Research ArticleCapital Gains Sensitivity of US BBB-Rated Debt to US TreasuryMarket Markov-Switching Analyses

Mariya Gubareva 12 and Ilias Chondrogiannis3

1ISCAL ndash Lisbon Accounting and Business School Instituto Politecnico de Lisboa Av Miguel Bombarda 201069-035 Lisbon Portugal2SOCIUSCSG-Research in Social Sciences and Management Rua Miguel Lupi 20 1249-078 Lisbon Portugal3School of Slavonic and East European Studies UCL-University College London 16 Taviton Street London WC1H 0BW UK

Correspondence should be addressed to Mariya Gubareva mgubarevaiscaliplpt

Received 19 May 2020 Accepted 22 July 2020 Published 26 August 2020

Guest Editor iago Christiano Silva

Copyright copy 2020 Mariya Gubareva and Ilias Chondrogiannis is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

We reexamine the relationship between credit spreads and interest rates from a capital gain perspective of bond portfolio Capitalgain sensitivity between US BBB-rated bonds and Treasury bonds is weak and positive in normal periods but strong and negativeduring recessions In the upward phase of business cycles changes in interest rates are fully reflected in the bond yields leavingspreads unchanged while in the downward phase rates and spreads move in opposite directions is alternation between twodistinct regimes reconciles a long-standing division in the literature We then discuss the efficiency of shorting Treasury bonds as ahedging strategy and policy suggestions

1 Introduction

e postcrisis monetary policy which resulted into uncer-tainty regarding further moves in risk-free interest ratesposes questions on the resilience profitability and stabilityof the international financial system especially in whatconcerns the financial health of banks and corporate firmse recently observed low borrowing costs have allowedfinancial institutions to clear their budgets but have alsolimited profits and raised concerns on whether therestructuring of balance sheets and the increase of capitalbuffers will be enough when interest rates eventually in-crease Such an increase is expected to apply pressure on thevaluation of assets particularly for companies with ongoinglegacy burdens erefore the historical relationship be-tween risk-free rates and asset valuation becomes a keyindicator both for companies and investors In addition thechoice of metrics used for assessing this sensitivity plays avital role in what effects can be captured

In this paper we employ capital gains of bond port-folio containing risk-free government and risky BBB-

rated nongovernment US securities to evaluate hedgingstrategies that consist of holding risky and selling risk-freeassets Our motivation for focusing the study on BBB-rated bond portfolios is based on their importance forinstitutional investors such as pension funds and in-surance companies as the investment guidelines of a vastmajority of them allow only very limited exposure to highyield debt securities while the BBB-rated bonds providemore attractive returns than higher quality investment-grade instruments We opt for studying bond portfolioinstead of single securities as for the institutional in-vestors the large aggregates represent the main focus oftheir activity contrary to cherry-picking of fixed-incomeexposures which in a first place may interest individualswith a rather restricted funding capacities is aspect alsoallows us to reduce the necessary computation capacityand still be able to produce valuable investment insightsBy using the capital gains metric to study the sensitivity ofrelatively risky US BBB-rated bonds to risk-free USTreasuries (UST) we show that the relationship betweencredit spreads and yields of risk-free assets is not constant

HindawiComplexityVolume 2020 Article ID 4159053 13 pageshttpsdoiorg10115520204159053

as suggested by earlier research (eg [1ndash3]) but insteadchanges between two distinct regimes

In the first regime which takes place during normalperiods of economic growth any changes in the yield of risk-free assets are fully mirrored by changes in the yield ofinvestment-grade BBB-rated bonds keeping credit spreadsstable In that context spread-to-rate sensitivity is nullFrom a capital gains perspective the sensitivity of capitalgains of US BBB-rated bonds to the capital gains of USTbonds is positive and equal to one

In the second regime whichmanifests during a recessionand a subsequent sharp recovery capital gains sensitivityturns to be negativee spreads of US BBB-rated bonds andthe yields of USTmove in opposite directions due to flight-to-quality behavior and a fall in interest rates us spread-to-rate sensitivity becomes negative

e large differences among previous results based onspread-to-rate sensitivity analysis and on different choices ofmodels and data produce inconclusive answers to questionson the actual dynamics between credit spreads yields andinterest rates e literature is broadly separated to papersthat identify a negative spread-to-rate relationship [2ndash5] andresearch that finds a weak positive or null spread-to-ratesensitivity [1 6ndash8]

For instance the Merton [4] structural model is based oncontingent claims and implies that the probability of defaultof a debt issuer is affected by changes in the interest rate Itimplies a permanently negative relationship between creditspreads and interest rates On the other hand Kamin andvon Kleist [6] suggest that changes in the interest rate arepassed fully or slightly augmented onto the yield of riskybonds in the context of emerging marketsere is a null or aslightly positive relationship between changes in bondspreads and changes in the risk-free ratee same pattern isalso documented by Eichengreen and Mody [7] for countryspread of emerging economies with respect to the US in-terest rate

Finding a common denominator between the twoaforementioned streams of research on interest rate sensi-tivity is challengingus to reconcile a pile of contradictoryresults we opt for a different capital gain approach anddemonstrate its capacity to provide clearer results over thelong-range observation intervals It is also worth mentioningthat the financial crisis caused a radical change in the real-world fundamentals and regulatory frameworks so to revisitold problems new techniques allowing for deeper insightsare needed Towards that direction capital gains sensitivityis able to capture long-term effects better than spread-to-ratesensitivity as the former directly focuses on the end-of-period bottom-line portfolio results while the latter isusually obtained by averaging of the daily sensitivity figureswhich are subject to greater computational uncertainty dueto smaller amplitudes of examined changes

A simple metric of the sensitivity of capital gains ratherthan the sensitivity of credit spread to risk-free interest rateshas been proposed to study the interest rate sensitivity of UScorporate debt [9] emerging market sovereign bonds [10]and emerging market corporate fixed-income securities [11]is paper represents an extension of this line of research to

the US BBB-rated debt enhanced by Markov-switchinganalysis for rigorous detection of sensitivity regime changes

e capital gains metric is defined as the change incapital gains of a portfolio containing only US BBB-ratedbonds over the corresponding change in capital gains of aportfolio containing only UST While the connection be-tween risk-free interest rates and risky bond yields un-avoidably underlines our work the main focus however lieson the profits or losses of bond portfolio is approach isnot widespread in the main stream of financial analysis andusually is employed when the effects of taxation and taxregulation on the performance of financial assets are dis-cussed because tax legislation usually differentiates betweencapital gains and interim payments

It is also worth noting that hedging mid- to long-termexposures usually classified as hold-to-collect and hold-to-collect-and-sell may differ from intraday short-term hedgemethodologies Additionally the differences across thenormal and distressed market regimes in the behavior ofrisk-free and risky bond portfolios suggest that hedginginterest rate risk by shorting UST or equivalently holding aninterest rate swap that receives a floating rate for a fixed rateis rather not a completely efficient strategy

is paper is further motivated by a series of recentreports from regulatory authorities that show a renewedpublic interest on the relationship between interest rate riskasset valuation and regulatory framework [12ndash14] Com-mittee on the Global Financial System (CGFS) 2018) ecommon denominator of these reports is to enhance aframework of interest rate risk management as after aprolonged period of historically low interest rates a generalovervaluation of assets may pause dangers for financialstability In that discussion the sensitivity of financial assetsto changes in yields spreads and capital gains plays a centralrole

e rest of the paper is structured as follows Section 2describes the data and methodology Section 3 provides theempirical results Section 4 presents the application ofMarkov-switching model to confirm the robustness of theempirical results Section 5 discusses the implications of thecapital gains sensitivity approach from the theoretical andpractical perspectives Section 6 concludes the paper

2 Data and Methodology

We use the monthly time series of blended yields and av-erage coupons to model the price dynamics of the US BBB-rated and UST bond portfolios on a period fromMarch 2001to August 2016 Our data come from yield and couponindices which are used to model prices and investigate thedynamics of annual capital gains of both US BBB-ratedbonds and UST securities e reason for limiting theanalysis to the US nongovernmental BBB bond portfolio andthe UST portfolio is rooted in the importance of BBB-ratedfixed-income exposures for institutional investors in gen-eral and insurance companies and pension funds in par-ticular due to their attractive risk-return attributes On theother hand UST bonds represent investment targets ofchoice for many investors as they are largely considered to be

2 Complexity

safe-haven investments which are also commonly employedas proxies for risk-free instruments used for designing di-verse hedge strategies In addition by limiting the number ofindices used in our research we are able to keep undercontrol the necessary computation capacity and producevaluable investment insights

For US BBB bonds we use the Citi Broad Investment-Grade US Credit BBB Yield to Maturity Local (Bloombergticker S200YL) and the Citi Broad Investment-Grade USCredit BBB Average Coupon Local (Bloomberg tickerS200CP)e constituent members of the pair of the S200YLand S200CP indices are the same and represent nongov-ernmental BBB-rated debt issued in the US which comprisescorporate financial and municipal issuers For UST secu-rities we employ the Citigroup indices Treasuries Yield toMaturity Local (Bloomberg ticker SA14YL) and TreasuriesAverage Coupon Local (Bloomberg ticker SA14CP) econstituent members of the SA14YL and SA14CP are identicand represent US Treasury bonds excluding Treasury BillsWe resort to yield and coupon indices to analyze capitalgains because there is no price index available with similarlength and characteristics and because the focus is onportfolios rather than individual assets Since interest is notreinvested a total return index is not necessary Althoughherein only the US BBB bonds and UST portfolio areemployed we consider that our findings will hold also formany other portfolios such as Corporates Financials MuniEmerging Market (EM) Corporates EM Financial and EMSovereigns among many others

Our methodology is based on [11] but the focus is on theUS market rather than emerging economies e averageprice of each portfolio can be calculated by discountingfuture cash flows of coupons and principals For simplicitywe assume annual coupon payments principal redemptionat maturity and a flat yield curve Bonds that reach maturityor are downgraded are removed from the indices whilenewly issued ones are added In essence we assume aportfolio that perfectly mimics the composition of the re-spective index and is continuously rebalanced according toany changes within that indexis assumption is frequentlyused to study risk minimization strategies for portfolioimmunization (eg [15]) In our case the continuousrebalancing happens on a monthly basis

e price PUST of a UST portfolio with an investmenthorizon (residual maturity) T average annual coupon CUSTface value FUST and yield yUST is

PUST 1113944T

i1

CUST

1 + yUST( 1113857k

+FUST

1 + yUST( 1113857T (1)

where yUST is the blended yield given by the SA14YL indexand CUST is the average coupon given by the SA14CP indexe nominal face value FUST is set to US$ 1000 million

Capital gain is defined as the difference between theinitial price and the final price of the entire portfolio ex-cluding interim coupon payments over a given period Inessence the initial price is the price the investor would haveto pay to purchase one fraction of the bond index while thefinal price is the price at the end of the given period As the

indices are continuously rebalanced our asset is continu-ously changing Since we do not aim to assess the efficiencyof rebalancing the associated transaction costs and costs ofcarry are ignored After the historical price series is con-structed the capital gain can be written as

CGUST(t H) PUST(t + H) minus PUST(t) (2)

where CGUST stands for the capital gains of the UST port-folio t is the initial date of the analyzed time interval and Hstands for a time horizon over which the capital gains areassessed e time horizon is one year since capital gains ofinvestment funds are transferred at the end of each calendaryear [16] e same approach is also applied for analyzingthe capital gains of the US BBB-rated bond portfolioSimilarly the price of the modeled US BBB-rated portfoliowith the same residual maturity as the UST portfolio is

PBBB 1113944T

i1

CBBB

1 + yBBB( 1113857k

+FBBB

1 + yBBB( 1113857T (3)

where CBBB stands for average annual coupon FBBB standsfor face value of a principal payment and yBBB is the blendedyield e yields and coupons are given by the respectiveindices S200YL and S200CP e face value is again US$1000 million Capital gains of US BBB-rated bond portfoliocan be written as

CGBBB(t H) PBBB(t + H) minus PBBB(t) (4)

with symmetric notation Equations (2) and (4) can be usedto construct a time series of capital gains CGUST and CGBBBwhich can be used to calculate the sensitivity of the relativelyrisky US BBB-rated bond portfolio capital gains to thecapital gains of the risk-free UST bond portfolio Followingthe earlier definition the sensitivity SBBBUST (t2 t1 H) is theratio of the capital gain of the corporate bond portfolio to thecapital gain of the UST portfolio over a period (t1 t2)

SBBBUST t2 t1 H( 1113857 CG t2 H( 1113857BBB minus CG t1 H( 1113857BBBCG t2 H( 1113857UST minus CG t1 H( 1113857UST

ΔCG t2 t1 H( 1113857BBBΔCG t2 t1 H( 1113857UST

(5)

e horizon H is moved forward by the number of daysequal to t2 minus t1 To make the capital gain-wise sensitivitymore representative we calibrate the rolling window oflength H so that it captures the more pronounced moves inthe capital gains time series of the modeled UST portfolioAn infinitesimal or even zero change in the capital gain ofthe UST portfolio amplifies the computational uncertaintyfor the ratio and a zero denominator may render it useless

It is important to remark that although the measurepractically uses bond face values and yields it can alsoprovide intuition for the dynamics of interest rates Interestrates and the yields of USTmove towards the same directionso an increase in the Fed rate pushes the yield curve for USTbonds upwards Changes in nominal rates (eg due to in-flation fiscal or monetary policy) are matched by changes inthe yield curve with a direct effect on the credit spread

Complexity 3

between government and corporate bonds Our method fitswell with a strategy that hedges interest rate risk by shortingthe USTportfolio and essentially assesses the profitability ofa long position in US BBB-rated bonds coupled with a shortposition in government bonds is intuition is useful forconnecting our findings with the wider literature that ex-plicitly uses interest rates and spreads

3 Empirical Results

31 Present Values and Capital Gains of Bond PortfoliosFigure 1 presents the historical time series of the yields andcoupons of the UST bonds and US BBB-rated bonds

For UST bonds the average coupon almost always exceedsthe corresponding yield rate with the exception of the two-year-long period preceding the global financial crisis namelyfrom the second half of 2005 until the end of the first half of2007 To a lesser extent the same holds for corporate bondsapart from the financial crisis period mid-2008 to mid-2010Notably the surge in the yields of corporate bonds during thefinancial crisis is well above the coupon curve signifying aperiod where these assets were traded at a great discount ischange in trend will become apparent later on

