00 01 04 a survey of contingent claims approaches to risky debt valuation

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debt. Much work still remains in sorting through the

numerous theories and finding those that can be sup-

ported empirically.

Because of the intricate properties of risky debt, a

model characterizing risky debt value will always require

a fair amount of complexity. The reason for this complexitylies in the number of factors driving risky debt value cou-

pled with the interaction of these factors. The first factor

driving corporate debt value is the required rate of return

on risk-free debt. An important question concerns how to

incorporate the risk-free rate into a risky debt valuation

model. Some models for valuing risky debt (and, by exten-

sion, deriving the term structure of credit risk1) assume the

default-risk-free rate is deterministic while others include

a model specification for a stochastic risk-free rate. This

survey will consider both types of approaches; however, the

risk-free rate models themselves will not be covered in

detail. Interested readers should refer to Chan, Karolyi,

Longstaff, and Sanders [1992], Vetzal [1992], and Back

[1997] for surveys of the risk-free term structure literature.

Traditionally, U.S. government securities have been used

as proxies for finding this r isk-free rate in the real world.

Ideally, the rate we would like to use is a corporate risk-

free rate closer to the rate earned on very high grade cor-

porate bonds. In this paper, the term risk-free raterefers to

the default risk-free rate. Other risks (e.g., liquidity) may

imply a different risk-free rate; however, the focus in this

survey is modeling credit r isky debt. Consequently, iden-

tifying the appropriate default risk-free rate is essential toproper model fitting and testing.

The second factor driving corporate debt value is

the probability the issuer will default on its obligation. This

factor tends to be the focus of most models of risky debt

value. A common approach to determining default prob-

abilities compares the value of an issuers assets with the

level of debt in the issuers capital structure. This approach

uses the firms market asset value as the fundamental fac-

tor determining the firms default probability. Duffie and

Singleton [1998] define this modeling framework as the

structural approach to risky debt valuation. BSM pre-

sent the classical version of this type of structural model.In this approach, default is assumed to occur when the

market value of assets has fallen to a sufficiently low level

relative to the issuers total liabilities. Essentially, the issuer

(more accurately, the issuers shareholders) receives an

option to default on its debt. The issuer will likely exer-

cise this option when its assets no longer have enough value

to cover its debt obligation. Different versions of the

model reflect varying assumptions about the constraints

governing when a firm can default. Merton [1974] assumes

default can occur only at the maturity date of the firms

outstanding debt. If the value of the firms asset are less than

the total debt, the debt holders receive the value of the

firm. Beginning with Black and Cox [1976], other authors

have extended this model to include certain kinds ofindenture conditions (e.g., safety covenants) effectively

allowing for default prior to the maturity of the debt. In

the case of debt issued with safety covenants, an issuer may

be forced into reorganization when its asset market value

falls too close to (or below) the principal value of its debt.

The key characteristic shared by structural models is their

reliance on economic arguments for why firms default.

Duffie and Singleton [1998] define a second,

related approach called reduced form. In this approach,

the time of default is modeled as an exogenously defined,

intensity process eliminating the need to have default

depend explicitly on the issuers capital structure. Note that

since the default process can be endogenously derived, any

structural model can be recast (with some modifications

to make the default stopping time inaccessible) as a

reduced-form model making the structural modeling

approach a special case of the reduced-form approach. The

strength of reduced-form models is also their weakness.

Divorcing the issuer from the intensity process enables

modeling default without much information about why

the issuer defaults. Herein lies the strength. Unfortu-

nately, data are poor and not well understood. Modeling

default without theoretical guidance runs the risk of bothignoring market information and drawing erroneous con-

clusions without the tools to discover the appropriate

explanation. Herein lies the weakness. Simply said:

Reduced-form models eliminate the need for an eco-

nomic explanation of default.

Jarrow and Turnbull [1995] introduce one of the

first reduced-form models where the default (stopping)

time is exponentially distributed. They extend this model

in Jarrow, Lando, and Turnbull [1997] by assuming the

default time follows a continuous-time Markov chain

with default occurring the first time the chain hits the

absorbing (default) state. They use S&P transition prob-ability matrices as input. In this case, the default process

is modeled as a finite state Markov process in the firms

credit ratings.

