheloantosupplementtheirincome,helptheirchildrengetonthepropertyladderorwhatever.Unlikeanormalloan,thisloanhasnofixedenddateandinvolvesnoregularinterestpayments.Instead,theloanendswhenthecustomerexitsthehouse,eitherbydeathorbygoingintoanursinghome,andtheamountowedontheloanaccumulatesovertimeuntiltheloanisrepaid.11Atthetimeofexit,thelendertakespossessionofthepropertyandsellsittorepaytheloan.Ifthereareanyproceedsleftover,thesearereturnedtothecustomerortotheirestate.Theterm“equityrelease”ismisleading,however.AsJefferyandSmith(2019,pp.6,55)pointout:

It cannot be repeated too often that ERM is a misnomer. The equity in apropertyisnotreleaseditisborrowedagainst.Thevalueofthepropertytotheownerbecomesgeared.....Ashousepricesmoveupanddown(!),theloanremainsunchangedinvalue.

JefferyandSmithareright,buttheterm“equityrelease”issowidelyusedthatwearestuckwithit,andtheAmericanalternative(“reversemortgage”)hasissuesofitsown.So“equityrelease”itis.TherearetwomaintypesofERM.Thefirst,knownasaLifetimeMortgage(LTM),isastraightforwardmortgageloan,wherethelenderhandsovertheloanamountatthetimethecontractismade.ThesecondisadrawdownERM,inwhichthecontractprovidesforamaximumpossible loanamount,but theborrowerhasdiscretionoverhowmuchtodrawdownagainstthismaximumandwhentodoso,subjecttotheconstraintthatthetotal amount drawn down cannot exceed the stipulated maximum. Typically, theborrowerinadrawdownERMwouldmakeadrawdownwhenthecontractismade,andthenmakemoreinlateryears,asandwhenheorshefeelstheneedtodoso.Inthisreport,wefocusmainlyonthevaluationissuesrelatingtoLTMERMs.Thesearesimpler,butwediscussdrawdowninChapter15.Sufficetonotethatthevaluationissuesaremuchthesame,exceptfortheadditionalcomplicationsintroducedbyadrawdownfacilityandhowthatmightbeusedbytheborrower.

10 Wearenotconcernedherewithothertypesofequityreleaseproductsuchhomereversions,inwhichtheborrowersellsallorpartoftheirpropertyatlessthanitsmarketvalueinreturnforatax-freelumpsum, a regular income, or both, but stays on in their home as a tenantwhopays no rent.Nor areweconcernedwithERMloansthatdonotincorporateNNEGs,sufficetonotethatmostincorporateNNEGsandthatallERMsissuedbymembersoftheEquityReleaseCouncildo. 11 Insomecases,theloancanalsoendbyearlyrepayment,butwedeferthatissuetoChapter16below.

11

TheERMloanwillbetakenoutassomelowproportionofthepropertyvalue–40%istypicalfora70yearold,butLoantoValueratios(LTVs)tendtobelowerforloweragesandhigherforhigherones–andthelenderisprotectedagainstanyriskoflossforaslongastheloanvalueisbelowthevalueofthehouse.Theloanratewillbefixedattheinceptionoftheloan.Thevalueofthecollateral,thehouse,willvarywiththehouse’smarketprice.Typically,housepriceshave risen in recent years andwemight (ormightnot!) expect them tocontinue to rise, but we would not usually expect the house price to rise at a rateexceedingtheloanrate.Inanycase,housepricesareuncertainandsometimesfall,soexpectationsoffuturehousepricesareunlikelytobeexactlyrealised.AtypicalcaseisshowninFigure2.1:

Figure2.1:LoanEquityinaTypicalEquityReleaseMortgage

Wewouldcertainlyexpecttheloanamount(showninblue)toriseovertime,andwewouldusuallyexpectthehouseprice(inblack)torisetoo,butevenso,wewouldexpecttheloanamounttoriseatafasterrateandeventually,ifthecustomerliveslongenough,theblueloanamountlinewillcrossovertheblackhousepriceline.Thereaftertheloanamountwillexceedthevalueofthehouse,i.e.,theloanwillgointonegativeequity.Ifthecustomerexitsthehousebeforethepointofnegativeequity(whichis21yearsinFigure2.1),thenthelenderwouldberepaidinfull.Ifthecustomerexitsafterthatpoint,theloanwouldexpireinnegativeequity, i.e.,thevalue of the property collateral would not be enough to cover the accumulated loanamount.IntheabsenceofaNNEG,thelendercouldsuetheborrowerortheirestate,buttheremighthavefewassetsleft,especiallyiftheborrowerwasmovingintoaretirementhome and any remaining assets were being used to finance their care. Most ERMsincorporateaNNEG,however,andinsuchcasesthenegativeequitybecomesalossbornebythelender.

12

AnotherwaytothinkabouttheERM-with-NNEGcontractisthatitgivesthelendertheminimumofthehouseprice(black)andloanamount(blue)lines.Thefactthatthelendergetstheminimumoftwovaluesindicatesthatthelenderisgrantingaputoptiontotheborrower.Thelender’spotentiallosswiththeNNEGinplaceisillustratedinFigure2.2,andlet’shenceforth assume for themoment that exit is due solely to death (althoughwewillrevisitthisassumptionlater):

Figure2.2:ERMLoanExpiresinNegativeEquity

Inthiscase,theborrowerdiesafter25yearsandthelendermakesthelossgiveninred,thedifferencebetweentheloanvalueandthehousepriceafter25years,relativetowhatthelenderwouldhavereceivedhadtherebeennoNNEGandthelenderbeenrepaidinfull.Naturally,thisloss(andwhetheranylossoccursatall)isuncertainbeforetheevent.Thetimingofdeathisuncertainandifthecustomerdiesearlythentherewouldbenolosstothelender.Butifthecustomerdieslaterthelendersuffersalossthatdependsamongother factors on the timing of death. Thus, the NNEG potentially exposes ERMs tolongevityrisk–theriskthatthecustomermightlivetoolong.ERMsarealsoexposedtohousepricerisk.ThisriskisillustratedinFigure2.3:

13

Figure2.3:TheImpactofaFallinHousePricesonNegativeEquity

Thehousepricemightbeloweratthetimeofdeaththanthelenderexpectedittobe.Figure3.3showsacasewherethehousepricedeclinesinsteadofrising.Ifthecustomerdiesafter25years,thenitisclearfromacomparisonofFigures2.2and2.3thatthelenderwillsufferalargerlossduetothehousepricefall.ERMsarethussubjecttohousepriceaswellaslongevityrisk.Fromtheborrower’sperspective,takingoutanERMloanmightbeasuitablechoiceforanolder individualorcouplewhoareassetrichbutcashpoor,e.g., theymighthaveaneedforcashorwishforahigherstandardoflivinginretirement.OnecanalsoimagineadditionalcircumstancesinwhichanERMmightbesuitable,e.g.,becausetheirchildrenmaybeaffluentorbecausetheydon’twant to leavetheirchildrenany inheritance,orbecausetheymayhavenochildrenanddon’twanttoleavetheirhousetoacats’home.Forsuchpeople,aregularmortgagewouldnotnormallybepracticalbecausetheywouldnolongerbeworkingandthereforenothavetheincometorepaysuchamortgage.From the lender’s perspective, an ERM loan offers a high loan rate and is highlycollateralised,at least tostartwith. Itsmaindownside is the impactof theNNEG, thevaluationofwhichisthecorefocusofthisreport.12Tohelpformanintuition,considerthatanequityreleasemortgagecanbebrokendownintotwocomponents.Thefirstcomponentistheloanamountwhichrollsupatahighloanorroll-uprate,whichcurrentlyaveragesatabout5.25%fornewERMloans,butthereisconsiderabledisparityaroundthataverage.Nowthisloaniscollateralisedbythehouse,andtheloanatinceptionwillbesomefractionofthehousevalue.Forexample,fora 70 year old, the loan to (house) value ratio will be around 40%. The highcollateralisationoftheloanmeansthatwecanregardtheloanasclosetorisk-free:whentheborrowerexitsthehome,thelenderwilltakepossessionandusethehometogetitsloanrepaid.Thereisnodangerofdefault.Thesecondcomponent istheNNEG.Sothe 12 ItisoftenclaimedthatERMportfolios(orexposuretoERMfirms)aresuitableforpensionfundsbecausetheyarelong-termassetsthatarecorrelatedwithlongevityrisk.However,thosewhomakesuchclaimsoftenoverlooktheexposureofERMportfoliostohousingrisk.Wewillhavemoretosayonthisissueinalaterreport.

14

valueoftheERMloantothelenderisthevalueoftheloan(whichwouldincludeexpectedprofitsontheloan,andwhichisnottobeconfusedwiththeloanamount)minusthevalueoftheNNEG.SupposenowthattheNNEGvalueisverylow.ThenitfollowsthatthevalueoftheERMtothelenderwillbeclosetothevalueoftheloancomponentoftheERM,becausetheNNEGvalueisverylowandsodoesn’treallymatter.TheERMisthenclosetobeingahighinterestloanthatiscertaintoberepaidinfull,whichisanattractivepropositiontothelender.Thus,ERMsarehighlyprofitabletothelenderiftheNNEGvalueisverylow.Atthe other extreme, if the NNEG value is very high, then an ERM loan might not beprofitable at all. A correct valuation of the NNEG is then essential to determine theprofitabilityoftheloan.EquityReleaseisaRapidlyGrowingSectorThegrowthoftheequityreleasesectorisapparentinthefollowingFigure,whichplotstheamountslentsince2000:

Figure2.4:GrowthoftheEquityReleaseSector

Source: Equity Release Council: http://www.equityreleasecouncil.com/document-library/equity-release-market-report-autumn-2018/equity-release-market-report-autumn-2018.pdf

Thebluelineplotstheamountslentandtheredlineplotsthenumberofnewcustomers.Thelattershouldbetreatedwithsuspicionasitincludesreturningcustomers,i.e.,thosewithdrawdownfacilitywhosubsequentlydrewdown.Thinking fromamacroperspective, the lendingbroadly reflects changes inUKhouseprices,i.e.,the2008peakisclearlyvisiblewithsubsequentdeclineandthenrecovery.Thefinalnewlendingamountiswellabovethepreviouspeakin2008.Wecouldattributethepost-crisisgrowthtoanumberoffactorsincludingpossiblystrongermarketingandTreasury‘approval’oftheproduct,orperhapsmorenaturallytoincreaseddemandfor

15

theproductasboomersreachretirementageandrealise that theycan’taffordanewwashingmachinebuthaveahouseworth£500k.Recentgrowthinthesectorhasbeenremarkable:

Since2016,theequityreleasemarkethasbeengrowingattheaveragerateof7.1 per cent each quarter. The amount of equity unlocked from people’shomesmore than doubled from £514m in the second quarter of 2016 to£1.08bninthefinalquarterof2018.13

The Equity Release Council’s latest (Autumn 2018) Market Report provides furtherdetails:

Atotalof38,912householdsaged55andoverusedequityreleaseproductsfrommembersofTheCounciltoaccesssomeoftheirpropertywealthduringH12018.Thisincluded21,490newplansagreedbycustomers,upby28%from16,805ayearearlier.A further 15,709 returning drawdown customersmadewithdrawals fromtheiragreedreservefundsbetweenJanuaryandJune,up25%yearonyear.…Thenumberofnewplansagreedin[2018]H1exceededtheentiresizeofthemarket in 2014 and represented an 81% increase since H1 2016. Activityamong new customers has increased during every half-year period in theinterveningtwoyears,asequityreleaseandhousingwealthhavetakenupamainstreampositionamongtheproductsandassetsthatformpartofmodernretirementplanning.Twothirdsofnewcustomers(65%)optedfordrawdownlifetimemortgagesinH1,while35%chose lumpsum lifetimemortgagesanda smallnumber(<1%)agreedhomereversionplans.(ERC2018,p.6,ouritalics)

TheECRreportoffersfurtherdetailaboutthegrowthofproductchoices:

Thegrowingbaseofequityreleasecustomersinrecentyearshasbeenmetwithagreaternumberofproductchoicesandflexibilities–helpingtomeethomeowners’increasinglycomplexneedsinlaterlife.AsofAugust2018,139productoptionswereavailabletoconsumers,morethandoublethenumber(58)seentwoyearsago.Therangeofproductoptionshasgrownbyover78%inthelastyearalonefrom78inAugust2017.Today’s equity release products also offer greater flexibilities thanks toongoingcompetitionandinnovationinthesector.Fourinfive(80%)productoptionsofferconsumersthechoicetomakead-hoc,penalty-freevoluntaryorpartialrepaymentsoftheirloan,upfrom68%ayearago.Therehasalsobeen

13L.Warwick-Ching“DramaticRiseinEquityReleasebyOver-55stoFundRetirement.”FinancialTimes4April2019.

16

anincreaseinproductsofferingfixedearlyrepaymentcharges(ERCs),from49%inAugust2017to51%inAugust2018.Almost half of product options (45%) offer downsizing protection whichallows customers to downsize to a smaller property and repay the loan,withoutincurringanyERC.Inheritanceprotection,whichallowscustomerstoring-fenceasectionoftheirhousingwealthasaguaranteedminimumamounttopassontothenextgeneration,regardlessofthetotalinterestaccrued–isofferedby46%ofproducts.(p.7)

Andaboutcustomerprofiles,e.g.:

TheaverageageofnewcustomersduringH12018wastheclosestseentodate across the twomain categories of lifetimemortgages.At just over68yearsold,theaveragenewlumpsumcustomerwasbroadlyinlinewiththatseenoverthelastthreeyears.Theaveragedrawdowncustomerwasalmosttwoyearsolderandjustshortoftheir70thbirthday.(p.9)

Sothesectorisdoinggreat,butthereisstilltheissueaboutNNEGvaluation.

17

ChapterThree:TheBasicsofNNEGandERMValuation 1.IntroductionThischapterexplainsthebasicsofNNEGandERMvaluation.2.ExitProbabilitiesThe NNEG valuation model has two key ingredients: a set of expected house-exitprobabilitiesandaputoptionpricingmodel.Thehouseexitprobabilities(orexitprobabilitiesforshort)refertotheprobabilitiesthattheborrowerwillexitthehouse(andhenceterminatetheloan)overeachofthenext1,2,3,…etcyears.Forthetimebeing,letusassumeawaythepossibilityofearlyrepaymentof the loanandassumethat theborrower isasinglemalewho isexpectedtoexit thehouseinabox.Undertheseconditions,theexitprobabilityforyear𝑡isequalto(3.1)𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏* = 𝑞* × 𝑆*where𝑞*isthemortalityrateforyear𝑡and𝑆*istheprobabilitythatanindividualalivenowwillsurvivetoyear𝑡.Notethat𝑆0 = 1and𝑆* = (1 − 𝑞*45)𝑆*45forall𝑡 > 0. 14Theexitprobabilitiesforamalejustturned70areshowninFigure3.1:

14 The‘𝑞’terminologyforthemortalityrateisstandardandalittleunfortunateintheNNEGcontext,where𝑞issometimesusedtorefertothenetrentalrateordefermentrate.Thereadershouldbearthisambiguityinmind,butthecontextshouldmakeitclearwhetheritisthemortalityrateorthedefermentratethatwearereferringto,anditismostlythelatter.

18

Figure3.1ExitProbabilitiesforMalesCurrentlyAged70

Notes: Exit probabilities are based on CBD-M5 model (Cairns et alia, 2006, 2009) cohortmortality rateprojectionsusingmaleEngland&Walesdeaths ratedata estimatedover ages55:89andyears1971:2017.Source:Life&LongevityMarketsAssociation.15

Thelefthand(low𝑡)exitprobabilitiesareclosetothelow𝑡mortalityratesandreflecttheearlyhighsurvivalprobabilities(i.e.,thatpeopleaged70haveahighprobabilityoflivingatleastafewyears),andthelater(high𝑡)exitprobabilitiesprimarilyreflectthefactthattheprobabilitiesoflivingtoextremeoldagearelowandapproachzerointhelimit.163.ValuationIssuesandthePutPricingModelThepresentvalue𝐸𝑅𝑀oftheEquityReleaseMortgageloancanbeconsideredtobethepresentvalue𝐿ofarisk-freeloan,onewhichisguaranteedtoberepaidinfull,minusthepresentvalue𝑁𝑁𝐸𝐺oftheNNEGguarantee:(3.2)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺Theoriginalloanamountgrowsattheloanrate(sometimescalledtherolluprate)𝑙fromitscurrentamountuntilthetimewhentheloanends.Therefore𝐿isgivenby(3.3)𝐿 = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏* × 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑙𝑜𝑎𝑛𝑎𝑚𝑜𝑢𝑛𝑡 × 𝑒(H4I)*]*

15 SeeA. J. G. Cairns,D. Blake andK.Dowd (2006) “ATwo-FactorModel for StochasticMortalitywithParameterUncertainty,”JournalofRiskandInsurance,73(4):687-718;andA.J.G.Cairns,D.Blake,K.Dowd,G.D.Coughlan,D.Epstein,A.Ong,andI.Balevich(2009)“AQuantitativeComparisonofStochasticMortalityModelsUsingDatafromEngland&WalesandtheUnitedStates,”NorthAmericanActuarialJournalVolume13(1):1-35.16WehavemoretosayonthederivationoftheexitprobabilitiesinChapter11.

19

where𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏*istheprobabilityofexitingthehouseinperiod𝑡and𝑟istherisk-freeinterestrate.17Thevaluationof𝐿isstraightforward.𝑁𝑁𝐸𝐺isthesumoftheproductsoftheexitprobabilitiesforeachfuturetime𝑡andthepresentvalueoftheNNEGguaranteeforeachfuturetime𝑡:(3.4)𝑁𝑁𝐸𝐺 = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏* × 𝑁𝑁𝐸𝐺*]* where𝑁𝑁𝐸𝐺*isthepresentvalueoftheNNEGguaranteeforperiod𝑡.Thequestionisthenhowtovalueeachoftheseindividual𝑁𝑁𝐸𝐺*termsandthencetheNNEGguarantee.RecallthattheNNEGgivesthecustomer(orthepersonactingforthecustomer)therighttorepaytheloanbypayingthelendertheminimumoftheloanvalueorthehousepriceatthetimeofdeath.Theright to repay theminimumof two futurevalues (oneofwhich, the futurehouseprice,isuncertain)atsomegivenfuturetimeimpliesaEuropeanputoptiongrantedbythelendertotheborrower.Sincethetimeofexerciseisuncertain,wecanthinkoftheNNEGasinvolvingaportfolioofsuchputoptions.18Inthecaseofourputoptionstheunderlyingvariableisaresidentialproperty(‘house’)ormoreprecisely,aforwardcontractonahouse,andweshouldthinkofahouseasanassetthatbearsacontinuousyieldintheformofa(net)rentalrate.Thisnetrentalyieldreflectstheusebenefitoflivinginthehouseortherentalincomewemightgetbyrentingthehouseout.A natural option pricingmodel to use in these circumstances is the Black ’76model(Black,1976).Black’76isanappropriatepricingmodelwhentheunderlyingisaforwardcontractwithamaturitycoterminouswiththatoftheoptionitself.Thismodelisanear-relativeofthefamousBlack-Scholesmodel(BlackandScholes,1973;Merton,1973).19TheBlack ’76 formula for theprice𝑝* of aEuropeanputoptionwithmaturity 𝑡 onaforwardcontractonacommoditybearingacontinuousyieldqisgivenbytheformula:

17 Notetheimplicitdistinctionherebetweentheloanamountorrolleduploanamount,ontheonehand,and𝐿,the(economic)valueoftheloan,ontheother.Theformeristheamountloanedplustheinterestaccumulated since the inception of the loan,whereas the latter is the value of the loan to the lender,includingtheexpectedprofitontheloan.AconcreteexampleofthedistinctionbetweenthetwoisgiveninTable3.1.Notetoothattheeconomicvalueoftheloanisnottobeconfusedwiththeaccountingbookvalueoftheloan,whichisanotherissueagain. 18 OnemightalternativelymodeltheNNEGasasingleAmericanoptionwithanearlyexercisefeaturebuttherewouldbenopointindoingso.Americanoptionsgetinterestingonlywhentheoptionisexercisedearly in theself-interestof theoptionholder,but thedecision toexerciseearlymakesnosense in thiscontext,becausesuchadecisionwouldbetantamounttotheborrowertakingtheirownlifetogetbackatthelender,presumablyburningdowntheirhouseintheprocess.19Thechoiceofoptionmodelisdiscussedinmoredetailintheappendixtothischapter.

20

(3.5)𝑝* = 𝑒4I*[𝐾*𝑁(−𝑑M) − 𝐹*𝑁(−𝑑5)]where𝑟istherisk-freerateofinterest,𝐾*isthestrikeorexercisepriceforperiod𝑡,𝐹*istheforwardhousepriceforperiod𝑡,thefunction𝑁(… ) isthevalueofthecumulativestandardnormaldistributionatthevaluespecifiedinbrackets,and𝑑5and𝑑Maregivenby:(3.6)𝑑5 = [𝑙𝑛(𝐹*/𝐾*) + 𝜎M𝑡/2]/(𝜎√𝑡)(3.7)𝑑M = 𝑑5 − 𝜎√𝑡where𝜎isthevolatilityoftheforwardhouseprice.20Thestrikeprice𝐾*isthentherolleduporaccumulatedloanamountbyperiod𝑡:(3.8)𝐾* = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑙𝑜𝑎𝑛𝑎𝑚𝑜𝑢𝑛𝑡 × 𝑒H*andtheforwardprice𝐹* ,thepriceagreednowtobepaidonpossessioninperiod𝑡,is:(3.9)𝐹* = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒(I4W)*where𝑞isthedefermentrate,namelythediscountrateappliedtothecurrenthousepricetogivethedefermentprice,thepricewewouldagreetopaytodaytotakepossessionofthehousein𝑡years’time.Thusthedefermenthouseprice𝑅*isgivenby:(3.10)𝑅* = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒4W*Thedifferencebetweentheforwardhousecontractandthedefermenthousecontractisthatwiththeforwardwesettlewhenwetakepossessionin𝑡years’time,butwiththedeferment contractwe settle today.21 Therefore, the deferment house price𝑅* is thepresentvalueoftheforwardprice,wherethepresentvalueisobtainedbydiscountingattherisk-freerate𝑟.Itisimportanttonotethatthedefermenthousepricewillbelessthanthecurrenthouseprice𝑆0becausethedefermentrate𝑞 > 0.Theforwardhouseprice𝐹*shouldnotbeconfusedwithfuturehousepricesorexpectedfuturehouseprices:

• Forwardpricesforfutureperiod𝑡areknown(orcanbeapproximated)nowandweneedtobeabletovalueoptionsusinginformationavailablenow.

20 Compared to the original Black-Scholes equation (Black and Scholes, 1973), we replace the spotunderlying,thecurrenthouseprice,withtheforwardhouseprice.Apointsometimesoverlooked,the𝑟termintheclassicBlack-Scholesformulasfor𝑑5and𝑑Malsodropsoutbecausetheunderlyingcontractispaidforatmaturityandnotatinception,andweassumethatarationalsellerwouldrequirecompensationforgrowthonthesumofmoneytheywouldhavereceivedifpaidupfront.Thisassumptionshouldnotbeconfusedwiththecommonmisconceptionthatthemodelassumesthattheunderlyinggrowsattherisk-freerate. 21 SeePRASS3/17(p.12,note2).

21

• Optionscannotbevaluedusingfuturehousepricesbecausefuturehousepricesarecurrentlyunknown.

• Optionsshouldnotbebasedonexpectedfuturepricesbecauseexpectationsoffutureprices,e.g.,futurehouseprices,donotappearintheBlack‘76optionpricingformula.

WeshouldalsokeepinmindthatalthoughtheoriginalBlack‘76articlediscussedoptionsonfutures,futurespricesarethepricesoffuturescontracts,aformofforwardcontract,notactualorexpectedfuturepricesofanysort.Amistaketobeparticularlyavoided–theonecommonamongUKERMactuaries–istoconfuseforwardandexpectedfutureprices.Thismistaketypicallymanifestsitselfintheinputting of an assumed expected house price inflation rate into (3.9) instead of theforwardrate𝑟 − 𝑞.To repeat: it is not the future or expected future value of a contract for immediatepossessionthatweuseintheoptionpricingequation,butratherthecurrentvalueofacontractforfuturepossession.Theapproachsetouthereisanexampleofwhatisknowninactuarialcirclesasa‘marketconsistent’ approach, which gives ‘market consistent’ valuations. We would define amarketconsistentvaluationasa‘fairvalue’valuationbasedontheIFRSdefinitionofafairvalueprice,namely

Thepricethatwouldbereceivedtosellanassetorpaidtotransferaliabilityinanorderlytransactionbetweenmarketparticipantsatthemeasurementdate22

Analternative(andforourpurposespracticallyequivalent)definitionisthatprovidedbyTimGordon:hedefinesamarketconsistentvaluationasonewhichisconsistentwithmodernfinancetheoryasthetermisusedinExley,MehtaandSmith(1997).23Weshalltakethevalidityofthisapproachasgivenforthetimebeing,butwewillprovideajustificationforitinlaterchapters.4.AValuationExampleWenowbuildanERMandNNEGvaluationmodelbasedonplausibleinputparametercalibrationvalues.Thebaselineparameterinputsare:

22 See,e.g.,https://www.iasplus.com/en/standards/ifrs/ifrs13. 23 SeeT.Gordon(1999)“ThePriceofActuarialValues.”PaperpresentedtotheStapleInnActuarialSociety(16February)andC.J.Exley,S.J.B.MehtaandA.D.Smith(1997)“TheFinancialTheoryofDefinedBenefitPensionSchemes,”BritishActuarialJournal3(4):835-966.

22

• Currentageofcustomer=70,atypicalageforERMs.24• Loantovalueratio=40%25• Risk-freerate𝑟=1.5%.• ERMloanrate𝑙 = 5.25%.• Defermentrate𝑞 = 4.2%.• Volatility𝜎 = 14.8%formalesaged70.

Allratesarein%p.a.Wewilldiscussthesecalibrationsinlaterchapters.Weassumeanillustrativehousepriceof£100which,combinedwiththeassumedloantovalueratioof40%,impliesaloanamount=£40.The death/exit probabilities are derived from projections of future mortality ratesobtainedusingtheM5versionoftheCairns-Blake-Dowdmortalitymodel(seeCairnsetalia,2006,2009)calibratedonEngland&Walesmalemortalitydatafortheperiod1971to2017 and spanning ages55 to89.Thedata are taken from theLife andLongevityMarketsAssociationdatabase(llma.org).TheM5-CBDmodelisparticularlysuitableforoldageprojectionsanditsgoodnessoffitandperformanceevaluationareassessedinCairnsetalia(2011)andDowdetalia(2010a,b).OurbaselineNNEGvaluationresultsareshowninTable3.1:

Table3.1:BaselineERM/NNEGValuationsCurrentHousePrice LoanAmount 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £40 £74.84 £32.19 £42.66Notes:𝐿isthepresentvalueoftheloancomponentoftheEquityReleaseMortgage,𝑁𝑁𝐸𝐺isthepresent value of the NNEG guarantee, and 𝐸𝑅𝑀 is the present value of the Equity ReleaseMortgage. Based on the baseline assumptions: male aged 70, 𝐿𝑇𝑉=40%,𝑟=1.5%, 𝑙=5.25%,𝑞=4.2%and𝜎=14.8%.Exit probabilities arebasedonM5-CBDmodelprojectionsusingmaleEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

Giventheageofthecustomer,theexpectedpresentvalue𝐿oftheperfectlycollateralisedloanis£74.84.𝑁𝑁𝐸𝐺isvaluedat£32.19andsothevalueoftheERM,𝐸𝑅𝑀,isequalto£74.84–£32.19=£42.66.Itissometimesconvenienttoreporttheseresultsintermsoftheratiosof𝐿,𝑁𝑁𝐸𝐺and𝐸𝑅𝑀totheloanamountasinTable3.2:

Table3.2:ValuationsDividedbyLoanAmount𝑳/Loanamount 𝑵𝑵𝑬𝑮/Loanamount 𝑬𝑹𝑴/Loanamount

187.1% 80.5% 106.6%Notes:AsperTable2.

24Implicitly,weareassumingasinglemalejustturned70.Inthecaseofasinglefemale,wewouldexpectdeath/exittooccursomewhatlater,whichwouldincreasethevalueoftheNNEG.Inthecaseofacouple,wewouldexpectevenlaterexit,whenthelongestlivingmemberofthecoupleexitsthehouse.25A40%LTVratiofora70-yearoldappearstobeapproximatelyinlinewithcurrentindustrypracticefornewERMloans.Wewillhavemoretosayonthissubjectinthenextchapter.

23

Weseethat𝐸𝑅𝑀,forexample,is106.6%oftheloanamount.5.SensitivitiesofValuationstoKeyInputParametersTable3showsthesensitivitiesof𝐿,𝑁𝑁𝐸𝐺and𝐸𝑅𝑀tochangesinkeyparameterinputs.Theseareexpressedinelasticityform,i.e.,wheretheelasticityoftherelevantoutputwithrespecttoachangeinaninputisthe%changeintheoutputdividedbythe%changeintheinput.

Table3.3:SensitivitiesofValuationsinElasticityFormElasticitywrt 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

𝑟 -0.27 -0.54 -0.07𝑙 0.94 1.89 0.23𝑞 0 0.57 -0.43𝜎 0 0.25 -0.19LTV 1 1.74 0.44

Notes:𝐿isthepresentvalueoftheloancomponentoftheEquityReleaseMortgage,𝑁𝑁𝐸𝐺isthepresent value of the NNEG guarantee, and 𝐸𝑅𝑀 is the present value of the Equity ReleaseMortgage. Based on the baseline assumptions: male aged 70, 𝐿𝑇𝑉=40%,𝑟=1.5%, 𝑙=5.25%,𝑞=4.2%and𝜎=14.8%.Exit probabilities arebasedonM5-CBDmodelprojectionsusingmaleEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

These results indicate that NNEG valuations are highly sensitive to changes in the𝑙 and LTV input parameter calibrations. It is also interesting to note that the ERMvaluationsaremuchlessso,becauseoftheoffsettingimpactsontheloanvalueandNNEGvaluations.

24

AppendixtoChapterThree:TheChoiceofOptionPricingModelInChapterThreewesuggestthatBlack’76isanappropriateputpricingmodelwhentheunderlying is a forward contractwithamaturity coterminouswith thatof theoptionitself.WhynotuseBlack-Scholes(BS)instead?AshallowbutcorrectresponseisthatBSisnotappropriatebecauseitisbasedontheassumptionthattheunderlyingdoesnotbearanyyield,whereasBlack’76isappropriatebecauseitallowsforsuchayield.Theyieldinthiscontextwouldbethenetrentalrate.However,thisdeficiencyofBSiseasilyfixed.Itiswell-knownthatBScanbeadaptedforanunderlyingthatpaysacontinuousdividendyield𝑞ifwereplace𝑆intheBSformulawith𝑆𝑒4W* .26If wemake this adaptation, then we can use BS because the underlying can also beinterpretedasa spotasset thatpaysacontinuousdividendyield. In thiscase,BSandBlack’76willbeequivalent.27Todemonstratethisequivalence,startwiththerelationshipoftheforwardtothespot:(3A.1)𝐹 = 𝑆𝑒(I4W)*(3A.1)implies(3A.2)𝑆 = 𝐹𝑒(W4I)*NowconsidertheBlack-Scholesputprice:(3A.3)𝑝 = 𝑒4I*𝐾 × 𝑁(−𝑑M) − 𝑒4W*𝑆 × 𝑁(−𝑑5)where(3A.4)𝑑5 = [ln(𝑆/𝐾) + (𝑟 − 𝑞 + 𝜎M/2)𝑡]/[𝜎√𝑡](3A.5)𝑑M = 𝑑5 − 𝜎√𝑡Substitute𝐹𝑒(W4I)*for𝑆toobtain:(3A.6)𝑝 = 𝑒4I*𝐾 × 𝑁(−𝑑M) − 𝑒4W*𝐹 × 𝑒(W4I)*𝑁(−𝑑5) =

26See,e.g.,J.C.Hull(2003)Futures,OptionsandOtherDerivatives,fifthedition,p.268.27 We could also price our puts by tweaking the Garman-Kohlhagen foreign currency option model(GarmanandKohlhagen,1983)orbyusinganappropriatespecialcaseoftheMargrabeoption,theoptiontoexchangeoneassetforanother(Margrabe,1978),buttheseapproachesarealsoequivalenttoBlack‘76.See M. B. Garman and S. W. Kohlhagen (1983) “Foreign Exchange Currency Options,” Journal ofInternational Money and Finance 2(3): 231-237; andW. Margrabe (1978) “The Value of an Option toExchangeOneAssetforAnother,”JournalofFinance33(1):177-186.

25

𝑒4I*[𝐾 × 𝑁(−𝑑M) − 𝐹 × 𝑁(−𝑑5)](3A.7) 𝑑5 = [ln(𝐹𝑒(W4I)*/𝐾) + (𝑟 − 𝑞 + 𝜎M/2)𝑡]/[𝜎√𝑡] =

[ln(𝐹/𝐾) + (𝜎M/2)𝑡]/[𝜎√𝑡](3A.8) 𝑑M = 𝑑5 − 𝜎√𝑡whichistheBlack76model!Theargumenthasbeenputtousthatsincetheforwardismorevolatilethanthespot(on account of the volatility of r and q impacting the forward rate), weshould prefer Black Scholes, which is priced off the volatility of the spot.The Black Scholes price will then be cheaper, and the NNEG will be lessexpensive.This objection is incorrect. Let’s assume that by BS we are referring to BS with theadjustmentforacontinuousdividendyield,i.e.,BSwith𝑆𝑒4W*astheunderlyingratherthanBSwith𝑆astheunderlying.28Inthatcase,ourresponsewouldbethatthisobjectionmustbe incorrectbecausebothmodelsareequivalent.Consideralsothe fundamentalmechanicsofoptionpricing.Theoptionpricereflectsthepayoffatexpirylessthecostofhedging,andthecostofhedgingresultsfromtheconstantrebalancingdrivenbychangesindelta,buttheoptiondeltasarethesameanddrivenbythe𝑑5term:

(3A.9)BSdelta=B76delta=−𝑒4I*𝑁(−𝑑5)Forbothmodels,the𝑑5termisdrivenbothbychangesintheassetprice,andbychangesinrandq. In theBlackScholesmodel this impact isexplicit,whereas inBlack76thisimpactisimplicitinthedefinitionofF.

28Asdiscussedalready,ifwearenotmakingthisassumption,thenwearedealingwithaversionofBSthatisnotappropriate.

26

ChapterFour:Loan-to-ValueRatiosTheLoan-to-Valueratioistheratiooftheloanamount,ortheoriginalloanamountplusrolledupinterest,tothepriceofthepropertythathasbeenmortgaged.TheRelationshipofLTVtoAgeERMlendershavetodecidehowmuchtoloanagainstthemortgagedproperty,andakeyfactortobeconsideredistheageoftheborrower.Giventhatthe loanrate is likelytoexceed any future house price inflation (hpi) rate, then the longer the loan lasts, thegreaterthelikelihoodthattheloanwillgointonegativeequity.Thereforelenderswillofferlowerloanstoyoungerborrowers.Anexampleofarulegoverningthisage-dependencyisthe‘age–40’rulesetoutbyHostyetalia(2007,p.31)andusedintheirNNEGanalysis:“Maximuminitialloantovalueratio(MLTV)15%atage55increasingby1%foreachyearofageto50%atage90(youngerageforjointlifecases).”OneformstheimpressionfromHostyetaliathatthisrulewasinuseintheperiodbeforetheywrote.(Afterall,ifitwasnot,thenwhywouldtheyciteit?)Ifthisviewiscorrect,thenitwouldappearthatLTVs/ageratioshaverisensincethenandthat lendersnowroutinelyofferlargerLTVsthantheyusedto,tolendersofthesameage.Ourevidencecomesfromtwosources.ThefirstisfromarecentspeechbyDavidRule,thePRA’sExecutiveDirectorofInsuranceSupervision(Rule,2018).Mr.Rulemakesthefollowingstatement:

Chart5plotsloan-to-valueratiosagainstageforequityreleasemortgagessoldbylifeinsurersin2017.Thepinkswatheshowsinsurers’risklimits.Thegreatmajorityofmortgagesarewithinriskappetites,butsomeloansexceedthelimits.Thismayormaynotbeaproblem.Forexample,itcouldbeexplainedbymedicalunderwritingofthemortgagethatjustifiesahigherloan-to-valueratio for a younger customer if their life expectancy is impaired. (Ouremphasis)

TheChart5hereferstoisreproducedbelow:

27

Hedoesn’tdefine“risklimits”butwemightinterprettheselimitsasthosesuggestedbyagoodpracticerulesi.e.,arecommendedltv/agerulewithsomediscretionaroundit.Thekeyphraseishereis“soldbylifeinsurersin2017,”sothesearenewlymintedERMsandnothistoricalonesThe relatively small number of points above the bounds seem to us to be thosewithimpairedlives,whowouldbeofferedbetterLTVtermsthanthoseinnormalhealth.Thelargenumberbelowwouldseemtoustoreflecttheimpactofun-drawdownbalancesindrawdownERMs.Thepinkriskboundssuggesttousthatfirmsareusingsomethingclosetoa‘modifiedage–30’rulethatgoesasfollows:

• Forages55to80:LTV=age/100–30.• Forages81to85:ΔLTV=Δage/100.• Forage86+:ΔLTV=0.

Wehavealsoseenexamplesfromcurrentindustrypracticethatareconsistentwiththis‘modifiedage–30’ruleandleadustobelievethatthisrule(orsomethingfairlyclosetoit)isnotabadapproximationtotheLTVsincurrentfreshmintedERMs.Thispointmade,therecanbeconsiderablevariationinLTVsacrossproducts(see,e.g.,LegalandGeneral(2019)orTunaru(2019,p.63,Table12)).29TheImpactofRollupandPastHousePriceGrowthonHistoricalLTVs

29LegalandGeneral“FlexibleLifetimeMortgage”(2019):https://www.legalandgeneral.com/adviser/files/retirement/literature-and-forms/product-summary/Q0052431.pdf

28

OnemustalsoconsiderthatafteranERMloanismade,LTVswillevolveafterwardsinlinewiththeloanrateandsubsequentlyrealisedhousepriceinflation.If𝐿𝑇𝑉(0)istheinitialLTV,after1yeartheLTVwillbe(4.1)𝐿𝑇𝑉 = 𝐿𝑇𝑉(0) × 𝑒H4fgh(5)where𝑙=(assumedconstant)loanrateandℎ𝑝𝑖(1)=housepriceinflationafterthefirstyear.After2years,theLTVwillbe(4.2)𝐿𝑇𝑉 = 𝐿𝑇𝑉(0) × 𝑒H4fgh(5) × 𝑒H4fgh(M)andsoforth.WecanformsomeideaofhowtheLTVsfromloansmadeinthepastwouldhavemovedbyasimplehistoricalsimulation.Let’sassumethatERMloansweremadeinthepastineachoftheyearsfrom2000to2017,andlet’sassumethattheseweremadeonthebasisofa30%LTVandaloanrateof7%,notingthatthesecalibrationswouldbereasonablefor this period. The next figure shows how the LTVs would have subsequently‘performed’.

Figure4.1:HistoricallySimulatedLTVs:2000to2017

Notes:Basedonloanrate=6%andLTV=40%whentheloanistakenout.

Tointerpretthefigure:anERMwitha30%LTVtakenoutin2000wouldnowhaveanLTVofabout39%iftheloanwerestillactivein2017;a30%-LTVloantakenoutshortlybefore the crisis would have an LTV of about 51% if the loan were still active; and(naturally!)a30%-LTVloantakenoutin2017wouldstillhaveanLTVof30%in2017.Weseethattheloansmostatrisk(inLTVterms)werethosetakenoutshortlybeforethecrisis.ThedrivingfactorbehindtheseLTVmovementswasthehpi:thestrongerthehpi,thelowerthegrowthoftheLTV.

29

We should however bear in mind that past performance is no guarantee of futureperformance.FutureLTVsDependonFutureHousePriceGrowthForanygivenLTV,itsfuturepathwilldependonfuturehouseprices.RemembertoothatiftheLTVexceeds100%whenexitoccurs,thentheNNEGwillbetriggeredandthelenderwilllosethedifferencebetweentherolled-uploanamountandthehousepriceatthattime.

Figure4.2:FutureHousePriceGrowthandthePathofLTVs

Notes:Basedonloanrate=6%andLTV=40%whentheloanistakenout.

TherapidratesofgrowthofthetoptwoLTVsplots(i.e.,thoseforhpi=0%inblueandforhpi=2%inred)arestriking.Inthefirstcase,LTVhits100%injustover15yearsandthesecondcase ithits100%in23years. Inthethird(hpi=4%,black)case,LTVhits100%in46years.Themessage is that lendersneedahigh futurehpi rate tokeep theirpotentialNNEGlossesdown.Forexample, ifhousepricesremainflatandtheborroweris70whenhetakesouthis loan, then there is a good chance thathewill live the15ormoreyearsrequiredtotriggertheNNEG.Bycontrast,ifhousepricesgrowat4%p.a.,thenthesameborrower is vanishingly unlikely to live the 46 years needed to trigger the NNEG.Consequently,anERMloanishighlyexposedtohousepricerisk.TheexposureofERMstohousepriceriskisfurtherillustratedbyasimplestresstest.Supposethat thesameborrowertakesout thesameERMloan,andthenthenextdayhousepricesfall40%.TheLTVis40%onthedaytheloanistakenout,butthe40%housepricefallimpliesthattheLTVjumpsto(1/(1-0.4))*40%=66.7%thenextday!Ifhousepricesthengrowatzeropercent,theLTVwillhit100%in8years;iftheygrowat2%theLTVwillhit100%in12yearsandiftheyriseat4%,theLTVwillhit100%in22years.

30

Thechancesofsuchanoutcomearepresumablysmall,butthepointistoillustratetheriskexposure.AmorelikelyadverseoutcomewouldbeaJapantypescenarioinwhichhousepricesdonotfallsuddenly,butinsteadfallsecularlyoveralongtime.UnderthisJapanscenario,LTVwouldhit100%injust13years.

31

ChapterFive:TheRisk-FreeRateFigure5.1showsaplotofrecentBankofEnglandspotrates:

Figure5.1:BankofEnglandSpotRates

Source:BankofEnglandUKnominalspotratecurve,downloaded6March2019.

Ifonewishedtouseasingle‘representative’spotrate,thenourearlierbaselinechoiceof1.5%isnotunreasonable.However,inamoresophisticatedanalysisthereisnogoodreasonnottousethewholespotratecurve,withspotratesformaturitiesgreaterthan40yearsset,e.g.,asequaltothe40-yearspotrate.

32

ChapterSix:TheLoanRateTheloanrateorrolluprateistheratethatthelenderchargesontheERMloan.Thisratewillbe fixed for the lifeof the loan,but therateappliedtodrawdownERMloanswilltypicallybe floating, in thesense that therateapplied toadrawdown typicallyvariesdependingonwhenthedrawdownsaremadeonaloan.InhisNNEGreport,Tunaru(2019,p.29)providesaniceplotoftheevolutionofERMratessince1999:

Figure6.1:EvolutionofERMrates

Notes:Tunaru(2019,Figure8).

Thisplotshowsthatloanrateswereabout7%in1999,thenroseshortlythereaftertojustover8%,beforetrendingdownwardstoabout5%in2018.IntheirAutumn2018marketreport,theEquityReleaseCouncilreport(p.7)thatasofJuly2018,theaverageloanrateacrossproductswas5.22%andthisvalueisconsistentwiththelatestvalueintheTunaruplot.Howmuchvariationthereisacrossproductsisdifficulttosay,butwehaveseencurrentratesthatvaryfrom4.15%AERto6.78%AER.ThelatestavailableloanratesquotedbytheERCandshownintheTunaruplotsuggestthatabaselineloanrateof5.25%mightbeappropriate.

