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AnempiricalevaluationofValueatRisk
MasterThesisIndustrialandfinancialmanagementUniversityofGothenburg,SchoolofBusiness,Economics&LawJanuary2009Instructor: ZiaMansouriAuthors: MartinGustafsson1982 CarolineLundberg1984
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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ABSTRACT
In lightoftherecentfinancialcrisis,riskmanagementhasbecomeaverycurrent issue.Oneofthe
most intuitiveandcomprehendibleriskmeasures isValueatRisk(VaR).VaRputsamonetaryvalue
on the risk that arises from holding an asset and is defined as the theworst loss over a target
horizonwithagiven levelofconfidence.Therearecurrentlynumerous techniques forcalculating
VaRonthemarketandnostandardissetonhowitshouldbedone.ThiscanmakethefieldofVaR
hardtooverlookforsomeonenotinitiatedintheworldofeconometrics.
InthispaperweexaminethreebasicwidelyusedapproachesusedtocalculateVaR.Theapproaches
examinedarethenonHistoricalSimulationapproach,theGARCHapproachandtheMovingAverage
approach. This paper has twomain purposes, the first is to test the different approaches and
comparethemtoeachotherintermsofaccuracy.Thesecondistoanalyzetheresultsandseeifany
conclusions can be drawn from the accuracy of the approach with respect to the return
characteristicsoftheunderlyingassets.TheaccuracyofaVaRapproach istestedbythenumberof
VaRbreaks that itproduces i.e. thenumberof times that theobserved asset returnexceeds the
predictedVaR.ThenumberofVaRbreaksisevaluatedwiththeKupiectestwhichdefinesaninterval
of VaR breaks in which the approachmust perform to be accepted. By the term asset return
characteristicswemean the statistical properties of the returns of the assets such as volatility,
kurtosisandskewness.
The study is conducted on three fundamentally different assets, Brent crude oil, OMXs30 and
Swedish threemonths treasurybills.Daily returndatahasbeencollectedstarting from January1st
1987 toSeptember30th2008 forall theassets.Observations spanning from1987 to1995willbe
usedashistoricalinputfortheapproachesandobservationsspanningfrom1996toSeptember30th
2008willbethecomparativeperiodinwhichtheapproachesperformancewillbemeasured.
Theresultsofthestudyshowthatnoneofthethreeapproachesaresuperiortotheothersandthat
more complex approaches do not guarantee more accurate results. Instead it seems that the
characteristicsoftheassetreturnsincombinationwiththedesiredconfidenceleveldeterminehow
well a certain approach performs on a certain asset.We show that the choice of VaR approach
shouldbeevaluatedindividuallydependingontheassetstowhichitistobeapplied.
Keywords: Value at Risk, return characteristics, historical simulation, moving average, GARCH,
normaldistribution,Brentoil,OMXs30,Swedishtreasurybills.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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GLOSSARYOFMAINTERMS
Autocorrelation:Isthecorrelationbetweenthereturnsatdifferentpointsintime.
Backtesting:Theprocessoftestingatradingstrategyonpriortimeperiods.
Fattails:Tailsofprobabilitydistributionsthatarelargerthanthoseofnormaldistribution.
GARCHapproach toVaR:Anapproach for forecastingvolatilities thatassumes thatvolatilitynext
perioddependsonlaggedvolatilitiesandlaggedsquaredreturns.
Historical simulation approach to VaR: An approach that estimates VaR from a profit and loss
distributionsimulatedusinghistoricalreturnsdata.
Kupiectest:IsavaluationmethodtoevaluateVaRresults.
Kurtosis:Measuresthepeakednessofadatasample.Ahighvalueofkurtosismeansthatmoreofthe
datasvariancecomesfromextremedeviations.
MovingaverageapproachtoVaR:Aparametricapproachthatassumesnormaldistributedreturns
anddisregardsthefactofvolatilityclustering.
Normaldistribution:TheGaussianorbellcurveprobabilitydistribution.
Outlier:Anextremerarereturnobservation.
Risk:Theprospectofgainandloss.Riskisusuallyregardedasquantifiable.
Skewness:Tellsuswhetherthedataissymmetricornot.
ValueatRisk(VaR):Themaximumlikelylossoversomeparticularholdingperiodataparticularlevel
ofconfidence.
Volatility:Thevariabilityofaprice,usuallyinterpretedasitsstandarddeviation.
Volatility clustering:High volatility observations seems to be clusteredwith other highvolatility
observationsandthesameistrueforlowvolatilityobservationsthatclusterwithotherlowvolatility
observations.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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LISTOFFIGURES
Figure1.5 Dispositionofthispaper.
Table2.3.2 Relatingstandarddeviationtochosenconfidencelevel.
Figure3.2.3a MLEestimationgraphforBrentOil.
Figure3.2.3b MLEestimationgraphforOMXs30.
Figure3.2.3c MLEestimationgraphforSTB3M.
Figure3.3 Graphshowinganegativerespectivelyapositiveskew.
Figure3.4 ProbabilitydensityfunctionforthePearsontypeVIIdistributionwithkurtosisofinfinity(red);2(blue);and0(black).
Table3.5 Statisticalcharacteristicsoftheassetreturns.Figure3.6a DailyreturnsforBrentoil.
Figure3.6b Histogramshowingthedailyreturnscombinedwithanormaldistributioncurve.
Figure3.7a DailyreturnsforOMXs30.
Figure3.7b Histogramshowingthedailyreturnscombinedwithanormaldistributioncurve.
Figure3.8a DailyreturnsforSTB3M.
Figure3.8b Histogramshowingthedailyreturnscombinedwithanormaldistributioncurve.
Table3.9 Resultsfromautocorrelationtest.
Table3.10 TargetnumberofVaRbreaksandKupiectestintervals.
Table4.1.1a Backtestingresultswithhistoricalsimulationapproach.
Figure4.1.1b VaRforBrentOil,HistoricalSimulation99.9%confidence19962008.
Table4.1.2 Backtestingresultswithmovingaverageapproach.
Table4.1.3 BacktestingresultswithGARCHapproach.
Table5.1 Summaryofbacktestingresults.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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TABLEOFCONTENTS
1.INTRODUCTION...................................................................................................................................7
1.1WHATISVALUEATRISK................................................................................................................7
1.2BACKGROUNDOFVaR...................................................................................................................8
1.3PROBLEMDISCUSSION..................................................................................................................8
1.4PURPOSE......................................................................................................................................11
1.5DISPOSITION................................................................................................................................11
2.THEORY..............................................................................................................................................12
2.1VALUEATRISKVaR....................................................................................................................12
2.2THENEEDFORVaR......................................................................................................................13
2.3VaRAPPROACHES........................................................................................................................14
2.3.1HISTORICALSIMULATIONAPPROACH..................................................................................14
2.3.2MOVINGAVERAGEAPPROACH............................................................................................15
2.3.3GARCHAPPROACH...............................................................................................................17
2.4THEUNDERLYINGASSETS............................................................................................................18
2.4.1BRENTOIL.............................................................................................................................18
2.4.2OMXs30................................................................................................................................18
2.4.3THREEMONTHSWEDISHTREASURYBILLS..........................................................................18
2.5BACKTESTINGWITHKUPIEC........................................................................................................19
3.METHOD............................................................................................................................................20
3.1ANALYTICALAPPROACH..............................................................................................................20
3.1.1QUANTITATIVEAPPROACH..................................................................................................20
3.1.2DEDUCTIVEAPPROACH........................................................................................................20
3.1.3RELIABILITY...........................................................................................................................21
3.1.4VALIDITY...............................................................................................................................21
3.2CALCULATIONOFVaR.................................................................................................................21
3.2.1HISTORICALSIMULATIONAPPROACH..................................................................................22
3.2.2MOVINGAVERAGEAPPROACH............................................................................................23
3.2.3GARCHAPPROACH...............................................................................................................23
3.3SKEWNESS...................................................................................................................................25
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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3.4KURTOSIS.....................................................................................................................................26
3.5THESOURCEOFDATA.................................................................................................................26
3.6BRENTOIL....................................................................................................................................27
3.7OMXs30.......................................................................................................................................29
3.8THREEMONTHSWEDISHTREASURYBILLS.................................................................................30
3.9AUTOCORRELATION....................................................................................................................31
3.10BACKTESTINGWITHKUPIEC......................................................................................................32
4.RESULTS&ANALYSIS.........................................................................................................................33
4.1BACKTESTINGRESULTS................................................................................................................33
4.1.1HISTORICALSIMULATIONAPPROACH..................................................................................33
4.1.2MOVINGAVERAGEAPPROACH............................................................................................36
4.1.3GARCHAPPROACH...............................................................................................................37
4.2EXACTNESS&SIMPLICITY............................................................................................................38
4.3VALIDITY......................................................................................................................................39
5.CONCLUSIONS...................................................................................................................................41
5.1CONCLUSIONS.............................................................................................................................41
5.2FURTHERRESEARCH....................................................................................................................43
REFERENCES..........................................................................................................................................44
LITERARYSOURCES............................................................................................................................44
INTERNETSOURCES...........................................................................................................................45
APPENDIX..............................................................................................................................................46
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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1.INTRODUCTION
Thischaptergivesabackgroundtothesubjecttreated inthisstudyValueatRisk. Itfurtherstates
theproblems associatedwith the calculationofValue atRisk and thepurposeof thispaper.The
chapterendswiththedispositionofthepaper.
1.1WHATISVALUEATRISK
Toanswerthisquestionwecanstartoffbyaskingourselvesthemorefundamentalquestionofwhat
risk is. Ineconomicstherearemanydifferenttypesofrisksuchas legalrisk,marketrisk,company
specificriskandsoon.Buthowdowedefinewhatrisk isandhowdowenumericallymeasure its
size? Infinance,risk isoftenthoughtofasthevolatilityorstandarddeviationofanassetsreturns.
Butallvolatilityreallytellsusishowmuchthereturnsofanassetvariesarounditsmean.Thislabels
bothpositive andnegativepricemovements as risk, althoughmost investorswould consider risk
somethingnegativeandupwardpricemovementsassomethingpositive(Jorion2001).
It isnotavery intuitivemeasureand canbehard tograsp for thosenot initiated in theworldof
finance.Whatthevalueatriskmeasurementdoesisthatitputsanumber,eitheramoneyvalueora
percent value of the worst loss over a target horizon with a given level of confidence (Jorion
2001:22).ValueatriskorVaRaswewilladdressitfromhereonthusconsistsofthreecomponents,a
confidencelevel,atimehorizonandavalue.Thetimecomponenttellsushowfarintothefuturewe
are looking,thefurtherwe lookthe largerthepotential loss.Theconfidence leveldetermineswith
whichcertainty themeasurement ismade,higherconfidence levelsmeanshigherpotential losses.