Duffie and Singleton [1998] introduce a slightly

different type of reduced-form model where the default

process is modeled as a stochastic hazard rate process

where the hazard rate indicates the conditional rate of the

arrival of default. Nothing is assumed about the factors

54 A SURVEY OF CONTINGENT-CLAIMS APPROACHES TO RISKY DEBT VALUATION SPRING 2000

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generating this hazard rate. Because these types of mod-

els do not require a causal relationship between firm

value and default, they are significantly more dependent

on the quality of the credit-spread data than the structural

models. Moreover, the parameters estimated in these

models will likely exhibit instability over time. The advan-tage of these models is they will likely fit a particular set

of credit spread data (regardless of the datas quality) and

price different derivatives and debt securities in an arbi-

trage-free, consistent manner at any particular point in

time. The parameters, however, will only be useful for

analyzing positions held for short periods of time (prob-

ably intra-day) making them suspect when used for mark-

ing to market (or to model) a portfolio of debt securities

or credit derivatives where the horizon of interest extends

over several days, weeks, months, or years. Reduced-

form models will also fall short when determining nor-

mative prices for debt and credit derivatives. Because

reduced-form models lack economic explanations for

value, these models will have difficulty determining where

a price ought to be.

Credit spread data tend to be noisy and unavailable

for certain tenors and credit qualities in the credit-risk

term structure. As a result, simple one factor reduced-form

models may have difficulty fitting the data. While struc-

tural models have economic reasoning to guide their

implementation and determine how best to fill holes in

the data, reduced-form models can only add parameters.

Consequently, the more sophisticated (and more com-plicated) reduced-form models include stochastic pro-

cesses for other factors (e.g., liquidity, loss-given-default)

possibly driving credit spreads. While empirical analysis

of actual defaults can provide clues regarding modification

of a structural model, mathematical tractability tends to

be the criterion for modification of a reduced-form

model. On the positive side, reduced-form models elim-

inate the need to specify the priority structure of a firms

liabilities and allow for exogenous assumptions regarding

observables. While reduced-form models are likely to fit

better any particular set of credit spread data (including

data full of noise) than structural models, they break thelink between the economics of firm behavior and the

event of default. Reduced-form models take the eco-

nomics out of the risky debt valuation problem.

The preceding discussion focuses on reduced-form

models in the context of straightforward debt valuation.

The default intensity process can be parameterized with

default probability and credit migration probability data

to arrive at models useful in the context of credit deriva-

tives, risky debt with complicated indentures, sovereign

debt, consumer debt, and other types of risky debt that

defy structural economic explanations of value. In these

cases, the reduced-form modeling expands our ability to

value instruments with credit risk.

In both reduced-form and structural models, athird factor affecting corporate debt value is the expected

loss-given default (LGD). The traditional specification

makes LGD a function of the firms asset value. Other

models assume it is related to the face value of debt or

assume it is stochastic. One would expect LGD to be

related to the debts collateralization and pr iority. Unfor-

tunately, the theory and empirical evidence regarding

LGD is sparse. Altman and Kishore [1996], Carty and

Lieberman [1996], and Carty, Keenan, and Shotgrin

[1998] have published the most comprehensive data on

historical LGD experience. Research into recovery rates

continues to constitute virgin territory.

In all these models, corporate debt value depends

on the terms and conditions written into the securitys

indenture. These terms and conditions define the bound-

ary conditions necessary to derive appropriate valuation

formulae. The challenge arises from analyzing each one

of these factors, understanding their interrelationships, and

coping with the ever increasing complexity of corporate

debt design.

To summarize, a structural model requires char-

acterization of the following:

1. Issuers asset value process.

2. Issuers capital structure.

3. Loss given default.

4. Terms and conditions of the debt issue.

5. Default risk-free interest rate process.

6. Correlation between the default-risk-free interest

rate and the asset price.

(A sampling of important extensions to the orig-

inal BSM structural models can be found in Black and

Cox [1976]; Brennan and Schwartz [1977, 1978, 1980];

Geske [1977]; Ingersoll [1976, 1977a, 1977b]; Leland[1994]; Leland and Toft [1996]; Longstaff and Schwartz

[1995]; and Zhou [1997].)

A reduced-form model requires characterization of

the following:

1. Issuers default (bankruptcy) process.

2. Loss given default (can also be specified as a

stochastic process).

SPRING 2000 THEJOURNAL OF RISK FINANCE 55




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3. Default-risk-free interest rate process.

4. Correlation between the default-risk-free interest

rate process and the default process.

(A sampling of important reduced-form models

can be found in Cathcart and El-Jahel [1998]; Duffieand Singleton [1998]; Jarrow and Turnbull [1995]; and Jar-

row, Lando, and Turnbull [1997].)

Let us now turn to a more detailed discussion of

these two families of models.

STRUCTURAL MODELS

BSM present closed-form solutions for the value

of debt and equity launching the contingent-claims

approach to valuing corporate securities. The economic

insight of BSMs model for risky debt valuation centers

on the relationship between a firms asset value and its obli-

gations. They point out that equity can be considered a

call option on the market value of the firms total assets

with a strike price equal to the book value of the firms

debt. By solving for the equity value and using the

accounting identity (in market value terms), total firm

assets = total debt + total equity, the value of corporate

debt can be determined. Risky debt can also be consid-

ered the combination of a default-free loan and a barrier

(if the firms asset value hits the default barrier, the firms

shareholders implicitly have the right to put the firm to

default) put option implicitly sold to the firm. Black andCox [1976] derive a closed form solution for risky debt

which accounts for this implicit barrier option.