33

ChapterSeven:NetRentalRateandDefermentRates:TheoryNetRentalRatevsDefermentRateLet’sstartbysomeclarifyingdefinitions.Thenetrentalrateistheratethatthelandlordreceives after deducting for void, management costs and maintenance costs. Thedefermentrateisthediscountrateappliedinthedefermentpriceformula,reflectingtheforegoneincomeoruseduringthedefermentperiod.Thesetworatesarefrequentlyruntogether,andasweshallshowlater,theyareinfactmathematicallyidentical,buttheyaredefineddifferently.Inthischapterweshallshowhowtoderiveonefromtheother,basedonthedefinitionsgiven.AFirstPrinciplesAnalysisAmoreprecisedefinitionofthedefermentrateisthediscountratethatwhenappliedtothefreeholdpriceofvacantpossessionresultsinthepriceofdeferredpossession.Thedeferment rate itself is not directly market observable, but it can be estimated as afunctionofmarketvariables.Themethodproposedhereusesnetrentalyields,asfollows.Let𝑑bethecurrentnetnominalannualrental,thecurrenttimebeingthebeginningoftheyear.(Weuse‘𝑑’herebecausetheapproachweareusingderivesfromthedividenddiscountmodel,where‘𝑑’isusedtoreferto(nominal)dividends.)‘Net’meansthegrossor headline rental paid by tenants, less the costs incurred by the lessor such asmanagement,maintenanceand theexpectedcostsofvoidoremptyperiodswhile thepropertyisbeingre-let.Thenweshallshowthat

(7.1)𝑑/𝑆 = 𝑞where𝑑isthenetrentalasabove,withthecurrenttimebeingthebeginningoftherentalyear;𝑆theestimated‘spotprice’,i.e.,thefreeholdvalueofvacantpossessionestimatedasthemarketvalueofanidenticalorsimilarpropertynotencumberedbyaleasehold;and𝑞thedefermentrateasdefinedabove.Then assuming that the value𝑆 of the property is the present value of all net rentalreceipts,which iswhat amarket participantwould reasonably assume, the followingequationholds:(7.2)𝑆 = 𝑑 × 𝑦 × (1 + 𝑦 + 𝑦M + 𝑦j … ) = 𝑑 × (𝑦 + 𝑦M + 𝑦j … ) = 𝑑 × 𝑦/(1 − 𝑦)where𝑦 = (1 + 𝑔)/(1 + 𝑟 + 𝜋),𝑟istheriskfreerate,𝜋istheriskpremiumrequiredbyinvestors inresidentialproperty,and𝑔 isgrowthofnet income (e.g.,dividendsornet

34

rental,notpropertyprice).30Soforanyrentalcashflowperiodweprojecttheyear-endnetrentalbytheexpectedgrowthrate𝑔,thendiscountbackbytheinvestorrequiredcostofequity𝑟 + 𝜋.Rearranging,itfollowsthat(7.3)𝑑/𝑆 = (1 − 𝑦)/𝑦Wenowwantthedefermentvalue𝑅m,i.e.,thevalueoftheproperty𝑆minusthelostnetrentalfor𝑛periods:31(7.4)𝑅m = 𝑑 × 𝑦 × (𝑦m + 𝑦mn5 … )Assumethatthereisnotermstructuretocostofequityorgrowth.(Theeffectoftermstructurewillbediscussedbelow.)Thefollowingequationisthentrue:(7.5)𝑅m = 𝑑 × 𝑦 × (𝑦m + 𝑦mn5 … ) = 𝑑 × 𝑦 × (1 + 𝑦 + 𝑦M … ) × 𝑦m = 𝑆𝑦mi.e.,(7.6)𝑅m = 𝑆𝑦mDefine𝑞asthediscretelycompoundeddiscountrateappliedtospot𝑆suchthat:(7.7)𝑅m = 𝑆𝑦m = 𝑆(1 + 𝑞)4mHence(7.8)𝑦 = 1/(1 + 𝑞).From(7.3),substituting1/(1 + 𝑞)for𝑦:(7.9)𝑑/𝑆 = (1 − 𝑦)/𝑦 = (1 + 𝑞)(1 − 1/(1 + 𝑞)) = (1 + 𝑞) − (1 + 𝑞)/(1 + 𝑞) = 𝑞andhence(7.10)𝑑/𝑆 = 𝑞which was to be proved. Therefore we can estimate the deferment rate 𝑞 usingobservablevaluesforthevalueofvacantpossession,𝑆,andavalueofthepreviousnetrentalamount𝑑0.Observethat(7.10)holdstruewhatevergrowthratewechoose,andwhateverinterestrateandriskpremiumarerequiredbyinvestors. 30(7.2)followsfromapplyingthediscountdividendmodel(e.g.,Gordon,1959)withpropertypricesandrentalstakingtheplaceofstockpricesanddividends.See,e.g.,https://en.wikipedia.org/wiki/Dividend_discount_model.31 I.e.wherelostrental= 𝑑 × (𝑦0 + 𝑦5 +⋯+𝑦m45)

35

Whichisreallystrange,too.Tunaru(2019,p.50)says“Forrisk-managementcalculationpurposesthen,itisveryimportanttohaveanaccuratemeasurementof𝑞.Lackofdataavailability and long-term horizon makes this exercise extremely difficult, if notpractically impossible.” This claim seems plausible.Who could predict unobservablessuch as dividend growth, risk premia and so on? Yetwe can calculate𝑞 by a simpleformulausingobservablevariables!TermStructureofGrowthInthederivationofequation(7.1)itwasassumedthatthereisnotermstructuretothecost of equity (𝑟 + 𝑃) or to growth (𝑔). Assuming a term structure would make adifferencetothedefermentrate.Decreasingthediscountrateforearlyperiods(saythefirst20years)orincreasingthegrowthrateforthesameperiodwouldhavetheeffectofincreasing𝑞,becauseitwouldmakethevalueoflostincomehigherasaproportionofthedefermentvalue.However,apronouncedtermstructuretoeitherriskpremiumorgrowthseemsunlikely.A forward rental rate is the rateonewouldpay to lease thepropertywitha forwardstartingdateforacertainperiod.Butwhywouldthemarketimaginethattheforwardratebetweenyears39-40 issignificantlydifferent fromthatbetweenyears40-41, forexample?Itisdifficulttoseewhatinformationwouldjustifysuchajump,andtheremustbeapresumptionthatweshouldn’t introduceadditionalcomplicating factorswithoutgoodreason. TermStructureofDefermentRateObservationsofdefermentratesusingleaseholdpricesshowatermstructure(ofwhichmoreinthenextchapter),withthedefermentrateforshortleaseshigherthanforlongleases.This effect arisesbecause thevalueof a short-term leaseapproaches thatof ashort-termrental,andtheshort-termrentalreflectsthegross,ratherthanthenetrentalyield.As the leasehold term increases, itsvaluewill approach that impliedby thenetrentalyield,whichcanbeshownasfollows.Let𝑞0betheshortterm(i.e.,annual)rentalrategrossofannualcosts𝑐suchasvoidrate,maintenance,shareofmanagementetc.Assumethefollowing(crude)model:(7.11)𝑞0 = (1 − 𝑐)𝐴 = 𝑞*(1 − 𝑐/𝑡)𝐴where𝐴 = (𝑦 + 𝑦M + 𝑦j … )and𝑞*isthegrosseffectiverentalrateoveraletof𝑡years.Then(7.12)𝑞* = 𝑞0(1 − 𝑐)/(1 − 𝑐/𝑡)Thusthegrossrentalconvergestothenetrentalastheleaseholdtermincreases.

36

Figure7.1givesanillustrativeplotofthegrossrentalvsleaseholdterm.

Figure7.1:GrossRentalvsLeaseholdTerm

Notes:basedonillustratevaluesofinitialgrossrental=5.5%and𝑐(costs)=25%.Forthechosencalibration(𝑞0=5.5%and𝑐=25%), thegrossrentalgoes frominitialvalue,5.5%towardsalong-runvalueof4.1%. TheDefermentRateandtheRealRisk-FreeRateInConsultationPaperCP7/19,thePRAproposed“totakeaccountofmovementsinrealrisk-freerateswhensettingthedefermentrate,”’ inordertopreventvariability intherealrisk-freeratecausingvariabilityintheforwardrate:

The PRAwould increase (reduce) the deferment rate if the review showsthere has been a material increase (reduction) in long-term real risk-freeinterestratessincethelastupdate.32

Inourderivationofequation(7.10)above,however,weshowedthatthedefermentrateisequaltothecurrentnetrentalyield,i.e.thenominalnetrentalpaymentdividedbythecurrentnominalhouseprice.Therealrisk-freeratedoesnotevenenterintoit!Sowhatisgoingonhere?WellitwouldappearthatthePRAproposalimplicitlydependsonsomeassumedrelationshiporequivalencebetweenthedefermentrateandtherealrateofinterest,butnofurtherdetailsaregiven.Diggingdeeper,itturnsoutthatthereisanequivalence,butonlyunderconditionsthatdonothold.Westartwiththedividenddiscountequation:(7.13)𝑞 + 𝑔 = 𝑟 + 𝜋 32CP7/19S2.4

37

where𝑞isthedefermentrate,𝑔thegrowthofnominalnetrental,𝑟theriskfreerateand𝜋theriskpremium.Nowassumefirstthatthepropertyinvestmentisriskfree,i.e.thatthereisnoriskpremium𝜋.Then(A1) 𝜋 = 0Substitutinginto(7.13)weobtain(7.14)𝑞 + 𝑔 = 𝑟Second,assumethat𝑔,theimputedgrowthinnetrental,isequaltothegeneralinflationrate𝑖:(A2) 𝑔 = 𝑖Hence(7.15) 𝑞 + 𝑖 = 𝑟Finally,assume(asstatedinCP7/19para2.5thatthenominalrate𝑟isthesumoftheexpectedgeneralinflationrateandtherealrate𝑟𝑟,arelationshipknownastheFisherEffect:(A3) 𝑟 = 𝑖 + 𝑟𝑟andwhere(A4)𝐸[𝑖] = 𝑖i.e., we assume that expected and actual inflation are the same. Substituting andsubtractingbothsidesof(A3):(7.16) 𝑞 + 𝑖 = 𝑟𝑟 + 𝑖Hence(7.17) 𝑞 = 𝑟𝑟Given those fourassumptions (A1-A4), it follows that thedefermentrateand therealinterestrateareidentical.Wewouldquestionthoseassumptions,however.First,apropertyportfolioisclearlynotriskfree. Aninvestmentinahousingportfolioiscommensuratetoaninvestmentinarisky index-linkedbond,asopposed toan investment inan index-linkedgilt,which isvirtuallyriskfree.Thisconsiderationsuggests𝜋 > 0,sothattheriskydefermentratewillbehigherthantherealrisk-freerate.Moreover𝜋mayvarythroughtime.

38

Second,whilerentalinflationandgeneralinflationarelikelytobecorrelated,theyarenotthesame.Nominalrentalswilltendtogouplinewithinflation,indeedrentalcostsarepartoftheUKConsumerPriceIndex,33butasFigure7.2belowshows,rentalinflationandCPIarefarfrom100%correlated.Therearelongperiods,suchas1991-2004,whenrentalinflationisconsistentlyhigherthanCPI.Figure7.2:UKConsumerPriceInflationversusUKRentalInflation,1970-2017

Source:OECD

Third,assumption(A3)abovedependsontheunobservablequantity,theexpectedfuturerateofinflation.Thisvariable,astheBankofEnglandmustknow,isdifficulttopredictwithanycertainty,andhenceisdifficulttomonitor.34Bycontrast,thenetrentalyield,which we proved above to be mathematically identical with the deferment rate, isrelativelysimpletoobserve.Hence,ifthePRAwantstomonitorthedefermentrate,itshouldmonitordevelopmentsinthenetrentalyield.35

33 HistoricaldatafromONS1988-2004canbefoundhere. 34ItispossiblethatthePRAintendstomonitormarketexpectedrealinterestusingthereturnonindex-linked gilts. However, returns on index-linked gilts have been negative in the 2010s, whereas it isimpossible for thedefermentrate tobenegative.TheCPalsonotes (S2.4) that thedefermentratewillalwaysremainpositive,inordertocomplywithPrincipleIIIofSS3/17,butgivenorationaleofwhythisshouldbeso.35Notethatthemathematicalequivalenceofthedefermentrateandnetrentalyieldalsodependsonthedividenddiscountmodel,butwithoutadditionalassumptionslike(A1)-(A4)above.

39

ChapterEight:NetRentalRateandDefermentRates:CalibrationWecandecomposethedefermentrate𝑞asfollows:(8.1)𝑞 = 𝑦 − 𝑣 − 𝑐 −𝑚where𝑦 isthegrossrentalyieldortheyieldpaidbythetenant,𝑣 isthevoidrate,𝑐 ismanagementcostand𝑚isthemaintenancecost.Definethemaintenancecost𝑚astherateofexpenditure(asapercentageofgrossrental)requiredtokeepthepropertyinperfectcondition(i.e.suchastoachievethebestsalesprice forapropertyof thatsize in thesamearea),anddefinethetenantmaintenanceshare(𝑠)astheproportionof𝑚thatthetenantislikelytospendonmaintenance.𝑠willtypicallyvarybetween0and100%.Forashortrental,𝑠willbeclosetozero,andforalongletwewouldexpect𝑠tobecloseto100%intheearlyyearsoftenancy,fallingovertime.Inthefinalyearsitmightfalltozero,evenforastandardtenancy,giventhelackofincentive tokeep in fullorder for the landlord’sbenefit.36ForanERM, itwouldseemunlikely that the ‘tenant’ at endof life, perhaps in the situationwhere theNNEGhadbitten,wouldhaveanyincentivetokeeptheproperty ingoodcondition,sowewouldexpect𝑠tofalltowardszerointhatcasetoo.Wenowusethefollowingcalibrations:

• Void:weusethestandard‘1monthin12’ruleofthumb,i.e.,𝑣 = (1/12) × 𝑦.37• Managementcost:followingTunaru(2019,p.32),weassumemanagementcost

𝑐 = 10% × 𝑦.• Maintenancecost:againfollowingTunaru(2019,p.32),weassumemaintenance

costs𝑚 = 15% × 𝑦.Thus,themaintenancecostbornebythelandlordandtobesubtractedfromthegrossrentalyieldis𝑚 = 15% × 50% × 𝑦 = 7.5% × 𝑦.Wethenhave(8.2)𝑞 = 𝑦 × (11/12 − 0.1 − 0.075) = 𝑦 × 0.7417.Thus,thenetis74.17%ofthegross.AgainfollowingTunaru(2019,p.31),wetake

36Infact,wecanalsoimagine𝑠 < 0.Soif𝑠 = 0reflectsnoactiveefforttokeepthepropertyincondition,𝑠 < 0reflectsadeterminedeffortbyoccupierstostriptheproperty(e.g.,oflightfittings,marblefireplaces,etc.)ortrashtheproperty!Ithappens.37Analternativeistouseempiricalvoiddata.Averagevoidperiodforlandlordsinprivaterentedsectorin the United Kingdom (UK) have varied from 2.4 weeks to 2.9 weeks. (Source:https://www.statista.com/statistics/421102/rental-properties-void-periods-in-the-uk/. Accessed 19March2019.)Ifwetakethemid-point,2.65weeks,thentheaveragevoidratebythismeasurewouldbe2.65/52=5.1%,ascomparedtothe‘ruleofthumb’voidrateof11/12=8.3%.

40

(8.4)𝑦 = 5.6%.Therefore(8.5)𝑞 = 74.17% × 5.6% = 4.15% ≈ 4.2%.Soweuse𝑞 = 4.2%asour‘bestestimate’.38

38Ifweusetheempiricalvoidrateof5.1%,thennetis77.4%%ofgrossandwewouldobtain𝑞 = 4.3%.

41

ChapterNine:DilapidationItisalawofnaturethatallpropertiesdepreciateovertime,eventheGreatPyramidofGiza.Westopdilapidationbymaintainingtheproperty.Thecostofmaintenanceisthusareflectionofthistendencytodilapidate:iftherewerenodilapidation,therewouldbenocostofmaintenance.Butwhereasdilapidation isanaturalprocess, thedecision tomaintainisachoice.Wecanchoosetomaintainourpropertyorwecanletitdilapidate.TheERMonomicsofDilapidation(I):BorrowerMoralHazard

It is usually the case that the owner has a long-term interest in the upkeep of theirproperty,anditisthenreasonabletopresumethatheorshewouldmaketheinvestmentsinthepropertythatarenecessarytocounterthedilapidationthatwouldotherwiseoccur.

Nowconsiderequityrelease.Naturally,thelenderwouldpreferthattheborrowerlookafterthepropertyasmuchastheywouldiftheystillhadalong-termvestedinterestinit,buttheborrowernolongerhasanysuchinterest.Inanycase,theborrowerwouldbecashpoor(becausewhyelsewouldheorshehavetakenoutanERMloan in the firstplace?)andashe/sheages,itwouldbecomemoredifficulttomaintainthepropertyeveniftheborrowerwishedtodoso.Thereisaclassicmoralhazardproblem.Ofcourse,thelendercanimposeconditionsaboutmaintenance,buthaslimitedmeansofmonitoringmaintenanceorenforcingthem.Soanysuchcovenantswouldhavelimitedeffectivenessandthereisstillamoralhazardproblem.Thelikelihoodthenisthattheborrowermightmakesomeinvestmentsinthepropertyearlyon(e.g.,anewkitchenorpatio)butastheyage,anysuchinvestmentswillbebasedon short-term considerations or become increasingly cosmetic (e.g., a paint job) orabsolutelynecessarytobeabletocontinuelivinginthehouse(e.g.,payingtofixawaterleak). By the time the lender takes possession of the property, it will likely havedepreciatedinvalue,relativetothevalueonemustassumeitwouldotherwisehavehad,and certainly relative to the value itwould have hadhad it been ownedby someoneyounger,withthemeans,abilityandincentivetomaintainitproperly.Consequently,anERMedpropertywilllikelydepreciateinvalueoverthelifetimeoftheERM,relativetothevalueofanotherwisesimilarpropertythathadnotbeenERMed.Thisdilapidationeffectisreasonablywellknown,butthereisnoconsensusonhowtodealwithit.ToquotePRACP48/16:

Opinionontheappropriateadjustment[neededforit]wasdivided,suggestionsincludedadjustmentstopropertyvalue,propertyvolatility,HPI,andamarginfordilapidation.Somefeltthatsystematicunderperformanceriskduetoadverseselectionshouldbeallowedforinthevaluation.(pp.25-26)

42

TheERMonomics ofDilapidation (II): StochasticDilapidation and LenderMoralHazardTherearealsootherfactorsinvolvedthatarenotsowellknown.Considerthefollowingfigure.39

Figure9.1:NNEGClaims:AvivaERF4Dataset

Source:Aviva:https://www.erfunding.co.uk/literature-library/erf4.html

ThechartshowscumulativeNNEGclaimsonERF4,asetofAvivaERMswhichstartsin2004.MostERMsstartwithaloantovalueoflowerthan50%,andpropertypriceshavegoneupinmostareasoftheUKsince2004,soittakesafewyearsforthecompoundloanamounttoreachcurrentpropertyvalueandreachNNEGterritory.SolowNNEGclaimsintheearlyyearsaretobeexpected.Butthereisasurpriseinstore.TheexerciseoftheNNEGwasinnocaseduetothecompoundloanvaluehittingthehousepriceindex.Avivahelpfullyprovide(i)and‘indexedhousevalue’atexit,togetherwith(ii)theoriginalvaluationand(iii)therealisedsalevalue.TheindexedvalueistheoriginalvalueprojectedbytheincreaseintheHalifaxindexsincethevaluation.Itturnsoutthatif all properties had followed the index,noNNEG would have been exercised,andallpropertieswouldhavebeensafelyoutofthemoneywhentheloanswererepaid.Instead, the exercisewas in all cases due to the underperformance, often a dramaticunderperformance,ofthepropertiesusedascollateral.Asanextraordinaryexample,considerthepropertythatcausedthelargeblipin2016.Itwasoriginallyvaluedat£1.2m,withanestimatedLTVof45%,i.e.,aloanvalueofabout£540,000.(Avivadonotprovideanexplicitloanrate,butweestimateabout7%basedon

39WethankT.PocockforpointingoutthisAvivaERMdataset: https://www.erfunding.co.uk/literature-library/erf4.html.

43

redemptionsandloanamountsatexit.)Theloanvalueatexitwas£1.4m,butthesalepriceofthehousewasonly£625,172,leavingaNNEGlossof£763,225.Inotherwords,whiletheHalifax indexwentup70%,withthe indexedhousevaluebeingover£2m–easilyenoughtocovertheloanvalueatexitof£1.4m–thepropertynotonlyfailedtofollowtheindex,butactuallyfellinvalue(byabout50%).Soitwasalsowith44%ofthepropertieswheretheNNEGwasexercised:nearlyhalfthepropertiesusedascollateralforequityreleasenotonlyfailedtomatchtheindex,butwereworthlessthanwhentheyfirstcollateralisedtheloan.40Whatishappeninghere?Well,weknowthataspeoplegetolder,theyarelessinclinedandlessabletomaintaintheirpropertytothestandardtheymaintainedearlier.Wealsoknowthattheyhavelessincentiveto,becausetheyhavemortgageditaway.Wediscussedtheseissuesalready.Evenso,themagnitudeoftheeffectissurprising.Itseemsthattheexerciseoftheputoptionappearstobeduetotheunderperformance,oftenadramaticunderperformance,ofthepropertiesusedascollateral.Evenmoresurprisingisthatfromtheperspectiveofthelender,theunderperformanceseemstohaveaconsiderablerandomelement,i.e.,wearedealingwithstochasticdilapidation!41ThenextFigureshowsscatterplotofthe‘achieved’valuesofERMedproperties,i.e.,theamounts that the lenderwasable to realiseafter theborrowerexited,expressedasapercentageoftheindexedvalue,basedontheSharedAppreciationMortgageSecurities(SAMS)originatedbyHBOSinthelate1990s:

40 One of our correspondentswho understands these datamuch better thanwe do suggests that thisexampleis“quiteclearlyafraudcase.”Hebacksupthisclaimwithapersuasiveanalysisandconcludesthattheonlynon-fraudexplanationhecanthinkofisifthesitehadbecomecontaminatedbyradioactivity.Wedefer to his judgment. Needless to say, firms should then either be modelling thepossibleimpactoffraudortheyshouldputsuitablemitigatingcontrolsinplace.41SeealsoCP7/19p.8:“Akeypointtodrawoutisthatlendersofequityreleasemortgagesareexposedtotheriskofindividualpropertyprices.Thisisbecauseinsurersprovideano-negative-equityguaranteetoeveryborrower.Modellingapproachesfocusedonhousepriceindicesdonotcapturealltherisks–aportfolioofoptionsisaverydifferentthingtoanoptiononadiversifiedindex.IndeedUKinsurershaveexperiencedanumberoftheseguaranteescrystallisinginrecentyearsdespitetherapidriseinUKhousepriceindicesoverthepastdecades.OnereasonisthatdifferentlocalitiesoftheUnitedKingdomhaveseenwidelyvaryinghousepriceinflation–anationalindexmaskstherangeofoutcomes….Anotheristhatsome properties may become dilapidated if elderly borrowers are unable to maintain the property.Willingness tomaintainmay be lowerwhere borrowers have limited or no equity remaining in theirproperties.Equityreleasecontractsgenerallyrequirepropertiestobemaintained.But,inpractice,lossesdooccurandcannotnecessarilyberecovered.”

44

Figure9.2:Indexedvs.AchievedHousePrices

Source:SAMS.Theachievementratesarerealdata;theindexedvalueistheoriginalvalueprojectedbytheincreaseintheHalifaxindexsincethevaluation.

Thereddotsarethescattershotoftheindividualachievementratesinthesample.Thedarkerbluerandom-lookinglineisasimulatedhousepriceindex.Whatjumpsoutisthattheachievedvaluesareallovertheplacerelativetotheindex.Onaverage the achieved value is about 94%of the index but there is a huge dispersionaround the index. The volatility of the index does not capture the volatility of thedispersionaroundtheindex!An insightful explanation is provided by one of our EUMAEUS correspondents. Toparaphrase:

Theachievementratesarelikebetasinstocks.Everyhouseisindividual.Butthedispersionofresultsaroundtheaverageandthevolatilityofthatdispersionarebothenormous.Theimpactofolderpeopleallowingtheirpropertiestosubsideinlaterlifeisthefirstreactionofmostpeople,butisinfactonlyasmallpartofwhatisgoingon.Now practical experience suggests that the difference in price between thecheapesthouseonastreetcomparedtothemostexpensivehouseonastreet–assumingthey’reallthesamesizeanddesign–shouldbenomorethanabout20%. That’s from our practical experience of taking ownership of propertiesarisingfromsecuritisationsandmanagingthesaleprocessourselvesratherthanrelyingonbankservicers.That’swhatitcoststofixanewbathroomandkitchen,take control of the gardens and generally paint the place before sale. So themaximumspreadofaseriesofsprucehousesandneglectedhousesshouldn’tbemorethanabout20%,andthisspreadshouldputa floorunderthe impactofdilapidation.Thetroubleisthatitdoesn’t.

45

Theproblemisthatnoneofthisprocessoccursinbankmanagedsales–allofwhichhappen“asis”oncethepropertyfallsintotheservicer’sownership.Noone inthechainofsalessetupbygeneralservicershasany incentivetotakecontrolofthevalueatsaleintheinterestsoftheowner.Everyonejustwantsthequickestpossiblesaleandtheremovalof“in-hand”propertiesfromthebalancesheet.Generalservicingispassiveandentirelyacceptingofanypriceaslongasthepropertyissold.Thenetoutcomeisthattheachievementratesinsecuritisationsthatneedactualhousesalesinthefuturetoachievetheircashflowsisgenerallydisappointing.Butnotbecauseofoldpeoplebecomingdisinterestedintheproperty.Themainreasonisthedislocationofactiveinterest[i.e.,incentivetosecureagoodprice]arisingfromsecuritisation.Theagents,lawyers,servicersandcashmanagersdonothavetheactiveinterest inmanagingtheactualhousesaleprocessthatanownerwouldhave.Thesecuritisationprocess itselfdetachesownership fromeffectivesalesmanagement.Sothelowachievementratesweseeinthedataaremostlyattributabletotheinvolvementoftheservicers,mostofwhomhavenoeconomicinterestinmaximisingsalesproceeds.42

Thereis,thus,asecondmoralhazardthatworkstotheadvantageofthelenderrelativetotheborrower.ERMlendersappeartoworkonthepresumptionthatERMloanswillusuallyexpireinpositiveequity,andthisassumptionisconsistentwiththeperformanceofmostERMloanstodate.Giventhispresumption,thenlendershavelittleincentivetomanage the properties they take possession of in order to find a good selling price.Instead, theirconcern issimplyto turnthepropertiesaroundtorecovertheamountsowed,i.e.,tomakeaquicksale,andanyproceedsleftoverarereturnedtotheborroweror their estate. If theborroweror their estate fetch apoorprice, then that is not thelender’s concern and the borrower or their estate has no control over the disposalprocess.Thissecondmoralhazardisanadditionalreasonforthe lowforERMedpropertiestofetchlowsalesprices.Wecanthinkofthissecondmoralhazardascreatinganeconomicdilapidationeffectinadditiontothelowsalespricescreatedbyphysicaldilapidation.

Thereisalsoathirdmoralhazard.Toquoteanotherofourcorrespondents:

theissuefrommyexperienceisthateverybodyhadanincentivetoovervaluethe property from the start, it’s not an issue of dilapidation over time …overvaluation isoneof thebiggestrisksand insurerswhohaveno lendingexperience are just clueless to this. I have seen the same issue in debtconsolidationmortgagesacrossEurope:ifthepropertyvaluedoesnotcomefrom an actual transaction, when you sell the property you always get asurprise!…In anormalpurchase the valueof theproperty is quite certainbecauseitisdeterminedbetweenawillingbuyerandawillingseller,butinERMthereisnosale,onlyanappraisal.Thelender,thebroker,theappraiser

42AsecondcorrespondentconfirmsthisanalysisandrefersustoaFitchUKRMBScriteriaaddendum(18-May-2018)thatstatesthataquick-saleadjustmentof17%ofpropertyvalueisappliedtohousesand25%toflats.Forilliquidproperties(i.e.inthetop5%and1%byprice)afurther10%or15%discountmaybeapplied.

46

andtheborrowerallhaveanincentivetoclosethedeal.Why?Tomeettheirbusiness plan volume. In these circumstances, history teaches that thepropertyvaluewillalwaysbeoverestimated,butthiswillnotappearinanyExcelspreadsheetandthereforeanymodellingwillbebasedoffunreliableLTVassumptions.

Inshort,theimpactofeconomicdilapidationassuchwouldappeartobedrownedoutbymultiplemoralhazards,whosenet impactmight easilybe confusedwithdilapidationunlessonedigsdeeperintotheunderlyingeconomiccausesoflowachievementrates.

Thesemoralhazards, in turn,reflectunderlyingagencyproblemswherebythosewithfiduciarydutiesofcaretowardsshareholdersinsteadputtheirowninterestsfirst.

StatisticalAnalysisoftheSAMSAchievementRatesTable 9.1 shows the main statistical features of the achievement rates in the SAMSdataset:

Table9.1:MainStatisticalFeaturesoftheSAMSAchievementRatesMean 94.3%

Standarddeviation 19%Skewness 0.30Kurtosis 3.82

Samplesize 1420Range [31%170%]

5%lowerbound 64%5%upperbound 125%

Source:SAMS.Weseethattheachievementratesarehighlydispersed,somewhatpositivelyskewedandsomewhatheavytailed.NotethattheresultsinTable9.1reflectthewholedataset,andinterpretationoftheseresultsismadedifficultbythefactthattheseloanshavedifferentdurations.Tomakeinterpretationeasier,Table9.2showsthesameresultsforthoseloansthatstartin1997andendin2017.Theseloanshaveanapproximatedurationof21years.

47

Table9.2:MainStatisticalFeaturesoftheSAMSAchievementRatesforLoansSpanning1997to2017

Mean 91.0%Standarddeviation 23.4%

Skewness 0.01Kurtosis 2.13

Samplesize 69Range [47%138%]

5%lowerbound 48%5%upperbound 129%

Source:SAMS.CalibratingtheDilapidationRateWecanthenusetheseresultstoestimatethedilapidationrate.

Forthe1420loansintheentireSAMSdataset,theaveragedurationis10.74yearsandtheaverageachievementrateis94.3%.Theaverageannualdilapidationrate𝑑isthen

(9.1)𝑑 = −(1/10.74) × ln(0.943) = 0.54%.Alternatively,forthe69loansintheSAMSdatasetthatstartin1997andendin2017,theaverage duration is about 21 years and the average achievement rate is 91.0%. Theaverageannualdilapidationrate𝑑isthen(9.2)𝑑 = −(1/21) × ln(0.91) = 0.45%.

So for ERMed, properties, there is an additional physical/economic/stochasticdilapidationeffectthatisnotpresentwhendealingwithtypicalnon-ERMedproperties,andtheassociatedaveragedilapidationrateisabout0.5%ayear.43

Finally,Table9.3showsthesemeandilapidationresultsalongwiththe5%lowerboundandthe5%upperboundforthedilapidationrateforthesecondsetofloans.

43Toquoteafurthercommentfromthepreviouscorrespondent.“Isthedifferenceinabsolute£explainablewith the costs of repairs and inefficiencies of an auction sale? Well, you can say that the sale isinefficient,orthatthepropertywasneverworththatmuch,especiallysincetheoriginalvaluewasnotanarm's lengthprice.There issomuch fraud into thosenumbers, IMHO.That isnotastochasticprocess,rather systematic over-valuation. If I was a mortgage broker, I would say to a customer that anovervaluationof20%+shouldgototallyunnoticed,andmaybewecanpushitto30-40%.Beyondthat,thelender probably has automated checks in place (Zoopla) though not in the 90swhen the SAMSwereoriginated.Thatfitswithyour0.5%pa,justfromadifferentperspective.”

48

Table9.3:MainStatisticalFeaturesoftheSAMSAchievementRatesforLoansSpanning1997to2017WholeSAMSdataset

Mean𝑑 0.54%Samplesize 1420SAMSloansspanning1997to2017Mean𝑑 0.45%

5%upperbound 3.5%5%lowerbound -1.21%Samplesize 69

Source:SAMS.Theseboundsreflectthe90%confidenceboundsforthedilapidationrate,andspan3.5%attheupperendto-1.21%atthelowerend.Thesmallsamplesizetellsusnottorelytooheavilyontheseresults,buttheyareindicative.44

Wearetemptedtoconcludethatthereisalotmoregoingonherethanmerephysicaldilapidation, although doubtless therewould be that too. The ‘other things going on’wouldappeartobetheeconomicconsequencesofthevariousmoralhazardswehavementioned. If these other factors dominate, as they appear to do, then the term‘dilapidation’ is only a small part of thepicture and, onemight say, is looking a littledilapidated.

44 AfulleranalysisoftheSAMSdatabasemight,e.g.,lookattheachievementratesanddurationsforeachloan, estimate 𝑑 for each, and thence we produce a sample of 𝑑 rates, from which we can obtain astatisticallystrongersenseofthem.

49

ChapterTen:VolatilitySupposethatallthestandardconditionsusuallyassumedinthederivationofBlack’76actuallyhold:completemarkets,continuoustradingandsoon.Insuchcircumstanceswewouldbeabletoobserveimpliedvolsfromoptionsmarketdata,andwewouldbaseourvolcalibrationontheseimpliedvols.Now suppose that we don't have these implied vols, e.g., because there is no tradedmarketinresidentialpropertyoptions.Ourstartingpointthenistoobtainavolatilityorsetofvolatilitiesbasedonhistoricalpropertyprices,e.g.,thosefromanhistoricalhousepriceindex.ExistingVolatilityEstimatesCP13/18states(p.9):

2.16ThePRAestimatedavalue for thepropertyvolatilityparameter fromanalysis of residential property price index data. Nationwide, Halifax andOfficeforNationalStatisticsindexdatawereanalysedandseveraltimeseriesmodelswerefittedtothequarterlylog-returnsofdatasetsoveravarietyofhistoricaltimeperiods.ThePRAselectedaparsimoniousmodelthatfittedthedata well, and extracted from the model the unconditional volatility forvarious holding periods, allowing for autocorrelation. Further adjustmentsweremadetoallowforconcentrationriskandbasisriskbetweenthechangesinpricesofindividualpropertiesandtheindex.ThePRA’scentralestimateisofa13%volatilityassumptionfortypicalholdingperiodsforERMs,althoughuse of alternative data choices gives a range of 13%-16%, andmaking anallowanceforparameteruncertaintygivesarangeof11%-18%.Estimatesforpropertyvolatilityprovidedto thePRAby firmsaregenerally in therange10%-15%.45(Ouremphasis)

Inhisreport,ProfessorTunarustates(pp.1,20)thattheNationwidehistoricalindexdatasuggest a range of volatility values between 3.85% to 6.5%.His Table 1 (p. 19) thensuggestsMaximumLikelihoodandMethodofMomentsestimatesofaround3.95%forbothNationwideandHalifaxdatasets,andbasedonthisevidence,heoptsforabaselinevolof3.9%.Itseemstousthatthesevolatilityestimatesareonthelowside,andhisownresults reports fordifferentUK regions and sampleperiodsbasedon theNationwidedatasetsuggestarangeofvolatilitiesspanning3.85%to6.5%,sowewouldsuggestitwouldhavebeenmoreprudenttohavepickedavaluesomewhereinthemiddleofthisrange,sayabout5%,butarguablyhigher.Butlet’sgowith5%asastartingpoint.

45 NotehoweverthatCP7/19issuesinApril2019proposestoabandonthe13%volatilitycalibrationandreplaceitwitharegularlyrevisedvolatilityestimatethefirstofwhichisseeminglyyettobedetermined.

50

VolEstimation:AFirstPassGeometricBrownianMotionToderivehistoricalvolatility,wecanuseanytimeseriesofpricessolongaspricesareavailableataconsistentsamplingperiod𝑘.Thissamplingperiodcanbedaily,weekly,monthlyoranyotherperiod.46Thevolatilityiscalculatedbycomputingthereturns(preferablylog-returns)ofthepriceseries,thencomputingthenon-annualisedstandarddeviation𝜎(𝑘)ofthereturns.Thisvolatilityisspecifictothefrequencychosen–returnsonthesameassetcanproducelowvolatilities if sampled daily, but higher volatilities if sampled less frequently, e.g.,annually.However,standardpricingformulassuchBlack-Scholes/Black’76assumethatthe returns have been sampled annually. To obtain the annualized vol or annualizedstandard deviation when returns are sampled at periods not equal to one year, we‘annualise’ the standarddeviationof returnsbymultiplyingby the square rootof thesamplingfrequency.Forexample,ifreturnsaresampledeveryworkingday,i.e.withafrequencyof250workingdaysayear,wemultiplythestandarddeviationofdailyreturnsbyroot250toobtainanannualisedvolatility.Ifreturnsaresampledeverymonth,wemultiplybyroot12,ifquarterly(astypicalforhousingindices)wemultiplybythesquarerootof4,andsoon.Thus,if𝜎(1)istheannualisedvol,and𝜎(𝑘)isthe(non-annualised)volderivedfroma𝑘yearsamplingperiod,then(10.1)𝜎(1) = 𝑛𝑜𝑛 − 𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑠𝑒𝑑𝑣𝑜𝑙 × (1/𝑘)0.z = 𝜎(𝑘) × (1/𝑘)0.zor(10.2)𝜎(𝑘) = 𝜎(1) × (𝑘)0.zTable10.1showsasetofillustrative𝜎forarangeof𝑘values:47

Table10.1:Illustrative𝝈foraRangeofResamplingPeriods𝒌 𝝈(𝒌)1 5%5 11.2%10 15.8%15 19.4%20 22.4%25 25%30 27.4%

Note:𝑘=resamplingperiod.Obtainedusingequation(10.2).

46 Exceptionscanbeallowed,suchasweekends,bankholidays,periodsofmissingdataetc,andtherearetechniquesforinfillingdataorcorrectingtimingerrors,butthisisawell-understoodandseparatesubjectwhichweshallpassover. 47 As the calculations underlying some of the tables in this chapter can be quite involved, we make available on our website an Excel workbook with the the relevant calculations for all the tables in this chapter.

51

So𝜎canvaryfrom5%to27.4%dependingonthesamplingperiod𝑘,giventhisrangeof𝑘.AutocorrelatedunderlyingThe square root rule depends on the assumption that the asset log-returns followBrownianmotion, the condition forwhich includes randomness, i.e. the returnatanytimeisindependentofwhateverhappensanytimeinthepastorthefuture.Howevertheevidence indicates that property prices are autocorrelated so the independenceassumption does not hold. In an autocorrelated process, a positive return yesterdayincreasesthechanceofapositivereturntoday,andanegativereturnyesterdayincreasesthechancesofanegativereturntoday.Consequently,thesquarerootruleapproachjustoutlinedwillnotdoandweneedtouseanalternativeapproachthattakesaccountofautocorrelation.48ApracticalwaytoimplementthisadjustmentisviatheHurstexponentapproachoutlinedinAppendix1tothischapter.WeestimatetheHurstexponent𝐻overourpropertyindexdatasetandobtain𝜎(𝑡)using(10.3)𝜎(𝑡) = 5% × 𝑡� where𝐻wouldtypicallylieintherangefrom0.7toover1.ForUKdata(source:DallasFed)𝐻 ≈ 0.82,so(10.4)𝜎(𝑡) ≈ 5% × 𝑡0.�MForpricingpurposesitisnotthematurity𝑡thatmattersbuttherehedgingperiod.However, as noted above, standard pricing formulas such Black-Scholes/Black ’76assumethatthereturnshavebeensampledannually.Ifwerehedgeevery𝑘years,wewouldthenusetheformula(10.5)𝜎 = 5% × 𝑘0.�M/𝑘0.z = 5% × 𝑘0.jMtoobtaintheannualisedvolatility𝜎thatwewouldinputintoourputpricingequation.49

48 Thetheoreticalsolutiontothisproblemwassetoutinanimportantpaper“Optionpricingandhedgingwithtemporalcorrelations”byLorenzoCornalba,Jean-PhilippeBouchaudandMarcPotters.(L.Cornalba,J.-P. Bouchaud and M. Potters (2002) “Option Pricing and Hedging with Temporal Correlations,”InternationalJournalofTheoreticalandAppliedFinance5(3):307-320).Thispaperprovidesafairlygeneralanalysisoftheimpactoftemporal(i.e.,auto-)correlationonoptionpricingandtheirconclusionsareclear.“IntheGaussiancase[theoneconsideredinBlack-Scholes],wefindthattheeffectof[auto-]correlationscanbecompensatedbyachangeinthehedgingstrategyandthereforeoptionsshouldbepricedusingthestandard uncorrelated Black-Scholes model (our italics).” The required change can be implemented bymeasuringvolatilityonthesametimescaleastherehedging,butthisqualificationmerelyamountstoanadjustmenttothevolatilitycalibration,ifeventhat. 49By‘annualisedvolatility’wemeanherethevolatilitywhich,whende-annualisedinthestandardwaybymultiplyingbyroot𝑘,willgivethesameresultastheempiricallyderivedvolatilityoftheautocorrelatedseries.

52

Aproofofequation(10.5)isgiveninAppendix2tothischapter.Table10.2showsasetofillustrative𝜎forarangeof𝑘values,assuming𝐻 = 0.82:

Table10.2:Illustrative𝝈foraRangeofResamplingPeriodsfor𝑯 = 𝟎. 𝟖𝟐𝒌 𝝈(𝒌) Annualised𝝈1 5.0% 5.0%5 18.7% 8.4%10 33.0% 10.4%15 46.1% 11.9%20 58.3% 13.0%25 70.0% 14.0%30 81.3% 14.8%

Note:𝑘=rehedgingperiod,H=Hurstexponent=0.82.𝜎(𝑘)isobtainedusingequation(10.4)with𝑡setto𝑘.Annualisedvolatilityisobtainedusingequation(10.5).

Wearenowlookingat𝜎intherangefrom5%to14.8%forthisrangeof𝑘,dependingonourchoiceof𝑘,i.e.,ourchoiceofhedgingperiod.VolatilityAroundtheIndexSofarwehaveonlydiscussedthevolatilityoftheindex,butthereisalsothevolatilityaroundtheindex.Thisadditionalvolatilitywouldincludetheimpactofregionalvariationaroundtheindex,buttherearefurthercontributoryfactorsaswell.Theseinclude,e.g.,the impact of changes in consumer relative demand for different types of property,expansions of nearby roads, the impact of newhousing estates, yuppification,middleclassflight,theopeningorclosingofagoodnearbyschool,andthestochasticdilapidationeffectsdiscussed in thepreviouschapter.Onewill recallFigure9.2 fromthepreviouschapter:

Figure9.2:Indexedvs.AchievedHousePrices

Source:SAMS

53

Recallthatthedarkerbluerandom-lookinglineisahousepriceindexandthereddotsareascattershotoftheindividualachievementratesinthesample.We immediately see that the achieved values aremuchmore volatile than the index.Aboveall,whenseekingtocalibratethevolatility,weneedtokeepinmindthatitisthatdispersionthatmatters,notthevolatilityoftheindexitself.Wecangetanevenbettersenseofthedispersionaroundtheindexfromthenextfigure:

Figure10.1:Indexedvs.AchievedHousePrices(II)

Source:SAMS

Itwould thenbehoveus to revise our earlier index-based volatility estimates to takeaccountofthisadditionalvolatility.WecouldstartwithourearliervolatilitiesreportedinTable10.2.Letuslabelthisvolatilityas𝜎�����whilstnotingthatitisdependentonthevalueof𝑘.Wenowobtain𝜎�� ,thevolatilityoftheachievementrate,asfollows:takearolling standard deviation of the achievement rate and divide by root time to get anannualisedvalue.WeassumeherethatthereisnoHurstorautocorrelationeffectfortheachievement rate, i.e., that the square root lawapplies.We found that the annualisedvolatilityvaluesvaryfrom7%to10%.Let’sworkwiththemiddlevalueof8.5%.Wethenhavetoassumeaplausiblecorrelationbetweentheindexvolandtheachievementratevol.Assumingzerocorrelation,which isnotunreasonable,we thenobtain the resultsreportedinTable10.3:

54

Table10.3:Illustrative𝝈foraRangeofResamplingPeriodsfor𝑯 = 𝟎. 𝟖𝟐𝒌 𝝈𝑰𝑵𝑫𝑬𝑿 𝝈𝑨𝑹 𝝈𝑰𝑵𝑫𝑬𝑿𝒂𝒏𝒅𝑨𝑹1 5% 8.5% 9.9%5 8.4% 8.5% 11.9%10 10.4% 8.5% 13.5%15 11.9% 8.5% 14.6%20 13.0% 8.5% 15.6%25 14.0% 8.5% 16.4%30 14.8% 8.5% 17.1%

Note:𝑘=rehedgingperiod,H=Hurstexponent=0.82,andtheassumedcorrelationbetweentheindex and the achievement rate is zero. The terms in the rightmost column are obtained byPythagoras.

The‘combined’or‘INDEXplusAR’volatilitynowvariesfrom9.9%for𝑘 = 1to17.1%for𝑘 = 30,butnotethatthesenumbersarelittlemorethaneducatedguesstimatesandthe‘true’numberscouldbehigher,e.g.,ifthecorrelationwerepositive.Thetableandthefigureshowthestrongeffectof‘stochasticdominance’,i.e.thetendencyofahigherconstituentvolatilitytodominatealoweronewhencorrelationisnegligibleor,inthiscase,zero.InsuchcasesthetotalvolatilityisgivenbyPythagoras.Forexample,ifthefirstvolatilityis10andthesecond1,thesumofsquaresis101,therootis10.05.Sothe lowervolatility,while10%of thehigherone,hasamarginal contributionofonly0.5%.Table10.3showsthatatamaturityofoneyear,theachievementrateof8.5%isdrivingthecombinedvolatilityof9.9%.Bycontrast,at30yearsitistherescaledindexvolatilityof14.8%thatisthedrivingfactor.InterestRateRiskasaFurtherContributortoVolatilityWe have hitherto assumed (as per Black-Scholes/Black ’76) that the interest rate isconstant,i.e.,thatthereisnointerestraterisk.Infact,interestraterisknotonlyexists,but arises from two sources. Consider the following components of the Blackmodel,reproducedhereinslightlysimplifiedfromChapter3:(10.6) 𝑝𝑢𝑡 = 𝑒4I*[𝐾 × 𝑁(−𝑑M) − 𝐹 × 𝑁(−𝑑5)] (10.7)𝑑5 = [𝑙𝑛(𝐹/𝐾) + 𝜎M𝑡/2]/(𝜎√𝑡)(10.8)𝑑M = 𝑑5 − 𝜎√𝑡(10.9) 𝐹 = 𝑆 × 𝑒(I4W)*

whereasusual:𝑝𝑢𝑡isthevalueofthe𝑡decrementput,𝐾isthestrikeprice,𝐹theforwardprice,𝜎theannualisedinputvolatility,𝑡thetimetoexpiryinyears,𝑟theinterestrate,𝑆the price of ‘spot’ possession of the property, and 𝑁(… ) is the cumulative normaldistributionfunction.

55

Theinterestrateterm𝑟appearsfirst(see(10.6))asadiscounttermwrappedroundthetermsrepresentingthefuturevalueoftheputoption,whichbringsthefuturevalue(i.e.,[𝐾 × 𝑁(−𝑑M) − 𝐹 × 𝑁(−𝑑5)])back topresent value.Here𝑟 plays the roleof anouterdiscountfactor.𝑟thenappearsagain(see(10.9)asaprojectiontermorinnerdiscountfactortakingusfromthespotprice𝑆totheforwardprice𝐹.Eachappearancegivesrisetointerestraterisk,butindifferentways.DiscountrateriskThefirstcanbecalleddiscountinterestraterisk.Thisriskcanbehedgedrelativelyeasily,thegistofitbeingtoswapfloatingintofixed.Amoredetailedexplanationgoesasfollows.Whenatradingdesksellsanoption,itplacesthepremiuminanaccountcalledthe‘hedgingaccount’.Thisaccountearnsinterestfromthefirm’scentralfundingdeskandtheinterestearnedwilltypicallybeclosetothefirm’soverallfundingrate.Tohedgetheriskarisingfromchangesinthisouterdiscountfactor,thedeskshouldmakeaninternalorexternalIRswapintoafixedratewithmaturityattheoptionexpirydate.Itcanthenbeshownthatthisswapguaranteesthat,withnootherchangetakingplaceinthemarket,thehedgeaccountwillearnthefixedrate𝑟intheouterdiscountfactor𝑒4I* .Thedemonstrationgoesasfollows.Let(10.10)𝑃 = 𝐹𝑉 × 𝑒4I*where𝑃istheoptionpremiumpaid,𝑟hereisthelongtermrateearnedontheoptionaccount,and𝐹𝑉isthefuturevalueoftheoptiongivenbytheundiscountedBlackformula.Nowsupposethelong-terminterestrate𝑟changes,butthereisnochangeintheforwardprice𝐹.Suchacircumstancewouldoccurwherethespotrate𝑆changedbyanamountΔ𝑆insuchawaythat𝐹remainedconstantundertheformulaconnecting𝑆with𝐹,i.e.(10.11) Δ𝑆 = 𝑆(𝑒4�I* − 1)whereΔ𝑟isthechangeindiscountrate.Inthissituation𝐹willremainthesameandhencethefuturevalue𝐹𝑉oftheputoptionwillalsoremainthesame.ThechangeΔ𝑃 inthevalueoftheoptionpremiumwillthenbeasimplediscountfunction:(10.12) 𝑃 + Δ𝑃 = 𝐹𝑉[𝑒4(In�I)* − 𝑒4I*]Assumingtheamount𝑃iscurrentlyheldinthehedgingaccount,wecouldreplicatethechangeΔ𝑃if 𝑃 were invested in a long dated zero-coupon bond with maturity 𝑡. Inpracticethesameeffectcanbeachievedbyinvesting𝑃atthefirm’sshort-termfundingrate,butswappingtheshort-termfloatingpaymentsintoazero-couponswap.

