a markov model for the term structure of credit … credit risk...markov process for credit ratings...

A Markov Model for theTerm Structure of CreditRisk Spreads

Robert A. JarrowCornell University

David LandoUniversity of Copenhagen

Stuart M. TurnbullQueen’s University

This article provides a Markov model for theterm structure of credit risk spreads. The modelis based on Jarrow and Turnbull (1995), with thebankruptcy process following a discrete statespace Markov chain in credit ratings. The pa-rameters of this process are easily estimatedusing observable data. This model is useful forpricing and hedging corporate debt with imbed-ded options, for pricing and hedging OTCderivatives with counterparty risk, for pricingand hedging (foreign) government bonds subjectto default risk (e.g., municipal bonds), for pric-ing and hedging credit derivatives, and for riskmanagement.

This article presents a simple model for valuing riskydebt that explicitly incorporates a firm’s credit rat-ing as an indicator of the likelihood of default. Assuch, this article presents an arbitrage-free model forthe term structure of credit risk spreads and theirevolution through time. This model will prove use-ful for the pricing and hedging of corporate debt with

We would like to thank John Tierney of Lehman Brothers for providingthe bond index price data, and Tal Schwartz for computational assistance.We would also like to acknowledge helpful comments received from ananonymous referee. Send all correspondence to Robert A. Jarrow, JohnsonGraduate School of Management, Cornell University, Ithaca, NY 14853.

The Review of Financial Studies Summer 1997 Vol. 10, No. 2, pp. 481–523c© 1997 The Review of Financial Studies 0893-9454/97/$1.50

The Review of Financial Studies / v 10 n 2 1997

imbedded options, for the pricing and hedging of OTC derivativeswith counterparty risk, for the pricing and hedging of (foreign) gov-ernment bonds subject to default risk (e.g., municipal bonds), andfor the pricing and hedging of credit derivatives (e.g. credit sensitivenotes and spread adjusted notes). This model can also be used forrisk management purposes, as it is possible to calculate the expectedcredit exposure profile over the life of a contract or a portfolio ofcontracts. To our knowledge, this is the first contingent claims modelto explicitly incorporate credit rating information into the valuationmethodology. Our model is an extension and a refinement of thatcontained in Jarrow and Turnbull (1995).

Previous models for the pricing of risky debt can be subdividedinto three classes. The first class of models views the firm’s liabili-ties as contingent claims issued against the firm’s underlying assets,with the payoffs to all the firm’s liabilities in bankruptcy completelyspecified. Bankruptcy is determined via the evolution of the firm’sassets in conjunction with the various debt covenants [e.g., Black andCox (1976), Chance (1990), Merton (1974), and Shimko, Tejima, andvan Deventer (1993)]. This approach is difficult to implement in prac-tice because all of the firm’s assets are not tradeable nor observable.Secondly, to utilize this technique, the complex priority structure ofthe payoffs to all of the firm’s liabilities need to be specified and in-cluded in the valuation procedure. This is a difficult task [see Jones,Mason, and Rosenfeld (1984)]. Furthermore, since this approach doesnot use credit rating information, it cannot be used to price creditderivatives whose payoffs depend directly on the credit rating [e.g.,credit sensitive notes and spread adjusted notes, see Das and Tufano(1995) for additional elaboration].

The second class of models views risky debt as paying off an ex-ogenously given fraction of each promised dollar in the event ofbankruptcy. Bankruptcy is determined when the value of the firm’sunderlying assets hits some exogenously specified boundary [e.g.,Hull and White (1991), Longstaff and Schwartz (1992), and Nielsen,Saa-Requejo and Santa-Clara (1993)]. This approach simplifies the firstclass of models by both exogenously specifying the cash flows to riskydebt in the event of bankruptcy and in simplifying the bankruptcy pro-cess. Although this approach simplifies computation by avoiding theneed to understand the complex priority structure of payoffs to allof the firm’s liabilities in bankruptcy, it still requires estimates for theparameters of the firm’s asset value, which is nonobservable; and itstill cannot handle various credit derivatives whose payouts dependon the credit rating of the debt issue.

The third class of models avoids these last two problems. This ap-proach, like the second, also views risky debt as paying off a fraction

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A Markov Model for the Term Structure of Credit Risk Spreads

of each promised dollar in the event of bankruptcy; but the timeof bankruptcy is now given as an exogenous process [e.g., Jarrowand Turnbull (1995) and Litterman and Iben (1991)]. This bankruptcyprocess is specified exogenously and does not explicitly depend onthe firm’s underlying assets. The advantage of this approach is that itallows exogenous assumptions to be imposed only on observables.All three classes of models are arbitrage-free.1 In fact, this modelingapproach includes the previous two as special cases where the pay-off process and/or the bankruptcy process are endogenously derived.Also, the third approach can easily be modified to include credit ratinginformation in the bankruptcy process, and therefore it can be used toprice credit derivatives dependent on the credit rating. This providesthe motivation underlying the choice of the model in this article.

This article takes the Jarrow and Turnbull (1995) model, and char-acterizes the bankruptcy process as a finite state Markov process inthe firm’s credit ratings. This new credit risk model has the followingcharacteristics:

1. Different seniority debt for a particular firm can be incorporatedvia different recovery rates in the event of default.

2. It can be combined with any desired term-structure model fordefault-free debt [e.g., Black, Derman, and Toy (1990), Cox, In-gersoll, and Ross (1985), or Heath, Jarrow, and Morton (1992)].

3. It utilizes historical transition probabilities for the various creditrating classes to determine the pseudo-probabilities (martingale,risk adjusted) used in valuation.

4. It can be utilized, as shown in Jarrow and Turnbull (1995), toprice and hedge options on risky debt or credit derivatives. Thepricing and hedging of vulnerable options is a special case of thisanalysis.

For implementation, we impose one simplifying assumption onthe interaction between the default-free term structure and the firm’sbankruptcy process. We assume that the two processes are statisticallyindependent under the pseudo-probabilities. Alternatively stated, theMarkov process for credit ratings is independent of the level of spotinterest rates (under the pseudo-probabilities). For investment gradedebt, this appears to be a reasonable first approximation in the his-torical probabilities, though for speculative grade debt, the accuracy

1 An analogy can be made between this class of models and those used to price equity options, thatis, Jarrow and Turnbull’s (1995) exogenous assumption on observables is equivalent to Black andScholes (1973), whereas Merton’s (1974) assumption on firm values is equivalent to the approachof Geske (1979).

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of the approximation deteriorates.2 The merit of this assumption is anoutstanding empirical issue. It is imposed as a simplifying assumptionto facilitate empirical investigation. If needed, it can be easily relaxed,as shown in Jarrow and Turnbull (1995). The extension of our creditclass model in continuous time to include spot rate–dependent prob-abilities can be found in Lando (1994).

An important application of this model is in the area of risk man-agement. In the recent (1993) Group of Thirty report,3 two of therecommendations address the issue of measuring current and poten-tial credit exposure, and aggregating credit exposures. The model canbe used to define the current credit exposure and to generate the dis-tribution of credit exposure over the life of a contract. Two commonlyused statistics can be computed: the maximum exposure and expectedexposure time profiles. That is, starting from a particular credit class,one can compute the probability of being in a given credit class aftera fixed time interval. For pricing purposes, it is necessary to use the“risk-neutral” probabilities. For risk management purposes, however,it is necessary to use both the “risk-neutral” and the empirical prob-abilities. This model can also be extended to portfolios of contractssuch as interest rate and foreign currency swaps.

An outline of this article is as follows. Section 1 describes the rele-vant features of the Jarrow and Turnbull model. Section 2 presents thediscrete time model. Section 3 presents the continuous time model.Section 4 concludes the article.

1. The Jarrow–Turnbull Model

The model we use is based on Jarrow and Turnbull (1995). We con-sider a frictionless economy with a finite horizon [0, τ ]. Trading canbe discrete or continuous (both cases are studied below). The un-derlying uncertainty is represented by a filtered probability space(Ä,Q,Fτ , (Ft )0≤t≤τ ). The details of this filtered probability space arespecified later. Traded are default-free zero-coupon bonds of all ma-turities, a default-free money market account, and risky zero-couponbonds of all maturities.

2 Moody’s Special Report (1992), Figures 15–18, show that 1-year default rates for investment gradebonds have had little variation over the time period 1970–1991. Although speculative grade 1-yeardefault rates exhibit more variation, on an absolute level, it is not large. The standard deviation of1-year default rates is highest for B-rated firms, and there it is only 5.04%. Longstaff and Schwartz(1992) provide some evidence inconsistent with this hypothesis.

3 The Group of Thirty is an international financial policy organization whose members includerepresentatives of central banks, international banks, and securities firms. They issued a reportentitled “Derivatives: Practice and Principles” in July 1993 to improve the supervisory and capitalrequirements of the off balance sheet risks related to derivatives trading activity.

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We assume that there exists a unique equivalent martingale mea-sure Q making all the default-free and risky zero-coupon bond pricesmartingales, after normalization by the money market account. Thisassumption is equivalent to the statement that the markets for default-free and risky debt are complete and arbitrage-free [see Harrison andPliska (1981)]. Sufficient conditions for the satisfaction of this assump-tion can be found in Jarrow and Turnbull (1995). Given information attime t , we denote conditional expectation and conditional probabilitystatements with respect to the equivalent probability measure by Et (•)and Qt (•), respectively.