Using BSMs framework, other authors have

extended the structural model by adding features to the

standard geometric Brownian motion characterization of

the firm value process (e.g., allowing the process to have

jumps). These modifications combined with more real-

istic boundary conditions (e.g., absorbing default bar-

rier) allow us to derive a valuation equation for more

realistic debt issues.2

Let us first outline the assumptions necessary for

deriving a BSM one-factor model of risky debt valuation:

1. Perfect Capital Markets: Perfect capital markets have

no transaction costs, no taxes, and no informational

asymmetries. Investors are price-takers.

2. Continuous trading.

3. The value of the firm behaves according to a stochas-

tic process where is the instantaneous expectedrate of return, and 2 is the instantaneous variance of

return on the firms assets. If applicable, payouts such

as coupons and dividends can also be defined per unit

time (locally certain and independent of the firms

capital structure).

4. is constant.

5. The instantaneous risk-free rate, r(t), is a knownfunction of time.

6. Management acts to maximize shareholder value.

7. Bankruptcy protection: Firms cannot file for

bankruptcy except when they cannot make required

cash payments. Perfect priority rules govern distri-

bution of assets to claimants at the time of liquidation.

8. Dilution protection: Unless all existing non-equity

claims are eliminated, no new securities other than

additional common equity can be issued. Equity

holders cannot negotiate arrangements on the side

with subsets of other claimants.

9. Perfect liquidity: Firms can sell assets as necessary to

make cash payouts with no loss in total value.

10. Boundary conditions: Indenture provisions such as

payouts and covenants determine the boundary con-

ditions for the partial differential equation that deter-

mines the value of the firms risky debt.

Given these assumptions, BSM demonstrate that

the value of a corporate liability is a function of the

firms value and time. This value satisfies a partial dif-

ferential equation that depends on the known sched-

ule of interest rates, the variance of firm value, and theterms outlined in the securitys indenture. Interestingly

enough, this approach to pricing results in the value of

corporate liabilities being independent of the equilib-

rium structure of risk and return.

Let us first consider a simple BSM model3 for a

firm with one class of equity and one class of zero-coupon

debt with face value, F. The firm pays no dividends.

Assume that the value, VA, of a firms assets4 can be char-

acterized by the following process (assumption 3 with the

firm making no payouts):5

dVA = VAdt + VAdz (1)

is the instantaneous expected rate of return onthe firms assets per unit time, 2 is the instantaneousvariance of the return on the firms assets per unit time,

and dz is a standard Wiener process.

Assume now that the value of the debt is a func-

tion only of the firms asset value, VA, the debts face

value, F, the risk-free interest rate, r, and the debts matu-





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rity of T,6 written as D(VA, F ,r, T). In order to derive this

valuation function, we need to make several more assump-

tions. First, assume that the price of a default-free discount

bond with the same tenor equals P(r, T) where r is the

prevailing risk-free rate of interest at the time of valuation.

For this discussion, we will assume that we know P(r, T)from the market rather than modeling its dynamics. (See

Longstaff and Schwartz [1995] and Briys and de Varenne

[1997] for derivations with stochastic processes assumed

for the risk-free interest rate.) Another key assumption

revolves around Modigliani and Millers theorem (see

Modigliani and Miller [1958]) regarding the indepen-

dence of a firms asset value and a firms capital structure.

We assume that changing the firms leverage does not

change the firms asset value.

Finally, we must specify the boundary conditions

for the debt contract. These conditions represent the spe-

cific nature of the debt contract and are important for

developing more realistic formulas (e.g., allowing for

default prior to maturity of the debt). In this model we

assume the debt holder receives whatever asset value

remains in the event of default and default can occur

only at the maturity of the debt. Otherwise, the debt

holder receives F. Moreover, the value of the debt can

never exceed the value of the firm and if there is no

value in the firm, the debt has no value. The boundary

conditions can be written as follows:

D(VA, T) = min(VA, F)D(V

A, t) V

A

D(0, t) = 0 (2)

Given the above assumptions, we solve the fol-

lowing partial differential equation to determine the value

of the debt:7

1/22V2AD

VAVA

+ rVAD

VA

- rD + Dt= 0 (3)

This partial differential equation can be solved

directly using standard techniques (such as Fourier trans-

forms or separation of variables). Merton [1974] bor-rows the results found in his earlier paper (Merton [1973])

and are found in Black and Scholes [1973]. He demon-

strates that by assuming equity to be a call option on the

value of the firm with strike price equal to the face value,

F, of the debt and using the identity that the debt value

equals the market value of assets less the market value of

equity, the same solution can be derived. BSM arrive at

a valuation for D in a formula similar to the following (the

function W(.) defines the Black-Scholes solution for the

value of a call option):

