September/October 2001 1

Default Parameter Estimation Using Market PricesRobert Jarrow

This article presents a new methodology for estimating recovery rates andthe (pseudo) default probabilities implicit in both debt and equity prices. Inthis methodology, recovery rates and default probabilities are correlated anddepend on the state of the macroeconomy. This approach makes twocontributions: First, the methodology explicitly incorporates equity pricesin the estimation procedure. This inclusion allows the separateidentification of recovery rates and default probabilities and the use of anexpanded and relevant data set. Equity prices may contain a bubblecomponent—which is essential in light of recent experience with Internetstocks. Second, the methodology explicitly incorporates a liquidity premiumin the estimation procedure—which is also essential in light of the largeobserved variability in the yield spread between risky debt and U.S. Treasurysecurities and the illiquidities present in risky-debt markets.

value-at-risk measure that successfullyintegrates market, credit, and liquidityrisk is the Holy Grail of a successful risk-management procedure. As I have previ-

ously argued (Jarrow 1998), arbitrage-free pricingtheory allows this construction, at least conceptu-ally. The remaining obstacles to a successful imple-mentation of this Holy Grail are the selection of aparticular parameterization of the general modeland the estimation of its parameters.

The available model structures are of twotypes–structural and reduced form. Structuralmodels are those that endogenize the bankruptcyprocess by explicitly modeling the assets and liabil-ity structure of the company (Merton 1974).Reduced-form models exogenously specify anarbitrage-free evolution for the spread betweendefault-free and credit-risky bonds (Jarrow andTurnbull 1995; Duffie and Singleton 1999).

Structural models have been successfullyimplemented in professional software.1 This partic-ular parameterization of the structural approachuses only equity prices and balance sheet data toestimate the bankruptcy process’s parameters. Theargument is that debt markets are too illiquid anddebt prices too noisy to be useful; hence, theyshould be ignored. Unfortunately, this implementa-tion of the structural approach ignores the possibil-ity of stock-price bubbles (e.g., for Internet stocks)and the misspecification that this omission implies.

In contrast, the existing literature on implementingreduced-form models concentrates only on debtprices while ignoring equity prices (Jarrow, Lando,and Turnbull 1997); Duffie and Singleton).

The two approaches seem to have partitionedthe market data: Structural models use only equityprices, and reduced-form models use only debtprices. This partitioning is artificial and unneces-sary. Both markets provide relevant informationabout a company’s default process and parameters,and both should be used. The purpose of this articleis to provide a new methodology for implementingreduced-form models that includes both debt andequity prices in the estimation procedure.

In particular, I present a methodology for esti-mating recovery rates and (pseudo) default proba-bilities implicit in debt and equity prices. Themethodology is quite general; it allows defaultprobabilities and recovery rates to be correlatedand dependent on the macroeconomy. Thus, theresulting reduced-form model integrates marketand credit risk with correlated defaults.

The article makes two contributions: First, asstated, the methodology explicitly incorporatesequity prices in the reduced-form estimation pro-cedure. For a fractional recovery rate, using debtprices alone allows the estimation of only theexpected loss—that is, the multiplicative product ofthe recovery rate times the (pseudo) default proba-bilities (Duffie and Singleton). The introduction ofequity prices enables one to separately estimatethese quantities. The procedure used to includeequity in the reduced-form model is commonlyused in portfolio theory (Duffie 1988). Simply

Robert Jarrow is a professor at Johnson Graduate Schoolof Management at Cornell University, Ithaca, New York.

A

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stated, the equity price is viewed as the presentvalue of future dividends and a resale value. Thefuture resale value is consistent with the existenceof equity price bubbles (Jarrow and Madan 2000).Given recent market experience with Internetstocks, such an inclusion is necessary for accurateestimation of bankruptcy parameters.

Second, because debt markets are notoriouslyilliquid, especially in comparison with equity mar-kets, the methodology explicitly incorporatesliquidity risk in the reduced-form model and theestimation procedure. Liquidity risk is introducedthrough the notion of a “convenience yield,” a well-studied concept in the literature on commoditiespricing that is consistent with an arbitrage-free butincomplete debt market. Liquidity risk introducesan important and necessary additional random-ness into the yield spread between risky-bondprices and U.S. Treasury securities. This additionalrandomness allows for the decomposition of thecredit spread into a liquidity-risk component and acredit-risk component. The liquidity-risk adjust-ment is needed to accurately estimate the bank-ruptcy parameters from credit spreads.

Model StructureThis section introduces the notation and economicstructure of the reduced-form model. The assump-tion is that markets are frictionless with no arbi-trage opportunities. Markets are not assumed to becomplete or perfectly liquid, nor are price bubblesexcluded.

A probability space underlies the economy inwhich P represents the “statistical,” “objective,” or“empirical” probability distribution. I will use theterm “statistical” probability distribution. The sta-tistical probability distribution is the probabilitydistribution that standard statistical proceduresdraw inferences about when using historical mar-ket prices. Alternatively stated, it is that probabilitydistribution generating the observed debt andequity prices in the economy.

Trading can take place anytime during theinterval Traded are default-free zero-coupon bonds of all maturities, equities, and risky(defaultable) zero-coupon bonds of all maturities.The following notation characterizes these pricesand the subsequent estimation procedure.

Let p(t,T) represent the time t price of a default-free dollar paid at time T, where Thedefault-free forward rates, f(t,T), are implicitlydefined by

(1)

where u is the variable of integration. The spot rateof interest is given by r(t) = f (t,t).

The notation for prices of the risky zero-coupondebt requires more structure than given so far. Con-sider a company issuing debt and equity to financeits operations. For the moment, suppose that its debttakes the form of zero-coupon bonds of perhapsdifferent seniorities (in the event of default).

Let v(t,T: i) represent the time t price of a prom-ised dollar of seniority i to be paid by this companyat time T, where . The debt is riskybecause if the company defaults prior to time T, thepromised dollar may not be paid.

Let τ represent the first time that this companydefaults ( is possible if the company does notdefault). The default time, τ, is a random variable.Let

(2)

denote the point process indicating whether or notdefault has occurred prior to time t. At this stage inthe analysis, the point process can be a generalstochastic process. Let λ (t) represent its randomintensity process. The time t intensity process, λ (t)∆,gives the approximate probability of default for thiscompany over the time interval (t, t + ∆).2

Without loss of generality, if default occurs, letthe zero-coupon bond of seniority i receive a frac-tional recovery of δi(τ)v (τ−,T : i) dollars, where0 ≤ δi(τ) and τ− represents an instant before default.After default, the debt is worth only a fraction of itspredefault value. The recovery fraction, δi(t), israndom. At this point, the fractional recovery rateassumption is without loss of generality becausethe recovery rate process is completely arbitrary.When a specific parameterization for the recoveryrate is imposed for empirical estimation, thisassumption becomes restrictive. Note that therecovery rate fraction, δi(t), completely specifies theseniority status of the debt issue. The greater theseniority of the debt issue, the larger the recoveryrate—everything else being constant.

This formulation is the standard structureimposed in reduced-form models. Now, considerthe formulation of equity prices. For analysis, think-ing of equity as the debt issue of “last” seniority isuseful. In this analogy, equity pays “coupons,”called dividends, and pays a liquidating payoff attime T∗ for .3 The time t value of thesepromised payments equals the value of the equity(per share) and is denoted by ξ (t). This procedureis standard for characterizing equity prices in port-folio theory (see Duffie). The equityholders receive

(0 T, ).

0 t T T .≤ ≤ ≤

p t T,( ) ef t u,( ) ud

t

T

∫–

,=

0 t T T≤ ≤ ≤

τ T>

N t( ) 1 t r≥( )=

1 if t τ≥0 otherwise

=

0 T∗ T≤ ≤

Default Parameter Estimation Using Market Prices

September/October 2001 3

these payments unless the company defaults. Ifdefault occurs, the equityholders get a fractionalrecovery payment on these promises equal toδe(τ)ξ (τ–), where δe(τ) ≡ 0.4

Now, some notation is needed for these divi-dend payments. The regular dividends are paid attimes 1, 2, . . ., T∗ and are denoted by Dt at time t.Assume these dividends are deterministic quanti-ties paid unless the company defaults prior to thedividend-payout date.5 This formulation implicitlydefines T∗ as that date to which this deterministicdividend assumption is true. For many stocks, T∗

will necessarily be set equal to a year (or less).The liquidating dividend is paid at time T∗

unless default occurs prior to that date. This divi-dend consists of a random payoff of L(T∗ ). Let S(t)represent the time t present value of this liquidatingdividend conditional upon no default prior to time t.

Finally, some evidence indicates that stockprices contain a “bubble” or “monetary value”component (see Jarrow and Madan). An exampleis the recent price growth of Internet stocks.6 So, letθ(t) represent this time t bubble component in thestock price.