56

ProjectionriskInthepreviousexampleweassumedthattheforwardpriceremainsconstantwhilethespotpricechanges,i.e.,ariseinlongterminterestrateswillforcethespotpricelower,whileafallintheinterestrateforcesthespotpricehigher.Thiseffectmightbeexplainedby a market expectation of unchanged future nominal rental cashflows, whosediscounted present value would fall or rise according to the long-term interest rateoperatingasadiscountfactor.Theoppositecasecanalsooccur,i.e.,wecouldhaveasituationwherethespotremainssteady,buttheforwardchangesduetothewayinwhichtheinterestrateoperatesasaprojectionfactor.Using the standard formula for the forward house price (i.e., (10.9)), and assumingconstant𝑞,thereturnontheforwardiscalculatedasfollows.(10.13)forwardreturn≈D𝐻𝑃 + (D𝐼𝑅 − D𝑞) × 𝑇 whereD𝐻𝑃 = ln((𝑆 + Δ𝑆)/𝑆) and𝑇 is thematurityof the forwardatanypoint in thehistoricaltimeseriesforthegivencombinationofinterestrate(𝐼𝑅),defermentrate(𝑞)andhousepriceindex(𝐻𝑃).50Aproofof(10.13)isprovidedinAppendix3tothisChapter.Anumberofpointsfollowfrom(10.13).CorrelationbetweeninterestrateandhousepriceindexFirst,given that thereare fourrisk factors (Index,achievementrate, interestrateanddefermentrate)impactingtheforwardprice,weneedtoconsidertheircorrelations.WehavealreadyassumedthatthecorrelationbetweentheIndexandtheachievementrateiszero.Thetablebelowshowsthecorrelationbetweenthe10yearinterestrate(whichwe take as a benchmark for thewhole term structure) and thehousing index, for 10representativecountries.

50 Notethatwecannothedgeawayinterestanddefermentratevolatilityontheassumptionthattheindexpriceislessvolatilethantheforward.AswehavepointedoutintheAppendixtoChapter3,therelevantvolatilityfortheBlack-Scholesoptionisnotthatofthespotpricealone,butratherthatofthe𝑑5termusedtodeterminetheoptiondelta.Thenumeratorinthe𝑑5termincludesbothinterestrateanddefermentrate–explicitlyinBlackScholes,implicitlyinBlack76.

57

Table10.4:Correlationbetween10YInterestRateandIndexCountry 𝝆𝑰𝑹,𝑰𝑵𝑫𝑬𝑿AUS10Y 0.22CAN10Y -0.05GER10Y 0.11ESP10Y 0.06FRA10Y 0.16GB10Y 0.10IRL10Y -0.03SWE10Y 0.14US10Y 0.03JP10Y 0.27

Source:OECD(10Yinterestrate)andDallasFed(HousePriceindices)ThetakeawaypointsfromthistablearethatthecorrelationsbetweeninterestratesandhousepriceindicesaregenerallylowandthatareasonablecorrelationfortheUKwouldbezero.51Correlationbetween𝑞andhousepriceindexEquation(10.13)alsoindicatesthatthedefermentrate𝑞isafurthersourceofvolatility.Takeequation(7.1),whichsaysthatthedefermentrateisequaltotherentalyielddividedby the house price, then replace the house price by the HP index and add a time 𝑡subscript.Wethenobtain:(10.14) 𝑞* = 𝑑*/𝐻𝑃* where𝑑* is the aggregate nominal rental.We have no time series data on aggregatenominalrentals,butwecanestimatetheirchangeusingrentalandhousepriceindices.DatafromOECDsuggestthatthedefermentrate𝑞isnotconstant(theannualvolatilityof𝑞fortheUKisoftheorderof0.3%)andthatchangesin𝑞arenegativelycorrelatedwithchangesintheindex.TheseeffectsareshowninFigure10.2:

51ThelowcorrelationsreportedinTable10.4areabitofasurprise,consideringthatalowerinterestrateimmediatelytransformsintohigheraffordabilityandtherefore-intheabsenceofnewsupply–intohigherhouseprices.OnereasoncouldbethatTable10.4 looksat10yrrateswhereasthekeydrivermightbeshort-term rates. Were the correlations between interest rates and house prices higher, the resulting ‘combined’ volatilities (of which more below) would be higher as well.

58

Figure10.2:UKNominalHouseandRentalIndicesandImpliedDefermentRate

Sources:OECD

AsDeancommentsinablogpostingfromearlierthisyear:52

WhenIworkedatthePRAonthepaperthatbecameCP13/18,Ihadassumedthatthedefermentratestaysroughlyconstant.Therationaleisthatifrentalsare expected to increase, this would increase the market value ofproperties,allotherthingsbeingequal.Butallotherthingsaren’tequal:thereisstrongevidencethatnominalrentalstrackpriceinflation,andalsostrongevidence that interest ratesanticipate inflation. So an increase in expectednominalrentalsshouldcorrelatestronglywithanincreaseintheinterestrateusedtodiscountthesamerentals,andtherentalyield,hencethedefermentrate,shouldremainroughlyconstant.53Iassumedthis,andIimaginethePRAassumedthistoo.

Apossibleexplanation for thevolatile𝑞 rateand thenegative correlationwithhousepricesmightthengoasfollows.Nominalhouseprices,whichintheoryshouldreflectthenetpresentvalueofallfuturenominal(net)rentalcashflows,tendquicklytoanticipate–perhapstoover-anticipate–futureupwardordownwardchangesinrentals.Nominalrentalsarestickyhoweverandrespondslowly.54Thusthelargefallinhousepriceswhichoccurredinthehousingrecessionoftheearly1990swasnotreflectedinrentalprices,whichcontinuedtoriseslowly,andso𝑞roseinthatperiod.Conversely,thesignificant

52“ItMoves.”TheEumaeusProject(14January2019):http://eumaeus.org/wordp/index.php/2019/01/14/it-moves/53 Reason,𝑞 = 𝑟 − 𝑔 using a derivation of the Gordonmodel,where𝑞 is deferment rate, 𝑟 is nominalinterest rate,𝑔 is expected growth in nominalrentals. If𝑔 rises, and𝑟is highly correlatedwith rentalinflation,𝑟willrisealso,andthetwoeffectswillcancelout.54 This effect is well known in the literature, although there is no consensus on the explanation. Forexample,CampbellandHercowitz(2009)findthat“movementsinU.S.houseprice-rentratioscannotbefullyexplainedbymovementsinsubsequentrentgrowth”(J.R.CampbellandZ.Hercowitz(2009)“WelfareImplicationsof theTransition toHighHouseholdDebt.” Journal ofMonetaryEconomics 56,1-16.For areview of the literature, see P. Gelain and K. Lansing “House Prices, Expectations, and Time-VaryingFundamentals,”FederalReserveBankofSanFranciscoWorkingPaper2013-03.

59

rise in house prices from the late 1990s to 2007was notably higher than the rise inrentals,so𝑞felloverthislaterperiod.Asanimportantaside,thiscombinationofavolatile𝑞thatisnegativelycorrelatedwithhousepriceshasaninterestingpolicyimplication.Ifhousepricesgoup,theloan-to-valueofanexistingequityreleasemortgagewillfall,whichwilldecreasethecostoftheNNEG.Atthesametime,thegraphabovesuggeststhedefermentratewillalsofall,whichwillmaketheNNEGevencheaper,giventhatthedefermentrateisthemaindriverofNNEGcost.Conversely,afallinhousepriceswillmaketheNNEGmoreexpensivebecauseofthefallitself,andwillthenmaketheNNEGevenmoreexpensivebecauseoftheimpliedriseinthe𝑞rate.Thecostoftheembeddedguaranteeisthusdoublygearedtothestateofthehousingmarket.Ouch!ThePRAwouldappeartobestillunawareofthisdoubleexposure,whichhasimplicationsforhowitshoulddesignitscapitalrequirementregimeforequityrelease.Butwedigress.ThisnegativecorrelationeffectisnotuniquetotheUK.Table10.5showsevidenceforastrongandconsistentnegativecorrelationbetween thedeferment rateand thehousepriceindexofourtencountries:

Table10.5:Correlationbetween𝑞andIndexCountry 𝝆𝒒,𝑰𝑵𝑫𝑬𝑿AUS -0.90CAN -0.95GER -0.79ESP -0.88FRA -0.84GB -0.82IRL -0.43SWE -0.80US -0.86JP -0.92

Source:OECD(10Yinterestrate)andDallasFed(HousePriceindices)VolatilityandtermstructureAnothercorollaryofequation(10.13)isthattheimpactofchangesininterestratesontheforwardreturnwilldecreaseastimepassesandthematurityshortens.Forexample,other things being equal, the impact of a 10bp change in interest rates on a 30 yearforwardwillbetochangetheforwardpriceby10x30=300bp,whichissignificant.Bycontrast,theeffectofthesamechangeonacontractwith3monthstomaturitywillbe10x3/12=2.5bp,whichisnotsignificant.Thischangingsensitivitythroughoutthelifeofthecontractmeansthatthevolatilitycausedbychangesininterest(anddefermentrates)is not constant. Instead, this volatility starts high and then falls towards zero as thecontractapproachesexpiry.

60

Wenowwishtodeterminetheaveragelifetimevolatilityofthecontract.If𝑋istheseriesofmaturitiesand𝑌istheseriesofreturns,itcanthenbeshownthatthevolatilityoftheproduct𝜎(𝑋𝑌)isthefollowingsimplefunction:(10.14) 𝜎(𝑋𝑌) = 𝜎(𝑌) × 𝑇/√3AproofisgiveninAppendix4.Thepointtonoteisthatthevolatilityoftheproductisnowdirectlyproportionalto𝑇.TotalForwardVolatilityFinally, we might consider the effect of all four risk factors (Index, IR,𝑞 and AR) in the forward rate (10.13) to give what wemight call the total forwardvolatility.WecandosobyestimatingacorrelationmatrixbetweenthefourriskfactorsasshowninTable10.6:

Table10.6CorrelationMatrixfortheFourMainRiskFactorsIndex IR 𝑞 AR

Index 1.00 0.00 -0.82 0.00IR 0.00 1.00 0.00 0.00𝑞 -0.82 0.00 1.00 0.00AR 0.00 0.00 0.00 1.00

Table10.7showsthevolatilitiesforthe4componentriskfactors:

Table10.7:VolatilitiesofComponentRiskFactorsComponentVolatility Value

𝜎����� 5%𝜎�� 8.5%𝜎�� 0.58%𝜎W 0.17%

WenextcombinethecorrelationsinTable10.6withthecomponentvolatilitiesinTable10.7toobtainthetermstructureforthetotalforwardvolatilityshowninTable10.8:

Table10.8:TermStructureofTotalForwardVolatility𝑡 TotalForwardVolatility1 10.0%5 10.6%10 12.2%15 14.2%20 16.5%25 19.0%30 21.6%

61

Ifweworkedwiththeseresults,wewouldapplya10%volatilitytotheputfordecrement𝑡 = 1,a10.6%volatilitytotheputfordecrement5,andsoon,andan21.6%volatilitytotheputfordecrement30.55ThistermstructuremeansthatinprinciplethereisnosinglevolatilityinputfortheseriesofputsthatconstitutetheNNEG.56Figure10.3showsaplotofthetotalforwardvolatilityoverahorizonofupto50years.

Figure10.3:VolatilityTermStructure

Notes:AsperTable10.6.

Noteonceagaintheeffectofstochasticdominance.Theinterestratevolatility(0.58%)anddeferment ratevolatility (0.17%)makeonlya smallmarginal contributionat theshortermaturities.Butrecall(seeequation10.14)thattheimpactsoftheinterestrateand thedeferment rate areproportional tomaturity.After about15years, these riskfactorsbecomethe‘strongest’drivingfactorsanddominatealmostentirelybyabout30years.Thisdominanceexplainswhythebluetotalvolatilitycurveincreasinglylookslikeastraightlinebeyondthosematurities.Forcomparison,thefigurealsoincludesaline(inred)representingthecontributionof𝑞andIRonly,thatis,thevolatilityoftheforwardintheeventsthat(a)theindexneverchangedand(b)therewasnoachievementratevolatility.The two linesmove increasingly inparallelbutdonot in fact converge, thereasonbeingthestrongnegativecorrelationbetweentheindexand𝑞.Ifthatcorrelationhadbeenzero,thetwocurveswouldhaveeventuallyconvergedandwewouldhavehadthecounterintuitiveconsequence that thevolatilityof the forwardat thesematuritieswouldhavenotbeendrivenbythehousingindexatall,butratherbytheinterestanddefermentrates!

55 Amorecomprehensiveapproachwouldtakeaccountofcorrelationunderlessfrequenthedgingandgiveamoredetailedconsiderationofthecorrelationbetweentheindexandtheachievementrate. 56 However, foranygiventermstructureandanygivenage,weshallshowinChapter27that it isstillpossibletoimputeasinglevolatilitythatgivesthesameNNEGvaluationasonewouldgetusingavolatilitytermstructure.However,thissinglevolatilitynumbermustbeconsistentwiththetermstructureandwillbeage-dependent.

62

ConclusionsThecalibrationofthevolatilityparameterisamoreinvolvedsubjectthaniscommonlysuggested.Wemusttakeaccountofnolessthanfourriskfactorsthataffectvolatility–thehousepriceindex,theachievementratesaroundthatindex,theinterestrateandthedefermentrate.WethenendupwithavolatilitytermstructureinwhichdifferentNNEGputshavedifferentvolatilities.Wealsofindthatthevariousvolatilityconstituentshavediffering impacts depending on the maturity, with the house price constituentsdominatingatshortermaturitiesandthedefermentrateandinterestrateconstituentsdominatingatthelongermaturities.

63

Appendix One to Chapter Ten: A Hurst Exponent Approach toAutocorrelationinPropertyPricesThe empirical evidence suggests that property returns are autocorrelated andautocorrelationhasimplicationsforvolatilityextrapolation.WecanhandlethisautocorrelationusingaHurstanalysis.AHurstExponentApproachtoAutocorrelationBy way of background, this problem appears to have first been observed by thehydrologistHaroldHurst(1880–1978)workingontheNileBasininthe1930s.57Hurstwas concerned to design an ideal reservoir that never overflows and never empties,basedonobservationsofdischargesfromthelake.Inanyyear𝑡therewillbeaninflux𝜉(𝑡)ofwaterintothereservoir,witharegulateddischarge⟨𝜉⟩¦fromthereservoir,where𝜏representsalongperiodoverwhichthereservoiroperates.Weneedtoestimatethestoragerequiredsuchthattheaverageamountofwaterreleasedovertheperiodequalstheaverageinflux,withoutthereservoiremptyingoroverflowingatanyintermediatetime.Theaverageinfluxis (10A1.1)⟨𝜉⟩¦ = (1/𝜏)∑ 𝜉(𝑡)¦

*¨5 whichshouldbeequaltotheamountreleasedperyear.Let𝑋(𝑡, 𝜏)betheaccumulateddepartureoftheinflux𝜉(𝑡)fromtheaverage⟨𝜉⟩¦:(10A1.2)𝑋(𝑡, 𝜏) = ∑ 𝜉(𝑢) −¦

©¨5 ⟨𝜉⟩¦.The difference between themaximum andminimum accumulated influx is called therange𝑅,whichalsorepresentstherequiredstoragecapacityforanidealreservoirthatneveremptiesnoroverflows.Thefigurebelow(fromFeder1988,p.151)sketchessuchareservoirwithinflux𝜉(𝑡),discharge⟨𝜉⟩¦andrange𝑅.Theheightofthedammustbeconsistentwiththisstoragecapacity.

57SeeH.E.Hurst,Long-TermStorage:AnExperimentalStudy (London,Constable,1965).The followingaccountisadaptedfromJ.Feder,Fractals,NewYork(Plenum1988),pp.149-151.

64

Figure10A1.1:ReservoirStorageCapacity

If influxesforsuccessiveannualperiodsareuncorrelatedthenitwouldbepossibletoworkouttherequiredrange,henceworkoutanappropriatereservoirdesign, foranyperiod𝜏simplybyapplyingasquarerootlaw,usingthefollowingrelation:(10A1.3) 𝑅 = 𝑆(𝜏/2)0.z where𝑆 isthestandarddeviationof𝑋(𝑡, 𝜏),ontheassumptionthatsuccessiveannualinflowswererandomevents.Empirical results, suchas fromthe flowrecordsofLakeAlbert,whichHurstwasengagedtoworkonin1938,weresomewhatdifferent.Hefoundthatreservoircapacitybasedontheempiricalresultswaslargerthanestimatesbasedonthesquarerootlaw.ExtensiveworkbyHurstdemonstratedthattheflowswerebetterexplainedbythefollowingfunction(10A14) 𝑅 = 𝑆(𝜏/2)� where𝐻,theso-calledHurstexponent,typicallyvariesempiricallyfrom0.7to1.58Wecanestimate𝐻byfittingapowerlawtothedata,i.e.,byplottingln(𝑅/𝑆)asalinearfunctionoflog 𝜏.ThisfittingprocessisillustratedinFigure10A.2:

58Originallycalled𝐾byHurst.Mandelbrotnamedit‘𝐻’forHurst.

65

Figure10A1.2:CalibratingtheHurstExponentUsingaLinearPlot

Notes:BasedonDallasFeddata.

Thefigureshowsplotsoftherescaledrangeln(𝑅/𝑆)againstln 𝜏foravarietyofnationalhousepriceindicesandforasimulatedGBMseries.TheslopeofeachlinegivestheHurstexponent,𝐻. If the timeseries isgeneratedbyarandomwalk(oraBrownianmotionprocess) then the slope or𝐻 exponent has a value of 0.5. The Figure shows that thevarious national house price series have𝐻 values between about 0.7 and 1, but thesimulatedGBMserieshasaslopeof0.5,aswewouldexpect. We can also estimate 𝐻 using a method recently developed by Ceballos and Largo(2017).59Weanalysedquarterlydataforhousingmarketsin21differentcountriesandfoundthesamephenomenonasHurst foundwith theNile,withresults for𝐻varying from0.69(Australia)to1.01(US).SeeTable10A1.1below:

Table10A1.1:HurstExponent(𝑯)bycountryCountry 𝑯 Country 𝑯 Country 𝑯Australia 0.69 Japan 0.82 Spain 0.89

NewZealand 0.78 S.Africa 0.84 Sweden 0.92Denmark 0.81 Italy 0.86 Germany 0.95Belgium 0.81 Finland 0.86 Luxembourg 0.95Canada 0.81 S.Korea 0.86 Ireland 0.99Norway 0.81 Switzerland 0.86 France 1.00UK 0.82 Netherlands 0.88 US 1.01

Datasource:DallasFed.TheestimationusestheCeballosandLargoadjustmentforsmallvaluesof𝜏.SowhydoestheHurstexponentmattertous?Theanswerisbecauseofitsimplicationsforvolatilityextrapolation.UnderGBM,wewouldextrapolatevolatilityusingthesquarerootrule: 59 See R. F. Ceballos and F. F. Largo (2017) “On the Estimation of theHurst Exponent Using AdjustedRescaledRangeAnalysis,DetrendedFluctuationAnalysisandVarianceTimePlot:ACaseofExponentialDistribution,”ImperialJournalofInterdisciplinaryResearch3(8):424:434.

66

(10A1.5)𝜎(𝑡) = 5% × 𝑡0.zwhere 𝑡 is the holding or rehedging period used to determine 𝜎, but underautocorrelation,weshouldextrapolateusing(10A1.6)𝜎(𝑡) = 5% × 𝑡� WeassumetheresultofParkinson(1980)showingthatthevolatilityofreturnsoveranyholdingperiod𝑡couldbeestimatedusingthehighandlowin𝑡.60SmithandJeffery(2019:24)alsofind,usingempiricaldatafromtheUKhousingmarket,thatthevolatilityofreturnsdependsonthechosenholdingperiod,andriseswithit.Whetherautocorrelationreallymatterstousdependshoweveronthechoiceofhedgingstrategyunderlyingtheoptionpricing.Inparticular,wewoulduse(10A1.6)toobtainaprojected𝑡-periodvolatilityifwearepricingtheoptionusingarehedgingstrategythatcallsforthesyntheticoptionorunderlyingpositiontoberebalancedevery𝑡years.Butinothercasesautocorrelationisnotanissueforus.Onesuchcaseiswhere𝑡 = 1,inwhichcase(10A1.5)and(10A1.6)coincide.

60M.Parkinson(1980)“TheExtremeValueMethod forEstimatingtheVarianceof theRateofReturn.”JournalofBusiness53,61–65.

67

AppendixTwotoChapterTen:ProofofEquation(10.5)Step#1:𝜎(𝑘)bydefinition is thevolatilityweempiricallyderivebysamplingreturnswithperiod𝑘,andtakingtherootmeansquare,withoutanykindofadjustment.Step#2:Toobtainthe‘annualised’value𝜎wedividebyroot𝑘.Thus(10A2.1)𝜎 = 𝜎(𝑘)/√𝑘where𝜎isthevalueweinputintoBlack‘76.Step#3:FromStep#1above,itfollowsthat𝜎(1)isthevolatilityweempiricallyderivebysamplingreturnswithperiod1.Step#4:AssumingGBM,then(10A2.2)𝜎(1) = 𝜎Step#5:WiththeHursteffect,(10A2.2)isfalse.Specifically,(10A2.3)𝜎(𝑘) = 𝜎(1) × 𝑘� where𝐻 ≠ 0.5.Step#6:FromStep#2above,(10A2.4)𝜎 = 𝜎(𝑘)/√𝑘 = 𝜎(1) × 𝑘�/√𝑘whichwastobeproved.

68

AppendixThreetoChapterTen:ProofofApproximation(10.13)Theforwardrate𝐹atanytime𝑡andforanymaturity𝑇isasfollows:(10A3.1) 𝐹*,« = 𝑆*𝑒(I(*,«)4W(*,«))« where𝑆*isthespotpriceattime𝑡,𝑟(𝑡, 𝑇)istheinterestrateofmaturity𝑇attime𝑡and𝑞(𝑡, 𝑇) is thedeferment rateofmaturity𝑇 at time 𝑡.With thepassageof time∆𝑡, theforward ratewill change as a result of changes in𝑆,𝑟 and𝑞, and of coursewith thepassageoftimeitself.Thus(10A3.2)𝐹*n∆*,«4∆* = 𝑆*n∆*𝑒­I(*n∆*,«4∆*)4W(*n∆*,«4∆*)®(«4∆*) Thisexpressionisfairlycomplex,butwecanmakeanumberofsimplifyingassumptionsasfollows.First,wecanassumethatthetermstructureofboth𝑟and𝑞iscontinuous.Wehave assumed throughout a flat term structure 𝑞, so it follows that 𝑞(𝑡 + ∆𝑡, 𝑇 − ∆𝑡)equals𝑞(𝑡 + ∆𝑡, 𝑇).Wecannotassumethat the termstructureof interest rates is flat,because it will usually slope upwards or downwards at any time. However, we canreasonably assume that changes in the term structure will make no significantcontributiontovolatility.Thatis,achangeover1monthtothe10yearinterestratewillnotbesignificantlydifferentfromthechangeinthe9year11monthinterestrate.Thus𝑟(𝑡 + ∆𝑡, 𝑇 − ∆𝑡)willbeapproximatelyequalto𝑟(𝑡 + ∆𝑡, 𝑇),forsmall∆𝑡.Hence(10A3.3)𝐹*n∆*,«4∆* ≈ 𝑆*n∆*𝑒­I(*n∆*,«)4W(*n∆*,«)®(«4∆*)

Theouterterm(𝑇 − ∆𝑡)canalsobeeliminated,asitrepresentsaconstantcarrythroughtime.Astimepasses,if𝑟isgreaterthan𝑞,theforwardpricewillgraduallyfall,orif𝑟islessthan𝑞,theforwardpricewillgraduallyrise.Butvolatilitycorrespondstothemeandifference fromtheaverage,whereasthecarrytermwillbeclosetotheaverageitself.Hence(10A3.4)𝐹*n∆*,«4∆* ≈ 𝑆*n∆*𝑒­I(*n∆*,«)4W(*n∆*,«)®« Weassumethatthedeterminantsofforwardvolatilityarethechangesinspot,interestrate and deferment rates alone, and that the passage of time is an insignificantcontributiontovolatility.Todeterminethevolatility,wemustfirstdeterminetheforwardpricereturn:(10A3.5)Forwardpricereturn=ln(𝐹*n∆*,«4∆*/𝐹*,«)Substitutingfromtheequationabove:(10A3.6)ln(𝐹*n∆*,«4∆*/𝐹*,«) =

ln¯𝑆*n∆*𝑒­I(*n∆*,«)4W(*n∆*,«)®«° − 𝑙𝑛[𝑆*𝑒­I(*,«)4W(*,«)®«] =

69

𝑙𝑛[𝑆*n∆*/𝑆*] + [𝑟(𝑡 + ∆𝑡, 𝑇) − 𝑟(𝑡, 𝑇) + 𝑞(𝑡, 𝑇) − 𝑞(𝑡 + ∆𝑡, 𝑇)] × 𝑇Now make the simplifying assumptions that 𝑟(𝑡 + ∆𝑡, 𝑇) − 𝑟(𝑡, 𝑇) = ∆𝑟* and 𝑞(𝑡 +∆𝑡, 𝑇) − 𝑞(𝑡, 𝑇) = ∆𝑞* .Wethenobtain:(10A3.7)forwardreturn≈D𝐻𝑃* + (D𝑟* − D𝑞*) × 𝑇whichwastobeproved,whereD𝐻𝑃 = ln((𝑆 + Δ𝑆)/𝑆).

70

AppendixFourtoChapterTen:ProofofEquation(10.14)Weneedtodeterminethevolatilityofatimeseriesofpricesforaforwardcontract,giventhat the maturity 𝑇 of the contract is constantly decreasing. Assume the followingstandardresultfortwoindependentvariables𝑋and𝑌:61(10A4.1) 𝑉𝑎𝑟(𝑋𝑌) = 𝑉𝑎𝑟(𝑋)𝑉𝑎𝑟(𝑌) + 𝑉𝑎𝑟(𝑌)𝐸[𝑋]M + 𝑉𝑎𝑟(𝑋)𝐸[𝑌]MLet𝑋betheseriesofmaturities,and𝑌bethechangesininterestrate∆𝑟*(ordefermentrate∆𝑞*).Assumethattheaverageinterestrateordefermentratechangeiszero,i.e.that𝐸[𝑌] = 0. (10A4.2) 𝑉𝑎𝑟(𝑋𝑌) = 𝑉𝑎𝑟(𝑋)𝑉𝑎𝑟(𝑌) + 𝑉𝑎𝑟(𝑌)𝐸[𝑋]M + 𝑉𝑎𝑟(𝑋)𝐸[𝑌]M =

𝑉𝑎𝑟(𝑌)[𝑉𝑎𝑟(𝑋) + 𝐸[𝑋]M]Thenwecan treat theseriesofmaturitiesasauniformdistribution fromthestartingmaturity 𝑇 down to zero. The variance 𝑉𝑎𝑟(𝑋) and the average 𝐸[𝑋]of a uniformdistributionovertheinterval(𝑥, 𝑦)areasfollows.(10A4.3)𝑉𝑎𝑟(𝑋) = (𝑦 − 𝑥)M/12 = 𝑇M/12(10A4.4)𝐸[𝑋] = 𝑇/2Substituting:(10A4.5)𝑉𝑎𝑟(𝑋𝑌) = 𝑉𝑎𝑟(𝑌)[𝑉𝑎𝑟(𝑋) + 𝐸[𝑋]M] = 𝑉𝑎𝑟(𝑌)[𝑇M/12 + 𝑇M/4] =

𝑉𝑎𝑟(𝑌) × 𝑇M/3(10A4.6) 𝜎(𝑋𝑌) = ±𝑉𝑎𝑟(𝑋𝑌) = 𝜎(𝑌) × 𝑇/√3whichwastobeproven.

61 L.A.Goodman(December1960).“OntheExactVarianceofProducts,”JournaloftheAmericanStatisticalAssociation55(292):708.

71

ChapterEleven:ModellingMortality1.IntroductionThischapterprovidesanintroductiontothemortalitymodellingissuesthatarisewithequityreleasevaluation.2.RealisedMortalityRatesConsiderthefollowingFigureshowingtherealisedmortalityratesofEngland&Walesmalesforages70andabove.

Figure11.1:RealisedMortalityRatesforMales70andAbove

Notes:BasedonCBD-M5model(Cairnsetalia,2006,2009)projectionsusingEngland&Walesmaledeathsratedataestimatedoverages55:89andyears1971:2017.Source:llma.org.

Byrealisedwemeanthemortalityratesexperienced for thesemalesover thesampleyears,1971:2017.62Weseethatthemortalityrateisalittleunder2%forage70andthenriseswithage,aswe would expect. However, there are two problems with the mortality rates in thisFigure.Thefirstisthattheyonlygouptoage89,whichisthemaximumageinoursampleagerange,andforNNEGvaluationwewantmortalityratesforhigheragesaswell.The

62Let𝐷bethenumberofpeopleofacertainagewhodieinacertainyearandlet𝐸bethenumberofcorrespondingexposuresorpeopleofthatageatriskofdyingthatyear.Thedeathrate𝑚 = 𝐷/𝐸andthemortalityrateis𝑞 = 1 − 𝑒4³(see,e.g.,Cairnsetalia,2009,p.3).Themortalityrateisamoreconvenientratetousethanthedeathratebecauseitismathematicallymoretractableandbecauseitisguaranteedtobelessthan100%,whereasdeathratescanexceed100%due,e.g.,toerrorsintheexposuresdata,whichmightbeduetomisreportedbirthdatesortheoccasionalunreportedhomicide.

72

secondisthattheserealisedmortalityratestakenoaccountofanyanticipatedlongevityimprovements.3.ProjectedMortalityRatesAtraditionalsolutiontothislatterproblemistousesomespeciallypreparedmortalitytables,whichareinessenceexperteducatedguessesaboutfuturemortalityrates.SuchtablescanbeobtainedfromtheContinuousMortalityInvestigation,forexample.63Thisapproachiseasiertoapplybutissubjectiveandlackstransparency.Amorescientificallygroundedandmoretransparentandthereforebetterapproach(althoughthereisabitofalearningcurve)istousemortalityrateprojectionsfromastochasticmortalitymodel.64OnesuchmodelistheCBD-M5modelmentionedearlier,whichwasdesignedspecificallyforoldagemortalityprojections.Figure11.2givestherealisedandprojectedfuturemortalityratesformalesjustturned70:65

Figure11.2:ProspectiveMortalityRatesforMalesAged70

Notes:AsperFigure11.1.

Wehaveshowntheprojectedmortalityratesouttoage100,butthemodelallowsustoprojectthemtoanyagewewish.66Weseethattheprojectedmortalityratesgrowata

63 See https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-mortality-and-morbidity-tables.64Thesemodelsnotonlyallowuserstoobtainlikelyprojectedmortalityrates,butalsoallowthemtoobtainstatisticallygroundedpredictionintervalsandscenarioanalyses,andtotakeaccountofrefinementslikeparameteruncertainty,individualdeathriskandtheimpactofBayesianpriorbeliefs.65Thesemortalityratesareknownascohortqrates,becausetheyfollowthecohortcurrentlyaged70astheyageovertimeandtheirmortalityratesreflecttheirincreasingage.66ForNNEGvaluation,weactuallyuseprojectedmortalityratesouttoage120,i.e.,formodellingpurposesweassume thatany individualswhoreach their121stbirthdayare thenautomaticallydispatched. It isimportanttotakeaccountoftheextremeoldage‘toxictail’whendealingwithlifetimefinancialproducts.

73

lowerratethantherealisedrates.Thisdifferencebetweenthetwoplotsillustratestheimpactofprojectedimprovementsonfuturelongevity.4.ExpectedSurvivorRatesThenextstepintheanalysisistoobtainthecorrespondingexpectedsurvivorrates,𝑆* ,i.e.,theprobabilitythatanindividualalivenowwillsurvivetoyear𝑡.Ifwelet𝑞*bethemortalityrateforyear𝑡,thenthefollowingholdsfor𝑆*:(11.1)𝑆0 = 1(11.2)𝑆5 = (1 − 𝑞5)(11.3)𝑆M = 𝑆5 × (1 − 𝑞5) = (1 − 𝑞5) × (1 − 𝑞M)etc.Figure11.3showstheexpectedsurvivorratesforthepeopleontheir70thbirthday:

Figure11.3:ExpectedSurvivorRatesforMalesAged70

Notes:AsperFigure11.1.

Theexpectedsurvivorratesfallfrom100%onday1downeventuallytowardszero.5.ExpectedMortalityRatesWetaketheexpectedmortalityrate(ofacohortofgivencurrentage)forfutureperiod𝑡tobetheproductofthemortalityratefor𝑡andtheexpectedsurvivorratefor𝑡,i.e.,theexpectedmortalityrateequals𝑞* × 𝑆* .Theexpectedmortalityratesforacohortofmalescurrentlyaged70areshowninthenextFigure:

74

Figure11.4:Figure3.1ExitProbabilitiesforMalesCurrentlyAged70

Notes:AsperFigure11.1.

whichisessentiallythesameasFigure3.1,ifweignorethepossibilityofhouseexitbygoingintoanursinghome.Weseethattheexpectedmortalityratesrisetopeakinthemidtolate80s,thenfallandeventuallygotozero.Now remember that these expectedmortality rates are theweights that apply to theNNEGputoptions.Thisweightingscheduletellsusthatthelowmaturityputsarenotsoimportant, because of the low probability of exercise, but the puts become moreimportantastheirmaturityrisesandtheonesthatareofmostsignificanceofthosewithmaturitiesofmaybe10toa littleover20years.Theputswith longermaturitieshavedecliningandeventuallyinsignificantimportance.6.ModelRiskinMortalityProjectionsAproblemwiththeseprojections isthattheyaredependentonanassumedmortalitymodel,M5.Theyarethereforeexposedtomortalitymodelrisk,i.e.,theriskoferrorfromtheuseoftheM5mortalitymodel.ThebestwaytoaddressthisissueistoconsideralternativemodelsandonethatwouldbesuitableisM7.ThismodelisanextensionofM5.Model5hastwoperiodeffects,butM7addsathirdperiodeffectandacohortoryearofbirtheffecttoM5.Furtherdetailscanbefoundin,e.g.,Cairnsetalia,2009.67Figure11.5showstheexpectedmortalityratesfor70yearoldmalesbasedonprojectionsfrombothmortalitymodels: 67Thosefamiliarwiththesemodelsmightbewonderingwhywedon’talsoincludemodelM6.Theansweristhatforreasonswedon’tquiteunderstand,M6givespotentiallyunreliableresultsforEngland&Walesfemales.Wethereforedropitfromconsideration.

75

Figure11.5:ExpectedMortalityRatesforMalesCurrentlyAged70:ModelsM5

andM7

Notes:AsperFigure11.1.

Theplotsaredistinctbuthavemuch thesameshape.Consequently,wewouldexpectthesemodelstogivefairlycloseNNEGvaluations.7.ModelRiskinMortalityProjectionsTable11.1showsthevaluationscorrespondingtoeachofthemortalitymodels:

Table11.1:BaselineNNEGandERMValuationsforMalesAged70CurrentHouse

PriceLoan

AmountModel 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £40 M5 £74.84 £32.19 £42.66£100 £40 M7 £74.29 £31.46 £42.83

Notes:𝐿isthepresentvalueoftheloancomponentoftheEquityReleaseMortgage,𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguaranteeand𝐸𝑅𝑀isthepresentvalueoftheEquityReleaseMortgage.Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%,𝑞=4.2%and𝜎=14.8%.ExitprobabilitiesarebasedonprojectionsoftheM5andM7variationsoftheCBDmodelusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.WeseeacertainamountofvariationintheloanvaluesandNNEGvaluations,butmuchlessvariationintheERMvaluationsduetotheoffsettingimpactsoftheloanvaluesandtheNNEGvaluationsontheERMvaluations.8.FemalesTable11.2givesthecorrespondingvaluationresultsforfemales:

76

Table11.2:BaselineNNEGandERMValuationsforFemalesAged70CurrentHouse

PriceLoan

AmountModel 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £30 M5 £80.79 £40.28 £40.52£100 £30 M7 £81.99 £41.84 £40.15

Notes:AsperTable11.1butforfemales.Theappropriatevolatilityforfemalesaged70is15.7%.ThefemaleloanvalueandNNEGvaluationsarehigherthanforthemales,aswewouldexpectfromhigherfemalelifeexpectancy.Thevaluationsfromthetwomodelsarealsoveryclose. 9.JointLivesManyERMloansaretocouplesratherthanindividuals.Intheory,oneshouldmodeltheexitprobabilitiesassociatedwithsuchloansinawaythattakesaccountofthelongevityprospectsofbothpartnersandthepointthatexitwilloccurwhenthelongestsurvivingpartnerexitsthehouse.Suchananalysisisalittleinvolved,however,andthestandardapproachistotreatsuchanERMloanasifitwerealoantotheyoungestpartner.Atypicalcasewouldbeacoupleinwhichthemaleis70andthefemale66,i.e.,sowehaveayoungerfemalebutbearinmindthatfemaleshavelongerlifeexpectancy.SomeresultsaregiveninTable11.3:

Table11.3:NNEGValuationsandLifeExpectanciesforaTypicalCoupleCurrentHouse

PriceLoan

AmountPartners 𝑵𝑵𝑬𝑮 Life

expectancy£100 £40 Maleaged70 £32.19 15.1£100 £40 Femaleaged66 £48.87 20.6

Notes:𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguarantee.Basedonthebaselineassumptions:𝐿𝑇𝑉=40%forthemaleand36%forthefemale,𝑟=1.5%,𝑙=5.25%,𝑞=4.2%and𝜎=14.8%forthemaleand17.3%forthefemale.ExitprobabilitiesandlifeexpectanciesarebasedonprojectionsoftheM5variationoftheCBDmodelusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89ThetableshowsNNEGvaluationsandassociatedlifeexpectancies,wheretheNNEGisthatofanERMloantoeachpartnerconsideredontheirown.TheNNEGvaluationsare£32.19fortheloantothe70yearoldmaleand£48.87fortheloantothe66yearoldfemale.BearinmindthattheNNEGforaloantoacouple,withexitdeemedtooccurwhenthelastremainingmemberexits,willalwaysbelargerthantheNNEGfromaloantoeitherindividualmemberalone.ThereasonisthatthelatterNNEGvaluationsdonotaccountfortherisk(tothelender)oftheotherpartnerexitinglaterthanthepartnertowhomtotheindividualloanwasmade.Thus,thetrueNNEGvaluationwouldbebiggerthaneitheroftheNNEGvaluationsshownintheTableanditisimmediatelyapparentthatonecouldgetamajorunder-estimationofthe𝑁𝑁𝐸𝐺ifonehadtreatedtheloanforNNEGvaluationpurposesasifitjustaloantoamaleaged70.

77

Eventhehigher(female66)NNEGof£48.87willunder-estimatethetrueNNEG,butifonelooksatthelifeexpectancies,itisalsoclearthatthefemaleisexpectedtolive5.5yearslongerthanhermalepartner.Itisthereforeunlikelythatthemalewilloutliveher,andonecanmakeaplausiblehandwavingargumenttotheeffectthattheerrorintheNNEGvaluationwillnottooimportant.68However,anysuchargumentreliesonthespecificsofthecaseathand–andinparticular,ontheyoungermemberhavinganotablylongerlifeexpectancy–andthatconsiderationwillnotalwaysapply.Forexample,ifthemaleandfemalewerereversedinage,thenthe66yearoldmalewouldhavealifeexpectancyof17.1yearsandhis70yearoldfemalepartnerwouldhavealifeexpectancyof18.4years.Thereisthenamuchbiggerriskoftheoldermemberofthecoupleoutlivingtheyoungerone,andtheNNEGvaluationerror,associatedwith treating the loan as if itwere a loan to the youngermember,will begreaterthaninthepreviouscase. 10.ImpairedLives ERMcompanieswillsometimesoffermoreattractivetermstoborrowerswithimpairedlives, i.e., thosewith reduced longevity prospects. A first pass atmodelling fair-valueNNEGguaranteesforimpairedlivesborrowersistoestablishtheimpactoftheirhealthconditionon theirexpected longevity, thenoffer themanLTVbasedon thatexpectedlongevity.Togiveasimpleexample,ifaborroweraged70hasthelifeexpectancyofan80yearold,thenthelendermightofferthemtheLTVi.e.loantermsofan80yearold,e.g.,sotheborrowermightgetaloanbasedonanLTVof40%insteadofthestandard30%.ForNNEGvaluationpurposes,then,wemighttreatthe70yearoldasifhewerean80yearold.Asafurtherrefinement,wemightalsotakeaccountofwhetherandifsohow,theirhealthconditionmightaffecttheprospectsforthelengthoftheirendoflifeperiodincare.

68TherearealsotwooffsettingeffectsonanyNNEGunder-valuation.First,thereisthepossibilitythatthesurvivingmemberofthecouplewillmoveoutafterfirstonehasdied.Second,thereissomeevidencefora‘heartbreak syndrome’ bywhich the death of the firstmember raises themortality rate of the other,althoughanecdotalevidencesuggeststhattheoppositecansometimesoccurtoo.

78

ChapterTwelve:Long-TermCareWehavehithertoassumedthatexitoccurswithdeath.Inreality,manyolderpeopleleavetheirhomestomoveintosomeformoflong-termcare(LTC)andthetimespentincarecanbesubstantial.LengthofTimeinLong-TermCareWeunderstandthataruleofthumbinthisareaisthatpeopleareexpectedtospendtwoyears in long-termcare.Orperhapsa little longer:“Onaverage,olderpeoplestay inaresidentialcarehomefor30months,”statesarecentIndependentAgearticlecitinganearlier(2009)reportbyLangBuisson.69ThesenumbersarecomparabletothosefromtheU.S.Forexample,ThomasDay(2010)statesthat,the“averagestayforelderlypatientswhodieinanursinghomeisjustshyof2years,”70whilstBradBreedingcitesU.S.reportsof2009and2010thatreportaveragelengths of stay in assisted living facilities of about 28 months and 29 monthsrespectively.71Butwhateveryoneagreesonisthatthelengthoftimeincare,andwhetheronewillneedtospendanytimeincareatall,arehighlyuncertain.TakingAccountofExpectedTimeinCareThedifferencebetweentheexpectedtimetodeath(i.e.,lifeexpectancy)andthelengthoftimetohouseexitcouldthenmakeamaterialdifferencetoequityreleasevaluations.Theproblemishowtotakeaccountofthisdifference.ThestandardactuarialpracticeintheUKistoobtainexitprobabilitiesfrom(conditional)mortalityprobabilitiesbyimposingloadingfactorsonthelatter.AloadingfactorinvolvesmultiplyingthosemortalityratesbyafactorthatreflectssomeLTC‘addon’.Forexample,for thepurposesofobtainingexitprobabilities frommortalityrates,wemightaddanadditional 2% to the latter. Our exit probabilities would then be 102% times the(conditional)mortalityrates.Hostyetalia(2007)offerascheduleofsuchloadingfactors: 69 See “Cost of average length of stay in a residential home is equivalent to 26 years’worth of familyholidays.” IndependentAge26Oct2017; LaingBuissonCareofOlderPeopleMarketReport, 27 edition,London:LaingBuisson.70T.Day(2010)“AboutNursingHomes.”https://www.longtermcarelink.net/eldercare/nursing_home.htm 71 B.Breeding(2016)“SoI’llProbablyNeedLong-TermCare,ButforHowLong?myLifeSite(6July).https://www.mylifesite.net/blog/post/so-ill-probably-need-long-term-care-but-for-how-long/

79

Table12.1:Hostyetalia(2007)MortalityLoadingFactors

Age Male(%) Female(%)≤ 70 2 3(70,80] 4 12(80,90] 5 13(90,100] 4 8

Source:Hostyetalia(2007,Table13).Theseloadingfactorsrisewithageandthenfallagainandreflectanintuitionabouttheexpectedlengthoftimespentincareoutsidethehome,dependingontheagewhenonegoesintocare.Thereareproblemswiththisapproach,however.OnewassetoutbyTunaru(2019,p.67):

Formulti-statemodellingconsidering the interactionbetween long-termcareentryandmortalityisparamountbecausethereissignificantlyhighermortalityexperiencedbylong-termcareresidentscomparedto“athome”mortalitymeansthattomaintainthesameaggregateassumptionformortalitybyagelighterthanaveragemortalityshouldbeassumedfor“athome”lives.

People incarewillbe lesswell thanpeopleof thesameagestillathomeandsotheirmortalityrateswillbehigher.Sincethepopulationmortalityratesareaveragesofthemortalityratesofpeople incareandpeopleoutsidecare, thenanyadjustmentfortheformerrequiresanoffsetting(butnotnecessarilyequallyoffsetting)adjustmentforthelatter. The problem here is that without further data on the numbers in care vs thenumbersoutsidecare,thenanysuchadjustmentsaredifficulttocarryout.Thereisalsoadeeperissue.Evenifwewereconfidentinourprojections,wehavenoreasontothinkthattheseloadingfactorsareanygoodinthefirstplace.Tocalculatethem‘properly’fromfirstprinciples,wewouldneed(a)reliableprojectionsforthemortalityratesofpeopleincare,(b)reliableprojectionsforpeopleoutsidecareand(c)reliableprojectionsoftherelativesizesofthesetwopopulations.Theproblemisthatwedon’thave any of these. Instead, we have are loading factors pulled out of thin air, i.e.,guesstimates,admittedlybylifeactuarieswithsomeintuitionforsuchissues,andthatisallwehave.Soweshouldn’tplacemuchrelianceontheHostyetalialoadingfactors.Fortunately,wedon’thaveto,becausethereisabetterwayanditissimpletoo.Supposewebelievethattheexpectedtimeinlong-termcareis2years.Ifwehavea70yearoldmale,thenhislifeexpectancyisabout15years,andweexpecthimtospendhislast2yearsincare,i.e.,weexpecthimtoexithishomein13years.Sowecanapproximatehistimetoexitbygivinghimthelifeexpectancyofa72yearold,andwecandothatbygivinghim theprojectedmortality ratesofa72yearold.Thus,we takeourNNEGandERMfunctions,andinputagesof72insteadof70.Table12.2givessomeillustratevaluations.