Finally,thevaluecomponentsimply is themonetaryvalue thatwerisk losingduringtheproposed
timehorizongiventheproposedconfidencelevel.
Forexamplea$1000,oneday,95percentconfidencelevelVaRvalueforastockmeansthatduring
thenextdaywecanbe95percentcertainthatthevalueofourholdingsinthisparticularstockwill
notdecreasebymorethan$1000.
AneventwherethereturnofanassetexceedstheestimatedVaRmeasureiscalledaVaRbreak.The
chosenconfidenceleveloftheVaRapproachdetermineshowmanyVaRbreaksanapproachshould
produceifperformingwell.TheaimofaVaRapproachisthereforenottoproduceasfewVaRbreaks
aspossible.AnaccurateVaRapproachproducesanumberofVaRbreaksascloseaspossibletothe
number of VaR breaks specified by the confidence level. Therefore if a VaR approachwith 95%
confidenceiscalculateditshouldproduceVaRmeasuresthatareexceededbythereturnoftheasset
fivepercentofthetimesitisapplied.ThusifVaRpredictionsaremadefor1000individualdaysthe
estimatedVaRshouldbeexceeded50times.AnydeviationfromtheexpectednumberofVaRbreaks
regardlessifitisnegativeorpositiveisasignofinaccuracyormissspecification.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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1.2BACKGROUNDOFVaR
Riskmanagement isan importantpart forall financial institutionsandalso forcompanies thatare
exposedtorisk.Duringthe lastdecadetherehasbeenarevolution inriskmanagementandVaR is
oneofthemeasuresthathavegottena lotofattention.Holton(2003)writesthattherootsofVaR
howeverdatebacktoasearlyas1922,atwhichtimeTheNewYorkStockExchangeimposedcapital
requirementsonmemberfirms.
ResearchonVaRdidhowevernot takeoffuntil1952when two researchers,MarkowitzandRoy,
almostsimultaneouslybutindependentofeachotherpublishedquitesimilarmethodsofmeasuring
VaR.According toHolton (2003), theywereworkingondevelopingameansofselectingportfolios
thatwouldbeabletooptimizetherewardforagiven levelofrisk.Hefurtherpointsoutthatafter
that ittookanother40yearsuntiltheVaRmeasurementbegantobewidelyusedamong financial
institutionsandcompanies.
AccordingtoFernandez (2003),past financialdisasterssuchastheone in1987andthecrisesthat
followedledtoadecisionbytheBaselCommitteethatallbanksshouldkeepenoughcashtobeable
tocoverpotential losses in their tradingportfoliosovera tendayhorizon,99percentof the time.
TheamountofcashtobekeptwastobecalculatedusingVaR.Pastcriseshasshownthatenormous
amounts of money can be lost over a day as a result of poor supervision and financial risk
management.TheVaRmeasurethereforegotitsbreakthroughbecauseofhistoricalmistakesinrisk
management.
Today,theutilizationofVaRiswidelyspreadinfinancialinstitutions,howeveritisnotaswidespread
in non financial firms. This can be explained by the fact that non financial firms do not usually
forecastprofitsand lossesonadailybasis.NeverthelessMauro(1999)pointsoutthatVaRcanbe,
andisused,innonfinancialfirmsthatareaffectedbyvolatilityinprices,particularlyinashorttime
horizon.
AnadvantageofVaRisthatitisameasurethatcanbeappliedtoalmostanyasset.Adisadvantage
however,isthatthereisalmostaninfinitenumberofwaystocalculateVaR,eachoftheapproaches
with its own advantages and disadvantages. This makes different VaR measurements hard to
comparewitheachotheriftheyhavenotbeencalculatedusingthesameapproach.
1.3PROBLEMDISCUSSION
ValueatRisk isameasurethat iswidespreadwithin financial institutionswherethe importanceof
strictriskmanagementhasbecomevital.AccordingtoJorion (2002)VaRhasbecomethestandard
benchmarkformeasuringfinancialrisk.Bankswithlargetradingportfoliosandinstitutionsthatdeal
withnumeroussourcesoffinancialriskhavebeenheadingtheuseofriskmanagement.Jorion(2001)
writesthattoputanumberonthepossiblemaximumlossunderaspecifictimehorizonhasformany
banksbecomeanobligation.Anexampleof this isLesleyDanielsWebster, formerheadofmarket
riskatChaseManhattanBank,whosawitasanecessity.Everymorninghereceiveda30pagereport
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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that summarized theVaRof thebank. Theneat little reportquantified the riskof all the trading
positionsofthebank.VaRisanimportantmeasurewithinriskmanagementbecauseitisanintuitive
riskmeasure as it puts amonetary number on the risk exposure a company faces.According to
Linsmeier&Pearson (1996) thestrengthsofVaR is that itprovidesasimplerandaccurateoverall
measure of themarket risk the company is taking. This is donewithout going into specific and
complexdetailsaboutthecompanyspositions.Thisisagreatadvantagesinceriskassessmentsare
mostoftenproducedforseniormanagementwithnoorlittleexperiencewitheconometrics.
Theredoesnot,however,appeartobeanyconsensusonhowtocalculateVaR.Todaythereseems
to be asmanyways to calculateVaR as there are practitioners using it. There are some general
approaches tochoose from,suchasparametric,semiparametricornonparametricapproaches.A
parametric approach means that you use a parameter or a model to describe reality, often
generalizing andmaking assumptions.Anonparametricapproachhowevermay forexampleonly
lookathistoricaldatatomakepredictionsaboutthefuture.Undereachofthesecategoriesthereare
manydifferentapproachesthatcanbeusedtocalculateVaRandeachandeveryoneofthemcanin
theirturnbeappliedindifferentwaysundervariousassumptionsorgeneralizations.Inthisstudythe
historicalsimulationapproachwillberepresentingthegroupofnonparametricapproaches.Moving
averageandGARCHwillberepresentingtheparametricapproaches.TheGARCHapproachconsiders
volatility clustering i.e. the fact that days with high volatility are often clustered together. The
approacheshavebeenchosenbecausetheyarethemostwidelyusedapproachesofcalculatingVaR.
Thereareseveralearlierstudiesonthesubject,howevermostofthemaimtodevelopnewandmore
advancedwaysofcalculatingVaR.LinsmeierandPearson (1996)concludes that there isnosimple
answertowhichVaRapproachisthebest.Theyalldifferintheirabilityofcapturingriskandfurther
theydifferintheireaseofimplementationandtheeaseofexplanationtoseniormanagement.They
state that the historical simulation approach is easy to implement but it does require a lot of
historical data. It is also an approach that is easy to explain to seniormanagementwhich is an
essential part of the purpose of theVaRmeasure. In another studymade byGoorbergh&Vlaar
(1999)themostimportantcharacteristicofstockreturnswhenusingVaRisvolatilityclustering.This
isaphenomenomthatcanbeeffectivelymodeledbymeansofGARCH.
Thestudiesmade in thepastarenotunanimous, theyproducecontradictory results.Forexample
Cabedo&Moya (2003) finds thatVaRs fromanautoregressivemovingaverageapproach (ARMA)
outperforms theGARCH.Costelloetal. (2008) finds that theopposite is true,GARCHoutperforms
the ARMA and they suggest that the conclusion drawn by Cabedo & Moya is based on the
assumptionofnormaldistribution.Thedifferentapproachesareappliedondifferentassetsandthe
approachesareoftenadaptedtotheassetstowhichtheyareapplied.Thisindicatethattheresults
fromdifferentstudiescanbehard to interpretandcompare.We therefore think that itwouldbe
interesting to conduct a study by using the threemostwidely used approaches. Theywill all be
appliedonthesamethreeassetstomaketheresultsmorecomparable.
TheVaRwillbe estimatedon adailybasisusing threedifferent confidence levels,95%,99% and
99.9%toseehowtheapproachesperformatdifferentconfidence levelsondifferentassets.Jorion
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
10
(2001)andmanyotherswithhimusestheseconfidence levelswhencalculatingVaR.Thedifferent
levels fitdifferentpurposesanddependson themanagements relation to risk.Choosingahigher
levelofconfidencewill result inahigherVaR.Thedifferencebetweenaconfidence levelof99or
99.9percentmightnotseembigbuthasalargeeffectontheVaRassessment.
Differentcharacteristicsoftheassetsreturns,i.e.thestatisticalpropertiesoftheassetsreturnssuch
asvolatility,kurtosisandskewness,mightaffectthecalculatedVaRmakingsomeapproachesmore
preferablewithspecificassets.This isespecially truewith theparametricapproaches thatassume
thereturnstobenormallydistributed.This leadstotheproblemthatthisstudy isfocusedon,the
connection between an assets return characteristics and its effect on the calculated VaR. Three
different assets have been chosen; Brent crude oil, Swedish stockmarket index (OMXs30) and
Swedishthreemonthtreasurybills(STB3M).Themotivesforchoosingtheseassetsarethattheyare
fundamentallydifferentassetswithfundamentallydifferentcharacteristics.Thereasonwhythisisso
important in this study is that it provides a better prerequisite to examine how the return
characteristicsaffectVaRapproaches.
VaRcanbeappliedonanyassetthathasameasurablereturn.Theassetsusedinthisstudyallhave
incommonthattheirreturnscaneasilybemeasured,theircharacteristicsdohoweverdiffer.Brent
oilhasmanyusesonebeingpetrolproductionandanotherisheatingpurposes.Oilcompaniessitting
on largereservesareverysensitivetochanges inoilpricesmakingVaRasuitableriskmeasurefor
assessingpotentialfuturelosses.Thesamegoesforanyonetradingwithoilregardlessofthembeing
buyersorsellers.Theoilmarketdiffersfromthestockmarketinthesenseofbuyers.Oilistradedin
hugeamountsby largeoilcompanies incontrasttostocksthatcanbetraded insmallquantitiesby
small investors. Stocks are perhaps themost common asset used in VaR calculations. The risks
connected to stock returns are quite easy to understand. Financial institutions invest in huge
portfoliosconsistingofvariousstockswithvaryingrisks.Themarketrisktheyfacemustbeevaluated
and according to Jorion (2002) this isbestdonebyusingVaR.When it comes to treasurybills it
should represent a stable and secure investment instrument. Fluctuating interest rates however
affects themarketvalueof thebills.Thismeans that themarketvalueof thebillscanvaryduring
theirholdingtimewhichmakesVaRausefulmeasureinordertoestimatepotentiallosses.