We then proceed to discount the future cash flows for thetwo portfolios based on the yield and coupon data eresulting bond prices (see Figure 2) demonstrate a similarpattern between mid-2003 and the outbreak of the financialcrisis in mid-2007 Since the second half of 2007 until the peakof the crisis a large flight-to-quality event is observed wherethe prices of safe assets increase and the prices of risky assetsdecrease [17] In 2009 this flight-to-quality stops and the gapbetween the two curves vanishes Notably the ranges of thepresent values of the two portfolios differ considerably erange relative to the UST portfolio (US$ 150 million) is nar-rower than the respective range of the present values of the USBBB-rated bond portfolio (US$ 250 million)

Capital gains for each portfolio can be directly calculatedfrom our earlier definition While the interpretation for theseparate UST and US BBB-rated bond portfolios isstraightforward we also discuss the capital gains of a USBBB-rated portfolio hedged by taking a short position inUST bonds In this case

CGhedged CGBBB minus CGUST (6)

Figure 3 represents the time behavior of yearly changesin present value for the US BBB-rated bond portfolio therisk-free UST portfolio and a portfolio of US BBB-ratedbonds hedged by shorting the respective USTportfolio Eachpoint represents the changes in present value having takenplace in the preceding year Between July 2007 and De-cember 2010 the period of the global financial crisis and itsimmediate aftermath the annual capital gains of the BBB-rated bond portfolio and UST bond portfolio move inopposite directions Hence the interest rate hedging of USBBB-rated debt with the short UST positions does notcompensate the negative impacts when such compensationis most needed

32 Capital Gain-Wise Interest Rate Sensitivity of US BBB-Rated Debt e main intuition lies in the interpretation ofcapital gains sensitivity across the portfolios we considernamely the US BBB-rated and UST portfolios and thehedged portfolio Instead of using average sensitivity overthe entire sample we examine the behavior of sensitivitywithin much shorter time intervals determined by localextrema Local minima and maxima are treated as turningpoints which separate upward and downward tendencies incapital gains dynamics

e tracking of large movements in UST capital gainsreduces the uncertainty in the denominator of equation (5)and hence increases the precision of sensitivity measure-ments e extrema are identified in the time series of theUST portfolio as in [18] is leads to 51 sufficiently largeand distinct movements both upwards and downwardswhich define the intervals we use and are split into threegroups according to the time of occurrence 18 movementstook place during the precrisis period (March 2002ndashMay2007) 8 movements belong to the crisis period (May2007ndashOctober 2010) and 25 movements happened duringthe postcrisis period (October 2010ndashAugust 2016) Sensi-tivity is calculated separately for each of the intervals eresulting pattern can be seen in Figure 4 along with averagesensitivity for each of the three phases

e bold line depicts the two different regimes Duringnormal economic conditions the average sensitivity ispositive while during the crisis it is negative and amplifiedWe clearly observe the time-varying behavior of capital gainssensitivity over each phase similar to [9 11] e plot showsthat sensitivity is mostly positive before 2007 and after 2011while it turns negative between that time interval isimplies a regime change during and slightly after the periodof the financial crisis From 2001 till 2007 the sensitivityremains positive from 2007 till 2011 it turns negative andfrom 2011 till 2016 it reverts back to positive ground Toproperly calculate the sign and value of sensitivity in a waythat is not affected by time variation we take the product ofcapital gains and duration (time interval) over which thesensitivity point estimates are given is yields a weightedsum of capital gains for each portfolio where the weight isthe fraction of time over which sensitivity (capital gains) wascalculatede average period sensitivity is thus the ratio ofthe weighted sum of capital gains for the US BBB-ratedportfolio divided by the UST equivalent In comparisonspread-to-rate sensitivity typically takes daily values whichare averaged over a longer period whereas our metric fo-cuses on start and end points

As discussed earlier the definition of our measure can beinterpreted as a gauge of hedging success when a longposition in the BBB-rated portfolio is balanced by a shortposition in the USTportfolio In terms of cash flows this canbe seen as either a permanent position or an interest rateswap that pays a fixed rate to receive a floating rate equal tothat of the corporate bond portfolio It becomes apparentthat all the gains from the short position in UST during theprecrisis period are wiped away during the crisis downturnand recovery As such an all-weather hedge or its equivalentin the form of a swap is inefficient

4 Complexity

Section 4 confirms our empirical observation of capitalgain-wise interest rate sensitivity regime changes by appli-cation of Markov-switching model

4 Markov-Switching Modeling andRegime Changes

Up to this point our empirical findings point towards astructural break in the time series we use is underlines afunctional discrepancy best represented by the contradictionbetween the Merton [4] model implying a permanentlynegative relationship between credit spreads and interestrates and the Kamin and von Kleist [6] approach which

does not identify such a relationship for a certain periodGiven the direct correspondence between interest rates andthe yields of government bonds it is straightforward toconnect the intuition of the previous section to the tworegimes To test the hypothesis of two alternating states onewhere the Merton model is valid and capital gains sensitivity(and conversely the relationship between credit spreads andinterest rates) is negative and one where capital gainssensitivity is close to zero (and the relationship betweeninterest rates and credit spreads is roughly one-to-one) as in[6 7] we estimate a simple Markov-switching model withtwo regimes using the time series of sensitivity and capitalgains of the hedged bond portfolio e figures in the

0123456789

1011

SA14YL indexSA14CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(a)

0123456789

1011

S200YL indexS200CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(b)

Figure 1 Yields and average coupons for the US BBB-rated (a) and UST bonds (b)

Complexity 5

previous section show a possible regime switch in the periodaverages but we also examine whether the two differentregimes may also have distinct volatilities

From the vast literature on Markov-switching and re-gime-switching models we select a set of simple applicationsin line with the arguments detailed in [19 20] For widerapplications on asset pricing portfolio optimization andstochastic volatility [21] contains a comprehensive literaturereview We employ a simple two-state dynamic regressionMarkov-switching model (DRMS) which is more suitable tothe monthly frequency of our data A DRMS specification

allows the probabilities to change instantaneously acrossstates since the realization of each state st does not dependon stminus1 but is drawn from a discrete probability distributionOn the other hand an autoregressive (AR) specificationwhere st follows a Markov process thus allowing thetransition probabilities to follow an AR process wouldintroduce autocorrelation in states since st would depend onstminus1 is is an unwelcome feature for our model for manyreasons Markov-switching AR (MSAR) models are bettersuited for quarterly or annual data and the introduction oflagged states multiplies the states In our case due to

850

900

950

1000

1050

1100

1150

PV BBB-rated portfolioPV UST portfolio

31

01

31

02

31

03

31

04

31

05

31

06

31

07

31

08

31

09

31

10

31

11

31

12

31

13

31

14

31

15

31

16

Figure 2 Present values of portfolios US BBB-rated bonds vs UST

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

BBB hedged by short UST

BBB long portfolioUST long portfolio

Figure 3 Yearly capital gains for the US BBB-rated UST and hedged portfolios

6 Complexity

autoregression there would be four possible regimes insteadof two without providing any additional intuition Also anMSAR specification would not allow the probabilities toadjust quickly enough to changes in states Regardlesspreliminary estimations of an AR specification yielded verysimilar results so we base our discussion on the DRMSmodels only

A general specification of the DRMSmodel that capturesall the subcases we consider is

Yt μLst + εst εst sim N 0 σ2st1113872 1113873 (7)

where t denotes time Y is the time series (sensitivity orportfolio capital gains) s denotes the state and is set to 1 or2 micro and σ denote the state dependent model parameters ofthe mean (intercept) and the standard deviation andinnovations ε are independent and identically distributedfollowing a normal distribution with state-dependentvariance We consider two variations one where themeans are state dependent and one where not only themeans but also the variances are state-dependent too enotation can be simplified in the case where only themeans are allowed to switch across states to σ2s = σ2(constant across both time and states) and remains asabove in the case where both means and variances are statedependent (μs and σ2s omitting t for brevity) Τhis leads tofour different estimations two for sensitivity and two forcapital gains e Akaike HannanndashQuinn and Schwarzinformation criteria show that the simplest model with nolagged terms is the most suitable one for all cases for bothsensitivity and capital gains hence we do not introducelags

41 Regime Switching in Sensitivity Table 1 reports the re-sults for capital gains sensitivity when only the means arestate dependent under a model specified as Yt μst + εtεtsimΝ(0 σ2) for states s 1 2 State 1 corresponds to theMerton model where s is negative and state 2 corresponds

to the Kamin and von Kleist approach where s is near zeroand positivee estimation verifies our earlier observationssince micro12 are both statistically significant with a negativevalue of minus241 and a positive value of 070 respectively eprobabilities for each state are 0838 and 0968 respectivelywhich show very high persistence for both regimes

e transition probabilities from state 1 (2) to state 2 (1)are 0162 (0032) which show a very low rate of change (seeTable 2) is indicates very clear breaks in the time seriesand high certainty over which state occurs which is a directoutcome of the DRMS specification

Figure 5 shows the transition probabilities compared tothe evolution of sensitivity during our period

Figure 5 evidences a dominance of the second state upuntil 2009 with a brief exception around 2003 and adominance of the first state between 2009 and 2011 State 1briefly reappears at the end of 2012 During the 2009ndash2012period any changes from state 1 to state 2 are instantaneousmeaning that the sensitivity reverts immediately back to itsinitial regime After 2012 the second state is always present

ese findings show that sensitivity and credit spreadsvary over time with two distinct regimes appearing whichcorrespond to two different theoretical approaches on therelationship between credit spreads and interest ratesDuring crises credit spreads and interest rates move inopposite directions following the Merton pattern of negativesensitivity During normal periods any changes in interestrates (risk-free yields) pass on nearly completely to the yieldsof corporate bonds which implies stable credit spreads andthe Kamin and von Kleist pattern

When regime changes are allowed in both the mean andthe variance the parameters and transition probabilities aregenerally similar (Table 3) e model specification followsequation (7) However the plot of the transition probabil-ities over the time series reveals that only state 2 manifeststhroughout our entire period ie there is not a switch fromone pattern to the other and only the Kamin and von Kleistrationale is validated e variance of state 1 (090) isconsiderably higher than the variance of state 2 (051) andslightly higher than the variance of the means-only regime-switching model (079) Moreover the means of the twostates are closer compared to the previous case One possibleexplanation is that the higher variance combined with thehigher mean of state 1makes the first state redundant despiteits high persistency e mean estimate of minus134 has astandard deviation of 02 which coupled with the standarddeviation of the state brings it well within the range of state 2We therefore conjecture that this similarity causes the modelnot to discriminate between states

42 Regime Switching in the US BBB Portfolio Hedged byShorting UST We can now proceed to the capital gains ofthe hedged synthetic portfolio where UST bonds are shortede capital gains of the hedged portfolio are calculated asfollows First the present values of the UST and BBBportfolios are calculated using equations (1) and (3) settingT 5 and FV 1000 Since the S200YL S200CP SA14YLand SA14CP indices are monthly we calculate monthly

ndash350ndash300ndash250ndash200ndash150ndash100ndash050

000050100150200250

310

320

02

310

320

03

310

320

04

310

320

05

310

320

06

310

320

07

310

320

08

310

320

09

310

320

10

310

320

11

310

320

12

310

320

13

310

320

14

310

320

15

310

320

16

On-the-move sensitivityPhase-averaged sensitivity

Figure 4 Phase-averaged sensitivity of US BBB-rated bonds2002ndash2016

Complexity 7

present values for the two portfolios We then calculate thedifference between the present value of month t and thepresent value of month tminus 12 or the difference over anannual interval e capital gain of the hedged portfolio istherefore the capital gain of the long position on BBBcorporate bonds minus the capital gain of the short positionon US Treasury bonds e generated capital gains spanfrom March 2002 till September 2016 More succinctlyformula (6) can be written as follows for H 12 months andP the present value (price) of a portfolio

CGhedged(t H) CGBBB minus CGBBB PBBB(t + H) minus PBBB(t)

minus PBBB(t + H) minus PBBB(t)

(8)

eCGhedged series is used to estimate different Markov-switching models whose results are reported in Tables 4 and5 We follow the same approach with sensitivity and employa DRMS model with no lags e pattern is similar to that ofsensitivity When the variance is common for the two statesand only the means change there are two major regimeswitches (Figure 6)

One of the changes which lasts longer takes place be-tween 2010 and 2011 which corresponds to the financialcrisis e other change takes place in 2004 is slightly lesspronounced and coincides with a minimum in the timeseries of UST capital gains It is worth noting that thisminimum of UST capital gains occurs within the one-year-long time interval (June 2003ndashJune 2004) when the federalfund target rate remains at its local minimum value of 1e transition probabilities are again very persistent asshown in Table 4 State 1 has a probability of 099 state 2 hasa probability of 0896 and the probability to move from state1 (2) to state 2 (1) is 001 (0104) Similarly the mean of thefirst state (minus1139) is negative and corresponds to theMertoncase while the mean of the second state (13479) is positiveand corresponds to the Kamin and von Kleist case (seeTable 4)

In its turn Table 5 demonstrates that when switches inboth the means and the variances are allowed the result issimilar to the previous section e means changemarginally but the much higher variance of one state(6901) causes it to dominate over the second state whichhas a variance similar to that of the means-only switch(4297)

Figure 7 shows how the probability of a state to be re-alized remains at 2 or equivalently how the other statepersists and there are no switches during the period On theother hand Figure 6 clearly shows two different states whenthe means are state dependent but the variance is not ismakes state-dependent variances inappropriate for ourpurposes Allowing for a change in variances leads to muchhigher estimates which cause one state to cover a muchwider range of potential values is makes the model un-informative and unable to pick the differences that interestthe paper As a result we base our results in the cases wherethe variance across states is common and the regime switchis in the mean

Table 1 Markov-switching regression in means only forsensitivity

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus2408 0163 minus1474 0000 (minus2729 minus2088)micro2 0701 0662 1060 0000 (0571 0831)σ 0791 0044 (0709 0882)p11 0838 0073 (0644 0937)p21 0033 0015 (0014 0077)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

Table 2 Transition probabilities between state 1 and state 2 for allmodels

Regime switch in means onlySensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0838 0162 State 1 0987 0013State 2 0033 0967 State 2 0104 0897

Regime switch in means and standard deviationsSensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0837 0163 State 1 0987 0013State 2 0111 0889 State 2 0087 0913e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

2

1

0

ndash1

ndash2

ndash3

2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 2018m1Date

State 1 one-step probabilitiesSensitivity of BBB bonds

Figure 5 Regime change in means only state plot for the sen-sitivity time series Probability with respect to state 1

Table 3 Markov-switching regression in means and standarddeviations for sensitivity

Parameter Estimate Sterror Z Pgt |z| 95 confidence

intervalmicro1 minus1345 0201 minus668 0000 (minus1739minus0951)micro2 0842 0142 594 0000 (0564 1119)σ1 0909 0097 (0739 1119)σ2 0516 0042 (0439 0606)p11 0837 0048 (0720 0911)p21 0111 0032 (0629 0190)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