Given this setup, one can easily see that the pershare equity value at time t is given by7

(3)

where v(t, j :e) represents a zero-coupon bond ofseniority e (equity) issued by this company and Djis the dividend paid at time j.

The equity value at time t is equal to the presentvalue of the liquidating dividend plus a bubblecomponent plus the present value of the regulardividend payments. The present value of the regu-lar dividend payments is seen to be equivalent to aportfolio of risky zero-coupon bonds of a particularseniority. The seniority is that of equity, with afractional recovery rate of δe (t). If default occurs,then the value of the equity drops to zero becausethe fractional recovery rate on the dividends (liqui-dating and regular) is assumed to be zero. Thebond’s default parameters are explicitly includedwithin this component of the equity’s price.

A special case of Equation 3—the announceddividend model—is worth mentioning because ithas been previously used in the option-pricing lit-erature. The dividend at time j, Dj, is zero unlessj = tx, where tx is the next ex-dividend date. Let tabe the announcement date of the next dividendpayment. Then, Equation 3 becomes

(4a)

and

. (4b)

The interpretation of Equations 4 is that the divi-dend is known and deterministic only after it isannounced (and prior to its payment). This modelis similar to a model used for the valuation of equityoptions with a known and discrete dividend (seeJarrow and Turnbull 1996). In this example, the dateT∗ is the same as tx but only for t ≥ ta; otherwise,T∗ = 0. This example clarifies the robustness of thedeterministic dividend assumption and the inter-action between the definition of T∗ and the specifi-cation of the regular dividends, Dt.

Risk-Neutral ValuationThis section presents the valuation formulas used inthe estimation of the bankruptcy parameters. Underthe assumption of no arbitrage, standard arbitragepricing theory implies that a probability distributionQ exists such that present values are computed bydiscounting at the spot rate of interest and thentaking an expectation with respect to Q.8 For exam-ple, using this characterization, one can write

, (5)

where Et(•) is conditional expectation with respectto Q at time t.

Equation 5 is the standard risk-neutral pricingrelationship satisfied by default-free zero-couponbonds. Applying this valuation methodology toprices of risky zero-coupon bonds and the liquidat-ing dividend produces

(6)

and

(7)

The risky-debt value is composed of two parts. Thefirst is the present value of the promised paymentin case of default. The second is the present value

ξ t( ) S t( ) θ t( ) Djv t j : e,( )j t≥

T∗

∑+ + if t τ<

0 if t τ ,≥

=

ξ t( )S t( ) θ t( ) Dtx

v t tx:e,( )+ + if t ta tx ) and t τ<,[∈

0 if t ta tx ) and t τ≤,[∈

=

ξ t( )S t( ) θ t( )+ if t ta tx ) and t τ<,[∉

0 if t ta tx ) and t τ≥,[∉

=

p t T,( ) Et er u( ) ud

t

T

∫–=

v t T :i,( ) Et δi τ( )v τ– T : i,( )er u( ) ud

t

τ

∫–1 τ T≤( )=

+1er u( ) ud

t

T

∫–1 T τ<( )

S t( ) Et L T∗( )er u( ) ud

t

T∗

∫–1

T∗ τ<( ) .=

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of the promised payment if default does not occur.The present value of the liquidating dividend issimilar. The only difference in these two expres-sions (Equations 6 and 7) is that L (T∗ ) is random forthe liquidating dividend whereas the promisedrisky-debt payment of $1 is not.

Risk-neutral valuation provides for no analo-gous expression for the bubble component, θ(t). Thereason is that one cannot write the bubble compo-nent as a discounted expectation (see Jarrow andMadan).9

Using a result from Duffie and Singleton (The-orem 1), under mild conditions, one can rewriteEquations 6 and 7 as10

(8)

and

(9)

where default has not occurred before or at time tand λ (u) is the intensity process under risk-neutralmeasure Q. I call this intensity process the “pseudoprobability of default.”

The importance of this simplification cannot beoverstated. The price of the risky zero-coupon bondcan again be written as an expected discountedvalue, but in this case, the discount factor is the spotrate of interest adjusted for the expected loss indefault, λ (t)[1 – di(t)]. A similar statement appliesfor the present value of the liquidating (random)dividend.

As pointed out by Duffie and Singleton, thepseudo probability of default always appears in thisvaluation formula for risky debt as part of a multi-plicative product. It is always multiplied by thefractional loss in default [1 – δi(t)]. Hence, debtprices allow one to estimate only the product of thepseudo probability of default times the fractionalloss, not the pseudo probability of default alone. Theintroduction of the equity valuation process as inEquation 3, in conjunction with Equations 8 and 9,overcomes this difficulty because the fractional lossfor equity is known (a priori) and equal to 1—that is,1 – δe(t) = 1 as δe(t) = 0.11 Thus, a joint estimation ofpseudo default probabilities and recovery rates, inwhich each is identified separately, is possible byusing both debt and equity prices. A procedure forthis joint estimation is discussed later.

The Liquidity PremiumThis section adds the liquidity premium into thepreceding model formulation (Equation 8). Liquid-

ity risk is an important consideration in the pricingof risky debt, and its inclusion is motivated by twoobservations. First, debt prices are difficult toobtain because of the sparsity of secondary-markettrading. In fact, at the time this article was written,the most frequent data available were monthlyobservations—the Wisconsin database (Warga1999). Compare this lack of frequency with theready availability of daily transaction data forequity prices. Second, a study by Schwartz (1998)indicated that, even for these monthly bond data,the number of outliers (measured relative to similardebt issues) is significant. One can attribute theseoutliers to the illiquidity in the market.

Corporate debt issues, analogous to Treasur-ies, can be used in repurchase agreements (repos)as collateral. Therefore, at times, particular corpo-rate bonds are in short supply, asking prices arehigh, and special repo rates are low (Rooney 1998).In these cases, one cannot buy the bond at reason-able prices and liquidity causes bond prices to be“too high.” Conversely, in times of credit scaresand high market volatility, corporate bonds (orparticular sovereigns—e.g., Russian bonds) can besold only at discount prices. In these cases, onecannot sell bonds at reasonable prices and liquiditycauses bond prices to be “too low.”

Although Duffie and Singleton suggested amodification of Equations 8 to incorporate liquidityrisk, they did not give a formal argument justifyingits inclusion. This section provides such a formaljustification based on a related argument used forconvenience yields in Treasury securities that iscontained in Jarrow and Turnbull (1997). The justi-fication is consistent with no-arbitrage opportuni-ties but an incomplete debt market.

Consider a market where one cannot syntheti-cally construct a particular credit-risky zero-couponbond (hereafter called “zero”) with price vl(t,T: i).The subscript l indicates that the market has a liquid-ity problem. Given an identical credit-risky zerowith no liquidity problems and price v(t,T:i), thefollowing no-arbitrage relationships hold:

v(t,T: i) ≤ vl(t,T: i) in a shortage so one cannotreadily buy, and

v(t,T: i) ≥ vl(t,T: i) in a glut so one cannotreadily sell.

The argument is simple. When one cannot synthet-ically construct the bond on the right side of thisexpression, the act of arbitrage cannot force equal-ity between the two prices. Thus, a function γi(t,T)exists such that

vl(t,T: i) = e–γi(t,T)v(t,T:i ). (10)

v t T: i,( ) Et 1er u( ) λ u( ) 1 δi u( )–[ ]+{ } ud

t

T

∫–

=

S t( ) Et L T∗( )er u( ) λ u( )+[ ] ud

t

T∗

∫–

,=

Default Parameter Estimation Using Market Prices

September/October 2001 5

In a shortage, when one cannot readily buy therisky bond, γi(t,T) ≤ 0 and the function –γi(t,T ) hasthe interpretation of being a positive convenienceyield obtained from holding (storing) the credit-risky zero. In the case of shortages of the riskybond, (special) repo rates are low and storing thebond provides benefits. This case is exactly analo-gous to positive convenience yields associatedwith storage of other commodities used in produc-tion (such as oil).

When a glut exists and one cannot readily sellthe risky bond, γi(t,T) ≥ 0 and the function –γi(t,T)is interpreted as a negative convenience yieldobtained from holding (storing) the credit-riskyzero. In this case, holding the bond in a portfolioproduces a negative externality, which is an implicitstorage cost exactly analogous to the negative con-venience yields associated with storage of spoilablecommodities.

For equity markets, liquidity costs are assumedto be zero. Here again, as for recovery rates, equityforms the base case against which debt’s bank-ruptcy parameters can be estimated.

Model of the Stock-Price BubbleFor simplicity, I model the stock-price bubble com-ponent as a random process that is proportional tothe present value of the liquidating dividend, as inthe following expression:

(11)

where µθ(u) ≥ 0 is the continuous return in the stockprice resulting from the bubble component.

Combined with Equation 11, Equation 3 can berewritten as

. (12)

Equation 12 represents a convenient decompositionof the stock price into its underlying components.