D(VA, F, r, T) = V

A- W(V

A, F, r, T)

D(VA, F, r, T) = V

AN(h

1) + FP(r, T)N(h

2) (4)

where

Equivalently, we can character ize the payoff to the

debtholder as min(VA(T)

, F) or the minimum of the value

of the firms assets at the maturity of the debt or the face

value of the debt. In other words, the debtholder receives

the face value of the debt unless the firm value falls to a

level below the face value. In this case, the debtholder

receives only what value is left in the firms assets. Recast-

ing the payoff as follows results in the same solution pre-

sented in Equation (4):

min(VA(T)

, F) = F - max(F - VA(T)

, 0) (5)

Notice that this characterization reflects the debt

payoff as a default-risk-free loan of the same amount plus

a short position in a put option on the firms assets with

strike price equal to the debts face value. The value of thedebt at any time prior to maturity will equal the value of

the default-risk-free loan less the value of this default

(put) option. The solution can then be written in the fol-

lowing form:

(6)

Notice that P(r, T)EL equals the value of a put

option in the Black-Scholes framework (see Black and

Scholes [1973] for derivation) written on VA

with strike

D V F r T P r T F EL V F r T

EL V F r T FN

V rT T F

T

V

P r TN

V rT T F

T

A A

A

A

AA

( , , , ) ( , )( ( , , , ))

( , , , )

(log log )

( , )

(log log )

=

= +

+ +

1

2

12

2

2

h

FP r T V T

Th h T

A

1

2

2 1

1

2=

= log( ( , ) / )





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price F expiring at time T. Moreover, the above equation

can be shown to equal Equation (4). (See Lando [1997]

for a discussion of this approach to the derivation.)

In the simplest sense, the value of the debt today

equals the expected value of payoffs at maturity. The

equation for debt presented earlier presents these payoffsin an intuitive way. The value of risky debt is the value

of otherwise similar, default risk-free debt less the pre-

sent value of the expected loss, EL (thus the strange vari-

able name for the put option value), given the company

defaults. The expected loss term can be divided into

two components:

1. The face value, F, of the debt multiplied by the risk

neutral probability of default (or the probability of

default if we lived in a risk-neutral world). Essentially,

this term represents the expected loss on the debt in

the case of no recovery.

2. The expected recovery in the event of default.

The key insight concerns the existence of risk-neu-

tral probabilities8 which allows us to discount future pay-

offs at the risk-free rate greatly simplifying our solution.

The difficulty with this formulation lies in empirically

finding all the necessary inputs.

Structural models rely on the economic argu-

ment that a firm defaults when its asset value drops to the

value of its contractual obligations. In this context, the

option pricing framework provides an elegant approachfor deriving the value of debt and equity. The mathe-

matical argument inside the normal distribution opera-

tor reflects this relationship. The risk-neutral probability

(note that the drift in the asset pr ice process in this result

equals the risk-free rate) of default equals the probabil-

ity of an event equal to the expected value of the firms

assets at maturity less the value of the debt obligation

divided by the volatility of the firms asset value. This

functional argument can be called the distance to default.

This distance to default is affected by the asset price pro-

cess, the nature of the firms debt obligations, and the

volatility of the firms asset value. More realistic modelscan be constructed by modifying the asset price process

(characterizes the expected growth rate of the firms asset

value and characterizes the firms asset volatility) or its

boundary conditions (characterizes the nature of the fir-

ms debt obligations). We now turn to a sampling of

these modifications.

An Even Simpler Structural Model

BSM presented the original structural version of

contingent-claim modeling of risky debt. In fact, we can

write down an even simpler structural model with simi-

lar characteristics. Again, we treat the firms debt as aderivative claim written on the underlying value of the

firm. The firm pays the face value of the debt with some

probability of no default. In this simpler model we make

an extra assumption that the firm pays some fraction of

the face value (instead of the asset value at maturity) in the

event of default. This binary option approach is not the

most realistic application of contingent-claims pricing;

however, the functional form resembles other more real-

istic approaches. More importantly, this simpler model is

more suitable for empirical testing.

We will begin with the case where default can only

occur at maturity and then modify the model to account

for default any time up to maturity. In the BSM model

presented above, the holders of the debt receive any value

in the firm up to the face value, F, of the debt. In this sim-

pler characterization, we fix the loss given default such that

it is a percentage of the debts face value. We will denote

this percentage as L. In other words, if the firm defaults

on its debt at any time up until the debts maturity date,

the debt holder receives (1 - L)F.