8 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

safe-haven investments which are also commonly employedas proxies for risk-free instruments used for designing di-verse hedge strategies In addition by limiting the number ofindices used in our research we are able to keep undercontrol the necessary computation capacity and producevaluable investment insights

For US BBB bonds we use the Citi Broad Investment-Grade US Credit BBB Yield to Maturity Local (Bloombergticker S200YL) and the Citi Broad Investment-Grade USCredit BBB Average Coupon Local (Bloomberg tickerS200CP)e constituent members of the pair of the S200YLand S200CP indices are the same and represent nongov-ernmental BBB-rated debt issued in the US which comprisescorporate financial and municipal issuers For UST secu-rities we employ the Citigroup indices Treasuries Yield toMaturity Local (Bloomberg ticker SA14YL) and TreasuriesAverage Coupon Local (Bloomberg ticker SA14CP) econstituent members of the SA14YL and SA14CP are identicand represent US Treasury bonds excluding Treasury BillsWe resort to yield and coupon indices to analyze capitalgains because there is no price index available with similarlength and characteristics and because the focus is onportfolios rather than individual assets Since interest is notreinvested a total return index is not necessary Althoughherein only the US BBB bonds and UST portfolio areemployed we consider that our findings will hold also formany other portfolios such as Corporates Financials MuniEmerging Market (EM) Corporates EM Financial and EMSovereigns among many others

Our methodology is based on [11] but the focus is on theUS market rather than emerging economies e averageprice of each portfolio can be calculated by discountingfuture cash flows of coupons and principals For simplicitywe assume annual coupon payments principal redemptionat maturity and a flat yield curve Bonds that reach maturityor are downgraded are removed from the indices whilenewly issued ones are added In essence we assume aportfolio that perfectly mimics the composition of the re-spective index and is continuously rebalanced according toany changes within that indexis assumption is frequentlyused to study risk minimization strategies for portfolioimmunization (eg [15]) In our case the continuousrebalancing happens on a monthly basis

e price PUST of a UST portfolio with an investmenthorizon (residual maturity) T average annual coupon CUSTface value FUST and yield yUST is

PUST 1113944T

i1

CUST

1 + yUST( 1113857k

+FUST

1 + yUST( 1113857T (1)

where yUST is the blended yield given by the SA14YL indexand CUST is the average coupon given by the SA14CP indexe nominal face value FUST is set to US$ 1000 million

Capital gain is defined as the difference between theinitial price and the final price of the entire portfolio ex-cluding interim coupon payments over a given period Inessence the initial price is the price the investor would haveto pay to purchase one fraction of the bond index while thefinal price is the price at the end of the given period As the

indices are continuously rebalanced our asset is continu-ously changing Since we do not aim to assess the efficiencyof rebalancing the associated transaction costs and costs ofcarry are ignored After the historical price series is con-structed the capital gain can be written as

CGUST(t H) PUST(t + H) minus PUST(t) (2)

where CGUST stands for the capital gains of the UST port-folio t is the initial date of the analyzed time interval and Hstands for a time horizon over which the capital gains areassessed e time horizon is one year since capital gains ofinvestment funds are transferred at the end of each calendaryear [16] e same approach is also applied for analyzingthe capital gains of the US BBB-rated bond portfolioSimilarly the price of the modeled US BBB-rated portfoliowith the same residual maturity as the UST portfolio is

PBBB 1113944T

i1

CBBB

1 + yBBB( 1113857k

+FBBB

1 + yBBB( 1113857T (3)

where CBBB stands for average annual coupon FBBB standsfor face value of a principal payment and yBBB is the blendedyield e yields and coupons are given by the respectiveindices S200YL and S200CP e face value is again US$1000 million Capital gains of US BBB-rated bond portfoliocan be written as

CGBBB(t H) PBBB(t + H) minus PBBB(t) (4)

with symmetric notation Equations (2) and (4) can be usedto construct a time series of capital gains CGUST and CGBBBwhich can be used to calculate the sensitivity of the relativelyrisky US BBB-rated bond portfolio capital gains to thecapital gains of the risk-free UST bond portfolio Followingthe earlier definition the sensitivity SBBBUST (t2 t1 H) is theratio of the capital gain of the corporate bond portfolio to thecapital gain of the UST portfolio over a period (t1 t2)

SBBBUST t2 t1 H( 1113857 CG t2 H( 1113857BBB minus CG t1 H( 1113857BBBCG t2 H( 1113857UST minus CG t1 H( 1113857UST

ΔCG t2 t1 H( 1113857BBBΔCG t2 t1 H( 1113857UST

(5)

e horizon H is moved forward by the number of daysequal to t2 minus t1 To make the capital gain-wise sensitivitymore representative we calibrate the rolling window oflength H so that it captures the more pronounced moves inthe capital gains time series of the modeled UST portfolioAn infinitesimal or even zero change in the capital gain ofthe UST portfolio amplifies the computational uncertaintyfor the ratio and a zero denominator may render it useless

It is important to remark that although the measurepractically uses bond face values and yields it can alsoprovide intuition for the dynamics of interest rates Interestrates and the yields of USTmove towards the same directionso an increase in the Fed rate pushes the yield curve for USTbonds upwards Changes in nominal rates (eg due to in-flation fiscal or monetary policy) are matched by changes inthe yield curve with a direct effect on the credit spread

Complexity 3

between government and corporate bonds Our method fitswell with a strategy that hedges interest rate risk by shortingthe USTportfolio and essentially assesses the profitability ofa long position in US BBB-rated bonds coupled with a shortposition in government bonds is intuition is useful forconnecting our findings with the wider literature that ex-plicitly uses interest rates and spreads

3 Empirical Results

31 Present Values and Capital Gains of Bond PortfoliosFigure 1 presents the historical time series of the yields andcoupons of the UST bonds and US BBB-rated bonds

For UST bonds the average coupon almost always exceedsthe corresponding yield rate with the exception of the two-year-long period preceding the global financial crisis namelyfrom the second half of 2005 until the end of the first half of2007 To a lesser extent the same holds for corporate bondsapart from the financial crisis period mid-2008 to mid-2010Notably the surge in the yields of corporate bonds during thefinancial crisis is well above the coupon curve signifying aperiod where these assets were traded at a great discount ischange in trend will become apparent later on

We then proceed to discount the future cash flows for thetwo portfolios based on the yield and coupon data eresulting bond prices (see Figure 2) demonstrate a similarpattern between mid-2003 and the outbreak of the financialcrisis in mid-2007 Since the second half of 2007 until the peakof the crisis a large flight-to-quality event is observed wherethe prices of safe assets increase and the prices of risky assetsdecrease [17] In 2009 this flight-to-quality stops and the gapbetween the two curves vanishes Notably the ranges of thepresent values of the two portfolios differ considerably erange relative to the UST portfolio (US$ 150 million) is nar-rower than the respective range of the present values of the USBBB-rated bond portfolio (US$ 250 million)

Capital gains for each portfolio can be directly calculatedfrom our earlier definition While the interpretation for theseparate UST and US BBB-rated bond portfolios isstraightforward we also discuss the capital gains of a USBBB-rated portfolio hedged by taking a short position inUST bonds In this case

CGhedged CGBBB minus CGUST (6)

Figure 3 represents the time behavior of yearly changesin present value for the US BBB-rated bond portfolio therisk-free UST portfolio and a portfolio of US BBB-ratedbonds hedged by shorting the respective USTportfolio Eachpoint represents the changes in present value having takenplace in the preceding year Between July 2007 and De-cember 2010 the period of the global financial crisis and itsimmediate aftermath the annual capital gains of the BBB-rated bond portfolio and UST bond portfolio move inopposite directions Hence the interest rate hedging of USBBB-rated debt with the short UST positions does notcompensate the negative impacts when such compensationis most needed

32 Capital Gain-Wise Interest Rate Sensitivity of US BBB-Rated Debt e main intuition lies in the interpretation ofcapital gains sensitivity across the portfolios we considernamely the US BBB-rated and UST portfolios and thehedged portfolio Instead of using average sensitivity overthe entire sample we examine the behavior of sensitivitywithin much shorter time intervals determined by localextrema Local minima and maxima are treated as turningpoints which separate upward and downward tendencies incapital gains dynamics

e tracking of large movements in UST capital gainsreduces the uncertainty in the denominator of equation (5)and hence increases the precision of sensitivity measure-ments e extrema are identified in the time series of theUST portfolio as in [18] is leads to 51 sufficiently largeand distinct movements both upwards and downwardswhich define the intervals we use and are split into threegroups according to the time of occurrence 18 movementstook place during the precrisis period (March 2002ndashMay2007) 8 movements belong to the crisis period (May2007ndashOctober 2010) and 25 movements happened duringthe postcrisis period (October 2010ndashAugust 2016) Sensi-tivity is calculated separately for each of the intervals eresulting pattern can be seen in Figure 4 along with averagesensitivity for each of the three phases

e bold line depicts the two different regimes Duringnormal economic conditions the average sensitivity ispositive while during the crisis it is negative and amplifiedWe clearly observe the time-varying behavior of capital gainssensitivity over each phase similar to [9 11] e plot showsthat sensitivity is mostly positive before 2007 and after 2011while it turns negative between that time interval isimplies a regime change during and slightly after the periodof the financial crisis From 2001 till 2007 the sensitivityremains positive from 2007 till 2011 it turns negative andfrom 2011 till 2016 it reverts back to positive ground Toproperly calculate the sign and value of sensitivity in a waythat is not affected by time variation we take the product ofcapital gains and duration (time interval) over which thesensitivity point estimates are given is yields a weightedsum of capital gains for each portfolio where the weight isthe fraction of time over which sensitivity (capital gains) wascalculatede average period sensitivity is thus the ratio ofthe weighted sum of capital gains for the US BBB-ratedportfolio divided by the UST equivalent In comparisonspread-to-rate sensitivity typically takes daily values whichare averaged over a longer period whereas our metric fo-cuses on start and end points

As discussed earlier the definition of our measure can beinterpreted as a gauge of hedging success when a longposition in the BBB-rated portfolio is balanced by a shortposition in the USTportfolio In terms of cash flows this canbe seen as either a permanent position or an interest rateswap that pays a fixed rate to receive a floating rate equal tothat of the corporate bond portfolio It becomes apparentthat all the gains from the short position in UST during theprecrisis period are wiped away during the crisis downturnand recovery As such an all-weather hedge or its equivalentin the form of a swap is inefficient

4 Complexity

Section 4 confirms our empirical observation of capitalgain-wise interest rate sensitivity regime changes by appli-cation of Markov-switching model

4 Markov-Switching Modeling andRegime Changes

Up to this point our empirical findings point towards astructural break in the time series we use is underlines afunctional discrepancy best represented by the contradictionbetween the Merton [4] model implying a permanentlynegative relationship between credit spreads and interestrates and the Kamin and von Kleist [6] approach which

does not identify such a relationship for a certain periodGiven the direct correspondence between interest rates andthe yields of government bonds it is straightforward toconnect the intuition of the previous section to the tworegimes To test the hypothesis of two alternating states onewhere the Merton model is valid and capital gains sensitivity(and conversely the relationship between credit spreads andinterest rates) is negative and one where capital gainssensitivity is close to zero (and the relationship betweeninterest rates and credit spreads is roughly one-to-one) as in[6 7] we estimate a simple Markov-switching model withtwo regimes using the time series of sensitivity and capitalgains of the hedged bond portfolio e figures in the

0123456789

1011

SA14YL indexSA14CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(a)

0123456789

1011

S200YL indexS200CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(b)

Figure 1 Yields and average coupons for the US BBB-rated (a) and UST bonds (b)

Complexity 5

previous section show a possible regime switch in the periodaverages but we also examine whether the two differentregimes may also have distinct volatilities

From the vast literature on Markov-switching and re-gime-switching models we select a set of simple applicationsin line with the arguments detailed in [19 20] For widerapplications on asset pricing portfolio optimization andstochastic volatility [21] contains a comprehensive literaturereview We employ a simple two-state dynamic regressionMarkov-switching model (DRMS) which is more suitable tothe monthly frequency of our data A DRMS specification

allows the probabilities to change instantaneously acrossstates since the realization of each state st does not dependon stminus1 but is drawn from a discrete probability distributionOn the other hand an autoregressive (AR) specificationwhere st follows a Markov process thus allowing thetransition probabilities to follow an AR process wouldintroduce autocorrelation in states since st would depend onstminus1 is is an unwelcome feature for our model for manyreasons Markov-switching AR (MSAR) models are bettersuited for quarterly or annual data and the introduction oflagged states multiplies the states In our case due to

850

900

950

1000

1050

1100

1150

PV BBB-rated portfolioPV UST portfolio

31

01

31

02

31

03

31

04

31

05

31

06

31

07

31

08

31

09

31

10

31

11

31

12

31

13

31

14

31

15

31

16

Figure 2 Present values of portfolios US BBB-rated bonds vs UST

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

BBB hedged by short UST

BBB long portfolioUST long portfolio

Figure 3 Yearly capital gains for the US BBB-rated UST and hedged portfolios

6 Complexity

autoregression there would be four possible regimes insteadof two without providing any additional intuition Also anMSAR specification would not allow the probabilities toadjust quickly enough to changes in states Regardlesspreliminary estimations of an AR specification yielded verysimilar results so we base our discussion on the DRMSmodels only

A general specification of the DRMSmodel that capturesall the subcases we consider is

Yt μLst + εst εst sim N 0 σ2st1113872 1113873 (7)

where t denotes time Y is the time series (sensitivity orportfolio capital gains) s denotes the state and is set to 1 or2 micro and σ denote the state dependent model parameters ofthe mean (intercept) and the standard deviation andinnovations ε are independent and identically distributedfollowing a normal distribution with state-dependentvariance We consider two variations one where themeans are state dependent and one where not only themeans but also the variances are state-dependent too enotation can be simplified in the case where only themeans are allowed to switch across states to σ2s = σ2(constant across both time and states) and remains asabove in the case where both means and variances are statedependent (μs and σ2s omitting t for brevity) Τhis leads tofour different estimations two for sensitivity and two forcapital gains e Akaike HannanndashQuinn and Schwarzinformation criteria show that the simplest model with nolagged terms is the most suitable one for all cases for bothsensitivity and capital gains hence we do not introducelags

