The first step in using market prices

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1 Strppng Coupons wth Lnear Programmng DAVID E. ALLEN, LYN C. THOMAS, AND HARRY ZHENG DAVID E. ALLEN s professor of fnance at the School of Fnance and Busness Economcs of Edth Cowan Unversty n Western Australa, Australa LYN C. THOMAS s professor of operatonal research n the Department of Busness Studes of the Unversty of Ednburgh n the U.K. HARRY ZHENG s a lecturer of operatonal research n the Department of Busness Studes at the Unversty of Ednburgh. The frst step n usng market prces to ft the parameters of models for the prce of bonds s to strp the bonds of ther coupons. Ths s because most bond prcng models really model the current term structures of spot rates of benchmark rsk-free and rsky securtes (Treasury and corporate bonds) that s, the prces of zero-coupon bonds. There are few zero-coupon bonds avalable n the market, however. Although Treasury STRIPS can be used to represent these theoretcal rsk-free spot rates, there are some problems wth ths approach. The man one s that the Treasury STRIPS market s less lqud than the Treasury coupon market, whch means that the observed rates on STRIPS reflect a premum for lqudty. It s thus necessary to extract spot rates from yelds of coupon bonds of dfferent maturtes, both n the Treasury and corporate bond markets. The standard methods of strppng coupons are bootstrappng (Fabozz [998]) or lnear regresson (Carleton and Cooper [976]). If for each perod there s one and only one coupon bond that matures, these technques generate a unque set of spot rates over the perods. If there are no bonds that mature for some perods, however, or f there are several bonds that mature at the same tme, then there are not unque answers, and n some cases the technques gve rse to rates wth unacceptable features, partcularly n the case of rsky bonds. Jarrow, Lando, and Turnbull [997], for example, use these methods to strp out the rsky zero-coupon bond prces, and pont out several msprcngs, such as fve-year AA zerocoupon bonds prced above fve-year AAA zero-coupon bonds, and four-year B zerocoupon bonds prced below fve-year B zerocoupon bonds. The authors attrbute these msprcngs to the nose of the data and the call features of some bonds. These msprcngs are bothersome, because t becomes dffcult to estmate the parameters n the credt bond prcng models. There has been a resurgence of nterest n such models, as they not only gve nvestors a clear ndcaton of current market perceptons of the rskness of partcular bonds, but are also a steppng stone to prcng many credt-senstve fxed-ncome dervatves, such as callable and putable bonds, caps and floors, and mortgage-backed securtes. Jarrow and Turnbull [998], for example, derve the default probabltes of rsky bonds by combnng a default process wth an nterest rate model. They apply the Black- Derman-Toy model to buld a recombned bnomal short rate tree, then combne t wth the default process to form a larger tree for credt-rsky bonds, and fnally obtan default probabltes by forward and backward nducton methods. To remedy the msprcng caused by bootstrappng, Thomas, Allen, and Morkel- Kngsbury [998] suggest usng lnear pro- JUNE 2000 THE JOURNAL OF FIXED INCOME

2 grammng to strp out rsky zero-coupon bond prces. Ths produces the same spot rates as the bootstrappng technque f there s one and only one coupon bond that matures for each perod, but s always able to ensure that, for the same maturty the hgher-rated zero-coupon bond s prced above the lower-rated zero-coupon bond, and for the same credt ratng the shorter-maturty zerocoupon s prced above the longer-maturty zero-coupon bond. Although the LP formulaton avods the dffcultes encountered n Jarrow, Lando, and Turnbull [997], other problems have crept n. The man one s that the gap between zero-coupon bond prces of dfferent credt ratngs wdens and then narrows as tme goes by, whch means the forward rate of hgher credt-rated zero-coupon