Let p(t, T ) be the time t price of a default-free zero-coupon bondpaying a sure dollar at time T where 0 ≤ t ≤ T ≤ t . We assumeforward rates of all maturities exist and that they are defined (1) in thediscrete time case by f (t, T ) ≡ − log(p(t, T + 1)/p(t, T )) and (2) inthe continuous time case by f (t, T ) ≡ −∂

∂T log p(t, T ). The default-freespot rate, denoted r (t), is defined by r (t) ≡ f (t, t). We do not specifyany particular stochastic process for spot rates. They can be modeleddirectly as in Cox et al. (1985), or indirectly via forward rates as inHeath et al. (1992). The money market account accumulates returnsat the spot rate and is denoted as either

B(t) = exp

(t−1∑i=0

r (i)

)in the discrete time case,

or B(t) = exp

(∫ t

0r (s)ds

)in the continuous time case.

Under the maintained assumption of arbitrage-free and completemarkets, we can write default-free bond prices as the expected, dis-counted value of a sure dollar received at time T , that is,

p(t, T ) = Et

(B(t)

B(T )

). (1)

Let v(t, T ) be the time t price of a risky zero-coupon bond promis-ing to pay a dollar at time t where t ≤ T ≤ τ . This promised dollarmay not be paid in full if the firm is bankrupt at time T .4 If bankrupt,the firm pays only δ < 1 dollars. The fraction δ, called the recoveryrate, can depend on the priority (seniority) of the risky zero-coupondebt relative to the other liabilities of the firm.

4 The term “bankrupt” should not be interpreted too literally. In our context it covers any case offinancial distress that results in the bondholders receiving less than the promised payment.

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The recovery rate δ is taken to be an exogenously given constant.This constancy is imposed for simplicity of estimation. As shown inJarrow and Turnbull (1995), it implies that the stochastic structure ofcredit spreads will be independent of the recovery rate, and depen-dent only on the stochastic structure of spot interest rates and thebankruptcy process. This assumption can easily be relaxed [see Dasand Tufano (1995)].5

Let τ ∗ represent the random time at which bankruptcy occurs.Then,

v(t, T ) = Et

(B(t)

B(T )(δ1{τ ∗≤T } + 1{τ ∗>T })

)(2)

where 1{τ ∗≤T } is the indicator function of the event {τ ∗ ≤ T }. Therisky zero-coupon bond’s price is seen to be the expected, discountedvalue of a “risky” dollar received at time T . Note that if bankruptcy hasoccurred prior to time t , it is assumed that claimholders will receive δfor sure at the maturity of the contract. This implies that the risky termstructure simplifies considerably as v(t, T ) = δEt (

B(t)B(T ) ) = δp(t, T ).

In other words, in bankruptcy, the term structure of the risky debtcollapses to that of the default-free bonds.

Next, we assume that the stochastic process for default-free spotrates {r (t)0≤t≤τ } and the bankruptcy process, as represented by τ ∗,are statistically independent under Q. This assumption is imposedfor simplicity of implementation. This assumption implies that thebankruptcy process (under the pseudo-probabilities) is uncorrelatedwith default-free spot interest rates. Under the additional structure im-posed below, it will also imply that the bankruptcy process (under theempirical probabilities) is uncorrelated with default-free spot interestrates. The reasonableness of this assumption is an outstanding empiri-cal issue. The relaxation of this assumption is discussed in Jarrow andTurnbull (1995) and implemented in Lando (1994). Whether or notthis generalization is required awaits empirical testing of the simplermodel.

Under this assumption, Equation (2) simplifies to

v(t, T ) = Et

(B(t)

B(T )

)Et (δ1{τ ∗≤T } + 1{τ ∗>T })

= p(t, T )(δ + (1− δ)Qt (τ∗ > T )), (3)

where Qt (τ∗ > T ) is the probability under Q that default occurs after

date T . The risky zero-coupon bond’s price is the default-free zero-

5 Random recovery rates δ(xT ), where xT is the state variable, are also easily fitted into the valuationmodel, as in Chapter 3 of Lando (1994).

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coupon bond’s value multiplied by the expected payoff (in dollars) attime T . As revealed in Equation (3), the evolution of the term struc-ture of risky debt is uniquely determined via specifying a distributionfor the time of bankruptcy, Qt (τ

∗ > T ), under the martingale proba-bilities. The contribution of our article is an explicit modeling of thisdistribution as the first hitting time of a Markov chain, with credit rat-ings and default being the relevant states. This modeling exercise isthe content of the remaining sections.

2. Credit Ratings and Default-Probabilities: The Discrete TimeCase

This section models the distribution of default time in a discrete trad-ing economy. Discrete trading is illustrated first for two reasons. First,because its mathematical simplicity facilitates understanding. Second,because the discrete time model, parameterized in terms of its con-tinuous limit (the next section), is the model formulation actually im-plemented on a computer. The theorems in this section parallel thosedeveloped in the continuous time section.

2.1 ValuationThe distribution for the default time is modeled via a discrete time,time-homogeneous Markov chain on a finite state space S ={1, . . . ,K }. The state space S represents the possible credit classes,with 1 being the highest (Aaa in Moody’s rankings) and K − 1 be-ing the lowest (C in Moody’s rankings). The last state, K , representsbankruptcy.

The discrete time, time-homogeneous finite state space Markovchain {ηt : 0 ≤ t ≤ τ } is specified by a K × K transition matrix6

Q =

q11 q12 · · · q1K

q21 q22 · · · q2K...

qK−1,1 qK−1,2 · · · qK−1,K

0 0 · · · 1

(4)

where qij ≥ 0 for all i, j , i 6= j , and qii ≡ 1 −∑Kj=1j 6=i

qij for all i. The

(i, j)th entry qij represents the actual probability of going from state ito state j in one time step. We assume for simplicity that bankruptcy(state K ) is an absorbing state, so that qK i = 0 for i = 1, . . . ,K − 1

6 With a slight abuse of notation, we denote the probability measure and the transition matrix asQ. No confusion should arise because the meaning is made clear by the context.

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and qK K = 1. This explains the last row in the transition matrix. InJarrow and Turnbull (1995) this assumption is relaxed.

The time homogeneity of this transition matrix is imposed for sim-plicity of estimation. Based on the evidence contained in Moody’sSpecial Report (1992), it appears to be more reasonable for invest-ment grade bonds, than it is for speculative grade bonds. Again, thevalidity of this assumption is an outstanding empirical issue.

Let qij (0,n) denote the n-step transition probability of going fromstate i at time 0 to state j at time n. It is well known that the n-stepK × K transition matrix, Q0,n whose (i, j)th entry is qij (0,n) satisfiesQ0,n = Qn, the n-fold matrix product of Q.

Estimates of this transition matrix, with a time step of 1 year, canbe obtained from Moody’s Special Report (1992) [see Lucas and Lon-ski (1992, Table 6) or Standard & Poor’s Credit Review (1993)]. Thenonzero entries of the historical matrix tend to be concentrated aroundthe diagonal, with movements of two credit ratings (or more) in a yearbeing rare or nonexistent.

Under the maintained assumption of complete markets and no ar-bitrage opportunities, the transition matrix from time t to time t + 1under the equivalent martingale probability is given by

Qt,t+1 =

q11(t, t + 1) q12(t, t + 1) · · · q1K (t, t + 1)

q21(t, t + 1) q22(t, t + 1) · · · q2K (t, t + 1)...

qK−1,1(t, t + 1) qK−1,2(t, t + 1) · · · QK−1,K (t, t + 1)

0 0 · · · 1

(5)

where qij (t, t+1) ≥ 0, for all i, j, i 6= j , qii(t, t+1) ≡ 1−∑Kj=1j 6=i

qij (t, t+1), and qij (t, t + 1) > 0 if and only if qij > 0 for 0 ≤ t ≤ τ − 1.Without additional restrictions, the martingale probabilities qij (t, t+1)can depend on the entire history of the process up to time t . Hence,under the martingale probabilities, the process need not be Markov.To facilitate empirical implementation, it is desirable to impose morestructure on these probabilities. In particular, we assume that the riskpremia adjustments are such that the credit rating process under themartingale probabilities satisfy7

qij (t, t + 1) = πi(t)qij for all i, j, i 6= j where (6)

7 One can always write qij (t, t + 1) = πij (t)qij where πij (t) can depend on the entire history ofthe process up to time t . The restriction in Equation (6) is twofold. First, πij (t) is independent ofj , and second, it is a deterministic function. The independence of j implies that moving from ito 1 receives the same risk premium as moving from i to K . This restriction may not be true inpractice.

488


πi(t) is a deterministic function of time such that

qij (t, t + 1) ≥ 0 for all i, j, i 6= j andK∑j=1j 6=i

qij (t, t + 1) ≤ 1 for i = 1, . . . ,K .

In matrix form, we can write this as

Qt,t+1 − I = 5(t)[Q − I ]

where I is the K × K identity matrix and 5(t) = diag(π1(t), . . . ,πK−1(t), 1) is a K × K diagonal matrix.

The last row in the transition matrix for Qt,t+1 in conjunction withEquation (6) implies that πK (t) ≡ 1 for all t . The proportionality ad-justments πi(t) have the interpretation of being risk premiums. Thesetransform the actual probabilities to those used in valuation. For lateruse, it is important to note that the restrictions imposed on the mar-tingale probabilities qij (t, t + 1) in Equation (5) imply that πi(t) ≥ 0for all i and t .

This assumption is imposed to facilitate statistical estimation sincethe historical transition matrix Q can be utilized in the inference pro-cess. The actual parameterization estimated is based on the continuoustime limit, and it is discussed in the next section. The implication ofthis assumption (combined with the previous structure) on the evo-lution of credit spreads is discussed later in this section.