Consider a heuristic decomposition of the

expected loss, EL, presented in Equation (6). The first

term represents the expected loss with no recovery (i.e.,risk-neutral probability of default times the face value of

the debt). The second term reduces the loss by the amount

of the expected recovery. We can consider an analogous,

but simpler, valuation approach where the debt is char-

acterized as a binary option. In this case, the boundary

conditions are slightly modified and become the only

two possible payoffs at maturity:

DT

= F, if VT

FD

T= (1 - L)F, if V

T< F (7)

In this simpler case, the expected payoff at matu-rity is the payoff in the case of no default times the prob-

ability of no default plus the payoff in the case of default

times the probability of default. The probabilities in this

case are still the risk-neutral probabilities. I will define the

risk-neutral probability of default as Q. The resulting

formula is analogous to Equation (6):

D(VA, F, T) = P(r, T)(F - LFQ) (8)





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In fact, most models of risky debt simplify to this

form. The primary difference lies in the characterization

of the expected loss term. With this simpler framework,

we can parameterize the formula in a way that is more eas-

ily empirically tested. A major disadvantage of this model

lies in the theoretical difficulty arising from the fact thatthe payoff in a state of bankruptcy may be greater than the

value of the assets available. Because this situation would

never occur in practice, this model overstates the amount

received in certain states of the world. Nonetheless, the

general pricing in the market will likely come close to this

formulation. If this approach represents any kind of truth

in debt valuation, this model will likely detect it. Simi-

larly, rejection of this simpler model will indicate this

entire approach may be in error.

In empirical terms, this re-casting of the problem

opens the door for testing a parsimonious model for risky

debt valuation. Contingent claims pricing models revolve

around formulations for Q. One approach is to assume we

can find the actual probability of default and then exploit

the relationship between actual probabilities and risk-

neutral probabilities to estimate this model. If we can

accurately determine the actual probabilities of default, we

can estimate a model which adjusts these amounts by

the market price of risk and some function of the time to

maturity to arrive at the suitable risk neutral probability.

We are interested in the actual probability, p, that the value

of the firms assets will be less than the face value of the

debt at maturity:

p = Pr[logVA(T) < logF|V

A(0) = V

A] (9)

Using Itos lemma on Equation (1) and integrat-

ing from time zero to time T provides the solution for the

actual default probability:

(10)

N(.) is the function for calculating the normal dis-

tribution. Note that the formula for the risk-neutral

probability will be similar to Equation (10) except that will be replaced by r, the risk-free rate.

Using some kind of factor pricing framework, we

can formulate a relationship between the expected return

on the firms assets and the overall expected return for the

p N

F V T T

T

A

= +

log log

1

2

2

market. Consider a CAPM world as a simple example of

this relationship (M

is the expected return on the mar-

ket and M

is the volatility of the market):

(11)

Returning to our formulation for the risk-neutral

probability, we can make substitutions for r and cov(rV, r

M)

( is the correlation of the return on the firms assets, rV,

and the return on the market, rM).9 By rearranging terms,

we can derive the argument inside the normal distribu-

tion operator presented in Equation (10). If we already

know p, we can apply the inverse of the normal distri-

bution function to return the argument inside Equation

(10). After manipulating the equations, we arrive at the

following formula for Q, the risk-neutral probability of

default (the key is creating a similar formula to (10) except

that is replaced by r, the risk-free rate):

(12)

Essentially, the actual probability of default is

adjusted upwards to reflect the compensation necessary to

motivate risk averse investors to buy an asset with price

sensitivity to overall market risk and time to maturity. is a parameter determined by the entire market and can

be interpreted as the reward per unit of market risk taken

(i.e., an overall market Sharpe ratio). derives from thesensitivity of the firms assets to the overall market risk.

Note that instead of the CAPM, a more sophisticated fac-

tor model can be used to determine the amount of vari-

ation in the return explained by the firms sensitivity to

certain market factors. The sensitivity parameter, , might

also be set equal to where R2 is the coefficient ofdetermination (measure of the goodness of fit for the

model) resulting from estimation of a suitable factor

model.10 Given proper specification of these parameters,

a risk-neutral probability of default can be estimated that

properly prices the debt despite the fact that the expected

payoff is being discounted at the risk-free rate. Rear-

ranging terms, we can derive a simple formula for the

term structure of credit spreads:

R2

Q N N p T= +

1( )

=

=

r r r

r

V M

M

M

M

cov( , )





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rD

- r = -1/T[log(1 - LQ)] (13)

Note that this formulation for Q does not assume the

default point to be an absorbing barrier. Fortunately, Equa-

tion (12) can be easily modified such that we can calculate

the (cumulative) risk-neutral probability of default given thata firm that hits the default barrier will not recover. Although

we can approximate an absorption probability (probability

of being absorbed into the default point barrier prior to the

maturity of the firms debt), this model retains the somewhat

controversial assumption that recovery will not occur until

the maturity date even if default occurs prior to maturity.

Any differences caused by this assumption will be absorbed

into the loss given default parameter. I would expect, how-

ever, the impact of this assumption to be minimal at worst.