bonds s hgher than that of lower credt-rated zerocoupon bonds. Such a result agan suggests that there are potental arbtrage opportuntes. We suggest a new lnear programmng formulaton to strp out rsky zero-coupon bond prces that resolves these problems. Frst, we use an extenson of the orgnal LP approach to strp Treasury bonds, whch works whatever the current date, coupon dates, and samplng dates. Then we ntroduce a new LP formulaton for strppng the coupons from rsky corporate bonds that ensures that the spreads ncrease over tme. We dscuss how the LP formulaton can be modfed to deal wth lqudty ssues and extend t to arbtrary tme ntervals between the samplng ponts at whch the zero-coupon bond prce s calculated. There s a connecton between the zero-coupon bond prces obtaned by the LP formulaton and the default probabltes that the market s mputng to the rsky corporate bonds. I. TREASURY STRIPS PRICES To derve pure dscount bond prces v 0 (t) of rskfree zero-coupon bonds payng at a set of prechosen tmes t = 0,,..., T, we use the observed market prces of N 0 bonds to solve the lnear programmng problem: a, b ³ 0 for =,..., N 0 and t = 0,,..., T, where P s the present value of the bond ; c (t) s ts cash flow at tme t; v 0 (0) = ; and m(t) s the mnmum expected forward rate from t to t +. The frst constrants seek to match the present value P to the dscounted cash flows c (t), and a and b are the msprcng errors. a s postve and b = 0 f the prce s too low ; b s postve and a = 0 f the prce s too hgh. The second constrant ensures that there s no msprcng wth respect to maturty. (When m(t) = 0, the constrant corresponds to sayng bonds of longer maturty should be prced lower than those of shorter maturty.) The cash flow c (t) s decded by the coupon payment, coupon date, samplng date, and current date. As an example, assume t, t 2,..., t T are fxed semannual samplng dates. A bond pays a coupon c every sx months wth a prncpal F and a maturty date before or at tme t T ; there s one cash flow n each samplng perod. Let v be the prce of the rsk-free zero-coupon bond payng at tme t. Assume a s the proporton of the tme between a coupon date and the next samplng date compared wth the tme between two samplng dates (therefore a s a number between 0 and ). Let b be the proporton of a samplng nterval between the current date (when the market prce of the bond s observed) and the next samplng date. There are two cases to consder:. There s no coupon payment between the current date and the next samplng date. 2. There s one coupon payment. In the frst case, we have a ³ b, and the relaton between the market prce and the future cash flows s approxmated by: P = P C + (a b)c N0 LP: Mnmze å ( a + b ) = subject to P + a = å c () t v () t + b v 0 (t) ³ [ + m(t)]v 0 (t + ) T 0 t= () = acv + cv cv T 2 + (c + af)v T + ( a)(c + F)v T (2) Here P s the present value, P C s the clean market prce, and (a b)c s the accrued nterest. We have splt each coupon payment and the prncpal nto two parts. One ac s pad at the prevous samplng date, and 2 STRIPPING COUPONS WITH LINEAR PROGRAMMING JUNE 2000

3 one ( a)c s pad at the subsequent samplng date. In the second case, we have a < b and the resultng equaton s P = P C + ( + a b)c = (a/b)c + [a + (b a)/b]cv + cv cv T 2 + (c + af)v T + ( a)(c + F)v T (3) Here ( + a b)c s the accrued nterest. Note that the two equatons are bascally the same except for the cash value at the present date and the cash pad out at t. In the case of a = 0 and b = (the current date s the samplng date, and the next coupon payment s on the next samplng date), we have a very smple equaton: P C = cv cv T + (c + F)v T (4) Ths set of tmngs leads to the specal case of LP where, f bond has coupon c and prncpal F, and c (t) = c for t =, 2,..., T c (T) = c + F (5) For more general current dates, we have that f a ³ b, then the cash flows are and c () = ac c (t) = c for t = 2,...,T 2 (6) c (T ) = c + af c (T) = ( a)(c + F ) For a < b wth t 0 the current date, the cash