HowdothedifferentcharacteristicsofthereturnsoftheunderlyingassetsaffectVaR?AnotherimportantaspectisthecomplexityoftheVaRapproachesthemselves.Wewilltakeacloser
lookon theperformancedifferencebetweenmorecomplexandsimplerapproaches tocalculating
VaR. Thiswill result in an evaluation ofwhether the extra effort put in to the use of themore
complexapproachesisworththewhileintermsofaccuracy.Theaccuracyoftheapproacheswillbe
measuredintermsofthenumberofVaRbreakstheyproduce
WhichoftheapproachesgivesthemostaccurateVaRmeasure?
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
11
1.4PURPOSE
Thepurposeof this study is tocompare threedifferentapproaches tocalculatingVaRnamely the
Historical Simulation approach, the Moving Average approach and the GARCH approach. The
approacheswillbeappliedon the threedifferentassetswith theconfidence levels95%,99%and
99.9%.Theperformanceandaccuracyoftheapproacheswillthenbeanalyzedwithrespecttoasset
returncharacteristicsandcomplexityoftheapproach.
1.5DISPOSITION
Introduction. The first chapter presents the subject examined in this
studyandgivesanintroductiontoVaRasameasureinriskmanagement
andhowitcanbeused.
Theory. In the upcoming chapter, the development and functions of
ValueatRiskaredescribedalongwiththechosenmethodsofcalculating
ValueatRisk. It isplacedbefore themethodchapter inorder togivea
better understanding for the approaches usedmaking the calculations
moreunderstandable.
Method.Thethirdchapterdealswiththemethodusedinthestudy.First,
the analytical approach is described and then we move on to the
methodsusedtocalculatetheValueatRisk.Thedataoftheassets,which
Value at Risk are calculated on, is also presented and their return
characteristicsareillustratedanddescribed.
Results&Analysis. In the fourth chapter, the results are presented in
formof theproducedValueatRiskestimations.Back testing ismade in
order to see how the methods have performed. It also contains the
analysiswheretheresultsareanalyzedtoseehowthecharacteristicsof
theassetreturnsaffectthecalculatedValueatRisk
Conclusions. In the fifth chapter, we present the conclusions that we
havedrawnfromthecalculationsanddiscussfurtherresearch.
Figure 1.5 Disposition of thispaper.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
12
2.THEORY
InthischapterwewillpresenttheriskmeasureValueatriskandthedifferentVaRapproachesthat
wewill be using in this thesis alongwith themathematicalmodels that they are based on. The
underlyingassetswillalsobepresented.
2.1VALUEATRISKVaR
Asmentionedearlier thedefinitionofVaRby Jorion (2001:22) is:VaR summarizes theworst loss
overatargethorizonwithagivenlevelofconfidence.Themostcommonapproachistheparametric
underwhichthemathematicsofVaRcanbedescribedbythefunctionbelow.
C $
The daily standard deviation times the number of standard deviations that corresponds to the
selectedconfidencelevel,C,timesthesquarerootofthetimehorizontimesthemonetarysizeofthe
investmentresultsintheVaR.InthisthesiswewillbecalculatingonedayVaRestimatessothetime
component can be excluded. Also, the monetary size of the investment is just there to put a
monetaryfigureontheriskandcanalsobeexcluded.Whatwehaveleftisthestandarddeviationof
historicalreturnsandtheconfidencelevel.
Ifchoosinga lowerconfidence level,as95percentor90percent, theVaRwilldecrease.Different
levelswillfitdifferentfirmsandpurposesandwillbechosenaccordingtothemanagementsrelation
torisk.Themoreriskaversethefirmisthehigherconfidencelevelwillbeselected.HoweverDowd
(1998) claims that VaR users and than in first hand banks will inmost cases prefer the lower
confidencevalueinordertodecreasethecapitalneededtocoverthepotentialfuturelosses.
InordertoprovideanoverviewofthemeasureVaR,asimpleillustrationisgiven.Supposethatthe
oilpricetodayis100USDperbarrelandthatthedailystandarddeviation()is20USD.Companies
thatbuylargeamountsofoilmightwanttoknowhowmuch,givenacertainconfidencelevel,they
canpossiblylosewhenbuyingtheoiltodaycomparedtotomorrow.Ifthechosenconfidenceinterval
is 99 percent, thatwouldmean that one day out of hundred the losswill be greater than the
calculatedVaR.This is truewhen thepricechange isnormaldistributedaround theaverageprice
change.
Valueduetoanincreaseintheoilprice:100 2.33 146.6USD
Valueduetoandecreaseintheoilprice:100 2.33 53.4USD
Thismeansthatwith99percentchancethelosswillnotbegreaterthan10053.4=46.6USDwhich
istheVaRforaconfidencelevelof99percent.
Often some assumption aremade inorder to calculate theVaR and ifwe suppose that thedaily
changesinoilpriceshasadensityfunctionthisfunctionismostoftenassumedtoberepresentedby
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
13
the normal distribution. The assumption has the benefit of making the VaR estimations much
simpler. However it has some disadvantages. The price changes do not always fit the normal
distribution curve and when more observations are found in the tails, normalbased VaR will
understate the losses that canoccur.A solution couldbe touseanotherdistribution that regards
fatter tails.The fatter tails that forexamplecomeswith the tdistributionmakeshigh lossesmore
commonaccordingtoDowd(1998)andthereforeitalsogiveshigherVaR.
When usingVaR, the assets have to be valued at theirmarket value. Thiswill not be a problem
accordingtoPenza&Bansal(2001)withassetstradedinaliquidmarket,wherepriceseasilycanbe
derived.ThisiscalledmarkingtomarketandDowd(1998)definesthetermasthepracticeofvaluing
andfrequentlyrevaluingpositionsinmarketablesecuritiesbymeansoftheircurrentmarketprices.
Financialassetsareoften tradedonadailybasisand therefore, it iseasier toobtain theircurrent
value.Forsomenonfinancialassets,thepricemayhavetobeestimated,whichmakethecalculations
moreuncertain.Theassetsusedinthisstudycausesnoprobleminthisarea,howeveronecanclaim
that the pricing does differ among them. Some claim to be able to predict the future price of a
specificstockbecauseofavailable informationconcerningthestock,whilefuturepricesonoilmay
beharder topredict.Reasons for thiscanbe the strong influenceOPEChason thepriceofoilby
changingtherelationofdemandandsupply.Sadeghi&Shavvalpour(2006)callattentiontotheneed
ofVaRasa tool forquantifyingmarketriskwithinoilmarkets.Futureassetpricesandreasons for
themareacomplexareahoweverasstatedbeforealltheassetsusedinthisstudyaretradeddaily,
makingitnoproblemtoderivetheneededmarketprices.
2.2THENEEDFORVaR
Holton(2002)putsthebreakthroughforVaRinthe1970sand1980s.Duringthistimemanychanges
tookplaceanddiversefinancialcrisesbecameafact.TheBrettonWoodssystemcollapsed in1971
makingtheexchangeratesflow.BecauseoftheoilcrisiscausedbyOPEC,oilpriceswentthroughthe
roofbygoingfromUSD2toUSD35.Alongwiththecrisisalsocamefinancialinnovationssuchasthe
proliferationofleverage.Beforethistime,theavenuesforcompoundingriskwerelimitedaccording
toHolton(2002).Withnewinstrumentsandnewformsoftransactions,newleveragewaysopened
up. Holton also brings up that when this happened, trading organizations sought new ways to
managerisktaking.Duringthistime,banksstartedtocreateameasureasaresultofmorecomplex
riskmanagement.Theriskshadtobeaggregatedsowhenfacingthisproblem,thewantedsolution
wasameasurethatcouldprovidethecompanieswithabetterunderstandingfortheriskexposure.
Itwas JPMorganwhobecame themost successful.Theycreated theRiskMetrics systemwhich is
mentionedby Jorion (2001). It revealed riskmeasures for300 financial instruments and thedata
characterizeavariancecovariancematrixofriskandcorrelationmeasuresthatprogress through
time.Thedevelopmentofthemeasuretookalongtimeandwasfinishedin1990.Fouryearslaterit
wasmadefreetothepublicsomethingthatcreatedbigattention.TheuseofVaRsystemincreased
enormously and apositive effecton the companies riskmanagementwere found. The important
contributionofRiskMetricswasthatitpublicizedVaRtoawideaudience.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
14
AnimportantlandmarkinriskmanagementthatmadeVaRknowntothemasswastheBaselAccord.
Itwasconcludedin1988andfullyimplementedintheG10countries1in1992.Thisismentionedby
Saunders (1999) and itwas a regulation on commercial banks to provide amore secure system
throughminimumcapitalrequirementsforbanksmarketsrisk.
2.3VaRAPPROACHES
Thethreedifferentapproachesused inthisstudywillbepresentedbelow.Theyallhavetheirown
advantages and disadvantageswhich affects the VaR values that they produce. The assumptions
madeby theapproachesdealswith the returncharacteristics indifferentways.Theeffectsof this
willbefurtherdiscussedintheanalysis.
2.3.1HISTORICALSIMULATIONAPPROACH
Anonparametricmethoddoesnotassumethereturnsoftheassetstobedistributedaccordingtoa
specificprobabilitydistribution. It isasimplewaytocalculatetheValueatRiskthat iswidelyused
justbecausethefactthatitignoresmanyproblemsandstillproducesrelativelygoodresults(Dowd
1998). The non parametric historical simulation approach used in the study is the historical
simulationapproach.
Theideaofthehistoricalsimulationapproachistousethehistoricaldistributionsofpricechangesin
order to calculate theVaR.Historicaldata is collected fora chosenperiodand then thehistorical
price changes are assumed to be a good assessment of future price changes. The function for
historicalsimulationcanbedescribedasbelow(Goorbergh&Vlaar1999).
|
The|istheVaRvalueattimet+1where istheinitialvalueoftheassetandisthep:th
percentile of each subsample, which means taking the asset return between time t and
preceding returns and calculating thepercentileof these values that corresponds toour selected
confidencelevel.
Themeritsofthisapproach,accordingtoDowd(1998),areitssimplicitywhichmakesitagreathelp
forriskmanagersandthedataneededshouldbeavailablefrompublicsources.Another important
benefitisthatitdoesnotdependontheassumptionaboutthedistributionofreturns.Whetheritis
thenormaldistributionornotthatgivesthebestestimatesisnotthequestionhere.Eventhoughthe
approach is indifferent towhichdistribution the returnshave is it important to remember that it
doesassumethedistributionofreturnstoremainthesameinthefutureasithasbeeninthepast.
