Erasmus University Rotterdam ErasmusSchoolofEconomics Global Yield Curve Arbitrage with Term Structure Modeling JiaweiXu 380392 Universitysupervisor: DickvanDijk Externalsupervisor: CasperZomerdijk Co-reader: MichaelGong Athesissubmittedinpartialfulfillmentoftherequirementsfor thedegreeof MasterinEconometricsandManagementScience Datefinalversion: October17,2017 Abstract This study investigates the profitability of trading strategies based on yield curve arbitrage op- portunities in international fixed-income markets. For constructing the yield curve, this pa- peradoptsthree-andfour-factorNelson-Siegelmodels,cubicB-splines,andkernelsmoothing methods as candidate models, and evaluates the model fit by their ability of identifying the mispricing of the government bonds with a generic yield curve arbitrage strategy that longs all the over-priced bonds and shorts all the under-priced bonds. Next the paper continues with the most feasible yield curve models selected and constructs a modified yield curve arbitrage strategywithimprovedstrategyspecificationsthatmaximizeexposuretothearbitrageopportu- nitieswhilelimitingeffectofotherriskfactors. Theresultsshowthatthereisnosuperioryield curvemodelgloballywhenthepricingpoweroftheyieldcurveisexaminedbutthefour-factor Nelson-Siegelmodelsarepreferredbymorecountries. Ayieldcurvearbitragestrategyisprof- itable for all countries and is able to generate a positive risk premium. A more refined strategy specificationisabletofurtherenhancetheperformancebyreducingvolatilitiesandeliminating exposurestootherriskfactors. Keywords: Fixed-incomearbitrage,governmentbonds,yieldcurvearbitrage,termstructure modeling,tradingstrategy Contents 1 Introduction 1 2 DataConstructionandAnalysis 5 3 YieldCurveFittingMethods 9 3.1 Yieldcurvemodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Modelevaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 YieldCurveModelSelection 20 4.1 Modelfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Economicvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Modelselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 YieldCurveArbitrage 35 5.1 Modifiedtradingstrategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.3 Sensitivityanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6 Conclusions 45 Appendices 50 AppendixA BootstrappingFama-Blissunsmoothedyields 50 AppendixB BayesianinferencefortheNelson-Siegelmodels 51 AppendixC EstimationofthecubicB-Spline 55 AppendixD Tables 58 AppendixE Figures 66 1 Introduction Fixed-incomearbitrage,asthenamesuggests,exploitsthemisvaluationsduringthepricingpro- cesses of fixed-income securities, and has become one of the most popular choices for hedge funds and arbitrageurs since the hedge fund crisis of 1998. Within the fixed-income arbitrage sector, yield curve arbitrage is one of the well-known arbitrage strategies that show good per- formanceandprofitability(seeDuarteetal.,2006). Essentially,ayieldcurvearbitragestrategy identifiesunder-andover-pricedgovernmentbondsalongtheyieldcurveandsetsupportfolios accordingly. Duarte et al. (2006) conclude that the better performance of the yield curve arbi- trage strategy boils down to the fact that this strategy requires most “intellectual capital” as it involvesmodelingtheyieldcurve. Thusforasuccessfulstrategyimplementation,modelingthe termstructureisasimportantasdesigningthestrategyspecifications. Thisresearchfollowsthepathoftheyieldcurvearbitrageliteratureandaimstogoonestep further by seeking for a trading strategy specification that harvests the potential mispricings of the government bonds, and exploring the profitability and significance of yield curve arbitrage as an anomaly. Along the way, this study also investigates different term structure models and intendstoidentifythemostfeasibleonesamongthem. Forthisstudy,thepriceinformationofthegovernmentbondsareobservedattheendofthe month,andtheyieldcurveisthenmodeledforsettingupthepositionsofthearbitrageportfolio atthebeginningofthenextmonth. Asneitherforecastingthetermstructurenorthedynamicsof thetermstructureovertimeisofinterest,affinetermstructuremodelsthatdescribethestylized time-series properties of the term structure (see, for example, Vasicek, 1977; Cox et al., 1985) are not considered for this research, and only the curve fitting category of the term structure