80

Table12.2:NNEGValuationsandLifeExpectanciesforaTypicalMale

CurrentAge 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴70 £31.19 £42.6672 £26.95 £44.33

Notes:𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguarantee.𝐸𝑅𝑀isthepresentvalueoftheERM.Basedonthebaselineassumptions:𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%,𝑞=4.2%and𝜎=14.8%forage70and14.1%forage72.ExitprobabilitiesandlifeexpectanciesarebasedonprojectionsoftheM5variationoftheCBDmodelusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.WeseethatthisageadjustmentmakesasubstantialdifferencetotheNNEGvaluationbutamuchsmalleronetotheERMvaluation.If we believe that the expected time in care is 2.5 years, then we can obtain ourapproximationsbytakingourNNEGtobeaverageoftheNNEGsforages72and73,andsoforth.Ideally,wemightwanttobuildsomefancymodel,butintheabsenceofdata,wecan’t.Inthemeantimethesimplisticapproachjustsuggestedcouldbethebestwecando.

81

ChapterThirteen:DelayedPossessionWehavehithertoimplicitlyassumedthattheloanisimmediatelyrepaidatthetimeofhouseexit.Inpractice,however,thereislikelytobesomedelay.Forexample,intheirNNEGstudy,Lietalia(2010)define𝛿astheaveragedelayintimefromthepointofexitto thesaleof thehouse.Theyuseabaselinecalibrationof𝛿=halfayear (p.16)andprovidesomesensitivityresults,buttheydonotprovideanyempiricaljustificationfortheir𝛿calibrations.Intheory,wetakeaccountoftheimpactofdelaysonourvaluationsifwehavegooddatathatallowsustoestimatetheexpecteddelay.Let’ssuppose,forexample,thatweexpecttheretobeahalfyeardelaybetweenhouseexitandhousesale.Wecanthenhandlethisdelaybytreatingtheborrowerasifheorshewerehalfayearyounger.Soiftheborrowerisactually70whentheloanistakenout,wetreattheborrowerasifheorshewere69andthenobtaintheNNEGorERMasequaltotheaverageoftheNNEGsorERMsfor69yearholdand70yearold.Theactuaryormodelercanpresumablyusetheirfirm’saccountdatabasetoobtainsomesenseofthedelayedexperiencedonpastERMloans.However,itmaynotalwaysbesostraightforward,andfortworeasons.Firsttherewillbedispersioninthe‘timetosale’statistics,andsomeofobservedtimestosalecanbelong.Inonecasewehaveseen,thetimetosalewasthreeyears.Asecondissueiswhethertheloanfixesatexitordeathandhasdelayedsettlementorkeepsrollinguntilsettlement.In the latter case, there is the potential for some serious abuse. One of ourcorrespondents,anexpertinERMpropertymanagementpractices,informsusthat

abankorbuildingsocietycanalsouselongtermassets(likeERMs)tohideitserrorsofvaluationatthebeginningofalongdatedbook...IhaveseensomenaughtyinstitutionsfiltertheharvestofERMassetsastheyrolloffbypostingthecashresultsofthosethatexpireatLTV<100%andjustallowingthe>100%onestorollonandon,withoutaddressingorcrystallisingthem.Sofroma reports and accounts perspective (where little of this level of analysis isexposedtothepublic)itlooksasthoughthecashisrollinginnicelyasplannedatinception,butthehorrors(theNNEGsthatshouldeatuptheaccumulatedcash accounting value of the assets) are also piling up unnoticed, becauseuncrystallised–dealwiththemlaterasitwere.Sothebooklookslikeit’sthebestcashgeneratingassetonthebalancesheet–cashflowsoaringandotherloansstillrollingup.Becausethere isnoexplicit“credit”effectonanERM,thereisnoobvious“default”ifno-onedecidestorecogniseit.Don’t forget thateven if someonedies, the recognitionof that factand themovementtosaleoftheunderlyingcollateralcanbedelayedforagestooasyoucanclaimtobenegotiatingwiththeheirsetcetc,andall thewhiletheassetisstillrollingupatthecompoundrate(eventhougheconomicallytheassetshouldhavebeencrystallisedmuchclosertoactualdeath/LTC)wheninfactitisgettingevermoreabove100%LTV.Butyoucanarguethatitisstillalegalclaimontheestate,evenifitsunlikelyevertobefullyrealised–youcan

82

saythatyouwereenmeshedinnegotiationswiththeestateandhopingforasuddendoublingof thehousingmarket. Ihaveseenactualcases in theUKwhere the time from both mortgagors final death date to my portfolioappraisaldateasputativebuyerhasbeengreaterthan7years–andSTILLtheassetshowsascurrent,nicelyrollingupetcetc,whenitpassed100%BEFOREtheydied,andwhenpresumablyitsnowfullofsquattersandfallingdown!Liftthecornerofthismatandallthelittlebugscomerushingout……….

Evenso,hesuggests,“abitoflaggingissmallbeercomparedtothewide[achievementrate]dispersionnomatterwhat.”

83

ChapterFourteen:CreditSpreadsWehavesofarassumedthatwearediscountingatrisk-free,i.e.,withnocreditspread.Noteveryoneagreeswiththisapproach,however.Therearetworeasonswhysomeonemightwanttoaddacreditspreadtothediscountrate. The first is because theywant to get lowerNNEG valuations, and the second isbecausetheybelievethatthereshouldbeacreditspread.Bothofthesearewrongandthefirstself-evidentlyso:thereisneveragoodreasonfortweakingcalibrationstogetdesired outcomes. But even if well-intended, it is still amistake to add in the creditspread.Agoodexampleof thecredit spreadargument isprovidedbyDavidLandat the IFoAsessionaleventonERMson11December2018.HeworksforRothesayLife,whoareamajorproviderofannuitiesandamajoruseroftheMatchingAdjustment.72TheyalsojustboughtawhackinggreatportfolioofEquityReleaseMortgages fromthetaxpayer.73 Itmakes sense that Rothesay would be interested in an approach that lowers NNEGvaluations.WethenwonderifRothesayareusingahigher-than-LIBORfundingratetocheapenthecostoftheirguarantees.Iftheyare,theyshouldn’tbe.In thediscussionat theDecember11event,Mr.Landaskedapointedquestion. If theworkingpartyhasn’tyet fixedtherightmethodofcalculatingtheforward, isn’t thataprettymajorsourceofpossibleerror?Nocoherentansweremerged,butLandraisedaninterestingpoint.Sowhatisgoingon?Well,ifwecan’tlowerthevalueofthenonegativeequityguaranteebyputtinginanoptimisticgrowthforecast,perhapswecantweakthefunding rate instead. He drops a hintwhen he suggests that there’s a large range ofpossiblefundingratesthatwemightconsiderandgoesontostatethat“ThePRAthinksthatyoucouldpossibly fundahouseatLIBORflat,whichseemsremarkablydifficult.”Theimplicationisthatweshouldbeaddinginaspread.Land’s argument is a seemingly plausible one and provides a much better case forloweringNNEGvaluationsthantheso-called‘realworld’approachthattheindustryaresoenamouredof.Itisstillwrong,butitiswronginamoreinterestingway.Nowweagreewithhimthat itwouldbeanunusual lenderwhomadearisky loanat‘LIBORflat’.Iftheyaretolendatall,theywouldchargeaspreadtocompensatefortheriskoflossfromdefault.Afterall,theborrowermightdefaultandleavethelenderwithalossbecauseinsufficientcollateralhasbeenposted.

72Seetheir2017SolvencyandFinancialConditionsReportathttps://www.rothesaylife.com/media/1183/rothesay-life-sfcr-2017.pdf73 See http://eumaeus.org/wordp/index.php/2018/09/28/doing-gods-work/ andhttp://eumaeus.org/wordp/index.php/2018/10/05/casting-magic-upon-daylight.The substance is thatin September 2018 Rothesay Life bought an £860m portfolio of equity release loans from UK AssetResolutionLimited(UKAR),agovernmentagencyestablishedon1October2010to‘facilitatetheorderlymanagement’oftheclosedmortgagebooksofbothBradford&BingleyandNRAM.(Source:OliverRalph,FT,27September2018,UKARwebsite).ThepreciseamountpaidbyRothesaywasnotdisclosed

84

Butconsiderthefollowing.Supposewedidaddaspreadandsuppose(becauseitistrueand easily verified) that adding the spreadproduced a lowerNNEG. You thenhave asituationwheretheworsethecreditqualityoftheborrower,thehigherthespread;andthehigherthespread,thecheapertheNNEG,becausetheNNEGisbasedonasetofputoptionspricedofftheforward.Sothespreadargumentdoesn’tpassthesnifftest.Let’strytodrawoutwhatLandmayhavehadinmind.Startwith:(14.1)𝑅 = 𝑆 × 𝑒4W*(14.2)𝐹 = 𝑆 × 𝑒4(In¶4W)*where𝑅 is thepriceof a reversion akadeferment contract,𝑆 is the spotpriceof theincome producing asset e.g., a property, 𝑞 is the discount rate aka deferment raterepresentingthepresentvalueoflostincomeoverthetermof𝑡years.𝐹isthepriceofaforwardcontract,𝑟theriskfreeand𝑠thefundingspreadoverLIBOR.IfweuseBlack76thentheunderlyingistheforwardprice𝐹ratherthanthedefermentprice𝑅.SincetheNNEGforanydecrement𝑡isaputoption,thehighertheforwardprice𝐹,themoretheoptionisoutofthemoney,andthelowerthecostoftheNNEGguarantee.SowecanmaketheNNEGatleastappearcheaperbyassuminganon-zerospread𝑠.Supposethenweareafirmholdingadefermentcontractforpossessionattandwewanttohedgeourriskbysellingthecorrespondingforwardcontract.Let’salsotakethecasemostfavourabletothecreditspreadargument,whichiswhereourcounterpartyhasnocollateralatall.Ifwepricethecontractperequation(14.2)above,itiscertainlytruewemaywanttochargeaspreadovertherisk-freerate𝑟tocompensatefortheriskthatthepurchaseroftheforwardcontractwilldefaultat𝑡.Sohowdowerepresentthevalueofourhedgedpositioninourbooks?Assumingnoriskofdefault,thepresentvalueofthepositionis(14.3)𝑅′ = 𝑅 × 𝑒¶*Butthereclearlyisariskofdefault,becausewejustassumedonewhenweappliedthespread𝑠!Wemustthereforereserveagainstthatrisk,butbyhowmuch?Wellifsisaspreadrepresentingthecostofdefault,i.e.,ifthedifferenceinpresentvalueattributableto𝑠 ispreciselythecostofdefault, thenthatsamedifferenceistheamountwewouldreserve.Consequently(14.4)Reserve= 𝑅 × 𝑒¶* − 𝑅Wemustthensubtractthereservefromourportfoliovalue:(14.5)Portfoliovalue= 𝑅¸ −reserve= 𝑅 × 𝑒¶* − (𝑅 × 𝑒¶* − 𝑅) = 𝑅Wearethenbackwherewestartedandthecorrectspreadiszero!Youcannotcheapenthevalueofadefermentcontractbyhedgingitinthederivativesmarket.Whateverincreasein

85

valueisattributabletoapplyingaspreadoverLIBORrepresentsacompensationforrisk,whichmustbesubtractedagainasareserve.Remembertoothatthiscaseistheonemostfavourabletothecreditspreadargument,that inwhich the counterparty provides no collateral. If the forward contract is fullycollateralised,on theotherhand, then therewouldbenocredit spreadbecause therewouldbeno credit risk. So either the spread goes in and then comesout again, or itdoesn’tgointostartwith.Andintheintermediatecasewherethecontractispartiallycollateralised,wehaveaweightedaverageofthesetwocasesandagainnospread.Wehaveonlyconsideredthecaseofaforward,butthesameappliestoanoption.Wehavejustshownthatwecannotraisetheforwardratebytweakingthefundingrate.Thefundingrateshouldbetheriskfreerate,notthehigherratethatmightbeofferedbyabank to, say, a buy-to-let investor. But if we cannot change the forward rate, then itfollowsthattheNNEGvaluationmustremainunchangedaswell.

86

ChapterFifteen:DrawdownDrawdownERMsdifferfromthe‘conventional’ERMs(oftencalledLifetimeMortgages,LTMs) that we have considered so far in that the borrower contracts to receive adrawdownfacilityratherthanaloantakenoutatanyonetime.Thisfacilitywouldgivetheborrower the rightbutnot theobligation todrawdownon the facility athis/herdiscretion,uptothemaximumamountofthedrawdownfacility.Typically,theborrowerwoulddrawdownan initialamountwhenthe facility issetupwithaviewtomakingsubsequentdrawdownslater.Theloancontractwouldspecifyacurrentdrawdownlendingrate,whichweunderstandwouldtypicallybealittlehigher(maybe5-6basispointshigher)thanthelendingrateonanLTMmortgage.Onceadrawdownismade, the lendingrateonthat loantranche isfixed.Thedrawdownlendingrate,liketheLTMloanrate,willchangeovertimeasthelenderperiodicallyadjuststherateinresponsetochangesinmarketconditionsanditsownlendingpolicy.TheexistenceofthedrawdownfacilitymakesDrawdownERMsmoreinvolvedthanLTMERMs,buttherehasbeenlittlediscussionofhowtohandlethedrawdownfacilityfromtheERMvaluationperspective.74One way tomodel valuations on drawdown ERMs and NNEGs would be to use pasthistoricaldata–whichthefirmwouldhave–onitsdrawdownexperience.Itsvaluationspecialistscouldthenusethesedatatoprojectthetimingoffuturedrawdownsandtheamountsdrawndowneachtime.Forexample,theymightanticipatethatafemaleaged70woulddrawdown50%oftheremainingfacilityatage74anddrawdowntherestofthefacilityatage78.Theycouldalsoforecastthedrawdownlendingratesatthesefuturetimes.ForERMandNNEGvaluationpurposestheycouldthentreatthedrawdownERMasequivalent toaportfolioof threeseparateLTMloansandwealreadyknowhowtovaluetheERMsandNNEGsofLTMloans:therewouldbeaninitialLTMloantakenoutnowwhentheborroweris70,asecondLTMloanthatisexpectedtobetakenoutin4years’timebythesamefemalewhensheis74andathirdLTMloanthatisexpectedtobetakenoutin8years’timebythesamefemalewhensheis78.Analternativewaytomodelvaluationsisjusttoassumethatthewholeofthedrawdownfacilityisdrawndowninonegowhenthecontractismade.Inthatcase,thedrawdownERMloanwouldbetreatedasequivalenttoasingleLTMloan.

74InitsCP7/19(section2.15to2.18),thePRAhasanextensivediscussionofdrawdownissuesthatis,nonetheless,quitesolution-lite.SeePRAConsultationPaper7/19“SolvencyII:EquityReleaseMortgages–Part2,”April2019.

87

ChapterSixteen:EarlyRepaymentManyERMcontractsallowtheborrowertorepayearlierundercertainconditions.Nowearlyrepaymentofmortgagesisanotoriouslyfiendishproblem.75Itiswellknown,forexample,thatprepaymentratesonmortgage-backedsecuritiesaresensitivetointerestratesandMBSmodellershavehadgreatdifficultiesgetting thismodelling ‘right’.Onemight however suppose (or at least hope) that the prospects of future repaymentsinducedbylowerinterestrateswouldnotbesuchaproblematcurrentlowinterestrates.TheusualapproachsuggestedintheNNEGliteratureistoworkwithassumedrepaymentrates, whichmight be either pulled out of thin air or based on historical repaymentbehaviour. Tunaru (2019, section 10.9) has a gooddiscussion of this subject, but notmuchisknown.76Historicalprepaymentratestendtobehighintheearlyyearsbutthentailoff.Suchratesmakesenseintuitively:intheearlyyear,borrowersmayregrettheloanandstillbeabletopayitback;inthelateryears,itbecomesincreasinglydifficulttopayoffearlyastheloanamountrollsupandtheirfinancialcircumstancesdeteriorate.Thebestwecandoisprobablytousehistoricalrepaymentratesasaproxyforfuturerepaymentprobabilities.HowtoHandletoEarlyRepaymentOnewaytomakeuseoftheseforNNEGandERMvaluationgoesasfollows.ImagineforthemomentthatanERMcontracthadnofacilityforearlyrepayment.ThenthevalueoftheNNEG,𝑁𝑁𝐸𝐺,andthevalueoftheERM,𝐸𝑅𝑀,wouldbewhattheywereinpreviouschapters,andwecanmoveon.Putitthisway:if𝑁𝑁𝐸𝐺m¹Iºg»¼isthevalueoftheNNEGwithoutthepossibilityofearlyrepaymentandsoforth,then(16.1)𝑁𝑁𝐸𝐺 = 𝑁𝑁𝐸𝐺m¹Iºg»¼(16.2)𝐸𝑅𝑀 = 𝐸𝑅𝑀m¹Iºg»¼Wenowintroducethepossibilityofearlyrepaymentandaskourselves:howdoesthepossibilityofearlyrepaymentaffectNNEGandERMvaluations?Let’s startwith a pre-existing ERM loan, and let its value be𝐸𝑅𝑀¹H½ . If the old/pre-existingERMloanisrepaid,thenacertainamount–𝑋,say–ispaidtothelenderandtheERMisextinguished.𝑋wouldincludeanyearlyrepaymentchargesstipulatedbytheERMcontract.The importantpoint for themodeller is thatwecaneasilyworkoutwhat𝑋mightbe. 75 Formore on these issues, see, e.g., JPMorganMBSPrimer, June 2006, or A. Davidson andA. LevinMortgageValuationModels:EmbeddedOptions,Risk,andUncertainty,OxfordUniversityPress,2014.76 Anotherproblem is that the ratingsagencies takeavery conservativeapproach to repayment.Theytypicallymodelmortgagesecuritiesusingayieldtomaturityapproach,andthenaccountforprepaymentas reinvestment at swapsminus 50 bps. This approachmakes prepayment appear very expensive forlenders.

88

Weneednowtomakeanassumptionaboutthelenderdoeswiththemoneyitreceives.ItcouldinvestinT-bill,holditincashandsoon,butlet’ssupposethatthelenderusesthemoneythatisrepaidtomakeanewERMloan,𝐸𝑅𝑀mº¾ .77Let’ssupposealsothatprocessofconvertingtheloanrepaymenttothepointwhereanew ERM loan is made costs the firm 𝑌, where 𝑌 would be the costs of marketing,distributionandsoon,netofanysuchchargestobeappliedtothenewcustomer.Again,themodellercanworkoutwhat𝑌mightbe.Consequently,theamountleftovertobeloanedtothenewcustomeris𝑋 − 𝑌.TomaketheloanamountexplicitinthevaluationofthenewERM,callthelatter𝐸𝑅𝑀mº¾|(𝑋 − 𝑌).Thus,𝐸𝑅𝑀mº¾|(𝑋 − 𝑌) is the value of thenewERM loan,where𝑋 − 𝑌 is the amountloaned at inception. Note that the input calibrations of the newERM loanmight andtypicallywouldbedifferentfromthoseoftheoldERMloan,e.g.,theloanratemighthavechanged.However,themodellerwouldhavetheinformationneededtoobtainthevalueofthenewERMloan.SowhenanERMloanispaidoffearly,theERMlenderlosesanassetworth𝐸𝑅𝑀¹H½ ,butacquires an assetworth𝐸𝑅𝑀mº¾|(𝑋 − 𝑌), andbyhypothesisweknowhow to obtain𝐸𝑅𝑀mº¾|(𝑋 − 𝑌).The‘transaction’mayormaynotbebeneficialtothelender,butthelenderhasnochoicebut to accept the loan repayment if the terms of original loan allow it, and we areassumingthatthebestthelendercandowiththemoneyrepaidistoinvestitintoanewERMloan.Exante,wedon’tknowwhethertheborrowerwillrepayearlyornot,butwecansupposethat the modeller has access to data on past loan repayment frequencies, e.g., if themodellerisanERMactuaryworkingforalargefirm,thenheorshewillhaveaccesstothefirm’sloanhistories,andcanthenestimatepastloanrepaymentfrequencies.Fromthese,themodellercanestimate𝜙,theprobabilitythatagivenERMloanwillberepaidoversomefutureperiod.ThemodellercanthenobtainthevalueoftheERMtakingintoaccountthepossibilityofearlyrepayment,e.g.:(16.3)𝐸𝑅𝑀 = 𝜙 × 𝐸𝑅𝑀mº¾|(𝑋 − 𝑌) + (1 − 𝜙) × 𝐸𝑅𝑀m¹Iºg»¼ 77Thisassumptionwillnotalwaysbeappropriate,however.Asoneofourreaderspointsout:“valuingoneERMassetontheassumptionthatyoucanoriginateanother,atwill,toorder,whenneeded,outofthinair,is going to struggle to pass IASB or SII orMAP constraints.Looks like falsely capitalising the value ofreinvestmentrisktome.”Hispointsarewell-taken.Ifre-investingtheproceedsfromanearlyrepaymentintoanewERMisanissue,thenoneneedstakeaviewonhowelsethoseproceedsshouldbeinvested.Forexample,wouldtheybeinvestedingilts,andsoon.Butthebasicapproachwesetoutherecaneasilybetweakedtohandlesuchalternatives.

89

ChapterSeventeen:Fees,ChargesandExpensesInprinciple,wemightwanttotakeaccountoffeesandcharges.Wesay“might”forgoodreason:iffees,chargesandexpensesarefairlyset,i.e.,theyareneitherexcessivenortoolow, then if those feesetc.arepaidbytheborrower, thentheyhaveno impactonthelenderandhencewouldbeirrelevantforNNEGvaluation.TheywouldalsobeirrelevantforERMvaluation,atleastfromthelender’sperspective.Whetherfeesandsofortharefairlysetisanothermatterandisbeyondthescopeofthisreport.Accordingly, in this chapterwesetoutwhat informationwehaveon feesandcharges,andleaveittopractitionerstodecideif,andifso,how,thesemightaffectNNEGvaluations.Forwhatitisworth,ourviewisthatthesearesecond-orderissuesrelativetothebigissueslikethecalibrationofthedefermentrateorvolatilities.Ontheotherhand,fees and chargeswouldbeof considerable interest toborrowersandwould certainlyrelevantwhendoingvalue-for-moneyanalyses.Thosehoweverarebeyondourremit.HostyetaliaonfeesandchargesHostyetalia(2007,p,29)reportthefollowingpolicyexpensessummary:

90

Ourunderstandingisthatthe‘distributionandsales’and‘marketing’chargesbothapply.Thus,thetotalchargefordistribution,salesandmarketingwouldbe3.5%oftheloan.Hosty et al. (2007, p. 31) also report a ‘specimen product specification’ that includessimilaritemsbutinadditionincludes:

• provider’slegalfee(£300)andpropertyvaluationfee(£1permille)toaddedtotheloan;

• property sales expenses equal to 2% of final property value: these would bechargedtotheborrowerunlesstheNNEGisactivated;

• additionalcostsineventofnegativeequityclaim:£500tocovercostsofadditionalvaluationandadministration;and

• earlyrepaymentcharges:“Marktomarketwith25%cap”,whateverthatmeans.

91

Thesechargesand feeswere in2007£s.According to theBankofEngland’s inflationcalculator the cumulative inflation since then has been 13.6%, so we would have tomultiplythesenumbersby1.136togettheirequivalentsattoday’spricelevel.78Hostyetalia(2008)providetheadditionalclarificationthat‘othercosts’arepassedtothecustomer,presumablywiththeexceptionofpropertysalesexpensesiftheNNEGisactivated,becausetodootherwisewouldviolatetheNNEG. ContemporaryFeesandChargesLoanrates:wehaveseencontemporary loanratesvarying from4.15%AERto6.78%AER.Itpaystoshoparound.Repaymentcharges:wehaveseencaseswhere thecharges torepay loanswereabout9.6%oftheamountowed.Foronelargefirmwehaveseenchargesandfeesthatinclude:

• Loanrate=5.35%AER• Startfee=£600• Advicefee=£995• Legalfee=£650• Endfee=£125.

Ourunderstandingisthatforthisfirm,allothercostsandexpenseswouldbepassedtotheborrower,unlessdoingsowouldviolatetheNNEG.Thereisalsotheissueofrepaymentcharges,andweareawareofonecontemporarycaseinwhichaborrowerfacedarepaymentchargeofabout16%oftherolled-uploanamountanditstruckusthatthischargewashigh.Therecouldalsobeotherfeesandcharges,butwehavebeenunabletodoasystematicsearch.A surprising example recently came to light, however. A recent Private Eye articlerevealedanicelittleschemeinvolvingAgeUK.79PotentialborrowerslookingattheAgeConcernwebsitewhoareinterestedinequityreleaseareencouragedto“dip[their]toeinthewaterwithhelpfromtheAgeCoUKEquityReleaseAdviceService,providedbyHubFinancialSolutionsLtd.”AstheEyearticlecontinues.“Hub,itturnsout,isownedby,er, Just Group. But its not just good business for Just. Age UK itself takes a handycommissionof “upto0.75percentof theamountadvancedundereachequityreleaseplansold,togetherwithacontributiontowardsmarkingsupport.””

78https://www.bankofengland.co.uk/monetary-policy/inflation/inflation-calculator79“JustReward.”PrivateEyeNo.149022February-7March2019,p.39.

92

ChapterEighteen:ScenarioAnalysisandStressTestingAscenarioanalysisisahypothetical‘whatif’exerciseinwhichweexaminewhatmighthappen to some variable of interest (e.g., a NNEG or ERM valuation) if some futurescenarioweretoplayout.Forexample,wemightexaminewhatwouldhappenaccordingtoourmodeliffuturehousepricesweretobehaveinaparticularway.Astresstestisascenarioanalysisinwhichthepositedscenarioisanadverseone(e.g.,alargedropinhouseprices).Type1HousePriceScenarioAnalysisorStressTestOnetypeofscenarioanalysis/stresstestistomodeltheimpactofanimmediateone-offhousepricefall.Weassumethathousepricesfall fiveminutesaftertheERMloanhasbeenmade.Thisexerciseiseasytocarryout.WefirstvaluetheNNEGorERMattheinitialhousepricevalue.Wethenre-valuethemimmediatelyafterthehousepricefallusingthenewLTV.Forexample,iftheinitialLTV=40%andhousepricesfallby50%,thenthenewLTVwillbe0.4 × 1/(1 − 0.5) = 80%. For this typeof stress testwedonotmakeanyprojectionsoffuturevariables,e.g.,futurehouseprices,otherthanthathousepricesfallshortlyaftertheERMloanismade.Table18.1givestheimpactofanimmediateone-offhousepricefallof50%.

Table18.1:ImpactofanImmediate50%FallinHousePrices 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Pre-stress £74.8 £32.2 £42.7Post-stress £74.8 £48.4 £26.4

∆𝑳 ∆𝑵𝑵𝑬𝑮 ∆𝑬𝑹𝑴Impactofstress £0 £16.2 -£16.2

%Impact 0 50.4 -38.1Notes:𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguaranteeand𝐸𝑅𝑀isthepresentvalueoftheEquityReleaseMortgage.Basedonthebaselineassumptions:maleaged70,𝑟=1.5%,𝑙=5.25%,𝑞=4.2%and𝜎=14.8%.Forthepre-stressscenario,weassumehouseprice=£100and𝐿𝑇𝑉=40%;forthestressscenario,weassumehouseprice=£50and𝐿𝑇𝑉=80%.ExitprobabilitiesarebasedonM5-CBDmodel projections using England&Walesmale deaths rate data spanning years1971:2017andages55:89.

Forthegivensetofcalibrations,𝑁𝑁𝐸𝐺risesby£16.2and𝐸𝑅𝑀fallsbythesameamount.𝑁𝑁𝐸𝐺increasesby50.4%and𝐸𝑅𝑀fallsby38.1%.Figure18.1showstheimpactonNNEGandERMvaluationsofarangeofimmediateone-offhousepricefalls.

93

Figure18.1:ImpactofaRangeofImmediateHousePriceFallsonNNEGandERM

Notes:AsperTable18.1.

Weseethatthegreaterthefallinhouseprices,thegreaterthefallinERMvaluation.Type2HousePriceScenarioAnalysisorStressTestInasecondtypeofexercise,weassumeahypotheticalrateofgrowthofℎ𝑝𝑖andshowhow,e.g., theERMvaluationwouldbehaveovertimeunderthepositedscenario.Thistypeofexerciseworksasfollows.WestartbyusingourERMvaluationmodeltoobtainthecurrentvalueoftheERM.ThenweobtaintheexpectedvalueoftheERMafteroneyear,whichwouldbe1yearexitprobabilitytimestheexpectedpayoffifexitoccursinyear1,plusthe1yearprobabilityofnoexittimestheERMvalueafteroneyear.ThislatterERMvalueisobtainedusingourERMvaluationmodel,buttakingaccountofthenewageafter1year(i.e.,1yearolder),thenewhousepriceafteroneyear(whichisequaltotheoldhousepricetimes𝑒fgh),andthenewLTVafteroneyear(whichisequaltotheoldLTVtimes𝑒H4fgh).WecarryoninlikemannerfortheexpectedvalueoftheERMaftertwoyearsandsoforth.Inpractice,wewouldoftenwanttocomparetwodifferentscenarios:abasescenario,whichmightbethescenarioweexpect,andastressoradversescenario.Forexample,wemightfollowJustGroupandassumeℎ𝑝𝑖 = 4.25%forourexpectedorbasescenario(seeChapter24below).Wethenpositsomestressscenario,e.g.,ℎ𝑝𝑖 = −1.7%,whichwastheaverageℎ𝑝𝑖inJapanover1990:2017.Figure18.2showsaplotofthesetwoscenarios:

94

Figure18.2:ExpectedERMValuationsUnder4.25%vs.JapanHousePriceGrowthScenarios

Notes:AsperTable18.1.

WeseeamuchlowerexpectedERMvaluationprojectionunderthestressscenario.Thelessonhereisthatifwewererelyingonℎ𝑝𝑖 = 4.25%butactualℎ𝑝𝑖turnsoutto-1.7%,then the large𝐸𝑅𝑀 increaseswewereexpectingwillnot comepass, and futureERMvaluationswilldeclinetozeroconsiderablymorequicklythanwehadexpected.ExpectedCashflowsunderType2HousePriceStressTestAnothertypeofstresstestisprojectcashflowsunderanassumedhousepricescenario.Figure18.3showtheprojectedcashflowsunderthesametwoscenarios.Figure18.3:ExpectedERMCashflowsUnder4.25%vsJapanHousePriceGrowth

Scenarios

Notes:AsperTable18.1.

95

Ifthe4.25%scenarioweretotranspirethensubsequentlyrealisedcashflowswouldbemuchlargerthaniftheJapanscenarioweretotranspire,andtheJapan-stylecashflowswouldbeevenlowerifwealsoexperienced,say,95%achievementrateinsteadofthe100%achievementrateassumedsofar:Figure18.4:ExpectedERMCashflowsUnder4.25%vsJapanHousePriceGrowth

Scenarios(II)

Notes:AsperTable18.1.

LongevityScenarioAnalysisorStressTestWecanalsocarryoutscenarioanalysesorstresstestsbasedonotherprojections.Animportantone in theequityreleasecontextwouldbea longevityscenarioanalysis, inwhichwepositsomechangetoexpectedlongevityandconsidertheimpactofthatchangeonNNEGorERMvaluations.Aneasywaytocarryoutsuchanexerciseistoassumeaparticularchangeinlongevity,say,weassumethatlongevitysuddenlyincreasesby3years.Wecanthenapproximatetheimpactofthisscenariobyreducingby3yearstheageoftheindividualinputtedintoourvaluationmodel,whilstkeepingotherparameters(andespeciallytheLTV)thesameastheywere.Table18.2givestheresultsofsuchanexercise.

96

Table18.2:Impactofa3YearIncreaseinLongevity 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Current £74.8 £32.2 £42.6Underscenario £82.8 £42.4 £40.4

∆𝑵𝑵𝑬𝑮 ∆𝑵𝑵𝑬𝑮 ∆𝑬𝑹𝑴Impactofscenario £7.9 £10.2 -£2.3

Notes:𝐿isthepresentvalueoftherisk-freeloancomponentoftheERM,𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguaranteeand𝐸𝑅𝑀isthepresentvalueoftheEquityReleaseMortgage.Basedon the baseline assumptions: house price=£100, male aged 70, 𝐿𝑇𝑉=40%,𝑟=1.5%, 𝑙=6%,𝑞=4.2%,and𝜎=14.8%.Forthescenario,weinputanageof67.ExitprobabilitiesarebasedonM5-CBD model projections using England & Wales male deaths rate data spanning years1971:2017andages55:89.Sourceofdata:llma.org.

Weseethatboth𝐿and𝑁𝑁𝐸𝐺risebycomparableamountsandlargelyoffsetintheirnetimpactonthevalueoftheERM.Itisalsointerestingtonotethatthesignoftheimpacton𝐸𝑅𝑀isnegative.AnincreaseinlongevitydecreasesthevalueoftheERM.ThisresultmeansthattheimpactofincreasedlongevityonthevalueoftheERMhasthewrongsigntofunctionashedgetoanannuitybook.Theliabilitiesofanannuitybookincreasewhenlongevityrises,sotofunctionasahedge,anassetmustalsoincreaseinvalue,butinthiscasetheERMdecreasesinvalueinstead.

97

ChapterNineteen:ThePRA’sGoodPracticeERMValuationPrinciplesIn its Supervisory Statement SS 3/17 published in July 2017, the UK PrudentialRegulationAuthoritysetoutcertaingoodpracticeprinciplesrelatingtoERMportfolios.TheseprinciplesincludetwothatimposeupperboundsonERMvaluations.PrincipleIIPrincipleIIstates:

TheeconomicvalueofERMcashflowscannotbegreaterthaneitherthevalueof an equivalent loan without an NNEG or the present value of deferredpossessionofthepropertyprovidingcollateral.

i.e.,(19.1)𝐸𝑅𝑀 ≤ 𝐿and𝐸𝑅𝑀 ≤ 𝑃𝑉(𝐹)wherePV(.)isthepresentvalueofthetermin(.).Proofthat𝐸𝑅𝑀 ≤ 𝐿Startwith(3.1)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺Weknowthat(19.2)𝑁𝑁𝐸𝐺 ≥ 0.If𝑁𝑁𝐸𝐺 > 0then(19.3)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺 < 𝐿.If𝑁𝑁𝐸𝐺 = 0then(19.4)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺 = 𝐿.Hence(19.5)𝐸𝑅𝑀 ≤ 𝐿whichwastobeproved.

98

Proofthat𝐸𝑅𝑀 ≤ 𝑃𝑉(𝐹)The Present Value (PV) period 𝑡 payoff to 𝐸𝑅𝑀*ismin[𝐿*, 𝑃𝑉(𝐹*)], where 𝐿* is thepresentvalueoftheloanassumingitmaturesin𝑡yearsandassumingthatthereisnoNNEGinvolved,and𝑃𝑉(𝐹*)isthepresentvalueoftheperiod𝑡forwardcontract,forall𝑡.But(19.6)min[𝐿*, 𝑃𝑉(𝐹*)] ≤ 𝑃𝑉(𝐹*)forall𝑡.Therefore(19.7)𝐸𝑅𝑀* ≤ 𝑃𝑉(𝐹*)forall𝑡.Hence(19.8)𝐸𝑅𝑀 ≤ 𝑃𝑉(𝐹)whichwastobeproved.PrincipleIIIPrincipleIIIstates:

Thepresentvalueofdeferredpossessionofapropertyshouldbelessthanthevalueofimmediatepossession

i.e.,(19.9)Defermenthousevalue<spothousevalue.OntheValidityofPRAPrincipleIII,i.e.,WhyDefermentPropertyValuesareLowerthanCurrentPropertyValuesAt the risk of belabouring the obvious (because wemust!), we provide a number ofalternativedemonstrationsofthevalidityofPrincipleIII.Demonstration#1 Comparethevalueoftwocontracts,onegivingimmediatepossessionoftheproperty,theothergivingdeferredpossessionwhenexitoccurs.Theonlydifferencebetweenthesecontractsisthevalueofforegonerights(e.g.,torentalincomeortouseoftheproperty)duringthedefermentperiod,andthevalueoftheseforegonerightsshouldbepositivefortheresidentialpropertiesusedascollateralforERMs.Itthenfollowsthatthepresent

99

valueofdeferredpossessionshouldbelessthanthevalueofimmediatepossession,i.e.,weobtainPrincipleIII.PrincipleIIIthusfollowsfromelementaryeconomics.Whywouldwenotpaylesstogetless?Demonstration#2 Asanalternativedemonstration,recall(3.10)𝑅* = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒4W*where𝑞isthedefermentrateand𝑅*isthedefermentprice,andnotethatthespothousevalueandthecurrenthousepricewillbeequal.Assuming(19.10)𝑞 > 0anditisreasonabletoassumethat(19.10)doeshold,then(19.11)𝑅* = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒4W* = 𝑠𝑝𝑜𝑡ℎ𝑜𝑢𝑠𝑒𝑣𝑎𝑙𝑢𝑒 × 𝑒4W*whichimplies(19.12)𝑅* < 𝑠𝑝𝑜𝑡ℎ𝑜𝑢𝑠𝑒𝑣𝑎𝑙𝑢𝑒Itisthenreasonabletosupposethatthedefermenthousevaluewillbeequalto𝑅*andPrincipleIIIfollows.Demonstration#3 Alongerandmorerigorousdemonstrationgoesasfollows:Let𝑞0,𝑞5,𝑞M,….bethesetofnetrentalratesforapropertyfromnow,period0,toforever.Thesenetrentalservicesaretheuse-benefitswegetfromlivinginaproperty(e.g.,thebenefitsofhavingaroofoverourheads)ortherentalincomeswecouldobtainbyrentingthepropertyout.Letusassumethattheseareallpositive.Afterall,zeroornegativerentalratesdonotmakemuchsense.LetAbethesetofthosenetrentalrates𝑞0,….forperiods0toforever.LetBbethesetofnetrentalrates𝑞* ,𝑞*n5….fromperiodsttoforever,where𝑡 ≥ 1.LetCbethesetofnetrentalrates𝑞0,...𝑞*45,forperiods0tot-1.AssumeforthemomentthatthepricesofA,BandCallexist.

100

Sincethesetsofrentalsarepositiveandhencevaluable, thenthepricesofA,BandCshouldeachbepositive.Bythelawofzeroarbitrage,thepriceofAshouldalsobeequaltothesumofthepricesofBandC.ButsincethepriceofCispositive,itmustfollowthatthepriceofB<thepriceofA,i.e.,thedefermentpricemustbelessthanthecurrentpriceandPrincipleIIIisestablished.Tochallengethisconclusion,itisnecessarytoarguethatsomeofthesepricesdonotexist.SincethepriceofAisthespotprice,thenthepriceofAclearlydoesexist,soonewouldhavetoarguethatthepricesofBand/orCdonotexist.Let’snotetobeginwiththattheempiricalbasisofanysuchclaimisarguable.Whilstitismanifestly obvious that the prices of B and/or C will rarely exist for some specificproperty, it isoftenpossible to inferproxyprices fordifferent typesofproperty fromcomparisonsoffreeholdandleaseholdpricesanditistheseproxypricesthatonewoulduseforvaluationpurposes.80Forexample,consideraleaseholdonaLondonflatwith99yearstorun.Thepriceofthisleaseholdwouldtypicallytradeatabout95%ofthepriceofavacantfreehold,andthecorrespondingfreehold,i.e.therighttoexclusivepossessionafter99years,wouldtradeat about 5%of the vacant value, and givesus theprice of possessiondeferredby99years.81Thefollowingchartshowsimplieddefermentprices–thedefermentpricesimpliedbyleaseholdprices,expressedasapercentageofthefreeholdvacantpossessionvalue82–forRICSprimecentralLondon2009andLeaseholdValuersLLP2017:

80Typically,onewouldobtaindefermentpricesforaparticularpropertybyapplyingrulesofthumbtothepricesfordifferentpropertytypes.Thesewouldtakeaccountofparticularfeaturesofapropertysuchaslocation,parkingavailability,thesizeofthegardenandsoon.81Seethe‘relativitygraphs’:http://www.graphsofrelativity.co.uk/.82Weassumethatthereisno‘marriagevalue’,i.e.thatthesumofthemarketvalueoftheleaseholdandfreeholdequalthevalueofvacantpossession.Foradiscussionofthisissue,seethenextfootnote.

101

Figure19.1:EmpiricalImpliedDefermentPrices

Notes: RICS = Royal Institute of Chartered Surveyors. Sources:http://www.graphsofrelativity.co.ukandLeaseholdValuersLLP.

Thesefallasthedefermenthorizonlengthensandarealwayslessthanthespotprice.83Butsupposeforthesakeofargumentandcontrarytotheevidencejustpresentedthatsomeofthesepricesdonotexistanddonothavenearapproximationsorproxiesintermsofothermarketprices.Inthissituation,wesimplyswitchthemetricfrompricestovaluesandwecanestablish thevalidityofPrinciple III inmuchthesamewayasbefore.Forexample,ifweassumethateachnetrentalratehasapositivevalue,thenitimmediatelyfollowsthateachofA,BandChaspositivevalue,sothevalueofBmustbelessthanthatofAandPrincipleIIIfollows.Indeed,evenifweassumethatthecurrentnetrentalrateispositiveandtheothersaremerelynon-negative,thenPrincipleIIIstillfollows.To challengePrinciple III, one is then lefthaving toargue thatnet rental ratesor thevaluesofnetrentalratesarenegative.Let’sconsiderpossibleexamples.Oneiswherethepropertyandthelandwhichitstandsarepollutedbeyondanyfeasiblerepair.Chernobylcomestomind:evenifthelandcouldberestoredtoausablestate,thecostsofdoingsowouldbeprohibitive. In thiscase,all𝑞0,𝑞5,….arenegativeandwillremain so. The property and the land itself would then be abandoned. This type ofsituationisrare,however.

83 Theseresultsareillustrationsonly,andotherfactorscomeintoplay.Forexample:(a)Currentleaseholdvaluesreflecttherighttoextendatamarketvalue,whereasERMborrowershavenosuchright(the‘lease’endswhen they exit into long term care or die, and the estate has no right of extension. (b)Wehaveassumednomarriagevalue–marriagevalue is theadditionalvaluean interest in landgainswhenthelandlord’s and the leaseholder’s separate interests are “married” into single ownership (see LawCommission,2018,p.23,n.62)-butaddingmarriagevaluewouldincreasetheimplied𝑞ratesfurther.and(c)ordinaryleaseholdstendnottoterminatewiththepropertyinruin,whereasthereisevidencethatveryoldERMborrowerstendtoneglecttheirproperty.

102

Acomparablecasewouldbewherethelandonwhichahouseisbuilterodesoffacliffintothesea.ButthecounterargumentisthatwewouldnotexpectmanylenderstobeERMingpropertiesthatcouldfallintotheseaafteracoupleofstorms. Alessrarecaseiswherethepropertyisuninhabitableandrepairwouldbeuneconomic,butthelanditselfisvaluable.PartsofDetroitcometomind.Onemightthensaythatthe(currentornearcurrent)netrentalproceedswerenegative,butthissituationwouldnotlastbecausethelanditselfisvaluable.Thepropertywouldbedemolished,perhapsafterbeingsoldoff,andthesiteredevelopedtorestoreapositivenetrentalstream.A third and more common case is where the property needs repair and repair iseconomically feasible. The propertymight not generate any current net rental, but itwouldberepairedandapositiverentalstreamrestored.Thissituationisnotuncommon,butisstillrelativelyinfrequent,inthatitdoesnotapplytomostpropertiesmostofthetime.Thegeneralcaseisthatmostpropertiesmostofthetimegenerateapositivenetrentalstream.Therefore,whenlookingforageneralruletoassessdefermentvalue,theonlysensibleruleistoassumeapositivenetrentalstream–andapositivenetrentalstreamimpliesthatthedefermentvaluewillbelessthanthecurrentpropertyvalue.Inshort,ifthepricesofA,BandCallexistandarepositive,thenthevalidityofPrincipleIIIfollowsfromzeroarbitrage.IfanyofthepricesofA,Band/orCdonotexist,however,thenwecanstillobtainPrincipleIIIbyswitchingovertoarationalvaluationargument,inwhich it suffices to argue that thevaluesofA,B andC are all positivebecause theunderlyingrentalshavepositivevalue.Demonstration#4:Thefiduciaryprinciple Thereisalsoanormativeargumentthatonecancallthe‘fiduciaryprinciple’.Evenwheremarketpricesdonotexist, accountingprinciples say that theaccountantshouldvalueeconomically similar assets in the same way and imply that valuationshouldreflectrationalinvestorpreferences.Theword‘should’or‘ought’appears,e.g.,inIFRS13B14a:“Cashflowsanddiscountratesshouldreflecttheassumptionsthatmarketparticipantswould use when pricing the asset or liability.” The fiduciary principle says that anaccountantorauditororsomeotherperson,whohasanobligationoftrusttowardsalessknowledgeable investor, must value an asset or liability as a rational knowledgeableinvestor(ormarketparticipant,orknowledgeable,willingindependentperson)would.Thisprincipleprovidesasafeguardagainstinterestedpartiescomingbackalongthelinesof “no arbitrage doesn’t apply here, so we can make up any price that benefitsmanagement,othernon-fiduciariesoranyoneelsewechoose.”Applyingthisprinciple,theaccountant,actuaryetc.mustacknowledgethatrentalserviceshavepositivevalueandthisacknowledgementsufficestoestablishPrincipleIII.

103

MisconceptionsAboutPrincipleIIIGuyThomasGuyThomastakesissuewithPrincipleIIIinarecentposting(Thomas,2018).Inhispiece,heacknowledgesthatthelossofforegonerights(e.g.,toincomeoruseoftheproperty)during the deferment period [i.e., the argument underlying Principle III] “appears areasonable argument” but even so, adds that “there are also reasonable counter-arguments.”Asheputit:

Housingtodayisownedmainlybyowner-occupiers.Theyhaveapreferencefora current interest to a deferred interest, because they need a roof over theirheads,theylikelong-termsecurityofoccupation,theylikebeingabletomaketheirownchoicesonextensionsandrepairs,etc.Inotherwords,theylikethepracticalandsentimentalbenefitsofhomeownership.Aminorityofownersarebuy-to-let landlords: they like understandable form of the investment, theunusual ability to finance it largely with borrowed money, and perhaps thedisengagementitfacilitatesfromthedistrustedpensionsandsavingsindustry.

Wewouldputitalittledifferently.Anyonewholivesinapropertygetsthe‘netrentalservices’ofthatproperty–theuse-valuebenefitsofaroofovertheirheadsandsoforth.Some people choose to obtain those benefits by buying their property and others byrentingthepropertytheylivein.Inthelattercase,thepropertyownergetsthebenefitofthe rent tenantspay, and inmostplausible situations, theownerwho rentsout theirpropertywillreceivearentthatmorethancoversthecostsofmaintainingtheirproperty.Thereareexceptionsaswehaveexplained,buttheseareunusual.