Given this structure, we can now compute the probability of defaultoccurring after date T , that is, Qt (τ

∗ > T ). Let qij (0,n) denote then-step transition probabilities of going from state i at time 0 to statej at time n. It is well known that the n-step K × K transition matrix,Q0,n, whose (i, j)th entry is qij (0,n), satisfies

Q0,n = Q0,1Q1,2, . . . , Qn−1,n. (7)

Lemma 1 (Probability of Solvency in Terms of Q). Let the firm bein state i at time t, denoted by ηt = i and define τ ∗ = inf{s ≥ t : ηs =K }, which represents the first time of bankruptcy. Then, the probabilitythat default occurs after time T is

Qit (τ∗ > T ) =

∑j 6=K

qij (t, T ) = 1− qiK (t, T ).

Proof. Since K is absorbing, the event (τ∗ > T ) is equivalent to ηt

not being in state K at time T , when starting in state i at time t . Usingthe Qt,T matrix gives the result.

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In Lemma 1 we make explicit the credit class dependence via asuperscript on the conditional probability Qi

t . Lemma 1 shows howto compute the probability of solvency at any future date T , startingfrom credit class i at time t . This solution requires the computation ofan n-fold matrix product.8

We can use these characterizations of the probability of solvencyto rewrite the risky zero-coupon bond’s valuation. Let vi(t, T ) be thevalue of a zero-coupon bond issued by a firm in credit class i at timet . Then9

vi(t, T ) = p(t, T )(δ + (1− δ)Qit (τ∗ > T )), (8)

where Qit (τ∗ > T ) is obtainable from Lemma 1.

Equation (8) provides a characterization of the credit risk spread.Given that the forward rate for the risky zero-coupon bond in creditclass i is defined by

f i(t, T ) ≡ − log(vi(t, T + 1)/vi(t, T )),

Equation (8) yields

f i(t, T ) = f (t, T )+ 1{τ ∗>t} log

([δ + (1− δ)Qi

t (τ∗ > T )]

[δ + (1− δ)Qit (τ∗ > T + 1)]

). (9)

This gives an explicit representation for the credit risk spread( f i(t, T )− f (t, T )) in terms of the recovery rate (δ) and the transitionmatrix for credit classes Q (as given in Lemma 1). Equation (9) showsthat credit spreads are always strictly positive in this model, exceptin bankruptcy. In bankruptcy, as in the discussion after Equation (2),f K (t, T ) = f (t, T ). Equation (9) also shows that the previous assump-tions, in conjunction, imply that the credit spreads ( f i(t, T )− f (t, T ))and ( f i(t, T )− f j (t − T )) are constants, except for random changesin credit ratings. Although this implication appears contrary to somepreliminary evidence [see Das and Tufano (1995)], its rejection awaitsa more thorough empirical investigation.

To get the spot rate, set T = t in Equation (9) and simplify

r i(t) = r (t)+ 1{τ ∗>t} log(1/[1− (1− δ)qiK (t, t + 1)]). (10)

In bankruptcy, r K (t) = r (t).

8 In the special case that 5(t) is a constant matrix, independent of t , Q0,n = Qn , where Qn is then-fold matrix product. Then, Qi

t (τ∗ > T ) = Qi

0(τ∗ > T − t), as the process is time homogeneous.

9 We can rewrite Equation (8) as

vi(t, T ) = p(t, T )[1− (1− δ)Qit (τ∗ ≤ T )].

The right side is the value of a default-free bond less the present value of the loss if default occurs.

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2.2 Options and hedgingFor practical applications, hedging jumps in credit ratings is essential.This section discusses how to hedge such jumps in both risky bondsand in (vulnerable) options on the term structure of credit risk spreads.To do so we must first extend our valuation methodology to applyto (vulnerable) options on the term structure of credit risk spreads.Afterwards, we analyze hedging these jumps in credit ratings.

As shown in Jarrow and Turnbull (1995), valuing (vulnerable) op-tions on the term structure of credit risk spreads is a straightforwardapplication of the martingale pricing technology. Given the randompayoff to a credit-risky claim, say CT at time T , its value at time t ,denoted Ct , is

Ct = Et (CT /B(T ))B(t). (11)

For example, a European call option with exercise date T and strikeprice X on the risky firm’s zero-coupon bond, maturing at time M ≥T , would have the random payoff CT = max[v(T ,M ) − X , 0], withtime t value given by Equation (11).

Valuation is thus transformed into an expected value calculation.Utilizing Lemma 1, simple formulas for risky coupon bonds, futureson risky bonds, options on risky debt, and vulnerable options canbe obtained (via direct substitution). Expanding the traded securitiesto include common stock, this method also allows us to price equityoptions and convertible debt under the above scenario [see Jarrowand Turnbull (1995) for details].

To hedge credit rating changes (including bankruptcy), as shown inJarrow and Turnbull (1995), the standard option hedging techniquesare applicable. Sufficient information to apply these techniques is acareful description of the evolution of forward rates for the risky andriskless zero-coupon bonds.

Define the function

φ(t, T , i)=

log

([δ+(1−δ)Qi

t (τ∗ > T )]

[δ+(1−δ)Qit (τ∗ > T+1)]

)for i = 1, . . . ,K−1,

0 for i = K .(12)

The change in the firm’s forward rate over [t, t + 1] is easily deducedfrom Equation (9). Let the firm’s time t credit rating be ηt = i, then

f ηt+1(t + 1, T )− f i(t, T ) = [ f (t + 1, T )− f (t, T )]

+ [φ(t + 1, T , ηt+1)− φ(t, T , i)] (13)

for ηt+1 ∈ {1, 2, . . . ,K } with pseudo-probabilities qiηt+1(t, t + 1). Thefirst component of the change in the risky firm’s forward rates is due

491


to changes in the default-free forward rate structure and a shorteningof time to maturity. Any desired term structure model can be applied[e.g., Cox, Ingersoll, and Ross (1985) or Heath, Jarrow, and Morton(1992)]. This component of the risk is hedged in the same manner asemployed in the literature on interest rate options.

The second component of the change in the risky firm’s forwardrates is due to changes in the default probability arising from an un-predictable change in credit class and a (predictable) shortening ofthe time to maturity. This risk has at most K different outcomes (ifqiηt+1(t, t+1) > 0 for all possible ηt+1). To hedge this credit class risk,in general, one needs at most K of this firm’s credit risky securities. Asthe entire term structure of the risky firm’s zero-coupon bonds trade,enough securities are available to implement this trading strategy. Thehedging procedure is analogous to that used in any multinomial (butMarkov) process, and a description of this technique can be found inJarrow and Turnbull (1995).

2.3 Fitting the credit class zero-curvesGiven estimates of the empirical transition matrix Q, and the risk pre-mium (π1(t), . . . , πK−1(t) for 0 ≤ t ≤ τ − 1), Equation (8) provides atheoretical pricing formula for risky zero-coupon bonds. This formulacan be utilized to identify arbitrage opportunities across the credit-risky term structures.

Alternatively, this methodology can be utilized to price and hedgeoptions on the risky debt (Section 2.2). This section presents a recur-sive procedure for selecting the risk premium (π1(t), . . . , πK−1(t) forall t) such that the theoretical pricing formula exactly matches a giveninitial term structure of credit-risky zeros (or credit class spreads).

From Equation (8), given estimates of the risky zero coupon bonds(vi(t, T )), the default-free zeros (p(t, T )), and the recovery rate (δ),the initial (time 0) credit class zero-curves will be matched if Qi

0(τ∗ ≤

T ) is selected such that

Qi0(τ∗ ≤ T ) = [p(0, T )− vi(0, T )]/[p(0, T )(1− δ)] (14)

for i = 1, . . . ,K and T = 1, 2, . . . , τ.

We next show how to select (5(t): t ∈ 0, 1, . . . , τ − 1) such thatEquation (14) is satisfied. The procedure is recursive.

Given the empirical transition matrix Q, we have

Q0,1 = I +5(0)(Q − I ).

From this matrix, and Lemma 1, we get

Qi0(τ∗ ≤ 1) = πi(0)qiK .

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Substitution into Equation (14) and algebra gives

πi(0) = [p(0, 1)−vi(0, 1)]/p(0, 1)(1−δ)qiK for i = 1, . . . ,K−1. (15)

Given that we have computed these risk premium, we can now returnto Equation (6) and compute Q0,1.

Given the empirical transition matrix Q and Q0,t (calculated fromthe previous step), we have

Q0,t+1 = Q0,t [I +5(t)(Q − I )].

From this matrix, and Lemma 1, we get

Qi0(τ∗ ≤ t + 1) =

K∑j=1

qij (0, t)πj (t)qjK .

In matrix form, after substitution into Equation (14) and algebra gives

Q0,t

π1(t)q1K...

πK (t)qK K

= [p(0, t + 1)− v1(0, t + 1)]/p(0, t + 1)(1− δ)

...

[p(0, t + 1)− vK (0, t + 1)]/p(0, t + 1)(1− δ)

.Assuming that Q−1

0,t exists (denote its entries by q−1ij (0, t)), the solution

to the matrix equation is

πi(t) =K∑

j=1

q−1ij (0, t)[p(0, t + 1)− vi(0, t + 1)]/[p(0, t + 1)(1− δ)qiK ].

(16)Given these risk premium, we can now compute Q0,t+1 via Equation(6).

In practical applications one should, of course, check that the re-sulting risk premium is nonnegative and the resulting matrix is indeeda probability transition matrix. If they are not, this indicates an incon-sistency of the data with the model. An illustrative computation inSection 3.4.4 clarifies this issue and provides an alternative estimationprocedure when an inconsistency is observed.

This completes the recursive procedure. Equations (15) and (16)will also prove useful for estimating the parameters of the continuoustime Markov chain model introduced in subsequent sections.