modelsareidealcandidatesformodelingtheyieldcurves. Termstructuremodelingwithcurve fitting methods has been studied intensively by fixed-income literature and great progress has been made over the last decades. Nelson and Siegel (1987) propose to fit the term structure withaparametricfunctionofthreeexponentialcomponentsthatarecapableoffollowingmany of the typically observed shapes that the yield curve assumes. Since then various extensions based on this framework have been explored to increase the flexibility. Bjo¨rk and Christensen (1999) and Svensson (1994) incorporate one extra factor to capture the yield movements in the short and medium maturities, respectively, and both their results show improvements in terms of yield curve fitting. Although with no restrictions regarding arbitrages imposed, the 1 Nelson-Siegel model and its extensions have been the workhorse of estimating the yield curve in financial practice due to their parsimony and good empirical performance. The Bank of International Settlements (BIS, 2005) reports that among the thirteen central banks that report their curve estimation methods to the BIS, nine use either the Nelson-Siegel or the Svensson model to construct zero-coupon yield curves. On the other hand, McCulloch (1971) estimates the discount factors using regression splines, making the spline principles a popular approach. FollowingthispathChambersetal.(1984)fittheyieldcurveitselfwithpolynomialsplines,and Vasicek and Fong (1982) show that exponential splines provide better yield curve estimations comparedtopolynomialsplines. Lintonetal.(2001)proposetousekernelsmoothingmethods for estimating the term structure and conclude that the estimations generated by their methods have sensible interpretations even under misspecification. Sediva and Marek (2015) show that yield curves estimated by kernel smoothing are comparable to those generated by parametric methods. ThisstudyadoptstheNelson-Siegelmodelsanditsextensions,cubicregressionsplines,and kernel smoothing for fitting the term structure. In order to overcome the numerical difficulties the Nelson-Siegel type of models suffer from due to the existence of non-linearity (see, for example,GimenoandNave,2006)andachievearobustfit,theestimationoftheNelson-Siegel model and its extensions is conducted using Bayesian inferences with Markov chain Monte Carlo(MCMC)simulationtechniques. The goodness-of-fit of the yield curves generated by different methods is afterwards eval- uated to select a most feasible yield curve model. Followed by most existing literature, the standard approach is to evaluate the models by means of their statistical properties (see, for example, Bliss, 1996; de Pooter, 2007) such as the (adjusted) R-squared or the pricing (yield) errors. Nevertheless, these measures are from a curve fitting point of view. For the purpose of evaluating term structure models that a trading strategy will be based on, one aspect that is probably even more relevant yet often neglected is the economic values behind the modeled curves. For example, any line that connects or goes through all the data points may give satis- factory statistics as it indeed produces no residuals, but it is not very likely a trading strategy ormonetarypolicybasedonthisyieldcurvemodelwillachieveconvincingresults. Thispaper proposestoevaluatetheyieldcurvemodelswithayieldcurvearbitragestrategythatdelvesinto howwellthemodeledcurvesidentifythemispricingsinthebonds. Theintuitionhereisthatif thefittedyieldcurvebyamethodissuperior,thatis,itcapturestheunderlying“true”termstruc- 2 turepreciselyovertime,anarbitrage-basedtradingstrategythatgoeslonginthebondsthatare undervaluedbythemarketandshortintheonesovervaluedshouldbeprofitable. Followingthis approach,agenericyieldcurvearbitragestrategyisformedtoconstructportfoliosthatlongsall the underpriced bonds and shorts the over-valued ones along the modeled yield curve, and the weightofeachbondintheportfolioisdecidedbythecorrespondingyielderror. Toavoiddura- tionbias,theweightsofthebondarealsoscaledaccordingtothecorrespondingdurations. and themostfeasibletermstructuremodelingmethodisthenselectedaccordingtotheperformance of this yield curve arbitrage strategy, with the criteria involving both the returns and the risks over the full sample period and the subsamples. As a byproduct, such a fixed-income