Implicit EstimationFor the joint implicit estimation of the recovery ratesand the (pseudo) default probabilities, additionalstructure needs to be imposed on both these quan-tities. Following Lando (1998), assume that thedefault process follows a Cox process in which λ (t)and δi(t) are predetermined functions of a vector ofobservable state variables, represented by X(t) for

. The vector X(t) is a multidimensional sto-chastic process. The state variables within X (t)

could include the spot interest rate, foreign curren-cies, GNP measures, or a market index.

Formally,

λ (t) = λ [t, X(t)] (13a)

and

δi(t) = δi[t, X(t)]. (13b)

Similarly, the liquidity discount, the bubble compo-nent of the stock price, and the present value of theliquidating value of the equity can depend on thesame X(t) state variables. That is,

γi (t,T) = γi [t,T, X(t)], (14a)

S(t) = S[t, X(t)], (14b)

and

µθ(t) = µθ[t, X(t)]. (14c)

Of course, prior to estimation, these deterministicfunctions need to be specified.

The estimation is performed at an arbitrarytime t by using cross-sectional and time series data.Given is a collection of observable prices of default-free zeros, p(t,T ), for various t and T and riskyzeros, vl (t,T : i) for various i, t, and T. Also availableare an observable equity price ξ (t), observable (pre-dictable) dividends D1, . . ., DT∗ , and observablestate variables X (t). Note that this procedure isconditioned on the fact that the company is not yetin default. With these observables, the left side ofthe following system of equations is determined:

vl(t,T: i) = vl{t,T:i,λ [t,X(t)], δi[t,X(t)], γl[t,T,X(t)]} (15a)

for various i and T and

(15b)

On the right side of these equations, the depen-dence of the risky debt and equity prices on the(pseudo) default probabilities (λ), the recovery rate(δi), the liquidity premium (γi), the bubble compo-nent (µθ), and the liquidating dividend [S (t)] ismade explicit. Notice that the equity prices do notdepend on a recovery rate or a liquidity premium.

The two systems of equations (Equations 15aand 15b) can be estimated in three stages. Stage Oneis to estimate the parameters in the system of equa-tions given by Equation 15a for the risky-debt prices.This system can be estimated cross-sectionally at aparticular time t. Here, as long as the number ofequations is at least as large as the number ofunknowns, the system can be inverted to obtainestimates of the parameters (a sum-of-squared-error-minimizing procedure may be necessary). As

θ t( ) S t( ) eµθ u( ) ud

0

t

∫1– ,=

ξ t( ) S t( )eµθ u( ) ud

0

t

∫Djv t j : e,( )

j t≥

T∗

∑+ if t τ<

0 if t τ≥

=

0 t T≤ ≤

ξ t( ) Djv t j:e λ t X t( ),[ ],,{ }j t≥

T∗

∑=

+ S t X t( ),[ ] eµθ u X u( ),[ ] ud

0

t

∫ .

Financial Analysts Journal

6 ©2001, AIMR®

indicated previously, however, the recovery rateand the default probability always appear as a prod-uct and are inseparable in this system.

In this estimation procedure, the prices ofrisky zero-coupon bonds were assumed to beobservable. But this is not usually the case.Instead, risky coupon-bearing bond prices areobservable. These procedure can be easily modi-fied, however, to incorporate this difference. Thismodification is Stage Two in the estimation proce-dure. There are two basic approaches: One is tofirst strip out the prices of the zeros from the pricesof the coupon bonds before applying the estima-tion procedure. Various techniques are availablefor this approach (see Schwartz). The alternativeis to apply the joint estimation procedure directlyto the prices of the coupon-bearing bond by usingthe fact that a risky coupon bond is a portfolio ofrisky zero-coupon bonds. In Equation 15a, the leftside would become the observable risky coupon-bearing bond and the right side would become asummation of the relevant zero-coupon bondsweighted by the coupon payments.

Stage Three is to estimate the parameters inEquation 15b for equity prices. For this estimation,the condition is that default has not yet occurred (i.e.,t < τ). This single equation can be estimated only byusing time-series analysis. The unknowns are the(pseudo) default probability, the bubble component,and the liquidating dividend. Obtaining a solutionrequires at least as many time-series observations ofξ(t) as there are unknown parameters in λ [t,T,X(t)],µθ[t, X(t)], and S[t, X(t)]. Then, given the estimatesfor the equity price’s default parameters, the recov-ery rates for the various seniority levels of the debtissues can be easily inferred from the debt-priceparameters estimated earlier.

An alternative to this three-stage procedure isa single-stage procedure that jointly estimates all ofthe parameters from the larger system of equationsby using coupon bonds and equity prices together.The difference between the two approaches is thatthe joint estimation procedure constrains theparameters to be identical in the two marketswhereas the three-stage procedure does not.

Next, I provide further description of thisestimation procedure for a special case of theformulation.

A Practical Empirical SpecificationTo estimate the system of equations represented byEquations 15, one still needs to specify the variousfunctions in Equations 13 and 14. These functionsare specified here, and without loss of generality,

we assume that the prices of risky zero-couponbonds are observable.

For a practical but realistic empirical specifica-tion of the reduced-form model, let there be twostate variables X (t) describing the system: (1) thespot rate of interest and (2) a general indicator ofthe health of the economic system—the cumulativeexcess return on a market index (as measured fromsome initial date).12

Now, an arbitrage-free evolution for thesestate variables needs to be specified. First, considerthe spot rate of interest, r(t). For illustration pur-poses, I use a single-factor model with determinis-tic volatilities that is sometimes called the extendedVasicek model (see Vasicek 1977 and Heath, Jar-row, Morton 1992). The term-structure evolution isdescribed by the evolution of the spot rate of inter-est under risk-neutral measure Q:

, (16)

where a = a mean-reversion parameter, a con-

stant that is not 0σr = volatility of the spot rate, where σr > 0

is a constant = a deterministic function of t

W(t) = a standard Brownian motion under Qinitialized at W(0) = 0.

In Equation 16, the spot rate of interest followsa mean-reverting process under the risk-neutralmeasure. As shown in Heath, Jarrow, and Morton(1992), to match an arbitrary initial forward-ratecurve, one must set

. (17)

Combined with Equation 16, the evolution for thespot rate of interest can be rewritten as

(18)

Note that the spot rate of interest is normally dis-tributed in the extended Vasicek model.

The second state variable is related to a marketindex, denoted M(t). The evolution for the marketindex is assumed to satisfy

dM(t) = M(t)[r(t)dt + σmdZ (t)], (19)

where the volatility of the market index, σm, isconstant and Z(t ) is a standard Brownian motionunder Q initialized at Z (0) = 0 that is correlatedwith W (t ) as dZ (t)dW (t ) = ϕrmdt with ϕrm (thecorrelation between the spot rate and the marketindex) a constant.13

dr t( ) a r t( ) r t( )–[ ] dt σrdW t( )+=

r t( )

r t( )f 0 t,( ) ∂f 0 t,( ) ∂ t σr

2+⁄ 1 e 2at––( ) 2a⁄[ ]+

a-----------------------------------------------------------------------------------------------------------=

r t( ) f 0 t,( )σr

2 eat–

1–( )2

2a2---------------------------------- σre

a t u–( )–W u( ) .d

0t

∫+ +=

Default Parameter Estimation Using Market Prices

September/October 2001 7

The market index follows a geometric Brown-ian motion with drift r(t) and volatility σm. The driftmust be the spot rate of interest under the risk-neutral measure. The evolutions of the market indexand the spot rate of interest are correlated, with

(20)

For subsequent use, note the market index pro-cess in its integral form:

. (21)

Given observation dates 1, 2, 3, . . ., t, Equation 21can be solved for Z(t ) as a function of Z(t – 1). Thissolution is given by

(22)

One sees here that Z(t) is a measure of the cumula-tive excess return per unit of risk (above the spot rateof interest) on the market index.14 Z(t) becomes thesecond state variable chosen because it is normallydistributed (as is the spot rate of interest underEquation 18).

Now, the assumption about the bankruptcyparameters and the recovery rate are imposed:

λ (t) = λ0 + λ1r(t) + λ2Z (t) (23a)

and

δi(t) = δi, (23b)

where λ0, λ1, λ2, and δi are constants.In this formulation, the (pseudo) probability of

default is assumed to be a linear function of the statevariables r(t) and Z(t). This assumption implies thatnegative default rates, λ (t) < 0, are possible. None-theless, given the tractability of the subsequentexpressions, this step is an acceptable first approx-imation. Its validity awaits empirical investigation.Also, the fractional recovery rate is assumed to bea constant, which is a first approximation that iseasily relaxed.