Dowd(1998)statesthatthismakestheapproachlessrestrictivebecauseweneitherhavetoassume
1 Belgium, Canada, France, Germany, Italy, Japan, theNetherlands, Sweden, the United Kingdom and theUnitedStates.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
15
thatthechangesare independentgivingtheapproachnoproblems inaccommodatingthefattails
thattormentnormalapproachestoVaR.
A possible disadvantage, argued by Jorion (2001), is that the approach requires a lot of data to
performwellathigherconfidencelevels.WhenestimatingVaRata99%confidencelevel,intuitively
at least100historicalvalueshave tobe inputted.Buteven then theapproachonlyproducesone
observation in the tail. Perhaps not enough historical data is available to produce a good VaR
estimate,aproblemthatontheotherhandcanoccurformostoftheVaRapproaches.Jorion(2001)
furtherargues that there isa tradeoff in theapproachwhen includingmoreandmorehistorical
values.Ononehand, theapproachbecomesmoreaccurateathigherconfidence levels,butat the
sametime,theriskofincludingoldvaluesthatarenotrelevantforfuturereturnsishigher.
The historical simulation approach also assumes that the past is identical to the future,making
historicalrisksthesameasfuturerisks,whichmightnotbethecaseiftherehavebeenextraordinary
eventsintherecentpast.Perhapsthisismostoftennotthecase,buttheimportanceofgettingdata
thattrulyreflectsthepastbecomescritical.Ifthechosendataperiodistooshort,itmightreflecta
moreor less volatileperiod thatdoesnot give a goodpictureof thehistorical volatility. Itmight
includeperiodsofunusualkindthatisnotrepresentative.
Goorbergh&Vlaar (1999)arguesthat it isnotgenerallypossibletomakehistoricalsimulationVaR
predictionsonwindowsizessmallerthanthereciprocaloftheselectedconfidencelevel.Thismeans
that forevaluating a99.9% confidence youwouldneed awindow sizeof at least
. 1000
observations.
The conclusions that can be drawn is that the approach does have both advantages and
disadvantages which might make it important to complement it with other tests that pick up
plausiblerisknotrepresentedinthehistoricaldata.
2.3.2MOVINGAVERAGEAPPROACH
Aparametricapproachassumes that theassets return followaprobabilitydistribution.Themore
naveapproach,herecalled themovingaverageapproach, isaparametricapproach thatassumes
thatthereturnsofassetsarenormallydistributed.Theapproachmeasuresthestandarddeviationof
pastreturnsandusesthistogetherwiththestandardnormaldistributiontodescribetheprobability
ofdifferentoutcomesinthefuture.Howeverthisapproachasmentionedabovedisregardsthewell
establishedphenomenonof volatility clustering.Dowd (1998) sees this as abigproblem because
there is a tendency for high volatility observations to be clustered with other highvolatility
observationsandthesameistrueforlowvolatilityobservationsthatclusterwithotherlowvolatility
observations.Thisphenomenahasbeenconfirmedbymanystudiessincethefirstobservationmade
byMandelbrot(1963).
Under thisapproach, the sample standarddeviation is firstcalculated fromanumberofhistorical
observationsusingthisfunction.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
16
1
1
Thenumberofobservationsisrepresentedbyn,standsforthedailylogpricefordayiand isthe
averagedailylogprice.
Calculating theactualVaRvalue isthendonebyretrievingthenumberofstandarddeviationsthat
corresponds to the selected confidence level from the normal cumulative distribution function
presentedbelow,andmultiplyingitwiththestandarddeviationofthehistoricalobservations(Jorion
2001).
Thefunctionbelowisthecumulativedistributionfunction(cdf)ofthestandardnormaldistribution.
This function calculates the probability that a random number drawn from the standard normal
distribution is less than x.The inverseof this functiongives the valueof x that is the limitunder
which a random value drawn from the standard normal distribution will fall with the inputted
probability(Lee,Lee&Lee2000).Erfstandsfortheerrorfunction.
12 1
2
Basedon this the confidence levels thatwehave chosen corresponds to the followingnumberof
standarddeviations(Penza&Banzal2001).
Confidencelevel Standarddeviations95,00% 1,64599,00% 2,32699,90% 3,090
Table2.3.2Relatingstandarddeviationtochosenconfidencelevel.
Animportantpartofthisapproachischoosingthenumberofdaystobasethevolatilityestimateon.
Ifthevolatilityoftheunderlyingassetisestimatedfromtoomanyhistoricalobservations,thereisa
riskof includingobservations thatare toooldand irrelevant.On theotherhand,ahistoricalvalue
based on too few valesmeans a risk of getting values that are too heavily influenced by recent
events. As a rule of thumb,Hull (2008)mentions that you should base your historical volatility
estimateonequallymanydaysasyouaretryingtoforecast intothefuture,butno lessthan30.In
thisstudy,thenumberofdayswouldthereforebe30.
Thedistributionofthereturnscanbedifferentforvaryingassets.Themostwidelyuseddistribution
isthenormaldistributionbecauseofitssimplicity,however,itdoesseldomexactlyfitthereturnsof
an asset. Dowd (1998) states that normality gives us a simple and tractable expression for the
confidenceintervalfortheVaRestimate,andwithanormalreturn,theVaRisjustthemultipleofthe
standarddeviation.Itisonlyabouttwoparameters,theholdingtimeandtheconfidencelevel.These
parameterscanvarydependingontheuser.Also,goingfromaVaRestimateforaspecificconfidence
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
17
leveltoanotheronecaneasilybedone justbychangingthenumbersfromthenormaldistribution
associatedwiththeconfidencelevel.
2.3.3GARCHAPPROACH
GARCHisnotreallyaVaRmethodinitselfbutratheramoreadvancedwaytocomputethestandard
deviationofpast returns that in turn is intended togivemorepreciseVaRestimates.TheGARCH
approach is considered to be semi parametric, meaning that it has both parametric and none
parametricproperties. It isparametric in thesense that theestimatedvarianceof theapproach is
usedwiththenormaldistribution,but it isatthesamenonparametricsincethe inputtedvariables
arerealreturnsandnotestimatedvolatilities.
TheGARCHapproachforvariance,h,lookslikethis(Jorion2001):
Where , and areweights. is theweight forhowmuch of todays variance influencesour
forecast, weights howmuch of yesterdays estimationweights into the forecast. Jorion (2001)
pointsoutthatthesumofandmustbelessthanoneandisusuallycalledthepersistenceofthe
approach. It iscalledpersistence, sinceduringamultiperiod forecast into the future, theweights
woulddecayastheyareboth lessthanoneandthecalculatedvalueofvariancewouldrevertback
towardstheestimatedlongtermvariance.
Thoughtheuseof itsnonparametric input,theGARCHapproachcantakesvolatilityclustering into
consideration.Thismeansthatittakesintoaccountthatlargechangesinpricesataspecifictimeare
usually followed by more large changes and vice versa. This is done by weighting historical
observationsandrecentobservationsunequally.
There isnowayofsolvingforthecorrectvaluesof,andmathematicallyso ithastobedone
numericallyusingmaximumlikelihoodestimation(MLE).Toestimatetheparameters,andthe
logarithmofthelikelihoodfunctionofthenormaldistributionisused(Jorion2001).
max 1
2
2
,andgives thevaluesofht,somaximizing this functionbyadjusting,andgivesus the
optimal values of , and . This optimization is done for each and every one of the historical
observationonwhichtheVaRmeasureistobebased.Theparameters,andarethenadjusted
togiveamaximumvalueofallofthoseobservationscombinedforthechosenwindowintime.
ItshouldbenotedthatastheMLEfunctionisderivedfromthenormaldistributionfunction,italso
assumesthatthereturnsoftheassetsarenormallydistributed.TheMLEfunctionmaytherefore, if
appliedtononenormallydistributedreturns,producequestionableresults(Jorion2001).
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
18
2.4THEUNDERLYINGASSETS
The underlying assets, on which our VaR calculations are based, are chosen because of their
fundamentallydifferenceswhichwouldindicatefundamentallyvaryingreturndistributions.Allthree
assetsshowdifferentpropertiesmakingtheminterestingforcomparisonaccordingtothepurposeof
thestudy.
Wheneverholdinganassetyou face the riskof theassetgainingor losing value in relation to its
purchaseprice.Thefieldofriskmanagement isallaboutassessingandmitigatingrisk.Aneffective
risk management is according to Saunders & Cornett (2007) central to a financial institutions
performance. This is also true for any company exposed to risk. As discussed in the problem
discussionVaRisausefulmeasureforallofourassets.
2.4.1BRENTOIL
Crudeoil isthemosttradedcommodityontheworldmarketandthemost importanthubsforthis
trading are New York, London and Singapore. The prices of the Brent oil, which is the world
benchmark for crudeoiland ispumped from theNorth Seaoilwells is theoilused in the study.
AccordingtoEydeland&Wolyniec(2003),abouttwothirdsoftheworldsupplyofoil ispricedwith
referencestothisbenchmark.
Theoilprice isveryvolatile,andareasonforthis isthe influenceofOrganizationofthePetroleum
ExportingCountries(OPEC).Thisisanoilcartelthatwascreatedin1960andnowconsistsoftwelve
membercountries.Theiroilproductionstandsfornearlyhalfoftheworldproduction,andbyraising
or lowering theirproduction, they can affect theprice. There aredifferentopinions aboutOPECs
influenceonthebalanceofdemandandsupply.Somepeopleclaimthattheypreventtheforcesof
themarkettosettherealpricewhileotherssaythattheycanonlyaffectthesupplyofoil,notthe
demand,makingtheforcesofthemarketthroughthedemandcreateanequilibriumprice.
2.4.2OMXs30
OMXs30 isanabbreviationofOMXStockholm30and is thedenominationof the30most traded
stocks on the Stockholm stock exchange. It is a capitalweighted index thatmeasures the price
developmentof thestocks.Thevalueof thestock for theseparatecompanies is thebasisof their
partofthe index. Theassemblyofthe index isconductedtwiceperyearandtherevenuesofthe
companiesthenworkasagroundforthecompanieselected.(www.omxnordicexchange.com)
2.4.3THREEMONTHSWEDISHTREASURYBILLS
TheSwedishtreasurybillsareissuedthroughtheSwedishNationalDebtOfficeandthebidsarethen
submittedtodealersauthorizedbythesame.Thebillsaretradedonthesecondarymarket,whichis
consideredtobeahealthymarketwhen lookingatthe liquidity.Thematurityvarieswithinayear,
however,thetreasurybillsusedinthisstudyarethethreemonthbills(STB3M).(www.riksbank.se)
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
19
2.5BACKTESTINGWITHKUPIEC
WhencalculatingtheVaR,one isonly interested inthe lefttailthatrepresentsthecaseswhenthe
returnsareworse thanexpected.Toevaluate themeasuresused,aback test canbe conducted.