2.4 DiscussionThis completes our analysis of the discrete time case. Some additionalresults can now be derived with respect to the shape and behavior ofcredit class spreads, but to avoid redundancy, these are only provided

493


for the continuous time case. The purpose of this section was to intro-duce the fundamental insights in a simple fashion and to introduce astructure suitable for computer implementation. Having accomplishedthis task, we quickly move on to the continuous time analysis.

3. Credit Ratings and Default Probabilities: The Continuous TimeCase

This section models the distribution for default time in a continuoustrading economy. The continuous time setting facilitates the derivationof additional insights due to the availability of stochastic calculus. Italso provides the parameterization of the bankruptcy process mostsuitable for estimation.

3.1 ValuationThe distribution for default time is modeled via a continuous time,time-homogeneous Markov chain on a finite state space S ={1, 2, . . . ,K }. As in the discrete time setting, these states representthe various credit classes, with state 1 being the highest and state Kbeing bankruptcy.

A continuous time, time-homogeneous Markov chain {ηt : 0 ≤ t ≤τ } is specified in terms of its K × K generator matrix

3 =

λ1 λ12 λ13 . . . λ1,K−1 λ1K

λ21 λ2 λ23 . . . λ2,K−1 λ2K

...

λK−1,1 λK−1,2 λK−1,2 . . . λK−1 λK−1,K

0 0 0 . . . 0 0

(17)

where λij ≥ 0 for all i, j , and

λi = −K∑j=1j 6=i

λij for i = 1, . . . ,K .

The off-diagonal terms of the generator matrix λij represent the tran-sition rates of jumping from credit class i to credit class j . The lastrow of zeros implies that bankruptcy (state K ) is absorbing.

494


The K ×K t -period probability transition matrix for η is given by10

Q(t) = exp(t3) =∞∑

k=0

(t3)k/k !, (18)

[see Karlin and Taylor (1975, p. 152)]. For the discrete time approxi-mation to this system, let [0, τ ] be divided into τ steps of equal length.Then the one-step transition matrix of Equation (4) is given by Q(1)via Equation (18). Denote the (i, j)th entry of the Q(t)matrix by qij (t).

By construction, state 1 is viewed as the best credit rating and K −1is the worst credit rating prior to the state of default. To be sure thatthe Markov chain used to model credit rating changes reflects the factthat “lower credit classes are riskier,” there exists a simple conditionon the generator matrix that one can check.

Lemma 2 (Credit Ratings Versus Risk). The following statementsare equivalent:

1.∑

j≥k qij (t) is a nondecreasing function of i for evey fixed k andt.

2.∑

j≥k λij ≤∑

j≥k λi+1, j for all i, k such that k 6= i + 1.

Proof. This follows by combining Proposition 7.3.2, Theorem 7.3.4,and the remark on p. 251 in Anderson (1991).

Note that a generator matrix, in which the (K−1)×(K−1) submatrixof 3 has the structure of a birth-death chain and where the defaultintensities in the K th column do not decrease as a function of rownumber, will satisfy the conditions of this lemma.

We assume that the generator matrix under the equivalent martin-gale probability is given by

3(t) ≡ U (t)3, (19)

where U (t) = diag(µ1(t), . . . , µK−1(t), 1) is a K × K diagonal matrixwhose first K − 1 entries are strictly positive deterministic functionsof t that satisfy∫ T

0µi(t)dt < +∞ for i = 1, . . . ,K − 1.

10 To get Equation (18) as the limit of the discrete time case, first divide 1 into n equal periodsof length (1/n). Define [I + 3/n] to be the transition matrix over each subperiod. Then, Q =[I +3/n]n . As n→∞, Q → exp(3).

495


For the more general case, the U matrix could be both history and timedependent. This is the analogous assumption to that preceding Equa-tion (6) in the discrete time case. The entries (µ1(t), . . . , µK−1(t), 1)are interpreted as risk premia, that is, the adjustments for risk thattransform the actual probabilities into the pseudo-probabilities suit-able for valuation purposes.

The K × K probability transition matrix from time t to time T forη under the equivalent martingale measure is given as the solutionto the Kolmogorov differential equations [see Cox and Miller (1972,p. 181)]:

∂Q(t, T )

∂t= −3(t)Q(t, T ) and (20a)

∂Q(t, T )

∂T= Q(t, T )3(T ) with the initial condition (20b)

Q(t, t) = I . (20c)

Let the (i, j)th entry of the Q(t, T ) matrix be denoted qij (t, T ).Under the assumption in Equation (19), the credit rating process is

still Markov, but it is now inhomogeneous. In the selection of this re-striction, we faced a trade-off between computability of the transitionmatrix Q(t, T ) and the ability to match any given initial term structureof credit risk spreads. Equation (19) was our compromise. A specialcase illustrates this reasoning. When

3 = diag(µ1, . . . , µK−1, 1)3

for strictly positive constants µ1, . . . , µK−1, the Markov process is timehomogeneous. Here, the solution to Equation (20) is easily computedas

Q(t, T ) = exp(diag(µ1, . . . , µK−1, 1)3(T − t)).

But this restriction does not enable the model to match any giveninitial term structure of credit risk spreads. Equation (19) is the simplestextension, which satisfies both requirements.

Analogous to Lemma 1 for the discrete time case we have

Lemma 3 (Probability of Solvency in Terms of Q). Let the firm bein state i at time t, that is, ηt = i and define τ ∗ = inf{s ≥ t : ηs = K }.Then

Qit (τ∗ > T ) =

∑j 6=K

qij (t, T ) = 1− qiK (t, T ).

The proof of this lemma is identical to that used in Lemma 1.

496


Conditional survival probabilities can be obtained via the expres-sion

Qit (τ∗ > T + 1|τ ∗ > T ) = Qi

t (τ∗ > T + 1)/Qi

t (τ∗ > T ). (21)

Given this lemma we can value the risky firm’s zero-coupon bondsas in Equation (8), that is,

vi(t, T ) = p(t, T )(δ + (1− δ)Qit (τ∗ > T )), (22)

where Qit (τ∗ > T ) is obtainable from Lemma 3.

Given that the forward rate for the risky zero-coupon bond in creditclass i is defined by f i(t, T ) ≡ −∂

∂T log vi(t, T ), Equation (22) yields

f i(t, T ) = f (t, T )− 1{τ ∗>t}

(1− δ)∂

∂TQi

t (τ∗ > T )

δ + (1− δ)Qit (τ∗ > T )

. (23)

In bankruptcy, f i(t, T ) = f (t, T ). We define the credit risk spread tobe [ f i(t, T )− f (t, T )].

To get the spot rate, let T → t in Equation (23). One obtains

r i(t) = r (t)+ (1− δ)λiKµi(t)1{τ ∗>t}, (24)

since ∂Qit (τ∗ > T )/∂T = −∂Qi

t (τ∗ ≤ T )/∂T = −λiKµi(t). In bank-

ruptcy, ri(t) = r (t).The spot rate on risky debt is seen to exceed the default-free spot

rate by a credit risk spread of (1− δ)λiKµi(t), where δ is the recoverrate and λiKµi(t) is the pseudo-probability of default. This expressionis analogous to that given in the discrete time case [Equation (10)], andit can also be found in Jarrow and Turnbull (1995).

3.2 Options and hedgingAnalogous to the discrete time case, option valuation is a straight-forward application of the martingale pricing technology and cor-responds to computing an expectation as in Equation (11). Conse-quently we need not repeat that discussion.

Hedging credit rating changes is also analogous to the discrete timecase. Similar to that case, given a specification of the stochastic processfor the risky forward rates, hedging follows the standard procedures[details are in Jarrow and Turnbull (1995)]. All that remains to be done,then, is to derive the stochastic process for changes in forward rates.This is the content of this section.

Let Nij (t) where i = 1, . . . ,K −1 and j = 1, . . . ,K be independentPoisson processes under the probability measure Q with intensities

497


λij . The counting process Nij (t) represents a credit class jump fromstate i to state j at time t if Nij (t) − Nij (t−) = 1 where Nij (t−) =limε→0 Nij (t − ε).

Under the martingale measure Q, Nij (t) has intensities λijµi(t) fori = 1, . . . ,K − 1 and j = 1, . . . ,K .

We can represent the Markov chain process {ηt : 0 ≤ t ≤ τ } as

ηt = η0 +K−1∑i=1

K∑j=1

∫ t

0( j − i)1{ηt−=i,ηt 6=K } dNij (t). (25)

Define the function

φ(t, T , i) =

−(1− δ)∂Qi

t (τ∗ > T )/∂T

δ + (1− δ)Qit (τ∗ > T )

for i = 1, . . . ,K − 1

0 for i = K .(26)

Then, letting η0 = i0 be the firm’s credit rating at time 0, we have

f ηt (t, T )− f i0(0, T ) = [ f, (t, T )− f (0, T )] (27)

+∫ t

0φ′(s, T , ηs−)ds +

K−1∑i=1

K∑j=1

×∫ t

0(φ(s, T , j)− φ(s, T , i))1{ηs−=i} dNij (s),

where φ′(s, T , ηs−) = dφ(s, T , ηs)/ds.The first component of the change in the risky debt’s forward rates

is due to changes in the default-free forward rates. Any desired default-free term structure model can be applied to hedge this risk as is donein the interest rate option literature [e.g., Heath, Jarrow, and Morton(1992)].

The second component is a smoothly varying term arising from theshortening of time to maturity. If the Markov chain is time homoge-neous, this is always negative.

The third component of the change in the risky debt’s forward ratesis due to jumps in the credit rankings. At any time s in state ηs− = i,there are at most K jump processes needed to be hedged, that is,Ni1(s), . . . ,NiK (s). These can be hedged in the standard manner usingat most K risky zero-coupon bonds (or any other traded claims subjectto this risk). A description of these hedging procedures can be foundin Jarrow and Turnbull (1995).

