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Mo deling Term Structures
of Defaultable Bonds
Darrell DuÆe
Stanford University
and
Kenneth J. Singleton
Stanford University and NBER
First Version: June, 1994
Current Version: February 4, 1999
Forthcoming: Review of Financial Studies
(^1) The authors are at The Graduate Scho ol of Business, Stanford Univer-
sity, Stanford CA 94305-5015. This pap er is a revised and extended ver- sion of the theoretical results from our earlier pap er \Econometric Mo del- ing of Term Structures of Defaultable Bonds," June, 1994. The empirical results from that pap er, also revised and extended, are now found in \An Econometric Mo del of the Term Structure of Interest Rate Swap Yields," Journal of Finance, Octob er, 1997. We are grateful for comments from many, including the anonymous referee, the Editor Ravi Jagannathan, Peter Carr, Ian Co op er, Qiang Dai, Ming Huang, Farshid Jamshidian, Jo e Langsam, Francis Longsta , Amir Sadr, Craig Gusta son, Michael Boulware, Arthur Mezhlumian, and esp ecially Dilip Madan; and from seminar participants at The University of Arizona, Boston University, Carnegie Mellon University, The Norwegian Scho ol of Economics and Business Administration, Queen's University, The University of Wiscon- sin, The University of California at Berkeley, The University of Chicago,
Abstract: This pap er presents convenient reduced-form mo dels of the valuation of contingent claims sub ject to default risk, fo cusing on applications to the term structure of interest rates for corp orate or sovereign b onds. Examples include the valuation of a credit-spread option.
Duke University, The University of California at San Diego, Cambridge University, Ecole des Hautes Etudes Commerciales (France), Universite de Montreal, McGill University, The University of Pennsylvania, Stan- ford University, The Fields Institute at the University of Toronto, The Oslo-Silivri Workshop on Sto chastic Pro cesses, The National Bureau of Economic Research, The Federal Reserve Bank of Atlanta Conference on Financial Markets in Coral Gables, The Lehman Brothers Credit Deriva- tives Forum, The Nikko Research Conference, The RISK Conference on Credit Risk, The Annual Meeting of the American Finance Asso ciation, and The Annual Meeting of the Western Finance Asso ciation. We are also grateful for nancial supp ort from the Financial Research Initiative at the Graduate Scho ol of Business, Stanford University. We are grate- ful for computational assistance from Arthur Mezhlumian and esp ecially from Michael Boulware and Jun Pan.
as well as for the e ect of losses on default. Pye (1974) develop ed a pre-cursor to this mo deling approach, in a discrete-time setting in which interest rates, default probabilities, and credit spreads all change only deterministically. A key feature of the valuation equation (1) is that, provided we take the mean-loss rate pro cess hL to b e given exogenously,^3 stan- dard term-structure mo dels for default-free debt are directly appli- cable to defaultable debt by parameterizing R instead of r. After developing the general pricing relation (1) with exogenous R in Sec- tion 2.3, sp ecial cases with Markov di usion or jump-di usion state dynamics are presented in Section 2.4. The assumption that default hazard rates and fractional recov- ery do not dep end on the value Vt of the contingent claim is typical of reduced-form mo dels of defaultable b ond yields. There are, how- ever, imp ortant cases for which this exogeneity assumption is coun- terfactual. For instance, as discussed by DuÆe and Huang (1996) and DuÆe and Singleton (1997), ht will dep end on Vt in the case of swap contracts with asymmetric counterparty credit quality. In Section 2.5, we extend our framework to the case of price-dep endent (ht ; Lt ). We show that the absence of arbitrage implies that Vt is the solution to a non-linear partial di erential equation. For exam- ple, with this nonlinear dep endence of the price on the contractual payo s, the value of a coup on b ond in this setting is not simply the sum of the mo deled prices of individual claims to the principal and coup ons. Section 3 presents several applications of our framework to the valuation of corp orate b onds. First, in Section 3.1, we discuss the practical implications of our \loss-of-market" value assumption, com- pared to a \loss-of-face" value assumption, for the pricing of non- callable corp orate b onds. Calculations with illustrative pricing mo d- els suggest that these alternative recovery assumptions generate rather similar par yield spreads, even for the same fractional loss co eÆ- cients. This robustness suggests that, for some pricing problems, one can exploit the analytical tractability of our \loss-of-market" pricing framework for estimating default hazard rates, even when \loss-of-
(^3) By \exogenous," we mean that ht Lt do es not dep end on the value of the defaultable claim itself.