Given these expressions, Appendix A showsthat the prices of the default-free zero-coupon bondand the risky zero-coupon bond can be rewritten as

(24)

and

(25)

where no default has occurred at or prior to time t,

, (26a)

(26b)

(26c)

(26d)

and

(26e)

To understand these pricing formulas, onemust first recognize that the randomness in theirvalues across time occurs for two reasons. The firstcause is randomly changing default-free rates.This cause enters through the µ1(t,T) term inEquations 24 and 25 and, in particular, through theterm involving the current forward rates,

. The second reason, which appliesonly to the risky debt, is the possibility of default,in which case the risky-bond price in Equation 25drops from vl(τ –,T : i) to vl(τ,T : i) = δivl(τ –,T : i).

For a better understanding of the randomnessarising from changing default-free rates, one cantransform these equations to the equation in Jar-row and Turnbull (2000) as follows. Appendix Ashows that

(27a)

where

cor dM t( )M t( )

--------------- dr t( ), ϕmdt.=

M t( ) M 0( )er u( ) u 1 2⁄( )σm

2t– σmZ t( )+d

0

t

∫=

Z t( ) Z t –1( )=

M t( )/M t –1( ) r u( ) u 1/2( )σm2 ud

t –1t

∫+dt –1t

∫–log

σm-------------------------------------------------------------------------------------------------------------------------------+

for t 1≥ and Z 0( ) 0.=

p t T,( ) Et er u( ) ud

t

T

∫–=

eµ1 t T,( )– σ1

2t T,( )/2+

=

vl t T : i,( ) eγi t T,( )–

=

E× t er u( ) λ0 λ1 r u( ) λ2Z u( )+ +[ ] 1 – δi( )+{ } ud

t

T

∫–

eγi t T,( )–

p t T,( )=

eλ0 1 – δi( ) T – t( )– λ1 1 – δi( )µ1 t T,( )– 2λ 1 1 – δi( ) λ1

21 – δi( )2

+ δ12

t T,( )/2+

×

eλ 2 1 – δi( )Z t( ) T – t( )– 1 λ1 1 – δi( )+[ ] λ 2 1 – δi( )ϕ

r mη t T,( ) T – t( )3λ22

1 – δi( )2/6+ +× ,

µ1 t T,( ) f t u,( ) ub u T,( )2 ud

2--------------------------

t

T

∫+d

t

T

∫=

σ12

t T,( ) b u T,( )2u ,d

t

T

∫=

b u t,( )σr 1 e

a t –u( )––[ ]

a----------------------------------------- ,=

η t T,( ) ρ v s,( ) vd

t

min s u,( )

∫ sd

t

T

∫

ud

t

T

∫=

σr

a3-----

1 ea T – t( )–

–[ ]–σr

a2-----

+=

e a T – t( )–× T – t( )σr

2a------

T – t( )2,+

ρ v s,( ) σrea s – v( )– .=

f t u,( ) udt

T

∫

µ1 t T,( ) c t T,( ) b t T,( )r t( )σr

------------------------- ,+=

Financial Analysts Journal

8 ©2001, AIMR®

(27b)

In Equations 27, c(t,T) and b(t,T) are deterministicfunctions of time. The randomness in Equation 27ais a result of spot interest rate r(t). SubstitutingEquations 27a and 27b into pricing Equation 24gives the valuation formula in Jarrow and Turnbull(2000). This substitution also shows that the pricesof default-free and risky zero-coupon bonds areMarkov in r(t). This Markov structure facilitatescomputation, and it is an advantage of using theextended Vasicek model.

For understanding the implications of Equa-tion 25 for the yield spread between prices of riskybonds and Treasury prices, the first step is toimplicitly define the yield spread at time t for bondT with a particular maturity, χ (t,T: i), as

(28)

Using Equation 27a for µ1 (t,T) and the definition ofthe yield spread to Treasuries given in Equation 28,

(29)

The yield spread consists of a term denoting liquid-ity risk, γi(t,T), and all the remaining terms denot-ing credit risk. The yield spread is random becausethe liquidity-risk component is random and thecredit-risk component contains the spot rate ofinterest, r(t), and the cumulative excess return onthe market index, Z(t), both of which are random.

To complete the empirical formulation, speci-fication is needed of the functional form for theliquidity discount, the bubble component, and theliquidating dividend as given in Equation 14. Thisis the task to which we now turn.

Assume that

(30)

where σL > 0 is a constant and wL(t) is a Brownianmotion under the martingale measure Q withdZ(t)dwL(t) = ϕmLdt and dW(t)dwL(t) = ϕrLdt, whereϕrL and ϕmL are constants.

In Equation 30, L(t) represents the time t liqui-dation value of the company’s assets less liabilities.This liquidation value can be viewed as the marketvalue of a portfolio containing the company’s assetsand liabilities. This portfolio’s value, if held by adefault-free entity, evolves through time accordingto Equation 30. For simplicity, this evolution isassumed to be a geometric Brownian motion underthe martingale measure with a drift rate equal to thespot rate, r(u). The value of this portfolio at time T ∗will be L(T ∗ ).

In the present case, however, L(t) is held by acompany that can default prior to time T ∗ . If thecompany defaults, then because of bankruptcycosts (lawyer’s fees, lost sales, etc.), the liquidationvalue declines by the fraction (1 – δe). The implica-tion is that the present value of the liquidationvalue to an equity holder in the risky company isless than or equal to the present value of the under-lying portfolio of assets and liabilities to a default-free agent; that is

(31)

Here, it can be shown (the proof is in AppendixA) that

(32)

Unfortunately, this expression for the present valueof the liquidating dividend has the unknown L(t) onthe right side. Therefore, this form of the presentvalue expression can provide no additional infer-ence about the default parameter process, λ (t),because there are more unknowns than observables.This insight motivates the following transformationof Equation 15b.

Let ∆ correspond to a discrete change in time.Taking logarithms of Equation 32 and subtractingtime t – ∆ from time t gives

c t T,( ) f 0 u,( ) b 0 u,( )2

2---------------------+

t

T

∫=

dub t T,( ) f 0 t,( ) b 0 t,( )2/2[ ]+{ }

σr----------------------------------------------------------------------------–× .

vl t T : i,( )p t T,( )

---------------------- eχ t T : i,( ) T – t( )–

.=

χ t T : i,( ) T – t( ) γi t T,( ) λ0 1 – δi( ) T – t( )+=

λ1 1 – δi( )c t T,( )+

2λ1 1 – δi( ) λ12 1 – δi( )2+[ ]σ 1

2 t T,( )2

-------------------------------------------------------------------------------------–

λ1 1 – δi( )b t T,( )r t( )σr

--------------------------------------------------+

λ2 1 – δi( )Z t( ) T – t( )+

1 λ1 1 – δi( )+[ ]λ 2 1 – δi( )ϕ rmη t T,( )–

T – t( )3λ22 1 – δi( )2

6--------------------------------------------- .–

L T ∗( ) L t( )er u( ) u –(1/2) σL

2 u + σL wL u( )dt

T ∗

∫dt

T ∗

∫dt

T ∗

∫,=

S t( )1 τ t>( ) Et L T ∗( )er u( ) λ u( )+[ ] ud

t

T ∗

∫

=

Et L T ∗( )er u( ) ud

t

T ∗

∫≤

L t( ) .=

S t( ) L t( )p t T∗,( )-------------------=

eλ1σ1

2 t T∗,( )– λ1σLϕrL b u T ∗,( ) u λ2ϕrm η t T ∗,( )– λ2σLϕmL T∗ – t( )2/2–dt

T∗

∫–×

v t T∗ :e,( ) .×

Default Parameter Estimation Using Market Prices

September/October 2001 9

(33)

where

Next, using the evolution of the liquidationvalue as given in Equation 30 produces

(34)

This evolution is under the martingale measure. Using Girsanov’s theorem, we can change the

martingale measure to the statistical probabilitymeasure. The change transforms the originalBrownian motion to a new Brownian motion underthe statistical measure and an adjustment for a riskpremium,

(35)

where is a Brownian motion under the sta-tistical measure and ΘL(u) is the liquidationvalue’s risk premium.