arbitrage strategy can serve as a foundation of finding a best yield curve arbitrage strategy that further exploitsandbenefitsfromthemispricingsofthebonds. TheresultsofthegenericstrategyshowthattheNelson-Siegelmodelsarepreferredbymost countries as a yield curve model, and within the Nelson-Siegel family, the four-factor models are with advantage. On top of providing a yield curve model selection, the results also give insight on some characteristics of the strategy performance. Firstly, such an arbitrage-based strategy is profitable in all countries. Moreover, the profit can be mostly attributed to periods when the market is with more turbulence or during crisis. Hu et al. (2013) conclude that the magnitudeofthegovernmentbondmispricingislargerduringcrisisfortheU.S.,asthearbitrage strategy is based on mispricings, this result is reasonable. To further understand the drives of arbitrage strategy returns, a market portfolio and two portfolios based on the yield curve slope and curvature changes are formed as common risk factors and the arbitrage strategy returns’ exposuretothesefactorsareexamined. Therisk-adjustedperformanceshowthatalthoughbeing exposedtotheriskfactors,especiallytotheyieldcurvepositions,thearbitragestrategyisableto generateapositivealphainallcountries,andthisriskpremiumbasedonarbitrageopportunities is significant for most countries. Finally, the Nelson-Siegel-model-based strategy shows more profitability in most countries in general, but it is also observable that the strategy with more flexible non-parametric methods (B-splines and kernel smoothing) has an advantage when the marketisvolatileandwithmuchmovements. With the selected modeled yield curve, this paper continues studying yield curve arbitrage by looking for improvements in the trading strategy settings. The yield curve arbitrage strat- egy for evaluating the yield curve fitting method is a generic starting point that considers only maximizing the exposure to the arbitrage opportunities along the yield curve. The results of 3 the risk-adjusted performance of the generic strategy, however, suggest that positioning neutral to the risk factors can enhance the performance such that the risk profile of the strategy can be improved. Moreover, lowered exposure to risk factors that influence the bond returns can also make the strategy itself more practically applicable. To improve upon the generic strategy, this researchconsidersthesettingofBeekhuizenetal.(2016)andconstructsamodifiedstrategythat groupstheavailablebondsintomaturitybucketsof1-3year,3-5year,5-7year,7-10year,10-15 yearand15+yearaccordingtheircorrespondingremainingtime-to-maturities. Foramaturity- bucket-based strategy specification, as each bond within the maturity bucket has similar matu- rity, the long- and short- positions are better hedged so the strategy is less exposed to market movements. Also, within a maturity bucket, the slope and curvature of the yieldcurve is much reducedsuchthattheimpactoftheyieldcurvechangeislessinfluential. Similarsub-portfolios asthegenericstrategyarethenconstructedwithinthematuritybuckets: eachsub-portfoliohas onedurationoflongpositionandonedurationofshortposition,andtheweightofeachbondis decided by the yield error. Finally the positions of bonds are scaled by the number of buckets enteringtradesuchthetotalpositionofthewholeportfolioisonedurationlongandoneduration short. The results show that this modified strategy is able to achieve at least as good profitabil- ity as the generic strategy with much lower volatilities. The exposures to other risk factors are much lowered at the same time. Sensitivity analysis with the holding period of the portfolios andincorporatingtransactioncostsalsoshowsthatthemodifiedstrategyismorerobustagainst alternativetradingstrategyspecificationsandisthusmorepracticallyimplementable. This study contributes to both academic literature and market practice in several ways. For academicresearch,thisstudycomparescomprehensivelyvariousversionsoftheNelson-Siegel modelsestimatedusingBayesiantechniqueswithdifferentnon-parametriccurvefittingmethods forthepurposeofstatictermstructureestimation. Thesampleforthisresearchcoversthemajor government bond market in the world, which is a wider range of countries than most of the existing literature. Moreover, evaluating the yield curve models with a fixed-income arbitrage strategythatexaminestheyieldcurvemodel’sabilityofidentifyingthemispricingsisinnovative for the fixed-income literature. Since the trading strategy and the returns of the portfolios are basedonarbitrageopportunities,thisstudyalsoenrichesthefactorinvestingliteraturewiththe arbitragefactor. Theremainderofthisthesisstartswiththedevelopingthedatainsection2. Theyieldcurve modelsconsideredinthisstudyareelaboratedandexaminedinsection3andsection4. Section 4 5 is dedicated for exploring possible improvements for the yield curve arbitrage strategy, and section6completeswithconclusionsanddiscussions. 