face" value is the more appropriate recovery assumption. For deep- discount or high-premium b onds, di erences in these formulations can b e mitigated by comp ensating changes in recovery parameters. Second we discuss several econometric formulations of mo dels for pricing of non-callable corp orate b onds. In pricing corp orate debt using (1), one can either parameterize R directly, or parameterize the comp onent pro cesses r , h, and L (which implies a mo del for R ). The former approach was pursued in DuÆe and Singleton (1997) and Dai and Singleton (1998) in mo deling the term structure of interest-rate swap yields. By fo cusing directly on R , these pricing mo dels combine the e ects of changes in the default-free short rate rate (r ) and risk- neutral mean loss rate (hL) on b ond prices. In contrast, in applying our framework to the pricing of corp orate b onds, Du ee (1997) and Collin-Dufresne and Solnik (1998) parameterize r and hL separately. In this way they are able to \extract" information ab out mean loss rates from historical information on defaultable b ond yields. All of these applications are sp ecial cases of the aÆne family of term- structure mo dels.^4 In Section 3.2 we explore, along several dimensions, the exi- bility of aÆne mo dels to describ e basic features of yields and yield spreads on corp orate b onds. First, using the canonical representa- tions of aÆne term-structure mo dels in Dai and Singleton (1998), we argue that the CIR-style mo dels used by Du ee (1997) and Collin- Dufresne and Solnik (1998) are theoretically incapable of capturing the negative correlation b etween credit spreads and U.S. Treasury yields do cumented in Du ee (1998), while maintaining non-negative default hazard rates. Several alternative, aÆne formulations of credit spreads are intro duced with the prop erties that hL is strictly p osi- tive and that the conditional correlation b etween changes in r and hL is unrestricted a priori as to sign. Second, we develop a defaultable version of the Heath, Jarrow, and Morton (1992) (HJM) mo del based on the forward-rate pro- cess asso ciated with R. In developing this mo del, we derive the counterpart to the usual HJM risk-neutralized drift restriction for
(^4) See, for example, DuÆe and Kan (1996) for a characterization of the aÆne
class of term-structure mo dels, and Dai and Singleton (1998) for a complete classi cation of the admissible aÆne term-structure mo dels and a sp eci cation analysis of three-factor mo dels for the swap yield curve.
the information available at time s, in the event of no default by s.
's denote the recovery in units of account, say dollars, in the
event of default at s.
rs b e the default-free short rate.
If the asset has not defaulted by time t, its market value Vt would b e the present value of receiving 't+1 in the event of default b etween t and t + 1 plus the present value of receiving Vt+1 in the event of no default, meaning that
Vt = ht e rt^ E tQ ('t+1 ) + (1 ht )e rt^ E Qt (Vt+1 ); (2)
where E tQ ( ) denotes exp ectation under Q, conditional on informa-
tion available to investors at date t. By recursively solving (2) for- ward over the life of the b ond, Vt can b e expressed equivalently as:
Vt = E (^) tQ
"T 1
X
j =
ht+j e ^
Pj k =0 rt+k^ 't+j +
Y^ j
`=
(1 ht+` 1 )
e ^
P (^) T 1 k =0 rt+k^ Xt+T
YT
j =
(1 ht+j 1 )
Evaluation of the pricing formula (3) is complicated in general by the need to deal with the joint probability distribution of ', r , and h, over various horizons. The key observation underlying our pricing mo del is that (3) can b e simpli ed by taking the risk-neutral exp ected recovery at time s, in the event of default at time s + 1, to b e a fraction of the risk-neutral exp ected survival-contingent market value at time s + 1 (\recovery of market value" or RMV). Under this assumption, there is some adapted pro cess L, b ounded by 1, such that
RMV: E sQ ('s+1 ) = (1 Ls )E sQ (Vs+1 ).