Goorbergh&Vlaar(1999)explainsthetestascountingthenumberofdaysintheevaluationsample
thathadaresultworsethanthecalculatedVaR.Anobservationwheretheactualreturnexceedsthe
VaRiscalledaVaRbreak.Thebacktestisimportanttouse,andbyperformingitandcomparingthe
differentapproaches,onecanensurethattheapproachesareproperlyformedaccordingtoCostello
&Asem&Gardner(2008).
Jorion(2001)statethatthenumberofVaRbreaksisexpectedtobethesameasoneminusthelevel
ofconfidence.So forasampleof100observationswherea95%confidenceVaR iscalculated,we
would expect five 100% 95% 100 5VaRbreaks tooccur. In the study this is called the
targetnumberofVaRbreaks. If therearemoreor lessVaRbreaks thanexpected, it isbecauseof
deficiencies intheVaRapproachortheuseofan inappropriateVaRapproach.Awidelyusedback
test is theKupiec test.This testuses thebinominaldistribution tocalculate theprobability thata
certainnumberofVaRbreakswilloccurgivenacertainconfidencelevelandsamplesize.TheKupiec
testfunctionis(Veiga&McAleer,2008):
|,
1
Thevariable is thenumberofVaRbreaks,N the sample sizeand corresponds to the levelof
confidencechosenfortheVaRapproach(makinga95%confidence levela5%probability inputfor
theapproach). Ifthesamplesize is inputtedand issettooneminusthe levelofconfidence,the
binomialfunctionproducesthelikelihoodthataspecificnumberofVaRbreaksistooccur.Byusing
thecumulativebinomialdistribution, it ispossibletocalculatean intervalwithinwhichthenumber
ofVaRbreaksmust fall, inorder for theapproach tobeaccepted.This isdoneby calculating for
whichvaluesof that thecumulativebinomialdistributionproducesprobabilities thatare in the
interval 2.5% 97.5% (which corresponds to a 95% test confidence). VaR approaches that
producevaluesofthatlieswithinthisrangecanthereforebeaccepted.Iftheapproachproduces
valuesofoutsidethisspan,theapproachisrejected.Arejectionmeansthattheconfidencelevel
thatweused intheVaRapproachdidnotmatchtheactualprobabilityofaVaRbreak.This inturn
indicatesthattheapproachisnotperformingwellandthatitshouldberejected.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
20
3.METHOD
In this chapter, themethods used in the study are presented. First, the nature of this study is
describedandthenthecalculationsofVaRareexplainedforthedifferentapproaches.
3.1ANALYTICALAPPROACH
This study takes an analytical approach and assumes an independent reality. This is a method
approach that iswidelyusedand thehistoryof itgoeswayback in time.Themostdistinguishing
characteristic concerning theprocedureof this approach is, in accordancewithArbnor&Bjerkne
(1994),itscyclicnature.Itstartswithfacts,endswithfactsandthesefactscanbethestartofanew
cycle.Themeaning istoextractgoodmodels inordertodescribetheobjectivereality.Themodels
withinthisapproacharemostoftenofquantitativecharacterandthisisalsotrueforthisstudy.Itis
naturalthatlogicandmathematicshasaparttoplayinthesearchforthebestmodeltoapply.
3.1.1QUANTITATIVEAPPROACH
In this study, the testsarebasedupona lotofhistoricaldata.Dailypricechangesare studied for
three different assets that are widely traded, this form of studymakes us apply a quantitative
approach.ThisapproachcomparedtoaqualitativeapproachisaccordingtoArbnor&Bjerkne(1994)
oftenmore about clear variables and can cover a larger number of observations,which fits our
compositionbetter.Itfurtherassumesthatwecanmakethetheoreticalconceptsmeasurable.Inthe
past, the quantitative approach has inmany caseswrongly been seen as the only real scientific
method, or as Holsti said in 1969, If you cant count it, it doesnt count. There are however
limitations connected to theapproachandHolme&Solvang (1991)pointsout that ifyouarenot
awareof them, the resultcaneasilybemisjudged.Numbersareoften seenas the truth,and this
trustcancreateproblemsmakingtheanalysisofthenumbersveryimportant.
Historicaldatahavebeencollectedinthisstudy,andbasedonthese,anumberofcalculationshave
beenmade in order to extract the VaR. The purpose is to be able to seewhich approach that
performsbest,andtheresultisbasedonthefiguresfromthecalculations.Howevertheresultisnot
justanumberitisalsoputintocontexttobettergetanunderstandingofitsmeaning.
3.1.2DEDUCTIVEAPPROACH
When taking a deductive approach one starts from an already existing theory and then either
strengthenorweakentheconfidenceofthetheory,meaningthatonestartsfromagenerallawand
thenmovestoaseparatecase.Inourstudy,weusethreewellknownapproachesforcalculatingVaR
and test their performance on assetswith different return characteristics. The purpose is to test
models,nottocreatenewones.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
21
Asaresultofthisstudy,thefunctionoftheapproacheswillperhapsbestrengthenedbecauseofa
specific character, however, the same approach applied on the same asset can turn out to be
unreliable for a certain confidence level. The approachs can from the start be rather simple,
however, by adding a dimension like the GARCH approach, one can make the approach more
complex andmaybebetter fitted to a certain asset.The testof the approacheswillbebasedon
backtesting, we will count the numbers of VaR breaks,meaning the number of times the loss
exceededthecalculatedVaR.ReasonsfortheactualnumberofVaRbreakswillbeanalyzed.
3.1.3RELIABILITY
Inordertodecidethereliabilityoftheresults,oneshouldbeabletodothetestanumberoftimesto
seeifthesameresultsareachieved.Arbnor&Bjerkne(1994)callsthisatestretest,anditishelpful
intellingwhetherthestudyanditsresultsarereliable.Allthedatausedinthestudytocalculatethe
VaRispublicinformation,makingthetestreproducible.
3.1.4VALIDITY
Themostimportantfactorwhenjudgingwhethertheresultsarejustifiableintheanalyticalapproach
isthroughthevalidity.Ifyoucannotanswerthequestionwhethertheresultsgivesatruepictureof
thereality,theresultscaneasilybecomemeaningless.Thecloserwegettothetruesituation,givena
specifieddefinition,thehigherthevalidity.Therelationbetweentheoryanddataisvital,andHolme
&Solvang(1991)pointsoutthatthevaliditycanbeenhancedbyacontinuouslyadaptionbetween
thetheoriesandthemethodsused intheexamination.This isdonebychoosingmethodsthathas
the ability to handle or show the impacts that different assets characteristics can have on the
measuredVaR.Arbnor&Bjerkne (1994)stresses the importanceof threekindsofvalidityand the
needforcombiningthem.Thesearethesurfacevaliditythatdealswiththereasonabilityassessment
in relation toearlier results, the internalvaliditydealingwith thequestion ifwehadexpected the
resultwehaveconcludedandfinallytheexternalvalidityandtherebytheusefulnessoftheresultin
otherareas.Thesequestionswillbefurtherdealtwithintheanalysis.
Itisfurtherimportanttohaveinmindthatanapproachofthekindusedinthestudycangiveyoua
numberoftheVaRbutitisalwaysanestimationofapossiblefutureloss.Givenacertainconfidence
leveltheVaRcanbecalculated,however,thenumberreceivedisnottheabsolutetruth,itcannotbe
guaranteedthatthefuturelosseswontbelargerthanexpected.
3.2CALCULATIONOFVaR
Withmanydifferentapproachesandmodels,thechoicethatVaRusersfaceisthechoiceofpicking
the rightone thatmatches theirpurposebest.Theapproachesshouldmakeestimates that fit the
futuredistributionofreturns. IfanoverestimationofVaR ismade,thenoperatorsendsupwithan
overestimateoftherisk.Thiscouldresultintheholdingofexcessiveamountsofcashtocoverlosses,
asinthecasewithbanksundertheBaselIIaccord.Thesamegoesfortheoppositeevent,whenVaR
hasbeenunderestimatedresultinginfailuretocoverincurredlosses.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
22
Inthisessay,wewillbecalculatingVaRforthreedifferentunderlyingassetsandcomparetheresults.
TheassetsthatwillbeusedforourcalculationsareBrentOil,OMXs30stockindex,andthreemonth
Swedish treasurybills.TheVaRapproaches thatwewillcalculateare thehistoricalsimulation, the
movingaverageapproach,assumingnormaldistribution,andtheGARCHapproach,assumingnormal
distribution.Whenusing theparametricapproaches,wedosuspect that the returns thatwehave
basedthecalculationsonaremost likelynotperfectlynormallydistributed. Infact,economictime
seriesrarelyare(Jorion2001).Theperformanceoftheseparametricapproacheswillthereforepartly
bedeterminedbyhowwell thenormaldistribution assumption fits the actualdistributionof the
returns.
The tool thatwehaveused toperformourcalculations isMicrosoftExcel.Excel isaveryversatile
toolthatcanbehelpfulifoneknowhowtouseitright.However,thedownsideofExcelisthatitis
muchslowerthansoftwarethat isspecificallydesignedforfinancialcalculations.Therearealsono
automated functions forcalculatinga lotof themoreadvancedoperations,oftenused in financial
timeseriesanalysis,suchasautocorrelationsetc.Thesemoreadvancedcalculationshavetobedone
manually, which can be very time consuming and also limits us somewhat when it comes to
computingmoreadvancedapproaches.
3.2.1HISTORICALSIMULATIONAPPROACH
CalculatingaVaRmeasureusing thehistorical simulationapproach isnotmathematicallycomplex
but can be trying since the calculations require a lotofhistoricaldata. The first task is topick a
historical time frame on which to base the simulation. Themore historical values we base our
calculationson themoreobservationswewillget in the tailsand thus themoreaccurate theVaR
measurewillbeathigherconfidencelevels.
Thedownsideisthataddingmorehistoricaldatameansaddingolderhistoricaldatawhichcouldbe
irrelevant to the future development of the underlying asset.Our simulationwill be based on a
slidingwindowoftheprevious2000observationswhichcorrespondsto8calendaryears.Wehave
selectedthisratherlargewindowforthehistoricalsimulationapproachaswewanttheapproachto
performwellatthe99%and99.9%confidence levels.Asargued inthetheorychapter itmakesno
sensetosetthewindowsizetoanylessthan1000observationsforthe99.9%confidencelevel,but
evenatthislevelwetheoreticallyonlyhaveoneobservationinthetail.Wethereforedecidedtoset
the window size to 2000 observations to enhance the performance of the approach at higher
confidenceintervals.