498


3.3 ExamplesThis section illustrates some important insights related to credit riskspreads via three examples. The first two examples illustrate the im-portance of taking into account the entire structure of the generatormatrix when calculating default probabilities and hence credit riskspreads. The third shows how to generate the model used in Jarrowand Turnbull (1995) as a special case of this article’s analysis.

Example 1. Consider the generator matrix

3 =−0.11 0.10 0.01

0.05 −0.15 0.10 0 0

In this example, there are three possible states. Examining the firstrow, the probability of staying in the first credit class over a smallperiod of time 1t is approximately 1 − 0.111t . The transition ratefrom the first credit class to the second credit class is 0.10, and therate of default from the first credit class is 0.01. One could then makethe inappropriate estimate that a firm that is in this credit class willhave a probability equal to 1− exp(−0.01t) of defaulting before timet since the holding time is exponential. This, of course, does not takeinto account the possibility of downgrading (occurring with rate 0.1)and subsequent default (occurring with rate 0.1 in the lower class).In Figure 1 we illustrate the difference by graphing the true defaultprobability as a function of time (the top curve) and the one basedon the rate of default directly from the highest credit rating.

It is important to estimate risks of downgrading and risks of defaultin lower rating classes to get an accurate estimate of default rates forthe top-rated firms. In Figure 2 we show the spreads [see Equation(23)] under the assumption that δ = 0, and it is clear that it is quitepossible for the spread to be decreasing over time. This will typicallyhappen to very low credit classes. Recall that the spread is reallya survival contingent spread: it reflects forward rates obtained if nodefault occurs before the maturity date. Hence, if a low-rated firm hassurvived long, it may have a lower spread simply because survivalincreases the probability that it has reached a higher credit rating.

Example 2. Now consider the generator matrix

3 =

−0.13 0.10 0.02 0.010.05 −0.16 0.1 0.010 0.05 −0.15 0.10 0 0 0

499


Figure 1A comparison of the true default probability as a function of time for credit class 1 andan exponential distribution based on the one-step default probability for credit class 1. Thiscomparison is for Example 1.

Figure 2The spreads (forward rate credit class i less the default-free forward rate) as a function of maturityare graphed for credit classes 1 and 2. These spreads are for Example 1. A zero recovery rate isassumed.

500


Figure 3Default probabilities as a function of time for credit classes 1, 2, and 3. These default probabilitiesare for Example 2.

In this example there are four possible states. From the first row, therate of default is 0.01. Note in this example that the two top creditclasses have the same risk of direct default (the rate is 0.01 for directdefault), but as Figure 3 illustrates, the probability of default for thesecond class (middle graph) is higher than that of the first class, simplybecause the second class is closer to the risky third class, whose highdefault probabilities are the top graph in Figure 3.

Example 3. If we change the last row of the generator matrix in Ex-ample 2 so that the rate of default is 0.01 (and change the diagonalelement correspondingly to −0.06), we see that the default proba-bilities, and therefore the spread structure, becomes independent ofcredit class. The lower graph of Figure 1 illustrates the default prob-ability as a function of time (an exponential distribution). It is equiv-alent to a bankruptcy process independent of credit classes, which isthe model analyzed in Jarrow and Turnbull (1995).

3.4 Parameter estimationTo utilize this model to price and hedge credit-risky bonds and op-tions, we need to estimate the parameters of the stochastic processgiven in Equation (23). This estimation procedure can be decomposedinto two parts. The first part estimates the parameters generating the

501


default-free forward rates, f (t, T ); the second part estimates the pa-rameters generating the credit risk spread [the second term in Equation(23)].

3.4.1 Estimation of default-free parameters. The parameter esti-mation problem for the stochastic process generating default-free for-ward rates is a well-studied issue in the realm of interest rate options[see Amin and Morton (1993) and Heath et al. (1992)]. Consequentlywe refer the reader to that literature for the relevant insights.

3.4.2 Estimation of the bankruptcy process parameters. Theparameter estimation problem for the stochastic process generatingthe credit risk spread involves estimating the recovery rate (δ), whichcan depend on the seniority of the debt, and estimating the generatormatrix 3 defined in Equation (19).

3.4.3 Estimating the recovery rate. An estimate of the recoveryrate (δ) can be obtained historically from past defaults, or implicitlyvia market prices.

Table 1 reproduces the historical recovery rates from Moody’s Spe-cial Report (1992). As can be seen, these recovery rates increase asthe seniority of the debt increases. These numbers vary across time.

An implicit procedure generates an estimate for δ (given estimatesfor all the other parameters) by setting the theoretical value equal toa market price for some traded derivative security and then inverting(most often numerically). The implicit procedure could utilize zero-coupon bond prices [Equation (22)], spot rates [Equation (24)], couponbonds, or options.

3.4.4 Estimating the generator matrix Λ. An estimate of the gen-erator matrix 3 can also be obtained either historically or implicitly.The implicit procedure estimates the generator matrix 3 implicitly us-ing market prices for the risky zero-coupon bonds. Implicit estimationof the generator matrix 3 would be preferred to historical estimationif it is believed that the historical transition matrix does not representthe actual transition matrix. This could be due to either nonstationar-ies in the bankruptcy process or untimely changes in the credit ratingsby the rating agency. For example, if credit rating changes lag the ac-tual changes in the default probabilities, then the historical transitionmatrix of changes in credit ratings will not be the relevant probabilitymatrix.

In an implicit estimation procedure for 3, (K −1)× (K −1) param-eters need to be estimated [see Equation (17)]. This is a large number

502


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91

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r$6

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33.6

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2).

503


of parameters to imply out from the data. We therefore suggest using adecomposition of the generator matrix under the martingale measureinto a product of the empirical generator matrix, which may be read-ily estimated, and a low-dimensional vector of time-dependent riskpremia. Prices observed in the markets and the empirical generatormatrix (historical or predicted) may then be used to imply out the riskpremia.

Alternative ways of simplifying the estimation of 3 could be torestrict the number of strictly positive transition probabilities betweenstates so that, for example, only jumps to the adjacent state can occur.This will reduce the number of parameters to 2(K − 1). Under thisrestriction it is also possible to relax the assumption on the risk premiaso that πi(t) becomes πij (t). Another approach would be to reducethe number of states to three classes: investment grade, speculativegrade, and default. However, if the true process of ratings with the K -dimensional state space is Markovian, it will typically not be the casethat this process on the smaller state space is Markovian. Applying afunction that is not one to one to a Markov chain destroys the Markovproperty.

We now concentrate on the historical approach. Our estimationprocedure is based on the zero-coupon bond prices and utilizes thedecomposition of the generator matrix 3 into the product of the em-pirical generator matrix 3 and the risk premia (µ1(t), . . . , µK−1(t))[see Equation (19)].

3.4.5 Estimation of the empirical generator matrix3. Estimatesof the empirical generator matrix 3 can be obtained from past obser-vations of credit rating changes [see Moody’s Special Report (1992) orStandard & Poor’s Credit Review (1993)]. From a statistical viewpoint,the Markov chain model we have specified is very convenient. If weobserve over a time period [0, T ] the exact times of credit class tran-sitions and defaults, a classical estimator of an off-diagonal elementλij of the generator matrix is given by

λij =Nij (T )∫ T

0 Yi(s)ds,

where

Nij (T ) = total number of transitions from i to j over the period [0, T ]

andYi(s) = number of firms in class i at time s.

For a derivation of this result, and for extensions to estimationof transition rates that are time dependent, have external covariates,

504


or are based on censored observations, see Andersen and Borgan(1985).

For this article we use data reported in Standard and Poor’s CreditReview (1993). These data do not include the exact timing of transi-tions, but they do include estimates of 1-year transition probabilitiesthat are obtained by observing the credit ratings of a fixed group offirms at the beginning and at the end of the year. We use these esti-mates to obtain an approximation to the generator matrix, and this,in turn, will be used to obtain transition probabilities Q(t) for every t .

Given the estimate of Q(1), we can estimate the generator matrix byassuming that the probability of making more than one transition peryear is small. In other words, every firm is assumed to have made ei-ther zero or one transition throughout the year. Under this hypothesis,it can be shown given λi 6= 0 for i = 1, . . . ,K − 1 that11

exp(3)≈

eλ1λ12(eλ1−1)

λ1· · · λ1K (eλ1−1)

λ1λ21(eλ2−1)

λ2eλ2 · · · λ2K (eλ2−1)

λ2λK−1,1(eλK−1−1)

λK−1

λK−1,2(eλK−1−1)

λK−1· · · λK−1,K (eλK−1−1)

λK−10 0 · · · 1

.

(28)To obtain our estimates of 3, we set Q(1) equal to the right side ofEquation (28) to obtain12

qii = e λi for i = 1, . . . ,K − 1 and (29a)

qij = λij (eλi − 1)/λi for i 6= j and i, j = 1, . . . ,K − 1. (29b)

The solution to this system is

λi = log(qii) for i = 1, . . . ,K − 1 and (30a)

λij = qij log(qii)/[qii − 1] for i 6= j and i, j, . . . ,K − 1. (30b)

A 1-year transition matrix is given in Standard and Poor’s CreditReview (1993) and is reproduced in Table 2 in the Appendix. It posesone problem for our analysis. Transitions to the “Not rated” (NR) cate-

11 See the Appendix for a proof. Since Q(1) = exp(3) ≈ I + 3, we could have used the approx-imation 3 = Q(1) − I , but we choose Equation (28) since it provided a better fit of exp(3) toQ(1) in our data.

