A hidden Markov chain model for the term structure of bond credit risk spreads

Transcription

1 A hidden Markov chain model for the term tructure of bond credit rik pread Lyn.C.Thoma Univerity of Edinburgh. David.E. Allen Edith Cowan Univerity Nigel Morkel-Kingbury Edith Cowan Univerity Abtract Thi paper provide a Markov chain model for the term tructure and credit rik pread of bond proce. It allow dependency between the tochatic proce modeling the interet rate and the Markov chain proce decribing change in the credit rating of the bond by their mutual dependency on a hidden Markov chain. Thi Markov chain can be thought of a the underlying economic condition. The model alo allow a new interpretation of rik premia ued in previou approache. It alo ue a linear programming approach to trip the bond of their coupon in uch a way a to guarantee there i no mi-pricing. Key Word Bond pricing; credit rating; credit pread; linear programming; Markov chain : rik premium JEL G2 Aet Pricing

2 I Introduction Corporate bond pricing model have been in exitence for twenty-five year but it i only recently that a pricing model, which incorporate a firm credit rating a an indicator of the likelihood of default, ha been developed. Thi i urpriing ince the rating of a company given by the maor international credit rating agencie i the mot widely available etimate of the credit rik involved in inveting in the firm bond. The firt model of bond price to incorporate credit rating (Jarrow, Lando, Turnbull, 977) aumed that the tochatic proce decribing the rating and poible bankruptcy of the firm wa independent of the tochatic proce giving future interet rate and hence the default-free bond price. Thi paper preent a generalization of thi model in which the two procee are dependent through their relationhip with the tochatic proce decribing the tate of the underlying economy. The model alo generalize the idea of rik premia adutment by reinterpreting them a belief that the future of the rating and bankruptcy proce i more extreme than it ha been hitorically. Thi paper alo introduce a procedure baed on linear programming for tripping out the zero-bond price for riky and rikle bond in a way that guarantee there i no mi-pricing. Since Merton ground breaking paper, Merton (974), there ha been a number of modeling approache to the price of riky debt. Duffee review, Duffee (996), and the paper by Jarrow et al, Jarrow (997), outline thee type of model. The firt model view the firm liabilitie a contingent claim againt the underlying aet and aume that bankruptcy and bond non-payment occur when the firm aet are exhauted. Thi wa the model introduced by Merton (974), but it lead to maller credit pread than thoe that actually occur. Black and Cox (976) aduted the model by defining bankruptcy to occur where liabilitie are ome fixed proportion 2

3 of the aet and thi lead to more realitic credit pread. Shimko, Teeima and van Deventer (993) generalized the model by allowing the rikle interet rate to follow a tochatic rather than a determinitic proce and that the interet rate tochatic proce wa correlated with the firm aet proce. Kim et al (993) allowed bankruptcy to be triggered at an exogenouly pecified aet value. Leland (994) and Leland and Toft (996) ued thi type of model, but with endogoneou condition to define when bankruptcy i declared, to examine how important i the maturity of the debt a well a the amount. The difficulty with thi approach i that it depend on knowledge of the firm aet, which are not tradable and are only partially obervable. Alo it ha to deal with the often complex priority tructure of a firm liabilitie. The econd approach aume that on bankruptcy, the firm will pay off a pre-pecified fraction of the rik-free value of the intrument where bankruptcy i again triggered when the firm aet firt reach ome pecified limit. Thi idea, firt developed by Hull and White (99) enable one to ignore the debt priority problem but till aume knowledge of the aet value tochatic proce. The Hull and White model aumed independence of the tochatic proce giving the firm aet value and the proce giving the rik-free value of the intrument, which i eentially the proce decribing the interet rate movement. Longtaff and Schwartz (995) generalized the model by allowing non-zero correlation between the two procee. Since in both thee model bankruptcy occur when the aet value, which i a continuou proce, hit a pre-pecified limit, firm never default unexpectedly. For thi reaon, Madan and Unal (994) and Lando (994), modeled the aet value a a ump proce o that the firm value can uddenly ump below the bankruptcy level. 3

4 The third approach ignore the aet value completely and again overcome the debt liability tructure by auming that on bankruptcy a given fraction of each promied dollar i paid off. Thi approach aume the bankruptcy proce i pecified exogenouly and doe not depend on the firm underlying aet. (e.g. Jarrow and Turnbull (995), Litterman and Iben (99)). Lando thei (994) wa the firt to ue the evolution of the firm credit rating a a model for the bankruptcy proce. In the econd eay in hi thei, he develop a continuou time Markov model in which he aume the bankruptcy proce and the proce that give the rik free bond price are independent. Thi model and a dicrete time equivalent model appear in the eminal paper by Jarrow et al (997) but both aume independence between the interet rate proce and the credit rating proce. Thi aumption mean their model can not take into account Duffee point (Duffee 996) that default are primarily driven by the buine cycle, which derive variation in the financial variable on which mot derivative are priced.. Benninga ( Chapter 7, 997) develop a imilar preadheet model for finding the expected return on a riky bond uing the probabilitie of default, the tranition probabilitie that the bond credit rating will move from one level to another and the percentage recovery on the face value of the bond. Our paper extend the model in the Jarrow, Lando, Turnbull (997) paper in two way. Firtly it allow dependency between the tochatic proce decribing interet rate and hence the rik-free bond price, and the tochatic proce decribing the movement in bond credit rating by linking them both to an underlying proce which decribe the tate of the economy. The tranition probabilitie between 4