Foraninsurer,ontheotherhand,thesepracticalandsentimentalbenefitsofacurrent interest inahousehavenorelevance.Themainpotentialbenefitofacurrent(asopposedtodeferred)interestisthepotentialincomefromletting.

True,andthispointappliestoanyownerwhorentsouttheirproperty.But a current interest also has several disbenefits [sic]: tenants need to bemanaged, houses need to be maintained, from time to time there are costs(Including possibly PR costs) of evicting tenants in arrears, and there is apossibility(throughexistingornewlegislation)thattenantsmightacquirenewrights.

Yes,therearecostsandriskstohavingtenants.Ifontheotherhandhousesarekeptvacant,thisgivesanothersetofproblems:counciltax,securityandmaintenancecosts,andpossiblyveryconsiderablePRcostsofowningsubstantialamountsofemptyhousing.

Yes,therearealsocostsfromkeepingpropertiesinvacantpossession.

104

These disbenefits are not fanciful; theirmateriality can be inferred from theobservablefactthatdespitetheexcellentlong-termperformanceofhousingasaninvestment,neitherinsurersnoranyotherfinancialinstitutionshaveshownanyenthusiasmoverthepastseveraldecadesforhousingasanassetclass.

These passages are a roundabout way of saying that there are benefits and costs ofowningpropertybutifanownerregardsthecostsasoutweighingthebenefits,thenthesensiblechoicefortheowneristosell.Thepropertywillthenendupinthehandsofanownerwhodoesvaluethebenefitsasmorethanthecosts–otherwisetheywouldn’thaveboughtthepropertyandsomeoneelsewould.The lackofenthusiasm(orotherwise)of financial institutions forhousingasanassetclassisanotherquestion.Hecontinues:

Socurrentinterestsinhousesareevidentlynotattractivetoinsurersandotherinstitutional investors. Deferred interest might well be more attractive,particularly if in the form of cash-settled financial contracts, so that all theproblems of current interests are permanently avoided. Even if a deferredinterest is not strictly preferred, the relative valuation of a deferred interestcomparedtoacurrentinterestseemsverylikelytobemuchhigherforaninsurerthanatypicalindividualowner.(Ouremphasis)

Nowiftherewereasubstantialmarketfordeferredinterests,themoneyweightof individuals’preferenceforcurrent interestsversus insurers’preferencefordeferredinterestswoulddeterminetherelativemarketpricesforthetwotypesofinterest(i.e.whatthePRAcallsthe‘defermentrate’).Butwehavethesameproblemaswiththehedgingarguments:themarketfordeferredinterestsdoesnotexistonanymeaningfulscale.(Ouremphasis)

Leaving aside that amarket for deferred interests does exist (see above), Thomas iscomparingonehypotheticalnon-marketvaluation (i.e., insurers’valuationsof currentpossession)againstanother(i.e.,theirvaluationsofdeferredpossession).Acomparisonof the relative valuations of spot and deferred possessionmade by a party that is exhypothesi not amajor player in themarket does not establish (a) anything about themarketpricesorplausiblevaluesforcurrentpossessionorthemarketpricesorplausiblevaluesfordeferredpossessionoranyrelationshipbetweenthem.Inanycase,nosuchcomparisonestablishes(b)thatdeferred,forwardorfuture‘interests’havethenegativevaluenecessarytounderminethevalidityofPrincipleIII.Tomakepoint(a)inadifferentcontext,supposewevalueatypicalstatelyhomeasbeingworth2timesthevalueofatypicalcastle,butthemarketvaluesatypicalstatelyhomeasbeingworth3timesthevalueofatypicalcastle.Ourviewsmightbesincerelyheld,buttheyareofnorelevanceifwedon’thaveanyportfoliosofcastlesorstatelyhomesandarenotinthemarkettradingthem.Becausewearenotinthemarkettradingthesethings,ourviewsabouttheirrelativevaluationshavenorelevancetoanyonebutourselves.Theonlyvaluationsthatmatterarethoseofthemarket.Ortogiveanothercomparison.WemightbelievethatHollywoodmoviesarerubbishandBollywoodmovies are great, or the other way round. Makes no difference. The only

105

inferenceswecouldreasonablydrawfromthatexpressionofrelativepreferencesisthatwewouldnevergotoseetherubbishmovies.Butalotofpeopledogotoseetherubbishmoviesbecausetheydon’tshareourgoodtaste,sotherubbishmovieshaveeconomicvalue.Thereforewemightrationallyinvestinthem,evenifwewouldneverwatchthem.Thepoint isnot toconfusesubjectivetastewitheconomicvalue,andwhat is thebestestimateoftheeconomicvalue?Itiswhatthemarketwillpay.Inanycase,nosuchcomparisonsestablishthatdeferred, forwardor future ‘interests’havethenegativevaluenecessarytounderminethevalidityofPrincipleIII.Thomas’sargumentalsorunsafoulofthefiduciaryprinciple:evenifanactuaryorauditorhasaprivateviewoftherelativevaluationsofdeferredandcurrentpossession,theyarestillrequiredtoreportthemarketvaluations,andonlythosevaluations.Inshort,thevalidityofPrincipleIIIcanbebuttressedbysoundeconomictheory,solidaccounting principles and abundant empirical evidence, but the counter argumentscannot.RaduTunaruonPrincipleIIITunaru also challengesPrinciple III. In his report he argues (p. 50) that the “prepaidforwardprice”,i.e.thedefermentprice,willalwaysbelowerthanthecurrenthouseprice(i.e.thefreeholdpriceofvacantpossession):

the idea that the prepaid forward price should always be lower than thecurrenthousepricecanbechallenged.Thisconditionwillworkobviouslyinnormal market conditions and for shorter maturities. However, in theaftermathoffinancialandeconomiccrises,conditionsmaybereversed.Forexample, in the aftermath of the subprime crisis house prices droppedsignificantly.Theusualquestionof“howmuchshouldyoupaytogetahouseinfiveortenyearstime?”shouldbereplacedwiththequestion“whatpricecanyougeton themarket tosignnowforpossessionofahouse in fiveorten years time?” Even if the house price market was depressed in theaftermath of the subprime crisis, the expectation of the market wouldnaturallybe that themarketwill recoverafter some timeand the forwardcurvewill be in contango. Thus, it is possible that themarketwill requireaprepaidforwardthatishigherthanthecurrenthouseprice.

Thisargumentisnottheoreticallydefensibleandwearenotawareofanyevidenceforit.Froma theoretical point of view,whywould a rational investorwho expected housepricestorecoverpaymoreforacontractforpossessionin(e.g.)5yearstimethantheywould pay for a contract for vacant possession?The prices of both contracts will beidenticalafter5years,soinbothcasestheinvestorwillbenefitfromthedesiredrecovery.Howeverwithvacantpossession,theinvestorwillbeabletousethepropertyorrentitout,whereaswithdeferredpossessionthereisnosuchbenefit.Therationalpreferencewouldbeforvacantpossession,andaswenotedabove,anaccountantwhowasvaluing

106

the contracts for thebenefitof shareholders isprofessionallyobligated to reflect thatrationalpreference.Froman empirical point of view,weknowof no evidence suggesting that longdatedfreeholdsintheLondonmarketweretradingabovethevacantpossessionpriceintheaftermathofthe2008collapse.A99yearleaseonaLondonapartmentwouldtypicallytradeat95-99%ofthevacantfreeholdvalue.Wehavenoevidencethatleaseholdvaluescollapsedtozero,orwentnegative,duringthecrisis.Perhapssuchevidenceexists,butgiventhetheoreticalunlikelihood,theonusshouldbeonTunarutoproduceit:strongclaims require strong evidence. He points (p.51) to the fact that short-dated EUREXfuturescontractstradedwithnegativedefermentratesforshortaperiodfollowingthecrisis,buttheserefertofuturesnotforward,anddonotrefertocontractsforlong-dateddeferred possession such as in the freehold and leasehold markets, which are theforwardsrelevanttothecaseofequityrelease.ConsideralsoevidencesuchasthefollowingfromaDailyMailarticlepublishedon24July2009.84Thearticlementions‘therecentlargepricefallsinthehousingmarket’andthenpointsout:

“Recent price reductions of up to 25 per cent in most areas mean anequivalentdropin thecostofextendingyour leaseorbuying the freehold,”saysAngusFanshawe,ofLondonestateagencyDouglas&Gordon.

But if Tunaru were right, the freehold (or the extension) should be getting moreexpensive,notless,givenmarketexpectationsofpricerisesinthefuture,no?

At the very posh end of the market, an apartment in Chelsea's exclusiveCadoganGardens has a 43-year lease and is on sale for £2.95million (02075813022,friendandfalcke.co.uk),butmaybeworthupto£4.5millioniftheleasewereextendedto99or125years.

Sothe43yearleaseholdhasapositivevalue,notthenegativeoneimpliedbyanegativedefermentrate.

…aflatinCentralLondonwithan88-yearleasewouldsellfor97percentofitsfullmarketpricebutthesameflatwith22yearsontheleasewouldsellforonly56percentandwithaten-yearleasefor32percent.

Soifthedefermentratewerenegative,wecouldbuyaflatwithvacantpossessionfor100%,carveoutan88yearleaseandsellfor97%,thenselltheremainingfreeholdforpossessionafter88yearsatover100%.Thatwouldbeanicetrade!All the relevant evidence indicates that deferment rates are positive as Principle IIImaintains.

84 https://www.dailymail.co.uk/property/article-1201766/Time-new-lease-life-Freehold-isnt-way-secure-future-profits.html

107

IFoAMisconceptionsonPrincipleIIIIn June 2016, the Institute and Faculty of Actuaries issued “DP 1/16: Equity ReleaseMortgages:IFoAResponsetothePrudentialRegulationAuthority,”itsofficialresponsetothePRA’searlierDiscussionPaperDP1/16,whichhadhadaskedforindustryviewsonERMs.Toquotefromthisresponse:

33.Forthesecondrelationshipinparagraph4.9[i.e.,PrincipleIII]tohold,intheory,thereneedstobeadeepandliquidmarket.OtherwisetheimplicationisthattheaveragevalueoftheHPI[HousePriceInflation]assumptionislessthanorequaltothediscountrateassumedinthevaluationoftheNNEG.Inpractice,theapproachtosettingtheHPIassumptionvariessignificantlyfromfirmtofirm.

Thereareseveralhowlershere:

• Mistake#1isthatforPrincipleIII“tohold,intheory,thereneedstobeadeepandliquidmarket.”ThevalidityofPrincipleIIIhasnothingtodowithadeepandliquidmarketandwehavejustshownthatitsvalidityholdsundergeneralconditions.

• Mistake#2istosuggestthatthe“averagevalueoftheHPIassumptionislessthanor equal to the discount rate assumed in the valuation of the NNEG.” Thisstatementisjustplainwrong.ThecorrectstatementisthatwecanassumeanyHPIwewant to, but the assumed value of the HPI is always irrelevant to thevaluationoftheNNEG.

Para35thengivessome illustrationsofcircumstances inwhichPrinciple IIIallegedlymightnothold:

• One is the claim that Principle III “is a statement of ‘value’ and applies to anyindividual.Howeverthisisnotnecessarilytrueintermsoftheexchangevalue.”Thisstrangestatementisanimaginativeadditiontotheeconomictheoryofvaluebutisunfortunatelyalsowrong.TheclaimthatthePrincipleIII“isastatementofvalueandappliestoanyindividual”istrue,butthecorollaryisthatitalsoappliesto all individuals including (and not excluding!) when they engage in trade atmarketorexchangevalues.

• Another is the claim that “in a negative yield curve scenario, the relationship(PrincipleIII)wouldfailasthepremisethatdeferralcouldleadtoalowerpresentvaluenolongerholds.”Thisstatementisaheadscratcherbutonecanseethatitmustbewrongbecausethedefermentprice(orvalue,makesnodifferencehere)isequalto𝑆0𝑒4W*andthisexpressiondoesnotincludeanyinterestrateoryield,negativeornot.Torepeat,PrincipleIIIdependsonlyonthe𝑞ratesbeingpositive(ormostlypositive)anditisdifficulttoimagineplausiblesituationswherethatwouldnotbethecase.

So how come the distinguished actuaries of the IFoA could make such mistakes? Apossibleclueisthatthecoveringletteropenswiththefollowingstatement:

108

The IFoA’sEquityReleaseMembers InterestGroup(ERMIG)andLifeBoardhavebeeninvolvedinthedraftingofthisresponse.Thecontributorstothisresponseincludememberswhoareactivelyengagedwithuseofequityreleaseassetsbylifeinsurers.(Myitalics)

TheIFoAhadalloweditselftobeusedasamouthpieceforERMindustrypractitionerstobroadcasttheirmisunderstandingoftheirproducts.But the authors of the IFoA official response to DP 1/16 are not alone inmisunderstanding these principles. Consider these passages from a recent DeloittecommuniquéonERMs:

Inourview,thethirdprinciple(thatfuturepossessionofapropertycannotbemorevaluablethancurrentpossession)islikelytoattractthemostfuturedebate.

ButPrincipleIIIisjustelementaryeconomics!

Very importantly, this principle implies that assumed future house pricegrowthcannotexceedthediscountrateappliedinthevaluation.…

Noitdoesnot.

ThePRAexpectstheretobeapositivevalueassociatedwithpossessionofaproperty.

Yes,obviously.The practical implication of this is that the assumed house price growthwithintheNNEGoptionpricingcalculationcannotexceedthediscountrate,asthiswouldimplythatfuturepossessionismorevaluable.Thisprinciple thereforeeffectivelysetsacapon firms’housepricegrowthassumptions.

Thesestatementsarenonsense.PrincipleIIIhasnoimplicationsaboutassumedfuturehousepricegrowth.YoucanmakeanyassumptionsaboutfuturehousepricegrowththatyoulikeandPrincipleIIIwouldbestillbevalid.

WewouldexpectfirmsinvestinginERMsandotherdirectinvestmentstoseeanincreasedlevelofscrutinyandquestioningfromthePRA,withthebarsetveryhighformanagement’sunderstandingofthevaluationofsuchinvestments.(Bulleyetalia,2017,ouritalics)

Theyareclearlyofftoaflyingstartonthatone.Theleadauthor,AndrewBulley,isapartnerinDeloitte’sCentreforRegulatoryStrategy.PriortojoiningDeloitte,Mr.BulleywasDirectorofInsuranceSupervisionattheBankofEngland.

109

To challenge Principle III is thus to make an egregious intellectual error and it isremarkable that the IFoA has not only failed to condemn any such challenge but hasexplicitlygivenititsimprimatur.ThissituationisanalogoustotheUK’stopmathematicalinstitute,theInstituteofMathematicsanditsApplications,takingtheofficialviewthatthevalidityof2+2=4isanopinion.Yousee,somemathematiciansareoftheopinionthat2+2=4butothershaveadifferentview,includingsomewhospeakfortheInstitute.Weappeartohavehereanothercaseof‘actuarialjudgment’goneawry.WeareremindedofsomecommentsmadeonthissubjectbyTimGordonalmosttwodecadesago(Gordon,1999).Hewrote(p.4)abouttheactuarialconvictionthat“actuarialjudgmentistheonlytechniqueforvaluinglong-termliabilities”but‘actuarialjudgement’producesananswerthat“variesenormouslydependingonwhichactuarycarriesoutthecalculation.”Hecontinued:

actuaries assume that judgmentalmethods are the onlymethods availablewhich give sensible answers. What is more, the judgement involved issomethingwhichapparentlyonlycomeswithyearsofexperience. Inotherwords,weclaimtoknowtheanswerbutcannottellanyoneelsehowtoderiveitinadvance.

Theexperiencedactuaryknows itwhen theysee it.Romanaugurshad thesameskillreadingchickenentrails.Ashecontinuedfurther:

The problem is that the difference that actuarial judgement can make tovaluationsusingthetraditionalapproachisenormous.Itmeansthat:•weareexposedtopressurefromclientsseekingtomoveanswersinthedirectionwhichfavoursthem,and•we losecredibilitybecauseweareunabletoexplainpreciselyhowwearriveatananswer.

Actuarial judgment can also lose credibility when it produces answers that aredemonstrablywrong.BoundsonERMandNNEGValuationsToreturntothemainstoryline,theimpactofthesetwoPrinciplesisillustratedinFigure19.2:

110

Figure19.2:IllustrationofPrinciplesIIandIII

Notes:Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%,𝑞=4.2%and𝜎=14.8%.ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

PrincipleIIimpliesthattheblue(𝐸𝑅𝑀*)linemustbebelowboththegreen(𝐿*)lineandthe red (deferred possession) line, and Principle III implies that the red (deferredpossession)lineshouldslopedownwards.ThereissomeinterestingintuitionunderlyingtheFigure:

• Forverylowhorizons,𝑁𝑁𝐸𝐺*isveryoutofthemoneyandprobabilityofexerciseis very low.Hence the value of the optionwill be negligible and𝐸𝑅𝑀*will beindistinguishablyclosetothevalueoftheloan𝐿* .

• Forlonghorizonsorhigh𝑡,theoptioniswellintothemoneyandtheprobabilityofexerciseishighandapproaching1.Therefore,the𝐸𝑅𝑀*lineconvergestothedeferredhousevaluelineforperiod𝑡.

Underlyingthesegraphsaresomeelegantmathematics.𝐸𝑅𝑀*isgivenby(19.13)𝐸𝑅𝑀* = 𝑒4I*𝐿* − 𝑒4I*[𝐿*𝑁(−𝑑M) − 𝐹*𝑁(−𝑑5)]

= 𝑒4I*[1 − 𝑁(−𝑑M)]𝐿* + 𝑁(−𝑑5)𝑒4I*𝐹*

where we have set the deferment price𝐷* = 𝑒4I*𝐹* . From the standard equivalence𝑁(−𝑥) = 1 − 𝑁(𝑥),wethenget(19.14)𝐸𝑅𝑀* = 𝑁(𝑑M)𝑒4I*𝐿* + 𝑁(−𝑑5)𝑒4I*𝐹*This expression is simpler and reflects the shapes of the curves clearly. As 𝑑M getspositive, −𝑑5gets negative, so 𝑁(𝑑M) goes to 1,𝑁(−𝑑5) goes to zero and 𝐸𝑅𝑀*approachesthepresentvalueoftheloan.As𝑑Mgoesnegative,it’stheotherwayround,sothetermontheleftdisappearsandthetermontherightapproachesthedefermentvalue𝑒4I*𝐹* .OneseestheseboundsatplayinFigure19.2.

111

Besidestheirmathematicalelegance,theseboundsimpliedbyPrinciplesIIandIIIhaveahelpfulpracticaluse:theyareaneasilycalculatedcross-checkonanyproposedERMorNNEGvaluation.ConsiderFigure19.3,which shows theupperbound for𝐸𝑅𝑀*madeexplicitandhighlightedinblue.

Figure19.3:ERMUpperBound

Notes:AsperFigure19.2

Asanaside,ifwestartwithafigurelikeFigure19.2andletthevolatilitygetsmall,thenit iseasy toshowthatFigure turns intoFigure19.3and themessage is that theERMvaluationapproachesthePrincipleIIupperbound.ButiftheERMvaluationapproachesitsupperbound,thenthecorrespondingBlack76’optionvaluationmustapproachthePrinciple IINNEGvaluation lowerbound, i.e., as𝜎 → 0, theBlack ‘76NNEGvaluationapproachesthePrincipleIINNEGlowerbound.Wecanobtainthe𝐸𝑅𝑀*upperboundastheminimumof𝑒4I*𝐿*and𝑒4I*𝐹* .NotethatthisupperboundcanbeestimatedusingonlyinformationaboutthecurrenthousepriceandLTV (which together giveus the current amount loaned), the risk-free rate𝑟, thenetrental𝑞,theloanrate𝑙andtheexitprobabilities.Forexample,inthebaselinecase,weestimate the𝐸𝑅𝑀 upper bound to be £46.7, which compares to our earlier baselineestimateof𝐸𝑅𝑀as£43.5.Soevenwithoutestimating𝐸𝑅𝑀orits𝑁𝑁𝐸𝐺orestimatingany underlying option model or calibrating any additional parameters, such as thevolatility,weimmediatelyknowthatanyproposedvalueofERMthatexceeds£46.7mustbewrong.But if we can estimate an upper bound for ERMwithout requiring an option-pricingmodelorrelyingonanyvolatilityparameters,thenby(3.2)𝐸𝑅𝑀 = 𝐿 − 𝑁𝑁𝐸𝐺wecanalsoestimatealowerboundfor𝑁𝑁𝐸𝐺onthesamebasis.Giventhat𝐿=£74.9inourbaselinecase,theupperbound𝐸𝑅𝑀estimateof£46.7impliesa𝑁𝑁𝐸𝐺lowerboundequalto£28.3.This lowerboundcomparestoourearlier𝑁𝑁𝐸𝐺estimateof£31.4.Soeven without estimating the 𝑁𝑁𝐸𝐺 or relying on any NNEG valuation model or any

112

volatilityestimatethatmightgointoanysuchmodel,weknowthatanyproposedNNEGvaluebelow£28.3mustbewrong.Tocuttothechase,giventhesevariousinputs–theassumedageandgender,theassumedhousepriceandLTV,theassumed𝑟,𝑞,and𝑙rates,andtheinputtedexitprobabilities–itisimpossibletogetaNNEGvalueanylowerthan£28.3whateveroptionpricingmodelonemightuseandregardlessofhowitmightotherwisebecalibrated.Attheriskofrepeatingourselves,wewouldstressthatthislowerboundNNEGvalueisnotdependentonBlack’76.TherecentInstitutereplytoCP13/18releasedon28Sep2018made a great deal of noise about how autocorrelation,mean reversion, lack ofGeometricBrownianMotion and so forthundermined the validity ofBlack ’76, and anumberofparticipantsattheLSEseminaron1October2018madesimilarpoints.Wewoulddisputethevalidityoftheseclaims–notleastbecausetheyoftenconfusesufficientwith necessary conditions for Black-Scholes type valuations to be valid, andwe havemoretosayontheseissuesinChapter20–buteveniftheseclaimswereallvalid,theydonotapplytothebounds-basedvaluationofferedhere,becausethatargumentisnotdependentonanyoptionpricingatall,Black’76orotherwise.Wealsohavehereahandycross-checkofanyproposedNNEGvaluation:ifanyproposedmodelbasedonthesameinputcalibrationsgivesalowerNNEGvaluationthaneitheroftheNNEGlowerbounds,thenitmustbewrong.Black’76vs.Principles-basedboundsresultsIt is interesting to compare our baseline NNEG valuation resultswith the resultswewould have obtained hadwe dispensedwith the option pricingmodel and used thePrinciples-basedboundsinstead:

Table19.1:BaselineERM/NNEGvs.PrincipleIIBoundsValuationsCurrentHouse

PriceLoan

Amount𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £40 £74.8 £32.2 £42.7CurrentHouse

PriceLoan

Amount𝑳 𝑵𝑵𝑬𝑮lower

bound𝑬𝑹𝑴upperbound

£100 £40 £74.8 £28.1 £46.8Notes:AsperFigure19.2.

TheseresultsindicatethatthebasicNNEGunder-valuationstoryobtainedearlierusingBlack’76stillholdstrueifweusethePrinciples-basedboundsinsteadofanyoptionpricingmodel.Thus,onecannotdismisstheNNEGunder-valuationstorybasedonarguments–rightorwrongmakesnodifference–aboutthevalidityofBlack’76appliedtoNNEGvaluation.Table19.2givesthecorrespondingresultsforthePrincipleIIIbounds:

113

Table19.2:BaselineERM/NNEGvs.PrincipleIIIBoundsValuationsCurrentHouse

PriceLoan

Amount𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

£100 £40 £74.8 £32.2 £42.7CurrentHouse

PriceLoan

Amount𝑳 𝑵𝑵𝑬𝑮lower

bound𝑬𝑹𝑴upperbound

£100 £40 £74.8 £13.2 £61.7Notes:AsperFigure19.2.

Figure19.4givesplotsofthePrincipleIINNEGlowerboundagainstdefermentrateandage,wherethelatterassumesthatLTVfollowsthe‘ageminus30’rule:

Figure19.4:𝑵𝑵𝑬𝑮andNNEGPrincipleIILowerBoundsvsDefermentRate

Notes:OtherwiseasperFigure19.2.

Figure19.5showsthecorrespondingERMupperbounds:

Figure19.5:𝑬𝑹𝑴andERMPrincipleIIUpperBoundsvsBorrowerAge

Notes:LTVfollowsthe‘ageminus30’rule.OtherwiseasperFigure19.2.

114

Figures19.6and19.7showthecorrespondingplotsforthePrincipleIIIbounds:

Figure19.6:𝑵𝑵𝑬𝑮andNNEGPrincipleIIILowerBoundsvsDefermentRate

Notes:OtherwiseasperFigure19.2.

Figure19.7:𝑬𝑹𝑴andERMPrincipleIIIUpperBoundsvsBorrowerAge

Notes:LTVfollowsthe‘ageminus30’rule.OtherwiseasperFigure19.2.

We see that for Principle II, the bounds are quite close to the Black ’76 valuations,especially in theplausible rangeof deferment rates.ThePrinciple III bounds are lessclose.

115

ChapterTwenty:TheMarketConsistentApproachIntroductionA‘marketconsistent’(MC)approachtovaluationisonewhichgives‘marketconsistent’valuations,anda‘marketconsistent’valuationcanbedefinedasa‘fairvalue’valuationbasedontheIFRSdefinitionofafairvalueprice,namely:

Thepricethatwouldbereceivedtosellanassetorpaidtotransferaliabilityinanorderlytransactionbetweenmarketparticipantsatthemeasurementdate.85

AlthoughitisoftenidentifiedwithBlack’76,theMCapproachdoesnotmandatetheuseofanyparticularoptionpricingmodel.Forexample,insteadofBlack’76,theMCapproachmightbeappliedwithanoptionpricingmodelthatallowsfor,e.g.,transactionscostsorwithaMonteCarlo(orstochastic)simulationmodel,provided(abigif)thatthemodelisfitforpurposeandthatthecalibrationsarereasonable.Theessenceoftheapproachismarket consistency, not adherence to any particular model. Any model is allowable,provideditmeetsthestandardsformarketconsistency.TheMCapproachissometimescalleda‘riskneutral’approachintheactuarialliterature.Thisalternative labelreflects theunderlyingderivativepricingmethodologybasedonrisk-neutralprobabilities.However,thislattertermisunfortunate,inthatitsuggeststothosewhodonotunderstandthismethodologythatitmustbeassumingthatinvestorsare actually neutral about risk and who would want to use a method that made aquestionableassumptionlikethat?Indeed,theargumentthatthe‘risk-neutral’approachis flawed because investors are not risk-neutral is probably the most commonmisconception one encounters against the approach, as if to suggestwe shouldwantsomealternative approach thatdoesnotmake this assumption.Butwe cannot stressoftenenoughthatthemarketconsistentor‘riskneutral’approachDOESNOTassumethatinvestors are risk-neutral. The truth of the matter is that the methodology worksregardlessofwhetherinvestorsarerisk-neutralornot;theirattitudetowardsriskisanentirely separate matter. The term ‘risk neutral’ has an unwarranted negativeconnotationwhenusedinbroadercirclesandisbestavoided.The MC approach is also subject to other misconceptions that have much confusedactuarial discussions. These arguments claim that you can’t use Black ’76 or Black-Scholesbecause:therearenoforwardcontractsoryoucan’tdoforwardcontracts,youcan’tshorttheunderlying,housepricesareautocorrelatedratherthanGBM,marketsareilliquidorincomplete,and,moregenerally,theassumptionsunderlyingthederivationofBlack-Scholesfamilymodelsdonotholdempirically.Iftheseassumptionsdon’thold,soitisclaimed,thenthesemodelswillbeunreliable.Butthisargumentislogicallyflawed.Yes,itisittruethatinitsusualclassical/textbookderivations,Black-ScholesorBlack’76makeawholebunchofassumptions,someofwhichareempiricallyfalse.However,theseassumptionsaresufficientratherthannecessaryforthemodel’sresultstobevalid,so

85 See,e.g.,https://www.iasplus.com/en/standards/ifrs/ifrs13.

116

youcan’tjustkickasidesomeoftheseassumptionsandconcludethatBlack’76resultsmust be wrong. To prove that those results are wrong, it is necessary to kick asideassumptionsthatarenecessaryfortheresultstohold,notconclusionsthataresufficientforthemtohold,andthatisamuchmoredifficulttask.Furthermore,todemonstratethatthesemodelsgivethe‘wrong’answers,onemustalsodemonstrate that somenecessary condition isnot only ‘wrong’, but also introduces amaterialerrortotheresultingvaluations.Eventhen,amodelcanstillbeuseful.AcaseinpointisBlack’76.Thismodelgivesusputvaluationsarebiased,butstilluseful,becauseitcanputalowerboundundertheputvaluation,i.e.,soifthemodeltellsusthattheputvalueis£X,thenweknowthatthetrueputvaluemustbeatleast£X.WeoftenencountercriticismsofBlack’76madebyactuarieswhothinkthatiftheycan,forwantofabetterwayofputtingit,knockthemodeloffitsperchthentheycantherebyjustify lower NNEG valuations as if by default to an alternative that produces lowervaluations. This view is seriouslymistaken. If Black ’76 does not hold, then the validalternativeisnotthediscountedprojectionorTunaruapproaches(andmoreontheseinlater chapters) but some other MC-consistent model that will deliver higher NNEGvaluationsthanBlack‘76.WesuspectthatmanyactuarialcriticsofBlack’76wouldbealotlesscriticaliftheyunderstoodthispoint.This point is a recurring theme in the literature that addresses option pricing in thepresenceoftransactionscosts.Thegeneralresultfromthatliteratureisthattransactionscosts increase option valuations above BS valuations or, in some cases, increase thevolatilitythatneedstobeinputtedintotheBSmodel,whichamountstothesamething.Thatliteraturealsoteachesusthatinsuchcircumstanceswemusttakeaccountofutilityorriskpreferenceswhich,inturn,makeforarangeofcorrectvaluations,i.e.,thereisnouniquelycorrectsolution.86Intruth,weshouldn’tputanymodelonapedestal,whetherthatmodelbeBlack’76orsomesuperioralternative,letaloneaninferiorone.Whatshouldgoonthepedestalisthepricingmethodology.Weshouldnotbethinking,“WeuseBlack‘76becauseitisthevalidmodel.” Instead, we should be thinking, “How do we apply the correct (that is, zeroarbitragerehedging)pricingmethodologyintheERMcontext?”Wethencometothefundamentalmethodologicalissuethatshouldbeattheforefrontofourthinking:weshouldspellouthowtherehedging-basedpricingmethodologywouldworkwhentheunderlyingisproperty.PutPricingfromFirstPrinciplesPutpricingundercontinuity 86 See,e.g.,H.E.Leland(1983)“OptionPricingandReplicationwithTransactionsCosts,”JournalofFinance40:1283-1301;S.D.HodgesandA.Neuberger(1993)“OptimalReplicationofContingentClaimsunderTransactionsCosts,”ReviewofFuturesMarkets8:222-239;orG.BarlesandH.M.Soner(1998)“OptionPricingwithTransactionsCostsandaNonlinearBlack-ScholesEquation,”FinanceandStochastics2:369-397.

117

InsteadofassumingthatBlack ’76holds, let’s firstassumethatweareworkingintheidealised theoretical world assumed by Black-Scholes, which is the same idealisedtheoreticalworldasthatassumedbyBlack’76.WewishtovalueaEuropeanputoptiononaforwardcontractonsomeassetandlet’ssupposethatwehaven’tevenheardoftheBlack’76putoptionpricingformula.Sowhatdowedo?The answer is that we construct a synthetic put, a series of dynamically rebalancedpositionsintheunderlyingcontractwhichreplicatesthepayoffoftheoptioncontractatexpiry.Theunderlyingcontractcanbeaforwardcontractinwhichwepayatmaturitytotakepossessionoftheassetatcontractmaturityoradefermentcontractinwhichwepaynowtoreceivethesameassetatmaturity.Asthepriceoftheunderlyingmovesaround,wecontinuouslyrebalanceoursyntheticput,toensurethatthesyntheticputalwayshasthesameexpectedpayoffasthe‘real’put,andwecontinuerebalancinguntilbothputsexpire.87Thevalueofthesyntheticoptionisthenthepayoffofthecontract(foraput,thedifferencebetweenthestrikepriceandtheunderlying,ifthelatterislower),minusthecostofhedging.ThevalueofthesyntheticwillbethesameasthevaluegivenbyBlack’76ifthehedgingiscontinuous(whichisimpossible to carry out in practice). So whatever the payoff, and whatever path theunderlying asset follows until expiry, the payoff minus the hedging cost will yieldapproximatelythesameamount,namelythetheoreticalBlack‘76optionpremium.88WeknowthatthesyntheticputvaluemustequaltheBlack’76putvaluebecausethispricingprocess is the same one that Fischer Black used to obtain his Black ’76 put pricingequation,includingtheimpossibleconditionthatthehedgingiscontinuous.PutpricingunderdiscontinuityThe assumption of continuous trading (like many other textbook assumptions) is ofcoursepurelytheoretical,becauseitappliesonlyatthelimitas𝑛 → ∞.Inpractice,onlydiscretehedgingispossible.Itthenturnsoutthatthevalueofthesyntheticoptionunderdiscrete hedging does depend on the path of the asset to expiry. To illustrate thisdependency,wesimulatedthevalueofadiscretelyhedgedat-the-moneyputoptionfor20,000 different paths, expressing the difference (or hedging error, HE) between thediscrete (H76)and thecontinuous (Black76)priceasapercentageof thecontinuousprice,i.e., (20.1)𝐻𝐸(𝑛) = (𝐻76 − 𝐵76)/𝐵76

87Thestandardtextbookaccountisthatweconstructourpositionfromacombinationofrisk-freebondsandtheunderlying.Aninstitutionaldeskwouldtakeaslightlydifferentapproach.Iftheoptionissold,thedeskwill receive a premiumwhich it investswith the firm’s own treasury department. Itwill receiveinterestbasedonthefirm’sownfundingratewhichreflectsitsexternalrating,lessaspreadtakenasprofitbythetreasury.Iftheoptionislongdated,thedeskmayalsohedgetheinterestrateriskbyaswapwithitsownswapdesk.Iftheunderlyingisafuturescontract,thenthecashpositionwouldneedtobecashsettled,i.e.,rebalancedtoreflectinterimprofitandlossontheunderlying.Iftheunderlyingisaforward,nosuchrebalancingisnecessary.88Weimplicitlyassumethatthevolatilityisapproximatelyconstant.

118

where𝑛isthenumberoftimesthatwehedge(sowerehedge𝑛 − 1times).Weassumeatimetoexpiryof1yearandmakeotherassumptions,includinganunderlyingcontinuousvolatilityof13%.Soif𝑛 = 52,werecalibrateeveryweek,if𝑛 = 250,werecalibrateeveryday,assuming250tradingdaystoayear,andsoforth.TheresultsareshowninFigure20.1:

Figure20.1:RehedgingErrorandHedgingFrequency

Notes:Basedon20,000simulationtrials,𝐹 = 100,𝐾 = 100,𝑟 = 0,𝑡 = 1and𝜎 = 13%.

Aswecansee,theaveragehedgingerror,thestandarddeviationofthehedgingerrorandthe confidence bounds around the hedging error converge to zero as𝑛 gets large, asoptiontheorysuggests.SomespecificresultsareshowninTable20.1:

Table20.1:RehedgingErrorandHedgingFrequencyn 50 100 250 500 1,000

AverageHE 1.54% 0.69% 0.32% 0.12% 0.04%StdHE 6.15% 4.41% 2.85% 2.05% 1.45%

Lowerbound -8.62% -6.29% -4.06% -3.01% -2.15%Upperbound 10.92% 7.32% 7.32% 3.09% 2.16%

Notes:AsperFigure21.1.‘HE’=hedgeerror.‘Lowerbound’=lowerboundof90%confidenceinterval.‘Upperbound’=upperboundof90%confidenceinterval.

Thechoiceof𝑛isasubjectivejudgment.With𝑛 = 50,i.e.rebalancingaboutweekly,thestandarddeviationis6.15%oftheat-the-moneyprice.Giventhatanoptiontraderwouldtypicallyquoteaspreadonimpliedvolatility,hecouldadjustthespreadtoimprovethechancesofmakingaprofitonthetrade.Forexample,ifthetraderestimatesvolatilityat13%,hewillquotea2%bidofferspreadinthemarket,meaninghebuysat12%,sellsat14%.WithanATMoption,oneyeartoexpiry,thatequatestoabid-offerspreadof4.78–5.58.Thedifferencefromthemidpriceof5.18expressedasapercentage,isthusabout7%.Giventhatthestandarddeviationoferrorisaroundthatnumber,itfollowsthatthetraderhasan84%probabilityofmakingaprofit, justforthatonetrade.Alternativelywithdailyhedging,𝑛wouldbearound250,implyingahedgingerrorofabout2.84%,inwhichcasethetraderwouldquoteacorrespondinglylowerbid-offerspread.

119

These results can easily be adjusted for different times to expiry. The two relevantvariablesare(i)thenumberofhedgingperiodsand(ii)thevolatilityoftheassetatthehedgingperiod. For example, supposewe arepricing a20 year option,with a chosenhedgingperiodof1month,andsupposethatthevolatilityof1monthreturnsis1%.(Notethatwearenotannualisingthevolatilityhere.Wemeanthatthestandarddeviationofmonthly returns is 1%.) Thenwe can pretendwe have a 1 year option divided into20 × 12 = 240periods,andthattheannualisedvolatilityoftheassetis1% × 2400.z ≈15.5%.This𝑛 = 240 case isclose to the𝑛 = 250 exampleabove, implyingastandarddeviationoferrorofabout2.5%.Thehedgingperiodiscrucialnotjustbecauseofthepricingerror,butalsobecauseoftheHurst (or autocorrelation) effect that we noted in Chapter 10. If the underlying isautocorrelated – another departure from Black-Scholes – then the longer the periodbetween rebalancing, the greater the impact of the Hurst effect, i.e. the greater therealised volatility comparedwith the theoretical volatility impliedby the square rootrule.Forexample,supposeour20yearoptionisonanunderlyingassetwhichishighlyautocorrelated,e.g.with𝐻 = 0.9,andthatwearerebalancingevery5years,ratherthaneverymonth.Thestandard(GBM)volatilitycorrectionformulaassumes(20.2)adjusted𝜎 = 1% × 600.z = 7.75%.HoweverwiththeHurstadjustment,theresultisverydifferent:(20.3)𝐻-adjusted𝜎 = 1% × 600.Å = 39.8%.Taking account of the Hurst effect (i.e., taking account of autocorrelation in theunderlying)willthenincreaseboththecostoftheoptionandthestandarddeviationofthehedgingerror.Thus,thisparticulardeparturefromtheBlack-ScholesworldwillleadtoahigheroptionpricethanthatgivenbyBlack’76.89DynamicReplicationIt is frequently claimed that without a complete liquid hedging asset, we cannotdynamicallyreplicateanoptionandsocannotderivetheBlack-Scholesformula.Toquotethe PRA agreeing with unnamed respondents who raise these objections, “The PRAagreeswiththerespondentsthattheattributesoftheresidentialpropertymarketdonotpermitthederivationoftheBlack-Scholesformulaviadynamicreplicationarguments”(PS31/182.28,p.9).Takenstrictlyandliterally,thisclaimisfalse.TheBlack-Scholesformulaisaderivationfromfirstprinciples,justasthePythagoreantheoremisaderivationfromfirstprinciples,anditisalwayspossibletoderiveit. 89WewouldemphasisehoweverthatnotalldeparturesfromaBlack-Scholesworldwillproduceputvaluesthat differ from the Black ’76 put value. For examples, see D. Buckner “Weird distributions #1.” TheEumeausProject17October2018;and“Weirddistributions#2.”TheEumeausProject17October2018.

120

Clearlysomethingelseismeant,butwhat?Istheclaimthatthetradercannotsuccessfullymakeamarket inanoptionthat iseither(i) impossibleor(ii)expensivetohedge?Inresponsetothefirstcase,itisalwayspossibletohedgeanoption.Justbecausetheredoesnotexistalistedmarket,doesnotmeanthatwecan’tvalueNNEGsandERMsusingsomemarket hedging approach. Forgive the triple negative. Our point is that hedgingpossibilitieswill alwaysexist.Onepossibility isanexternalhedge, i.e., todynamicallyhedgetheERMintheOTC(overthecounter)market.Foreachmaturityandeverysooften,theERMprovidercouldestimatethenumberofpropertiesexpectedtoexitatthatmaturity, the average price of such properties, then agree to deliver properties of anequivalent standard to an external hedge provider. Most investment banks make abusinessofsucharrangementsforaprice.Therearenon-trivialimplementationissues,but theyaremanageableones,andwouldbe lessdifficult than those involved in, say,pension buyouts or buyins, in which there has been a flourishingmarket for over adecade.Asforthesecond,i.e.expensivetohedgecase,ofcoursethepriceofthehedgemightbehigh,butahighpricearguesforanequivalentlyhighvaluationfortheNNEG,relativetothevaluationwemightobtainusingavolatilitybasedonobservedmarketmid-prices.OristheobjectionbeingmadethatwecannotuseBlackorBlack-Scholesasanaccountingbasis for theoption?Thereare tworeplies to thisobjection.The first,whichweshalladdressinthissection,isthatwecanusefirstprinciplestoestablishanupperboundforthevalueoftheERM,oracorrespondinglowerboundfortheNNEG.Thesecond,whichweshalladdressinasubsequentsection,isthatwecanusethestandardoptionmodeltoprovideaP/Lexplanationforthepayoffoftheoption,eveniftheoptionisunhedged.Recall how we established in Chapter 19 that we do not require dynamic hedgingargumentsor complete liquidmarketsoranoptionpricingmodelorevenavolatilityestimatetoestablishupperboundsvaluefortheERMandcorrespondinglowerboundvaluationsfortheNNEG.SimpleargumentssufficeasperthePRA’sPrinciplesIIandIII.Torecap,ifafreehold,i.e.acontractfordeferredpossessionatacertainmaturity,isonthemarketforacertainprice,thenarationalinvestorwouldnotpaymorethanthatpriceforanERMatthatsamematurity,andaccountingprinciplesrequirethefirmtoreflecttheconsiderationsthatarationalinvestorwouldmake(seeChapter26formoreontheaccounting issues). Similarly, a rational investor would not pay more for deferredpossession than for spot. Sowehave lowerboundson theputvalueand these lowerboundscorrespondtothevalueswewouldgetunderBlack’76forlimitingcasesoftheinputs,i.e.,where𝜎 ⟶ 0and𝑞 ⟶ 0.TheformerlimitingcaseisthelowerboundunderPrincipleIIandthelatterlimitingcaseisthelowerboundunderPrincipleIII.90TheimpactoftheseboundswasshowninFigure10.3.Figure20.2givesthesamefigurebutwiththeBlack’76𝐸𝑅𝑀*basedon𝜎 = 3%superimposedonit.

90WewouldalsopointoutthatthePRA’sprinciples(PrinciplesIIandIII)areconsistentwiththestandardoptionpricingmodel,and,indeedtheyunderlieit.PrinciplesIIisabouttheminimum-of(A,B)conditionthatunderliesalloptionmodels,whereasPrinciplesIIIisabouthowtheforwardpriceisthebackboneoftheoption.

121

Figure20.2:Black’76ERMValuationandPrincipleIIERMUpperBound

Notes:𝐿*and𝐸𝑅𝑀*aretheloanvalueandERMvaluesfordecrement𝑡.Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%,𝑞=4.2%and𝜎=3%.ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

Asthevolatilitygetssmall,the𝐸𝑅𝑀*curveapproachestheERMupperboundand𝑁𝑁𝐸𝐺*approachesitslowerbound.Table20.2showstheBlack’76NNEGandERMvaluationscomparedtothePrincipleIIbounds.

122

Table20.2:Black’76NNEGandERMValuationsComparedtoValuationsBasedonthePrincipleIIBounds

Volatility 𝑵𝑵𝑬𝑮/𝑵𝑵𝑬𝑮𝑳𝑩 𝑬𝑹𝑴/𝑬𝑹𝑴𝑼𝑩0.00001% 100% 100%

1% 100.03% 99.98%2% 100.22% 99.87%3% 100.54% 99.68%4% 100.99% 99.41%5% 101.57% 99.06%6% 102.29% 98.62%7% 103.16% 98.10%8% 104.18% 97.50%9% 105.34% 96.79%10% 106.34% 96.01%11% 108.07% 95.15%12% 109.63% 94.21%13% 111.30% 93.21%14% 113.08% 92.14%15% 114.95% 91.01%

Notes:𝑁𝑁𝐸𝐺isthevalueoftheNNEGguarantee,𝑁𝑁𝐸𝐺𝐿𝐵isthevalueoftheNNEGPrincipleIIlowerbound,𝐸𝑅𝑀isvalueoftheEquityReleaseMortgageand𝐸𝑅𝑀𝑈𝐵isthevalueoftheERMPrincipleIIupperbound.Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%,𝑞=4.2%and𝜎asindicated.ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

EachlineshowstheratiooftheBlack’76NNEGvaluetotheNNEGlowerbound,andtheratio of the Black ’76 ERM value to the ERMupper bound, for a given volatility. Thevolatilitiesrangefrom1%to15%.Weseethattheratiosareremarkablycloseto100%,especiallyforthelowervolatilityvalues.Forexample:

• forvolatilitiesofupto5%,thediscrepancybetweentheBlack’76ERMvaluationandtheERMupperbound<1%;

• forvolatilitiesofupto11%,thediscrepancybetweentheBlack’76ERMvaluationandtheERMupperbound<5%;and

• forvolatilitiesofupto15%,thediscrepancybetweentheBlack’76ERMvaluationandtheERMupperbound<10%.91

Soifyou(a)don’thaveanoptionpricingmodelthatyoulikeor(b)don’tlikethedynamichedgingassumptionsofBlack‘76,thesolutionissimple:don’tworryaboutthedynamichedgingstuffandjustusetheboundsinstead!RememberthattheboundsareBlack76aswell,butwithvolatilitysettoanarbitrarilylowvalue.

91Oneinterpretationoftheseresultsisintermsoftolerancelevels.Ifweapplyatolerancelevel,andthattolerancelevelis10%,say,thenwewouldbecomfortablewiththeERMboundvaluationsprovidedwefeltthatthe‘true’volatilitywasnomorethan15%,andsoon.