41 Regime Switching in Sensitivity Table 1 reports the re-sults for capital gains sensitivity when only the means arestate dependent under a model specified as Yt μst + εtεtsimΝ(0 σ2) for states s 1 2 State 1 corresponds to theMerton model where s is negative and state 2 corresponds

to the Kamin and von Kleist approach where s is near zeroand positivee estimation verifies our earlier observationssince micro12 are both statistically significant with a negativevalue of minus241 and a positive value of 070 respectively eprobabilities for each state are 0838 and 0968 respectivelywhich show very high persistence for both regimes

e transition probabilities from state 1 (2) to state 2 (1)are 0162 (0032) which show a very low rate of change (seeTable 2) is indicates very clear breaks in the time seriesand high certainty over which state occurs which is a directoutcome of the DRMS specification

Figure 5 shows the transition probabilities compared tothe evolution of sensitivity during our period

Figure 5 evidences a dominance of the second state upuntil 2009 with a brief exception around 2003 and adominance of the first state between 2009 and 2011 State 1briefly reappears at the end of 2012 During the 2009ndash2012period any changes from state 1 to state 2 are instantaneousmeaning that the sensitivity reverts immediately back to itsinitial regime After 2012 the second state is always present

ese findings show that sensitivity and credit spreadsvary over time with two distinct regimes appearing whichcorrespond to two different theoretical approaches on therelationship between credit spreads and interest ratesDuring crises credit spreads and interest rates move inopposite directions following the Merton pattern of negativesensitivity During normal periods any changes in interestrates (risk-free yields) pass on nearly completely to the yieldsof corporate bonds which implies stable credit spreads andthe Kamin and von Kleist pattern

When regime changes are allowed in both the mean andthe variance the parameters and transition probabilities aregenerally similar (Table 3) e model specification followsequation (7) However the plot of the transition probabil-ities over the time series reveals that only state 2 manifeststhroughout our entire period ie there is not a switch fromone pattern to the other and only the Kamin and von Kleistrationale is validated e variance of state 1 (090) isconsiderably higher than the variance of state 2 (051) andslightly higher than the variance of the means-only regime-switching model (079) Moreover the means of the twostates are closer compared to the previous case One possibleexplanation is that the higher variance combined with thehigher mean of state 1makes the first state redundant despiteits high persistency e mean estimate of minus134 has astandard deviation of 02 which coupled with the standarddeviation of the state brings it well within the range of state 2We therefore conjecture that this similarity causes the modelnot to discriminate between states

42 Regime Switching in the US BBB Portfolio Hedged byShorting UST We can now proceed to the capital gains ofthe hedged synthetic portfolio where UST bonds are shortede capital gains of the hedged portfolio are calculated asfollows First the present values of the UST and BBBportfolios are calculated using equations (1) and (3) settingT 5 and FV 1000 Since the S200YL S200CP SA14YLand SA14CP indices are monthly we calculate monthly

ndash350ndash300ndash250ndash200ndash150ndash100ndash050

000050100150200250

310

320

02

310

320

03

310

320

04

310

320

05

310

320

06

310

320

07

310

320

08

310

320

09

310

320

10

310

320

11

310

320

12

310

320

13

310

320

14

310

320

15

310

320

16

On-the-move sensitivityPhase-averaged sensitivity

Figure 4 Phase-averaged sensitivity of US BBB-rated bonds2002ndash2016

Complexity 7

present values for the two portfolios We then calculate thedifference between the present value of month t and thepresent value of month tminus 12 or the difference over anannual interval e capital gain of the hedged portfolio istherefore the capital gain of the long position on BBBcorporate bonds minus the capital gain of the short positionon US Treasury bonds e generated capital gains spanfrom March 2002 till September 2016 More succinctlyformula (6) can be written as follows for H 12 months andP the present value (price) of a portfolio

CGhedged(t H) CGBBB minus CGBBB PBBB(t + H) minus PBBB(t)

minus PBBB(t + H) minus PBBB(t)

(8)

eCGhedged series is used to estimate different Markov-switching models whose results are reported in Tables 4 and5 We follow the same approach with sensitivity and employa DRMS model with no lags e pattern is similar to that ofsensitivity When the variance is common for the two statesand only the means change there are two major regimeswitches (Figure 6)

One of the changes which lasts longer takes place be-tween 2010 and 2011 which corresponds to the financialcrisis e other change takes place in 2004 is slightly lesspronounced and coincides with a minimum in the timeseries of UST capital gains It is worth noting that thisminimum of UST capital gains occurs within the one-year-long time interval (June 2003ndashJune 2004) when the federalfund target rate remains at its local minimum value of 1e transition probabilities are again very persistent asshown in Table 4 State 1 has a probability of 099 state 2 hasa probability of 0896 and the probability to move from state1 (2) to state 2 (1) is 001 (0104) Similarly the mean of thefirst state (minus1139) is negative and corresponds to theMertoncase while the mean of the second state (13479) is positiveand corresponds to the Kamin and von Kleist case (seeTable 4)

In its turn Table 5 demonstrates that when switches inboth the means and the variances are allowed the result issimilar to the previous section e means changemarginally but the much higher variance of one state(6901) causes it to dominate over the second state whichhas a variance similar to that of the means-only switch(4297)

Figure 7 shows how the probability of a state to be re-alized remains at 2 or equivalently how the other statepersists and there are no switches during the period On theother hand Figure 6 clearly shows two different states whenthe means are state dependent but the variance is not ismakes state-dependent variances inappropriate for ourpurposes Allowing for a change in variances leads to muchhigher estimates which cause one state to cover a muchwider range of potential values is makes the model un-informative and unable to pick the differences that interestthe paper As a result we base our results in the cases wherethe variance across states is common and the regime switchis in the mean

Table 1 Markov-switching regression in means only forsensitivity

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus2408 0163 minus1474 0000 (minus2729 minus2088)micro2 0701 0662 1060 0000 (0571 0831)σ 0791 0044 (0709 0882)p11 0838 0073 (0644 0937)p21 0033 0015 (0014 0077)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

Table 2 Transition probabilities between state 1 and state 2 for allmodels

Regime switch in means onlySensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0838 0162 State 1 0987 0013State 2 0033 0967 State 2 0104 0897

Regime switch in means and standard deviationsSensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0837 0163 State 1 0987 0013State 2 0111 0889 State 2 0087 0913e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

2

1

0

ndash1

ndash2

ndash3

2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 2018m1Date

State 1 one-step probabilitiesSensitivity of BBB bonds

Figure 5 Regime change in means only state plot for the sen-sitivity time series Probability with respect to state 1

Table 3 Markov-switching regression in means and standarddeviations for sensitivity

Parameter Estimate Sterror Z Pgt |z| 95 confidence

intervalmicro1 minus1345 0201 minus668 0000 (minus1739minus0951)micro2 0842 0142 594 0000 (0564 1119)σ1 0909 0097 (0739 1119)σ2 0516 0042 (0439 0606)p11 0837 0048 (0720 0911)p21 0111 0032 (0629 0190)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

8 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

between government and corporate bonds Our method fitswell with a strategy that hedges interest rate risk by shortingthe USTportfolio and essentially assesses the profitability ofa long position in US BBB-rated bonds coupled with a shortposition in government bonds is intuition is useful forconnecting our findings with the wider literature that ex-plicitly uses interest rates and spreads

3 Empirical Results

31 Present Values and Capital Gains of Bond PortfoliosFigure 1 presents the historical time series of the yields andcoupons of the UST bonds and US BBB-rated bonds

For UST bonds the average coupon almost always exceedsthe corresponding yield rate with the exception of the two-year-long period preceding the global financial crisis namelyfrom the second half of 2005 until the end of the first half of2007 To a lesser extent the same holds for corporate bondsapart from the financial crisis period mid-2008 to mid-2010Notably the surge in the yields of corporate bonds during thefinancial crisis is well above the coupon curve signifying aperiod where these assets were traded at a great discount ischange in trend will become apparent later on

We then proceed to discount the future cash flows for thetwo portfolios based on the yield and coupon data eresulting bond prices (see Figure 2) demonstrate a similarpattern between mid-2003 and the outbreak of the financialcrisis in mid-2007 Since the second half of 2007 until the peakof the crisis a large flight-to-quality event is observed wherethe prices of safe assets increase and the prices of risky assetsdecrease [17] In 2009 this flight-to-quality stops and the gapbetween the two curves vanishes Notably the ranges of thepresent values of the two portfolios differ considerably erange relative to the UST portfolio (US$ 150 million) is nar-rower than the respective range of the present values of the USBBB-rated bond portfolio (US$ 250 million)

Capital gains for each portfolio can be directly calculatedfrom our earlier definition While the interpretation for theseparate UST and US BBB-rated bond portfolios isstraightforward we also discuss the capital gains of a USBBB-rated portfolio hedged by taking a short position inUST bonds In this case

CGhedged CGBBB minus CGUST (6)

Figure 3 represents the time behavior of yearly changesin present value for the US BBB-rated bond portfolio therisk-free UST portfolio and a portfolio of US BBB-ratedbonds hedged by shorting the respective USTportfolio Eachpoint represents the changes in present value having takenplace in the preceding year Between July 2007 and De-cember 2010 the period of the global financial crisis and itsimmediate aftermath the annual capital gains of the BBB-rated bond portfolio and UST bond portfolio move inopposite directions Hence the interest rate hedging of USBBB-rated debt with the short UST positions does notcompensate the negative impacts when such compensationis most needed

32 Capital Gain-Wise Interest Rate Sensitivity of US BBB-Rated Debt e main intuition lies in the interpretation ofcapital gains sensitivity across the portfolios we considernamely the US BBB-rated and UST portfolios and thehedged portfolio Instead of using average sensitivity overthe entire sample we examine the behavior of sensitivitywithin much shorter time intervals determined by localextrema Local minima and maxima are treated as turningpoints which separate upward and downward tendencies incapital gains dynamics

e tracking of large movements in UST capital gainsreduces the uncertainty in the denominator of equation (5)and hence increases the precision of sensitivity measure-ments e extrema are identified in the time series of theUST portfolio as in [18] is leads to 51 sufficiently largeand distinct movements both upwards and downwardswhich define the intervals we use and are split into threegroups according to the time of occurrence 18 movementstook place during the precrisis period (March 2002ndashMay2007) 8 movements belong to the crisis period (May2007ndashOctober 2010) and 25 movements happened duringthe postcrisis period (October 2010ndashAugust 2016) Sensi-tivity is calculated separately for each of the intervals eresulting pattern can be seen in Figure 4 along with averagesensitivity for each of the three phases

e bold line depicts the two different regimes Duringnormal economic conditions the average sensitivity ispositive while during the crisis it is negative and amplifiedWe clearly observe the time-varying behavior of capital gainssensitivity over each phase similar to [9 11] e plot showsthat sensitivity is mostly positive before 2007 and after 2011while it turns negative between that time interval isimplies a regime change during and slightly after the periodof the financial crisis From 2001 till 2007 the sensitivityremains positive from 2007 till 2011 it turns negative andfrom 2011 till 2016 it reverts back to positive ground Toproperly calculate the sign and value of sensitivity in a waythat is not affected by time variation we take the product ofcapital gains and duration (time interval) over which thesensitivity point estimates are given is yields a weightedsum of capital gains for each portfolio where the weight isthe fraction of time over which sensitivity (capital gains) wascalculatede average period sensitivity is thus the ratio ofthe weighted sum of capital gains for the US BBB-ratedportfolio divided by the UST equivalent In comparisonspread-to-rate sensitivity typically takes daily values whichare averaged over a longer period whereas our metric fo-cuses on start and end points

As discussed earlier the definition of our measure can beinterpreted as a gauge of hedging success when a longposition in the BBB-rated portfolio is balanced by a shortposition in the USTportfolio In terms of cash flows this canbe seen as either a permanent position or an interest rateswap that pays a fixed rate to receive a floating rate equal tothat of the corporate bond portfolio It becomes apparentthat all the gains from the short position in UST during theprecrisis period are wiped away during the crisis downturnand recovery As such an all-weather hedge or its equivalentin the form of a swap is inefficient

4 Complexity

Section 4 confirms our empirical observation of capitalgain-wise interest rate sensitivity regime changes by appli-cation of Markov-switching model

4 Markov-Switching Modeling andRegime Changes

Up to this point our empirical findings point towards astructural break in the time series we use is underlines afunctional discrepancy best represented by the contradictionbetween the Merton [4] model implying a permanentlynegative relationship between credit spreads and interestrates and the Kamin and von Kleist [6] approach which

does not identify such a relationship for a certain periodGiven the direct correspondence between interest rates andthe yields of government bonds it is straightforward toconnect the intuition of the previous section to the tworegimes To test the hypothesis of two alternating states onewhere the Merton model is valid and capital gains sensitivity(and conversely the relationship between credit spreads andinterest rates) is negative and one where capital gainssensitivity is close to zero (and the relationship betweeninterest rates and credit spreads is roughly one-to-one) as in[6 7] we estimate a simple Markov-switching model withtwo regimes using the time series of sensitivity and capitalgains of the hedged bond portfolio e figures in the

0123456789

1011

SA14YL indexSA14CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(a)

0123456789

1011

S200YL indexS200CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(b)

Figure 1 Yields and average coupons for the US BBB-rated (a) and UST bonds (b)

Complexity 5

previous section show a possible regime switch in the periodaverages but we also examine whether the two differentregimes may also have distinct volatilities

From the vast literature on Markov-switching and re-gime-switching models we select a set of simple applicationsin line with the arguments detailed in [19 20] For widerapplications on asset pricing portfolio optimization andstochastic volatility [21] contains a comprehensive literaturereview We employ a simple two-state dynamic regressionMarkov-switching model (DRMS) which is more suitable tothe monthly frequency of our data A DRMS specification

allows the probabilities to change instantaneously acrossstates since the realization of each state st does not dependon stminus1 but is drawn from a discrete probability distributionOn the other hand an autoregressive (AR) specificationwhere st follows a Markov process thus allowing thetransition probabilities to follow an AR process wouldintroduce autocorrelation in states since st would depend onstminus1 is is an unwelcome feature for our model for manyreasons Markov-switching AR (MSAR) models are bettersuited for quarterly or annual data and the introduction oflagged states multiplies the states In our case due to

850

900

950

1000

1050

1100

1150

PV BBB-rated portfolioPV UST portfolio

31

01

31

02

31

03

31

04

31

05

31

06

31

07

31

08

31

09

31

10

31

11

31

12

31

13

31

14

31

15

31

16

Figure 2 Present values of portfolios US BBB-rated bonds vs UST

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

BBB hedged by short UST

BBB long portfolioUST long portfolio

Figure 3 Yearly capital gains for the US BBB-rated UST and hedged portfolios

6 Complexity

autoregression there would be four possible regimes insteadof two without providing any additional intuition Also anMSAR specification would not allow the probabilities toadjust quickly enough to changes in states Regardlesspreliminary estimations of an AR specification yielded verysimilar results so we base our discussion on the DRMSmodels only