5 tate in the two procee (interet rate and credit rating) vary depending on the tate of the economy proce. Thi extenion i Model of ection two. Lando (994) in the third eay of hi thei uggeted a bond pricing model baed on urvival analyi where the probability of a bond defaulting i given by a proce whoe parameter can include interet rate information, although no empirical calculation are made uing thi model. The interet rate information affect the probability of the bond credit rating changing but if the rating doe change it doe not affect the probability of which rating it will move to. Our model (Model ) ha a more indirect connection between interet rate and credit rating but one that allow more general interaction. We alo invetigate how good the model bond price fit with empirical data. Two different verion of thi extenion which allow connection between interet rate and credit rating change are conidered and one of them allow a reinterpretation of the rik premium idea uggeted by Jarrow, Lando and Turnball (997). In order to get their model to give price for the riky bond that agreed with actual value, they modified the hitorically derived tranition matrix between rating, P A by making it a mixture of thi and the identity matrix, namely π(t) P A + (-π(t))i. They interpret the π(t) a rik premium but unfortunately ome of the rik premium are negative and other are very large. An alternative interpretation i that thi i an example of the mover-tayer Markov chain model (Frydman et al 985 ) where there i a heterogeneou et of bond, ome of which will never change their rating (the tayer) and other of which are moving around. The view that the rating of bond will never change i a very optimitic one becaue it guarantee that there will be no default. An equally extreme but peimitic view i that all bond of the given rating 5

6 will default at the next period. Recently Kiima and Komoribayahi (998) alo identified that thi might be a better choice of rik premium than that uggeted by Jarrow et al (997). Model S ( S for ubective) of ection two allow for the pricing of the bond to reflect the market relative belief between the hitoric movement of credit rating and both thee two extreme alternative. The rik premium can then be interpreted a the belief the market put on the extreme riky (or rikle) future cenario. Empirical work on bond pricing require one to calculate the zero-coupon price for riky bond from the bond in the market almot all of which have coupon. Longtaff and Schwartz (992) take average of bond price and coupon rate and average maturity over a number of month and find the bet regreion fit. Jarrow et al (997) take the average price and average coupon rate for each combination of bond rating and maturity period and then ue thee value to olve a triangular ytem of equation to get zero coupon bond price. However both method can lead to mipricing. The zero coupon bond price need not decreae with maturity or lower credit rating. One can overcome thee difficultie and ue the full detail of each bond in the market rather then average value by uing linear programming to find the zerocoupon bond price which minimie the l -average error and enure that no mipricing can occur. In the next ection, the bond pricing model which allow for dependency between credit rating movement and interet rate movement are introduced. Section three decribe the linear programming method of enuring no mi-pricing when tripping out the coupon from riky and rikle bond. Section four decribe the empirical data ued to determine the parameter of the bond pricing model outlined 6

7 in ection two. The reult in term of bond price and rik premium of uing thi data in the model of ection two are alo dicued. Section five draw ome concluion. 2. Model of bond price The bond price are modeled a a dicrete time trading economy both becaue dicrete time implifie the mathematic and becaue the credit rating information i ummarized in dicrete time format (ee Standard and Poor (997a)). There are three interconnected procee which make up the model. Firtly the underlying economic condition, E t, are modeled a a dicrete-time time homogenou Markov chain with two tate {G(Good), B(Bad)}. Let g Prob{E t+ G E t G} b Prob{E t+ B E t B} (2.) The interet rate proce over time period t,,2 T i a generalization of the lattice Markov chain model outlined in Plika (Plika, Ch.6, 997). To be precie let I t denote a tochatic proce with initial value I and tate pace I {,, T}. The tranition probabilitie atify. P{I t+ n+ I t n, E t G} p g (t, n) for n I P{I t+ n I t n, E t G} p g (t, n) for n I (2.2) P{I t+ n+ I t n, E t B} p b (t, n) P{I t+ n I t n, E t B} p b (t, n) for n I for n I Although I t i neither time homogeneou nor a Markov chain, the proce (I t, t, E t ) i both a Markov chain and time homogeneou. The proce I t give knowledge of the pot interet rate r. If I t n, E t G, then the pot interet rate i r t (n,g) while if I t n, E t B, the interet rate i r t (n,b). Moreover it implie knowledge of future interet 7