flows are c (0) = (a /b)c c () = [a + (b a)/b]c c (t) = c for t = 2,..., T 2 (7) and c (T ) = c + af c (T) = ( a)(c + F ) The proof of Equatons (2) and (3) s gven n the appendx, wth some other cases. We use 3 Treasury bonds n the market on February 7, 2000, wth maturty dates up to the second half of Data come from Datastream. Snce coupons are pad semannually, we choose a sx-month tme nterval for a perod, and set the samplng dates on May 5 and November 5. Exhbt s the strpped Treasury zero-coupon bond prces v(t) on February 7, 2000, gven by LP. We apply prcng Equatons (2) and (3) to model the cash flows and market prces. The last column s the observed U.S. STRIPS prces on the same day from Datastream. The total error between the market prces and the estmated prces s 2.3, and the total market value of these bonds s 763. So the relatve error s less than 0.03%, a very good ft. The results are exactly the same for several dfferent mnmum forward rates m(t) from 0 E XHIBIT U.S. Treasury Zero-Coupon Bond Prces Date LP Prce Yeld (%) U.S. STRIPS 05/5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ JUNE 2000 THE JOURNAL OF FIXED INCOME 3

4 to 0.03, whch mples that the choce of m(t) s farly robust. Comparng the result wth the observed U.S. STRIPS prces, we see they are very close. II. RISKY ZERO-COUPON BOND PRICES Suppose bonds are classfed accordng to ther rskness nto ratngs from to M. The bond rated has the hghest qualty and the lowest default rsk, and the bond rated M has the lowest qualty and the hghest default rsk. Suppose there are N bonds observable n the market. Bond has present value P, maturty date T, cash flows c (t) for t =, 2,..., T, and credt ratng d(). Defne c (t) = 0 for = T +,..., T where T s the longest-maturty date among N bonds. Suppose for the class of bonds wth credt ratng j the prce of a bond strpped of ts coupons payng at date t s v j (t) for t =,..., T. To construct these term structures of spot rate curves of credt-rsky bonds, assumng we have already calculated the zero-coupon Treasury bond prces v 0 (t), t =, 2,..., T, we can formulate and solve the lnear programmng problem: LP2: Mnmze å ( a + b ) v j (t + ) v j+ (t + ) ³ v j (t) v j+ (t) a, b ³ 0 N0 = subject to P + a = å c () t v () t + b for =,..., N, j = 0,..., M, and t = 0,,..., T, where v j (0) =. The nequaltes v j (t + ) v j+ (t + ) ³ v j (t) v j+ (t) are used to characterze these bond propertes: The prce of a longer-maturty bond s cheaper than that of a shorter-maturty bond, and the prce of a hgher-rated bond s hgher than that of a lower-rated bond. The frst condton s satsfed by rewrtng the constrant as v j+ (t) v j+ (t + ) ³ v j (t) v j (t + ) and repeatedly applyng t from ratng j to 0 usng the fact that v 0 (t) v 0 (t + ) ³ 0. The second condton s satsfed by repeatedly applyng the constrant from tme 0 to t snce v j (0) v j+ (0) = = 0. The constrant actually conveys more nforma- T d() t= (8) ton. It says that the forward rates of hgher-rated bonds are lower than those of lower-rated bonds. Ths wll become clear when we study default probabltes of credt-rsky bonds later. We downloaded the lst of U.S. ndustry corporate bonds on February 7, 2000, from Datastream, whch provdes nformaton on S&P ratng, amount ssued, amount outstandng, next call date, and last date prce changed, as well as all standard bond nformaton. We use twenty-sx AA bonds, thrty-two A bonds, and thrty-two BBB bonds wth maturty up to November 5, 2005 (sx years), after excludng bonds that are unrated, or have call optons embedded, or have dfferent ssung amount and outstandng amount, or have not been traded for at least two months, or are of market value less than 00,000. The last two crtera try to remove bonds whose prces may be rrelevant because of llqudty. We do not nclude AAA, BB, or B bonds ether, as there are relatvely few such bonds avalable. Exhbt 2 shows the corporate pure dscount bond prces derved usng the lnear programmng model wth the Treasury pure dscount bond prces