12 The notation qij should not be confused with the pseudo-probabilities qij defined in the discretetime section.

505


Table 2Average 1-year transition probabilities, 1981–1991

Rating at the end of year

Initialrating AAA AA A BBB BB B CCC D NR

AAA 0.8746 0.0945 0.0077 0.0019 0.0029 0.0000 0.0000 0.0000 0.0183AA 0.0084 0.8787 0.0729 0.0097 0.0028 0.0028 0.0000 0.0000 0.0246A 0.0009 0.0282 0.8605 0.0628 0.0098 0.0044 0.0000 0.0009 0.0324BBB 0.0006 0.0041 0.0620 0.7968 0.0609 0.0151 0.0017 0.0043 0.0545BB 0.0004 0.0020 0.0071 0.0649 0.7012 0.0942 0.0115 0.0218 0.0970B 0.0000 0.0017 0.0027 0.0058 0.0451 0.7196 0.0380 0.0598 0.1272CCC 0.0000 0.0000 0.0102 0.0102 0.0179 0.0665 0.5729 0.2046 0.1176

Standard and Poor’s Credit Review (1993), Table 10.

Table 3Modified transition probabilities


Initialrating AAA AA A BBB BB B CCC D

AAA 0.8910 0.0963 0.0078 0.0019 0.0030 0.0000 0.0000 0.0000AA 0.0086 0.9010 0.0747 0.0099 0.0029 0.0029 0.0000 0.0000A 0.0009 0.0291 0.8894 0.0649 0.0101 0.0045 0.0000 0.0009BBB 0.0006 0.0043 0.0656 0.8427 0.0644 0.0160 0.0018 0.0045BB 0.0004 0.0022 0.0079 0.0719 0.7764 0.1043 0.0127 0.0241B 0.0000 0.0019 0.0031 0.0066 0.0517 0.8246 0.0435 0.0685CCC 0.0000 0.0000 0.0116 0.0116 0.0203 0.0754 0.6493 0.2319D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

No-rating category eliminated.

gory are listed, but no estimates of subsequent defaults or reentranceto the rating categories are reported.13 We choose to eliminate thispart of the sample by redefining the transition probability from statei to j (both states different from NR) as being

qij = fraction of firms going to j from i

fraction of firms going to a state different from NR. (31)

This is the modified transition matrix reported in Table 3. Usingthe estimation procedure described above we get an estimate of thegenerator matrix 3, reported in Table 4.

3.4.6 Estimation of the risk premium. Estimates of the risk pre-mia (µ1(t), . . . , µK−1(t)) can be obtained from the zero-coupon bondprices [Equation (22)].

13 As noted in the Standard & Poors Credit Review (1993), the majority of the transitions to an NRoriginate from issuers repaying outstanding debt or bringing it below a limit of $25 million, butsome are caused by insufficient information being provided by the issuer and subsequent defaultsdo occur.

506


Table 4Generator matrix estimated from modified transition probabilities in Table 3


Initialrating AAA AA A BBB BB B CCC D

AAA −0.1154 0.1019 0.0083 0.0020 0.0031 0.0000 0.0000 0.0000AA 0.0091 −0.1043 0.0787 0.0105 0.0030 0.0030 0.0000 0.0000A 0.0010 0.0309 −0.1172 0.0688 0.0107 0.0048 0.0000 0.0010BBB 0.0007 0.0047 0.0713 −0.1711 0.0701 0.0174 0.0020 0.0049BB 0.0005 0.0025 0.0089 0.0813 −0.2530 0.1181 0.0144 0.0273B 0.0000 0.0021 0.0034 0.0073 0.0568 −0.1929 0.0479 0.0753CCC 0.0000 0.0000 0.0142 0.0142 0.0250 0.0928 −0.4318 0.2856D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Using an analytic approximation to the forward equation [Equation(20b)], we get that over a small time period 1t14

Q(t, t +1t) ≈ I + 3(t)1t = I + U (t)31t . (32)

Letting 5(t) ≡ U (t) and Q − I ≡ 31t in Equations (15) and (16),yields an estimation procedure for the risk premia U (t). U (t) in thisapproximation is assumed to be right continuous over [t, t+1t). Thisestimation procedure will be such that the model prices given viaEquation (22) will exactly match the initial credit-risky zero-couponbond price curves.

3.4.7 Survival probabilities and spreads under risk neutrality.We can illustrate some main features of the model by looking at therisk-neutral case, that is, the case where the pricing measure and theempirical measure are the same.

From the estimated generator matrix, we have calculated the sur-vival probabilities for firms of the various credit ratings as a functionof time. In other words, if a firm is in a given credit class today, andif its rating evolves according to the Markov chain specified by theempirical generator matrix, what is the probability of having no de-faults within the next t years? The result is shown in Figure 4. Notethat with the modified empirical transition matrix specified in Table 3,we could have obtained estimates for survival probabilities for integeryears in the future by simply calculating powers of that matrix andreading off the entry in the default column. But we are interested indefault probabilities for every time to maturity, and in particular for

14 This approximation comes from the forward equation [Equation (20b)]. For small 1t ,

∂Q(t, T )/∂T∣∣

T=t+1t≈ [Q(t, t +1t)− I ]/1t and Q(t, t +1t)3(t +1t) ≈ Q(t, t)3(t) = 3(t).

Substitution into Equation (20b) gives Equation (32).

507


Figure 4The survival probabilities (no defaults in the next t years) for the different credit classes as afunction of time. These graphs are based on the generator matrix given in Table 4.

periods shorter than 1 year. Our estimate of the generator matrix pro-vides us with estimates of these probabilities. As expected, the defaultprobabilities for every fixed time are lower for the better rated firms.

In Figures 5 and 6 we have graphed forward spreads as definedin Equation (23) for investment grade and speculative grade bonds,respectively. We have set the recovery rate equal to zero in orderto give the graphs an alternative interpretation as hazard rates: Thehazard rate corresponding to credit class i at time t is the rate ofdefault at time t for a firm that is in class i at time 0 and which hasnot defaulted up to time t . The spread at time 0 is the credit spread onthe spot rate. If we set the recovery rate to δ > 0, these spot rate creditspreads are multiplied by 1 − δ. For a value of δ equal to 0.67, thiswould bring the extremely high spreads on speculative rates downin a more reasonable range, but it would also lower the already verylow spreads on the investment grade bonds.

3.4.8 An illustrative estimation of the risk premia. This sectionillustrates the computation of the risk premia (µ1(t), . . . , µK−1(t)) fort = 0, 1, . . . , τ utilizing the procedure described above. This exampledemonstrates the feasibility of performing the described computations

508


Figure 5The spreads (forward rate credit class i less the default-free forward rate) in basis points as afunction of maturity for credit classes AAA, AA, A, and BBB. These spreads assume risk neutralityand a zero recovery rate. They are based on the generator matrix given in Table 4.

Figure 6The spreads (forward rate credit class i less the default-free forward rate) in basis points as afunction of maturity for credit classes BB, B, CCC. These spreads assume risk neutrality and azero recovery rate. They are based on the generator matrix given in Table 4.

509


and provides some preliminary insights useful for designing a morethorough empirical investigation of the proposed model.

The computations were performed using bond index price data forFriday, December 31, 1993. The corporate bond price data was kindlyprovided by Lehman Brothers, and it is included in Table 5.

Table 5 contains a matrix whose columns correspond to variousmaturity ranges, and whose rows correspond to credit classes. Withineach cell of the matrix is given (1) the number of bond issues repre-sented in the cell, (2) the market value weighted coupon payment forthe cell, and (3) the (bid) “yield to worst” (market value weighted)for the cell. The yield to worst is the bond’s yield, except whena bond is callable. When callable, the yield to worst is the mini-mum of the following two numbers: (1) the yield computed basedon the bond’s maturity date, and (2) the yield computed based on thebond’s first call date. Obviously this is an incomplete adjustment toincorporate a bond’s call provision into the calculation of the bond’syield. As such it can introduce significant error into the followingprocedure. In essence the computation of yield to worst can placea callable bond’s yield in the wrong maturity cell. This observationerror is important to keep in mind when evaluating the subsequentresults.

The first step in the computation of the risk premium from thiscorporate bond data is to strip out the risky zero-coupon bond prices.Our procedure for doing this is described in the Appendix. The resultsare provided in Table 6. Also included in Table 6 are default-freezero-coupon bond prices. These are the U.S. Treasury strip prices asreported in the Wall Street Journal for December 31, 1993. Figures 7and 8 contain the graphs of these zero-coupon bond prices.

Occasional “mispricings” can be observed in these various zero-coupon bond price curves. Indeed, whenever the credit class termstructures cross, potential arbitrage opportunities exist. For example,at the maturity of 1 year, the AA zero is priced above the AAA zero-coupon bond, and the B zero-coupon bond is priced above the BA-zero coupon bond. These crossings are inconsistent with the previousmodel structure. For the purposes of this section, we attribute thesemispricings to noise in the data (most likely due to the yield-to-worstissue previously discussed). The noise in the data is particularly both-ersome when the number of issues in a cell are small, such as in thecase of the B-rated bonds in the first two maturity classes (only oneissue each).

These mispricings will generate risk premium (µ1(t), . . . , µK−1(t))for t = 0, . . . , τ that do not satisfy the conditions imposed in Equation(6) [using 5(t) = U (t) from Equation (32)], which guarantee that Q is

510


Tab

le5

Leh

man

Bro

ther

sco

rpo

rate

bo

nd

ind

exd

ata

for

Dec

emb

er3

1,

19

93

Mat

urity

Qual

ity0–

1.75

1.75

–2.2

52.

75–3

.25

3.25

–4.7

54.

75–5

.25

5.25

–6.5

6.5–

7.5

9.5–

10.5

10.5

–18

AA

A1.