8 rate o that if I t n, E t G, pot interet rate next period will be one of r t+ (n+,g), r t+ (n+,b), r t+ (n,g) and r t+ (n,b). The tranition probabilitie defined in (2.2) are the conditional rik neutral tranition probabilitie for the proce. The third proce R t decribe the evolution of the credit rating of the bond. Aume there are M+ poible rating level (,,2, M) where i the rating given to rik-free government bond. Riky corporate bond have a rating from (mot ecure i.e. AAA in S and P ) to M (leat ecure C-grade in S and P ), with M correponding to bankruptcy. R t i a dicrete time proce which i almot a Markov chain ince the tranition probabilitie are defined by P{R t+ k R t, E t G} p G k(t) P{R t+ k R t, E t B} p B k(t) (2.3) with Σp G k(t) Σp B k(t) for all t. Note that p G (t) p B (t) and p G MM(t) p B MM(t) for all time t. Thu (R t, t, E t ) a well a (R t, I t, t, E t ) are finite tate tationary Markov chain. Unlike Jarrow et al (997) the rating proce R t and the interet rate proce (I t, t) are not now independent, but are related through their mutual dependency on the economic condition proce E t. If we aumed E t ha only one poible tate then thi model reduce to the Jarrow model though no pecific form of the interet proce i ued there. Taking there to be only one economic tate for the proce E t reduce the interet rate proce (I t, t) to the lattice interet rate model detailed in Plika (Plika 997). Having defined the evolution of the economic variable, it i now poible to define and calculate the bond price in the model. Let Z t(n, E, ) be the time t price of a zero-coupon bond promiing to pay a dollar at time when the bond rating i at time t and the interet and economic condition then are I t n and E t E, where E i either G, B or ome ditribution of 8

9 belief over the two poibilitie. One feature of dicrete time model i that everal event occur in the ame time period. One can chooe arbitrarily what the order of thee event will be. We aume that Z t(n, E, ) i the price of the bond at the beginning of period t, when the bond i redeemed at the end of period. During any period, we aume all change of tate occur toward the end of the period after the redemption date for that period, with firt change in interet rate I t, then change in rating R t and finally change in the economic condition E t. If a company default, it i aumed that a fraction f of the face value of the bond will be repaid. In order for the dicounted zero coupon bond price to be free of arbitrage opportunitie, then they mut be martingale, and o the price at any period mut be the expected value of future bond price under the rik neutral probabilitie. Uing the equence of event within a period decribed above, thi martingale requirement lead to the equation Z t (n, E, ) + rt (n, E) k E p k (t){pe (t, n) + ( pe (t, n)) c [ ez (n +, E, k) + ( e) ( Z (n +, E, k) )] t+ c [ ez (n, E, k) + ( e)z (n, E, ] for t T, n t, M- (2.4) where E G or B and if E G, e g and E c B while if E B, e b, E c B. Alo following the ordering of event within a period, defined above, one get t+ t+ t+ k) } Z t t(n, E, ) + U Q( W if M and Z t(n, E, M) f t,, n, E. (2.5) At t, aume I, and E i either G, B or a ditribution (p, -p) over (G, B). The price of bond at time t can be ued to identify the price of zero-coupon bond Z (, E, ) by uing method dicued more fully in ection three of thi paper. Thu the model appear to have 3+2T(T+) + 2T(M-)(M-2) parameter g, b, 9

10 f, 2T(T+) parameter of the form p g (t,n), p b (t,n), r t (n,g), r t (n,b), and 2T(M-)(M-2) of the form p G k(t), p B k(t) given * % S W S W. N MN MN Ideally the model will atify TM+ contraint in that it hould cloely fit the zero coupon bond price Z (, E, ), for, T, and, M- and at time t atifie (2.5). Since there are more parameter than contraint one could expect to impoe other condition on the parameter. However there i le freedom than eem the cae. If for example the tranition matrice p G k(t) and p B k(t) are aumed to be tationary and given by pat hitory, there hould appear to be more than enough other parameter - 2T(T+) to atify (M+)(T+) condition. However, the number of parameter p g (t,n), r t (n,g) etc increae linearly with t, o there are only 3 parameter at t to et the time- price and only 8 parameter at t to et the time- 2 price. Thi i not enough to define the M+ bond price given for each t-time for the early t-time. What i important i that the interet-rate parameter p g (t,n), r t (n,g) etc. are more than enough to define the rik-free bond price Z (, E, ), and can help define good approximation to the riky bond price Z (, E, ). Thi reflect the fact that there are an infinite number of future tochatic evolution of interet rate which give the current yield curve for rikle government bond, but there i information in the yield tructure of the riky bond which help define which evolution i being aumed by the market. Model : Simple Model One obviou way of implifying the number of parameter in Model i to aume the rating tranition are tationary and the interet rate tranition are tate independent. Alo, one can aume the underlying economic tate of the ytem only affect the