Extracting the VaR measure from the historical data simply requires us to choose the desired
confidence level andpickout then:thobservation in thehistoricaldata that corresponds to that
confidencelevel.Forexample,a95%confidencelevelmeansthatweareinterestedintheworst5%
oftheobservations. Ifweforexampleareusing1000observations,thiswouldmeanthatthe95%
VaRwouldbethe50thworstobservation(1000 5% 50.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
23
ThePERCENTILEfunctionweused inExcelcalculatesthen:thpercentileofthevalues inachosen
dataset.Ifthedesiredpercentileisnotamultipleofthenumberofvaluesinthedataset,Excelwill
doalinearinterpolationbetweenthetwoclosestvaluestofindthedesiredvalue.
3.2.2MOVINGAVERAGEAPPROACH
Underthisapproach,wecontinuouslymeasuredthestandarddeviationofreturnsoverawindowof
thelast30days.Thisstandarddeviationwasthenmultipliedwiththenumberofstandarddeviations
extractedfromthestandardnormaldistributionthatcorrespondstotheselectedconfidence level.
Inotherwords,thestandarddeviationchangesasthewindowmovesalongbutthemeanisassumed
tobezero.
3.2.3GARCHAPPROACH
ThechallengewiththeGARCHapproach istoestimatetheparameters,and.Asdescribed in
the theory chapter, theseparametershas tobe solved fornumerically,usingmaximum likelihood
estimation. The literature that we have studied describes the MLE function but provides little
practicalinformationonhowtoimplementit.Themathbehindmaximumlikelihoodestimationcan
becomplicatedtounderstandforthosenotfamiliarwithbusinessstatistics.Manyanalyststhatdo
these types of calculations use preconfigured software and seldom have to engage in the
mathematicsthatliesbehindthecalculations.
TheMLEformula itself isnotcomplicatedandwas implementedforeachandeveryobservation in
ourstudy.MaximizingthefunctionwasdoneusingtheSOLVERfunction inExcel.Thefirstproblem
thatweencounteredwashowtodecidehowmanyhistoricalobservationsthatweshouldbaseour
MLEon.At first,wehadplanned tobase theMLEonamovingwindowof250values,matchinga
year in tradingdays,but thisapproachproducedvery inconsistent resultsoftensettingoneof the
parameters,or,closetozeroandtheotherclosetoone.Theconclusionwedrewwasthatthe
MLEwasbasedon too fewvalues tobeconsistent.Asmentioned in the theorychapter, theMLE
functionalsoassumes that the returnsarenormallydistributed.Thus, thesmaller thesample that
theMLEestimation isbasedon, the larger the risk that thosevaluesmightbe significantlyparted
fromthenormaldistribution.
Welookedthroughagreatdealofarticlesandbooksinhopeoffindingsomekindofdocumentation
onhowmanyobservationstoincludeinourMLEbutwewereunabletofindanyrecommendations
togoby.Wethereforemadeourownestimationofanappropriatetimeframeonatrialanderror
basis.Thiswasdonebyconstructingamacro inExcel thatwouldmaximize theMLE function, first
basedonthe250firstobservationsandthenadd100observationsatatimeeachtimereestimating
theparameters,anduntilwehad testedallof theobservations.Themacro isshown in the
appendix1.Theresultingvaluesof,andareshowninthegraphs3.2.3a,bandc.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
24
Figure3.2.3aMLEestimationgraphforBrentOil
Figure3.2.3bMLEestimationgraphforOMXs30.
Figure3.2.3cMLEestimationgraphforSTB3M.
Thegraphsshowthevaluesofandinblueandredontheleftverticalaxisandthevaluesofin
greenontherightverticalaxis.Thehorizontalaxisshowsthenumberofobservationsincludedinthe
estimate.Sincethesumofandmustbelessthanoneandthereforemustgodownfortogo
up.Thisiswhythevaluesdivergeduringthefirstestimationstolaterleveloutorbecomestable.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
25
Onecanclearlyseehowvaluesof,andstabilizeasweaddmoreandmorevaluestotheMLE.
We also interpret these results as an indication of to what degree the returns are normally
distributedornot.
By lookingatthegraphs,weseethatataround2000observationstheMLEhasstabilizedforallof
our three assets, which lead us to conclude that 2000 observations is a suitable number of
observationstobaseourMLEon.
Afterdeciding theappropriatenumberofobservations tobaseourMLEon,we then recalculated
theMLEfunction forourdatasets,nowwitha lagof2000observationsfrom19960101onwards.
Newgraphshavebeenplottedforthevaluesof,and,andtheycanbefoundinappendix3.
After thevaluesof,andhadbeenestimated, theywere inputted into theapproachand the
resultswillbeshownintheresultsandanalysischapter.
3.3SKEWNESS
Theskewnesstellsuswhetherthedataissymmetricornot.Normallydistributeddataisassumedto
besymmetricallydistributedarounditsmean,i.e.ithasaskewof0.Adatasetwitheitherapositive
or negative skew therefore deviates from the normal distribution assumptions. This can cause
parametricapproaches, inourcase themovingaverageapproachand theGARCHapproach, tobe
lesseffectiveifassetreturnsareheavilyskewed,sincetheseapproachesassumethatthereturnsare
normally distributed. The result can be an overestimation or underestimation of the VaR value
dependingontheskewoftheunderlyingassetsreturn(Lee,Lee&Lee2000).
Figure3.3Graphshowinganegativerespectivelyapositiveskew.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
26
3.4KURTOSIS
Thekurtosismeasuresthepeakednessofadatasampleanddescribeshowconcentratedthereturns
arearoundtheirmean.Ahighvalueofkurtosismeansthatmoreofthedatasvariancecomesfrom
extremedeviations.ThekurtosisofthenormaldistributionsalsomentionedbyLee,Lee&Lee(2000)
asmesokurtic is three soaswith the skewnessanydeviations from thenormaldistributioncause
problems for theparametricapproaches.Ahighkurtosismeans that theassets returns consistof
more extreme values than modeled by the normal distribution. This positive excess kurtosis is
accordingtoLee&Lee&Lee(2000)calledleptokurticandlowerkurtosismeaninganegativeexcess
iscalledplatykurtic.LowkurtosisproducesVaRvaluesthataretoosmall.
3.5THESOURCEOFDATA
Thedataused in thestudyare timeseries thatreflects thehistoricalpricechanges for thechosen
assets.Theperiod,whichthecalculationsarebasedon,stretchesfrom19870101to20080930for
allthethreeassets.However,thedatafrom1987to19951231hasbeencollectedtoenableusto
baseourcalculationsthatrequirehistoricaldataon.
Different sources have been used to collect the data.Daily oil prices are taken from the Energy
InformationAdministration (www.eia.doe.gov),whoprovidesofficialenergy statistics from theUS
government.TheOMXs30datacomesfromOMXNordicExchange(www.omxnordicexchange.com),
and the Treasury bill data can be found on the homepage of the Swedish central bank
(www.riksbank.se).Thecharacteristicsofthedifferentassetsaredescribedbelow.
Wehavecalculatedsomecharacteristickeyfiguresaboutthedatasetsinordertobetterbeableus
to compare them. The key figures thatwe have calculated are skewness, kurtosis, volatility, and
Figure3.4ProbabilitydensityfunctionforthePearsontypeVIIdistributionwithkurtosisofinfinity(red);2(blue);and0(black).
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
27
average price change. Table 3.5 shows some of the statistical characteristics of each dataset,
measuredduringtheentireperiodof19872008.
BrentOil OMXs30 3MSTB
Skewness: 0.83 0.02 3.77
Kurtosis: 19.14 7.28 182.51
Dailyvolatility: 2.32% 1.47% 1.49%
Annualvolatility: 36.78% 23.37% 23.66%
Averagelogpricechange: 1.63% 1.03% 0.58%
Table3.5Statisticalcharacteristicsoftheassetreturns.
3.6BRENTOIL
Figure3.6aDailyreturnsforBrentoil.
The oil prices are themost volatile prices in this study,with a 2.32% daily volatility and 36.78%
annualvolatility.Thekurtosis is19.14,whichmake it farnarrowerwithmuch fatter tails than the
normaldistribution.
TheskewnessoftheBrentoilreturnshaveanegativeskewof0.83,whichindicatesthatthereturns
areskewedtotheright.Thismeansyetanotherdeparturefromtheassumptionsofnormalityinthe
parametricapproachescausingthesetoperformlesseffective.Thedailyaveragelogpricechangeis
1.63%,whichbasicallymeansthatoverthemeasuredtimeperiodtheaveragereturnwas1.63%.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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Ingraph3.6b,wehavefittedanormaldistributionprobabilitydistributioncurveagainstahistogram
ofthereturns.Themostvisuallyprominentdeviationfromthenormaldistributionassumptionisthe
kurtosis, which can be seen from the middle bars of the histogram rising above the normal
distribution.Therearealsoseveraloutlierstorightand leftofthehistogram(whichcanbehardto
seebecauseofthescaleofthepicture)which layoutsidetheboundsoftheboundsofthenormal
distributioncurve.
Figure3.6bHistogramshowingthedailyreturnscombinedwithanormaldistributioncurve.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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3.7OMXs30
Figure3.7aDailyreturnsforOMXs30.
Sincethereturnsofthe indexcorrespondstoawelldiversifiedportfolio,theyarenotveryvolatile.
Theaveragevolatilityis1.47%dailyand23.37%annually.Thekurtosisis7.28,whichsuggestthatthe
distributionofthevaluesisnarrowerthanthenormaldistributionandwithsomewhatfattertails.It
ishowevernotashighasitwasfortheBrentoil,whichmightindicateabettercompatibilitywiththe
parametricapproaches,atleastatlowerlevelsofconfidence.
Theskewness isonly 0.02,whichandtheaveragedaily logpricechange is1.03%.Apartfromthe
kurtosisthenormaldistributioncurveisaprettynicefitonthesereturns.
Therearehowever largeoutliers in thatarenot clearlyvisible in thegraphwhich layoutside the
confinementsofthenormaldistributioncurve.Thereisalsonosignificantsignofautocorrelationas
shownbythetableinthepreviouschapter.
Figure3.7bHistogramshowingthedailyreturnscombinedwithanormaldistributioncurve.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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3.8THREEMONTHSWEDISHTREASURYBILLS
Figure3.8aDailyreturnsforSTB3M.
The returnsof the treasurybillshave themostextremecharacteristicsofall thedatasets. Ithasa
strongpositive skewnessof3.77, and a very large kurtosisof182.51 suggesting apoor fit to the
normaldistributionwhichalsocanbeobservedfromthehistogram infigure3.8b.Theaverage log
pricechangeis0.58%butfromfigure3.8awecanseethatvolatilityvariesfromlowtoextreme.The
mostextremereturnsarefromtheearly1990swhenthebankingcrisisoccurredinSweden.