144

1320

314

123

282.

8.41

57.

592

7.45

99.

009

9.38

38.

076

8.99

07.

369

8.13

83.

4.35

14.

508

4.97

45.

493

5.03

45.

843

5.84

16.

277

6.24

0A

A1.

1310

1746

1435

2422

662.

7.08

66.

463

7.25

06.

753

8.41

97.

267

8.47

68.

133

8.74

03.

4.23

34.

555

4.73

15.

394

5.34

75.

670

5.98

96.

381

6.14

4A

1.70

5861

153

3913

981

7117

92.

7.79

17.

289

7.87

97.

898

8.01

27.

726

9.04

16.

743

8.19

13.

4.28

64.

689

4.94

45.

403

5.52

35.

899

6.18

76.

480

6.71

9B

AA

11.

2213

3787

2992

5156

892.

8.35

77.

628

8.40

98.

585

8.63

58.

734

9.09

57.

961

8.84

73.

4.87

05.

006

5.18

15.

944

6.18

26.

514

6.70

17.

102

7.03

7B

A1.

21

715

836

2228

582.

10.9

759.

450

8.88

58.

808

11.1

949.

288

7.59

39.

272

9.63

63.

7.29

16.

023

6.22

47.

494

7.43

37.

531

9.15

58.

455

8.57

9B

1.1

14

2111

4634

6150

2.10

.000

9.50

09.

987

7.76

311

.551

10.3

0910

.715

9.43

48.

186

3.5.

428

7.80

79.

981

11.6

089.

418

9.28

99.

676

9.71

29.

969

CA

A1.

10

12

23

35

12.

15.0

000.

000

11.3

752.

475

9.88

611

.976

13.1

292.

823

11.0

003.

8.57

10.

000

11.1

7015

.085

15.2

1013

.425

11.8

0610

.276

10.4

07

With

inea

chce

llis

liste

d:

1.N

um

ber

ofis

sues

2.Coupon

mar

ketw

eigh

ted

3.Y

ield

tow

ors

tFo

rth

eB

A,B

,CA

Aqual

ity,th

e10

.5–1

8co

lum

nis

real

ly10.5+

with

no

term

inal

dat

epro

vided

.

511


Tab

le6

Stri

pp

edze

ro-c

ou

po

nb

on

dp

rice

sfo

rD

ecem

ber

31

,1

99

3

Mat

urity

Qual

ity1.

002.

003.

004.

005.

006.

007.

008.

009.

0010

.00

11.0

012

.00

13.0

014

.00

GO

VT

96.9

6992

.656

87.8

4482

.719

77.7

1973

.000

68.2

5063

.781

60.1

7656

.570

52.9

6549

.359

45.7

5442

.148

AA

A95

.830

*91

.549

86.3

53*

80.4

20*

78.2

40*

70.4

27*

66.4

9762

.076

57.6

5553

.234

50.4

1247

.589

*44

.766

*41

.944

*A

A95

.939

91.4

5987

.001

80.7

8976

.804

71.3

5765

.660

61.2

7556

.891

52.5

0750

.128

47.7

4945

.370

42.9

91A

95.8

9091

.218

86.4

4880

.780

76.1

3370

.322

64.7

0160

.581

56.4

6052

.340

48.8

9845

.455

42.0

1338

.571

BA

A1

95.3

5690

.684

85.8

9479

.033

73.6

1967

.742

62.5

7058

.060

53.5

5049

.040

46.1

9443

.347

40.5

0037

.654

BA

93.2

0489

.057

83.4

7374

.439

69.3

9664

.232

52.2

3749

.377

46.5

1743

.658

40.3

9537

.133

33.8

7130

.608

B94

.851

*85

.060

74.5

2563

.482

*64

.000

59.0

0052

.190

47.9

0443

.617

39.3

3135

.905

32.4

78*

29.0

52*

25.6

26*

CA

A92

.106

82.2

6672

.426

56.5

5347

.551

46.6

6347

.626

*44

.633

41.6

4138

.648

35.8

8533

.122

30.3

5827

.595

*Indic

ates

apote

ntia

lm

isprici

ng.

512


Figure 7Stripped zero-coupon bond prices on December 31, 1993, versus time to maturity for Treasuries,AAA, and AA rated debt. A face value of $100 is assumed. Treasuries are —–, AAA are – – –, andAA are - - - - - -.

a probability matrix. The relevant condition is that the risk premiumis nonnegative, µi(t) ≥ 0, for all credit classes i and times t .15

To illustrate this outcome we compute the risk premia using the(unconstrained) procedure as described in Section 2.3. Needed for this

15 Strictly speaking, the condition is that µi(t) > 0 for all credit classes and times (where thegenerator matrix has strictly positive values). However, imposing strict positivity will not allowus to solve the minimization problem. Therefore we do not impose this strict positivity condition(see footnote 16).

513


Figure 8Stripped zero-coupon bond prices on December 31, 1993, versus time to maturity for A, BAA,BA, B, and CAA rated debt. A face value of the $100 dollars is assumed. A are ——, BAA are – ––, BA are - - - - - -, B are – - – - –, and CAA – - - – - - –.

computation are an estimate of the recovery rate (δ) and an estimateof the Q matrix. We set δ = 0.3265, which is the weighted averagevalue for 1991 as given in Table 1. We set Q equal to the generatormatrix given in Table 4 [as required by Equation (32)], with one smallmodification. In order that the risk premia µi(0) for i = 1, . . . ,K−1 arewell-defined for credit class i via Equation (15), we need to guaranteethat the one-step default probability qiK 6= 0. This is not satisfied bythe default probabilities in the first two rows of Table 4. For this

514


reason we replace the zero default probabilities qiK for these tworows with 0.0001 (and reduce the diagonal element in row i by anequal amount). This is the smallest significant number possible withinTable 4.

The results of this computation are given in Table 7. As noted,many of the risk premia are negative, indicating the existence of arbi-trage opportunities, data errors, or model rejection. We believe thesemispricings are most likely due to the noise introduced into the com-putation of the zero-coupon bond prices due to the yield-to-worstissue previously discussed.

Given that an unconstrained estimation procedure for (µ1(t), . . . ,µK−1(t)) for t = 0, . . . , 14) indicates mispricings, we next computethe best values for the risk premia consistent with no arbitrage, where“best” is defined to mean minimizing the sum of squared errors of theactual prices from the model prices. This procedure uses all the data,including the mispriced zero-coupon bonds.

The constrained estimation technique follows a recursive proceduresimilar to that described in Section 2.3. We now briefly describe thisprocedure.16

At step 0, we find such that (µ1(0), . . . , µK−1(0)) such that∑K−1i=1 [vi(0, 1) − vi(0, 1)observed]2 is minimized subject to µi(0) ≥ 0

for i = 1, . . . ,K − 1, where vi(0, 1) is determined from Equation (3)and Qi

0(τ∗ > 1) in Equation (3) is determined as Equation (15) in the

text.Given the optimized values for (µ1(0), . . . , µK−1(0)), to prepare for

the next step in the procedure, we compute the one-step transitionmatrix Q0,1, using Equation (6).

At step t+1, we enter with Q0,t . We compute (µ1(t+1), . . . , µK−1(t+1)) such that

∑K−1i=1 [vi(0, t+1)−vi(0, t+1)observed]2 is minimized sub-

ject to µi(t+1) ≥ 0 for i = 1, . . . ,K−1, where vi(0, t+1) is determinedfrom Equation (3) and Qi

0(τ∗ > t + 1) in Equation (3) is determined

as Equation (16) in the text. In particular,

Qi0(τ∗ > t + 1) = 1−

K∑j=1

qij (0, t)µj (t)qjk .

Given the optimized values for (µ1(t + 1), . . . , µK−1(t + 1)), wethen compute Q0,t+1 using Equation (6) in order to prepare for thenext step.

16 To satisfy the strict positivity condition in footnote 15, one could add the restriction that µi(0) ≥0.0001 for all i. The estimates obtained in this case would be similar to those subsequentlyreported.

515


Tab

le7

Ris

kp

rem

iaco

mp

ute

du

sin

gan

un

con

stra

ined

pro

ced

ure

on

Dec

emb

er3

1,

19

93

Mat

urity

Qual

ity0.

001.

002.

003.

004.

005.

006.

007.

008.

009.

0010

.00

11.0

012

.00

13.0

0

AA

A17

4.29

419.

87−2

61.9

3−4

29.9

855

4.71

4745

.25−2

5111.8

726

9.69

649.

3910

76.3

45.

44−1

318.

545.

93−6

4.38

AA

157.

6842

9.97

−255.9

6−4

28.9

946

6.93

4744

.94−2

4906.0

926

7.70

646.

6210

71.2

68.

5583

4.32

16.9

033

.24

A16

.51

44.6

9−3

0.55

−50.

2055

.26

492.

50−2

546.

5427

.57

65.8

410

9.38

−0.2

2−3.3

80.

60−0.4

1B

AA

15.

045.

44−7.7

4−1

0.18

15.5

494

.54

531.

555.

7313

.60

22.6

1−0.2

3−0.9

7−0.1

1−0.2

1B

A2.

11−2.3

1−0.0

52.

4014

.62

7.40

−106.1

21.

002.

403.

93−0.0

3−0.1

60.

00−0.0

2B

0.43

1.10

2.02

2.15

−7.9

30.

100.

90−0.0

60.

240.

260.

590.

791.

071.

41CA

A0.

260.

360.

411.

121.

64−2.4

0−3.3

40.

010.

120.