11 probability of change in the interet rate and not the interet rate level. Thi correpond to keeping g, b, f a in model and defining p g (t,n) p g (t); p b (t,n) p b (t) n,t ( + r t (n,g)) ( + r t (n,b)) ( + r t ())/c(t) n n,t,, k, t (2.6) The c(t) can be interpreted a meaure of the volatility of the time t pot interet rate. One advantage of thi model i that one can obtain the baic interet rate level r t () a an analytic expreion of the other parameter, g, b, c(t), p g (t) and p b (t). Before proving thi reult note that the definition of Z t(n,e,) and the aumption in (2.6) mean that we can define z t (n) by z t (n) Z t n c(t) n t(n,e,) c(t) z t () + r (n, E) + r () t t (2.7) Alo define the vector Z t(n,) W Q* Q% V W (2.8) V Lemma i): Define the following 2x2 matrice: P t g(-p g (t) + p g (t)c(t+) c()), (-g)(-p g (t) + p g (t)c(t+) c()) (-b)(-p b (t) + p b (t)c(t+).. c()), b(-p b (t) + p b (t)c(t+) c()) (2.9) then Z t(n,) P z (n)z (n) (2.) Π t ii) z () Π 2 Z (, E,) + r () Z (, E,) eπ e P P (2.) where e (,) if E G and e (,) if E B. Proof:

12 2 i) can be proved uing backward induction on t in Z t (n,e, ) of (2.) tarting with t. We will concentrate on Z t (n,g,) a the proof for the other component i the ame. For t, Z (n,g,) ) (n Z (n,g) r + Aume (2.) i true for Z k(n,) for k t+. By (2.4) Z t(n,g,) z t (n)(p g (t)(gz t+(n+,g,) + (-g)z t+(n+,b,)) + (-p g (t))(gz t+(n,g,) + (-g)z t+(n,b,)) ( ) ( ) ( ) ( ) (2.2) (n)]} z (n)) z ( g) (- (n) z (n)) z ( (t))[g p ( )] (n z )) (n z ( g) (- ) (n z )) (n z ( (t)[g (n){p z - t - t g - t - t g t P P P P Since z (n+) c()z (n) for t+, -, (2.2) become ( ) ( ) (n) (n)z z P (n) (n)z z P ) ( ) ( )... ( ) ( ) ( )( ( ) ( )... ( ) ( ) ( ( t t Π Π n z c t c t p t p g c t c t p t p g t g g g g ii) From (2.) we have that the ratio of Z (,E,) and Z - (,E,) i () () () () (),) (,,) (, 2 2 P e P e z z z z z E Z E Z 2 P e P e Hence (2.) follow. The lemma implie that under model once the 3T parameter p g (t), p b (t), c(t), t,.t- are given the parameter r t () can be choen to enure the zero-rik bond price are met. We will aume the economy tranition probabilitie g and b are given

13 by etimating from hitoric data. Thu all that remain to be fixed in the model are the rating tranition matrice p G k(t) and p B k(t). Thi paper invetigate two different way of defining the rating tranition matrice. Model H: Hitorical Data In thi the tranition are etimated from the actual tranition in rating in the pat and the tranition matrice are aumed to be time-independent. Thu one define the tranition matrice P AG,(P AB ) actual good (actual bad) - from actual hitoric data o that P G k(t) p AG k, p B k(t) p AB k, t,, k M (2.3) It i clear that thi model cannot hope to obtain all the riky bond price completely accurately, a there are T(M-) riky bond price and now only 3T + parameter available, namely p g (t), p b (t), c(t), and f. An alternative approach i to aume that the market doe not accept that the hitoric movement in rating are the one that will occur in the future. The market view i a mixture of belief, ome baed on hitoric movement, ome on more extreme view of the movement. We conider two extreme poition: catatrophe and no change. The catatrophe view (C) i that in the coming year all riky bond of all rating will default. Thi correpond to a tranition matrix P C M M 3

14 The no change poition (NC) i the lazy view that all bond will keep the ame rating in the coming year and correpond to a tranition matrix P NC I. Model S Subective Rating Aume that in the good time period, the market take a the rating tranition matrix, a mixture of the hitoric rating change in good time and the extreme view that there will be no change. In the bad time period, the market view i a mix of the hitoric rating change in bad period and the extreme view that all bond will default. We aume that the ratio of the mixture can differ for different bond rating and for different time period. Thi lead to the definition: P G k(t) π G (,t) p NC k + (-π G (,t)) p AG k P B k(t) π B (,t) p C k + (-π B (,t)) p AB k. (2.4) The value π G (,t), π B (,t) could then be conidered a type of rik premium meaure. In thi cae, it i how much weight the market put on the extreme view of the future. In fact, one can reinterpret the rik premium which Jarrow et al (997) introduced into their paper a -π G (,t) in thi formulation if one aume the only underlying tate i G. Thi may explain why they end up with negative rik premium in their calculation. They have only allowed for the market to have a more optimitic view than the hitoric one of the future. Thi reinterpretation however only make ene if π G (,t) wherea Jarrow et al (997) allow value greater than. Model S eek to allow the market to have both a more optimitic and a more peimitic view of the future than wa the hitoric average and for implicity retrict optimitic view to good year and peimitic view to bad year.. 4