as reference v 0 (t). The last three columns are the yeld spreads between the Treasury bonds and the corporate bonds (n bass ponts). The ncreasng gap between rsk-free and rsky bond prces ndcates the ncreasng default rsks over longer terms. The relatve errors of LP prces and observed market prces are 0.25% for AA bonds, 0.4% for A bonds, and.6% for BBB bonds. The ncreased errors may be partly due to the rpple effects of hgher-rated bond prcng errors. We do not need to calculate the Treasury bonds and the corporate bond prces separately, but can calculate ther zero-coupon prces n the same LP problem. Ths ncorporates the constrants of LP and LP2 nto LP3: Mnmze å ( a + b ) v 0 (t) ³ [ + m(t)]v 0 (t + ) v j (t + ) v j+ (t + ) ³ v j (t) v j+ (t) a, b ³ 0 N0 + N = subject to P + a = å c () t v () t + b T d() t= (9) 4 STRIPPING COUPONS WITH LINEAR PROGRAMMING JUNE 2000

5 E XHIBIT 2 Rsky Zero-Coupon Bond Prces and Yeld Spreads Bond Prces Yeld Spreads (bp) Date Treasury AA A BBB AA A BBB 05/5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ for =,..., N 0 + N, j = 0,,..., M, and t = 0,,..., T. III. LIQUIDITY ISSUES The objectve of a lnear programmng model s to mnmze the sum of all under/over errors. Such a formulaton ndcates all bonds are treated equally. Yet the ssue amount of each bond may be qute dfferent, from hundreds of thousands of dollars for a corporate bond to tens of mllons of dollars for a Treasury bond. Ths has a sgnfcant mpact on the lqudty of ndvdual bonds. If the amount outstandng of a bond s small (the bd-ask spread tends to wden (to compensate for possble llqudty), whch may result n hgher/lower bond prces than for other more lqud bonds. Therefore, we should treat each bond dfferently dependng on ts lqudty. One way to do ths s to use the amount outstandng nformaton of all bonds n the market, whch s readly avalable from fnancal nformaton servces such as Datastream. If some bonds have much lower amounts outstandng than other bonds, we may treat them as llqud and remove them from the data set. Ths approach s easly mplemented by settng a threshold value and removng any bonds whose amount outstandng s below that value. Ths s the method we have used so far. For the Treasury bonds, the cutoff pont s set to be $0 mllon, whch s less than the amount outstandng of most Treasury bonds. For corporate bonds the cutoff pont s $00,000. Ths approach retans most lqud bonds whle removes some possbly llqud bonds. The dsadvantage of ths approach s how to choose a threshold value. Ths problem can be easly solved by reformulatng the LP model. Instead of the smple sum of under/over errors of the objectve functon, we can use the weghted sum of under/over errors. The weght of a bond s the proporton of ts amount outstandng to the total amount outstandng of all bonds n the market. To wrte out ths dea mathematcally, suppose there are N bonds to be used to derve pure dscount bond prces, and bond has amount outstandng M. Then the objectve functon s defned as Mnmze å w ( a + b ) (0) where weghts w = M /M and M = M M N. The obvous advantage of ths approach s that we do not need to set a threshold value to remove possble llqud bonds. If a bond has a smaller amount outstandng, ts weght s also small compared to other bonds. Snce weghts act as penalty costs n the objectve functon, the LP model wll try to mnmze errors of the bonds wth greater weghts and pay less attenton to those wth smaller weghts. Ths n turn removes the effect of bonds wth small amounts outstandng. Exhbt 3 shows the Treasury pure dscount bond prces usng the weghted LP model and all relevant Trea- JUNE 2000 THE JOURNAL OF FIXED INCOME 5