Figure3.8bHistogramshowingthedailyreturnscombinedwithanormaldistributioncurve.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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3.9AUTOCORRELATION
Intimeseriesanalysis,autocorrelation,orserialcorrelationas it isalsocalled, isameasurementof
whether the data correlateswith itself over time.Autocorrelation in financial time series data is
measuredbyregressionoftheobservedreturnswithalaggedversionofthemselves.Regressingthe
returnswith themselves like thiscanbedescribedas testing if it ispossible todescribe returnsof
todayasalinearfunctionofthereturnsfromyesterday.Theresultingresidualsfromtheregression
are then tested for autocorrelation. The presence of autocorrelation means that the applied
approachwillhaveapoorfittotheactualdatawhichleadstheanalysttoconcludethatthereturns
oftodaycannotbeaccuratelydescribedasa linearfunctionofthereturnsofyesterday.Thusthere
must be other factors besides the historical returns that effect todays returns and thereby
approachesthattrytopredictfuturereturnsbylookingatthepastwillbelesseffective(Tsay,2002).
Wehavechosentoperformaone laggedDurbinWatson(DW)testonourdatatosee ifthereare
anyfirstorderautocorrelation.TheDWtestusesthefollowingformulatocalculateitsstatistic:
Theparameterdenotes the residuals from the regression.TheDW statistic canassumeavalue
between0and4.ValuesoftheDWstatisticlargerthan2indicatenegativeautocorrelationwhereas
valueslessthantwoindicatepositiveautocorrelation.Avaluecloseto2indicatesthattheresiduals
areprobablynotautocorrelated(Tsay2002).
Whentestingforautocorrelation,aconfidence interval iscalculatedaround2withinwhichtheDW
statisticmustfallinorderfortheresidualstobeconsiderednottobenotcorrelatedwiththegiven
degree of significance. Harvey (1990) has shown that for large sample sizes, this interval is
approximatelynormallydistributedwithameanoftwoandavarianceof
.
We have used the statistical softwareprogramR to test our time series for autocorrelation. The
autocorrelation tests results have been compiled in table 3.9 and some additional regression
statisticscanbefoundinappendix2.
RESULTSFROMAUTOCORRELATIONTESTAsset No. Lag Autocorrelation dL DW dU PBRENTOIL 5454 1 0,000660287 1,955455 1,998650 2,044545 96,40%OMXs30 5475 1 0,000758127 1,955540 1,997565 2,044460 94,80%STB3M 5468 1 0,009438294 1,955512 1,980684 2,044488 36,80%
Table3.9Resultsfromautocorrelationtest.
ThepvalueisaprobabilityratiothatRcalculateswhichdonatestheprobabilitythattheDWstatistic
wouldoccurrandomly.AlowpvaluethusindicatesthattheDWstatisticinquestionisveryunlikelyif
theapproachiscorrect.Thenullhypothesisis:
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
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Thenullhypothesiswillbe rejected if thep value is less thanoneminus the selected confidence
intervalwhichinourcaseis95%.Thusifpwouldhavebeenlessthan5%,thenullhypothesiswould
havebeenrejected,whichwouldhavemeantthatthereturnswereautocorrelated.Asstatedbythe
pvaluesandtheDWstatisticsinthetable,thereisnosignofautocorrelationintheresiduals.
3.10BACKTESTINGWITHKUPIEC
InordertomeasurewhichoftheVaRapproachesthathasbeenmostsuccessful,wewillbacktest
theVaRvaluesagainsttheactualreturnsofeachday.Intheeventthattheactuallossonaparticular
dayislessthantheprojectedVaRvalue,wesaythataVaRbreakhasoccurred.
TheexpectednumberofVaRbreaksisoneminustheselectedlevelofconfidencetimesthenumber
ofobservations.Thenumbershavebeen roundedoff to thenearest integer.Wehavecompileda
tableofthedesirednumberofVaRbreaksintable3.10.
DESIREDNO.OFVaRBREAKSANDKUPIECTESTINTERVALS
95% 99% 99.9%
Asset No. MINTARGET MAX MIN TARGET MAX MIN TARGET MAX
BrentOil 3259 139 163 187 22 33 43 0 3 6
OMXs30 3215 137 161 184 22 32 43 0 3 6
STB3M 3225 137 161 185 22 32 43 0 3 6
Table3.10TargetnumberofVaRbreaksandKupiectestintervals.
ThenearerthedesirednoofVaRbreaksanapproachproducesthebettertheapproachisconsidered
toperform.WhethertheproducednumberofVaRbreaksisgreaterorsmallerthatthetargetvalue
makesnodifference, it istheabsolutenumberofVaRbreaksthattheapproachdeviateswiththat
determineshoweffectiveitis.
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4.RESULTS&ANALYSIS
Inthischapter,wewillconcludeourresultsandthereafteranalyzetheminrelationtothepurposeof
thestudy.Theanalysiswillbedivided,treatingtheapproachesseparately, inordertogetabetter
overviewoftheapproachesandtheirresults.
4.1BACKTESTINGRESULTS
Thebacktestingresultsarepresentedforthethreedifferentassetsunderthethreeapproaches.The
outcomes depend on the characteristics of the asset, and by examining them, the result can be
explained.
4.1.1HISTORICALSIMULATIONAPPROACH
Thebacktestingresultsforthethreeassets,calculatedusingthehistoricalsimulationapproach,are
shown in table4.1.1a.The table shows theminimumandmaximumvaluesallowedby theKupiec
test,thetargetnumberofVaRbreaksandtheresultingnumberofVaRbreaksfromthesimulation.
VaRbreaksthatarewithintherangeallowedbytheKupiectestaremarkedgreenandthosethatare
notaremarkedred.
KUPIECTESTHISTORICALSIMULATION,2000DAYLAG 95% 99% 99.9%
Asset No. MIN TARGET RESULT MAX MIN TARGET RESULT MAX MIN TARGET RESULT MAXBrentOil 3259 139 163 209 187 22 33 37 43 0 3 7 6OMXs30 3215 137 161 216 184 22 32 54 43 0 3 5 6STB3M 3225 137 161 109 185 22 32 24 43 0 3 7 6
Table4.1.1aBacktestingresultswithhistoricalsimulationapproach.
Theresultsshowthattheapproachperformspoorlyonthe95%confidence levelwhile itproduces
betterresultsonthehigherconfidence levels.Onthe95%confidencelevel,theapproachproduces
toomanyVaRbreaks,whichindicatethattheapproachoverestimatestheVaROnthe99%levelthis
approachperformsequallycomparedtotheGARCHapproachandoutperformsthemovingaverage
approach.Whenitcomestothehighestlevelthisapproachistheonlyonethatcomesevencloseto
beingacceptedbytheKupiectest.
This approach looks at the latest2000daily returnswhen calculating the volatility,which in turn
makes theapproach react slowly tonew informationand changes in thedaily returns.There isa
tradeoff between new information and old historical information that we presented in earlier
chapters.Ifashorterintervalwouldhavebeenchosen,thiswouldhaveincreasedtheimpactofnew
observations,puttinghigherfocusonrecentmarketconditions.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
34
Table4.1.1ashowstheVaRestimatedbyhistoricalsimulationatthe99.9%confidencelevelforBrent
Oil.Outofall theassets,thehistoricalsimulationapproachperformsbestresultswithBrentoil. It
doeshoweverseem that theapproachgenerallyunderestimates theValueatRisk, thusproducing
toomany VaR breaks. The characteristics of Brent oil showed that the normal distribution, and
thereforetheparametricapproaches,wouldmakeapoorfitandthatthehistoricalsimulationwould
suittheassetbetter.Brentoilisthemostvolatileoftheassetsbutthevolatilityisratherpredictable,
ormore correctly, highbut stabile. The reason towhyhistorical simulation performsbetterwith
Brentoil isbecauseof itsstability.Asdiscussedearlierthehistoricalsimulationapproachdoesnot
assume the returns to followaspecificprobabilitydistribution,but itdoesassume that the return
distributionisthesameinthepast,inourcase2000days,asitwillbetomorrow.Thestabilityofthe
Brentoilreturnsisthereforethereasonforthegoodresults.
Whentestingtheaccuracyofanapproachitisequallyimportantwithunderestimationsasitiswith
overestimations.Theresultsforthetwootherassetsareaboutthesamewiththedifferencelyingin
overestimationsoftheValueatRiskfortheSTB3MandunderestimationsofthevalueforOMXs30.
Thevolatility for theseassets isnotashighas forBrentoilbut itchangesmoredramaticallyover
time. If a shorter time period had been chosen, the results for STB3Mwould have looked very
different ignoringthegreatvolatilitychangesthatappearedaround1993.Thehistoricalsimulation
would in thatcasehaveproducedbetterresultssince the fluctuations inreturnswouldhavebeen
less.
Figure4.1.1bVaRforBrentOil,HistoricalSimulation99.9%confidence19962008.
The look of the graph 4.1.1b is characteristic for historical simulationVaR estimates. The sudden
jumpsupanddownandtheplateausinbetweensignaltheentranceandexitofanextremehistoric
return that affects theVaR as long as it remains in thewindow. The longwindow thatwe have
chosen seems to be the cause to why the approach overestimates the VaR for STB3M and
underestimatesitforBrentOilandOMXs30.Atthe95%,confidencelevelthelargewindowchosen
seemstoberesultingintoofewVaRbreaksforSTB3M.Itwouldseemthatthelargewindowistoo
large for the95%confidence level,adding toomanyextremeobservations in the tails.Thiscauses
theapproachtooverestimatetheVaR,andproducetoofewVaRbreaks.ForBrentoilandOMXs30,
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
35
the opposite seams to apply. The large window puts toomuch weight on themore frequently
occurringsmallerreturnsanddrawsthecenterofgravityintheapproachawayfromtheinteresting
outliers,causingtheapproachtoproducetolowVaRandthustomanyVaRbreaks.Thesameseems
toapplyatthe99%confidencelevel,buttheeffectsseemmilder,perhapsbecausethewindowsize
ismoreappropriatehere.
As canbe seen in table4.4.1a theapproach is rejectedby theKupiec testat the95% confidence
level.Webelievethisisduetothewindowsizeof2000observationsbeingtoolargeforthehistorical
simulation approach at the 95% confidence level. The approach could probably have performed
betterhadwechosenamoreappropriatewindowsize.