15 3. Uing Linear Programming to trip out coupon Model of bond price take zero-coupon bond a their baic entity, wherea mot bond have coupon which involve part payment during the life of the bond, a well a the redemption value to be paid on maturity. Thu there i a need to trip out the coupon and calculate what the market price of the bond implie about the value of a bond that will ut pay unit at time t. Some author (Longtaff, Schwartz (995 )) take the average bond price, coupon rate and maturity each month for over a given time period and fit a regreion line. The data however will include the change over time in market entiment and o doe not reflect the poition at a given time. Jarrow, Lando and Turnbull (997) plit bond into clae depending on their credit rating and their maturity. For each cla the average market price and average yield were taken to be the value for bond of that rating and maturity. Solving a triangular ytem of equation gave the zero-coupon bond price. However, there wa ome mipricing of their bond with their calculated zero-coupon bond price not necearily increaing a the credit rating improved nor decreaing a maturity increaed. Alderon and Zivney (994) report imilar example of mi-pricing in unk bond and they how that reported bond yield depend on which invetment trategie are aumed. One can et up the problem of tripping out the coupon to get zero-coupon bond price from bond with coupon a a linear program, in the following way. Aume bond are given one of M credit rating,, 2, M- (credit rating M correpond to default). Suppoe there are N bond and all have maturity and coupon 5

16 payment within the next T period. Aume bond i i N ha a current market price of p i, a credit rating of d(i) and the coupon and redemption payment involve a payment of c i (t) in period t, t, T. Let R {i d(t) }, M- be the et of bond with rating. Let the preent value of a zero-coupon, -rated bond which pay one unit at time t, be v (t),, M-, t, T. Ideally one ha S L W 7 F W Y L GL W L K (3.) However, one cannot guarantee thi will occur, o intead one require S L + D L 7 F WY W + E W L GL L where a i, b i are the above or below error in the market price. (3.2) Thu one can find v (t), M-, t, T by olving the following linear program, LP. Minimie N (ai + bi) i T pi + ai ci(t)vd(i) (t) + bi t i, KN v (t) (+ m(t))v(t + ), KM, t, KT (3.3) v (t) v + (t), KM 2, t, KT (3.4) ai bi v (t) i, KN,, KM, t, KT. where m(t) i the minimum poible interet rate in period t. (3.3) guarantee that the zero-bond price atify the obviou financial maturity requirement. If m(t), (3.3) reduce to the requirement that the price of bond decreae with increaing maturity. Condition (3.4) enure that there i no mipricing on credit rating o the bond with the bet (lowet) credit rating have the highet price. 6

17 The linear program ha MT+2N variable and N+M(T-) + (M-)T contraint. Such program can be olved by dedicated linear programming olver or by olver in preadheet package uch a EXCEL. The olver in EXCEL 97 can only deal with 2 variable, o only le than bond can be dealt with at a time.. However, one can olve the problem a a et of neted linear program if one ha more than bond, by olving firt the price of the -rated bond only, dropping contraint (3.4) and applying (3.2) only to i ε R. Then olve for the v (t) only replacing (3.4) by v (t) v (t), t, T where v (t) wa obtained from the previou linear program and (3.2) only hold for i ε R. Repeating thi procedure for all the rating in turn correpond to olving LP but with an obective function M Minimie L (ai i R + bi ) where L i i an order of magnitude greater then L i+ for i, M-2. The linear program LP give the price that bet fit the actual bond data in the ene of minimiing the average abolute error, while alo enuring there i no mi-pricing. 4. Data and reult of an example uing US bond price The model of ection two and three were applied to data on US bond price and credit rating obtained from DATASTREAM and Standard and Poor (Standard and Poor 997a, 997b) repectively. 64 of the US Treaury Bond which make up the DATASTREAM US yield curve data et in 995 and 996 were taken a the rikle 7

18 bond. Their market price on 3 rd July 996 wa taken -the data being choen a an example of a mid-week, mid-year, pre-holiday period. The et of riky bond atified three criteria. They were in the DATASTREAM databae of US indutrial and US financial bond; their market price and S&P rating for 3 rd July 996 were available; there were no callable date. The extra option that being callable give a bond i more difficult to trip out of the price than the coupon. There were 78 uch bond in total (7 rated AAA, 24 rated AA, 6 rated A, 68 rated BBB, 2 rated BB, 6 rated B). DATASTREAM doe not uually record the price of C-rated peculative bond but there were 8 bond in the et that moved from C to invetment grade or vice vera during the year (Standard and Poor 997b) and hence we were able to obtain the 996 market price when they were C-rated. Year RISKLESS AAA AA A BBB BB B C Year to2 22+ RISKLESS AAA AA A BBB BB B NA C NA NA NA NA NA TABLE Zero-coupon bond price The linear program developed in ection three wa applied to the 25 bond in total, and the zero-coupon price for each individual year 996 to 2, one common price for year 2 to 22, and one common price for all year beyond 22 were 8

19 calculated. It wa aumed that there would be no bankruptcie in the ret of 996 o v (996) wa aumed contant for all rating. The reult are given in table where there wa no price available for C-rated zero-coupon price beyond 27 becaue there were no uch bond with maturity beyond thi date. The yield curve for the variou rated bond are given in Figure. Figure 2 graph the pread for riky bond of their yield compared with thoe of the rikle bond. The curve are not mooth and the pread of differently rated bond converge and then eparate at everal point but the general hape eem reaonable. The highet pread i for the C-rated bond of early maturity and in general the pread for thee bond decreae with time. The lowet pread are for the AAA-rated bond of early maturity and thee pread lowly increae with time. Note that the Figure :Bond Price from Linear Programming.2.8 Price Year RISKLESS AAA AA A BBB BB B C 9