6 E XHIBIT 3 Bond Prces Usng Weghted LP Models Date LP Prce Weghted LP U.S. STRIPS 05/5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ sury bonds. We note that 90% of prcng errors are caused by 30% of the most llqud bonds. The prces derved from the two LP models are remarkably close, whch may be because a threshold value s used n the orgnal LP model. IV. SAMPLING INTERVALS So far we have dealt wth sx monthly ntervals between the samplng dates, but we mght want to have fner samplng dates for the near future and sparser ones for the dstant future. The general prcng equatons can be extended to allow for ths as follows. For each cash flow c k at tme s k, we can fnd two adjacent samplng dates t n and t n+ such that s k les n between. Defne t - s n+ k a k = tn+ - tn () Then the dscount factor v ~ at tme s can be approxmated k k as v ~ k = a k v n + ( a k )v n+ (2) The present value of all cash flows s the sum of ckv ~ k, whch then leads to a prcng equaton. Suppose the samplng perods are sx months for the frst fve years, and then one year for the next ten years. If a bond has three years to maturty, no change s requred. If a bond has ten years to maturty, then n the frst fve years, the contrbutons of each cash flow to ts adjacent samplng dates are a and a, respectvely. From year sx, there are two cash flows n each nterval. The contrbutons of the frst cash flow to ts adjacent samplng dates are ( + a)/2 and ( a)/2, respectvely; those of the second cash flow are a/2 and (2 a)/2, respectvely. Ths approach can smplfy dervaton of dscount factors for bonds coverng very long perods. For the same bond data as above, we use semannual ntervals for years 2000 to 2003, and annual ntervals for years 2004 to The results, gven n Exhbt 4, are very smlar to the equal samplng perod results. V. DEFAULT PROBABILITIES We have descrbed a way of constructng theoretcal Treasury and corporate pure dscount bond prces from the observed coupon bond prces. The yeld spread between Treasury STRIPS and corporate strps represents the premum of several rsk factors, e.g., default rsk, lqudty rsk, sector rsk. To smplfy matters, we assume the yeld spread s due purely to default rsk. Ths assumpton obvously exaggerates the default rsk, but t makes calculaton of default probabltes easer, and at least t gves E XHIBIT 4 Bond Prces Usng Varyng Samplng Intervals Semannual Date Prce Yeld (%) Prce 05/5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ /5/ STRIPPING COUPONS WITH LINEAR PROGRAMMING JUNE 2000

7 the upper bound of the rsk perceved by the market. Suppose the Treasury STRIPS prces v 0 (t) and the corporate zero-coupon bond prces v (t) are gven, where s the credt ratng. If a company defaults before ts bond matures, then a proporton d (the recovery rate) of the face value, dscounted by the remanng years to maturty, s gven to bondholders. Ths s the assumpton that most authors make (Jarrow and Turnbull [998], for example). (We assume the recovery rate s the same for all bonds, whether AAA bonds or C bonds, for smplcty. Ths assumpton can be relaxed to make d credt-ratng dependent.) Denote Q and k P as the cumulatve default and k survval probabltes of a bond currently rated at the end of perod k, respectvely, and let q and k p be the margnal k default and survval probabltes n perod k. Then, because f a t maturty zero-coupon bond does not default at all t s worth v 0 (t), whle under the assumpton above f t does default t s worth dv 0 (t), one has or v (k) = ( Q k )v 0 (k) + Q k dv 0 (k) Q k æ v( k) ö = ç - - d è v ( k) ø (3) for k =, 2,.... The other probabltes can be easly computed usng the relatons P k = Q k p k = P k /P k 0 q k = p k (4) for k =, 2,..., where P =,.e., a rsky bond s not n 0 default at tme 0. Exhbt 5 lsts default probabltes derved wth Equatons (3) and (4) for an example n Jarrow and Turnbull (998). The recovery rate d s assumed to be 0.4. The result s the same as that Jarrow and Turnbull [998] derve by buldng an nterest rate tree as well as a default tree. The sgnfcance of these recursve formulas s twofold. They provde a quck way to compute default probabltes, and they llustrate the ndependence between default probabltes and nterest rate models. The two ssues are decoupled. E XHIBIT 5 An Example by Jarrow and Turnbull [998] k v 0 (k) v (k) q k Q k E XHIBIT 6 Default Probabltes of