Calculating the absorption probability requires us to mod-

ify the probability formula as follows:

ap = 1 - Pr[logVA(T)

> logF, inf logVA(t)

> logF|VA(0)

= VA]

for all t [0, T] (14)

The twist in this formula results from the added

condition that no default requires not only that the asset

value exceed the default barrier at maturity but also that

the minimum asset value over the time to maturity never

hits the default barrier. If the asset value process has no

drift, then the absorption probability can be easily calcu-

lated as follows:

ap = 1 - Pr[logVA(T)

- logF > 0|VA(0)

= VA] -

Pr[logVA(T)

- logF < 0|VA(0)

= VA] (15)

Notice that I have been able to rewrite the absorp-

tion or cumulative probability in terms of the probabil-

ity at horizon previously defined as p. By substitution, we

arrive at the following convenient relationship:

ap = 2p (16)

If the asset value process actually has drift, the

relationship is not quite so simple. However, the difficultyassociated with independently estimating the drift term

and the fact that under most circumstances the simple for-

mula is approximately equal to the more complicated

formula support a strong case for ignoring the more gen-

eral formula for absorption probabilities. The cost in

terms of modeling accuracy will be small.11 The follow-

ing formula for determining Q is exactly true for firms

whose asset value processes have no drift and approxi-

mately true for most of those firms whose asset value pro-

cesses have (small) non-zero drift:

(17)

In the analysis of actual term structures of credit

spreads, the slope of the curves suggest that over time the

market price of risk may differ across term. In order to test

this hypothesis, the following adjustment can be made to

the above equation:

(18)

By adding the parameter, we can test the possibility

that the market price of risk scales as a function of term.

Ifends up equal to 1/2 then credit spread behavior is con-

sistent with asset returns following a normal distribution. If

exceeds (falls below) 1/2 then the market price of risk

increases (decreases) with term. This parameterization allows

for the presence of a term structure of the markets price

of risk. Some market participants have suggested a term

premium is attached to longer dated loans and bonds imply-

ing a different price of risk at longer time horizons.

Extensions of the Structural Model

This framework can be extended to more com-

plicated debt securities by reflecting a securitys payouts

and indenture provisions in the model definitions. Exten-

sions of this modeling framework can be made by mod-

ifying the assumptions governing each aspect of the

valuation problem:

1. Asset value process.

2. Default-risk-free rate process.

3. Conditions triggering default including both char-

acterization of the default barrier and assumptionsgoverning the reasons for default.

4. Characterization of LGD.

For example, Merton [1974] presents a formula-

tion for a callable coupon bond with no sinking fund.

First, the asset value process must be modified to reflect

coupon payouts:

Q N N pT

T T=

+

21

2

1

Q N N p T=

+

21

2

1





9/18

dVA

= (VA

- C)dt + VAdz (19)

where C equals the amount of the instantaneous coupon

payment due on the outstanding debt.

Merton demonstrates that corporate debt with

value, D(VA, t), satisfies the following partial differentialequation and boundary conditions:

1/22V2AD

VAVA

+

(rVA

- C)DVA

+ Dt- rD + C = 0 (20)

The boundary conditions will also be different

for a callable, coupon bond:

D(0, t) = 0

D(VA, T) = min(V

A, F)

D(VA

, t) = k(t)P

DVA(V

A, t) = 0

where F is the outstanding bond principal at time t, k(t)

is the call price schedule per unit principal, and T is the

maturity date of the bond. The upper free boundary,

VA(t), corresponds to the optimal call barr ier at or above

which the firm will call the bonds. Solving this modified

partial differential equation provides a formula for callable,

coupon-bearing debt.

The problem of allowing default prior to maturity

is remedied by Black and Cox [1976] who introduce an

absorbing barrier to reflect the presence of net worth orsafety covenants. In this way, the asset value can be mod-

eled such that it can be absorbed into the default barrier

(the value at which the firm can no longer meet its con-

tractual obligations). Valuation becomes a first-passage-

time problem i.e., this type of model determines the

probability of the first time the asset value passes through

the default barrier.

One criticism of traditional BSM models focuses

on the fact that in the context of the model, default can

never occur by surprise. Therefore, as time to maturity

goes to zero, credit spreads should also approach zero. In

practice, we see non-zero credit spreads for nearly allcorporate debt regardless of maturity. Zhou [1997] pre-

sents one solution to this problem with a version of a BSM

model that allows for a jump process to periodically shock

the asset price process. These jumps create discontinuities

in the path followed by the firms asset value. In this way,

short-dated risky debt can be shown to require a signif-

icant credit spread. The jump version of a structural

model presents one possible explanation. This model

assumes investors demand a price for both general credit

risk and credit risk arising from jumps.