A general specification of the DRMSmodel that capturesall the subcases we consider is

Yt μLst + εst εst sim N 0 σ2st1113872 1113873 (7)

where t denotes time Y is the time series (sensitivity orportfolio capital gains) s denotes the state and is set to 1 or2 micro and σ denote the state dependent model parameters ofthe mean (intercept) and the standard deviation andinnovations ε are independent and identically distributedfollowing a normal distribution with state-dependentvariance We consider two variations one where themeans are state dependent and one where not only themeans but also the variances are state-dependent too enotation can be simplified in the case where only themeans are allowed to switch across states to σ2s = σ2(constant across both time and states) and remains asabove in the case where both means and variances are statedependent (μs and σ2s omitting t for brevity) Τhis leads tofour different estimations two for sensitivity and two forcapital gains e Akaike HannanndashQuinn and Schwarzinformation criteria show that the simplest model with nolagged terms is the most suitable one for all cases for bothsensitivity and capital gains hence we do not introducelags

41 Regime Switching in Sensitivity Table 1 reports the re-sults for capital gains sensitivity when only the means arestate dependent under a model specified as Yt μst + εtεtsimΝ(0 σ2) for states s 1 2 State 1 corresponds to theMerton model where s is negative and state 2 corresponds

to the Kamin and von Kleist approach where s is near zeroand positivee estimation verifies our earlier observationssince micro12 are both statistically significant with a negativevalue of minus241 and a positive value of 070 respectively eprobabilities for each state are 0838 and 0968 respectivelywhich show very high persistence for both regimes

e transition probabilities from state 1 (2) to state 2 (1)are 0162 (0032) which show a very low rate of change (seeTable 2) is indicates very clear breaks in the time seriesand high certainty over which state occurs which is a directoutcome of the DRMS specification

Figure 5 shows the transition probabilities compared tothe evolution of sensitivity during our period

Figure 5 evidences a dominance of the second state upuntil 2009 with a brief exception around 2003 and adominance of the first state between 2009 and 2011 State 1briefly reappears at the end of 2012 During the 2009ndash2012period any changes from state 1 to state 2 are instantaneousmeaning that the sensitivity reverts immediately back to itsinitial regime After 2012 the second state is always present

ese findings show that sensitivity and credit spreadsvary over time with two distinct regimes appearing whichcorrespond to two different theoretical approaches on therelationship between credit spreads and interest ratesDuring crises credit spreads and interest rates move inopposite directions following the Merton pattern of negativesensitivity During normal periods any changes in interestrates (risk-free yields) pass on nearly completely to the yieldsof corporate bonds which implies stable credit spreads andthe Kamin and von Kleist pattern

When regime changes are allowed in both the mean andthe variance the parameters and transition probabilities aregenerally similar (Table 3) e model specification followsequation (7) However the plot of the transition probabil-ities over the time series reveals that only state 2 manifeststhroughout our entire period ie there is not a switch fromone pattern to the other and only the Kamin and von Kleistrationale is validated e variance of state 1 (090) isconsiderably higher than the variance of state 2 (051) andslightly higher than the variance of the means-only regime-switching model (079) Moreover the means of the twostates are closer compared to the previous case One possibleexplanation is that the higher variance combined with thehigher mean of state 1makes the first state redundant despiteits high persistency e mean estimate of minus134 has astandard deviation of 02 which coupled with the standarddeviation of the state brings it well within the range of state 2We therefore conjecture that this similarity causes the modelnot to discriminate between states

42 Regime Switching in the US BBB Portfolio Hedged byShorting UST We can now proceed to the capital gains ofthe hedged synthetic portfolio where UST bonds are shortede capital gains of the hedged portfolio are calculated asfollows First the present values of the UST and BBBportfolios are calculated using equations (1) and (3) settingT 5 and FV 1000 Since the S200YL S200CP SA14YLand SA14CP indices are monthly we calculate monthly

ndash350ndash300ndash250ndash200ndash150ndash100ndash050

000050100150200250

310

320

02

310

320

03

310

320

04

310

320

05

310

320

06

310

320

07

310

320

08

310

320

09

310

320

10

310

320

11

310

320

12

310

320

13

310

320

14

310

320

15

310

320

16

On-the-move sensitivityPhase-averaged sensitivity

Figure 4 Phase-averaged sensitivity of US BBB-rated bonds2002ndash2016

Complexity 7

present values for the two portfolios We then calculate thedifference between the present value of month t and thepresent value of month tminus 12 or the difference over anannual interval e capital gain of the hedged portfolio istherefore the capital gain of the long position on BBBcorporate bonds minus the capital gain of the short positionon US Treasury bonds e generated capital gains spanfrom March 2002 till September 2016 More succinctlyformula (6) can be written as follows for H 12 months andP the present value (price) of a portfolio

CGhedged(t H) CGBBB minus CGBBB PBBB(t + H) minus PBBB(t)

minus PBBB(t + H) minus PBBB(t)

(8)

eCGhedged series is used to estimate different Markov-switching models whose results are reported in Tables 4 and5 We follow the same approach with sensitivity and employa DRMS model with no lags e pattern is similar to that ofsensitivity When the variance is common for the two statesand only the means change there are two major regimeswitches (Figure 6)

One of the changes which lasts longer takes place be-tween 2010 and 2011 which corresponds to the financialcrisis e other change takes place in 2004 is slightly lesspronounced and coincides with a minimum in the timeseries of UST capital gains It is worth noting that thisminimum of UST capital gains occurs within the one-year-long time interval (June 2003ndashJune 2004) when the federalfund target rate remains at its local minimum value of 1e transition probabilities are again very persistent asshown in Table 4 State 1 has a probability of 099 state 2 hasa probability of 0896 and the probability to move from state1 (2) to state 2 (1) is 001 (0104) Similarly the mean of thefirst state (minus1139) is negative and corresponds to theMertoncase while the mean of the second state (13479) is positiveand corresponds to the Kamin and von Kleist case (seeTable 4)

In its turn Table 5 demonstrates that when switches inboth the means and the variances are allowed the result issimilar to the previous section e means changemarginally but the much higher variance of one state(6901) causes it to dominate over the second state whichhas a variance similar to that of the means-only switch(4297)

Figure 7 shows how the probability of a state to be re-alized remains at 2 or equivalently how the other statepersists and there are no switches during the period On theother hand Figure 6 clearly shows two different states whenthe means are state dependent but the variance is not ismakes state-dependent variances inappropriate for ourpurposes Allowing for a change in variances leads to muchhigher estimates which cause one state to cover a muchwider range of potential values is makes the model un-informative and unable to pick the differences that interestthe paper As a result we base our results in the cases wherethe variance across states is common and the regime switchis in the mean

Table 1 Markov-switching regression in means only forsensitivity

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus2408 0163 minus1474 0000 (minus2729 minus2088)micro2 0701 0662 1060 0000 (0571 0831)σ 0791 0044 (0709 0882)p11 0838 0073 (0644 0937)p21 0033 0015 (0014 0077)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

Table 2 Transition probabilities between state 1 and state 2 for allmodels

Regime switch in means onlySensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0838 0162 State 1 0987 0013State 2 0033 0967 State 2 0104 0897

Regime switch in means and standard deviationsSensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0837 0163 State 1 0987 0013State 2 0111 0889 State 2 0087 0913e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

2

1

0

ndash1

ndash2

ndash3

2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 2018m1Date

State 1 one-step probabilitiesSensitivity of BBB bonds

Figure 5 Regime change in means only state plot for the sen-sitivity time series Probability with respect to state 1

Table 3 Markov-switching regression in means and standarddeviations for sensitivity

Parameter Estimate Sterror Z Pgt |z| 95 confidence

intervalmicro1 minus1345 0201 minus668 0000 (minus1739minus0951)micro2 0842 0142 594 0000 (0564 1119)σ1 0909 0097 (0739 1119)σ2 0516 0042 (0439 0606)p11 0837 0048 (0720 0911)p21 0111 0032 (0629 0190)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

8 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

Section 4 confirms our empirical observation of capitalgain-wise interest rate sensitivity regime changes by appli-cation of Markov-switching model

4 Markov-Switching Modeling andRegime Changes

Up to this point our empirical findings point towards astructural break in the time series we use is underlines afunctional discrepancy best represented by the contradictionbetween the Merton [4] model implying a permanentlynegative relationship between credit spreads and interestrates and the Kamin and von Kleist [6] approach which

does not identify such a relationship for a certain periodGiven the direct correspondence between interest rates andthe yields of government bonds it is straightforward toconnect the intuition of the previous section to the tworegimes To test the hypothesis of two alternating states onewhere the Merton model is valid and capital gains sensitivity(and conversely the relationship between credit spreads andinterest rates) is negative and one where capital gainssensitivity is close to zero (and the relationship betweeninterest rates and credit spreads is roughly one-to-one) as in[6 7] we estimate a simple Markov-switching model withtwo regimes using the time series of sensitivity and capitalgains of the hedged bond portfolio e figures in the

0123456789

1011

SA14YL indexSA14CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(a)

0123456789

1011

S200YL indexS200CP index

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

(b)

Figure 1 Yields and average coupons for the US BBB-rated (a) and UST bonds (b)

Complexity 5

previous section show a possible regime switch in the periodaverages but we also examine whether the two differentregimes may also have distinct volatilities

From the vast literature on Markov-switching and re-gime-switching models we select a set of simple applicationsin line with the arguments detailed in [19 20] For widerapplications on asset pricing portfolio optimization andstochastic volatility [21] contains a comprehensive literaturereview We employ a simple two-state dynamic regressionMarkov-switching model (DRMS) which is more suitable tothe monthly frequency of our data A DRMS specification

allows the probabilities to change instantaneously acrossstates since the realization of each state st does not dependon stminus1 but is drawn from a discrete probability distributionOn the other hand an autoregressive (AR) specificationwhere st follows a Markov process thus allowing thetransition probabilities to follow an AR process wouldintroduce autocorrelation in states since st would depend onstminus1 is is an unwelcome feature for our model for manyreasons Markov-switching AR (MSAR) models are bettersuited for quarterly or annual data and the introduction oflagged states multiplies the states In our case due to

850

900

950

1000

1050

1100

1150

PV BBB-rated portfolioPV UST portfolio

31

01

31

02

31

03

31

04

31

05

31

06

31

07

31

08

31

09

31

10

31

11

31

12

31

13

31

14

31

15

31

16

Figure 2 Present values of portfolios US BBB-rated bonds vs UST

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

BBB hedged by short UST

BBB long portfolioUST long portfolio

Figure 3 Yearly capital gains for the US BBB-rated UST and hedged portfolios

6 Complexity

autoregression there would be four possible regimes insteadof two without providing any additional intuition Also anMSAR specification would not allow the probabilities toadjust quickly enough to changes in states Regardlesspreliminary estimations of an AR specification yielded verysimilar results so we base our discussion on the DRMSmodels only

A general specification of the DRMSmodel that capturesall the subcases we consider is

Yt μLst + εst εst sim N 0 σ2st1113872 1113873 (7)

where t denotes time Y is the time series (sensitivity orportfolio capital gains) s denotes the state and is set to 1 or2 micro and σ denote the state dependent model parameters ofthe mean (intercept) and the standard deviation andinnovations ε are independent and identically distributedfollowing a normal distribution with state-dependentvariance We consider two variations one where themeans are state dependent and one where not only themeans but also the variances are state-dependent too enotation can be simplified in the case where only themeans are allowed to switch across states to σ2s = σ2(constant across both time and states) and remains asabove in the case where both means and variances are statedependent (μs and σ2s omitting t for brevity) Τhis leads tofour different estimations two for sensitivity and two forcapital gains e Akaike HannanndashQuinn and Schwarzinformation criteria show that the simplest model with nolagged terms is the most suitable one for all cases for bothsensitivity and capital gains hence we do not introducelags

41 Regime Switching in Sensitivity Table 1 reports the re-sults for capital gains sensitivity when only the means arestate dependent under a model specified as Yt μst + εtεtsimΝ(0 σ2) for states s 1 2 State 1 corresponds to theMerton model where s is negative and state 2 corresponds

to the Kamin and von Kleist approach where s is near zeroand positivee estimation verifies our earlier observationssince micro12 are both statistically significant with a negativevalue of minus241 and a positive value of 070 respectively eprobabilities for each state are 0838 and 0968 respectivelywhich show very high persistence for both regimes

e transition probabilities from state 1 (2) to state 2 (1)are 0162 (0032) which show a very low rate of change (seeTable 2) is indicates very clear breaks in the time seriesand high certainty over which state occurs which is a directoutcome of the DRMS specification

Figure 5 shows the transition probabilities compared tothe evolution of sensitivity during our period

Figure 5 evidences a dominance of the second state upuntil 2009 with a brief exception around 2003 and adominance of the first state between 2009 and 2011 State 1briefly reappears at the end of 2012 During the 2009ndash2012period any changes from state 1 to state 2 are instantaneousmeaning that the sensitivity reverts immediately back to itsinitial regime After 2012 the second state is always present

ese findings show that sensitivity and credit spreadsvary over time with two distinct regimes appearing whichcorrespond to two different theoretical approaches on therelationship between credit spreads and interest ratesDuring crises credit spreads and interest rates move inopposite directions following the Merton pattern of negativesensitivity During normal periods any changes in interestrates (risk-free yields) pass on nearly completely to the yieldsof corporate bonds which implies stable credit spreads andthe Kamin and von Kleist pattern

When regime changes are allowed in both the mean andthe variance the parameters and transition probabilities aregenerally similar (Table 3) e model specification followsequation (7) However the plot of the transition probabil-ities over the time series reveals that only state 2 manifeststhroughout our entire period ie there is not a switch fromone pattern to the other and only the Kamin and von Kleistrationale is validated e variance of state 1 (090) isconsiderably higher than the variance of state 2 (051) andslightly higher than the variance of the means-only regime-switching model (079) Moreover the means of the twostates are closer compared to the previous case One possibleexplanation is that the higher variance combined with thehigher mean of state 1makes the first state redundant despiteits high persistency e mean estimate of minus134 has astandard deviation of 02 which coupled with the standarddeviation of the state brings it well within the range of state 2We therefore conjecture that this similarity causes the modelnot to discriminate between states