Using this change of measure produces

(36)

w h e re t h e e r ror t e r m s , , for all t are independent identically

and normally distributed terms with zero meanand variance of σL

2∆.Substitution of Equation 35 into Equation 33

yields

(37)

For estimation purposes, the excess return onequity can be written as

(38)

(The proof is in Appendix A.) In this nonlinear regression equation for the

excess return on the equity, the coefficients give thedefault parameters. To complete this estimation, amodel for the risk premium and the bubble compo-nent is required. For example, if one assumes thatthe risk premium can be approximated by using acapital asset pricing model and that the bubblecomponent can be approximated with the varianceof the stock price, then

(39)

and the system is easily estimated with only twoextra parameters, β0 and β1, which denote, respec-tively, the systematic risk of the market portfolioand the risk premium for the bubble component.

logξ t( ) T ∗

j t≥Djv t j : e,( )∑–

ξ t – ∆( ) T ∗j t – ∆≥

Djv t – ∆ j : e,( )∑–------------------------------------------------------------------------------------

log L t( )L t – ∆( )------------------=

µθ u( ) u log ψ t T ∗,( )ψ t – ∆ T ∗,( )-----------------------------+d

t – ∆

t

∫+

v t T ∗ : e,( )v t – ∆ T ∗ : e,( )----------------------------------log ,+

ψ t T*,( ) λ1σ12 t T ∗,( )– λ1σLϕrL b u T ∗,( ) ud

t

T∗∫–=

λ2ϕrmη t T ∗,( )–λ2σLϕmL T ∗ t–( )2

2--------------------------------------------- .–

log L t( )L t – ∆( )------------------ r u( ) u

12---

σL2 ∆–d

t – ∆

t

∫=

σL wL t( ) wL t – ∆( )–[ ] .+

wL t( ) wL t( )=

ΘL u( ) u,d

0

t

∫+

wL t( )Q

log L t( )L t – ∆( )------------------ r u( ) σLΘL u( )+[ ] ud

t – ∆

t

∫=

12---

σL2∆– ε t – ∆( ) ,+

ε t – ∆( ) σL w[ L t( )≡wL t – ∆( ) ]–

logξ t( ) T∗

j t≥Djv t j : e,( )∑–

ξ t – ∆( ) T∗j t – ∆≥

Djv t – ∆ j : e,( )∑–------------------------------------------------------------------------------------ r u( ) ud

t – ∆

t

∫–

σLΘL u( ) 12---

σL2

– µθ u( )+ ud

t – ∆

t

∫=

log ψ t T ∗,( )ψ t – ∆ T ∗,( )-----------------------------+

log v t T ∗ : e,( )v t – ∆ T ∗ : e,( )----------------------------------+

ε t – ∆( ).+

logξ t( ) T ∗

j t≥Djv t j : e,( )∑–

ξ t – ∆( ) T ∗j t – ∆≥

Djv t – ∆ j : e,( )∑–------------------------------------------------------------------------------------

r t – ∆( )∆–

λ0∆– λ1b t – ∆ T ∗,( )2

2------------------------------- ∆ log p t T ∗,( )

p t – ∆ T ∗,( )----------------------------–

–≈

λ12 b t – ∆ T ∗,( )2

2------------------------------- ∆ λ 1σLϕrLb t – ∆ T ∗,( )∆–+

λ2 Z t( ) T ∗ – t( ) Z t – ∆( ) T ∗ –t – ∆( )–[ ]–

λ22 T ∗ –t( )2∆

3------------------------------- λ2σLϕmL T ∗ – t( )∆–+

σLγL t – ∆ X t – ∆( ),[ ] 12---

σL2– µθ t – ∆ X t – ∆( ),[ ]+

∆+

ε t – ∆( ) .+

σLγL t – ∆ X t – ∆( ),[ ] 12---

σL2– µθ t – ∆ X t – ∆( ),[ ]+

∆

β0 log M t( )M t – ∆( )-------------------- β1 log ξ2 t( )

ξ2 t – ∆( )--------------------- ,+=

Financial Analysts Journal

10 ©2001, AIMR®

Finally, one can model the liquidity discountas a first-order Taylor series approximation for amore general function of the state variables:

γi(t,T) = γi0 + γi

1r(t) + γi2Z (t). (40)

Combining all of these empirical specificationsinto Equations 15a and 38 gives the following systemof equations, which contains both cross-sectionaland time-series observations:

(41a)

for various i, T, and t and

(41b)

for various t.Equation 41a is the debt-pricing equation,

whereas Equation 41b is the equity-pricing equa-

tion. The solution to this system of equations can beobtained via a nonlinear regression. The solutiondepends on the initial forward rate curve, f(0,T); theterm-structure evolution parameters, a and σr; themarket index parameters, ϕrm and σm; and the liq-uidating dividend parameters, ϕrL, ϕmL, and σL.Additional parameters to be estimated are thedefault process coefficients, λ0, λ1, and λ2; the recov-ery rate, δi; the liquidity discount coefficients, γi

0, γi1,

and γi2; and the bubble/risk premium coefficients,

β0 and β1. This system of equations needs to beestimated through the use of both cross-sectionaland time-series data.

ConclusionI presented a new procedure for implicit estimationof a liquidity premium, the recovery rate, and the(pseudo) default probabilities using debt andequity prices. This new procedure is quite general.It allows the default process to be correlated acrosscompanies and to depend on the state of the mac-roeconomy. It allows debt markets to be illiquid andequity markets to contain bubbles. Its empiricalevaluation, however, awaits subsequent research.

Although the procedure formally estimates thepseudo or risk-neutral default intensities, if thereduced-form model is properly specified, there isgood reason to believe that the statistical and risk-neutral default intensity functions are equal—which they will be if default risk is idiosyncraticafter properly conditioning on macroeconomicvariables (see Jarrow, Lando, and Yu 1999). Thishypothesis of idiosyncratic default risk is intuitivelyplausible; its validation awaits subsequent research.The methodology presented here is consistent withthis properly conditioned reduced-form model.

vl t T : i,( )

e γ0i– γ1

ir t( ) γ2

iZ t( )+ += p t T,( )

e× λ 0 1 – δi( ) T – t( )– λ 1 1 – δi( )µ1 t T,( )– 2λ1 1 – δi( ) λ 12

1 – δi( )2+ σ1

2t T,( ) /2+

e× λ2 1 – δi( )Z t( ) T – t( )– 1 λ1 1 – δi( )+[ ]λ 2 1 – δi( )ϕrmη t T,( ) T – t( )3λ2

21 – δi( )2

/6+ +

logξ t( ) T*

j t≥Djv t j : e,( )∑–

ξ t – ∆( ) T*j t – ∆≥

Djv t – ∆ j : e,( )∑–

------------------------------------------------------------------------------------------

r t – ∆( )∆–

λ0∆– λ1b t – ∆ T*,( )2

2------------------------------ ∆ log p t T*,( )

p t – ∆ T*,( )---------------------------–

–≈

λ12 b t – ∆ T*,( )2

2------------------------------ ∆ λ 1σLϕrLb t – ∆ T*,( )∆–+

λ2 Z t( ) T ∗ –t( ) Z t – ∆( ) T ∗ –t ∆–( )–[ ]–

λ22 T*– t( )2∆

3----------------------- λ2σLϕmL T ∗ –t( )∆–+

β0log M t( )M t – ∆( )-------------------- β1log ξ2 t( )

ξ2 t – ∆( )---------------------+ +

ε t – ∆( )+

Default Parameter Estimation Using Market Prices

September/October 2001 11

Appendix A. ProofsIn this appendix, I provide the derivations of Equations 24, 25, 27a, and 27b and lay out the computation ofthe equity model, Equation 38.

Derivation of Equations 24 and 25. From Equation 21,

(A1)

where

ρ(v, s) = σr e–a(s – v)

and

Define

(A2)

After changing the order of integration, a direct computation yields

(A3)

and

(A4)

Substitution gives

(A5)

A direct computation gives

(A6)

r s( ) f t s,( ) b t s,( )2

2----------------- ρ v s,( ) W v( ) ,d

t

s

∫+ +=

b t s,( ) ρ t v,( ) vdt

s

∫=

σr1 e

a s – t( )––

a------------------------- .=

X1 r s( ) sdt

T

∫≡ f t s,( ) sdt

T

∫=

b t s,( )2sd

2------------------------

t

T

∫+

ρ v s,( ) W v( )d s.dt

s

∫t

T

∫+

b t s,( )2 sd2

------------------------t

T

∫b v T,( )2 vd

2--------------------------

t

T

∫=

ρ v s,( ) W v( )d sdt

s

∫t

T

∫ b v T,( ) W v( ).dt

T

∫=

r s( ) sdt

T

∫ f t s,( ) sdt

T

∫=

b v T,( )2vd

2--------------------------

t

T

∫+

b v T,( ) W v( ).dt

T

∫+

µ1 t T,( ) Et r s( ) sdt

T

∫≡ f t s,( ) sdt

T

∫=

b v T,( )2vd

2--------------------------

t

T

∫+

Financial Analysts Journal

12 ©2001, AIMR®

and

(A7)

Define Following Parzen (1962, p. 81),

(A8)

But

(A9)

Now,

(A10)

Substitution and simplification yield

(A11)