Rsky Bonds Margnal Default Cumulatve Default Probabltes Probabltes Date AA A BBB AA A BBB 5/05/ // /05/ // /05/ // /05/ // /05/ // /05/ // For the Treasury and corporate zero-coupon bond prces derved n Exhbts and 2, we can quckly compute the cumulatve and margnal default probabltes. The results are shown n Exhbt 6. Margnal or cumulatve default probabltes that are negatve or greater than clearly ndcate that there are msprcngs of zero-coupon bonds. The LP formulaton ensures that ths wll not happen. VI. CONCLUSION We have shown that lnear programmng can be used to strp coupons for both Treasury and corporate bonds. The advantages of the LP approach are that there s no msprcng and the spread structure s bult nto the model. Real data can be easly analyzed snce the LP formulaton works whatever the current date, coupon dates, and samplng dates. The weghted LP model can be used to deal wth data that may nclude some less lqud bonds. Fnally, default probabltes of rsky bonds perceved by the market can be easly calculated wthout relyng on any JUNE 2000 THE JOURNAL OF FIXED INCOME 7

8 nterest rate models. If there are two cash flows n the future, then A PPENDIX Proof of Equatons (2) and (3) There are only two cases need to be consdered: ) there s no coupon payment between now and the next samplng date, and 2) there s a coupon payment between now and the next samplng date. The samplng dates (from the next one) are labeled as t =, 2,..., T. The frst case corresponds to a ³ b. The present value of all cash flows of a bond s equal to the sum of c ~ (t)v ~ (t) over all t from to T, where c ~ (t) s the cash flow, and v ~ (t) s the dscount factor at tme t. Usng lnear nterpolaton, we can wrte the value of v ~ (t) as a combnaton of repayments at t and t + v ~ (t) = av t + ( a)v t+ for t =,..., T. Therefore the prncpal value PV satsfes PV = c ~ ()v ~ () c ~ (T )v ~ (T ) = c(av + ( a)v 2 ) (c + F)(av T + ( a)v T ) = acv + cv cv T 2 + (c + af)v T + ( a)(c + F)v T The accrued nterest s taken n the market to be (a b)c, and the general equaton for bond prces s gven by MP + AI = PV where MP s the market (clean) prce of the bond. Substtutng the accrued nterest and the present value nto the equaton, we obtan the frst prcng equaton. The second case corresponds to a < b. The present value of all cash flows of a bond s equal to the sum of c ~ (t)v ~ (t) over all t from 0 to T. The dscount factors v ~ (t) can be computed the same as above for t =,..., T. In computng v ~ (0), however, we must remember that the tme nterval from now to the frst cash flow s b a, and the tme nterval from now to the next samplng date s b, so lnear nterpolaton gves ṽ = a b + - a v b b 0 If there are three or more cash flows n the future, then The accrued nterest s equal to ( b + a)c. Substtutng everythng nto the general equaton for bond prce gves the second prcng equaton. ENDNOTE a éb- a ù PV = c + ê c + a( c + F) ú v + ( - a)( c + F) v b ë b û a æ b- a ö PV = c + ç c + a cv + cv b è b ø The authors are grateful to Datastream for the data and to Ngel Morkel-Kngsbury for advce on extractng the data. REFERENCES 2 cv + ( c + af) v + ( - a)( c + F) v T-2 T- T Carleton, W., and I. Cooper. Estmaton and Uses of the Term Structure of Interest Rates. Journal of Fnance, 3, No. 4 (976), pp Fabozz, F. Valuaton of Fxed Income Securtes and Dervatves. New Hope, PA: Frank J. Fabozz Assocates, 998. Jarrow, R., D. Lando, and S. Turnbull. A Markov Model for the Term Structure of Credt Rsk Spreads. Revew of Fnancal Studes, 0, No. 2 (997), pp Jarrow, R., and S. Turnbull. Credt Rsk. In C. Alexander, ed., Rsk Management and Analyss. New York: John Wley & Sons, 998, pp Thomas, L., D. Allen, and N. Morkel-Kngsbury. A Hdden Markov Chan Model for the Term Structure of Bond Credt Rsk Spreads. Workng paper, Centre for Fnancal Markets Research, Unversty of Ednburgh, 998. If there s only one cash flow n the future, then a b PV = c+ F + - a ( ) ( c+ F) v b b 8 STRIPPING COUPONS WITH LINEAR PROGRAMMING JUNE 2000