In addition to modifying the assumptions govern-

ing a firms asset process, other authors have presented

models that relax the assumption that default-risk-free

interest rates are deterministic. Shimko, Tejima, and VanDeventer [1993] (STV) point out that the BSM frame-

work can be combined with a stochastic process for the

default-risk-free interest rate as long as the instantaneous

variance of the return of the risk-free zero-coupon bond

depends only on time to maturity. General equilibrium

models of the default-risk-free rate such as the ones intro-

duced by Cox, Ingersoll, and Ross [1985] and Longstaff

and Schwartz [1992] do not meet this criterion. The

mean-reversion model of Vasicek [1977] does. STV

develop a two-factor model combining the BSM frame-

work with Vasiceks model of the risk-free rate. An impor-

tant characteristic of this model specification involves the

correlation between the asset value factor and the risk-free

interest rate factor. Assuming a non-zero asset-interest cor-

relation introduces a fair degree of complexity that is yet

to be justified empirically. (See the discussion below

reviewing empirical work in this area.)

Others who have developed two factor models

include Longstaff and Schwartz [1995] (their model is

hereafter referred to as the LS model) who derive a sim-

ple version of this type of model with an exogenous

threshold value at which financial distress occurs. LS also

assume a Vasicek model for the default r isk-free interestrate process. If the firms asset value hits a threshold value

(in this case the face value of the debt which equals one)

over the life of the debt, then the firm is forced into

bankruptcy and the debtholder receives (1 - L) or the

recovery amount given default. Kim, Ramaswamy, and

Sundaresan [1993] change the bankruptcy trigger from

asset value to cash flows and build a model that incorpo-

rates a CIR model of the default-risk-free term structure.

(See Cox, Ingersoll, and Ross [1985] for an exposition of

the general equilibrium Cox-Ingersoll-Ross or CIR

model of the term structure; the notable characteristic of

this model is that the instantaneous volatility of the inter-est rate is a function of the square root of the interest rate

level.) In their model, a firm will default if its cash flows

are unable to cover its interest obligations. Unfortunately,

this more complicated framework results in a partial dif-

ferential equation with no known closed-form solution.

Other more complicated characterizations of these mod-

els where default can occur prior to maturity (the option

to default can be considered a barrier option) are presented





10/18

by Ericsson and Reneby [1995] and Briys and deVarenne

[1997]. After simplification, the form of the valuation

equation for many of these models resembles the one in

Equation (6). Unfortunately, these models are even more

difficult to implement empirically. If we increase the fac-

tors and the complexity, we increase the number ofparameters to be estimated.

Another focus of modification concerns specifi-

cation of the asset value that tr iggers bankruptcy. In

most structural models, this value is assumed to be

exogenous. Leland [1994] endogenizes the value of

assets that triggers bankruptcy by introducing taxes and

bankruptcy costs as factors in determining the optimal

asset value at which a firm should declare bankruptcy.

Leland and Toft [1996] extend this model to derive a

term structure of credit spreads. Similar to other mod-

ified BSM models, the model is mathematically elegant,

but empirically awkward. An important question in the

context of this type of endogenous model is whether

incorporating taxes and bankruptcy costs into a struc-

tural model will really make a difference. A pr ior i I

would assert these factors have second or third order

impact. Given the noisy bond pricing data available

and the lack of good data on taxes and bankruptcy

costs, empirically detecting the influence of these fac-

tors will be difficult.

The preceding examples demonstrate the ease

with which this modeling framework can be modified.

The cost of these modifications is tractability. Themore realistic the model becomes, the more complex

is the resulting valuation equation. In some of the

more extreme cases, we are unable to find closed-

form solutions. In these cases we must rely on numer-

ical solutions which can be unintuitive and

computationally expensive. Even in the cases where we

can find closed-form solutions, we may lose clarity

regarding the factors driving value. More often than

not, however, we end up with equations characterized

by numerous parameters that are difficult to estimate.

Finding the appropriate balance between realism and

tractability requires assumptions and approximations.

Empirical research can illuminate the aspects of these

models that can be simplified and maybe even ignored.

(Again refer to Exhibits 1 and 2 for a summary of the

defining characteristics found in key structural models

published in the finance literature.)