Substituting RMV into (3) leaves
Vt = (1 ht )e rt^ E Qt (Vt+1 ) + ht e rt^ (1 Lt )E tQ (Vt+1 )
= E (^) tQ
e ^
PT 1 j =0 Rt+j^ Xt+T
where
e Rt^ = (1 ht )e rt^ + ht e rt^ (1 Lt ): (5)
For annualized rates but time p erio ds of small length, it can b e seen that Rt ' rt + ht Lt ; using the approximation of ec^ , for small c, given by 1 + c, Equation (4) says that the price of a defaultable claim can b e ex- pressed as the present value of the promised payo Xt+T , treated as if it were default free, discounted by the default-adjusted short rate Rt. We will show technical conditions under which the approxima- tion Rt ' rt + ht Lt of the default-adjusted short rate is in fact precise and justi ed in a continuous-time setting. This implies, under the assumption that ht and Lt are exogenous pro cesses, that one can pro ceed as in standard valuation mo dels for default-free securities, using a discount rate that is the default-adjusted rate Rt = rt + ht Lt instead of the usual short rate rt. For instance, R can b e param- eterized as in a typical single- or multi-factor mo del of the short rate, including the Cox-Ingersoll-Ross (1985) (CIR) mo del and its extensions, or as in the HJM mo del. The b o dy of results regard- ing default-free term-structure mo dels is immediately applicable to pricing defaultable claims. The RMV formulation accommo dates general state-dep endence of the hazard-rate pro cess h and recovery rates without adding com- putational complexity b eyond the usual burden of computing the prices of riskless b onds. Moreover, (ht ; Lt ) may dep end on, or b e correlated with, the riskless term structure. Some evidence consis- tent with the state-dep endence of recovery rates is presented in Fig- ure 1, based on recovery rates compiled by Mo o dy's for the p erio d 1974 through 1997.^5 The square b oxes represent the range b etween the twenty- fth and seventy- fth p ercentiles of the recovery distribu- tions. Comparing senior secured and unsecured b onds, for example, one sees that the recovery distribution for the latter is more spread out and has a longer lower tail. However, even for senior secured
(^5) These gures are constructed from revised and up dated recovery rates as rep orted in \Corp orate Bond Defaults and Default Rates 1938-1995," Mo o dy's Investor's Services, January, 1996. Mo o dy's measures the recovery rate as the value of a defaulted b ond, as a fraction of $100 face, recorded in its secondary market subsequent to default.
the assumption of indep endence of ht and rt. In allowing for state-dep endence of h and L, we do not mo del the default time directly in terms of the issuer's incentives or ability to meet its obligations (in contrast to the corp orate-debt pricing liter- ature b eginning with Black and Scholes (1973) and Merton (1974)). Our mo deling approach and results are nevertheless consistent with a direct analysis of the issuer's balance sheet and incentives to de- fault, as shown by DuÆe and Lando (1997), using a version of the mo dels of Fisher, Heinkel, and Zechner (1989) and Leland (1994) that allows for imp erfect observation of the assets of the issuer. A general formula can b e given for the hazard rate ht in terms of the default b oundary for assets, the volatility of the underlying asset pro cess V at the default b oundary, and the risk-neutral conditional distribution of the level of assets given the history of information available to investors. This makes precise one sense in which we are prop osing a reduced-form mo del. While, following our approach, the b ehavior of the hazard-rate pro cess h and fractional loss pro cess L may b e tted to market data and allowed to dep end on rm-sp eci c or macro economic variables (as in Bijnen and Wijn (1994), Lundst- edt and Hillgeist (1998), McDonald and Van de Gucht (1996), and Shumway (1996)), we do not constrain this dep endence to match that implied by a formal structural mo del of default by the issuer. Our discussion so far presumes the exogeneity of the hazard rate and fractional recovery. There are imp ortant circumstances in which these assumptions are counterfactual, and failure to accommo date endogeneity may lead to mispricing. For instance, if the market value of recovery at default is xed, and do es not dep end on the pre-default price of the defaultable claim itself, then the fractional recovery of market value cannot b e exogenous. Alternatively, in the case of some OTC derivatives, the hazard and recovery rates of the counterparties are di erent and the op erative h and L for discounting dep ends on which counterparty is in the money.^6 While
(^6) This would b e the case with a swap or forward contract b etween counterpar-
ties A and B of di erent credit quality. As the market value of the contract to counterparty A changes from p ositive to negative, the exp ected loss rate applied to the swap switches from that of counterparty B to that of counterparty A. Using the framework develop ed in this pap er, DuÆe and Huang (1996) present several numerical examples of the consequences of this switching for swap prices.