2 Data Construction and Analysis The bond data used for this research is retrieved from Barclays database1. More than 80 char- acteristics of 78,676 bonds from all over the world are given on a monthly basis, starting from September1988toOctober2016. To keep the bonds included in the analysis as homogeneous as possible, bonds bearing dif- ferentcharacteristicsthatmayinfluencetheyieldpatterns(forexample,creditriskorilliquidity) arenotdesirable. Forthispurpose,Ionlyconsidertreasuriesissuedinlocalcurrencybygovern- ments from Australia (AU), Canada (CA), Germany (DE), the U.K. (UK), Japan (JP), Sweden (SE),andtheU.S.(US),asthegovernmentbondmarketsofthesecountriesarerelativelylarge, safe,liquidanddeveloped2. Theremainingdatafromthesesevencountriesarealsogiventreat- ments for the same purpose of keeping homogeneity. Following existing literature (see, for example, Diebold and Li, 2006; Gu¨rkaynak et al., 2007; Diebold et al., 2008), the procedure is asfollows 1. Calculate the all-in dirty price as the sum of price and accrued interest as the bond price isonlygivenasthecleanprice. 2. Exclude floating rate or deferred-coupon and floating coupon frequency bonds3, bonds with option features4, bonds extended beyond the original redemption date, bonds with non-regular-bond security types5, bonds that changed their characteristics (coupon type, 1Therearetwoversionsofthedataavailableforeachmonth,onefromthestatisticsuniversereflectingthedaily changesofthebondsandanotherfromthereturnuniversepresentingthecharacteristicsofthebondsattheend ofeachmonth. Thosetwoversionshaveidenticaldataforthemajorityofcases,exceptionshappenwhenabond dropsoutorenterstheindexinthemiddleofthemonth,whenthishappensitwillbedocumentedinthestatistics universe,butwillnotbeusedforreturncalculationofthatmonth,causingdifferenceinthosetwoversionsofthe data. This study focuses on the return universe, as it may happen that a bond the portfolio holds drops out of the index during the holding period for some reason, and the return universe in this case will document its last observedinformation,whichcanbeusedforrebalancingwiththeassumptionthatthepositionisclearedwhenit islastobserved,thestatisticsuniverse,however,willnolongerhaveanyinformationforthisbondattheendofthe month,whichcausesproblemsforthestrategyconstructionandreturncalculation. 2About51.91%oftheobservationsbelongtothesesevencountries. 3Coupon type of ‘Euribor Floater’, ‘GBP Floater’, ‘Pure Libor Floater’, ‘Step-up Once’, ‘Step-up Multiple’, ‘CompoundcouponpaidatMaturity’,‘PartlyPaid’,‘Pay-In-Kind’andundefinedcoupontype. 4Put/calltypeof‘Clean-upCallFeature’,‘Callable,Refundable’,‘Callable,NeverRefundable’,‘EuropeanCall (CallableOnlyonScheduledDates)’,‘PutablewithaFixedSchedule’,andundefinedput/calltype. 5Security type of ‘Collateralized Strip’, ‘Credit Default Swap’, ‘Global Issue’, ‘Locally Issued Bonds’, ‘RE- TAIL’,‘TreasuryInflationProtectionSecurities’andundefinedsecuritytype. 5 security type, and currency) over time, and bonds with market value smaller than 50 million. About28.13%oftheobservationsareleftafterthisprocedure. 3. Excludeobservationswithnot-availableprice,observationswithnon-trader-quotedprice6, observations with zero or not-available amount outstanding, observations of bonds with yield-to-maturity lower than -1% or higher than 50%, observations with zero modified durationandobservationsofbondsmaturinginlessthanthreemonths,whereamonthis definedas30.4375daysasinDieboldandLi(2006)7. About25.38%oftheobservations areleftafterthisprocedure. 