14−0.0

2−0.0

3−0.0

3−0.0

2D

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

516


The recursive procedure stops when t + 1 = 14.The constrained estimates for the risk premia are contained in Ta-

ble 8. The theoretical risky zero-coupon bond prices consistent withthese risk premia are contained in Table 9. The standard error of theestimate and the percentage error (standard error/average price in thecolumn) are provided under each column in Table 6. The percentageerror is seen to be the smallest for the shortest maturity zeros, andincreases as time to maturity increases. This is to be expected as pric-ing errors compound in the computing procedure. The dollar pricingerror for maturities less than 4 years is no more than $1.0826 per $100face value bond. For maturities exceeding 4 years, the pricing error isat most $3.7908 per $100 face value bond. These risky zero-couponbonds and the risk premia (µ1(t), . . . , µK−1(t)): t = 1, . . . , 14) in Ta-bles 8 and 9 provide the necessary inputs for computing the valuesof options on risky debt and credit derivatives.

The previous tables demonstrate the simplicity of obtaining the pa-rameter estimates for the Markov model. They also provide additionalinsights into numerous estimation issues that need to be addressedin a more thorough investigation. First, the data underlying the esti-mates for the risky zero-coupon bonds need to be filtered to removeall imbedded options. This filtering will remove the yield-to-worst biaspresent in the previous results. Second, to get a more homogeneousset of bond prices, partitioning by industry groups may be a goodidea. Third, time-series observations are needed to (1) verify the timeconsistency of the estimations and (2) for testing trading strategies tosee if the “mispricings” are due to arbitrage opportunities, data errors,or model misspecification.

4. Conclusion

This article develops a Markov model for the term structure of creditrisk spreads, extending the work of Jarrow and Turnbull (1995). Thebankruptcy process is modeled via a discrete state space, contin-uous time, time-homogeneous Markov chain in credit ratings. Thisbankruptcy processes’ parameters can be estimated from observabledata. An illustrative example is provided. The validation or rejectionof this model awaits subsequent research.

Appendix A

Proof of Equation (28). This matrix can be derived using a probabilisticargument. Under this hypothesis, we can approximate the evolutionof the firm’s credit rating over a year by assuming that if the firm leavesstate i before the year is over for state j , it is absorbed in state j . To

517


Tab

le8

Ris

kp

rem

iaco

mp

ute

du

sin

gco

nst

rain

edo

pti

miz

atio

no

nD

ecem

ber

31

,1

99

3

Mat

urity

Qual

ity0

12

34

56

78

910

1112

13

AA

A8.

6580

8.65

808.

6580

8.65

800.

0000

8.11

908.

6580

8.65

808.

6580

8.65

800.

0000

0.00

000.

0000

0.00

00A

A9.

5785

9.57

850.

0000

9.57

850.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

00A

8.53

248.

5324

8.53

248.

5324

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

BA

A1

5.04

041.

8159

0.00

001.

5447

0.00

000.

0000

0.00

000.

0000

1.17

705.

8445

0.00

000.

0000

0.00

000.

0000

BA

2.11

170.

0000

0.00

000.

0000

0.00

000.

0000

3.12

670.

4607

3.95

263.

9526

0.00

000.

0000

0.00

000.

0000

B0.

4307

0.98

651.

8181

2.58

750.

0000

0.00

000.

0000

0.00

000.

0000

0.44

560.

0309

0.02

010.

0244

0.02

44CA

A0.

2607

0.35

300.

4034

1.10

680.

6978

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

00

518


Tab

le9

Arb

itra

ge-f

ree

risk

yze

ro-c

ou

po

nb

on

dp

rice

sfo

rD

ecem

ber

31

,1

99

3

Mat

urity

Qual

ity1

23

45

67

89

1011

1213

14

AA

A96

.921

592

.501

487

.122

581

.251

376

.249

270

.864

665

.163

760

.865

457

.074

752

.743

949

.363

45.9

904

42.6

181

39.2

471

(1.0

825)

(0.9

524)

(0.7

695)

(0.8

313)

(−1.

9908

)(0

.437

6)(−

1.33

33)

(−1.

2106

)(−

0.58

03)

(−0.

4901

)(−

1.04

9)(−

1.59

86)

(−2.

1479

)(−

2.69

69)

AA

96.9

064

92.0

011

86.7

370

80.9

128

75.9

209

70.5

794

64.9

239

60.6

421

56.8

693

52.5

555

49.1

8745

.826

642

.466

539

.107

6(0

.967

4)(0

.542

1)(−

0.26

4)(0

.123

8)(−

0.88

31)

(−0.

7776

)(−

0.73

61)

(−0.

6329

)(−

0.02

17)

(0.0

485)

(−0.

941)

(−1.

9224

)(−

2.90

35)

(−3.

8834

)A

96.4

118

91.5

915

86.1

866

80.2

087

75.2

303

69.9

297

63.9

945

59.7

643

55.9

816

51.7

065

48.3

897

45.0

821

41.7

748

38.4

689

(0.5

218)

(0.3

735)

(0.2

614)

(−0.

5713

)(−

0.90

27)

(−0.

3923

)(−

0.70

65)

(−0.

8167

)(−

0.47

84)

(−0.

6335

)(−

0.50

83)

(−0.

3729

)(−

0.23

82)

(−0.

1021

)B

AA

195

.356

90.3

754

84.8

356

78.7

447

73.7

927

68.6

292

62.1

318

58.0

049

54.2

150

50.0

835

46.8

659

43.6

595

40.4

533

37.2

49(0

.000

0)(−

0.30

86)

(−1.

0584

)(−

0.28

83)

(0.1

737)

(0.8

872)

(−0.

4382

)(−

0.05

51)

(0.6

650)

(1.0

435)

(0.6

719)

(0.3

125)

(0.0

467)

(−0.

405)

BA

93.2

0487

.605

481

.086

673

.911

768

.989

363

.735

957

.252

653

.436

549

.873

946

.104

243

.138

140

.184

237

.230

334

.278

7(0

.000

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519


make our argument, we define τij to be independent exponentiallydistributed random variables with intensities λij . Let τi ≡ minj τij , then

τi is exponentially distributed with parameter − λi ≡K∑j=1j 6=i

λij

and prob(τi = τij | τi ≤ 1) = prob(τi = τij ) = λij/ − λi [see Resnick(1992; Exercise 4.45 and Proposition 5.1.1)]. Our hypothesis impliesthat

qii = prob(τi > 1) = eλi and

qij = prob(going to state j before time 1)

= prob(τi = τij and τi ≤ 1)

= prob(τi = τij | τi ≤ 1)prob(τi ≤ 1) = λij

−λi(1− eλi ).

This completes the probabilistic argument. We now provide a secondproof.

The hypothesis, in analytic form, is that

(i) λijλjk ≈ 0 for i 6= j and j 6= k, and(ii) For each i,

∑∞n=0(

∑nm=0 λ

mi λ

n−mj )/n! is independent of j .

Roughly, in probabilistic terms, (i) states that starting in state i, it isunlikely to jump to j and then k, in a small time period; (ii) states thatstarting in state i, given that one jumps out, the likelihood of jumpingto state j (in a small time period) is proportional to λij .

Now, exp(3) =∑∞n=03n/n!.

Performing the matrix multiplication term by term, and then summing,one gets that the diagonal elements are eλj for j = 1, . . . ,K − 1. Thisfollows from (i) alone. The off-diagonal elements (i, j) for i 6= j are

λij

[ ∞∑n=0

(n∑

m=0

λmi λ

n−mj

)/n!

].

This also follows from (i) alone. But since the rows of exp(3) sum toone, we get

eλi +K∑j=1j 6=i

λij

( ∞∑n=0

(n∑

m=0

λmi λ

n−mj

)/n!

)= 1,

hence the result, since λi ≡ −∑K

j=1j 6=iλij . This completes the argument.

520


Appendix B

Procedure for stripping risky zero-coupon bond pricesWe use the maturities T = 1, 2, 3, 4, 5, 6, 7, 10, and 14 years. Theseare the midpoints of the maturity ranges given in Table 5.

Given are

c = coupon,

yiT = yield to worst for the ith credit class, and

T = maturity.

Let BiT be the bond’s price.

To strip out the zeros, we first compute BiT by

BiT =

T∑j=1

c

(1+ yT ) j+ 100

(1+ yT )T.

This assumes the coupon payment is made once per year and theface value is 100. Then, using a triangular system of equations, wecompute vi(T ) recursively as follows:

Bi1 = (c + 100)vi(1) compute vi(1)

Bi2 = cvi(1)+ (c + 100)vi(2) compute vi(2)

...

Bi7 = cvi(1)+ cvi(2)+ · · · + (c + 100)vi(7) compute vi(7),

missing T = 8, 9. To determine these we used linear interpolation.Let

vi(8) ≡ (2/3)vi(7)+ (1/3)vi(10)

vi(9) ≡ (1/3)vi(7)+ (2/3)vi(10).

Bi10 = cvi(1)+ · · · + cvi(7)+ c[(2/3)vi(7)+ (1/3)vi(10)]

+ c[(1/3)vi(7)+ (2/3)vi(10)]

+ [c + 100]vi(10) compute vi(10),

missing T = 11, 12, 13. To determine these let

vi(11) ≡ (3/4)vi(10)+ (1/4)vi(14)

vi(12) ≡ (1/2)vi(10)+ (1/2)vi(14)

vi(13) ≡ (1/4)vi(10)+ (3/4)vi(14).

521


Bi14 = cvi(1)+ · · · + cvi(10)+ c[(3/4)vi(10)+ (1/4)vi(14)]

+ c[(1/2)vi(10)+ (1/2)vi(14)]

+ c[(1/4)vi(10)+ (3/4)vi(14)]

+ [c + 100]vi(14) compute vi(14).

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a markov model for the term structure of credit … credit risk...markov process for credit ratings...

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