20 Fig2:Spread from LP price a of Spread (Bai point*) C B AAA A BBB B C Year linear programme derive the ame value for the price of C-rated bond with a greater than twelve year maturity a for B-rated bond of the ame maturity becaue there were no example of the former for the linear programme to ue. The data for the rating proce wa obtained from the Standard and Poor Rating Performance (S&P 997a) which give the number of bond making each poible annual rating tranition for the year The deciion on which were the good and bad year in the underlying proce ued two et of data. Firtly the annual rating tranition were invetigated and % downgrading +% default -% upgrading taken a a meaure of the rating change in that year. The year were then ordered according to thi meaure and the highet rated were taken a good. Secondly, after examining Dow Jone Index long-term and hort-term interet rate, US unemployment data, US CPS Indutry Production, leading indicator and yield pread, we claified the year a good or bad. With one change, the two equence agreed with one another with the rating equence lagging one year behind the ubective economic equence. Thi lagged economic equence wa thu ued and 8, 84, 87, 92, 93, 94, 96 were claified a good year and 82, 83, 85, 86, 88, 89, 9, 9, 95 claified a bad. Totaling the annual bond rating change for thee two equence 2

21 eparately and then tranlating into percentage led to the annual tranition matrice for good and bad year repectively given in table 2. In thi table the row repreent the bond rating at the beginning of a year and the column repreent the bond rating at the beginning of the next year, while the value are the probability of uch a tranition in good and bad year. Good AAA AA A BBB BB B C DEFAULT AAA AA A BBB BB B C Bad AAA AA A BBB BB B C DEFAULT AAA AA A BBB BB B C Table 2: Credit rating tranition matrice for good and bad year One can check uing χ 2 tet that the tranition for a given row are ignificantly different in the two matrice. Looking at the pattern of good and bad year in the equence which give thee tranition matrice and adding in year 98 and 997 both of which are claified a good year enable u to etimate g and b. There are 5 time a bad year i followed by a bad year and 4 time it i followed by a good year. Good year were followed by good year 4 time and by bad year 4 time. Thi data lead to the etimate g4/8 and b5/9. We aume that in mid-996 when the bond price are taken it i not yet clear if the economic condition in 996 are good or bad and o we aume that ince 2

22 995 wa a bad year the chance 996 i bad i 5/9. Hence we aume E (4/9, 5/9). Table 3:Survival Probabilitie fo r 2 - ta te m odel tarting in Good State Probabilitie AAA AA A BBB BB B C 4 7 Year and the price of -rated t-maturity zero-coupon bond in 996 i taken a 4/9Z t (,G,)+ 5/9Z t (,B,). The effect on the rating tranition of a hidden underlying 2-tate model of the economy can be hown by looking at the urvival probabilitie ( i.e. the probabilitie of not defaulting ) after t period for bond rated now. Figure 3 how the reult of thi for the 2-tate model tarting in a good year.one can do imilar calculation for both the 2-tate model tarting in a bad year and the -tate model where one calculate the tranition matrix from all year put together The one-tate urvival probabilitie all lie below the two-ate urvival probabilitie when the current year i aumed good and are all above the two-tate urvival probabilitie when the current year i bad, The larget difference in the three graph occur in the early year of B and C rated bond. Of coure eventually the urvival probabilitie will become for all rating in both model ince in both cae the rating proce i an aborbing Markov chain. 22

23 All the parameter in the rating and the underling economic procee have been defined above from hitorical data. The only parameter left to be defined in Model H are thoe decribing the interet rate proce -c(t), p g (t), p b (t) for all t up to T- and f, the fraction of the face value repaid if a bond i defaulted. We will concentrate on the time interval , o eek to build a model that matche the July 3 rd 996 bond price for zero-coupon bond maturity at the end of each of thee year. Our ordering of the event during a year mean that no bond will default during the ret of 996 and hence the price of bond maturing in thi period i the ame for all credit rating. Thi follow from (2.5) and the dicuion preceding it. One poible approach i to chooe reaonable value for the interet rate parameter and find the value of f that give the bet match with the price obtained in Table 2. A an example we choe c(t), p g (t).6, p b (t).4 for all t and then find the value of f which minimie the mean quare error (MSE) over the 8 different rating (T-bill, AAA, AA, A, BBB, BB, B, C) and the -year of the model zero-coupon bond price compared with the actual bond price. The bet value i f.363 with error.2 and the implied rik free interet rate and bond price are given in Table c(t) p g (t) p b (t) r()% g.5 b.56 f.36 MSE.2 Table 3: Parameter value for Model H with optimal f and choen c,p. One can eek a better fit by optimizing over the interet rate parameter a well a f. There are limit on the parameter the p g (t), p b (t) mut be probabilitie and c(t) mut be le than or equal to in order that the actual interet rate r t (n,e) are 23