(1) (and (4)) apply with price-dep endent hazard and recovery rates, this dep endence makes the pricing equation a nonlinear di erence equation that must typically b e solved by recursive metho ds. In Section 2.5 we characterize the pricing problem with endogenous hazard and recovery rates and describ e metho ds for pricing in this case. One can also allow for \liquidity" e ects by intro ducing a sto chas- tic pro cess as the fractional carrying cost of the defaultable instru- ment.^7 Then, under mild technical conditions, the valuation mo del (1) applies with the \default-and-liquidity-adjusted" short-rate pro- cess R = r + hL +:
In practice, it is common to treat spreads relative to treasury rates rather than to \pure" default-free rates. In that case, one may treat the \treasury short rate" r ^ as itself de ned in terms of a spread (p er- haps negative) to a pure default-free short rate r , re ecting (among other e ects) rep o sp ecials. Then we can also write R = r ^ + hL + ^ , where^ absorbs the relative e ects of rep o sp ecials and other de- terminants of relative carrying-costs.
2.2 Continuous-time Valuation
This section formalizes the heuristic arguments presented in the pre-
ceding section. We x a probability space ( ; F ; P ), and a family
fFt : t 0 g of -algebras satisfying the usual conditions. (See, for
example, Protter (1990) for technical details.) A predictable short- rate pro cess r is also xed, so that it is p ossible at any time t to invest one unit of account in default-free dep osits and \roll over" the pro ceeds until a later time s for a market value at that time of exp
R s
t ru^ du
.^8 At this p oint, we do not sp ecify whether rt is de- termined in terms of a Markov state-vector, an HJM forward-rate mo del, or by some other approach.
(^7) Formally, in order to invest in a given b ond with price pro cess U , this as-
sumption literally means that one must continually make payments at the rate `U. (^8) We assume that this integral exists.
that the pro cess which is 0 b efore default and 1 afterward (that is, t = (^1) ftT 0 g ) can b e written in the form
dt = (1 t )ht dt + dMt ; (8)
where M is a martingale under Q. One may safely think of ht as the jump arrival intensity at time t (under Q) of a Poisson pro cess whose rst jump o ccurs at default.^9 Likewise, the risk-neutral conditional probability, given the information Ft available at time t, of default b efore t + 1, in the event of no default by t, is approximately ht for small time intervals.^10 We will rst characterize, and then (under technical conditions) prove the existence of the unique arbitrage-free price pro cess U for the defaultable claim. For this, one additional piece of information is needed: the payo X 0 at default. If default o ccurs at time t, we will supp ose that the claim pays
X 0 = (1 Lt )Ut ; (9)
where Ut = lims"t Us is the price of the claim \just b efore" default,^11 and Lt is the random variable describing the fractional loss of market value of the claim at default. We assume that the fractional loss pro cess L is b ounded by 1 and predictable, which means roughly that the information determining Lt is available b efore time t. Section 2. provides an extension to handle fractional losses in market value that are uncertain even given all information available up to the time of default. As a preliminary step, it is useful to de ne a pro cess V with the prop erty that, if there has b een no default by time t, then Vt is the market value of the defaultable claim.^12 In particular VT = X and Ut = Vt for t < T 0.
(^9) The pro cess f(1 t )ht : t 0 g is the intensity pro cess asso ciated with
, and is by de nition non-negative and predictable with
R (^) t 0 hs^ ds^ <^1 almost surely for all t. See Bremaud (1980). Artzner and Delbaen (1995) showed that, if there exists an intensity pro cess under P , then there exists an intensity pro cess under any equivalent probability measure, such as Q. (^10) This is true in a limiting sense, for example, if h is right continuous. (^11) We will also show that the left limit Ut exists. (^12) Because V (! ; t) is arbitrary for those! for which default has o ccurred b efore t, the pro cess V need not b e uniquely de ned. We will show, however, that V is uniquely de ned up to the default time, under weak regularity conditions.