Theapproachdoes,however,performbetteratthetwohigherconfidencelevels,wheretheselected
windowsizeisbettersuited.Sincethehistoricalsimulationapproachisnonparametric,ittakesallof
theoutliersof the returns intoaccount thatare ignoredby theparametricapproaches,using the
normaldistribution.This isalsooneof the reasons towhy theapproachperformsbetterat these
higherconfidencelevelscomparedtotheothertwoapproachesaswewilldiscoverlater.
Byweighing all the returns equally, it takes time for the approach to react to fluctuations in the
returns.ThismeansthatoldextremeoutliersinthereturnsaffectthecalculatedVaRforalongtime.
WhendealingwithreturnsthatsufferfromextremeleptokurtosisasisthecasewiththeSTB3M,one
endsupwithanaverageVaRthatisconsiderablyhigherthantheaveragereturn,inotherwordsthe
fewextremeoutliershaveatoobigimpactontheVaRvaluegivenacertainwindowsizeandlevelof
confidence.Thehistoric simulationapproach isgenerally considered toperformwellwith returns
thatareleptokurtic,i.e.hasfattails,butthereseemstobealimitwhentheleptokurtosisbecomes
toomuchalsoforthehistoricalsimulationapproachtohandle.Wethereforethinkthatthesizeof
thehistoricalwindowhastobechosenwithrespectnotonlytotheconfidencelevelbutalsotothe
kurtosis.
Thehistorical simulation approach assumes that thedistributionof returnsdoesnot changeover
time.Withoutthisassumption,therewouldbenoreasonatalltolookatthepastreturnsinhopeof
predictingthefuture.Inotherwords,onecontributingcauseoferrorintheapproachcouldbethat
thedistributionofreturnsisnotstationary.
Over all this approach performs best on high confidence levels, and this is because the chosen
historicalwindow size is better suited for these confidence levels. The reason that the approach
performs relativelybetterathighconfidence intervalscompared to theother threeapproaches is
thatitconsiderstheextremevaluesthatfallsoutofthenormaldistribution.Thiscanclearlybeseen
inthehistogramsinchapterthreeshovingthedailyreturnsandthenormaldistribution.ForBrentoil
andtheOMXs30,asignificantnumberofobservationscanbefoundinthetails.FortheSTB3M,most
observationsfallinthemiddleofthenormaldistribution,causingtheapproachtooverestimatethe
VaR.HistoricalsimulationcanberecommendedwhenestimatingVaRathighlevelsofconfidencefor
returnsthatarestationaryandnottooleptokurtic.
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
36
4.1.2MOVINGAVERAGEAPPROACH
ThemovingaverageapproachisthemostbasicapproachforcalculatingVaRandithasbeenaround
thelongest.Eventhoughitiselementary,itperformsquitewellatlowerconfidencelevelsandunder
therightconditions.Thebacktestresultsareshownintable4.1.2.
KUPIECTESTMOVINGAVERAGEAPPROACH,30DAYLAG 95,00% 99,00% 99,90%Asset No. MIN TARGET RESULT MAX MIN TARGET RESULT MAX MIN TARGET RESULT MAXBrentOil 3259 139 163 178 187 22 33 48 43 0 3 13 6OMXs30 3215 137 161 181 184 22 32 65 43 0 3 21 6STB3M 3225 137 161 170 185 22 32 96 43 0 3 47 6
Table4.1.2Backtestingresultswithmovingaverageapproach.
ThemovingaverageapproachunderestimatestheVaRathigherconfidencelevelsduetothefatter
tailofthereturnsthanassumedbythenormaldistribution.Theapproachseamstoperformwellon
lowerconfidencelevelswherethenormalityassumptionismorevalid.Consequently,theKupiectest
rejectsthemovingaverageapproachatthehigherconfidencelevelsandacceptstheapproachatthe
95%level.
Asdiscussedinthemethodchapter,thecharacteristicsoftheOMXs30returnsgiveanindicationthat
theassumptionofnormalityisnotthatunreasonable.TheOMXs30returnsarenotthatskewedand
thekurtosis isnot thatgreat.Despite this, theapproachperformsbetteron theBrentoil returns
eventhoughthecharacteristicsofthosereturns looktobefurtherfromthenormalityassumption.
This canperhapsbeexplainedby lookingat theplotof the returnsof theassets.The returns for
BrentoilismoreskewedandleptokurticthanthoseoftheOMXs30,butthereturnsoftheBrentoil
aremorestablewhereastheOMXs30returnsshowclearsignsofvolatilityclustering.
The extreme volatilitypeaksof the STB3M returnsmakes forpoor compatibilitywith themoving
average approach at higher confidence levels, which can be seen in table 4.1.2. The approach
producesapproximately thedoubleamountofVaRbreaks, compared to theother twodata sets.
Thisisaresultoftheoutliersoftenappearalonewithoutapriorriseinvolatilitythedaysbeforethe
VaRbreak,as isusualwhenvolatility is clustered.This isalso the reason for the returnsbeing so
leptokurtic.Thismeansthattheapproachhasnowayofforecastingthesesuddenrises involatility
and thus a VaR break occurs. The day after the VaR break when the outlier passes in to the
measurement window the VaR measure rises significantly. Since the VaR break was a single
occurrenceitisnotfollowedbyadditionalextremeoutcomes.
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4.1.3GARCHAPPROACH
Table4.1.3showsthebacktestingresultsoftheGARCHapproach.
KUPIECTESTGARCH 95,00% 99,00% 99,90%Asset No. MIN TARGET RESULT MAX MIN TARGET RESULT MAX MIN TARGET RESULT MAXBrentOil 3259 139 163 156 187 22 33 43 43 0 3 16 6OMXs30 3215 137 161 161 184 22 32 47 43 0 3 17 6STB3M 3225 137 161 95 185 22 32 38 43 0 3 19 6
Table4.1.3BacktestingresultswithGARCHapproach.
TheGARCHapproachproducesgood results for the95percentconfidence levelwithexceptionof
theSTB3M.Theresultsforthe99%confidence levelarealsoreasonable.Theapproachproducesa
bittoomanyVaRbreaksasexpectedfromaapproachbasedonthenormaldistribution.Thebetter
performanceoftheGARCHapproachonthe99%confidencelevel,comparedtothemovingaverage
approach, is as we interpret it solely due to the GARCH approaches more advanced way of
estimating the daily variance. The GARCH approach does however also suffer from the same
deficiency as themoving average approachwhich is that it assumes the returns to be normally
distributed. This becomes apparent at the 99.9% confidence level, where the GARCH approach
constantlyproducedtomanyVaRbreakstobeacceptedbytheKupiectest.
TheKupiectestrejectstheapproachforalloftheassetsatthe99.9%level.Atthe99%level,theonly
asset forwhich theapproachwas rejectedwas theOMXs30.Thiswassurprisingat first,since the
GARCHapproachperformedsowellontheOMXs30atthe95%confidencelevel.Thereasontowhy
the approach produced too many VaR breaks is probably that the assumption of normality is
acceptable at lower confidence levels but not at higher ones. The further out in the tail of the
distributionof returnswecome, the lessacceptable theassumptionofnormalitybecomes.This is
howeverconflictedby the fact that theKupiec testaccepted theapproachat the99%confidence
level for the Brent Oil and STB3M returns. This can be explained by the observation that the
approach produces fewerVaR breaks than the target for these returns at the 95% level, i.e. the
approach produces to high VaR values. At the 99% level itwould seem that the assumption of
normality that generally causes the approach to produce toomany VaR breaks is countered by
another error in the approach, which cancels both errors out, resulting in the approach being
acceptedbytheKupiectest.
Thenextquestion is to figureoutwhat the error in the approach is that results in the approach
producingtoofewVaRbreaksatthe95%confidencelevelforBrentOilandSTB3M.Wereasonthatit
is yet again the assumption of normality that is throwing our calculations off. The maximum
likelihoodestimationformulathatweusedtoestimatethevaluesof,andalsoassumedthat
the returns that itwasevaluatingwerenormal.As argued in themethod chapter, the returnsof
BrentoilandSTB3Mexhibitedthecharacteristicsleastcompatiblewiththenormaldistribution.We
AnempiricalevaluationofValueatRiskGustafsson&Lundberg
38
thereforesuspectthattheassumptionofnormalityhascausedtheMLEfunctiontoproducevalues
of,and thatare slightlyoff theiroptimum,which in turncouldbe thecauseof theGARCH
functionproducingtoofewVaRbreaks.Thisargumentissupportedbylookingatthegraphsofthe
valuesof,andinappendix3forallofourthreeassets.Thegraphsshowhowthevaluesof,
and changeasweestimate themwitha rollingwindowof theprevious2000observations.The
values for theOMXs30 returns are stablewhereas the values for the two other assets fluctuate
somewhat,indicatingthattheMLEfunctionmightnotfunctionproperlywiththereturnsasaresult
oftheassumptionsofnormality.
Thesetwoerrorsbothspringfromthenormalityassumption,butseemtobychanceaffecttheVaR
inoppositedirection,causingtheapproachtoperformbetterthanexpectedatthe99%confidence
level.SoeventhoughtheapproachisacceptedbytheKupiectestforBrentoilandSTB30atthe99%
confidencelevelwehaveourreservationsofwhethertheoutcomereallyisaresultoftheapproach
performingwellorifitisjustarandomoccurrence.
Altogether,we feel that theGARCHapproachdoesagood jobofhandlingvolatilityclusteringand
estimatingavarianceforecasts.Wedo,however,feelthattheMLEfunctionthatwehaveuseddoes
notworkproficientlyandthatitwouldhavetobeadjustedtobetterfittheassetsreturnsinorderto
furtherincreasetheperformanceoftheGARCHapproach.
According to Goorbergh & Vlaar the most important return characteristic to account for when
calculatingVaRisvolatilityclustering.Wefindthatvolatilityclusteringindeedisimportantbutdonot
agree that it is themost important characteristic.Aswe have shown in this chapter themoving
averageapproachperformswellatthe95%confidence leveleventhough itdoesnottakevolatility
clustering into consideration. In our opinion the most important return characteristic is the
distribution of the returns and howwell theymatch the distribution assumed by the approach.
Volatilityclusteringisaccordingtoourresultsthesecondmostimportantcharacteristic.
4.2EXACTNESS&SIMPLICITYThe tradeoff between exactness and simplicity is an interesting but hard dilemma to solve or to
decide about. People will argue that if amodel do not provide exact and accurate answers or
estimations,theyarenotuseful.Somemodelshavetoprovideresultsornumbersthat istoa100
percentcorrect,otherwise itcan result in severeoutcomes.A straight forwardexample ismodels
usedintheconstructionofairplanes