42 Regime Switching in the US BBB Portfolio Hedged byShorting UST We can now proceed to the capital gains ofthe hedged synthetic portfolio where UST bonds are shortede capital gains of the hedged portfolio are calculated asfollows First the present values of the UST and BBBportfolios are calculated using equations (1) and (3) settingT 5 and FV 1000 Since the S200YL S200CP SA14YLand SA14CP indices are monthly we calculate monthly

ndash350ndash300ndash250ndash200ndash150ndash100ndash050

000050100150200250

310

320

02

310

320

03

310

320

04

310

320

05

310

320

06

310

320

07

310

320

08

310

320

09

310

320

10

310

320

11

310

320

12

310

320

13

310

320

14

310

320

15

310

320

16

On-the-move sensitivityPhase-averaged sensitivity

Figure 4 Phase-averaged sensitivity of US BBB-rated bonds2002ndash2016

Complexity 7

present values for the two portfolios We then calculate thedifference between the present value of month t and thepresent value of month tminus 12 or the difference over anannual interval e capital gain of the hedged portfolio istherefore the capital gain of the long position on BBBcorporate bonds minus the capital gain of the short positionon US Treasury bonds e generated capital gains spanfrom March 2002 till September 2016 More succinctlyformula (6) can be written as follows for H 12 months andP the present value (price) of a portfolio

CGhedged(t H) CGBBB minus CGBBB PBBB(t + H) minus PBBB(t)

minus PBBB(t + H) minus PBBB(t)

(8)

eCGhedged series is used to estimate different Markov-switching models whose results are reported in Tables 4 and5 We follow the same approach with sensitivity and employa DRMS model with no lags e pattern is similar to that ofsensitivity When the variance is common for the two statesand only the means change there are two major regimeswitches (Figure 6)

One of the changes which lasts longer takes place be-tween 2010 and 2011 which corresponds to the financialcrisis e other change takes place in 2004 is slightly lesspronounced and coincides with a minimum in the timeseries of UST capital gains It is worth noting that thisminimum of UST capital gains occurs within the one-year-long time interval (June 2003ndashJune 2004) when the federalfund target rate remains at its local minimum value of 1e transition probabilities are again very persistent asshown in Table 4 State 1 has a probability of 099 state 2 hasa probability of 0896 and the probability to move from state1 (2) to state 2 (1) is 001 (0104) Similarly the mean of thefirst state (minus1139) is negative and corresponds to theMertoncase while the mean of the second state (13479) is positiveand corresponds to the Kamin and von Kleist case (seeTable 4)

In its turn Table 5 demonstrates that when switches inboth the means and the variances are allowed the result issimilar to the previous section e means changemarginally but the much higher variance of one state(6901) causes it to dominate over the second state whichhas a variance similar to that of the means-only switch(4297)

Figure 7 shows how the probability of a state to be re-alized remains at 2 or equivalently how the other statepersists and there are no switches during the period On theother hand Figure 6 clearly shows two different states whenthe means are state dependent but the variance is not ismakes state-dependent variances inappropriate for ourpurposes Allowing for a change in variances leads to muchhigher estimates which cause one state to cover a muchwider range of potential values is makes the model un-informative and unable to pick the differences that interestthe paper As a result we base our results in the cases wherethe variance across states is common and the regime switchis in the mean

Table 1 Markov-switching regression in means only forsensitivity

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus2408 0163 minus1474 0000 (minus2729 minus2088)micro2 0701 0662 1060 0000 (0571 0831)σ 0791 0044 (0709 0882)p11 0838 0073 (0644 0937)p21 0033 0015 (0014 0077)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

Table 2 Transition probabilities between state 1 and state 2 for allmodels

Regime switch in means onlySensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0838 0162 State 1 0987 0013State 2 0033 0967 State 2 0104 0897

Regime switch in means and standard deviationsSensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0837 0163 State 1 0987 0013State 2 0111 0889 State 2 0087 0913e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

2

1

0

ndash1

ndash2

ndash3

2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 2018m1Date

State 1 one-step probabilitiesSensitivity of BBB bonds

Figure 5 Regime change in means only state plot for the sen-sitivity time series Probability with respect to state 1

Table 3 Markov-switching regression in means and standarddeviations for sensitivity

Parameter Estimate Sterror Z Pgt |z| 95 confidence

intervalmicro1 minus1345 0201 minus668 0000 (minus1739minus0951)micro2 0842 0142 594 0000 (0564 1119)σ1 0909 0097 (0739 1119)σ2 0516 0042 (0439 0606)p11 0837 0048 (0720 0911)p21 0111 0032 (0629 0190)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

8 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

previous section show a possible regime switch in the periodaverages but we also examine whether the two differentregimes may also have distinct volatilities

From the vast literature on Markov-switching and re-gime-switching models we select a set of simple applicationsin line with the arguments detailed in [19 20] For widerapplications on asset pricing portfolio optimization andstochastic volatility [21] contains a comprehensive literaturereview We employ a simple two-state dynamic regressionMarkov-switching model (DRMS) which is more suitable tothe monthly frequency of our data A DRMS specification

allows the probabilities to change instantaneously acrossstates since the realization of each state st does not dependon stminus1 but is drawn from a discrete probability distributionOn the other hand an autoregressive (AR) specificationwhere st follows a Markov process thus allowing thetransition probabilities to follow an AR process wouldintroduce autocorrelation in states since st would depend onstminus1 is is an unwelcome feature for our model for manyreasons Markov-switching AR (MSAR) models are bettersuited for quarterly or annual data and the introduction oflagged states multiplies the states In our case due to

850

900

950

1000

1050

1100

1150

PV BBB-rated portfolioPV UST portfolio

31

01

31

02

31

03

31

04

31

05

31

06

31

07

31

08

31

09

31

10

31

11

31

12

31

13

31

14

31

15

31

16

Figure 2 Present values of portfolios US BBB-rated bonds vs UST

ndash200

ndash150

ndash100

ndash50

0

50

100

150

200

250

01-0

3-20

01

01-0

3-20

02

01-0

3-20

03

01-0

3-20

04

01-0

3-20

05

01-0

3-20

06

01-0

3-20

07

01-0

3-20

08

01-0

3-20

09

01-0

3-20

10

01-0

3-20

11

01-0

3-20

12

01-0

3-20

13

01-0

3-20

14

01-0

3-20

15

01-0

3-20

16

BBB hedged by short UST

BBB long portfolioUST long portfolio

Figure 3 Yearly capital gains for the US BBB-rated UST and hedged portfolios

6 Complexity

autoregression there would be four possible regimes insteadof two without providing any additional intuition Also anMSAR specification would not allow the probabilities toadjust quickly enough to changes in states Regardlesspreliminary estimations of an AR specification yielded verysimilar results so we base our discussion on the DRMSmodels only

A general specification of the DRMSmodel that capturesall the subcases we consider is

Yt μLst + εst εst sim N 0 σ2st1113872 1113873 (7)

where t denotes time Y is the time series (sensitivity orportfolio capital gains) s denotes the state and is set to 1 or2 micro and σ denote the state dependent model parameters ofthe mean (intercept) and the standard deviation andinnovations ε are independent and identically distributedfollowing a normal distribution with state-dependentvariance We consider two variations one where themeans are state dependent and one where not only themeans but also the variances are state-dependent too enotation can be simplified in the case where only themeans are allowed to switch across states to σ2s = σ2(constant across both time and states) and remains asabove in the case where both means and variances are statedependent (μs and σ2s omitting t for brevity) Τhis leads tofour different estimations two for sensitivity and two forcapital gains e Akaike HannanndashQuinn and Schwarzinformation criteria show that the simplest model with nolagged terms is the most suitable one for all cases for bothsensitivity and capital gains hence we do not introducelags

41 Regime Switching in Sensitivity Table 1 reports the re-sults for capital gains sensitivity when only the means arestate dependent under a model specified as Yt μst + εtεtsimΝ(0 σ2) for states s 1 2 State 1 corresponds to theMerton model where s is negative and state 2 corresponds

to the Kamin and von Kleist approach where s is near zeroand positivee estimation verifies our earlier observationssince micro12 are both statistically significant with a negativevalue of minus241 and a positive value of 070 respectively eprobabilities for each state are 0838 and 0968 respectivelywhich show very high persistence for both regimes

e transition probabilities from state 1 (2) to state 2 (1)are 0162 (0032) which show a very low rate of change (seeTable 2) is indicates very clear breaks in the time seriesand high certainty over which state occurs which is a directoutcome of the DRMS specification

Figure 5 shows the transition probabilities compared tothe evolution of sensitivity during our period

Figure 5 evidences a dominance of the second state upuntil 2009 with a brief exception around 2003 and adominance of the first state between 2009 and 2011 State 1briefly reappears at the end of 2012 During the 2009ndash2012period any changes from state 1 to state 2 are instantaneousmeaning that the sensitivity reverts immediately back to itsinitial regime After 2012 the second state is always present

ese findings show that sensitivity and credit spreadsvary over time with two distinct regimes appearing whichcorrespond to two different theoretical approaches on therelationship between credit spreads and interest ratesDuring crises credit spreads and interest rates move inopposite directions following the Merton pattern of negativesensitivity During normal periods any changes in interestrates (risk-free yields) pass on nearly completely to the yieldsof corporate bonds which implies stable credit spreads andthe Kamin and von Kleist pattern

When regime changes are allowed in both the mean andthe variance the parameters and transition probabilities aregenerally similar (Table 3) e model specification followsequation (7) However the plot of the transition probabil-ities over the time series reveals that only state 2 manifeststhroughout our entire period ie there is not a switch fromone pattern to the other and only the Kamin and von Kleistrationale is validated e variance of state 1 (090) isconsiderably higher than the variance of state 2 (051) andslightly higher than the variance of the means-only regime-switching model (079) Moreover the means of the twostates are closer compared to the previous case One possibleexplanation is that the higher variance combined with thehigher mean of state 1makes the first state redundant despiteits high persistency e mean estimate of minus134 has astandard deviation of 02 which coupled with the standarddeviation of the state brings it well within the range of state 2We therefore conjecture that this similarity causes the modelnot to discriminate between states

42 Regime Switching in the US BBB Portfolio Hedged byShorting UST We can now proceed to the capital gains ofthe hedged synthetic portfolio where UST bonds are shortede capital gains of the hedged portfolio are calculated asfollows First the present values of the UST and BBBportfolios are calculated using equations (1) and (3) settingT 5 and FV 1000 Since the S200YL S200CP SA14YLand SA14CP indices are monthly we calculate monthly

ndash350ndash300ndash250ndash200ndash150ndash100ndash050

000050100150200250

310

320

02

310

320

03

310

320

04

310

320

05

310

320

06

310

320

07

310

320

08

310

320

09

310

320

10

310

320

11

310

320

12

310

320

13

310

320

14

310

320

15

310

320

16

On-the-move sensitivityPhase-averaged sensitivity

Figure 4 Phase-averaged sensitivity of US BBB-rated bonds2002ndash2016

Complexity 7

present values for the two portfolios We then calculate thedifference between the present value of month t and thepresent value of month tminus 12 or the difference over anannual interval e capital gain of the hedged portfolio istherefore the capital gain of the long position on BBBcorporate bonds minus the capital gain of the short positionon US Treasury bonds e generated capital gains spanfrom March 2002 till September 2016 More succinctlyformula (6) can be written as follows for H 12 months andP the present value (price) of a portfolio

CGhedged(t H) CGBBB minus CGBBB PBBB(t + H) minus PBBB(t)

minus PBBB(t + H) minus PBBB(t)

(8)

eCGhedged series is used to estimate different Markov-switching models whose results are reported in Tables 4 and5 We follow the same approach with sensitivity and employa DRMS model with no lags e pattern is similar to that ofsensitivity When the variance is common for the two statesand only the means change there are two major regimeswitches (Figure 6)

One of the changes which lasts longer takes place be-tween 2010 and 2011 which corresponds to the financialcrisis e other change takes place in 2004 is slightly lesspronounced and coincides with a minimum in the timeseries of UST capital gains It is worth noting that thisminimum of UST capital gains occurs within the one-year-long time interval (June 2003ndashJune 2004) when the federalfund target rate remains at its local minimum value of 1e transition probabilities are again very persistent asshown in Table 4 State 1 has a probability of 099 state 2 hasa probability of 0896 and the probability to move from state1 (2) to state 2 (1) is 001 (0104) Similarly the mean of thefirst state (minus1139) is negative and corresponds to theMertoncase while the mean of the second state (13479) is positiveand corresponds to the Kamin and von Kleist case (seeTable 4)

In its turn Table 5 demonstrates that when switches inboth the means and the variances are allowed the result issimilar to the previous section e means changemarginally but the much higher variance of one state(6901) causes it to dominate over the second state whichhas a variance similar to that of the means-only switch(4297)

Figure 7 shows how the probability of a state to be re-alized remains at 2 or equivalently how the other statepersists and there are no switches during the period On theother hand Figure 6 clearly shows two different states whenthe means are state dependent but the variance is not ismakes state-dependent variances inappropriate for ourpurposes Allowing for a change in variances leads to muchhigher estimates which cause one state to cover a muchwider range of potential values is makes the model un-informative and unable to pick the differences that interestthe paper As a result we base our results in the cases wherethe variance across states is common and the regime switchis in the mean

Table 1 Markov-switching regression in means only forsensitivity

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus2408 0163 minus1474 0000 (minus2729 minus2088)micro2 0701 0662 1060 0000 (0571 0831)σ 0791 0044 (0709 0882)p11 0838 0073 (0644 0937)p21 0033 0015 (0014 0077)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

Table 2 Transition probabilities between state 1 and state 2 for allmodels

Regime switch in means onlySensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0838 0162 State 1 0987 0013State 2 0033 0967 State 2 0104 0897

Regime switch in means and standard deviationsSensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0837 0163 State 1 0987 0013State 2 0111 0889 State 2 0087 0913e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

2

1

0

ndash1

ndash2

ndash3

2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 2018m1Date

State 1 one-step probabilitiesSensitivity of BBB bonds

Figure 5 Regime change in means only state plot for the sen-sitivity time series Probability with respect to state 1

Table 3 Markov-switching regression in means and standarddeviations for sensitivity

Parameter Estimate Sterror Z Pgt |z| 95 confidence

intervalmicro1 minus1345 0201 minus668 0000 (minus1739minus0951)micro2 0842 0142 594 0000 (0564 1119)σ1 0909 0097 (0739 1119)σ2 0516 0042 (0439 0606)p11 0837 0048 (0720 0911)p21 0111 0032 (0629 0190)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

8 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

autoregression there would be four possible regimes insteadof two without providing any additional intuition Also anMSAR specification would not allow the probabilities toadjust quickly enough to changes in states Regardlesspreliminary estimations of an AR specification yielded verysimilar results so we base our discussion on the DRMSmodels only

A general specification of the DRMSmodel that capturesall the subcases we consider is