24 monotonically increaing in n, which enure that the interet rate r t (n,e) reflect the underlying ordering in the interet rate pace I.. To avoid the modeling partially collaping to a determinitic one we will in fact impoe.5 p g (t), p b (t).95. and to enure volatility i not too great we require c(t).5. Finally to enure conitency with the aumption underlying the derivation of the real zero-coupon price we will aume the rik-free pot interet rate in all year are at leat %. Uing the nonlinear olver in Excel give the parameter value in Table 4 a the one that minimie MSE with a MSE value of c(t) p g (t) p b (t) r()% g.5 b.56 f.4387 MSE.856 Table 4:Parameter value for model H with optimal f, c and p. The reult of table 4 ugget a model where the interet rate are expected to rie coniderably in the year and in all year except 96 the chance of interet rate riing i higher if it i good year than if it i a bad year. In ection two it wa pointed out that one could not match all the bond price with the hitorical model H but a nearer fit may be poible if one allowed ubective view of the future rating tranition a uggeted in Model S. Initially one would expect that with the extra flexibility that the rik premium parameter, π G (,t), π B (,t) give one could match the real price exactly. The time -rik premium π G (,), π B (,) can be ued to match the time- bond price Z (,E,), then the time-2 rik premium can be defined to get the time-2 maturity bond and o on. In each cae the bond price i a linear function of the correponding rik premium and o the olution can be 24

25 obtained by olving linear equation or by linear programming. However ince (2.5) implie the -maturity bond price are the ame for all rating, the rik premium π G (,t) in the good tate ha no effect if it i defined in thi way unle there i a chance there i an immediate tranition to the default tate M. From Table 2 one can ee that there i no uch chance of default for AAA and AA bond. A econd problem i the tability of uch a calculation. The time- rik premium are et by the time- bond price but they in turn are a factor in all the longer maturity bond. Any error in the time- bond price i then reflected in the time- premium and the time-2 premium have to correct for thee if they want to match the time 2 bond price. Thu any error grow a the rik premium eek to compenate for error in earlier rik premium. π G AAA AA A BBB BB B C π B AAA AA A BBB BB B C.47 c(t) p g (t) p b (t) r()% g.5 B.5556 f.363 MSE.565 Table 5: rik premium uing interet rate date of table 3 An alternative i to give up the advantage of linearity and ue all the rik premia to match all the bond price in one go. Since bond price of maturity t depend 25

26 on product of the rik premium for all time up to t thi i a non-linear problem. One can olve the problem uing non-linear algorithm including the one in Excel. Thi i the approach we adopt here. Table 5 how the rik premia that arie if one ue the data for the interet rate proce given by Table 3. Uing rik premia reduce the Mean Square Error over the 88 price to.565. Table 6 how the reult when the data of table 4 which wa the interet rate parameter that minimized mean quare error the mot were ued. In thi cae the adding of rik premium bring the MSE down from.8564 to G AAA AA A BBB BB B C π B AAA.45 AA A BBB.883 BB B.7839 C.2724 c(t) p g (t) p b (t) r()% g.5 B.5556 f.4387 MSE.533 Table 6: Rik premium uing rik data of Table 4 Intead of firt finding the interet rate data that bet fit the bond price tructure and then finding the bet rik premium for thi interet rate data, one could eek to optimize over interet rate data and the rik premium at the ame time to try and find a 26

27 good fit to the bond price tructure. Table 7 give the reult of doing exactly that and lead to a fit where the Mean Square Error i.98. Mot of thi error - the total quare error over the 88 bond price i.744 i in the fitting of the B and C rated bond. There were not many of thee in the original ample and their zero-coupon price are the mot upect ince they are not underpinned by price of lower rated bond. The parameter can be choen o that the total quare error over the 66 BB and higher rated bond price i.2739 which correpond to a mean quare error of.34. π G AAA AA A BBB.56 BB B C π B AAA AA A BBB BB B C c(t) p g (t) p b (t) r()% g.5 b.5556 f.572 MSE.98 Table 7: Rik premia when optimizing over both rik and interet rate parameter Comparing the rik premia in Table 5,6 and 7 the only noticeable feature in table 5, where all but one of the interet rate parameter were fixed i that the premia ugget the market i overly optimitic about the urvival of C-rated bond. Thi overoptimim of the market i much more marked in Table 6 where the interet rate parameter are thoe that bet fitted the bond price. 38 of the 7 poible π G (,t) are 27