123

YoucanthenrestassuredthatbyusingtheboundvaluationinsteadofpureBlack’76theerrorinyourvaluationswillbesmallifBlack’76isanywhereclosetobeingthe‘correct’model.Theonlycircumstanceinwhichtheremightbea largererrorworthnothingisthat inwhichthe‘correct’modelisonethatgivesmuchhigherNNEGvaluationsthanBlack’76.SoforthosecriticsofBlack’76whofeelthatBlack’76NNEGvaluationsaretoohigh,webearnewsofgreatjoy:throwthemodelinthebinandusetheboundsinstead,butlet’shearnomorecomplaints.Evenifdynamichedgingvialiquidmarketsisimpossible,andsoitisimpossibleforBlack’76togivetheexactcostofhedging,wecanalwaysapplyanMCapproachusingthePRAboundsandtheseboundsdonotdependonanyimpossibleconditions.Indeed,we could also use Black ’76 too, butnot on the grounds thatwe aremakingassumptionsaboutliquidcompletemarketsthatweknowarefalsebutnecessaryforthevalidityofthemodel.Instead,wecoulduseBlack’76merelyonthegroundsthatBlack’76givesus results close to thebounds.Comparewith thepracticeof surveying. It isimpossibleinpractice,perhapsevenintheory,tomeasuretheexactlengthoftheradiusofacircle,oritscircumference.Ourmeasuringinstrumentwillneverbeaconstantlengthduetotheeffectofheat,andtheexactlengthoftheinstrumentatthesub-atomiclevelisimpossibletoquantify.Norforthesamereasonisitpossibletocompareitslengthexactlywiththeobjectmeasured.Nonethelesswearecomfortableapplyingtheformulac=2pr,eventhoughtheformulaistrueonlyinthe‘continuous’worldofpuregeometry.Wearecomfortable because the mathematical ratio is a good approximation to the ratiodeterminedbypracticalmeasurement,towhateverdegreeofprecisionwewant.92PricingBeyondtheBoundaryIntheprevioussection,weshowedhowasimpleboundaryapproachcouldbeusedasabasis for accounting valuations of the NNEG and ERM. This approach required noassumptionsaboutthepossibilityofdynamichedgingatall.Butasweshallnowargue,itisevenpossibletoaccountforthecostsofdynamichedgingusingapurelynotionalapproachtohedging.Wecansimplyassumethattheproviderhascreateaninternalhedgebycreatingtwoseparatedesks–anoptiondeskandatradingdesk–whichtradeinternallywitheachother.Theoptiondesk(likeanyoptiondesk)wouldcreatetwoseparateaccounts,anoptionaccountandahedgingaccount.TheoptionaccountwouldfacetheexternalmarketandwouldbevaluedusingthestandardBlack76approach.Thehedgingaccountwouldbeopenedwiththepremiumfortheoption.TheoptiondeskwouldcalculatetheBlack‘76deltas of the option (i.e. the sensitivities of the option to changes in the underlyingforward contract) at each decrement, and the delta-adjusted number of properties 92 Indeed, the method of Archimedes was precisely to start with a discrete approach to determining p, by approximating a circle to a polygon then adding sides to the polygon, on the assumption that as the number of sides approached infinity, the perimeter of the polygon would approach that of a circle.

124

forecasttoexit,ateachdecrement,thenagreetodelivertheindexedvalueofthatnumberof properties to the trading desk, at each exit date. It would thus sell correspondingamountsofproperty forward(i.e. fordeliveryandsettlementat theexitdates) to thetrading desk. The prices agreedwould be by agreement between the desks, perhapsbasedonnation-wideorregionalpriceindices.Thefigurebelowshowsillustrative(spot)valuesofthehedgevsdecrementyear.

Figure20.3:SpotValueofDeltaHedgeAgainstDecrement

Notes:OurcalculationsbasedonNationwideIndex.

Thefigureplotsthevalueofthedeltahedgeforeachdecrementnumber(orthetimetomaturity)ofeachputintheNNEG.Forlowdecrementnumbersthedeltaiszerobutthenrisesaccordingtoabellshapepatterntopeakatdecrement25,andthenfallsthereaftertowardszero.Forexample,theoptiondeskcouldagreetodeliverthevalueof22averagepropertiesintheWestMidlandsregioninyear15.ThespotpricecouldbedeterminedbyreferencetotheNationwide index.Assumeweareat theendofQ12019,when theaverageWestMidlands property is worth £189,263, so 22 such properties would now be worth£4,163,782.93Assuminganetrentalyieldof4%inthatarea,andthatthefirm’sfundingrateisclosetoriskfree,say1.5%.ThentheagreedforwardpriceFwouldbe:(20.4)F=£4,163,782× 𝑒((5.z%4Ê%)×5z)=£2,861,723In15years’timethetradingdeskwouldpaytheagreedpriceof£2,861,723,butatthesame timewouldbepaid the valueof 22WestMidlandsproperties, according to theIndex.Thus,supposingtheIndexhaddoubledinvalueoverthatperiod,thevaluepaid

93Inprinciple,propertiescouldbebrokendownbytypeinalargenumberofdifferentways,e.g.,bytypeofproperty(houseorflat,pricerange,propertysize,gardensize, ifthereisagarden,parkingfacilities,location,etc.).Weglossoverthoseissueshere,butonecouldfinetunethehedgetotakeaccountofthemifonewishedto.

125

wouldbe£4,163,782x2=£8,327,564,hencethetotaltransferwouldbe£8,327,564-£2,861,723=£5,465,841.94Assumingthattheexpectednumberofpropertiesareinfactdeliveredbytheborrowers’estatesthroughtheERMagreement,andsolongastheappreciationinthedeliveredstockmatchestheappreciationintheIndex,theoptiondeskwillhavebrokeneven.Nowtheoptiondeskstillbearstheriskthatthevalueofthepropertiesdeliveredatexitwillnotmatchthevalueofthehedgeposition.Underthearrangementdiscussed,thedeskpromisestodelivertheindexedvalueof22properties,nottherealisedor‘achieved’value.Buttherecouldbeanumberofreasonswhythevaluepromisedcoulddifferfromthevaluerealised.Thereistheriskthatthepredictednumberofpropertiesreceivedonexitdoesnotmatchtheactualnumber–forexampleifborrowerslivedlongerthanexpectedoriftheyexitedatsomeearlierorlaterdate.Thereistherisk,giventhattheprovidercanestimateonlythenumberofpropertiesthatexit,butnottheidentityoftheproperties,thattheiraveragequalitydoesnotmatchtheaveragequalityoftheindex.Evenso,muchofthelatterriskwouldbepickedupandoffsetatotherdecrements.Forexample,if22extrapropertiesexitedinyear16,ratherthaninyear15,thentheonlylosswouldbeinthecostofcarry.Likewiseifthequalityofpropertieswerebelowaverageinyear15,andiftheoverallbookdidreflectaveragequality,wewouldexpectaboveaveragepropertiestoexitinyear16,oratsomepointinthefuture.Aswenotedinchapter10,therewouldbebasisriskfromthe‘slippage’betweenthevaluesofindividualpropertiesandthevalueoftheindex,butthedynamichedgingstrategywouldtakecareof(thebulkof)theperiod-to-periodbasisriskoverthelifetimeoftheNNEG.Infact,thatisitspurpose.TheAppendix to this chapterprovides amoredetailedworkingofhowsuchahedgemightwork.Apossibleobjectiontothistwo-desksargumentisthatsuchinternaltradingisartificialand does not represent reality.95 Our response is that firms do sometimes use sucharrangementsbuttheyarebestunderstoodasaccountingdevices.Theaggregatep/lofthetwo desks will always add up to the external reported profits of the firm, becausewhateverprofit ismade internallybyonedeskwillmatchexactlya lossmadeby theother.Theredoesn’tevenhavetobeamannedtradingdesk–theaccountscouldsimplybemadeuponthebasisofthearrangementabove.Theaccountingarrangementissimplyaformofp/lexplanation,assigningtoonebooktheprofitsorlossesaccruingfromindexand basis volatility, and assigning to the other book whatever accrues from purelyproprietarytradingpositions.Thesamereplycouldbemadetotheobjectionthatnoproprietarytradingdeskwouldbelikelytoacceptamandatefromseniormanagementtotakeontradesatcostfromtheoption desk. Perhaps not, but given that the arrangement is merely an accountingexercisetoexplainthedifferentsourcesofprofitandloss,therewouldn’thavetobeanytraders.Middleofficestaffcouldsimplyperformthework,forabasicsalary.Torepeat,thetwodesksarrangement isanaccountingdevice,but fromourperspective, thekey 94Thedifferenceof£2,861,723representingprincipalplusinterestonthepaymentof£2,285,132thattheoptiondeskwouldhavereceivedifthetradingdeskhadpaidthemupfrontforthevalueofthedeferment. 95 We have encountered this objection in correspondence.

126

pointisthatitisausefuldevicetoexplainhowafirmcanimplementadynamichedgeagainstanoptionposition.Figure20.4plotstheBlack ’76vs.rehedgeputvaluesbasedonahistoricalsimulationoverNationwidehousepricedatafortheperiod1971:Q1to2018Q1.Thetimeunitsherearequarters,so𝑘 = 1heremeansrehedgingeveryquarter.

Figure20.4:Black’76vsRehedgePutValues:HistoricalSimulationfor𝒌 = 𝟏

Notes:IllustrativeputvaluesimulationbasedonNationwideUKpropertydatafor1971Q1:2018Q1. WeseethatthesyntheticputvaluescloselytracktheBlack’76putvalues,implyingthattherehedgingapproachgivesmuchthesameresultsasBlack’76.Or,puttheotherwayround,Black’76wouldgivemuchthesameresultsastherehedgingapproach.Black’76asaNNEGLowerBoundTopullthediscussiontogether,thereareanumberofreasonstothinkthatBlack’76canunder-estimatetheNNEGputvalue.Theseinclude:(a)ThepresenceoftheHursteffectforcaseswhere𝑘 > 1. (b)The fact thatanyrehedgingapproachwoulduseadiscreteratherthancontinuousn(see,e.g.,thepositivemeanhedgeerrorinFigure20.1(a).(c)Theexistenceoftransactionscosts(seefootnote2).(d)Thelikelihoodofbasisrisk,whichmightcreatetheneedforareservefundtoabsorbpossiblelossesor,ifwewishtobookontheliabilitysideofthebalancesheetinsteadoftheassetsside,thebasisriskmightcreatetheneedforacapitalrequirementtocovertheriskofloss.Thepriceoftheoptionisthenthecostofthesyntheticplusthecostofthereserveorcapitalrequired.Thecostof therequiredreserveor requiredcapital isadeepsubject,however,andwedonotexamineitfurtherinthisreport.

127

ConclusionTheMCapproachcanbeimplementedusing(a)thePRAprinciplesbounds,(b)Black’76or(c)somedeltarehedgingmethod.TheprinciplesboundswillproducevaluationsthatarelowerthanthoseofBlack’76andBlack’76willproducevaluationsthatarelowerthan those of the rehedging approach, but there are tradeoffs between ease ofimplementation and accuracy. The first is easiest to implement and makes the leastcalibrationdemands,butistheleastaccurate;thethirdisthemostaccuratebutmakesthemost calibration demands and is the least easy to implement; and Black ’76 liesbetweenthetwo.However,thedifferencesbetweenthethreeapproachesaresmallandanyofthethreewouldbereasonable.Again,weemphasise that theMCapproachdoesnot require thatweuseBlack ’76ordependonanyassumptionsthatareimpossibletohold.

128

AppendixtoChapterTwenty:AWorkedHedgeExampleThisAppendixgoesthroughanillustrativeexercisetoshowthepracticalstepsonemightgothroughtodeltahedgeanERMexposuretoapropertypriceindex.Tofollowthecalculations,werefertothecellentriesintheExcelspreadsheet“Chapter20 Appendix ERM Example” under the tab “Hedge example”. Note that the precisecalibrationsareunimportant.Whatmattersisthehedgecalculationprocess. AssumetheproviderissuesoracquiresabookofERMson8,100propertiesscatteredthroughouttheUK(seeB5).96Theaverageageisaround70(seeF7)andtheaverageloanto value is around 40% (see B11). In practice such uniformity is unlikely, but themanagementof amoredispersedbook is straightforward.With theaveragepropertyvalued at £213,000 (see B6) at the beginning of 2019,97 there is an implied a totalcollateralvalueofaround£1,723m(seeB8),and(withaverageLTVof40%)atotalloanvalueof£689m(seeB7).Weassumeriskfreerateof1.5%(seeB9),defermentrate2%(seeB10),loanrateof5%(seeB12)andinputvolatilityof13%(seeB13).Theimpliedinternalrateofreturnis3.2%(seeB14),comprisingtherisk-freerate,aNNEGspreadof1.8%(seeB15)andanexpectedexcessreturnof1.7%(seeB16). InrealitytheproviderwillmarktheERMstooriginationvalueandusetheexcessreturnasaspreadoverrisk-freetodiscountthepensionliabilitiesviaaMatchingAdjustment.Forsimplicity,weassumethattheliabilitiesarevaluedcorrectlybydiscountingatriskfree,butthattheassetsarebookedatadayonegain. Thesecalibrationsgivealoanvalue,i.e.,L,equalto£1,316m(seeE12).Listhevalueofprojectingtheamountlent(weightedbylongevity)ateachdecrementattheloanrate,thendiscountingatrisk-free,ontheassumptionthatthereisnoNNEGcost.ThepresentvalueoftheNNEGontheseassumptionsis£388m(seeE13),givinganetERMvalueof£928m(seeE14).Hencethereisadayoneprofitof£928m-£689m=£239m(seeE15).Theriskwillbegreatestaroundyears14to18(seeM36:M40).Belowthatyear,theERMportfoliowillconsistincreasinglyofnon-defaultableloans.Abovethatyear,therewillbefewerborrowers.Soconsiderdecrementyear16. Theexitratewillbeabout4.55%atyear16(seeV38),meaningthatabout369(~4.55%x8,100,seeAG38)ofthe8,100propertieswillbedelivered.Wedon’tknowwhichofthe8,100propertiesthesewillbe,onlythat369willbedrawnfromthegeneralpopulationof properties. The delta of the ERM is 29.2% at that decrement (see AH38), whichcorresponds to 107 properties (see AI38). Assuming the average current value ofpropertiesof£213k (seeB6), anda20Y forwardpriceof90%of spot (seeE16), theforwardpriceofeachpropertywillbe£192,453(seeE17).Assumingtheforwardpricefallsby£1,thevalueoftheyear20ERMfallsby£107,i.e.107propertiestimes£1(see

96 One of the Aviva ERF books (#2) started with exactly this number. 97 Source, Nationwide HP index

129

E18).Itfollowsthattheforwardpriceriskcanbeeliminatedbyselling107propertiesforwardtothetradingdesk.ThefallinvalueoftheERMwillbeapproximatelyoffsetbytheriseinvalueofthepositionheldwiththetradingdesk.Ofcoursewiththetradingdeskholding an equal andopposite position, i.e. being long the forward, the impact of thehedgeonthefirm’soverallp/lwillbezero,butasarguedabove,theinternaltradedeviceis simply an exercise in explaining p/l, to apportion the p/l in an appropriate waybetweenthosetakingoptionrisk,andthosetakingpurelylinearhedgepositions.Onewouldapplysimilaranalysistotheotherdecrements.Thoughhighsimplified,thisspreadsheetshowsthattheprocessofdeltahedginganERMexposuretoapropertypriceindexisactuallyquitestraightforward.

130

Chapter Twenty-One: Misconceptions About the Market ConsistentApproachTheMarket Consistent approach has come in for considerable criticism amongst theactuarialprofession.Criticssometimesclaimthatyoucan’tuseBlack’76orBlack-Scholesbecause these models are derived on the basis of a bunch of assumptions that areempiricallyinvalid.Iftheseassumptionsdon’thold,itisclaimed,thenweshouldn’tusethesemodels.Actuarialcriticsoftenwriteas ifoptionspractitionerswereunawareofthese issues, but in fact, the ‘holes in Black-Scholes’ are well-known to optionspractitionerswhoareexpertinworkingtheirwayroundthem(‘traders’lore’).FischerBlack,thesameBlackasinBlack‘76,oncewroteanarticleentitled“Howtousetheholesin Black-Scholes” (1988) in which he set out no less than ten of these unrealisticassumptions:

• Thestock’svolatilityisknownanddoesn’tchangeoverthelifeoftheoption.• Thestockpricechangessmoothlyandneverjumpsupordown.• Theshort-terminterestrateneverchanges.• Anyonecanborroworlendasmuchastheywantatasinglerate.• An investorwhosells the stockor theoptionshortwillhave theuseofall the

proceedsofthesaleandreceiveanyreturnsfrominvestingtheseproceeds.• Therearenotradingcosts.• Aninvestor’stradesdonotaffectthetaxespaid.• Thestockpaysnodividends.• Aninvestorcanexercisetheoptiononlyatexpiration.• Therearenotakeoversorothereventsthatcanendtheoption’slifeearly.

Hethenshowedhowonemightworkaroundtheselimitations.“Sincetheseassumptionsaremostlyfalse,”heconcluded,“weknowtheformulamustbewrong.Butwemaynotbe able to find any other formula that gives better results in a wide range ofcircumstances”(1988,p.67).98Blackconcedestoomuch.Heisrighttoacknowledgethattheseassumptionsaremostlyfalse,butitdoesnotfollowthatthefalsityoftheseassumptionsmakesthemodelinvalid.IfpropositionAimpliespropositionB,thenestablishingthatAisfalsedoesnotestablishthatBisfalse.TheargumentthatAimpliesBsonot-Aimpliesnot-Bisthelogicalfallacyofdenyingtheantecedent.Let’ssaythatweclaimtohaveprovedthattheMoonismadeofgreencheese.Youthendisproveourproof.However,theMoonmightstillbemadeofgreencheeseevenifourproofisinvalid.ThepointisthatweneedtobecarefultodistinguishsufficientandnecessaryconditionsforBlack-Scholestohold.Yes,itisittruethatinitsusualclassical/textbookderivations,Black-Scholesmakesvariousassumptions,someofwhichareempiricallyfalse.However,

98 SeealsoT.Gordon“ThePriceofActuarialValues,”StapleInn1999.

131

mostoftheseassumptionsaresufficientratherthannecessaryforthemodeltobe‘valid’inthesenseweputforwardinChapter20,namelythatitprovidesarobustestimateofpayoff minus cost of discrete hedging. So you can’t just kick aside some of theseassumptionsandconcludethattheBlack-Scholesresultsmustbewrong.Forastart,therearemultipleways toobtainBlack-Scholes– amongstothers, thereare themartingaleapproach,thebinomialapproach,CapitalAssetPricingModel(CAPM)andutility-basedapproaches99–andthesewillbebasedondifferentsetsofsufficientconditions.Toshowthatthemodelis‘wrong,’onewouldthenhavetoestablishwhichparticularassumptionswerenecessary,andthendemonstratethatoneormorenecessaryconditionswerenotonlywrong,butalsointroducedmaterialerrorsintotheresultingvaluations.Anysuchanexerciseismoreinvolvedthanmerelyassertingthatsomeparticularassumptionisempiricallyfalse.Black-ScholesisRobusttoNon-RandomnessBlack-Scholesworkswhentherealisedprofitorlossfromreplicatingtheoptionmatchestheassumptionsusedwhenpricingtheoption.Asexplainedinthepreviouschapter,westart by computing the ‘delta’, or sensitivity of the option price to changes in theunderlyingprice,thentakeapositionequalandoppositetothedelta,sothatchangesinthepriceoftheoptionareoffsetbychangesinthehedgeposition.Sincethedeltaitselfchangessomewhatwiththeunderlyingprice(aneffectknownasgamma),wemightalsooccasionallyre-hedge,butthatisaseparateissue.It turnsoutthattheaccuracyofBSissurprisinglyrobusttoanumberofassumptionscommonlymadetoobtaintheBSformula.OnesuchassumptionisGBM,solet’stakeasimulatedGaussiandistribution,butsortitinorder of magnitude, so that while it is still Gaussian, it is no longer random, so the‘Brownian’bitof‘GeometricBrownianMotion’nolongerapplies.WethengettheplotsshowninFigure21.1:

99Formoreonthese,see,e.g.,J.Andreasen,B.JensenandR.Poulson(1998)“EightValuationMethodsinFinancialMathematics:TheBlack-ScholesFormulaasanExample,”MathematicalScientist23:18-40.

132

Figure21.1:SyntheticvsBlack-ScholesPutOption:GeometricNon-BrownianMotion

AswecanseefromthegreenlineintheFigure,thepricefallsthroughoutthefirsthalfoftheseriesandthenrises.Theblue line shows theprice of a put optionprice struck at 90, computedusing thestandardBlack76optionformula.Asyouexpect,thepricerisesastheunderlyingfallsbelow95,thenfallsbackastheunderlyingincreases,endingatzeroastheoptionexpiresoutofthemoney.Theredlineshowsthevalueofthesyntheticorreplicatingoptionconstructedusingthedeltaoftheput.The(short)deltastartsat18%,sincetheunderlyingpricebeginsat95,risestonearly100%astheputisincreasinglyinthemoney,butthenfallsbacktozeroattheend.Thehedgeisnotperfectbutitisclose.Theimplicationisthat,whiletheassumptionofrandomnessmaybeasufficientcondition forBlack-Scholes tohold, it isbynomeansnecessary.Black-ScholesRobusttoMeanReversionInitsreplytoCP13/18,theIFoAstates(p.10)that

Using theBlack-Scholes formula inpricingNNEGwill affect thecostof theguarantee, sinceallowance isnotmade for the featuresofmeanreversion,momentumandjumpsdescribedabove.UndergeometricBrownianmotionthevolatilityincreaseswiththesquarerootoftimewhileforothermodelsitdoesnot;thevalueforlongtermderivativessuchasNNEGcouldmateriallydifferfromthatassumedundertheBlack-Scholesmodel.100(Ouremphasis)

100“CP13/18:SolvencyII–EquityReleaseMortgages:IFoAResponsetoPrudentialRegulationAuthority.”28September2018.

133

Let’shavealookatthismeanreversionclaim.Figure21.2plotsBlack-Scholesagainstaputvalueunderamean-revertingprocess:

Figure21.2:SyntheticvsBlack-ScholesPutOption:GBMvsMean-ReversionProcess

Thechartaboveshows(greenline)anunderlyingassetthatfollowsthepriceseries95,96,95,96…i.e.theserieshasameanof95.5towhichitcontinuallyreverts.Thebluelineshowsthepriceofaputoptionstruckat90,modelledbythestandardBSoptionformula.The red line shows the value of a hedge, constructedfrom a derivative of the sameformula(‘delta’)andtheyendupinapproximatelythesameplace.WeseethatBSworkswell,eventhoughtheseriesismeanrevertingandisnotdrawnfromanythingthatevenresemblesanormaldistribution.SoitistruethatunderaGBMprocessthevolatilityincreaseswiththesquarerootoftimewhilstforamean-revertingprocessitdoesnot,butthisdifferencedoesn’tmatterhere.Thelongerthesamplingperiodforthestrangedistributionabove,thelowerthesampledvolatility: the prices series is going exactly nowhere. Yet the standard option pricingmodelstillworkswell.Doadifferentsimulationoftheunderlyingpathandyouwouldgetsimilarresults.

134

Black’76isUnderminedbyAutocorrelationinHousePricesAnotherexampleisautocorrelationinhouseprices.Onp.14ofhisreport,Tunaruwrites:

GBMas adatageneratingprocess forhouseprices is totally inappropriatebecause it ignores serial correlation and stickiness of prices, as well asclusteredvolatilityanddownwardjumps.

WeagreewithhimthatGBMdoesnotprovideanempiricallyaccuratedescriptionoftheempiricalhousepriceprocess,butwewouldstillarguethatBlack’76whichassumesGBMstillprovidesagoodframeworktovalueNNEGsandERMs.Let’sleaveasideissuesofpricestickiness,clusteredvolordownwardjumps,whichreallypointtothecalibrationofthevolatilityparameterfedintoBlack ’76,andfocusontheserialcorrelationorautocorrelationofhouseprices.TheissueiswhetherautocorrelationunderminesthevalidityofBlack’76.It is true that many random variables are autocorrelated and that that the usualderivations of Black ‘76 assume that the underlying random variable is notautocorrelated.However, recall that the question of howautocorrelationmight affectoption pricing was addressed in the Cornalba- Bouchaud-Potters article that wediscussedinChapter10andtheirconclusionswereclear.“IntheGaussiancase[theoneconsidered in Black-Scholes], we find that the effect of [auto-] correlations can becompensatedbyachangeinthehedgingstrategyandthereforeoptionsshouldbepricedusingthestandarduncorrelatedBlack-Scholesmodel”(ouritalics).Thekeyistoensurethat the volatility is measured on the same time scale as the rehedging but thisqualificationmerelyamountstoanadjustment,ifany,tothevolatilitycalibration,andBSstillholds.ThesameargumentcanthenmadeforBlack’76.Wediscussedtheconnectionbetweenrehedgingandvolatility inChapters10and21.Theessentialpointisthatthevolatilityshouldcorrespondtothatoftherehedgingperiod.Supposethenthatanoptionisbeingrehedgedevery5years.Ifthemonthlyvolatilityis1%,thenunderGBMwewouldinput(21.2)adjusted𝜎 = 1% × 600.z = 7.75%intoouroptionpricingequation,butiftheunderlyingisautocorrelatedwith𝐻 = 0.9,say,wewouldinputthefollowingH-adjustedvolatility(21.3)𝐻-adjusted𝜎 = 1% × 600.Å = 39.8%.Black’76isstillvalidinthepresenceofautocorrelationprovidedweusetheappropriatevolatility,i.e.,(21.3)insteadof(21.2).

135

‘NoShort-Selling’ArgumentThe ‘noshorting’argument isoftenwheeledout toclaimthat theusualarbitrage-freeargumentsunderpinningBlack’76don’tapply.CriticsclaimthatthevalidityofBlack’76dependsonano-arbitrageargumentthatitselfdependsonbeingabletoselltheforwardcontract,butthenclaimthatno-arbitragedoesnotapplybecause,e.g.,onecannotselltheforward.AnexampleisgivenbyTunaru:

In our opinion, the formula (10) [i.e. Black ‘76] simply does not apply forhousepricesand this isunrelated to theGaussiandistributionassumptionbehind the GBMmodel. The forward contract on a house price cannot becalculatedas in(11)[i.e.accordingtoourequation(3.10): forwardprice=spotprice× 𝑒(I4W)*], simply imitating theno-arbitrage formula for a stockpayingdividend,wherethedividendyieldisreplacedbythenetrentalrate.Thatformulacannotworkbecausecurrentlywecannotshortsellthevalueofahouse.Hence, theno-arbitrageprincipledoesnot applyhere to lock in theforward price as in the case of corporate stock. (Tunaru, 2019, p. 14, ouremphasis)

Butofcourseyoucansellahouse!‘Can’justmeans‘itispossible’andtherearenolegalrestrictionsagainstdoingso.Asweexplainedinthepreviouschapter,anyonecanenterinto an agreement whereby one party agrees to deliver an individual property ofspecified size, location and standard to the other party, at an agreed date and for anagreedprice.TheOver-the-Counter(OTC)marketmakesthesetradesallthetime.Ifnoindividualpropertyisavailable,thepartiescouldcontracttobuyorsellthevalueof,say,theHalifax index, times some currencymultiplier, at some agreed date in the future.Partieswhowishtomakesuchtradescanusuallyfindinvestmentbankerswillingtobetheircounterparties.Another version of the ‘no shorting’ argument is that the usual arbitrage-freeassumptions don’t apply, because you can’t (delta) hedge your property exposure.Howeverthisversionisalsofalse,becauseitisactuallyquitestraightforwardtodelta-hedgeapropertyexposureandexplainedinthepreviouschapter.Torecap.Youhaveacollectionofpropertieswhichwillbecomingontoyourbooksattimesandamountscorrespondingtoyourlongevitymodelling.Youworkoutyourdeltasto themarket across thedecrements and sell the rights to futurepossession to someinterested counterparty. So youagree todeliver a fixed amountofproperties at eachdecrementyear,accordingtoapre-agreedstandard(nbedrooms,y locationetc),withcontractually agreed penalties for late delivery or non-delivery. The only differencebetweenthisarrangementandastandardreversionarycontractisthattheidentityofthepropertiesisnotknowninadvance.Forexample,supposeyouexpecttoreceive50fourbedroompropertiesintheSouthEastinyear24,andyourdeltais70%.Thenyoucommittodeliver50x70%=35propertiesinthatyear,inreturnforcash.Ifitturnsoutonly,say,30propertiesexit,thenyoudeferthedeliveryoftheremaining5untilthefollowingyear,withanagreedpenalty,perhapsoneyear’srentalequivalent,orsettlethewholeamountforcash.Thekeyisworkingoutthecashamountthehedgerwouldbewillingtopayinreturnforreceivingthe35properties.

136

Indeed,sometimesyoudon’tevenneedhedgingtocarryoutvalidBlack’76valuation.Consideradeep-out-of-the-moneyoption,illustratedbytheleft-hand-sideofFigure21.3,where𝐸𝑅𝑀* isequal to𝐿* , thePVof the loanvalue.Orconsideradeep-in-the-moneyoption, illustratedby theright-hand-sideofFigure21.3where𝐸𝑅𝑀*approachesor isequaltothePVoftheforwardat𝑡.

Figure21.3:ERMValuationandForwardHousePriceVolatility

Notes:𝐸𝑅𝑀* is the value of the Equity ReleaseMortgage for decrement 𝑡 with the relevantvolatilityand𝐿*isthevalueoftheloanfordecrement𝑡.Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%and𝑞=4.2%.ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

Inbothcases,thevalueof𝑁𝑁𝐸𝐺*doesnotdependonwhethertheoptioncanbehedged.Yes,youmightsay,butyouhavenotshownthatthisargumentalsoappliestothecentralregion of the Figurewhere the value of𝐸𝑅𝑀* is below theminimumof the loan anddeferredhouseprice curves.This objection is true, butnowallow the volatility to gotowardszeroandthecentralregiondisappears.Black‘76isthenvalidacrosstheentirehorizon spectrum. It follows that, atmost, any concerns about thepossible impact ofimperfecthedgingopportunitiesontheoptionvaluemusttranslateintothecalibrationofthevolatilityparameter,soBlack‘76isstillvalidoratleastapproximatelysoprovidedonehandlesthevolatilitycalibrationwithappropriatecare.Ah,youmight respond, the redupperbound (ordefermenthouseprice) curve in theabove figurestilldependson theability toshort-sell theright todeferredpossession.However,thisresponseisnotcorrect.Recalling(3.10)𝑅* = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡ℎ𝑜𝑢𝑠𝑒𝑝𝑟𝑖𝑐𝑒 × 𝑒4W*we can always obtain a deferment price if we have the spot price and a reasonableestimateofthedefermentrate𝑞.Theformeriseasytoascertainandwediscussedthecalibration of the deferment rate in Chapters 8. Even though 𝑞 is unobservable, theaccounting fiduciary principle comes into play (see Chapter 26 below), that not only

137

allows but mandates that even when a market price or other relevant entity isunobservable,aninstrumentshouldbepricedinawaythatreflectstheassumptionsthatmarketparticipantswouldusewhenpricingthe instrument.Marketparticipantsarebydefinition independent of one another, knowledgeable or have a reasonableunderstandingoftheinstrument,andwillingandabletoenterintoatransaction.101Inpricingadeferment,amarketparticipantwouldthereforeconsiderthecostoflosingtheincomeforthedefermentperiod,andwouldadjust thepriceof immediatepossessionaccordingly (see also SS 3/17, p.12). That this exercise involves some judgment goeswithoutsaying,butthatjudgementhastobereasonableandthereareboundsonwhatareasonablejudgementcanbe. ‘RentalRateisIrrelevantBecauseaHouseisaConsumptionAsset’Afinalargumentisthatthestandardforwardvaluation(e.g.,equation(3.9)whichstatesthatforwardprice=currenthouseprice× 𝑒(I4W)*)isinappropriatebecausetherentalrateisirrelevanttothosewhopurchasehousestoliveinthem.ToquoteTunaruagain(p.30):

Theconceptofrentalyieldhasbeenintroducedintorealestatevaluationbyanalogywiththelinkbetweendividendsandshareprices.However,itcanbearguedthatthebuyerofahouseisnottheequivalenttoaninvestorbuyingahouseasaninvestmentasset.Forthemajorityofbuyers,housesplaytheroleof a consumptionasset andnot thatof an investmentasset.102There isnoevidence that rental yieldsaredriving futurehouseprices so theexpectedhouse prices at various future long horizons cannot be determined withgrowthmodels in thesamewayexpectedsharepricesmaybedeterminedwithgrowthmodelslinkedtodividends.

OneresponseisthattheERMvaluationquestionappliestoinstitutionalinvestors,namelyERMlenders,whowanttoacquireresidentialpropertyexposure,andwhoarenotusingthepropertyasaconsumptionasset,butratherasaninvestmentasset.Ourpointisthatit is the institutional investorswhomatterhere,because theyare theoneswho issueNNEGs,nottheretailinvestors.In any case, why would the ordinary buyer approach the valuation of a deferredpossessionanydifferentlyfromaninstitutionalbuyer?Supposewearecurrentlylockedintoaleaseholdbutarelookingtomoveintoapropertyin2yearstimeafterourtenancyagreementrunsout.Youarelookingtomoveoutofyourpropertyatthesamedate,butneedmoneynow.Soweagreetopayyoumoneynowinordertopossessthepropertyat

101 SeeIFRS2013,para87onunobservableinputs,andAppendixAondefinitions. 102Arelatedargumentwehaveheardisthatwhiletheutilityofnothavingtopayrentalmayberelevantto1sttimebuyerstryingtoescaperentalintoowner-occupier,oncepeoplehaveahomeoftheirown,theydon’t think about rental yield when looking at moving house “up the ladder.”This type of argumentfocussesonhowpeople thinkof their ownhome, i.e. as somewhere to live, andnot as an investment.However,wewouldarguethatthepsychologyofindividualhomeownersisirrelevant,ifonlybecausetherelevantowneristheERMlender,whohasapileofpropertywithrightstodeferredpossessionontheirbooks.

138

that future date, but the price we pay now would not be the price of immediatepossession.Whyshouldwepaythefullpriceofthepropertynowifwecannotmoveinorrent itout fora full twoyears?Andhowwewouldvalue thedeferredpossession?Answer:atsomediscounttothevalueofcurrentpossession,wherethediscountwouldbedrivenbytherentalrate.103Tunaru writes that “there is no evidence that rental yields are driving future houseprices”butthatclaimisirrelevantevenifitistrue.Itisnotfuturehousepricesweareconcernedwith,butratherthepricenowofacontractforpossessioninthefuture.Wearecomparingthecurrentpricesoftwodifferentcontracts,oneforimmediate,theotherforfuturepossession,andwearenotspeculatingaboutfuturehouseprices,whichareirrelevanthere.Ofcourse,anticipatedfuturepricesmayberelevantifitisthedecisionthatisdeferred.Dowebuynowordowewaitfortwoyears?Ifweexpectpricestorise,wewillpayforimmediateordeferredpossession.Ifweexpectthemtofall,wewilldeferourdecisiontobuy,whetherthatbeimmediatepossessionordeferredpossession.Decisionisdifferentfrompossession,however.Ifwepaynowfordeferredpossession,wearealreadyexposedtofuturehouseprices.Ifweexpectpricestofall,wewillpayneitherforimmediatenordeferredpossession.Rather,wewillwait,i.e.wewilldeferourdecision.Allthesepointsfollowfromelementarypricingeconomics.

103Orstillanotherversionof the ‘householder isdifferent’argument.AndrewRendellat theARCERMLaunchEventon28thFebruary2019arguedthatwemustconsidertheutilitytotheoccupieroflivinginthehouse,withoutsayinghowwemightmeasurethatutility.Butwhatifweaskyoutovacateyourhousefor2years,soyounolongerhavethatutility?Whatisthecostofit?Surelythecostofrentingfor2years.

139

ChapterTwenty-Two:TheDiscountedProjectionApproachThestandardapproachusedbyERMactuaries in theUK is theDiscountedProjection(DP)approach,sometimesalsocalledthe‘realworld’approach.ThisapproachisbasedontheuseofaprojectionoffuturehousepricegrowthtovaluetheNNEG.Inparticular,it replaces the forwardhouse price as the underlying in theMC approachwith some‘expected’ future house price. Equivalently, it replaces the forward rate 𝑓 in the MCapproachwithsomeassumedrateoffuturehousepricegrowthℎ𝑝𝑖.Since𝑓 = 𝑟 − 𝑞,replacing𝑓byℎ𝑝𝑖gives(22.1)ℎ𝑝𝑖 = 𝑟 − 𝑞.Givenalsothatℎ𝑝𝑖hasbeenspecifiedand𝑟canbeeasilycalibratedfromthespotratecurve,(23.1)impliesthatwecanbackoutthefollowingimplied𝑞:(22.2)𝑞 = 𝑟 − ℎ𝑝𝑖.Togiveanexample,ifweset𝑟 = 1.5%andusethe4.25%ℎ𝑝𝑖assumedrecentlyusedbyJustGroup104thenwecanbackout𝑞as(22.3)𝑞 = 1.5% − 4.25% = −2.75%.Thenegativesigninfrontofthe‘2.75%’ontheright-handsideof(23.3)isnotatypo.Forthiscalibration,andindeedforanycalibrationinwhichℎ𝑝𝑖exceedstherisk-freerate,theDPapproachproducesanegative𝑞netrentalrate.OnecanimplementthisDPapproachbytakinganotherwisesoundMC(e.g.,Black’76)calibrationandreplacingtheforwardratewithanassumedfuturehousepricegrowthrateℎ𝑝𝑖.Soonereplacesthe𝑞rateonewouldotherwiseuse(e.g.,asensible𝑞rateofaround3%ormore)withanimplied𝑞rateequalto𝑟 − ℎ𝑝𝑖(e.g.,a𝑞rateofabout-2.75%asintheJustexample).To illustrate the impact of this approach, Table 22.1 shows the NNEG and relatedvaluationsforourbaselinecalibration,obtainedusingeachapproach:

104 Thefirmreportedusingthisnumberinbothits2016and2017AnnualReports(seepp.163and110respectively).Thesamenumberalsoappearsinits2018H1results(p.18).

140

Table22.1:BaselineERM/NNEGValuations:MarketConsistentvs.DiscountedProjectionApproaches

Approach 𝑳 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴Marketconsistent £74.84 £32.19 £42.66

Discountedprojection £74.84 £4.37 £70.47Notes:𝐿isthepresentvalueoftheloancomponentoftheEquityReleaseMortgage,𝑁𝑁𝐸𝐺isthepresent value of the NNEG guarantee, and 𝐸𝑅𝑀 is the present value of the Equity ReleaseMortgage. Based on the baseline assumptions: male aged 70, 𝐿𝑇𝑉=40%,𝑟=1.5%, 𝑙=5.25%,𝑞=4.2%fortheMCapproachand𝑞=-2.75%fortheDPapproach,and𝜎=14.8%.ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

Inthiscase,whichisnotuntypicalofotherswehavelookedat,theDPapproachgivesNNEGvaluationsthatareclosetoanorderofmagnitudelowerthanthoseproducedbytheMCapproach.TheresultisaconsiderableovervaluationoftheERM,inthiscaseby57.3%.Butaskyourself:dotheDPvaluationsevenlookright?Ifyoubelievethem,thenyouhavetobelievethatthe‘true’𝑁𝑁𝐸𝐺isonly4.37/74.84=5.8%of𝐿.ThisNNEG/𝐿ratiolooksawfullylowwhenyouconsiderthespreadbetweentheloanrateandtherisk-freerate,whichis5.25%-1.5%=3.75%.Iftheloanhassolittlerisk,thenwhyisthespreadsohigh?HereweseeinanutshellthevaluationproblemsentailedbytheuseoftheDPapproach.OriginsandOverviewoftheDiscountedProjectionApproachSowheredoesthisapproachcomefrom?In2005,theIFoApublishedareportonNNEGvaluation(IFoA,2005).ThisreportconfirmedthatitwasreasonabletouseBlack-Scholesmethodologywhenseekingtoobtaina‘marketconsistent’NNEGvaluationwhilstnotingthatit“isnotwithoutitsdifficultiesandshortcomings.”Wehavenoargumentwiththatassessment.However, italsonotedthat,“Othersmayhowever,prefertoapproachtheassessmentof theNNEGusingmoreofa “realworld”stochasticmodellingapproach,”whateverthatmightbe,andtheydidnotexplain.AndsowehavethejuxtapositionofBSasareasonableapproachto ‘marketconsistent’NNEGvaluation,versusanalternativeunspecified ‘real world’ approach that is pulled out of thin air and gives a differentvaluation.Twoyearslater,theIFoAissuedanotherreportonNNEGvaluation,Hostyetalia(2007).ThisreportstartedwithsomeconcernsaboutthedeclineofprofitabilityanditsimpactonthedevelopmentoftheERMmarket:

the competitive environment that has driven product innovation has …resultedinlowerproductmargins.Thisisallgoodfortheconsumer,butitisincreasinglydifficultforproviderstoreachtargetreturnsoncapital,andthisisdeterringsomeprospectivenewentrants.Oneofthepurposesofthispaperistoinvestigatetheprofitabilityoftypicalschemesinthemarketatpresent,andsotoaddressthequestionofwhethercompetitionhasforcedthemarket

141

tofunctionatnon-profitablelevels.…Wewillaimtoprovidearationalpricingmethodologywhichcanbeadoptedbyanyorganisationactiveinthemarket,andwehopethatthiscansupportthemarketasitexpandsoverthecomingyears.There is now concern that providersmay not be able to offer a productprofitably at currentmargins. Some competitive pressure is clearly a goodthing,asitwillforceproviderstofindmoreefficientwaysofprovidingtheirproduct to consumers. In the equity releasemarket, toomuch competitivepressuremaybeabadthing.(pp.1-2,ouremphasis)

To cut to the chase: their main concern is that overly high NNEG valuations mightunderminetheabilityoffirmstomeettheirprofittargets.Wearesuretheyareright,butthequestionis,iftheirconcernsarecorrect,howtoreconciletheircommercialconcernswiththeirobligations(underactuarialandaccountingstandards,seeChapters25and26below)toprovideunbiasedandfairvaluevaluations.If therewasn’t a conflict between these twoobjectives (i.e., profits andunbiased/fairvalue),thenwewouldn’tevenbehavingthisconversation.ThefactthatERMindustryleaders emphasise the conflict between the two objectives and their preference forcommercialconsiderationsoverunbiased/fairvaluationindicatesthattheseobjectivesare inconflict.Otherwise theywouldargue forunbiased/fairvaluationsandwouldn’tneedtoarguefortheprimacyofcommercialconsiderationsinstead.Theythenexaminewhattheyunderstandthe“marketconsistent”approachtobe.Theydo not define the term “market consistent” however and the nearest we get to anexplanationisthatthisapproachisbasedonan

approximate market consistent basis similar to the pricing of options onstocks.…Themainchallengewithamarketconsistentbasisisthefactthatthereisnounderlyingmarkettospeakof.Accordinglywehavetriedtocreateaproxymarketconsistentbasisusingtechniquesthatarestandardinsimilarmarkets,specificallyBlackScholesstylemodelling.(p.26)

Thecounter-argumentisthatthereisalwaysanunderlyingmarket!Almostallpropertytransactions are forwards, admittedly, short-maturity forwardsbut there areno legalbarrierstolongermaturityforwardsandERMfirmscouldalwaysapproachinvestmentbanksforquotes.WhetherERMfirmswishtotradeatthoseratesisanothermatter.Wetaketheirpointabout‘proxy’valuations,butaswediscussedinChapter9,wecanobtainempiricallygroundedproxyvaluationsfromvaluationsintheleaseholdmarket.Hostyetaliathenexplainwhattheymeanbyproxyvaluation:

Usingariskneutralbasis,housepriceinflationshouldbelinkedtothereturnon long term risk free instruments (i.e. government stocks) less anassumptionforrentalincome(netofexpenses).(p.26,ouremphasis)

Butthiswayofproxyingvaluationsisbasedonahowlerofanerror.Theerroristhatitconfusesthefuturepriceofspotpossessionwiththecurrentpriceofdeferredpossession.

142

Thiserrorishugelymaterial,sincetheonevariable(thefuturepriceofspotpossession)usuallygoesupovertime,whilsttheother(thecurrentpriceofdeferredpossession)fallswithmaturity.Toillustratethemagnitudeofthisdifference,considertheplotsinFigure22.1:

Figure22.1:FuturevsDefermentHousePrices

Notes:Basedoncurrenthouseprice=£100,ℎ𝑝𝑖=4.25%and𝑞=4.25%.

Theblueplotgivesthecorrectpricetouse intheoptionpricingformula, i.e., thespotprice of deferred possession. For amaturity of, say, 15 years, and our recommendeddeferment rate of 4.2%, the deferment house price (the price of a spot contract forpossessionin15years)is£53.3.TheredplotgivestheincorrectpricethattheHostyetaliaargumentimpliedshouldbeused,i.e.,thefuturehouseprice.Thisparticularplotisbasedonanassumed4.25%hpirate.Theexpectedfuturepricein15yearsis£189.3.So based on these calibrations, the Hosty et alia argument implies that we use anunderlyingvalueof£189.3intheoptionpricingequationwhenthecorrectvalueis£53.3.Ifyoubelieve that theHostyapproach is right, thenyouarebelieving thatanasset,aforwardworth£53.3,isactuallyworth£189.3,inwhichcaselet’sdoatrade.Thus,therootissuewiththeDPapproachisamisinterpretationoftheforwardcontractthatleadstoalargeoverestimationofthecontracts’value.The Hosty approach then produces a NNEG of £3.00 if we make our other baselineassumptionswhenBlack’76correctlyappliedwouldgiveusaNNEGof£31.42.TheHostyapproachthusleadstoaNNEGvaluationthatis9.5%oftheBlack’76valuation.TheHostyetaliauseoftheincorrectterm‘housepriceinflation’insteadthecorrectterm‘forwardhouseprice’suggeststhattheyconsiderthatthefuturehouseprice(orhpirate,dependingontheformulationonewishestouse)shouldgointotheBSorBlack’76model,butneitherthefuturehousepricetheexpectedfuturehousepricenortheexpectedhpiratebelonginthosemodels.Youcaninputthemifyouinsist,butyoushouldn’t,becausethemodelgivesyouhaveno leave to.Thesevariablesare irrelevant inanyBS-familyoptionpricemodel.Instead,BSor,moreprecisely,Black’76tellsusthattheunderlying

143

variablethatshouldgointotheoptionpricingequationistheforwardprice,inthiscase,theforwardhouseprice.Theuseofterm‘housepriceinflation’inthiscontextsuggestsaseriousmisunderstandingofhowBSoptionpricingworksandleadstoamajorunder-estimationoftheNNEGvalue.ButHostyetaliagoontomakeplainthattheydonotliketheMCapproach:

Inrealitytheabsenceofanunderlyingmarketmeansthatthisproxymarketconsistentapproachisonlyoflimitedacademicvalue…(p.27,ouritalics)

By“absenceofanunderlyingmarket”theymeantheabsenceofaliquidmarketinwhichtheoption canbehedgedusing e.g. a zero-arbitrage trading strategy.But aswehaveexplained in Chapter 20, it is perfectly feasible to apply the MC approach in the UKpropertymarketcontext,andwhateverthedifficultiesofdoingso,thosedifficultiesinnowaygiveHostyetalialicencetoreplacetheforwardhousepriceintheoptionpricingequation with some guess-estimate of the future house price. The “only of limitedacademicvalue”jibeispresumablymeanttosuggestthattheMCapproach–or“proxymarketconsistentapproach”astheyputit–isofnopractical‘realworld’useandperhapstohintthatpractitionersshouldbelookingforamore‘realworld’-friendlyalternative?Again,wedisagree.TheMCapproachisnotonlyfeasible,buthasnofeasiblealternative.ThentheymakeafurthercriticismoftheMCapproach:

For providers attempting to price the NNEG on amarket consistent basisthereisinsufficientproductmargininordertoprovideacompetitiveproductunlesstheyhavestrongcompetitiveadvantagesinoneormoreoftheothercostareas.(p.30)

Whetherornotthisclaimistrue,thisstatementbegsthecentralissue,i.e.,whethertheMC-based valuations are reliable, and Hosty et al. provide no convincing scientificgroundstoquestionthem.SoHostyetalia’smainobjectiontoMCvaluationboilsdowntoitgivingvaluationsthattheydon’tlike.ButremembertheproblemsthatEquitableLifegotinto20yearsagowhenitwasdiscoveredtohavebeenundervaluingitslong-termguarantees!Section7.3.2examinestheirpreferredalternative,an“insurancepricingbasisusing“realworld''assumptions.”Whattheseassumptionsmightbetheydonotexplain;nor,dotheyexplainhowthis“realworld”approachmightbeconsistentwithaveryun-realworldnegativenetrentalrate.Infact,theydon’texplainwhattheir“realworld”approachevenis.Section7.3.2consistsofonly143wordsandisherereproducedalmostinfull:

7.3.2“Realworld:assumptions”Thealternativemethodwehaveusedistocalculatetheoptioncostusing“realworld”basis.Themethodologywehaveusedisasfollows:- Usethelognormalmodelasbefore(withsamevolatility).