σ12

t T,( ) vart r s( ) sdt

T

∫≡

b v T,( )2v.d

t

T

∫=

X2 Z u( ) u.d

t

T

∫≡

µ2 t T,( ) Et X2( )≡ Et Z u( )[ ] udt

T

∫=

Z t( ) udt

T

∫=

Z t( ) T – t( );=

σ22

t T,( ) vart X2( )≡ 2 u – t( ) ud vdt

v

∫t

T

∫=

T – t( )3

3-----------------;=

σ12 t T,( ) covt X1 X2,( )≡ Et X1X2( ) Et X1( )Et X2( )–=

Et r s( ) s Z s( ) sdt

T

∫dt

T

∫ f t s,( ) sb v T,( )2

vd2

--------------------------t

T

∫+dt

T

∫ Z t( ) T – t( )–=

Et r s( )Z u( )[ ] sd u f t s,( ) sb v T,( )2

vd2

-------------------------t

T

∫+dt

T

∫ Z t( ) T – t[ ] .–dt

T

∫t

T

∫=

Et r s( )Z u( )[ ] Et f t s,( ) b t s,( )2

2----------------- ρ v s,( ) W v( )d

t

s

∫+ + z v( ) Z t( )+dt

u

∫

=

ϕrm ρ v s,( ) v f t s,( ) b t s,( )2

2-----------------+ Z t( ).+d

0

min s u,( )

∫=

f t s,( ) b t s,( )2

2-----------------+ Z t( ) ud sd

t

T

∫t

T

∫ Z t( ) T – t( ) f t s,( ) b t s,( )2

2-----------------+ sd

t

T

∫=

Z t( ) T – t( ) f t s,( ) sb v T,( )2 vd

2-------------------------

t

T

∫+dt

T

∫ .=

σ12 t T,( ) ϕrm ρ v s,( ) vdt

min s u,( )

∫ sdt

T

∫

udt

T

∫

.=

Default Parameter Estimation Using Market Prices

September/October 2001 13

Given that (X1, X2) is bivariate normal, we have (see Hogg and Craig 1970)

(A12)

where A and B are arbitrary constants.

µ1 ≡ Et(X1),

µ2 ≡ Et(X2),

σ12 ≡ vart(X1),

σ22 ≡ vart(X2),

and

σ12 ≡ covt(X1, X2).

Then, for Equation 24, using Equation A12, we get

(A13)

where

A = –[1 + λ1(1 – δi)].

This gives the desired result.Next, for Equation 25,

(A14)

where

A = –[1 + λ1(1 – δi)]

and

B = – λ2(1 – δi).

Equation A12 gives the desired result.

Derivation of Equation 27a and 27b. Under the spot rate model of Equation 18, it can be shown thatthe arbitrage-free forward-rate process is given by

(A15)

where

We are interested in evaluating the following integral of forward rates:

(A16)

Et eAX1 BX2+ eµ1A µ2B σ1

2A

2 2σ12AB σ22B

2+ +( )/2+ +=

Et e1 λ1 1 – δi( )+[ ] r u( ) ud

t

T

∫–

Et eAX1 =

eµ1A σ12A

2/2+ ,=

Et e1 λ1 1 – δi( )+[ ] r u( ) u λ2 1 – δi( )– Z u( ) ud

t

T

∫dt

T

∫–

Et eAX1 BX2+ ,=

f t u,( ) f 0 u,( ) α v u,( ) v ρ v u,( ) W v( ) ,d0

t

∫+d0

t

∫+=

α v u,( ) ρ v u,( ) ρ v s,( ) s.dv

u

∫=

f t u,( ) udt

T

∫ f 0 u,( ) u α v u,( ) vd0

t

∫ udt

T

∫+dt

T

∫=

ρ v u,( ) W v( )d0

t

∫ u.dt

T

∫+

Financial Analysts Journal

14 ©2001, AIMR®

Given the definitions of α (v,u), ρ(v,u), and b(v,u), the following facts can be proven by direct computation:

(A17)

and

(A18)

Using the first of these facts, we can show that

(A19)

But we know from Equation 18 that

(A20)

Using this observation and Equation A18, we have

(A21)

Direct substitution of these observations produces

(A22)

Substitution of this integral into the definition of µ1(t , T), together with the fact that

, gives Equations 27a and 27b.

Equity Model Computations. From Equations 9 and 30, we have

(A23)

Using Equation 23 gives

(A24)

To evaluate the expectation, we use Equation A12 with the following identifications:

α v u,( ) vdt

u

∫b t u,( )2

2------------------=

ρ v u,( ) W v( )d0

t

∫ udt

T

∫ b t T,( ) ρ v t,( ) W v( ).d0

t

∫=

α v u,( ) vd0

t

∫ udt

T

∫ α v u,( ) vd0

u

∫ u α v u,( ) vdt

u

∫ udt

T

∫–dt

T

∫=

b 0 u,( )2

2------------------- u

b t u,( )2

2------------------

t

T

∫–d u.dt

T

∫=

r t( ) f 0 t,( )–b 0 t,( )2

2-------------------– ρ v t,( ) W v( ).d

0

t

∫=

ρ v u,( ) W v( )d

0

t

∫ ud

t

T

∫b t T,( ) r t( ) f 0 t,( )– b 0 t,( )2/2–[ ]

σr-----------------------------------------------------------------------------------.=

f t u,( ) udt

T

∫ f 0 u,( ) ub 0 u,( )2

2-------------------- u

b t u,( )2

2------------------ ud

t

T

∫–dt

T

∫+dt

T

∫=

b t T,( ) r t( ) f 0 t,( )– b 0 t,( )2/2–[ ]σr

------------------------------------------------------------------------------------ .+

b t u,( )2 u/2d

t

T

∫ b u T,( )2 u/2d

t

T

∫=

St Et L T∗( )er u( ) λ u( )+[ ] ud

t

T ∗

∫–

L t( )e1/2( ) σL

2ud

t

T ∗

∫–Et e

λ u( ) u σL wL u( )dt

T ∗

∫+dt

T ∗

∫–.= =

St L t( )e1/2( ) σL

2u λ0 ud

t

T ∗

∫–dt

T ∗

∫–Et e

λ1r u( ) u λ2Z u( ) u σL wL u( )dt

T ∗

∫+dt

T ∗

∫–dt

T ∗

∫–.=

A 1,≡

x λ1 r u( ) u λ2 Z u( ) udt

T ∗

∫–dt

T∗

∫–≡ λ1X1 λ2X2 ,–=

B 1,≡

y σL wL u( ).dt

T∗

∫≡

Default Parameter Estimation Using Market Prices

September/October 2001 15

The expectation is , where

and

But

(A25)

Next,

(A26)

because But

(A27)

because

eµx µy 1/2( )σx

2 σxy 1/2( )σy2+ + + +

µx λ1µ1 t T ∗,( )– λ2Z t( ) T∗ t–( ),–=

µy 0,=

σx2 λ1

2σ12 t T ∗,( ) 2λ1λ2σ12 t T∗,( )

λ22 T* t–( )3

3---------------------------,+ +=

σy2 σL

2 T* t–( ),=

σxy covt λ1 r u( ) u λ2 Z u( ) u, σL wL u( )dt

T ∗

∫dt

T∗

∫–dt

T∗

∫–=

λ1σLcovt r u( ) u, wL u( )dt

T ∗

∫dt

T ∗

∫– λ2σLcovt Z u( ) u, wL u( )dt

T∗

∫dt

T∗

∫ .–=

covt r u( ) u, wL u( )dt

T ∗

∫dt

T ∗

∫ covt b u T∗,( ) W u( ), wL u( )dt

T ∗

∫dt

T ∗

∫=

ϕrL b u T∗,( ) udt

T∗

∫ .=

covt Z u( ) u, wL u( )dt

T∗

∫dt

T ∗

∫ Et Z u( ) u wL u( )dt

T∗

∫dt

T ∗

∫ ,=

Et wL u( )d

t

T ∗

∫ 0.=

Et Z u( ) u wL u( )dt

T ∗

∫dt

T∗

∫ Et Z u( )– Z t( )– Z t( )+[ ] u wL u( )dt

T ∗

∫dt

T ∗

∫

=

Et Eu Z v( ) wL v( )dt

T∗

∫dt

u

∫ udt

T ∗

∫

=

Et Z v( ) wL v( )dt

u

∫dt

u

∫ udt

T ∗

∫

,=

Eu wL u( )d

u

T ∗

∫ 0.=

Financial Analysts Journal

16 ©2001, AIMR®

Finally,

(A28)

Thus,

(A29)

Substitution of these results into the expression for S (t) gives

. (A30)

From Equation 25, we have

(A31)

Substitution gives

(A32)

Next, we derive the expression for the excess return on equity. From Equation 12 and Equation A30, ifwe take natural logarithms and then the difference from time t – ∆ to time t, we obtain

(A33)

In the following identifications, terms of order ∆p for p ≥ 2 are omitted:

λ0[(T∗ – t) – (T∗ – t – ∆)] = λ0∆. (A34)