28 non-zero including the C-rated one for all time t while only 8 of the 7 π B (,t) are non-zero. So on balance the price reflect a market that i much more likely to accept there will be no change in bond rating than one that i worrying that they will default. The interet rate parameter how the market expect a large rie in interet rate in the period. When a in Table 7, one allow both the interet rate and the rik premium to be moving at the ame time to find a bet fit to the price, one get a fit where the error decreae by 6%. However what happen i that the rik premium eek to decribe not ut the pread between the differently rated bond but alo the term tructure of all bond. Thu the interet rate parameter in thi cae ugget an interet rate tructure that i eentially determinitic and flat. All the uncertainty in it ha been tranlated into a much more complex rik premium tructure. One can recover the model with only one underlying economic tate ( which i akin to the Jarrow model ( Jarrow et al 997)) by etting g and tarting in tate E G. The reult of doing thi and finding the bet fit to the bond price over f, c(t) and p g (t), p b (t) are given in Table 8. Comparing the reult with the comparable 2-tate model in Table 4 one find the MSE i now.23 wherea the 2-tate model ha a MSE of.856 which i 3% lower c(t) p g (t) r()% g f.4669 MSE.23 Table 8: Parameter value for -tate model with optimal f, c and p The improvement the 2-tate model make over the -tate model i even more marked in the reult with rik premia, where we follow the Jarrow model and only allow the more optimitic extreme view in the good tate. π G i the probability in our 28

29 model that the market i taking the no change view in the good tate. Table 9 give the rik premia value and all the other parameter which bet fit the bond price in the - tate model obtained by taking g. In thi cae the mean quare error (MSE) of the - tate model i.3 while that for the 2-tate model i.98 a cut in the error of 8% Year AAA AA.2 A.2576 BBB BB B.44 C c(t) p g (t) r()% g F MSE.3 Table 9: Rik premia for -tate model optimizing over all parameter 5. Concluion The previou ection develop a hidden Markov chain model for the term tructure of credit rik pread further extending the idea in Lando (9994), Jarrow and Turnbull (995) and Jarrow, Lando and Turnbull (997). Thi model allow dependency between the rating proce and the interet rate proce through their oint dependency on a tate of the economy proce. The paper alo provide a reinterpretation of the idea of rik premia introduced therein a the chance the market view of the rating change i more extreme than ha been the cae in the pat. The paper alo ue linear programming to provide a way of tripping the coupon for bond in uch a way a to 29

30 minimie the mean abolute error and at the ame time enure there i no mi-pricing of the zero-coupon bond price Acknowledgement Thi paper wa written while LT wa a Viiting Profeor at Edith Cowan Univerity. We wih to acknowledge the financial upport of the Univerity that made thi viit poible. Reference Alderon M.J.,, Zivney T.L., (994), On computing bond return the evaluation of low-grade debt. J. Financial Reearch, 7, Benninga S., (997), Financial Modeling, MIT Pre, Cambridge, Ma. Black F., Cox J., (976), Valuing corporate ecuritie; ome effect of bind indenture proviion, J. Finance, 3, Duffee G.R., (996), On meauring credit rik of derivative intrument, J. Banking and Finance, 2, Frydman H., Kallberg J.G., Kao D-L, (985), Teting the adequacy of Markov chain and mover-tayer model a repreentation of credit behavior, Operation Reearch 33, Hull J., White A., (99), The Impact of default rik on option price, Working paper, Univerity of Toronto Jarrow R.A., Lando D, Turnbull S.M.,(997) A Markov model for the term tructure of credit rik pread, Review of Financial Studie,, Jarrow R.A., Turnbull S., (995), Pricing derivative on financial ecuritie ubect to credit rik, J. Finance, 5, Kiima M., Komoribayahi K., ( 998), A Markov chain model for valuing credit rik derivative, Journal of Derivative, 5,

31 Kim I., Ramawamy K., Sundarean S., (993), Doe default rik in the coupon affect the valuation of corporate bond? A contingent claim model, Financial Management, 22, 7-3 Lando D., (994), Three Eay on contingent claim pricing, Ph.D. thei, Cornell Univerity. Ithaca. Leland H.E., (994), Corporate Debt Value, Bond Covenant, and Optimal Capital Strucutre, Journal of Finance, 49, Leland H.E., Toft K.B., (996), Optimal capital tructure, endogenou bankruptcy and the term tructure of credit pread, J. Finance, 5, Litterman R., Iben T., (99), Corporate Bond Valuation and the term tructure of credit pread, Financial Analyi Journal, Spring Longtaff F.A., Schwartz, E.S., (995), A imple approach to valuing riky fixed and floating rate debt, J. Finance, 5, Madan D.B., Unal H., (994), Pricing the rik of default, Working Paper, Wharton Buine School, Philadelphia. Merton R.C., (974), On the pricing of corporate debt: the rik tructure of interet rate, J. Finance, 2, Plika S.R., (997), Introduction to Mathematical Finance, Dicrete time model, Blackwell, Oxford. Shimko D., Teeima N., van Deventer D., (993), The pricing of riky debt when interet rate are tochatic, J. Fixed Income, September, Standard and Poor (997a), Standard and Poor Rating Performance 996 Stability and Tranition, New York. Standard and Poor (997b), Standard and Poor Rating Roundup; Corporate, Financial Intitution and Sovereign Rating Change for 996, New York. 3

32 Table 3:Survival Probabilitie for 2-tate model tarting in Good State Probabilitie AAA AA A BBB BB B C Year

33 Fig2:Spread from LP price a of Spread (Bai point*) C B AAA A BBB B C Year 33

34 Figure :Bond Price fromlinear Programming.2.8 Price Year RISKLESS AAA AA A BBB BB B C 34