2.3 Exogenous Exp ected Loss Rate
From the heuristic reasoning used in Section 2.1, we conjecture the continuous-time valuation formula
Vt = E (^) tQ
exp
Z T
t
Rs ds
X
where
Rt = rt + ht Lt : (11)
In order to con rm this conjecture, we use the fact that the gain pro cess (price plus cumulative dividend), after discounting at the short-rate pro cess r , must b e a martingale under Q. This discounted gain pro cess G is de ned by
Gt = exp
Z t
0
rs ds
Vt (1 t )
Z t
0
exp
Z s
0
ru du
(1 Ls )Vs ds : (12)
The rst term is the discounted price of the claim; the second term is the discounted payout of the claim up on default. The prop erty that G is a Q-martingale and the fact that VT = X together provide a complete characterization of arbitrage-free pricing of the defaultable claim. Let us supp ose that V do es not itself jump at the default time T 0. From (10), this is a primitive condition on (r; h; X ) and the infor-
mation ltration fFt : t 0 g. This means essentially that, although
there may b e \surprise" jumps in the conditional distribution of the market value of the default-free claim (X ; T ), h, or L, these sur- prises o ccur precisely at the default time with probability zero. This is automatically satis ed in the di usion settings describ ed in Sec- tion 2.4.1, since in that case Vt = J (Yt ; t), where J is continuous and Y is a di usion pro cess. This condition is also satis ed in the jump- di usion mo del of Section 2.4.2 provided jumps in the conditional distribution of (h; L; X ) do not o ccur at default.^13
(^13) Kusuoka (1998) gives an example in which a jump in V at default is induced
however, for the more general case, treated in Section 2.5, in which h or L may dep end on the value of the claim itself.
2.4 Sp ecial Cases with Exogenous Exp ected Loss
Next, we sp ecialize to the case of valuation with dep endence of ex- ogenous r , h, and L on continuous-time Markov state variables.
2.4.1 A Continuous-Time Markov Formulation
In order to present our mo del in a continuous-time state-space setting that is p opular in nance applications, we supp ose for this section that there is a state-variable pro cess Y that is Markovian under an equivalent martingale measure Q. We assume that the promised contingent claim is of the form X = g (YT ), for some function g , and
that Rt = (Yt ), for some function^15 ( ). Under the conditions of
Theorem 1, a defaultable claim to payment of g (YT ) at time T has a price at time t, assuming that the claim has not defaulted by time t, of
J (Yt ; t) = E Q
exp
Z T
t
(Ys ) ds
g (YT ) Yt
Mo deling the default-adjusted short rate Rt directly as a function of the state variable Yt allows one to mo del defaultable yield curves analogously with the large literature on dynamic mo dels of default- free term structures. For example, supp ose Yt = (Y 1 t ; : : : ; Ynt )^0 ; for some n, solves a sto chastic di erential equation of the form
d Yt = (Yt ) dt + (Yt ) dBt ; (15)
where B is an fFt g-standard Brownian motion in R n^ under Q, and
where and are well b ehaved functions on R n^ into R n^ and R nn^ ; resp ectively. Then we know from the \Feynman-Kac formula" that,
(^15) For notational reasons, we have not shown any dep endence of on time
t, which could b e captured by including time as one of the state variables. Of course, we assume that and g are measurable real-valued functions on the state space of Y , and that (14) is well de ned.
under technical conditions,^16 (14) implies that J solves the backward Kolmogorov partial di erential equation
D ;^ J (y ; t) (y )J (y ; t) = 0 ; (y ; t) 2 R n^ [0; T ]; (16)
with the b oundary condition
J (y ; T ) = g (y ); y 2 R n^ ; (17)
where
D ;^ J (y ; t) = Jt (y ; t) + Jy (y ; t)(y ) +
trace [Jy y (y ; t) (y ; t) (y ; t)^0 ] : (18)
This is the framework used in mo dels for pricing swaps and corp orate b onds discussed in Section 3.