144

- Abestestimateof4.5%p.a.forHPIinthefuture(seeSection4.4).Thisisthenthemeanreturnunderthemodel.

- Wehaveassumedthatarealworlddiscountrateof4.75%perannum.- Wehavenotassumeda“meanreversion'' so that therandomwalk in

each future period is applied independently of the position is [sic]preceding periods. The authors acknowledge that use of a “meanreversion”approachisequallyvalid.…

Ascanbeseen[fromTableshown],theresultingcostsaresignificantlybelowthoseassessedusingourproxymarketconsistentbasis.

Sonotawordofexplanationastowhyweshouldregardthis‘realworld’approachasreliable,butthephrasethatjumpsoutis“Abestestimateof4.5%forHPIinthefuture,”i.e.,theRWapproachisbasedonaguessaboutfutureHPI!Basedonthelimitedinformationprovided,their‘realworld’approachwouldappeartobe similar to the MC approach, but with the forward house price replaced by someassumedexpectedfutureHPI.Wenowseetheseedgerminate.The2005IFoAreportintroducedtheTrojanHorseofhouse price inflation, but at least did the calculations correctly. This error could beforgivenasaninnocuousterminologicalone,exceptthatthepassagequotedopensthedoortofull-scalemisuseandseemstoconfirmthattheHostyetalia2007‘realworld’valuationapproachisbasedonexactlythaterror.TheinclusionofHPIisnolongerameremislabeling,butabedrockprincipleoftheRWapproach.Tospellitout,HPIisnowakeyinputinitsownright.Whichpointsconfirmthatthisapproachisinconsistentwithoptionpricingtheoryandthereforewrong.Section 7.3.3 clarifies the authors’ views on which approach is to be preferred. Wereproducepartofithere:

7.3.3Marketconsistentorrealworld?On our proxymarket consistent approachwe have derived a cost for theNNEG which would render the product non-profitable, whilst real worldmodellinghasproducedasignificantlylowercost.

Thisstatementhasmajorrepercussions.IfNNEGvaluationsonaMCbasiswouldmakeERMsunprofitableandifthereisnojustifiablealternativetoNNEGvaluationsonanMCbasis,thendoesn’tthatmaketheERMsectorunprofitable?Andifthesectoronlyappearstobeprofitablebecausethe‘realworld’NNEGvaluationsmakeitappearso,thendoesn’tthatmeanthattheprofitsthatthefirmshavebeenmakingmayhavebeenmoreapparentthanreal?TheimportanceofcommercialconsiderationsasareasonforpreferringthisapproachwasconfirmedbyTomKennyatthe28February2019StapleInnlauncheventfortheTunaru report. Mr Kenny was the chair of the event, and is Director of Actuarial &

145

Underwriting,RetirementLendingatJustGroupplcinhisdayjob:“clearlyifwemovedownapurelymarketconsistentroute…it’sgoingtobeextremelyexpensive,”hesaid.However,theissueisnotwhetherthefirms’NNEGvaluationswouldgoupiftheyusedanotherapproach.The issue iswhether firmsareusingtherightapproach inthe firstplace.IffirmsareusingavaluationapproachthatgreatlyundervaluestheirNNEGs,thentheyhavegreatlyunder-estimatedtheircostsandthosecostsarealreadybeingbornebyfirms and their investors, regardless of whether firms acknowledge that fact. Firmsshouldbefacinguptothisprobleminsteadofdenyingit.Fortheirpart,analystsshouldbewonderinghowbigthisproblemmightbeandaskingthemselvesaboutthepotentialimpactonfirms’financialconditions.Under-valuedcostsmeanhiddenlossesandover-estimatedcapital,potentiallyonalargescale.Imagine if Bosch are under-valuing the guarantees they issue with their washingmachines.Theirmanagementthendiscoverthatthecostsofreplacingorrepairingtheirwashingmachinesaregoingtobehigherthantheyhadexpected,buttheydon’tyetknowhowmuchhigher.Theproblemmightonlybeasmallproblembutthenagainitmightnot. Sowhat is themost appropriate response from themanagementwhen they areinformed of it? Should they deny it on the grounds that theywouldn’t like it if theirguaranteesturnedouttobemorecostlythantheyhadthoughtorshouldtheylookintothe issuewithaview to fixing theproblembefore it getsanyworse?Wewouldhavethoughtthattheanswertothatquestionwasobvious,butthenwhywouldtheanswerbeanydifferentifitwasERMsratherthanwashingmachineswhoseguaranteeswerebeingunder-valued? And if the losses involved might potentially be on a large scale, thendoesn’tthatreinforcetheneedtoaddresstheproblemasamatterofsomeurgency,lestapotentiallylargeproblemgrowintoalargerproblemdowntheroadifnothingisdoneaboutit?ProblemswiththeDPApproachConsiderthattheDiscountedProjectionapproach:

• hasneverbeenconvincinglyjustifiedbythosewhoadvocateit;• is beingpromotedbypractitionerswith a vested commercial interestwhoare

promoting it for openly commercial reasons and are dismissive of the onlyapproachthatisscientificallyrespectablebecausetheydonotlikethevaluationsitproduces;

• hasnotbeenendorsedbyarecognisedindependentexpert;• doesnotappearinthecorpusofrecognisedscientificresearchjournalsthatare

subjecttorigorouspeer-review;105and 105 Admittedly,Hostyetalia(2007)waslaterpublishedintheBritishActuarialJournal,butitisnotclearwhetherBAJarticles(orforthatmatter,anyarticlesandreportspublishedbytheIFoA,e.g.,suchastheTunaru report) are subject to “rigorouspeer review”and theonly thing that is clearabout the reviewprocess,whateverthatmightbe,isthatitisunclear.ToquotefromtheBAJ’swebsite,“BritishActuarialJournalcontains thesessional researchprogrammeof the InstituteandFacultyofActuariesalongwithtranscriptsofthediscussionsanddebates.ItalsocontainsPresidentialaddresses;memoirsandpapersofinteresttopractitioners.”Nothinghereaboutrigorousscientificpeerreview.

146

• is contradicted by alternative approaches such as Black ’76 that are used andtaughtallovertheworldandhavebeenpublishedintoptieracademicjournals,albeitthattheirapplicationsaresometimesstillcontroversial.

There are two root problemswith theDP approach. The first is that it is based on aforecast, in this case a forecast of futurehouseprice growth.However, it is a seriousmistaketobaseanoptionvaluationonaforecast,becausebasicoptionpricingtheory(e.g.,Black-ScholesorBlack’76)tellsusthatthevalueofanoptionisnotdependentonanyforecast,letalonesomeguessvaluepulledoutofthinair.Instead,theoptionshouldbepricedusingcurrentvariablesonly–admittedlysubjecttosomejudgementsaboutcalibrationbutexpertsknowhowtohandlethesecalibrationissues.Itfollowsthatanyapproachthatdoesdependona forecastmustbe invalidandwhena firmsaysthat itbasesitsNNEGvaluationsonaforecast–anyforecast–thenweknowthatthefirmmustbegettingitwrong.Thesecondrootproblemisthat it iswrongonprincipletoreplacetheforwardhousepriceintheputpricingequationwiththeexpectedfuturehouseprice(orequivalently,toreplacetheforwardratewiththeexpectedfuturehousepriceinflationrate).Thiserrorcan produce results that are known to be impossible. As Table 22.2 shows, the DPapproachcanproduceresultsthatviolatetwodifferentsetsofimpossibilitybounds:thePrincipleIIboundsandPrincipleIIIboundsexaminedinChapter19above):Table22.2:BaselineERMandNNEGValuations:DiscountedProjectionValuations

vsBoundsApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Discountedprojection £4.37 £70.47 𝑵𝑵𝑬𝑮lowerbound 𝑬𝑹𝑴upperbound

PRAPrincipleIIbounds £28.09 £46.75PRAPrincipleIIIbounds £13.15 £61.69

Notes:𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguarantee,and𝐸𝑅𝑀isthepresentvalueoftheEquityReleaseMortgage.Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%, 𝑞=-2.75% for the DP approach and 𝑞=4.2% for the bounds, and 𝜎=14.8%. ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

TheDPvaluationsviolateallthesebounds.TheDPNNEGvaluationsfallbelowthelowerbounds,andtheDPERMvaluationsexceedtheupperbounds.Sowhatdotheseviolationsactuallyimply?Well,theboundsarespecifiedintermsoftheforwardand loanvaluedecrements𝐹*and𝐿* .Assumingthesetobecorrect(andwhyshouldn’t we?) then if ERM decrements 𝐸𝑅𝑀* violate their bounds, then the ERMvaluationwillbeimpossiblyhighandtheNNEGvaluationimpossiblylow.SincetheseDPvaluationsareknowntobeimpossible,thennoauditorcansignoffonthembecausefairvalueprinciplesdonotallowimpossiblevalues.Inshort,ifwewantedaonesentenceassessmentofthevalidityoftheDPapproach,allwe need to know is that it produces valuations that violate bounds that cannot beviolated.Itishardtoseehowanyapproachcangetmuchmorewrongthanthat.

147

Thus, thecorrectapproach is to startwith the forwardprice=𝑆𝑒(I4W)* (seeequation(3.9), which then gives us the discounted forward price or deferment price = 𝑆𝑒4W*(equation(3.10)).TheDPapproachincorrectlytreatsthe‘forwardprice’astheprojectionprice=𝑆𝑒fgh×* ,which then gives the ‘discounted forward price’ or ‘discounted projection’ price =𝑆𝑒(fgh4I)* .TheDPisapproachiswrongonprincipleand(asnotedinChapter3)confusesthefuturepriceofspotpossessionwiththecurrentpriceoffuture(=deferred)possession.Itwillgivethewronganswersingeneralexceptinthespecialcasewhereℎ𝑝𝑖 = 𝑟 − 𝑞.

148

ChapterTwenty-Three:TheTunaruReport On 19 February 2019, the Actuarial Research Centre106 published Professor RaduTunaru’smuchawaitedreportonNNEGvaluation.107HisreporthadbeencommissionedbytheAssociationofBritishInsurersandtheIFoAinthehopethatitwouldresolvesomeoftheNNEGcontroversy.TheIFoApressreleaseaccompanyingtheTunarureportreassuredthepublicthatthe“valuations arising from insurers’ current models and bases are sufficient,” as if tosuggestthattheTunarureportputsanyconcernsaboutthesemodelstorest.Tunarudoesnosuchthing.Instead, Tunaru offers an approach that is free of the DP approach’s most glaringweaknesses – the reliance for option pricing on forecasts and on absurdly low andtypicallynegativenetrentalrates–butgeneratesNNEGvaluationsthatareofthesameorderofmagnitudeasthatapproach.Thereforeittoofailsthesnifftest,inthatitproducesNNEG/𝐿ratiosthataretoolowtojustifythespreadbetweenlendingandrisk-freerates.Nonetheless,wecanseewhytheindustrywouldwelcomeit.Fromtheirpointofview,Tunaru offers another way to skin the cat without getting the flak about negativedefermentrates.ToquotefromthetranscriptoftheStapleInnlauncheventon28February2019:

GarethMee(EY):ExcellentnewresearchpublishedtodayonEquityReleaseMortgages.Reallymovestheindustryforwardsintermsofunderstandingandriskmanagement.

Itseemstoushoweverthatthecelebrationmightbeprematureandwearenottheonlyones to have our doubts. Similar concerns have been expressed by Tony Jeffery andAndrewSmithintheirequityreleasereporttotheSocietyofActuariesinIrelandon28March2019,andbyanumberofactuarieswhohavecorrespondedwithusprivately.ExcessiveParameterisation:theARMA-EGARCHModelProfessorTunaruopenshisreportwithanoft-citedquotefromGeorgeBox:“Allmodelsarewrongbutsomeareuseful.”108Heusesthisquoteasawaytointroducehispreferredmodel,theARMA-EGARCHmodel–ortogiveititsfullname,theAutoregressiveMovingAverageExponentialGeneralisedAutoregressiveConditionalHeteroskedasticmodel.109

106 Disclosure: one of us (Dowd) is currently working with the ARC on its project on the modelling,measurementandmanagementoflongevityandmorbidityrisk.107 R.S.TunaruandE.Quaye“UKEquityReleaseMortgages:areviewoftheNoNegativeEquityGuarantee.”ActuarialResearchCouncilandInstituteandFacultyofActuaries. 108G.E.P.Box(1976)“ScienceandStatistics,”JournaloftheAmericanStatisticalAssociation71(356):791-799.109 See, e.g., R. F. Engle (1982) “Autoregressive Conditional Heteroscedasticity with Estimates of theVarianceofUnitedKingdomInflation,”Econometrica50(4):987–1007;T.Bollerslev(1986)“Generalized

149

Thismodelhasbeenaroundforalongtimehoweverandiswellknowntotime-serieseconometricians and applied economists, who use it principally to model short-termvolatilitydynamicsinassetmarkets.Itisnotobvioushoweverwhyonewouldwanttoapplyamodelofshort-termvolatilitydynamicstoalong-termproblemsuchasNNEGvaluation.Justas‘is’doesnotimply‘ought’,‘can’doesnotimply‘should’.110TheARMA-GARCHmodeladdsneedlesscomplexitytoasetofproblemsthatthatarealreadymuchmisunderstood.The ARMA-EGARCH has a lotmore parameters than Black ’76, and on the subject ofchoosingbetweendifferentmodelsProfessorBoxwentontostate

Sinceallmodelsarewrongthescientistmustbealerttowhatisimportantlywrong.Itisinappropriatetobeconcernedaboutmicewhentherearetigersabroad.(Box,1976,p.792)

Thekeyphraseis“importantlywrong”,asopposed,e.g.,totriviallywrong.Hispointwasthenwhenchoosingbetweenmodels,itisadvisabletogowiththemoreparsimoniousoneunlessthereisgoodreasontothecontrary.ToquoteBoxagain(1976,p.792):

Since all models are wrong the scientist cannot obtain a "correct" one byexcessiveelaboration.OnthecontraryfollowingWilliamofOccamheshouldseekaneconomicaldescriptionofnaturalphenomena.Justastheabilitytodevisesimplebutevocativemodelsisthesignatureofthegreatscientistsooverelaborationandoverparameterizationis

nottoberecommended.Sowhenconsideringaddingnewparameterstothemodel,oneshouldestablishaclearpurposeindoingso.Ataminimum,onewouldhavetoshowthattheexistingsimplemodelwasinadequateinsomerespectandthatfixingthisinadequacyrequiresaddingatleastonemoreparameter.ButTunaruoffersnosuchdemonstration.Let’strytoreconstructanargumentthatmightunderliehisthinking.ApointthatTunarustresses repeatedly is that property prices are autocorrelated, sowe get at least oneadditionalparameterfromthat.InourearlieranalysisthisparametertooktheformoftheHurstExponent,butitcouldequallytaketheformofanautocorrelationcoefficient,aparametermorefamiliartoeconomists.Weprefertoworkwiththeformerbecauseitleads more naturally to the solution to the problems posed by autocorrelation (seeChapter10above),butinprincipleonecouldworkwitheither.Evenso,itstilldoesnotfollowthattheBlack’76modelneedstobereplacedbysomealternativemodelthatcanhandleautocorrelation,suchastheARMA-EGARCHmodel.Why?BecausewecanstilluseBlack’76inanautocorrelatedcontext,i.e.,becauseBlack’76canhandleautocorrelationtoo. As we have stated before (see Chapter 10), we can handle an autocorrelatedunderlying inBlack ’76 provided thatwe adjust the volatility calibration fed into themodel.Soautocorrelationdoesexistanddoeshaveanimpact,butdoesnotrequirethat

AutoregressiveConditionalHeteroskedasticity,”JournalofEconometrics31(3):307–327;andD.B.Nelson(1991)“ConditionalHeteroskedasticityinAssetReturns:ANewApproach,”Econometrica59(2):347–370.110ThismodelisnotnewtotheNNEGliterature.ItsfirstappearanceinanacademicjournalisJ.S.-H.Li,M.R.HardyandK.S.Tan(2010)“OnPricingandHedgingtheNo-NegativeEquityGuaranteeinEquityReleaseMechanisms,”JournalofRiskandInsurance77(2):499–522.

150

wereplaceBlack’76withsomeothermodelwithmoreparametersthanBlack‘76.TheBox/Occam’s Razor principle then kicks in andwe are led to the conclusion thatweshould go with Black ’76 combined with our proposed autocorrelation ‘fix’. Putdifferently,wedon’tneedtothrowtheBSoutwiththebathwater;wejustneedtogetthevolatilitycalibrationright.TheARMA-EGARCHmodelfailstheBoxtest.InconsistentonMarketConsistencyProfessor Tunaru repeatedly criticizes the ‘risk-neutral’ pricing methodology thatunderpinsstandardoptionpricingformulas,e.g.,

“Intheabsenceofanunderlyingmarket,liquidandfreeofcounterpartyrisk,itisnotpossibletohaveadirectrisk-neutralapproach.”(p.11)“Unfortunately,Black(1976)modelcannotbeappliedinthecurrentcontextfortheNNEGmarketsincethereisnofutureshousepricecontractcurrentlytradedintheUK.”(p.14)“Inouropinion, the[Black ‘76] formula…simplydoesnotapply forhousepricesandthisisunrelatedtotheGaussiandistributionassumptionbehindtheGBMmodel.Theforwardcontractonahousepricecannotbecalculatedas in the [Black ‘76] simply imitating the no-arbitrage formula for a stockpayingdividend,wherethedividendyieldisreplacedbythenetrentalrate.Thatformulacannotworkbecausecurrentlywecannotshortsellthevalueofahouse.Hence,theno-arbitrageprincipledoesnotapplyheretolockintheforwardpriceasinthecaseofcorporatestock.”(p.14)

WehavedealtwiththeseandothercriticismsofBlack ’76(or theMCapproachmoregenerally)inChapter22.Asanalternative,hethenproposes(pp.16-17)analternativerisk-neutralpricingrulebasedonanEsschertransform.111There is an inconsistency here. Having repudiated option pricing theorybecause ofincomplete markets, inability to take short positions etc, he then recommends theEsschertransformrisk-neutralpricingrule.Howevertheargumentsfortheexistenceofany risk-neutral pricing rule still hangonmuch the same completemarket / costlessshortingassumptionsherejectselsewhere.Tospellitout,TunarudismissesBlack’76becauseitreliesonthestandard‘risk-neutral’pricingmethodology,whichreliesonabunchof‘unrealistic’assumptionsthathedoesnotlike.HetheninvokeshispreferredEsschertransformpricingmethodology,butthatpricingmethodology relies onmuch the same set of assumptions thathehas already

111 See F. Esscher, (1932) “On theProbability Function in the CollectiveTheory ofRisk,”SkandinaviskAktuarietidskrift15(3):175–195;orH.U.GerberandE.S.W.Shiu“OptionPricingbyEsscherTransforms,”TransactionsoftheSocietyofActuaries46:99–191.

151

rejected. But if those underlying assumptions can be invoked to justify his preferredmodel,thentheycanalsobeinvokedtojustifyBlack’76,andinthatcasehisargumentsagainstBlack ’76 fall away.On theotherhand, if those arguments cannotbe invokedbecause markets are incomplete, then Tunaru can’t invoke those arguments for theARMA-GARCHmodeleither.Inthatcasehehasnopricingmethodology.Thebottomlineisthateitheryoubelieveintheapplicabilityofa ‘risk-neutral’pricingapproachbasedonmarketcompletenessoryoudon’t.Ifyoudo,youcanuseitandifyoudon’t,thenyoucan’tuseit.Butyoucannotrejectonepricingapproachbecauseitreliesonassumptionsyoudon’tlikeandthenturnaroundandproposeanalternativepricingapproachthatreliesonthesameassumptionsthatyouhavejustrejected.InsufficientVolatilityTherearealsoissuesaboutTunaru’scalibrationandlet’sstartwiththevolatility.TunarubeginsbysettingoutsomeMaximumLikelihood(MLE),GeneralisedMethodofMoments(GMM)andMethodofMoments(MM)volatilities.ThesehereportsinhisTable1:

Notes:Tunaru(2019,Table1).

We would describe these estimates as referring to annualized spot volatilities, asopposed,e.g.,toanyforwardvolatilities.Onthebasisoftheseestimates,heproposesabaselinevolatilityof3.9%.He then presents a set of comparable results for alternative sample periods and UKregionsbasedontheNationwidequarterlyhousepriceindex,overafullsampleperiodof1971to2018.Theseresultsindicatethat“arangeofvaluesbetween3.85%to6.5%seemsrepresentativeforGBMvolatilityparameter”(p.1,referringtohisTable2).Oneisimmediatelystruckbythefactthathisbaselinevolatilityisjustnexttotheminimumofthe volatility range: one usually puts baseline parameters somewhere in themiddle,becauseotherwisetherangebecomesredundant.Itisthenstrange,inourview,thathedoesnotrevisehisbaselinevolatilityestimateupwardsinlightoftheresultsinhisTable2.Andthat,essentially,isit,asfarashisvolatilityanalysisisconcerned.He goes on to argue against de-smoothing for house prices. De-smoothing is analternative approach to our Hurst exponent analysis for dealing with autocorrelated

152

houseprices.112Curiously,however,heacceptsthatdesmoothingcanbeusefulforCRE.He then explains, “This point [why one should not de-smooth for house prices] isimportantsinceintheNNEGliterature10%volatilityistakenasindicativefortheUK.Based on the results in Table 2 we can see that a value of 10% is already a veryconservative stressed upwards estimate.” (p. 21) Elsewhere (p. 1) he elaborates bystatingthat“10%or13%isthenmoreofastressedscenariovalue.”Inshort,hetakesthespotvolatilityof3.9%andrecommendsthatthisvolatilityestimatebeusedasthesinglevolatilityinputforallNNEGputs,andheregardsafigureof10%oroverasmoreofastressedvalue.ImplementingthislowvolatilityintohisNNEGmodelthenhelpsproducethe(very)lowNNEGvaluationsthathereports.Oneerrorinthistreatmentisthatitconfusesthevolatilityofindividualhouseswiththevolatilityofanindex.AswehaveseeninChapters9and10(andseeespeciallyFigure9.2reproducedbelow),thevolatilityofhousepricesismuchgreaterthanthevolatilityoftheindex:

Figure9.2:Indexedvs.AchievedHousePrices

Source:SAMS.

SoitseemstousethatTunarushouldhavepickedahigherspotvolthanthe3.9%thatheselectedbasedonhisTable2resultsandthenmadeafurtheradjustmenttoallowforvolatilityaroundtheindex.Butevenwithoutdoinganythingespeciallyambitious,hecouldhavefollowedthePRAinitsanalysisunderpinningsection2.16inCP13/18andusedsomesquarerootruleor,better, someHurstExponent extrapolation rule, to get fromhis spotvolatility (whichwouldhavebeen>3.9%)tosomehighervolatility.Inthiscontext,itisalsointerestingthatthePRAreportsthatthevolatilitiesprovidedtoitbyfirmswere“generallyintherange10%-15%,”sohisbaselinevolatilityiswayoutof line with those estimates too, and what makes that all the more odd is that he 112Formoreonde-smoothing,see,e.g.,P.BoothandG.Marcato(2004)“TheMeasurementandModellingofCommercialRealEstatePerformance,”BritishActuarialJournal10(1):5–61.

153

acknowledgesthathisbaselinecalibrationsarealso“selectedbasedondiscussionswithexpertsworkingonERMsandusingpublicavailabletables fromLegal&General, JustGroupandEquityReleaseCouncil”(p.27).Ourpointisthedisconnectbetweenthefirmsreporting 10% to 15% to the PRA and Tunaru going for 3.9% based on advice frompresumablymuchthesamesetofexperts.TheRentalYieldandtheTunaruMultiplierAfinal issueisTunaru’sunusualwayofestimatingrentalyields.Tunarustarts(p.31)withanestimateofthemeangrossrentalyieldofjustunder5.2%.Hethennotesthatlessthan20%ofpropertiesarerentedoutandconcludes:

Thismeansthataroughcalculationwouldgiveatotalrentalyield,weightedby the 20% representing the actual rentingmarket, of 1.03% (5.1776%×20%)perannum.(Tunaru,2019,p.32)

This conclusion does not follow, however. As Andrew Smith observes in privatecorrespondence:

Th[is]rentalyieldanalysisjustseemswrongtome-arguing,asfarasIcantell,thatonly20%ofthemarketisrented,sotherentalyieldonthemarketasawholeisafifthoftheyieldonrentedproperties.Thatimpliesthatowner-occupiersplacenovalueontheirrighttooccupytheirownproperty.Ican'tseeanyperspectivefromwhichthismakessense.(Ouremphasis)

ItisalsoapparentfromthediscussionatStapleInnthattherewaslittlesupportfortheTunarumultiplierthereeither.Tobefair,Tunaruacknowledgesthepoint.Inafootnotehewrites:

Ithasbeendebatedwithotheracademicsandmarketpractitionerswhoarenotentirelyconvincedabouttheweightingbeingapplied.(Tunaru,2019,p.32,note16)

Wearenotentirelyconvincedeither.Oneofthesewashisowncolleague,DanAlai,whoraisedtheissueataKentseminaron28January2019:

I was wondering why you multiply the rental yield by the proportion ofpropertiesthatarerentedout.Inotherwords,whyis5.1776%dividedby5.Ijustdonotseehowitisrelevantwhetherotherpropertiesarebeingrentedoutornotindeterminingtheappropriaterentalyieldforacertainproperty.(QuotedinTunaru,2019,p.74)

Dr.Alaiiscorrect.Theeconomicrental–thatis,theusevalue,thevalueofthe‘roofoverone’s head,’ etc. – is still enjoyed by someone (or potentially enjoyable even if thepropertyisvoid)regardlessofwhetherthepropertyisrentedoutornot.Whatmattersisthattheeconomicrentalisvaluable,notwhetherthepropertyisactuallyrentedout.Evenifthereisanowner-occupier,thepropertystillhasusevalueandthebestestimatefor

154

themarketvalueofthatusevaluecomesfromtherentsforsimilarpropertiescurrentlyprevailinginthepropertyrentalmarket,notthoserentsmultipliedby20%.113Rememberthatwhatwearetryingtodohereisusetherentalyieldonrentedpropertiesto come up with an approximate calibration for the implied rental yield on ERMedproperties.Theproportionofpropertiesthatarerentedout is irrelevant. Imagine, forexample,thatIhavejustERMedmyhouse,buttheidenticalhouse(plusgardenetc)nextdoorhasjustbeenrentedoutatagivenrentalrate.ThenIcanestimatethevalueoftherental services onmy house from the actual rental on the house next door, andmyestimateof thevalueof thoserentalserviceswillbe100%of thevalueofnextdoor’srental.Itisassimpleasthat.In any case, the claim, indeed the whole report, completely fails to engage with therationalegivenbyCP13/18(para3.16,p.19),thattheonlydifferencebetweenacontractforimmediatepossessionandonefordeferredpossession,isthevalueofforegonerights(e.g.torentalincomeoruseoftheproperty)duringthedefermentperiod.Youwillpayless for deferred possession because you will lose the income that you could get byrentingthepropertyoutor,alternatively,youwilllosetheusebenefitthatyoucouldgetbylivingintheproperty.Whywouldyourobyourselfbypretendingthatyouhaveonlylost20%ofthatincomeoruse?Youhavelost100%ofwhatyouhavelost.Ifthisargumentisnotclearthenconsiderthefollowingreductioargument:byTunaru’slogic,iftheproportionofrentedpropertiesweretofalltosay0.001%,thentherentalyieldtobeusedinERMcalculationswouldbe5.1776%×0.001%=0.005%,effectively0%.Arentalyieldofaboutzerocannotbecorrect.Why?Becausetherentalyieldonmyhousecannotbevirtually0%oftherentalyieldontheidenticalhouserentedoutnextdoor.Whywouldyoupaythesameforapropertythatyoucouldnottakepossessionofforanother20years,whenyoucouldbuyasimilarpropertynowandhavetheuseofitforthesame20years?AndiftheTunaruargumentpushedtoitslimitsgivesan(evenmore)obviouslyincorrectresult,thentheargumentitselfmustbefaulty.Tunaruthensubtractsabout36%ofgrossrentaltogetthenetrental–wehavenostrongargumentwiththatcalculation–andarrivesatanetrentalrateof1.06%×64%=0.66%.Ourbestestimatewas4.2%.

113Othershavesaidmuchthesamething.ToquoteDavidRule’s‘DearCEO’letterof3April2019:“Oneofthe most financially significant parameters is the deferment rate, which the research estimates byconsidering rental yields. Several commentators have already challenged the research’s judgement tomultiplytherentalyieldbyafactor(currently20%)representingtheproportionofpropertiesrentedout–toitscredit,theresearchhighlightsachallengetothisjudgementmadebyanacademicreviewer.ThePRA’sownviewisthatthechallengesarewell-founded,thejustificationforthe20%factorisnotpersuasive,andthatitisnecessarytoconsiderthebenefitsofowner-occupationonpropertiesthatarenotrentedout(suchasthoseonwhichERMsarewritten).”(Ouremphasis)https://www.bankofengland.co.uk/-/media/boe/files/prudential-regulation/letter/2019/solvency-ii-equity-release-mortgages-part-2-apr-19.pdf See also Turnbull (2019), who suggests that the Tunurumultiplier “isbest leftbehind” (C.Turnbull, “On theActuarialTreatmentofEquityReleaseMortgages.”https://www.linkedin.com/pulse/actuarial-treatment-equity-release-mortgages-craig-turnbull/ 13 June2019).

155

Summary:TunaruDoesNotWorkTheTunaruapproachproducesNNEGvaluationssimilar to thoseproducedbytheDPapproach,butwithouttherelianceonforecastsorincrediblenetrentalratesthatarewellbelowzeropercent.But ithasmajorweaknesses: (1) Itusesanoverlyparameterizedmodel, when a simpler model such as Black ‘76 would have sufficed. Given theassumptionsTunarumakesandespeciallyaboutautocorrelationinhouseprices,Black’76 would work, provided the volatility calibration was appropriate to theautocorrelation. (2) His arguments against Black ’76 also undermine his ownARMA-GARCH model as well. (3) His recommended volatility is way too low and (4) hisrecommendednetrental/defermentrateisbasedonanelementaryerrorandisonefifthofwhatitshouldbe.StripawaythefirsttwoerrorsandhemayaswellhaveusedBlack’76.StripawaythesecondtwoerrorsandhewouldhaveendedupwithBlack’76basedonourbaselinecalibrations.Quaye, Research Assistant

156

ChapterTwenty-Four:JustGroup’sDefermentRateCalibrationsInthepreviouschapterwestatedthatJustGrouphaduseda4.25%hpiassumptionwhenvaluingitsNNEG.WhenwemadeapresentationonequityreleasetotheLondonSchoolofEconomicson1October2018oneequityreleaseanalyst intheaudiencesuggestedthatwehadmadethenumberup.114Thefactsareeasilyverified,however: the firmreportedusingthisnumber inboth its2016and2017AnnualReports(seepp.163and110respectively)andtherelevanceofthisnumberinthesereportsisalsoclear,becauseitistheIFRSreportsthatshareholderswouldbe interested in. The samenumber also appears in its 2018H1 results (p. 18).Howeverintheir2017SFCR,thefirmreportsanexplicit𝑞rateof0.5%(p.54).Thelatteraccompanies an almost £1 billion hit to their balance sheet that is offset behindtransitionals(Buckner,2018a,b).Page 83 of the firm’s 2017 Solvency and Financial Condition report reconciles thestatutory with the regulatory balance sheet. The almost £1bn figure appears as thechangein‘othervaluationdifferences’,fromend2016toend2017.However,thisalmost£1billionlossdoesnotmakeasignificantimpactoncapitalbecauseitislargelyoffsetbyanincreaseinthePRAtransitionalarrangement,whichisanentryontheassetsideoftheregulatorybalancesheetthatcanbeusedtocreateextraregulatorycapital. It isapuzzlewhythislatteritem(whichismeanttobeslow-movinganddecliningovertime)shouldhave increasedsomuchover justoneyear. Ithasbeenput tous that the firmincreasedthisitemmerelytohidethehittoitscapital,butwefinditdifficulttobelievethatareputablefirmlikeJustwouldhaveresortedtosuchdissimulation,sotheremustbesomeinnocentexplanationthatweareunawareof.Thesituationfor2016isrelativelystraightforward.Toquoteits2016AnnualReport:

Whencalculatingthevalueoftheno-negativeequityguaranteeonthelifetimemortgages,certaineconomicassumptionsarerequiredwithinthevariantoftheBlack-Scholesformula.…In the absence of a reliable long-term forward curve for UK residentialproperty price inflation, the Group has made an assumption about futureresidentialpropertypriceinflation.Thishasbeenderivedbyreferencetothelong-termexpectationoftheUKretailpriceinflation,“RPI”,(consistentwiththeBankofEnglandinflationtarget)plusanallowancefortheexpectationofhousepricegrowthaboveRPI(propertyriskpremium) lessamargin foracombination of risks including property dilapidation and basis risk. Thisresultsinasinglerateoffuturehousepricegrowthof4.25%.(p.163)

Thenatural reading is that theyareusing theBlack76 formula (which takes forwardprices,notspot)usinganhpiof4.25%.ThisreadingsuggeststhattheyaretakingtheBlackforwardrate,whichshouldbeequaltoriskfreeminusthedefermentrate:

114 TheseminarisreportedinDowd,2018b.

157

(24.1)𝑓 = 𝑟 − 𝑞andreplacingit(incorrectly)withtheforecastℎ𝑝𝑖,i.e.,(24.2)ℎ𝑝𝑖 = 𝑟 − 𝑞as per the bad old ‘discounted projection’ approach that no-one should be using.Rearranging,wegetanimplied𝑞:(24.3)𝑞 = 𝑟 − ℎ𝑝𝑖Ifweassumethat𝑟 = 1.5%thenweget(24.4)implied𝑞 = 1.5% − 4.25% = −2.75%.Wecanthencriticisethisimplied𝑞valueasmakingnosense.Thesituationfor2017ismoreinvolved,however.Their2017AnnualReportstates

Thereturnonequityreleaseassetsisadjustedtoallowfortherisksassociatedwiththeseassets–namely,thepotentialshortfallresultingfromtheNo-NegativeEquityGuarantee(“NNEG”).TheGroupcalculatestheshortfallinrespectoftheNNEGusingavariantoftheBlack-Scholesoptionpricingmodel.Inputsrequired(e.g.currenthouseprices,futurehousepricegrowthandhousepricevolatility)arederivedfromavailablemarketdata.(p.51)Intheabsenceofareliablelong-termforwardcurveforUKresidentialpropertyprice inflation, the Group has made an assumption about future residentialpropertypriceinflationbaseduponavailablemarketandindustrydata.Theseassumptionshavebeenderivedwithreferencetothelong-termexpectationoftheUKretailpriceinflation,“RPI”,(consistentwiththeBankofEnglandinflationtarget)plusanallowancefortheexpectationofhousepricegrowthaboveRPI(property risk premium) less a margin for a combination of risks includingpropertydilapidation andbasis risk.An additional allowance ismade for thevolatilityof futurepropertyprices.Thisresults inasinglerateof futurehouseprice growth of 4.25%, with a volatility assumption of 12% per annum.(ibid,p.110)

Thispassage is consistentwith thepreviouslyquotedpassed from their2016AnnualReport.Goingthroughthesamecalculationsasbefore,wethengetthesameimplied𝑞 =1.5% − 4.25% = −2.75%.Sofar,sosame.Thenthefunstarts.Intheir2017SFCR,thefirmclaimedtobeusingadefermentrateof0.5%:

Asat31December2017,theBoardconsiderstheMatchingAdjustmentintheGroup’sbalancesheetinrespectofLTMnotessatisfiestheprinciplesofSS3/17givingrisetoanimpliedpropertyvolatilityof12%andapositivedefermentrateof0.5%onariskneutralbasis.(2017SFCR,p.54)

158

Theissuethenishowtoreconcilethisimplied𝑞 = −2.75%withtheexplicit𝑞 = 0.5%thattheyalsoclaimtobeusing.Thedifferencebetweenthetwoisenormous.Furthermore,theaverage𝑞ratewearediscussinghereisaslow-movingvariable,whichcannotmovemuchfromoneperiodtoanother.Ajumpof325basispointsfrom-2.75%inoneyearto0.5%inthenextyearishighlyimplausible,evenleavingasidethefactthatboth𝑞ratesarewayoutoflinewiththeempiricalevidencesetoutinChapter9.Now for the awkward bit. In its 2018H1 results (p. 18) the firm offers the followingtreatmentofanimpliedHPIvsan‘actual’orexplicitHPI:

Implied HPI=actual HPI – volatility/dilapidation – effect of capitalrequirement–effectofsecuritisation=4.25%-3%-1.5%-1.4%=-1.65%

whichtheyroundto-1.7%.So the firm has gone from an explicit ℎ𝑝𝑖 = 4.25% to an implicit ℎ𝑝𝑖 = −1.7%, adifferenceofcloseto600basispoints!ThederivationoftheimpliedHPIisalsoproblematic.Theadjustmentfor ‘volatility’ isodd, given that the Black formulaalreadyincludes an explicit treatment of volatility,namelythedirectinputforvolatility(whichthefirmtellsusis12%).Ifthefirmisusinga 4.25% hpi input for the forward rate calculation and a volatility input for the putvaluation, then it would be wholly incorrect to include an additional volatility‘adjustment’aswell.Likewiseitisspurioustoincludethecapitalrequirement,becausethecalculationisfortheamountofcapitalavailable,notthecapitalrequired.TheNNEGcalculationisaninputdeterminingtheamountofcapitalavailableonly.Sothereappearstobesomeobviousdoublecounting.The‘effectofsecuritisation’itemisalsostrangeandwehavenoideawhatitisorwhyitisthere.Wenowrearrange(24.5)as(24.6)𝑟 = 𝑞 + ℎ𝑝𝑖andsubstitute𝑞 = 0.5%andℎ𝑝𝑖=-1.7%into(24.6)toobtain(24.7)𝑟 = 0.5% − 1.7% = −1.2%!Soinmakinganexplicitassumptionof𝑞 = 0.5%andgoingfromexplicitℎ𝑝𝑖 = 4.25%toan implicitℎ𝑝𝑖 = −1.7%, the firmisalso implyinganastonishing𝑟 = −1.2%,but thatcan’tberighteither.What seems tohavehappened is this: In2016, the firmusedanexpectedhpi rateof4.25%tomodelitsNNEG,equivalenttousinganimplicitqrateof-2.75%orthereabouts.Aswe have repeatedly stated, this approach ismanifestlywrong, because the q rateshouldbemuchhigher.In2017,thefirmagainusedthe4.25%expectedhpirateof4.25%tomodelitsNNEG,butthistimeit introducedaseriesof(mostlyinappropriate)extraitemsdrivingawedgebetweenthishpirateandanimpliedhpiratethatisonlyconsistent

159

with the firm’s assumed𝑞 rate of 0.5% if 𝑟 = −1.2%, which confirms that the firm’sreconciliationofitsexplicitandimpliedHPIratesmakesnosense.Thederivationoftheimpliedexpectedhpirateandthecorresponding0.5%𝑞rateisthustotallyhalf-baked.Table24.1summarisesthefirm’sNNEGvaluationapproachesfor2016and2017intermsoftheir(a)explicitparameterassumptions,(b)their impliedparametersand(c)theirerrors.

Table24.1KeyParametersofJust’sNNEGValuation:2016vs2017 2016 2017Explicitparam ℎ𝑝𝑖 = 4.25% ℎ𝑝𝑖 = 4.25%

𝑞 = 0.5%Implicitparam 𝑞 = −2.75% ℎ𝑝𝑖 = −1.7%

𝑟 = −1.2%Error 𝑞 ≪ 0 𝑟 = −1.2%

Inshort,thefirmhasahighlyoriginalapproachtoitsNNEGmodellingthatdefeatsoureffortstomakesenseofit.

160

ChapterTwenty-Five:ActuarialStandardsTheIFoAhaveafairamountofmaterialontheirwebsiteaboutactuarialstandardsandregulation.Herearesomequotesfromtheirwebsite(buttheitalicsareours):https://www.actuaries.org.uk/about-us“UnderourRoyalCharterwehaveadutytoputthepublicinterestfirst”https://www.actuaries.org.uk/upholding-standards“Weregulateactuariesinthepublicinterest.”https://www.actuaries.org.uk/about-us/stepping-out-shadows“One thirdof the UK public say they understand our traditional role in navigatingfinancialrisk,butwebelievethatit’sourethicsandprofessionalismthatsetusapart.”Weagree.https://www.actuaries.org.uk/about-us/our-brand“Our vision is for the Institute and Faculty of Actuaries (IFoA) to serve the public byensuringthatwherethereisuncertaintyof futureoutcomes,actuariesaretrustedandsoughtafterfortheirvaluedanalysisandauthority”“Integrity

• Weare:Doingtherightthing fortheorganisation,ourmembers,theprofessionandthepublicinterest

• Bybeing:o Honesto Accountable,ando Professional.”

https://www.actuaries.org.uk/upholding-standards/standards-and-guidance/actuaries-codeTheprinciplesoftheActuaries’Codeinclude,andwequote:

1. Integrity:memberswillacthonestlyandwiththehigheststandardsofintegrity2. Competence and care: members will perform their professional duties

competentlyandwithcare3. Impartiality: members will not allow bias,conflicts of interest, or the undue

influenceofotherstooverridetheirprofessionaljudgement4. Compliance: members will comply with all relevant legal, regulatory and

professionalrequirements,takereasonablestepstoensuretheyarenotplacedinapositionwheretheyareunabletocomply,andwillchallengenon-compliancebyothers

Theconflictsofinterestspagestates:“AsoneofthefivekeyprinciplesoftheActuaries’Code,impartialityisplacedinsharpfocusinthecontextofprofessionalconflictsofinterest,actualorperceived”

161

ToquotetheindustrymanualonNNEGvaluation,Hostyetalia(2007):“ForprovidersattemptingtopricetheNNEGonamarketconsistentbasisthereisinsufficientproductmargininordertoprovideacompetitiveproduct…”(p.30.).Section7.3.3explainsthatunder amarket consistent approach the product would not be profitable, whilst thediscountedprojection(akarealworld)model“hasproducedasignificantlylowercost”andisthereforetobepreferred.Ourpointisthatwecannotruleoutthepossibilitythatsomepeoplemightperceiveaconflictofinteresthere.On the question of honesty, there is also a legal standard namely,what any ordinarypersonwouldreasonablyregardasdishonest.LordLaneCJsetoutthewellknown‘GhoshTest’ortwolimbapproachtotheissueofdishonestyinRvGhosh1982.Thetestis(i)whetheraccordingtotheordinarystandardsofreasonableandhonestpeoplewhatwasdonewasdishonest,and(ii)Ifso,didthedefendantrealisethatwhatwasdonewasbythosestandardsdishonest.JusticeCookeappliedthetestinthewell-knowncaseagainstTomHayes,thetraderwhowas convicted for the manipulation of LIBOR.115 Cooke ruled that the standard fordishonestyisabsolute,andcannotchangebyreferencetomarketstandardsormarketethos, standard practice in an industry or any common understanding amongstemployees.Thereisnoauthorityforthepropositionthatobjectivestandardsofhonestyaretobesetbyamarket.Quitethecontrary:

Thehistoryofthemarketshaveshownthat,fromtimetotime,marketsadoptpatternsof behaviourwhich aredishonestby the standardsof honest andreasonablepeople;insuchcases,themarkethassimplyabandonedordinarystandardsofhonesty.Eachofthemembersofthiscourthasseensuchcasesand the damage caused when a market determines its own standards ofhonesty in this way. Therefore to depart from the view that standards ofhonestyaredeterminedbythestandardsofordinaryreasonableandhonestpeople is not only unsupported by authority, but would undermine themaintenanceofordinarystandardsofhonestyandintegritythatareessentialtotheconductofbusinessandmarkets.