4. Excludeobservationswithdataerrors: observationswherethebondispricedbystalepric- ing8, observation with uncleaned price9, observations with a issuing date later than the maturing date, bonds issued by corperates while classified as government bonds, obser- vationswithyieldsdifferinggreatlyfromyieldsatnearbymaturities10,andobservations withexcessmonth-to-dayreturnperdurationmorethan150basispoints. About21.63% oftheobservationsareleftafterthisprocedure. After the data cleaning, there are 4,111 bonds with 105,802 observations left for the use of modeling the term structure of the seven countries. An overview of how the data is distributed across countries is summarized in Table 1. One can observe that when looking at sample size, Japan and the U.S. are the markets with larger number of bonds while Australia and Sweden have relatively less bonds. The number of bonds available per month are not stable and vary substantiallyforallcountries,thusnumericalproblemsmayappearwhentheyieldcurvemodel tobeestimatedhasalargenumberofparameters,especiallyforAustraliaandSweden. 6Onlytrader-quotedpriceisthetradedpriceofthebond. 7Suchaspecificationisaweightedaverageofallpossiblenumbersofdaysinamonth: 30.4375 = 7 ×31+ 12 4 ×30+ 1 (3 ×28+ 1 ×29). Similarly,ayearisdefinedas365.25daysthroughoutthisstudy. 12 12 4 4 8Thisisdonebycheckingifthepricestaysthesameupto4decimalsfortwoconsecutivemonths. 9Thisisdonebycheckingifthereisashootupofthecleanpriceandazeroaccruedinterestwhentheobservation dateisnotaroundacouponpaymentday. 10Thisisdonebycheckingifthereisanextremeandunexplainableoutlierofthebacked-outFama-Blissspot ratesforthecorrespondingbondobservationused,seethetextbelow. 6 Australia Canada Germany theU.K. Japan Sweden theU.S. #ofbonds 52 153 284 184 894 44 2500 #ofobs. 2923 8339 12368 7204 52684 2565 64197 Avg. #ofbondspermonth 14.03 42.63 56.523 32.73 127.20 11.586 145.90 Min. #ofbondspermonth 9 25 44 24 43 8 97 Max. #ofbondspermonth 22 50 82 46 152 15 202 Table1: Numberofbondsandobservationsavailablepercountry. Thefirstandthesecondrowshowthe totalnumberofbondsandobservationsavailableforeachcountry. Thethirdtofifthrowshowtheaverage, theminimum,andthemaximumnumberofbondsobservedpermonthforeachcountry,respectively. Table 2 provides summary statistics of the (cleaned) bond data. Only the results for U.S. bonds are presented in the text for demonstration as the U.S. is one of the largest and most developedgovernmentbondmarkets,andsimilartablesforbondsfromtheothercountriescan be found in the Appendix. The bonds are grouped and analyzed according to their maturities for the ease of illustration. For the U.S., most market shares are given to the long-term bonds, although more than a half of the bonds available are with maturity of less than five years. The yield of the bonds show that longer term bonds in general have higher yield as well as lower volatilities, which is in line with the stylized fact that the yield curve general takes an upward shape and that the long end of the yield curve is less volatile. Note that most of the bonds with 10 to 15 years till maturity are earlier issued as 30 or longer term bonds, which explains the olderageofthe10-15yearmaturitybucket. Avg. Avg. Avg.Maturity Avg.Market Avg.Age Avg. Std. Skew. Kurt. JB Maturity #Bonds Duration (year) value(bln.) (year) Yield(%) Yield(%) Yield Yield Stats. US(09/88-10/16) 0-1year 36.08 0.55 0.56 21.20 2.28 3.34 2.64 0.26 1.95 19.18 1-3year 53.68 1.76 1.85 19.85 2.75 3.71 2.60 0.18 1.90 18.88 3-5year 34.91 3.62 4.00 18.28 2.85 4.20 2.42 0.12 1.95 16.27 5-7year 17.01 5.14 6.02 19.21 3.93 4.54 2.26 0.13 2.03 14.18 7-10year 14.31 6.86 8.52 26.24 2.96 4.84 2.11 0.14 2.10 12.48 10-15year 8.20 8.41 12.64 12.49 14.29 4.99 2.09 0.30 2.14 13.25 15+year 25.73 12.96 23.78 20.06 6.11 5.47 1.83 0.15 2.11 12.32 Table2: SummarystatisticsforgovernmentbondsoftheU.S.,fromSeptember1988toOctober2016, groupedaccordingtothematurityofthebonds. Column2tocolumn6containthecharacteristicsofthe bondsinthematuritybucketsovertime,measuredbytheaverage: numberofbonds,duration,maturity, marketvalueandageofthebonds;column7tocolumn11reportthestatisticsoftheyieldsofthebonds in the maturity buckets: mean, standard deviation, skewness, kurtosis and Jarque-Bera statistic. The resultsforyieldsareonannualbasis. Next the spot rates are extracted from the filtered data using the method documented by Fama and Bliss (1987) for bootstrapping the unsmoothed zero-coupon yields. As measuring the zero-coupon rates precisely is a crucial foundation for modeling the yield term structure, 7