Et Z v( ) wL v( )dt

u

∫dt

u

∫ udt

T∗

∫

Et Z v( ) wL v( )dt

u

∫dt

u

∫ udt

T ∗

∫

=

ϕmL u t–( ) u.dt

T ∗

∫=

covt Z u( ) u, wL u( )dt

T ∗

∫dt

T ∗

∫ϕmL T∗ t–( )2

2-------------------------------- .=

St L t( )eλ0 T ∗ t–( )– λ1µ1 t T ∗,( )– λ2Z t( ) T∗ t–( )– 1/2( )λ1

2σ12

t T∗,( ) λ1λ2σ12 t T∗,( ) λ 22

T∗ t–( )3/6+ + +=

e

λ– 1σLϕrL b u T∗,( ) u λ2σLϕm L T ∗ t–( )2/2 (1/2)σL2

T∗ t–( )+–d

t

T ∗

∫×

v t T ∗ : E,( )p t T∗,( )

-------------------------eλ1σ1

2t T∗,( )– λ2σ12 t T ∗,( )–

eλ0 T∗ t–( )– λ1µ1 t T∗,( )– λ2Z t( ) T ∗ t–( )– 1/2( )λ 1

2σ12

t T∗,( ) λ1λ2σ12 t T∗,( ) λ22

T∗ t–( )3/6+ + +.=

St L t( )v t T ∗ : E,( )

p t T∗,( )--------------------------e

λ1σ12

t T∗,( )– λ 2σ12 t T∗,( )– λ1σLϕ rL b u T ∗,( ) u λ2σLϕmL T ∗ t–( )2/2–dt

T ∗

∫–

.=

logξ t

T ∗j t≥

Djv t j : e,( )∑–

ξ t – ∆T∗j t – ∆≥

Djv t – ∆ j : e,( )∑–------------------------------------------------------------------------- log

Lt

Lt – ∆-----------

µθ u( ) u λ0 T∗ t–( ) T∗ t– ∆–( )–[ ]–dt – ∆

t

∫+=

λ1 µ1 t T ∗,( ) µ1 t ∆ T∗,–( )–[ ]– λ2 Z t( ) T∗ t–( ) Z t – ∆( ) T∗ t– ∆–( )–[ ]–

12---

λ12 σ1

2 t T∗,( ) σ12 t – ∆ T∗,( )–[ ] λ 1λ2 σ12 t T∗,( ) σ12 t – ∆ T∗,( )–[ ]+ +

λ22 T∗ t–( )3 T∗ t– ∆–( )3–

6--------------------------------------------------------- λ1σLϕrL b u T∗,( ) u b u T∗,( ) ud

t – ∆

T∗

∫–dt

T∗

∫–+

λ2– σLϕmLT∗ t–( )2 T∗ t– ∆–( )2–

2--------------------------------------------------------- .

Default Parameter Estimation Using Market Prices

September/October 2001 17

Next, we have

(A35)

(A36)

(A37)

(A38)

(A39)

(A40)

Combined, these identifications produce

(A41)

which completes the derivation of Equation 38.

λ1 µ1 t T ∗,( ) µ1 t – ∆ T∗,( )–[ ] λ 1 logp t T∗,( )

p t – ∆ T∗,( )----------------------------–

b v T ∗,( )2vd

2------------------------------

t – ∆

t

∫+

=

λ1 logp t T∗,( )

p t – ∆ T ∗,( )----------------------------–

b t – ∆ T∗,( )2∆2

-----------------------------------+

;≈

12---

λ12 σ1

2 t T∗,( ) σ12 t – ∆ T ∗,( )–[ ] 1

2---

λ12 b v T∗,( )2 vd

t – ∆

t

∫=

λ12 b t – ∆ T∗,( )2∆

2------------------------------------------ ;≈

λ1λ2 σ12 t T∗,( ) σ12 t – ∆ T∗,( )–[ ] λ 1λ2ϕrm ρ v s,( ) vd sd u ρ v s,( ) vd sd udt – ∆

min s u,( )

∫t – ∆

T

∫t – ∆

T

∫–dt

min s u,( )

∫t

T

∫t

T

∫=

λ1λ2ϕrm ρ v s,( ) vd sd u ρ v s,( ) vd sd udt – ∆

min s u,( )

∫t – ∆

T

∫t – ∆

T

∫–dt – ∆

min s u,( )

∫t

T

∫t

T

∫≤

λ1λ2ϕrm ρ v s,( ) vd sd u λ1λ2ϕrm ρ v s,( ) vd sd udt – ∆

t

∫t – ∆

t

∫t – ∆

t

∫≤dt – ∆

min s u,( )

∫t – ∆

t

∫t – ∆

t

∫=

λ1λ2ϕrm ρ t – ∆ t,( )∆3;≈

λ22 T∗ –t( )3 T∗ –t – ∆( )3–

6------------------------------------------------------ λ2

2 2 T∗ –t( )2∆ T∗ –t( )2∆2– ∆3–6

-------------------------------------------------------------------------=

λ22 T∗ –t( )2∆

3------------------------ ;≈

λ1σLϕrL b u T ∗,( ) u b u T∗,( ) ud

t – ∆

T ∗

∫–d

t

T ∗

∫ λ1σLϕrL b u T∗,( ) ud

t – ∆

t

∫=

λ1σLϕrL b t – ∆ T∗,( )∆;≈

λ2σLϕmLT∗ –t( )2 T∗ –t – ∆( )2–

2------------------------------------------------------ λ2σLϕmL

2 T∗ –t( )∆ ∆2–2

-------------------------------------=

λ2σLϕmL T∗ –t( )∆.≈

logξ t

T ∗j t≥

Djv t j : e,( )∑–

ξ t – ∆T ∗j t – ∆≥

Djv t – ∆ j : e,( )∑–------------------------------------------------------------------------- log

Lt

Lt – ∆-----------

µθ u( ) u λ0∆–dt – ∆

t

∫+=

λ1 – logp t T∗,( )

p t – ∆ T∗,( )--------------------------- b t – ∆ T∗,( )2 ∆

2---+

–

λ2 Z t( ) T∗ –t( ) Z t – ∆( ) T∗ –t – ∆( )–[ ]–

λ12

b t – ∆ T∗,( )2 ∆2--- λ2

2T∗ –t( )2 ∆

3--- λ1σLϕrL b t – ∆ T∗,( )∆–+ +

λ2σLϕmL T∗ –t( )∆.–

Financial Analysts Journal

18 ©2001, AIMR®

Notes1. See Jarrow and Turnbull (2000) for a review.2. The intensity process is defined under the risk-neutral prob-

ability. This statement will become clear after the nextsection, “Risk-Neutral Valuation.”

3. A convenient approach is to think of the liquidating payoffas the present value of all future dividends paid over thetime period (T∗ , ∞).

4. In fact, as the subsequent analysis will show, what is reallybeing assumed here is that δe(τ) is the minimal recoveryrate. Under this interpretation, all the subsequent recoveryrates will be relative to δe (τ).

5. As will be seen later, if the future dividends are random,they are included within the S(t) component—that is, thetime t present value of the liquidating dividend conditionalupon no default prior to time t.

6. For example, Money Magazine in April 1999 gave Yahoo’sP/E as 1,176.6 (p. 169).

7. Equation 3 is a simple no-arbitrage restriction that thepresent value of the sum of multiple cash flows equal thesum of the present values of the cash flows.

8. See Jarrow and Turnbull (1995). “No arbitrage” guaranteesthe existence but not the uniqueness of a probability mea-

sure Q. Without any additional hypotheses about the econ-omy, the uniqueness of Q is equivalent to markets beingcomplete (see Battig and Jarrow 1999). In incomplete mar-kets, equilibrium (requiring additional hypotheses) guar-antees the uniqueness of Q. The uniqueness of Q is essentialfor estimation.

9. This insight implies that the techniques for inferring theasset’s volatility, by using Merton’s model of risky debt, aremisspecified in the presence of bubbles.

10. The mild condition is that the value of the debt and equitynot jump at the time of default. Given the fractional recov-ery rate process, this assumption is reasonable.

11. In the event that δe(τ) is not zero, the estimated fractionalloss for equity will be the ratio of [1 – δi(τ)]/[1 – δe(τ)].

12. Higher-dimensional systems can be easily accommodated,but this extension is left to subsequent research.

13. The assumption that M(t) earns the riskless return under Qimplies that the economy is also arbitrage free with respectto inclusion of an additional traded asset, the market index.

14. The variable Z(t) can be estimated using past observationsof M(t).

ReferencesBattig, R., and R. Jarrow. 1999. “The Second FundamentalTheorem of Asset Pricing—A New Approach.” Review ofFinancial Studies, vol. 12, no. 5 (December):1219–35.

Duffie, D. 1988. Security Markets: Stochastic Models. San Diego,CA: Academic Press.

Duffie D., and K. Singleton. 1999. “Modeling Term Structures ofDefaultable Bonds.” Review of Financial Studies, vol. 12, no. 4(October):197–226.