Actuarial standards are summarised in the following fourdocuments, fromwhichwereproducekeypassages.ThefirstisAPSX1:

115https://www.judiciary.gov.uk/wp-content/uploads/2015/12/r_-v_tom_alexander_william_hayes_redacted_approved.pdf

162

APSX1:ApplyingStandardstoActuarialWork116“8.2.Membersmustbeabletojustifythestandardsapplied(and/ornotapplied)totheirActuarialWork,ifreasonablycalledupontodoso.9.1.AfailuretocomplywiththisAPSmayresultinafindingofmisconductintermsoftheIFoA’sDisciplinaryScheme.”APSX2:ReviewofActuarialWork117ThesecondisAPSX2,whichappliestothereviewofactuarialwork,andwhichisrelevanttotheIFoA/ABIworkingparty’sworkonNNEGvaluation:“1.3.In considering for the purposes of paragraphs 1.1 and 1.2whether and towhatextentWorkReviewshouldbeappliedtoapieceofwork(includingwhetherandtowhatextentWorkReviewshouldbeintheformofIndependentPeerReview),Membersshouldhaveregardtoalloftherelevantcircumstances,includingthefollowing:1.3.1.thedegreeofdifficultyofthepieceofworkanditscomplexity1.3.2.thesignificanceofthepieceofwork,includinganyfinancial,reputationalorotherconsequencesfortheperson(s)forwhomtheworkisproduced1.3.3.whether the circumstancesof thepieceofworkmake itmore likely that errorscouldbemade1.3.4.thereasonableexpectationsoftheperson(s)forwhomtheworkisproduced;1.3.5.theextenttowhichjudgementand/oranalysisisrequired1.3.6.theapplicationofotherqualityassurancecontrolstothepieceofwork;”Wehavetriedandtriedtoelicitinformation–hardinformation,asopposedtoboilerplatewaffle–aboutthequalityassuranceprocessesusedbytheIFoA/ABIworkingparty’sonNNEGvaluation,butno-oneinanypositionofresponsibilitywilltakeresponsibility,eveninprivate,letaloneinpublic.Whatwastheindependentscrutinyprocess,whoweretheseniorfiguresintheIFoAorActuarialResearchCouncilwhosignedoff,etc?Soasregardsthequalityassuranceinthiscase,wecan’tworkoutwhatitwasandno-oneintheknowwilltellus.“1.3.7.thedesirabilityofassuringpublicconfidenceinthequalityoftheworkinquestion.”

116https://www.actuaries.org.uk/documents/aps-x1-applying-standards-actuarial-work117https://www.actuaries.org.uk/documents/aps-x2-review-actuarial-work

163

Sothequestionishowfailingtoanswerconcretequestionsaboutthequalityassuranceprocesshelpstoassurepublicconfidenceinthequalityoftheworkinquestion.Technical Actuarial Standard 100: Principles for Technical Actuarial Work,FinancialReportingCouncilDecember2016.118“TechnicalActuarialStandard100:Principles forTechnicalActuarialWork (TAS100)promoteshighqualitytechnicalactuarialwork.ItsupportstheReliabilityObjectivethat“usersforwhomactuarialinformationiscreatedshouldbeabletoplaceahighdegreeofrelianceonthat information’srelevance,transparencyofassumptions,completenessandcomprehensibility, including the communication of any uncertainty inherent in theinformation.Howuserscanplace“ahighdegreeofreliance”onvaluationsproducedbyanapproachwhichproducesvaluationsthatareimpossibleandclosetoanorderofmagnitudetoolow?1.Judgementshallbeexercisedinareasonedandjustifiablemanner;materialjudgementsshall be communicated to users so that they are able to make informed decisionsunderstandingthemattersrelevanttotheactuarialinformation.”Whatexactlyis“reasonedandjustifiable”abouttheDPapproach?Andinwhatsensearedecisionsbasedonimpossiblevaluationstobeconsideredinformed?2.Datausedintechnicalactuarialworkshallbeappropriateforthepurposeofthatworksothatuserscanrelyontheresultingactuarialinformation.“appropriate”,“rely”…2.1Datashallberelevantforthepurposeofthetechnicalactuarialwork.”ActuariesareusingassumptionsabouthpitopricetheforwardintheirNNEGvaluationmodels,buthpiisirrelevant.See,e.g.,PRASS3/17(p13,para3.17)whichstates:“Itisimportanttonotethatviewsonfuturepropertygrowthplaynoroleinpreferringonecontract over the other. Investors in both contractswill receive the benefit of futureproperty growth (or suffer any property depreciation) because they will own theproperty at the end of the deferment period. Hence expectations of future propertygrowthareirrelevant…”Soinwhatsenseisanirrelevantvariablerelevant?“3.Assumptionsused,orproposedforuse,intechnicalactuarialworkshallbeappropriateforthepurposeofthatworksothatuserscanrelyontheresultingactuarialinformation.”

118https://www.frc.org.uk/getattachment/b8d05ac7-2953-4248-90ae-685f9bcd95bd/TAS-100-Principles-for-Technical-Actuarial-Work-Dec-2016.pdf

164

Howcanitbeappropriatetouseanincorrectapproachthatdependsonanirrelevantvariable?Andhowareresultsbasedonaninappropriateassumptionaboutanirrelevantvariablereliableforusers?“Technical Actuarial Standard 200: Insurance,” Financial Reporting CouncilDecember2016119ThefourthisTAS200,whichappliestoinsurance.“8.Measures,assumptionsandjudgementsusedtoderiveanyestimatesdescribedas“bestestimate”, “central estimate” or other similar terms shall be neither optimistic norpessimisticandshallnotcontainadjustmentstoreflectadesiredoutcome.”Seetheconflictofinterestdiscussionaboveontheimportanceofprofitabilityconcerns.Finally,someadviceonwhattheIFoAmightdo:

Ifwemakemistakeswewanttoputthingsright.Bymonitoringanyconcernsraised, including any formal complaints, and by taking prompt correctiveactionwherenecessary,weseektolearnfromwherethingshavegonewrongandimprovethestandardofourserviceforfutureusers.120

ThatquotecomesfromtheIFoA’sdocument“Puttingthingsright.”

119 https://www.frc.org.uk/getattachment/c866b1f4-688d-4d0a-9527-64cb8b1e8624/TAS-200-Insurance-Dec-2016.pdf120https://www.actuaries.org.uk/documents/putting-things-right.

165

ChapterTwenty-Six:AccountingStandardsThischaptersetsouttheaccountingbasicsastheyapplytoequityreleasevaluation.FairValueForUK-recognisedinsurancefirms,thevaluationoftheregulatorybalancesheetissetoutintheSolvencyIIregulationsastransposedintoUKlaw.Thisprocessisstrictlyrulesbased. The valuation of the statutory balance sheet, by contrast, is governed byaccountingstandardssuchasIFRS(‘InternationalFinancialReportingStandards’)whichtend to bemore principles-based. Undermodern accounting standards such as IFRS,valuationsmustbebasedontheprincipleof‘fairvalue’.IFRSdefinesa“fairvalue”priceas:

Thepricethatwouldbereceivedtosellanassetorpaidtotransferaliabilityinanorderlytransactionbetweenmarketparticipantsatthemeasurementdate.121

IFRSdoesnotdefine“fair,”buttheassumptionisthatamarketparticipant,i.e.someonewhoisindependent,knowledgeable,ableandwillingtoenterintothetransaction,wouldnot be duped into anunfair transaction. So currentmarket pricesmust deemed fair,becauseamarketparticipantwouldnotbedupedintobuyingatgreaterthanthemarketprice, or be duped into selling at less than themarket price. Consequently fair valueequalsmarketprice,wherethemarketpriceexists.Butwhatisfairvalueifthemarketpricedoesnotexist?TheanswercomesfromtheLevel1/Level2/Level3fairvaluehierarchy.Level1fairvalueisthemarketprice,wherethemarketpriceexists.WherenoLevel1fairvaluesexist,i.e.wheretherearenomarketprices,IFRSusesLevel2fairvalues:thesearethepricesofrelatedinstrumentsthatcanbeusedasproxiesforunobservable values. An example in the equity release context would be the use ofleaseholdandfreeholdmarketpricesasproxiesforthevaluesofthenotional“leasehold”grantedtotheequityreleaseborrowerwhenanERMistakenout. WherenoLevel2pricesareavailable,IFRSusesLevel3ormark-to-modelfairvalues,i.e.,Level3 involves theuseof amodel toobtain fair values.However, themodel and itscalibrations should still reflect “the assumptions that market participants would usewhen pricing the asset or liability, including assumptions about risk.” In the equityreleasecontext,thenaturalexampleisaNNEGmodel.Suchamodel,whichisbydefinitionmark tomodel,would require under Level 3 to be calibrated using assumptions thatmarket participantswouldmake. One such assumptionwould be Principle II that no

121See,e.g.,https://www.iasplus.com/en/standards/ifrs/ifrs13.

166

valueoftheERMcanexceedthevalueofforwardcontract(see(19.1)above);anotherisPrincipleIII, thatthedefermenthousevaluemustbe lessthanthecurrentspothousevalue,reflectingthepoint thatamarketparticipantwouldwantcompensation for theincomeorusethatwaslostthroughdeferment.Whateverlevelisused,theunderlyingprincipleisalwaysthesame.ToquoteFinancialReportingStandard102:

2.2Theobjectiveoffinancialstatementsistoprovideinformationaboutthefinancialposition,performanceandcashflowsofanentitythatisusefulforeconomicdecision-makingbyabroadrangeofuserswhoarenotinapositiontodemandreportstailoredtomeettheirparticularinformationneeds.122

Thisinformationshouldenableuserstotakeaneutralandobjectiveviewofthecompanyandensurethattheyarenotbeingcheated.Qualitative of this information include understandability, substance over form,completeness,comparabilityandtimeliness,aswellas(quotingFRS102):

• Relevance:2.5Theinformationprovidedinfinancialstatementsmustberelevanttothedecision-makingneedsofusers. Informationhasthequalityofrelevancewhenitiscapableofinfluencingtheeconomicdecisionsofusersbyhelpingthemevaluate past, present or future events or confirming, or correcting, their pastevaluations.

• Materiality: 2.6 Information is material—and therefore has relevance—if itsomissionormisstatement,individuallyorcollectively,couldinfluencetheeconomicdecisionsofuserstakenonthebasisofthefinancialstatements.

• Reliability:2.7Theinformationprovidedinfinancialstatementsmustbereliable.Informationisreliablewhenitisfreefrommaterialerrorandbiasandrepresentsfaithfullythatwhichiteitherpurportstorepresentorcouldreasonablybeexpectedtorepresent.Financialstatementsarenotfreefrombias(ienotneutral)if,bytheselection or presentation of information, they are intended to influence themakingofadecisionorjudgementinordertoachieveapredeterminedresultoroutcome.

• Prudence: 2.9 The uncertainties that inevitably surround many events andcircumstancesareacknowledgedbythedisclosureoftheirnatureandextentandby the exercise of prudence in the preparation of the financial statements.Prudenceistheinclusionofadegreeofcautionintheexerciseofthejudgementsneeded inmaking theestimatesrequiredunderconditionsofuncertainty,suchthat assets or income are not overstated and liabilities or expenses are notunderstated. However, the exercise of prudence does not allow the deliberateunderstatementofassetsorincome,orthedeliberateoverstatementofliabilitiesorexpenses.Inshort,prudencedoesnotpermitbias.

122FinancialReportingStandard102TheFinancialReportingStandardapplicableintheUKandRepublicofIreland.”FinancialReportingCouncil,September2015.https://www.frc.org.uk/getattachment/e1d6b167-6cdb-4550-bde3-f94484226fbd/FRS-102-WEB-Ready-2015.pdf

167

TheaccountantisthenhiredbythemanagementofthecompanytodrawupaccountsonthebasisoftheseprinciplesandinaccordancewithIFRSrulesandexistinglaw(e.g.,theCompaniesAct).Theseaccountswouldbeapprovedbythedirectors,whoaredeemedtohavepreparedtheaccounts,andthenpresentedtotheauditorsideallyforsign-off.“Anauditorisanindependentlyqualifiedpersonwhoisappointedtogiveshareholdersanindependent,professionalandinformedopiniononthefinancialstatementspreparedbythedirectors”123andthe

auditor’s objectives are to obtain reasonable assurance aboutwhether thefinancialstatementsasawholearefreefrommaterialmisstatement,whetherdue to fraud or error, and to issue an auditor’s report that includes theauditor’s opinion. Reasonable assurance is a high level of assurance ….Misstatements canarise from fraudorerrorandare consideredmaterial if,individuallyorintheaggregate,theycouldreasonablybeexpectedtoinfluencethe economic decisions of users taken on the basis of these financialstatements.124(Ouremphasis)

ImprovingonFairValue?Thereareanumberofcommonobjectionstofairvalue.Thefirstrelatestotheissueofwhetherpeopleshouldseekto‘improve’onthefairvalue/marketvalueprice.Theshortansweris“no.”Forexample,actuariessometimesclaimthatcurrentmarketvaluesshouldbeignoredbecausetheyarecurrentlytoolowortoohigh,relativetotheactuary’sjudgementofwhatthe“long-term”priceshouldbe.AsDavidWilkieonceputit,"Theactuaryis…sayingthatthemarkethastemporarilygotitwrong,butthat,induecourse,itwillgetitright.”125Sothe actuary is suggesting that themarket price should be replaced by a non-marketvaluationbasedonactuarialjudgementor,ifyouwanttoputitthatway,byanimplied(notevenexplicit!)“actuarialforecast.”Objection#1:Atleastiftheforecastwereexplicitwecouldscrutinisethemethodologyonwhichitisbasedandcometoaninformedviewofitsmerits.However,allwehavetogo on here is a nebulous ‘actuarial judgement’. Basing a critique of market pricevaluationsonan“Iknowit’swrong”gutfeelingiswrongonprinciple.126 Objection#2: It isdoubtful thatareliable forecastexists.Wecanpredicteclipses, thereactionofhydrogenandoxygentoaflame,theaccelerationduetogravityandsoforth, 123 http://www.corplaw.ie/blog/bid/337442/The-Responsibility-Of-Auditors.Deanlinkdoesn’twork. 124 FRC“DescriptionoftheAuditor’sResponsibilitiesfortheAuditoftheFinancialStatementsApplicablefor Audits of Financial Statements for Periods Commencing on or after 17 June 2016.”https://www.frc.org.uk/auditors/audit-assurance/auditor-s-responsibilities-for-the-audit-of-the-fi/description-of-the-auditor%E2%80%99s-responsibilities-for125DavidWilkieonp.549oftheDiscussionofA.C.L.DysonandC.J.Exley,(1995)“PensionFundValuationandInvestment.”BritishActuarialJournal1(5):965-977.126 Itlacksobjectivityandisscientificallyindefensible:“weclaimtoknowtheanswerbutcannottellanyoneelsehow toderive it inadvance”asTimGordoneloquentlyput it. SeeT.Gordon (1999) “ThePriceofActuarialValues.”Paperpresented to the Staple Inn Actuarial Society (16 February).

168

butsciencehasn’tfoundawaytopredictthepathofmarketprices.Theproblemisthatthemarketpriceofanassetitselfinvolvesaforecast,bythemarket,offuturecashflows,sointryingtopredictwherethemarketwillbeinayear’stime,wearetryingtoforecastaforecast.Insteadoftryingtopredicttheresultofthenextelection,itisliketryingtopredictwhattheTimeswillpredictittobe.Goodluckonthat.Also, either themarket price is the best forecast, or it is not. If the former,we can’timproveonit.Ifthelatter,wehavetoforecastwhatthebadforecastwillbeinayear’stime.Butwhichbadforecastdowechooseandhowweselectit?Togototheheartofthematter,wecanbeprettyconfidentthatthemarketpricewillchangeall the time,but theproblem is thatwedon’tknowhow themarketpricewillchangefromoneperiodtothenext.Themarketvaluationmightnotbeverygood,butit’sthebestwehave. Objection#3:Evenifwehadperfectforesight,suchasGodmighthave,wewouldstillhavenoleavetomarkthevalueofanassettoanythingotherthanthecurrentmarketprice.Itmaybethatthemarketisinsomesense‘wrong’.Clearlythemarketpricemustbe‘wrong’mostofthetime,becauseitischangingallthetime.Evenso,ifwemarkanassetonafirm’sbooksathigherthanthemarketonthegroundsthatwehaveperfectforesight, or better judgment than the market, then we are defraudingprospectiveshareholdersofthefirm,becausetheywouldpaymoreforsharesthantheywouldhavepaidhadwemarkedthesharestomarket.Ifwemarkthevalueatlowerthanthe market price, because our flawless judgment values it at less, then we aredefraudingexistingshareholders,becausetheirshareswouldbevaluedatlessthantheywouldhavebeenhadwemarkedthesharestomarket. IfGodwereanaccountant,Hewouldnotvalueanassetdifferentlyfromitsmarketvalue,despitebeingomniscient,forGodisalsoPerfectlyGood,andsowouldnotgetinvolvedinfalseaccounting.Wellclearly,ifevenGodwouldnotsuperimposeHisJudgementoverthatofthemarket,thenthereisn’tmuchofacaseforanyoneelsetosuperimposehisorherjudgementoverthatofthemarketeither. TheLiquidityPremiumFallacyTheargumentisoftenmadethatifafirmholdsanilliquidasset,abond,say,andintendstoholdthatassettomaturity,thenthefirmisentitledtomarkuptheassetto‘capture’theliquiditypremium127.Oneproblemwiththisargumentisthatallweknowinpracticeisthattheassethasaspreadovertherisk-free,butwhetherthatspreadisariskspreadoraliquiditypremium,orsomuchofoneandsomuchoftheother,wedonotknow.Weonlyobservethespread,

127Notethatthisnotionofaliquiditypremiumisquitedifferentfromtheonefamiliartoeconomists.Theeconomicnotionofaliquiditypremiumrefersthepricedifferencebetweentwootherwisesimilarassets,whereoneassetisliquid(i.e.,easytosell)andtheotherisnotliquidormayprovetobedifficulttosellinthefuture.

169

nottheliquiditypremium,andanyclaimwemightmakeaboutthesizeoftheliquiditypremiumismerelyahypothesis.A counterargumentwe have sometimes encountered is that liquidity premiums havebeenreliablyestimatedinapaperbyWebberandChurmpublishedinthe2007Q4Bankof EnglandQuarterly Bulletin.128 This paper presents a chart purporting to show thatmodelledcorporatebondspreadsareabout50%oftheobservablespread,fromwhichtheyinferthattheresidualis‘illiquidityrisk’thatcouldbecapturedbyholdingtheassettomaturity.However,wehavedoneourown reconstructionandwe found that afterusing adifferent leverageparameter, almost all thedifferencebetweenmodelled andobservedspreaddisappears.129Soitappearsthattheyhadmisspecifiedtheequationandthenmisidentifiedtheresidualfromtheirmisspecifiedequationas‘liquidityrisk’.Butforthesakeofargument,let’streattheliquiditypremiumasifWebberandChurmhadcorrectlycaptureditandlet’sevengranttheargumentthatthefirmisjustifiedinmarking up the asset to capture that premium. Therefore, when a prospectiveshareholder comes topurchase the shares,hepays thepremium.However,when thesameindividualbuysthesameassetinthemarket,hewillpayless,becausethemarketpriceoftheassetismarkeddownbythesamepremium.Sincethemarketpriceisthefairvalueoftheasset,heispayingtoomuchwhenhebuystheshareandisthereforebeingdefrauded.An example of this ‘liquidity premium argument’ is to be found in remarksmade byAndrewRendellatthe28FebruaryStapleInneventfortheTunarureport:

…ifyouhaveacorporatebond,istheeconomicworthtotheinsurerthesameasitistoeverybodyelse,arguablyitisn’t,andthereasonforthatbeingthatatypicalmarketparticipantwillputadiscounttothepricethattheywouldbepreparedtopayforit,becausethatcorporatebondhasrisksaroundliquidity,andithasrisksaroundpricevolatilityoverthedurationoftheasset.Theinsurersays,“WellIdon’tcareaboutthat,becauseI’mgoingtoholdmyassettomaturity,andthereforeIdon’tneedthatdiscount,sothecorporatebondisworthmoretomethanitistoatypicalparticipant.”…SothequestionthenishowdoesthatmapthroughtotheERM,inparticularthepropertysideofERMs,so…what’stheeconomicworthofthatproperty?…youarenotgoingtoseeanycashoutofthatassetforsometime,youcan’tsellit,sotomanyinvestorsthatwouldbequiteasignificantdisadvantage,butmaybelesssotoaninsurerthathaslongtermliabilitiesanditcanjustwaitforthatvaluetoemerge.130

Thereare threeproblemshere. [1]Mr.Rendell talksabout theeconomicworthto theinsurer,butwhoistheinsurer?Theinsurerisn’taperson.Theinsurerisacompanywhich 128L.WebberandR.Churm(2007)“DecomposingCorporateBondSpreads.”BankofEnglandQuarterlyBulletin2007(Q4):533-541.129SeeD.Buckner(2019)“FromthePostbag–MertonModelPartII.”TheEumaeusProject(19June). 130https://youtu.be/DdLRqcIvR20?t=3794

170

hasshareholders,soweshouldbeaskingabouttheeconomicworthtotheshareholders.Sowhatistheeconomicvaluetotheshareholdersofanassetwhereyouhaveto“waitforthatvaluetoemerge”asheputsit?Well,theeconomicvaluetoshareholdersisjustthesameasiftheyweren’tshareholdersatall,butwerebuyingintheopenmarket.Ifthemanagement of the insurance company decides that a prospective shareholdermustpaynowforthevaluethatwillemergelater,sothattheycangetawhackinggreatbonus,then they have defrauded the prospective shareholder,whomust paymore than themarketprice.[2]Asfor“waitingforthevaluetoemerge”,themanagement(orexistingshareholders)haven’twaitedatall.Theyhavecrystallised thevaluenowandtherebyrobbedfutureshareholdersoftherewardstheywerewaitingpatientlyfor.Thosewhobelieve in “waiting for the value to emerge,” shouldwait for the value to emerge. [3]Finally,itisthecurrentorprospectiveshareholderorinvestor,notjustthecompany,towhomtheaccountanthasafiduciarydutytoreportafairvalue.Ifwestillhaven’tpersuadedyou,considerthefollowingargument.Webuytherighttopossessionofapropertywhoseestimatedvacantvalueis£1m,i.e.afreeholdencumberedbyaleasehold.Theleaseisfor99years.Basedonavailabledata(e.g.fromtheLandTribunal)thevalueofthedefermentwouldbeabout£30,000.Wesell thesamerighttoyoufor£1mbecauseyouthinkthere isamassive illiquiditypremiumthatcanbemarkedupnow.Wehavethenmadeanimmediatemassiveprofit(£970k)fromthisdeal.Wehavejustearnedanilliquiditypremiumthatwouldhaveotherwisehavetakenus99yearstoearn,and you have paid away the same thing! You will earn nothing from the liquiditypremium,whichhasallgonetous.Whyshouldyoupaythepremiumtosomeoneelse?Youmay argue that it is differentwith an insurance company, because an insurancecompanyisthereforthelongtermandhasthepatiencetowait99years.Maybeso,butwhywouldtheinsurancecompanymarkupnowaliquiditypremiumthatmustbeearnedover99years?Youcanonlyobtaintheliquiditypremiumafter99years,andifyousellthe asset, youonly get the currentmarketpricewhichdoesnot include any liquiditypremium.Butaboveall,whywouldanyonepay£1mwhentheycouldpay£30kinsteadforthesameassetinthemarket?The fallacy is not in asserting that there may be an illiquidity premium that can be‘captured’byholdingtheassettomaturity.Theexistenceandextentofanysuchpremiumisanempiricalmatterand(astheWebber-Churmexampleshows)isdifficulttopindown.Instead,thefallacyliesinthebeliefthataliquiditypremiumtobeearnedinthefutureshouldbemarkedupnow,abovethecurrentlyprevailingmarketpriceoftheasset.Orconsider this finalargument.Suppose thereexistsabond typeasset thatoffersanilliquiditypremium.Solet’ssetupacompanywhereweborrowlongdatedliabilitiesatriskfreeandinvesttheproceedsintheselong-datedilliquidassetswiththeir(certain)illiquidity premium. Persuade shareholders/PRA etc that an illiquidity premium thatexiststhat‘itcan’tbearbitragedout’.Createapileofequitybydiscountingliabilitiesatriskfree+premium,payyourselfalotofdividendsorsellthecompany,andretiretothebeach.

171

Congratulations!Youhavejustarbitragedouttheilliquiditypremium,whichnolongerexists. The‘Buyn’Hold’FallacyArelated(andcommon)argumentisthataninsurercanmarkuphigheryieldingassetsbecausetheinsureris(a)goingtoholdtheassetstomaturity,so(b)willnotbeexposedtochangesinmarketvalue,and(c)willrealisetheadditionalreturn.Thisargumentmightseemplausiblebutquicklyunravelsonexamination:First,howdoweknowthattheinsurerwillholdtheassetstomaturity?Wedon’t.Itmayintendtoholdthemtomaturity,butitcouldbeforcedtoselllater.Whoknows?Second,evenifwegrantthattheassetwillbeheldtomaturity,itis(usually)falsetoclaimthattheholdingentityisnotexposedtochangesinthemarketvalueoftheasset.Iftheassetisabond,thebondmightdefaultbeforematurity.Ifitisanequity,thematurity(oftheannuity)maycoincidewithadownturninthemarket,andifitisanERM,maturitycouldbeintheaftermathofaJapanstylehousingdecline.Theonlyassetsthatarenotsubject to riskof loss (innominal terms)wouldbegilts,butgiltsdon’toffera returnhigherthanrisk-free.Thustheargumentisinternallycontradictory:ifanassetoffersahigherreturnthanrisk-free,thenthatreturnmustberisky,sothereisriskoflossatmaturity;butifthedoesnotofferhigherthanrisk-free,becauseitisrisk-free,thenthereisnohigherreturntorealise.Soyoucanhave(b)oryoucanhave(c),butyoucan’thaveboth(b)and(c).PrudentialRegulationandFairValueAnother class of objections to fair value is that fair value does not apply becauseprudentialregulatorshavedifferentobjectivesorareworkingtodifferentstandards.Asan example, consider the following letter of 19 March 2019 that we wrote to HansHoogervorst,thechairoftheInternationalAccountingStandardsBoards(IASB):“DearMrHoogervorst,Oneofus(Buckner)wrotetoyouon9January2019raisingtheconcernthatsomelifeinsurance firms are valuing embedded guarantees in a way radically different fromacceptedfinancialtheory,anddifferent(inourview)fromthewaytheywouldandshouldbevaluedunderbroadIFRSprinciples,i.e.howamarketparticipantwouldvaluethem.TheIASBresponseon4March2019tothisconcernwasapuzzlingone.Thereplyclaimsthat ‘differing valuations for prudential purposes are largely driven by the differentobjectivesofthetwomeasurementbases’.

172

Prudential regulators focus onmeasures of regulatory capital that absorblosses, whereas accounting standard setters are concernedwith “financialinformationaboutthereportingentitythatisusefultoexistingandpotentialinvestors, lendersandothercreditors inmakingdecisionsaboutprovidingresourcestotheentity”(IFRSConceptualFrameworkOB2).Thosedifferentobjectives and approaches to measurement will sometimes give rise todifferentvaluationsforthesameinstrument.

Wehaveworkedformorethan40yearsbetweenusintheareasofcapitalmeasurementandcapitalmanagementandtheviewthattherearedifferingvaluationsforprudentialpurposes is news to us. To be sure, prudential regulators are concerned with theadequacyofanygivenamountofcapital.Ifoneinsurerhas£10bnoflowqualityassets,another insurer has the same amount of high quality assets, and both have £9bn ofpension liabilities at the same duration, then both have the same amount of capital,namely£1bn.131Buttheadequacyofthatamountofcapitalisadifferentmatter,andtheobjectiveoftheprudentialregulatoristoassesscapitaladequacybymeansofestablishedtechniques such as value at risk, stress testing etc. There is no equivalent of capitaladequacyorcapitalrequirementinIFRS,however.Theissueraisedintheletterof9Januarywasnotabouttheadequacyofcapital,butratherabouttheamountofcapital,andinparticular,abouthowthatamountismeasured.Inthiscasetheobjectivesofregulatorsandofaccountingstandardsettersareidentical,namelytovalueassetsandobligationsasamarketparticipantwouldvalue them,andweareastonishedthattheIASBwouldgoonrecordtoclaimanydifferent.Insurersevenprovideareconciliationofregulatoryandstatutorybalancesheetsintheirfinancialreports,fromwhich it is clear that, while there are minor differences, the purported objective ofmeasurementisthesame,i.e.arm’slengthvaluation,fairvaluemeasurementetc.…TheIASBresponsegoesontosaythataccountingstandardsettersareconcernedwith“financialinformationaboutthereportingentitythatisusefultoexistingandpotentialinvestors,lendersandothercreditorsinmakingdecisionsaboutprovidingresourcestotheentity.”Butcanyoupleaseexplainhowisitusefultomarketparticipantsiffirmsvalueassets andobligations inways that amarketparticipantwouldnot?How thenwouldinvestorsknowwhetherthereportedvalueswerefair/useful/reasonableorevenlegal,ornot?Ifvaluationsarehigherthanthosethatamarketparticipantwouldplaceonthem,thenprospectiveinvestorsarebeingunfairlydisadvantaged;ifvaluationsarelowerthanthosethatamarketparticipantwouldplaceonthem,thenexistinginvestorsarebeingunfairlydisadvantaged.Theonlyfairwaytoavoideitherpartybeingdisadvantagedistomakethesamevaluationsthatamarketparticipantwouldmake.Faithfully,etc”Wearestillawaitingareply.

131Weleaveoutdetailssuchasregulatorycapitalincludingsubordinateddebt.Suchdetailsareirrelevantinthepresentcase.

173

TheFiduciaryPrinciple Wethencomeback to the fiduciaryprinciple.Evenwheremarketpricesdonotexist,accountingprinciplessaythattheaccountantshouldvalueeconomicallysimilarassetsinthesamewayandimplythatvaluationshouldreflectrationalinvestorpreferences.Toquote,e.g.,IFRS13B14a:“Cashflowsanddiscountratesshouldreflecttheassumptionsthatmarketparticipantswouldusewhenpricingtheassetorliability.”Anaccountantorauditor or some other person, who has an obligation of trust towards a lessknowledgeable investor,must value an asset or liability as a rational knowledgeableinvestor(ormarketparticipant,orknowledgeable,willingindependentperson)would.This principle applies regardless of whether the accountant, auditor or whoever hasprivateviewsaboutvaluationthatdifferfromfairvaluevaluations.Italsoappliesevenifheorshehassuperiorknowledgeofthefuture:evenifonehadaperfectcrystalball,oneisnotallowedtouseittoprovidevaluationsthatdifferfromfairvalueones.FalseAccountingThose who report valuations are also required to desist from false accounting, thepracticeofwhichisacriminaloffence.Tobeprecise,theoffenceoffalseaccountingisasubclassoftheoffenceoftheft132andiscreatedbysection17oftheTheftActof1968whichstates:

17.-(1)Whereapersondishonestly,withaviewtogainforhimselforanotherorwithintenttocauselosstoanother,-• destroys, defaces, conceals or falsifies any account or any record or

documentmadeorrequiredforanyaccountingpurpose;or• infurnishinginformationforanypurposeproducesormakesuseofany

account, or any such record or document as aforesaid, which to hisknowledge is or may be misleading, false or deceptive in a materialparticular;

heshall,onconvictiononindictment,beliabletoimprisonmentforatermnotexceedingsevenyears.(2)Forpurposesofthissectionapersonwhomakesorconcursinmakinginanaccountorotherdocumentanentrywhichisormaybemisleading,falseordeceptive in a material particular, or who omits or concurs in omitting amaterial particular from an account or other document, is to be treated asfalsifyingtheaccountordocument.(Ouremphasis)

Weemphasisetwopointsaboutthisdefinition.Thefirstisthattobeguiltyoftheoffenceoffalseaccountingitisnotenoughtoreportinformationthatisormightbemisleading. 132TheoffenceoftheftisdefinedunderSection1oftheTheftActof1968:“Apersonisguiltyoftheftifhedishonestlyappropriatespropertybelongingtoanotherwiththeintentionofpermanentlydeprivingtheotherofit…”Seehttp://www.legislation.gov.uk/ukpga/1968/60/pdfs/ukpga_19680060_en.pdf.

174

One must also do so dishonestly, i.e., knowing that one’s behaviour is dishonest byreasonablestandards,andonemustdosowiththeintentofpersonalgainortodeprivesomeoneelseofwhatislawfullytheirs.Theotherpointisthatthephrase“concealsorfalsifiesany account orany record or documentmade or required forany accountingpurpose…”ishighlyencompassing,andwouldincludepublicaccounts,internalbooksandotherdocumentssuchas,e.g.,spreadsheetsandinternalmemos.Italsopotentiallyincludesknowinglymakingon-the-recordstatementsthatareanythingbutthefulltruth.ConsideralsothefollowingNationalFraudandCybercrimeReportingCentrestatementonfalseaccountingfraud:

False accounting fraud happens when company assets are overstated orliabilities are understated in order to make a business appear financiallystrongerthanitreallyis.False accounting fraud involves an employee or an organisation altering,destroyingordefacinganyaccount;orpresentingaccountsfromanindividualor an organisation so they don’t reflect their true value or the financialactivitiesofthatcompany.…Someexamplesoffalseaccountingfraudinclude:• anemployeemakinginflatedexpensesclaims• acustomeroranemployeefalsifyingaccountsinordertostealmoney• anemployeeusing falseaccounting tocoverup lossesbuiltup through

tradingorfraudulentactivity.…• attheextremeendofthescale,thefraudmaymeanthatacompanyhas

incurredseriousfinanciallossesand/oristradingwhileinsolvent.133

Therearealsorelatedoffencessuchasconspiracy(wheretwoormorepeopleplantoengageinanotheroffence,e.g.,falseaccounting)134andaiding,abetting,counsellingandprocuringthecommissionofanotheroffence,e.g.,falseaccounting.135

133/www.actionfraud.police.uk/a-z-of-fraud/false-accounting-fraud134http://serious-crime-solicitors.co.uk/conspiracy.php135http://www.legislation.gov.uk/ukpga/Vict/24-25/94/section/8

175

ChapterTwenty-Seven:RecommendationsforGoodValuationPractice AnyreasonableapproachtoNNEGandERMvaluationmustuseamodelthat is fit forpurposeandbebasedonreasonable(i.e.,plausibleanddefensible)calibrations.TheserequirementsnarrowthefieldtosomeformofMCapproach.ThealternativestoanMCapproacharetheDPandTunaruapproaches,butneitherofthesemeettherequirementsofbeingfitforpurposeorbeingreasonablycalibrated.WecanimplementanMCapproachusingarehedgingalgorithm,Black’76orthePRA’sPrincipleIIbounds.Thelattertwoaretheeasiesttoimplementanddifferencebetweenthesetwoare:

• Thebounds-basedNNEGvaluationswillbe lowerthantheBlack ’76valuationsandthebounds-basedERMvaluationswillbehigherthanERMvaluationsbasedonBlack‘76.

• TheBlack ’76 valuationswill depend on the volatility calibration, but bounds-basedvaluationscanbeobtainedwithoutusinganyvolatilitycalibration.

WeemphasisehoweverthatanyoftheseMCapproachesisreasonable.Onthecalibration,ourrecommendedcalibrationsare:

• Loantovalueratiobasedon ‘ageminus30’rule.Thiscalibration is justified inChapter4.

• Risk-freerate𝑟=1.5%.ThiscalibrationisjustifiedinChapter5.• ERM loan rate 𝑙 = 5.25% for current conditions. This calibration is justified in

Chapter6.• Defermentrate𝑞 = 4.2%.ThiscalibrationisjustifiedinChapter8.

Thevolatilitycalibrationismoreinvolved.Inprincipleonecoulduseanyofarangeofvolatilities,butthekeyistochoosevolatilitycalibrationsthatareconsistentwithone’sunderlying position on the rehedging frequency, explicitly if one uses a rehedgingapproachtovaluetheNNEG,andimplicitlyifoneusesBlack’76.AssumingoneusesBlack’76,theninprincipleonewouldusetheBlack’76putswithvolatilitiesforeachdecrementdrawnfromthevolatilitytermstructureweidentifiedinChapter10.Toputthevaluationformulasintomathematics,andusingobviousnotation:(27.1)𝑁𝑁𝐸𝐺 = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏* × 𝑁𝑁𝐸𝐺*]* = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏* × 𝑝𝑢𝑡*]* where(27.2)𝑝𝑢𝑡* = 𝑒4I*[𝐾*𝑁(−𝑑*M) − 𝐹*𝑁(−𝑑*5)](27.3)𝑑*5 = [𝑙𝑛(𝐹*/𝐾*) + 𝜎*M𝑡/2]/(𝜎*√𝑡)(27.4)𝑑*M = 𝑑*5 −𝜎*√𝑡

176

where𝜎*isthevolatilityoftheforwardhousepriceatmaturity𝑡.However, sinceweare all used toworkingwith a single volatility rather thana termstructure,itisconvenienttouseasinglevolatilityforallputs.Theuseofasinglevolatilityisacceptable,providedoneusesasinglevolatilitycalibrationthatisappropriatetotheborrower’s age and gender. To obtain such a calibration, one could use the expectedvolatilityobtainedbyweightingeachvolatilitybyitsexitprobordecrementprobability.Theformulafortheexpectedvolatility𝜎º isthen(27.5)𝜎º = ∑ [𝑒𝑥𝑖𝑡𝑝𝑟𝑜𝑏* × 𝜎*]* Table27.1showstheseexpectedvolatilitiesagainstborrowerage:formales21.7%forage55,14.8%forage70(andhenceourearlierbaselinesinglevolatilityrecommendationformalesaged70)andsoon.Theseexpectedvolatilitiesgiveresultsthatareveryclosetotheresultsthatonewouldhaveobtainedhadoneusedthefullvolatilitytermstructure.

Table27.1:ExpectedVolatilitiesforDifferentAges

BorrowerAge ExpectedVolatility(Males)

ExpectedVolatility(Females)

55 21.7% 22.9%60 19.1% 20.0%65 16.8% 17.8%70 14.8% 15.7%75 13.2% 13.9%80 12.0% 12.5%85 11.2% 11.5%90 10.7% 10.8%

Notes:Basedontheassumptions:𝐿𝑇𝑉basedon‘age-30’rule,𝑟=1.5%,𝑙=5.25%and𝑞=4.2%.ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesdeathsratedataspanningyears1971:2017andages55:89.

Table 27.2 shows the resultingNNEG and ERM valuations for the Black ’76 and PRAPrincipleIIvaluationapproachesforaborroweraged70:

Table27.2:ERMandNNEGValuations:MaleAge70ValuationApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Black’76(usingvoltermstructure) £32.3 £42.5Black’76(usingexpectedvol) £32.2 £42.7PRAPrincipleIIbounds £28.1 £46.8

Notes:𝑁𝑁𝐸𝐺isthepresentvalueoftheNNEGguarantee,and𝐸𝑅𝑀isthepresentvalueoftheEquityReleaseMortgage.Basedonthebaselineassumptions:maleaged70,𝐿𝑇𝑉=40%,𝑟=1.5%,𝑙=5.25%and𝑞=4.2%.ExitprobabilitiesarebasedonM5-CBDmodelprojectionsusingEngland&Walesmaledeathsratedataspanningyears1971:2017andages55:89.

Table27.3givesthesamevaluationsforage70aspercentagesoftheinitialloanamount,£40:

177

Table27.3:ERMandNNEGValuationsasPercentagesofLoanAmount:MaleAge70

ValuationApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴Black’76(usingvoltermstructure) 80.8 106.3Black’76(usingexpectedvol) 80.4 106.7PRAPrincipleIIbounds 70.2 116.9

Notes:AsperTable27.2.Table27.4givesthesameresultsasinTable27.3foragesspanning55to90:

Table27.4:ERMandNNEGValuationsasPercentagesofLoanAmount:MaleAge

70ValuationApproach 𝑵𝑵𝑬𝑮 𝑬𝑹𝑴

Age55Black’76(usingvoltermstructure) 235.2 91.2Black’76(usingexpectedvol) 235.9 90.5PRAPrincipleIIbounds 213.3 113.1

Age60Black’76(usingvoltermstructure) 170.8 96.9Black’76(usingexpectedvol) 170.8 96.7PRAPrincipleIIbounds 154.0 113.7

Age65Black’76(usingvoltermstructure) 120.0 102.1Black’76(usingexpectedvol) 119.9 102.1PRAPrincipleIIbounds 106.9 115.3

Age70Black’76(usingvoltermstructure) 80.8 106.3Black’76(usingexpectedvol) 80.4 106.7PRAPrincipleIIbounds 70.2 116.9

Age75Black’76(usingvoltermstructure) 51.8 109.0Black’76(usingexpectedvol) 51.3 109.6PRAPrincipleIIbounds 43.2 117.7

Age80Black’76(usingvoltermstructure) 31.6 110.2Black’76(usingexpectedvol) 31.1 110.7PRAPrincipleIIbounds 24.7 117.1

Age85Black’76(usingvoltermstructure) 18.4 110.0Black’76(usingexpectedvol) 17.9 110.5PRAPrincipleIIbounds 13.0 115.3

Age90Black’76(usingvoltermstructure) 10.3 108.9Black’76(usingexpectedvol) 10.0 109.2PRAPrincipleIIbounds 6.4 112.9

Notes:Asindicated,otherwiseasperTable27.2.Figures27.1and27.2givethecorrespondingplots.

178

Figure27.1:NNEG/LoanRatiosVsBorrowerAge

Notes:Asindicated,otherwiseasperTable27.2.

Figure27.2:ERM/LoanRatiosVsBorrowerAge

Notes:Asindicated,otherwiseasperTable27.2.

Onenoticesthattheratiosof𝑁𝑁𝐸𝐺toloanamountfallsharplywithage.However,whatismostsignificantisthelowratiosof𝐸𝑅𝑀toloanamount.Eventheupperboundsarebelow120%foranyborrowerage.ForBlack’76theyarebelow100%foragesunder65andbarelytouch110%forthepeakprofitabilityagerangewhichis75to85.Wemightaddthatthesenumbersaregrossofoperatingcosts.TheseresultssuggestthatERMloanstoyoungerborrowersareloss-makingandthatERMloanstoolderborrowersarelessprofitablethaniscommonlythought.These results suggest that lenders might do better if they stop lending to youngerborrowersandthequestionthenariseswhetherekingoutlow-returnshort-termERMloans to older borrowers is the best possible use of shareholders’ capital. As far asshareholdersareconcerned,itwouldappearthattheterm‘equityrelease’maybemoreappropriatethananyonehasyetrealised.

179

Chapter Twenty-Eight: Recommendations for Governance andDisclosurePoorvaluationpracticesareendemicintheequityreleasesector,buttheIFoAhasdonenothing to condemn such practices. Indeed, the IFoA itself is on record as havingendorsedanumberoferrorsonNNEGvaluation.Thisishardlyahealthystateofaffairs.TheIFoAneedstoputthissituationright,andpromptly.Werecommendthatitissueacorrectivestatementalongthefollowinglines:

TheIFoAacknowledgesandregretsthatseriouserrorshavebeenmadebyanumber of equity release actuaries in the valuation of No-Negative EquityGuarantees.TheIFoArecognisesthatthereareavarietyofpossibleapproachestoNNEGandERMvaluation,butitaffirmsthemarketconsistencyprinciple:anyvalidapproachmustbemarketconsistent.

TheIFoAregardsapproachesthatarenon-marketconsistentasnotmeetingtechnical actuarial standards. In particular, it regards the so-called ‘RealWorld’ or ‘Discounted Projection’ approach as inherently fallacious andadvisesthatpractitionersshouldrefrainfromusingit.

Theregulator,thePRA,needstotakeactiontoo.WerecommendthatthePRAissueanewPolicyStatementstatingthatequityreleaseposesaprudentialproblemandconfirmingthatitregardsonlyMCapproachesasbeingconsistentwithgoodactuarialpractice.ThePRA should also make clear what it regards good practice to be. Besides marketconsistency,goodpracticeshouldcover issuessuchasacceptablemodellingpractices,plausiblecalibrationandtheimportanceofdisclosureandtransparency.Inaddition,thePRAshouldmakeclearthatgoodpracticevaluationmethodsshouldnotbeswayedbynotionsofprofitability.Iffirmscan’tmaketheprofitstheywant,orcan’tmakeprofitsatall,thentheyarefreetoexittheindustry,butthatshouldn’tbethePRA’sconcern.ThePRA’sconcernshouldbetoensurethatthevaluationsaredoneproperly.Toquoteanauthoritativesourceonthisveryissue:

Nomancanservetwomasters:foreitherhewillhatetheone,andlovetheother;orelsehewillholdtotheone,anddespisetheother.YecannotserveGodandMammon.(Matthew6:24)

RegardingtheproposalsinCP3/19,thePRAshouldmonitorthenetrentalrateratherthanthereal interestrate,and itshoulddisclose itsmethodologysothat it isopentooutsidescrutiny.

180

The PRA would still face the problems associated with firms’ gaming the regulatorysystem,soitneedstotakeactiontominimisefirms’scopeforgaming.Itcoulddosointhefollowingways:

• ThePRAshouldimposeamarketconsistencyrequirementonthedefermentrate,withfirmstoprovideaplausibledefenceoftheirassumptions.Withthecurrentrateofgrossrentalyields(2019around5%)wewouldfindanyrateoflessthan3%difficulttojustify.

• Firmsshouldoffer thePRAaplausiblemarket-baseddefenceof theirvolatilityanalysis and assumptions, demonstrating consistency with observed values ofachievement dispersion, interest rate and deferment rate volatility, andcorrelation.

Weunderstand that thePRAalreadyrequires firms to reportNNEGandERMboundsbasedonitsPrinciplesIIandIII,andwillnotacceptanyvaluationsthatviolateanyofthesebounds.Thirdly,wehavesomeadviceforaccountingandauditpracticebasedonourevidencetotheBrydonReviewonthequalityandeffectivenessofaudit:136

• Auditors shouldbe encouraged to ensure, as far aspossible, that shareholdershavetheinformationtheyneedtodeterminetheadequacyofafirm’scapital.

• Firmsshouldberequiredtoreportallthemajorriskfactorsthatmightimpedethequalityofcapital.Somefirmsalreadydoso,butitisdoubtfulthattheycapturethe whole truth. In our experience firms will often choose to ignore or tounderstatethemostmaterialrisks.

• Auditorsshouldbeencouragedtodeterminewhetheranymaterialrisktocapitalhasbeenomittedfromthefinancialstatements.Thistaskshouldnotbedifficult.Giventhatitiseasytoidentifybookswithexcessorunusualreturns,theauditorshouldbydefaultdeclaresuchbookshighrisk,andcheckthattheriskhasbeencommunicatedclearlyinthestatements.

• Riskshidden in thematuritystructureshouldnotbeconcealed in the financialstatements. Many firms report the structure of debt up to 5 years, with anaggregate of debt longer than 5 years. Yet the sensitivity to debt is (roughly)proportionate to the term! Our research has uncovered some staggering risksensitivitiesconcealedinthisway.

• Material risks should not be concealed in some ‘other’ category. Carillion’saccountsrepresentedanearlypaymentfacilityas‘othercreditors’,meaningitwasnotincorporatedinadebttoearningsratiopresentedtolenders.137Eumaeushasreviewedanotherfirmwhichreports‘othervaluationdifferences’ofnearly£1bn,morethanhalfofitsreportedcapital!138

Asfortheindustry,goodluck,butitisessentialtogetthevaluationsright.

136 https://www.gov.uk/government/publications/the-quality-and-effectiveness-of-audit-independent-review137‘Carillion’,Business,EnergyandIndustrialStrategyandWorkandPensionsCommittees2018,p.43138Seee.g.http://eumaeus.org/wordp/index.php/2018/10/19/past-and-present-tens.