Yt μLst + εst εst sim N 0 σ2st1113872 1113873 (7)

where t denotes time Y is the time series (sensitivity orportfolio capital gains) s denotes the state and is set to 1 or2 micro and σ denote the state dependent model parameters ofthe mean (intercept) and the standard deviation andinnovations ε are independent and identically distributedfollowing a normal distribution with state-dependentvariance We consider two variations one where themeans are state dependent and one where not only themeans but also the variances are state-dependent too enotation can be simplified in the case where only themeans are allowed to switch across states to σ2s = σ2(constant across both time and states) and remains asabove in the case where both means and variances are statedependent (μs and σ2s omitting t for brevity) Τhis leads tofour different estimations two for sensitivity and two forcapital gains e Akaike HannanndashQuinn and Schwarzinformation criteria show that the simplest model with nolagged terms is the most suitable one for all cases for bothsensitivity and capital gains hence we do not introducelags

41 Regime Switching in Sensitivity Table 1 reports the re-sults for capital gains sensitivity when only the means arestate dependent under a model specified as Yt μst + εtεtsimΝ(0 σ2) for states s 1 2 State 1 corresponds to theMerton model where s is negative and state 2 corresponds

to the Kamin and von Kleist approach where s is near zeroand positivee estimation verifies our earlier observationssince micro12 are both statistically significant with a negativevalue of minus241 and a positive value of 070 respectively eprobabilities for each state are 0838 and 0968 respectivelywhich show very high persistence for both regimes

e transition probabilities from state 1 (2) to state 2 (1)are 0162 (0032) which show a very low rate of change (seeTable 2) is indicates very clear breaks in the time seriesand high certainty over which state occurs which is a directoutcome of the DRMS specification

Figure 5 shows the transition probabilities compared tothe evolution of sensitivity during our period

Figure 5 evidences a dominance of the second state upuntil 2009 with a brief exception around 2003 and adominance of the first state between 2009 and 2011 State 1briefly reappears at the end of 2012 During the 2009ndash2012period any changes from state 1 to state 2 are instantaneousmeaning that the sensitivity reverts immediately back to itsinitial regime After 2012 the second state is always present

ese findings show that sensitivity and credit spreadsvary over time with two distinct regimes appearing whichcorrespond to two different theoretical approaches on therelationship between credit spreads and interest ratesDuring crises credit spreads and interest rates move inopposite directions following the Merton pattern of negativesensitivity During normal periods any changes in interestrates (risk-free yields) pass on nearly completely to the yieldsof corporate bonds which implies stable credit spreads andthe Kamin and von Kleist pattern

When regime changes are allowed in both the mean andthe variance the parameters and transition probabilities aregenerally similar (Table 3) e model specification followsequation (7) However the plot of the transition probabil-ities over the time series reveals that only state 2 manifeststhroughout our entire period ie there is not a switch fromone pattern to the other and only the Kamin and von Kleistrationale is validated e variance of state 1 (090) isconsiderably higher than the variance of state 2 (051) andslightly higher than the variance of the means-only regime-switching model (079) Moreover the means of the twostates are closer compared to the previous case One possibleexplanation is that the higher variance combined with thehigher mean of state 1makes the first state redundant despiteits high persistency e mean estimate of minus134 has astandard deviation of 02 which coupled with the standarddeviation of the state brings it well within the range of state 2We therefore conjecture that this similarity causes the modelnot to discriminate between states

42 Regime Switching in the US BBB Portfolio Hedged byShorting UST We can now proceed to the capital gains ofthe hedged synthetic portfolio where UST bonds are shortede capital gains of the hedged portfolio are calculated asfollows First the present values of the UST and BBBportfolios are calculated using equations (1) and (3) settingT 5 and FV 1000 Since the S200YL S200CP SA14YLand SA14CP indices are monthly we calculate monthly

ndash350ndash300ndash250ndash200ndash150ndash100ndash050

000050100150200250

310

320

02

310

320

03

310

320

04

310

320

05

310

320

06

310

320

07

310

320

08

310

320

09

310

320

10

310

320

11

310

320

12

310

320

13

310

320

14

310

320

15

310

320

16

On-the-move sensitivityPhase-averaged sensitivity

Figure 4 Phase-averaged sensitivity of US BBB-rated bonds2002ndash2016

Complexity 7

present values for the two portfolios We then calculate thedifference between the present value of month t and thepresent value of month tminus 12 or the difference over anannual interval e capital gain of the hedged portfolio istherefore the capital gain of the long position on BBBcorporate bonds minus the capital gain of the short positionon US Treasury bonds e generated capital gains spanfrom March 2002 till September 2016 More succinctlyformula (6) can be written as follows for H 12 months andP the present value (price) of a portfolio

CGhedged(t H) CGBBB minus CGBBB PBBB(t + H) minus PBBB(t)

minus PBBB(t + H) minus PBBB(t)

(8)

eCGhedged series is used to estimate different Markov-switching models whose results are reported in Tables 4 and5 We follow the same approach with sensitivity and employa DRMS model with no lags e pattern is similar to that ofsensitivity When the variance is common for the two statesand only the means change there are two major regimeswitches (Figure 6)

One of the changes which lasts longer takes place be-tween 2010 and 2011 which corresponds to the financialcrisis e other change takes place in 2004 is slightly lesspronounced and coincides with a minimum in the timeseries of UST capital gains It is worth noting that thisminimum of UST capital gains occurs within the one-year-long time interval (June 2003ndashJune 2004) when the federalfund target rate remains at its local minimum value of 1e transition probabilities are again very persistent asshown in Table 4 State 1 has a probability of 099 state 2 hasa probability of 0896 and the probability to move from state1 (2) to state 2 (1) is 001 (0104) Similarly the mean of thefirst state (minus1139) is negative and corresponds to theMertoncase while the mean of the second state (13479) is positiveand corresponds to the Kamin and von Kleist case (seeTable 4)

In its turn Table 5 demonstrates that when switches inboth the means and the variances are allowed the result issimilar to the previous section e means changemarginally but the much higher variance of one state(6901) causes it to dominate over the second state whichhas a variance similar to that of the means-only switch(4297)

Figure 7 shows how the probability of a state to be re-alized remains at 2 or equivalently how the other statepersists and there are no switches during the period On theother hand Figure 6 clearly shows two different states whenthe means are state dependent but the variance is not ismakes state-dependent variances inappropriate for ourpurposes Allowing for a change in variances leads to muchhigher estimates which cause one state to cover a muchwider range of potential values is makes the model un-informative and unable to pick the differences that interestthe paper As a result we base our results in the cases wherethe variance across states is common and the regime switchis in the mean

Table 1 Markov-switching regression in means only forsensitivity

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus2408 0163 minus1474 0000 (minus2729 minus2088)micro2 0701 0662 1060 0000 (0571 0831)σ 0791 0044 (0709 0882)p11 0838 0073 (0644 0937)p21 0033 0015 (0014 0077)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

Table 2 Transition probabilities between state 1 and state 2 for allmodels

Regime switch in means onlySensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0838 0162 State 1 0987 0013State 2 0033 0967 State 2 0104 0897

Regime switch in means and standard deviationsSensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0837 0163 State 1 0987 0013State 2 0111 0889 State 2 0087 0913e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

2

1

0

ndash1

ndash2

ndash3

2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 2018m1Date

State 1 one-step probabilitiesSensitivity of BBB bonds

Figure 5 Regime change in means only state plot for the sen-sitivity time series Probability with respect to state 1

Table 3 Markov-switching regression in means and standarddeviations for sensitivity

Parameter Estimate Sterror Z Pgt |z| 95 confidence

intervalmicro1 minus1345 0201 minus668 0000 (minus1739minus0951)micro2 0842 0142 594 0000 (0564 1119)σ1 0909 0097 (0739 1119)σ2 0516 0042 (0439 0606)p11 0837 0048 (0720 0911)p21 0111 0032 (0629 0190)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

8 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

present values for the two portfolios We then calculate thedifference between the present value of month t and thepresent value of month tminus 12 or the difference over anannual interval e capital gain of the hedged portfolio istherefore the capital gain of the long position on BBBcorporate bonds minus the capital gain of the short positionon US Treasury bonds e generated capital gains spanfrom March 2002 till September 2016 More succinctlyformula (6) can be written as follows for H 12 months andP the present value (price) of a portfolio

CGhedged(t H) CGBBB minus CGBBB PBBB(t + H) minus PBBB(t)

minus PBBB(t + H) minus PBBB(t)

(8)

eCGhedged series is used to estimate different Markov-switching models whose results are reported in Tables 4 and5 We follow the same approach with sensitivity and employa DRMS model with no lags e pattern is similar to that ofsensitivity When the variance is common for the two statesand only the means change there are two major regimeswitches (Figure 6)

One of the changes which lasts longer takes place be-tween 2010 and 2011 which corresponds to the financialcrisis e other change takes place in 2004 is slightly lesspronounced and coincides with a minimum in the timeseries of UST capital gains It is worth noting that thisminimum of UST capital gains occurs within the one-year-long time interval (June 2003ndashJune 2004) when the federalfund target rate remains at its local minimum value of 1e transition probabilities are again very persistent asshown in Table 4 State 1 has a probability of 099 state 2 hasa probability of 0896 and the probability to move from state1 (2) to state 2 (1) is 001 (0104) Similarly the mean of thefirst state (minus1139) is negative and corresponds to theMertoncase while the mean of the second state (13479) is positiveand corresponds to the Kamin and von Kleist case (seeTable 4)

In its turn Table 5 demonstrates that when switches inboth the means and the variances are allowed the result issimilar to the previous section e means changemarginally but the much higher variance of one state(6901) causes it to dominate over the second state whichhas a variance similar to that of the means-only switch(4297)

Figure 7 shows how the probability of a state to be re-alized remains at 2 or equivalently how the other statepersists and there are no switches during the period On theother hand Figure 6 clearly shows two different states whenthe means are state dependent but the variance is not ismakes state-dependent variances inappropriate for ourpurposes Allowing for a change in variances leads to muchhigher estimates which cause one state to cover a muchwider range of potential values is makes the model un-informative and unable to pick the differences that interestthe paper As a result we base our results in the cases wherethe variance across states is common and the regime switchis in the mean

Table 1 Markov-switching regression in means only forsensitivity

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus2408 0163 minus1474 0000 (minus2729 minus2088)micro2 0701 0662 1060 0000 (0571 0831)σ 0791 0044 (0709 0882)p11 0838 0073 (0644 0937)p21 0033 0015 (0014 0077)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

Table 2 Transition probabilities between state 1 and state 2 for allmodels

Regime switch in means onlySensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0838 0162 State 1 0987 0013State 2 0033 0967 State 2 0104 0897

Regime switch in means and standard deviationsSensitivity BBB hedged by short USTState fromto State 1 State 2 State fromto State 1 State 2State 1 0837 0163 State 1 0987 0013State 2 0111 0889 State 2 0087 0913e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

2

1

0

ndash1

ndash2

ndash3

2002m1 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 2018m1Date

State 1 one-step probabilitiesSensitivity of BBB bonds

Figure 5 Regime change in means only state plot for the sen-sitivity time series Probability with respect to state 1

Table 3 Markov-switching regression in means and standarddeviations for sensitivity

Parameter Estimate Sterror Z Pgt |z| 95 confidence

intervalmicro1 minus1345 0201 minus668 0000 (minus1739minus0951)micro2 0842 0142 594 0000 (0564 1119)σ1 0909 0097 (0739 1119)σ2 0516 0042 (0439 0606)p11 0837 0048 (0720 0911)p21 0111 0032 (0629 0190)e Merton case corresponds to state 1 the Kamin and von Kleist casecorresponds to state 2

8 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

It must be stressed that such effects can be identified onlyif averaging over a very long sample is avoided e averagesensitivity for the entire period is slightly below zeromeaning the strong negative sensitivity in the second periodcancels out the low positive sensitivity in the first and thirdperiods is may lead to flawed or counterintuitive resultsparticularly if the effect of one regime is very strong or ifstrong effects in all regimes negate each other It is evenpossible to conclude that there is no sensitivity relative toeither yields or capital gains is finding also provides anillustration on why long spread-to-rate sensitivity averagesfailed to capture the dynamics depicted by our measure

5 Discussion and Implications

51 e Relationship between Credit Spreads Interest Ratesand Yields e estimation results provide sufficient evi-dence on the time variation in the relationship betweencredit spreads and yields or interest rates ey suggest acompromise between the pattern identified by Merton [4] ofa negative relationship and the findings of Kamin and vonKleist [6] of no strong relationship between spreads andinterest rates We show that on the contrary the rela-tionship is not constant and alternates between two phaseseach of which corresponds to one of the two approacheseKamin and von Kleist intuition (state 2) appears in normalperiods while the Merton intuition (state 1) appears duringdistress times During normal periods changes in interestrates are fully passed on to the yields of BBB-rated bondsleaving credit spreads unaffected During crises on the otherhand the increased uncertainty leads to a flight-to-qualityfrom the riskier corporate bonds to the safe risk-free USTbonds which pushes the respective yields to opposite di-rections After the end of a crisis the flight to quality stopsand there is a reversion to the regime of the normal periodOur findings contradict literature that supports a constantrelationship (eg [3]) both on the basis of constant sensi-tivity over time and on the effect of business cycles on creditspreads

Capital gains of the hedged US BBB portfolio exhibit tworegime changes which coincide with the two highest capitalgains of the UST Long portfolio (see Figure 4 for a jointpresentation) A potential answer to what triggers regimechanges may lie in expectations on the risk-free rate asviewed through the Fed target rates Between June 2003 andJune 2004 the federal fund rate reached a minimum value of1 (see Figure 8) At the same time a maximum in USTcapital gains was realized A similar situation occurred in2009 which coincides with another regime change

e same discussion can be held in terms of defaultprobabilities In state 1 changes in risk-free interest rates(yields) have a significant opposite effect on US BBB bondyields while in state 2 a change in risk-free interest rates isreflected by changes in US BBB bond yields of the same sizeand magnitude During the precrisis period (state 2) theaverage capital gains sensitivity of US BBB-rated bonds isclose to 1 averaging at 084 particularly between 2004 and2007erefore the response of US BBB bonds to changes inthe yields of risk-free assets on a capital gains basis is slightly

Table 4 Markov-switching regression in means only for the USBBB portfolio hedged by shorting UST bonds

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus11397 4038 minus282 0005 (minus19312minus3483)micro2 134794 14953 901 0000 (105486 164102)σ 47034 2556 (42281 52321)p11 0988 0009 (0951 0997)p21 0104 0068 (0027 0326)State 1 is the Merton case state 2 is the Kamin and von Kleist case

Table 5 Markov switching regression in means and standarddeviations for the US BBB portfolio hedged by shorting UST