Heath, D., R. Jarrow, and A. Morton. 1992. “Bond Pricing andthe Term Structure of Interest Rates: A New Methodology forContingent Claim Valuation.” Econometrica, vol. 60, no. 1(January):77–105.

Hogg, R., and A. Craig. 1970. Introduction to MathematicalStatistics. 3rd ed. New York: Macmillan.

Jarrow, R. 1998. “Current Advances in the Modeling of CreditRisk.” Derivatives: Tax, Regulation, Finance, vol. 3, no. 5 (May/June):196–202.

Jarrow, R., and D. Madan. 2000. “Arbitrage, Martingales andPrivate Monetary Value.” Journal of Risk, vol. 3, no. 1: 73-90.

Jarrow, R., and S. Turnbull. 1995. “Pricing Derivatives onFinancial Securities Subject to Credit Risk.” Journal of Finance,vol. 50, no. 1 (March):53–85.

———. 1996. Derivative Securities. Cincinnati, OH: South-Western.

———. 1997. “An Integrated Approach to the Hedging andPricing of Eurodollar Derivatives.” Journal of Risk and Insurance,vol. 64, no. 2 (June):271–299.

———. 2000. “The Intersection of Market and Credit Risk.”Journal of Banking and Finance, vol. 24, no. 1 (January):271–299.

Jarrow, R., D. Lando, and S. Turnbull. 1997. “A Markov Modelfor the Term Structure of Credit Risk Spreads.” Review ofFinancial Studies, vol. 10, no. 2 (April):481–523.

Jarrow, R., D. Lando, and F. Yu. 1999. “Default Risk andDiversification: Theory and Applications.” Working paper,Cornell University.

Lando, D. 1998. “On Cox Processes and Credit Risky Securities.”Review of Derivatives Research, vol. 2, no. 2/3:99–120.

Merton, R.C. 1974. “On the Pricing of Corporate Debt: The RiskStructure of Interest Rates.” Journal of Finance, vol. 29, no. 2(May):449–470.

Parzen, E. 1962. Stochastic Processes. San Francisco, CA: Holden-Day.

Rooney, M. 1998. “Credit Default Swaps (TransferringCorporate and Sovereign Credit Risk).” Global Fixed IncomeResearch (October). Merrill Lynch.

Schwartz, T. 1998. “Estimating the Term Structures of CorporateDebt.” Review of Derivatives Research, vol. 2, no. 2/3:193–230.

Vasicek, O. 1977. “An Equilibrium Characterization of the TermStructure.” Journal of Financial Economics, vol. 5, no. 2(November):177–188.

Warga, A. 1999. Fixed Income Data Base. University of Houston,College of Business Administration (www.uh.edu/~awarga/comp.html).

Financial Analysts Journal September/October 2001 Vol. 57, no. 5, pp. 75-92

Default Parameter Estimation using Market Prices

Corrigendum

by Robert Jarrow1

1 Thanks are expressed to Clive Saunders for pointing out these typos and errors.

1

Given are the corrected equations. For most of the equations given, some signs were reversed.

∆∆∆∆ξ

ξ

∆

)t(r)e:j,t(vD)t(

)e:j,t(vD)t(log

j

*T

tj

j

*T

tj −−

−∑−−

∑−

−≥

≥ (38)

).t())]t(X,t()2/1())t(X,t([

)t*T(2/)t*T(

)]t*T)(t(Z)t*T)(t(Z[*)T,t(b

2*)T,t(b

*)T,t(p*)T,t(plog

2*)T,t(b

2LLL

mLL222

2

2rLL1

221

2

10

∆ε∆∆∆µσ∆∆γσ

∆ϕσλ∆λ

∆∆λ∆∆ϕσλ

∆∆λ∆

∆∆λ∆λ

θ

−+−−+−−−+

−+−−

+−−−−−−+

−−

−

+

−++≈

( ) ( ) ( ) ( )( )( )( ) ( ) [ ] ( ) 6/1tT)T,t(111)tT)(t(Z)1(

2)T,t(112)T,t(1)tT(1

)t(Z)t(rl

2i

22

3rmi2i1i2

21

2i

21i11i1i0

i2

i1

i0

e

e

)T,t(pe)i:T,t(v

δληϕδλδλδλ

σδλδλµδλδλ

γγγ

−−+−−++−−−

−+−+−−−−−

−−−

•

•=

(41.a)

∆λ∆∆

∆∆ξ

ξ

∆

0

j

*T

tj

j

*T

tj )t(r)e:j,t(vD)t(

)e:j,t(vD)t(

log +≈−−

−−−

−

∑

∑

−≥

≥ (41.b)

)t())t(/)t(log())t(M/)t(Mlog(

)t*T(2/)t*T(

)]t*T)(t(Z)t*T)(t(Z[*)T,t(b

2*)T,t(b

*)T,t(p*)T,t(plog

2*)T,t(b

2210

mLL222

2

2rLL1

221

2

1

∆ε∆ξξβ∆β

∆ϕσλ∆λ

∆∆λ∆∆ϕσλ

∆∆λ∆

∆∆λ

−+−+−+

−+−−

+−−−−−−+

−−

−

+

−+

[ ] ).t(Z2)s,t(b)s,t(fdv)s,v(

)t(Z)v(dZ)v(dW)s,v(2)s,t(b)s,t(fE

))u(Z)s(r(E

2)u,smin(

trm

u

t

s

t

2t

t

++

=

+

++

=

∫

∫∫

ρϕ

ρ (A9)

2

.)v(dW)t,v()T,t(bdu)v(dW)u,v(t

0r

T

t

t

0∫∫ ∫ =

ρ

σρ (A18)

2/)t*T(du*)T,u(b

6/)t*T(*)T,t(*)T,t()2/1()t*T)(t(Z*)T,t()t*T(t

2mLL2

*T

trLL1

3221221

21

212110

e

e)t(LS

−−∫−

−+++−−−−− •=

ϕσλϕσλ

λσλλσλλµλλ

(A30)

.2/])t*T()t*T[(

]du*)T,u(bdu*)T,u(b[6/])t*T()t*T[(

*)]T,t(*)T,t([*)]T,t(*)T,t([)2/1(

)]t*T)(t(Z)t*T)(t(Z[*)]T,t(*)T,t([

)]t*T()t*T[(du)u(LL

log)e:j,t(vD

)e:j,t(vDlog

22mLL2

*T

t

*T

trLL1

3322

12122121

21

21

2111

0

t

tt

t*T

tjjt

*T

tjjt

∆ϕσλ

ϕσλ∆λ

∆σσλλ∆σσλ

∆∆λ∆µµλ

∆λµ∆ξ

ξ

∆

∆θ

∆

∆∆

+−−−−

∫−∫−+−−−+

−−+−−+

+−−−−−−−−

+−−−−+

=

−−

−

−

−−

−≥−

≥∫

∑

∑

(A33)

∆λ∆λ 00 )]t*T()t*T[( −=+−−− (A34)

].2/*)T,t(b*)T,t(p

*)T,t(plog[

]2/dv*)T,v(b*)T,t(p

*)T,t(plog[*)]T,t(*)T,t([

21

t

t

21111

∆∆∆

λ

∆λ∆µµλ

∆

−−

−

−≈

−

−

−=−− ∫− (A35)

.2/*)T,t(b

dv*)T,v(b)2/1(*)]T,t(*)T,t([)2/1(

2

t

t

221

21

21

21

∆∆

λ∆σσλ∆

−−≈

−=−− ∫− (A36)

(A38) .2/)t*T(

6/])t*T(3)t*T(3[6/])t*T()t*T[(22

2

32222

3322

∆λ

∆∆∆λ∆λ

−−≈

−−−−−=+−−−

3

.*)T,t(b

du*)T,u(b]du*)T,u(bdu*)T,u(b[

rLL1

t

trLL1

*T

t

*T

trLL1

∆∆ϕσλ

ϕσλϕσλ∆∆

−−≈

∫−=∫−∫−− (A39)

.)t*T(2/])t*T(2[2/])t*T()t*T[(

mLL2

2mLL2

22mLL2

∆ϕσλ∆∆ϕσλ∆ϕσλ

−−≈−−−=+−−−

(A40)

.)t*T(*)T,t(b2/)t*T(2/*)T,t(b

)]t*T)(t(Z)t*T)(t(Z[

]2/*)T,t(b*)T,t(p

*)T,t(plog[

du)u(LL

log)e:j,t(vD

)e:j,t(vDlog

mLL2

rLL122

222

1

2

21

0

t

tt

t*T

tjjt

*T

tjjt

∆ϕσλ∆∆ϕσλ∆λ∆∆λ

∆∆λ

∆∆∆

λ

∆λµ∆ξ

ξ

∆θ

∆

∆∆

−+−+−−−−

+−−−−−

−+

−

++

++

=

−−

−

∫∑

∑

−−

−≥−

≥

(A41)

4