2.4.2 Jump-Di usion State Pro cess
Because of the p ossibility of sudden changes in p erceptions of credit quality, particularly among low-quality issues such as Brady b onds, one may wish to allow for \surprise" jumps in Y. For example, one can sp ecify a standard jump-di usion mo del for the risk-neutral
b ehavior of Y , replacing D ;^ in (16), under technical regularity, with
the jump-di usion op erator D given by
D J (y ; t) = D ;^ J (y ; t) + (y )
Z
Rn
[J (y + z ; t) J (y ; t)] dy (z );
where : R n^! [0; 1 ) is a given function determining the arrival
intensity (Yt ) of jumps in Y at time t, under Q, and where, for each y , y is a probability distribution for the jump size (z ) of the state variable. Examples of aÆne, defaultable term-structure mo dels with jumps are presented in Section 3.
(^16) See, for example, Friedman (1975) or Krylov (1980).
have several numerical examples of an application of this framework to defaultable swap rates. For cases of endogenous dep endence of the risk-neutral mean- loss-rate hL on the price of the claim, not necessarily based on a Markovian state-space, DuÆe, Schro der, and Skiadas (1996) provide technical conditions for the existence and uniqueness of pricing, and explore the pricing implications of advancing in time the resolution of information.
2.6 Uncertainty ab out Recovery
We have b een assuming that the fractional loss in market value due to default at time t is determined by the information available up to time t. An extension of our mo del to allow for conditionally uncertain jumps in market value at default is due to Schonbucher (1997). A simple version of this extension is provided b elow for completeness. Supp ose that at default, instead of (9), the claim pays
X 0 = (1 `)ST 0 ; (22)
where is a b ounded random variable^18 describing the fractional loss of market value of the claim at default. It would not b e natural to require that 0, as the onset of default could actually reveal, with non-zero probability, \go o d" news ab out the nancial condition of the issuer. Given limited liability, we require that 1. It can b e shown that there exists a pro cess L such that Lt is the exp ectation of the fractional default loss given all current informa- tion up to, but not including, time t. To b e precise, L is a predictable pro cess, and LT 0 = E (` j FT 0 ). With this change in the de nitions of X 0 and Lt , the pricing for- mula (10) applies as written, with R = r + hL, under the conditions of Theorem 1. The pro of is almost identical to that of Theorem 1.
3 Valuation of Defaultable Bonds
An imp ortant application of the basic valuation equation (10) with exogenous default risk is the valuation of defaultable corp orate b onds.
(^18) Here, ` is FT 0 -measurable.
We discuss various asp ects of this pricing problem in this section, b e- ginning with the sensitivity of b ond prices to the nature of the default recovery assumption. We argue that the tractability of assumption RMV may come at a low cost in terms of pricing errors for b onds trading near par even if, in truth, b onds are priced in the markets assuming a given fractional recovery of face value. Then, maintain- ing our assumption RMV, we present several \aÆne" mo dels for pricing defaultable, non-callable b onds, giving particular attention to parameterizations that allow for exible correlations among the riskless rate r and the default hazard rate h. Additionally, we derive the default-environment counterparts to the HJM no-arbitrage con- ditions for term-structure mo dels based on forward rates. Finally, we discuss the valuation of callable corp orate b onds.
3.1 Recovery and Valuation of Bonds
The determination of recoveries to creditors during bankruptcy pro- ceedings is a complex pro cess that typically involves substantial ne- gotiation and litigation. No tractable, parsimonious mo del captures all asp ects of this pro cess so, in practice, all mo dels involve tradeo s regarding how various asp ects of default (hazard and recovery rates) are captured. To help motivate our RMV convention, consider the following alternative recovery-of-face value (RFV) and recovery-of- treasury (RT) formulations of 't :
RT: 't = (1 Lt )Pt , where L is an exogenously sp eci ed fractional recovery pro cess and Pt is the price at time t of an otherwise- equivalent, default-free b ond (Jarrow and Turnbull (1995)).^19
RFV: 't = (1 Lt ); the creditor receives a (p ossibly random) frac- tion (1 Lt ) of face ($1) value immediately up on default (Bren- nen and Schwartz (1980) and Du ee (1998)). Under RT, the computational burden of directly computing Vt from (3), for a given fractional recovery pro cess (1 Lt ), can b e
(^19) In the case of zero-coup on b onds, the recovery-of-treasury assumption can b e reinterpreted as one in which the creditor receives the fraction (1 Lt ) of par, which is paid at the maturity date of the original defaultable b ond. In the case of coup on b onds, recovery up on default includes a fraction of the promised p ost-default coup on payments and not just of the face amount.