Parameter Estimate Sterror z Pgt |z| 95 confidence

intervalmicro1 minus13258 3812 minus347 0001 (minus20745minus5771)micro2 119439 18024 663 0000 (84112 154766)σ1 42970 2556 (38241 48283)σ2 69012 9987 (51969 91644)p11 0987 0009 (0949 0997)p21 0087 0057 (0023 0280)Merton case state 1 Kamin and von Kleist case state 2

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 6 Regime change in means only state plot for the capitalgains hedged US BBB portfolio 32002ndash92016

300

200

100

0

ndash100

ndash200

Capi

tal g

ains

2002 2004 2006 2008 2010 2012 2014 2016 2018

022

02

018

016

014

Prob

abili

ties

BBB hedged by short USTState 1 one-step probabilities

Figure 7 Regime change in means and standard deviations stateplot for the capital gains of the UST-hedged US BBB portfolio32002ndash92016

Complexity 9

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

lower than 1-to-1 and changes in the yield of risk-free assetspassing on to the yield of corporate bonds are slightly re-duced According to [6] the probability of default is notaffected by changes in the interest rate and hence a change inthe latter does not result in a significant response On thecontrary between 2007 and 2010 (state 1) sensitivity be-comes muchmore volatile and dips in negative territoryeaverage sensitivity is clearly negative at a level of minus147which signifies a negative and amplified response of creditspreads to changes in interest rates According to [4] whichis based on contingent claims such a negative responseimplies that the probability of default of a debt issuer isaffected by changes in the interest ratee increase in creditspreads signifies a higher credit risk and probability ofdefault

is in turn is a signal for the creditworthiness of debtissuers After 2010 the intuition of Kamin and von Kleistemerges again with a positive average sensitivity of 078

Although the time span of our analysis ends in August2016 there is an interesting period ahead with environ-ments with prolonged low interest rates that had fosteredrisk-taking up to the coronavirus crisis Such risk-takingscenario has been faced by advanced economies for sometime and then later on the eve of the pandemic-fueledcrisis by some developing countries As the impacts of thecoronavirus crisis from the point of view of the interest-rate-based finance are commensurate with those of the2007ndash2008 global financial crisis we consider the interestrate sensitivity of the BBB-rated nongovernmental USbonds during the coronavirus recession to be negative andamplified It is so because the US Treasury yields arediminishing towards the all-time low while the precor-onavirus bubble in the BBB-rated debt outstanding makesthe yields of the nongovernmental relatively risky BBB-rated bonds climb us in accordance with our back-on-the-envelope estimates our conclusions hold in this en-vironment However thorough rigorous investigation ofthe low interest ratesrsquo influence and the impacts ofcoronavirus on the interest rate sensitivity is desirable andwill be addressed in our further research We also positthat during the initial recovery from the coronavirus crisislows the interest rate sensitivity will remain negative thistime because of the increase in UST treasury yieldsmdashdue

to the economic recoverymdashaccompanied by the decreasein BBB-rated yieldsmdashdue to improving business con-juncture and hence diminishing probabilities of defaulte sensitivity plot around the coronavirus crisis will bequalitatively similar to that in Figure 4 but shifted alongthe time scale to the right by roughly thirteen yearsseparating the two major crisis events

52 Sensitivity and Hedging during the Business Cyclee performance across diverse asset categories changesaccording to the phases of the business cycle [22 23] esephases are typically split into early- mid- and late-cyclephases and recession During the mid- and late-cycle phasesof economic expansion interest rates tend to be higher toprevent the economy from overheating During the reces-sion and early-cycle phases interest rates are kept lower tostimulate growth Notably these phases match those ob-served in the results for sensitivity and capital gains

e first and third period of our sample where spreadsare generally constant can be related to the mid and latephases of the business cycle both empirically and theoret-ically During those periods of normal to moderate eco-nomic growth default probabilities are low andcreditworthiness is only moderately by changes in the risk-free rate During the second period however the inverserelationship can be related to the downside of the businesscycle where a recession (crisis) and the first stage of recoveryhave taken place e observed flight-to-quality puts pres-sure on the prices of UST bonds which are coupled by acentral bank trying to stimulate the economy by reducinginterest rates Either of these factors or their joint occur-rence pushes the yields of the risk-free assets upwardsFigure 9 demonstrates these points along with the observedyields of US BBB-rated securities vs UST yields e di-vergence between the two yield time series matches thebehavior of credit spread with relation to the risk-free in-terest rate and the phase of the business cycle

At the same time the deteriorating economic conditionslow business performance and increased uncertainty makecorporate debt riskier which pushes US BBB-rated yieldsupwards us companies face more difficulties in servicingexisting debt and issuing new one at an acceptable rate

000

100

200

300

400()

500

600

700

800

03-0

1-20

0003

-01-

2001

03-0

1-20

0203

-01-

2003

03-0

1-20

0403

-01-

2005

03-0

1-20

0603

-01-

2007

03-0

1-20

0803

-01-

2009

03-0

1-20

1003

-01-

2011

03-0

1-20

1203

-01-

2013

03-0

1-20

1403

-01-

2015

03-0

1-20

1603

-01-

2017

03-0

1-20

18

Figure 8 US federal fund effective rate

10 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

Prices of corporate bonds move downwards and prices ofgovernment bonds move upwards e movement from thesecond to the third period of our sample can be seen as thereturn to a ldquonormalrdquo state after an increase in interest ratesfrom the central bank Tighter monetary policy coupledwith an improvement of economic conditions reduces the

premia of corporate bonds since the yields of risk-free assetsincrease and default risk decreases Capital losses are ob-served in USTportfolios while positive capital gains occur incorporate bond portfoliose capital gains sensitivity of theUS BBB-rated bonds during the recovery phase changesfrom negative to slightly positive

Old normal regime Distress crisis and recovery New normal regime

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (084)

Spreads mostlyinsensitive to ratespositive price-wisesensitivity (078)

Credit spreads tighten when the risk-free rates increase

and vice versanegative price-wise sensitivity (ndash147)

Risk-free rateCorporate yield

(a)

0

2

4

6

8

10

12

S200YL index

SA14YL index

31

019

101

31

029

102

31

039

103

31

049

104

31

059

105

31

069

106

31

079

107

31

089

108

31

099

109

31

109

110

31

119

111

31

129

112

31

139

113

31

149

114

31

159

115

31

16

(b)

Figure 9 Spread-to-risk-free rate dynamics compared to observed yields (a) Negative spread-to-rate sensitivity in distress compared to nullsensitivity in ldquonormalrdquo times (b) US BBB-rated (blue) and UST (red) bond yields

Complexity 11

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

Our findings for the downward phase of the businesscycle agree and expand on [3] but contradict their con-clusions on the upward phase ey find both a constantrelationship and no connection of their results to thebusiness cycle While we agree with a negative relationshipbetween spreads and interest rates during the recession andearly-cycle phases we interpret the periods of economicexpansion described in [3] as periods of sharp recovery fromrecession We present evidence that under the normal re-gime of the mid-cycle and the late-cycle expansion thenegative relation between interest rates and credit spreadsturns to insensitivity

Under these circumstances the dilemma of an investorin US BBB-rated bonds is a choice between a short positionin UST bonds or equivalently an interest rate swap thatreceives a floating rate for a fixed rate Our findings suggestthat such a hedge is meaningful and profitable in the upwardsection of the business cycle where spreads are relativelystable If this position however is maintained during arecession or an early recovery stage any profits will likely beeliminated swiftly as suggested by our results [14] In thedownward part of the business cycle an interest rate swapthat receives a fixed rate for a floating rate would be morebeneficial Hedging therefore must be dynamic and includeat least some degree of rebalancing as the business cycleprogresses As a result we are able to show that a businesscycle approach to interest rate hedging can add value as partof intermediate-term hedge strategies

6 Conclusion

To study the comparative dynamics of US BBB-rated debtand UST bonds we propose a comprehensive measure ofsensitivity based on capital gains of the representativeportfolios e approach puts direct emphasis on profitsor losses incurred by a portfolio due to changes in theunderlying yields While compared to assessing spread-to-rate sensitivity metrics the capital gains measure re-veals itself as a more suitable for the long-run investmentperspective as this measure by construction is primarilyfocused on the profits or losses of bond portfolio on ayear-on-year basis

We reconcile two opposing strings of literature thatassume a constant relationship between credit spreads of USBBB-rated bonds and the yields of risk-free UST by showingthat the relationship is not permanent but changes over timebetween two different regimes e first regime correspondsto periods of high and moderate growth in the businesscycle during which credit spreads have little to no reactionto changes in interest rates and risk-free yields [6] esecond regime corresponds to periods of recession and sharprecovery from it when credit spreads and interest rates movein opposite directions with amplified effect [4] Our pro-posed capital gain-based metric is able to shed light on thedynamics of spreads and yields and explain a long-standingtheoretical contradiction

Apart from the theoretical contribution our findingsalso have practical value for portfolio management andpolicy regulation A sensible hedging strategy for bond

portfolio would be to hold a long position in governmentbonds and a short position in risk-free bonds Note thatholding a short position in UST in such a synthetic portfoliois largely equivalent to a fixed-for-floating interest rate swapHowever under time-varying sensitivity such a position iseconomically inefficient since losses in an economicdownturn may be so severe that they may cancel out thegains obtained in normal times erefore for long-terminvestments dynamic positions and a proper reassessmentof fundamentals are crucial A portfolio investor that con-siders the phase of the economic cycle would not always relyon shorting USTas a hedge strategy but would alter or evenreverse crossing a crisis his exposure to risk-free UST in-struments along the business cycle

e implications for policy makers and portfolio in-vestors alike lie in the proper timing of trends A policymaker on the other hand should expect that other positiveeffects notwithstanding a reduction in interest rates during arecession may put additional pressure on corporate yieldsduring a time of low economic performance and perceivedhigh default risk Our remarks contribute to the ongoingdiscussion of exposure to credit risk and risk assessmentunder the Basel III capital accord namely Pillar IImethodologies

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by IPL (Instituto Politecnico deLisboa) and by FCT IP the Portuguese national fundingagency for science research and technology under theproject UIDB045212020

References

[1] M Boulkeroua and A Stark ldquoOn the determinants of thesensitivity of the yield spread of corporate bonds to changes inthe level and slope of the yield curverdquo in Proceedings of the IVworld finance conference pp 118ndash167 Rhodes Greece July2013

[2] D Sraer and V Haddad ldquoe banking view of bond riskpremiardquo Society for Economic Dynamics vol 814 2016

[3] B Dupoyet X Jiang and Q Zhang ldquoA new take on therelationship between interest rates and credit spreadsrdquoTechnical report Working paper of Florida InternationalUniversity Miami FL USA 2018 httpfacultyfiuedusimdupoyetbcredit_spreads_heteroskedasticitypdf

[4] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo e Journal of Finance vol 29no 2 pp 449ndash470 1974

[5] I Loncarski and P G Szilagyi ldquoEmpirical analysis of creditspread changes of US corporate bondsrdquo International Reviewof Financial Analysis vol 24 pp 12ndash19 2012

12 Complexity

[6] S B Kamin and K von Kleist e Evolution and Determi-nants of Emerging Markets Credit Spreads in the 1990s Bankof International Settlements Basel Switzerland 1999 httpsssrncomabstract850104

[7] B Eichengreen and A Mody ldquoLending booms reserves andthe sustainability of short-term debt inferences from thepricing of syndicated bank loansrdquo Journal of DevelopmentEconomics vol 63 no 1 pp 5ndash44 2000

[8] J Hilscher and Y Nosbusch ldquoDeterminants of sovereign riskmacroeconomic fundamentals and the pricing of sovereigndebtlowastrdquo Review of Finance vol 14 no 2 pp 235ndash262 2010

[9] M Gubareva and R Borges ldquoSwitching interest rate sensi-tivity regimes of US Corporatesrdquo e North AmericanJournal of Economics and Finance 2019 In press

[10] M Gubareva ldquoHistorical interest rate sensitivity of emergingmarket sovereign debt evidence of regime dependent be-haviorrdquo Annals of Economics and Finance vol 19 no 2pp 405ndash442 2018

[11] M Gubareva and M R Borges ldquoBinary interest rate sensi-tivities of emerging market corporate bondsrdquo e EuropeanJournal of Finance vol 24 no 17 pp 1569ndash1586 2018a

[12] Basel Committee on Banking Supervision ldquoStandards in-terest rate risk in the banking book Bank for InternationalSettlementsrdquo 2016 httpswwwbisorgbcbspubld368pdf

[13] European Banking Authority (EBA) ldquoGuidelines on themanagement of interest rate risk arising from non-tradingactivitiesrdquo 2018 httpwwwebaeuropaeu

[14] Committee on the Global Financial System (CGFS) ldquoFi-nancial stability implications of a prolonged period of lowinterest ratesrdquo CGFS Paper No 61 httpswwwbisorgpublcgfs61pdf 2018

[15] H G Fong and O A Vasicek ldquoA risk minimizing strategy forportfolio immunizationrdquo e Journal of Finance vol 39no 5 pp 1541ndash1546 1984

[16] I Welch Corporate Finance An Introduction Student ValueEdition Plus Myfinancelab Student Access Kit Prentice-HallUpper Saddle RiverNJ USA 2008

[17] M Gubareva andM R Borges ldquoTypology for flight-to-qualityepisodes and downside risk measurementrdquo Applied Eco-nomics vol 48 no 10 pp 835ndash853 2016

[18] M Gubareva and M R Borges ldquoRethinking economic capitalmanagement through the integrated derivative-based treat-ment of interest rate and credit riskrdquo Annals of OperationsResearch vol 266 no 1-2 pp 71ndash100 2018

[19] A Ang and G Bekaert ldquoRegime switches in interest ratesrdquoJournal of Business amp Economic Statistics vol 20 no 2pp 163ndash182 2002

[20] A Ang and A Timmermann ldquoRegime changes and financialmarketsrdquo Annual Review of Financial Economics vol 4 no 1pp 313ndash337 2012

[21] M Guidolin ldquoMarkov switching models in empirical fi-nancerdquo in Missing Data Methods Time-Series Methods andApplications pp 1ndash86 Emerald Group Publishing LimitedBingley UK 2011

[22] A Mikaelian ldquoCycle-adjusted capital market expectationsunder black-litterman framework in global tacticalasset allocationrdquo Review of Business and Economics Studiesvol 1 no 1 pp 89ndash99 2013

[23] M M Andreasen T Engsted S V Moslashller and M SanderldquoBond market asymmetries across recessions and expansionsnew evidence on risk premia (6-9-2017)rdquo CREATES Re-search Paper 2016-26 httpsssrncomabstract2834898 orhttpdxdoiorg102139ssrn2834898 2017

Complexity 13