E X H I B I T 1Categorization of Structural Models I

Reference Asset Value Default Risk-Free Rate

Black and Scholes [1973]; Merton [1974] dVA

= VAdt + V

Adz dr = rdt

Black and Cox [1976] dVA

= ( - )VAdt + V

Adz dr = rdt

Leland [1994]; Leland and Toft [1996] dVA

= ((VA, t) - )dt + V

Adz dr = rdt

Shimko, Tejima, and Van Deventer [1993] dVA

= VAdt +

1V

Adz

1dr = (- r)dt +

2dz

2

Kim, Ramaswamy, and Sundaresan [1993] dVA

= VAdt +

1V

Adz

1a dr = (- r)dt +

2dz

2

Longstaff and Schwartz [1995] dVA

= VAdt +

1V

Adz

1dr = (- r)dt +

2dz

2

Briys and de Varenne [1997] dVA

= rVAdt +

1(dz

2+ V

Adz

1)b dr = (t)((t) - r)dt +

2(t) dz

2

Zhou [1997] dVA

= ( - )VAdt +

1V

Adz

1+ ( - 1)dJc dr = (- r)dt +

2dz

2

Note: I have standardized all notation to be consistent with this article. Consequently, the variables used may differ from the ones originally pre-

sented in the referenced articles. The mathematical relationships are the same.aV

Ais the net cash outflow from the firm resulting from optimal investment decisions.

bThis characterization of the stochastic process is under the risk-neutral probabilities to highlight the correlation assumption.cdJ is a Poisson (jump) process with intensity parameterand jump amplitude equal to > 0. Note that the expected value of is + 1.

r12

r




11/18

REDUCED-FORM MODELS

Structural models begin with an economic argu-

ment about why a firm defaults (e.g., the firms value does

not cover its obligations). These economic models pro-

vide the framework to derive a relationship between debtprices (or credit spreads) and market variables. In their

solved form (with adjustments to make the default

time inaccessible), structural models can be shown to be

a special case of reduced-form models. However, it is not

necessary to make a structural argument as to why a firm

defaults to derive and estimate a reduced-form model.

Reduced-form models rest on the assumption that default

is an unpredictable event governed by an intensity-based

or hazard-rate process (see Duffie and Singleton [1998]).

Using notation similar to the earlier sections, we can

present reduced-form models in a format similar to the

structural models. For example, the value of a zero-

coupon bond issued by a firm with one class of equity (no

dividends) and one class of debt can be valued as follows:

D(F, r, T) = P(r, T)(F - LFQ(* < T)) (21)

The difference between our previous structural

characterization and this reduced-form characterization of

the model lies in the specification for Q. In this model,

Q indicates the risk-neutral probability the (unpredictable)event of default occurred at a time *, which happenedto precede the maturity of the debt. The timeof default

is assumed to follow a stochastic process governed by its

own distribution that must be parameterized by an inten-

sity or hazard rate process. This default or stopping time

is inaccessible i.e., it jumps out at you (from nowhere).

Most extensions to reduced-form models focus on more

sophisticated characterizations of the hazard rate process.

Similar to the structural model framework, many exten-

sions explore assumptions surrounding recovery rates,

default-free interest rates processes, and contract bound-

ary conditions.

Jarrow and Turnbull [1995] present one of the

first reduced-form models using the simple assumptions


E X H I B I T 2Categorization of Structural Models II

Reference Default Barrier Recovery

Black and Scholes [1973]; Merton [1974] F VA(T)

Black and Cox [1976] LFe-r(T - t); ABa LFe-r(T - t)

Leland [1994]; Leland and Toft [1996] V*A(t)

(, T, , ); ABb (1 - L)V*A(t)

Shimko, Tejima, and Van Deventer [1993] F VA(T)

Kim, Ramaswamy, and Sundaresan [1993] c/; AB min[(1 - L(t))P(r, t, c), BA(t)

]c

Longstaff and Schwartz [1995] K; AB (1 - L)F

Briys and de Varenne [1997] LFP(t, T)d; AB LFP(t, T)

Zhou [1997] K; AB (1 - L)F

aAB denotes an absorbing barrier. The firm can enter into default prior to maturity if its asset value hits this barrier at any

time up until maturity.bIn this model, represents bankruptcy costs as a fraction of the value of the firm in bankruptcy. represents the firmstax rate. In this model, the default barrier is endogenous.cUpon default, debtholders receive either the total value of the firm or a fraction of an otherwise similar default risk-free

bond whichever is less.dDesignates the price of a risk-free bond based on the stochastic process behind the risk-free interest rates. This default bar-

rier extends Black and Cox [1976] to a stochastic interest rate environment.




12/18

of constant LGD and exponentially distributed default-

time. The default-time distribution is parameterized by a

hazard rate or default intensity, h.12 They further assume

the default-free rate process, the hazard rate process

(defined by h), and the LGD function are mutually inde-

pendent. In this framework, risky bonds can be modeledas foreign currency bonds denominated in promised

dollars. The exchange rate equals 1 if default has not

occurred and equals the recovery rate if it has. This frame-

work allows for a variety of specifications for the default

risk-free rate process making the model quite flexible

along this dimension. Unfortunately, the assumption of a

constant default intensity is unrealistic. While this speci-

fication implies default is a Poisson arrival making the

model easier to estimate, firms will likely have different

default intensities depending on the time horizon being

considered (e.g., strong firms would be expected to have

low default intensities in the near future with increases in

the distant future as potential for future problems become

more pronounced).

In response to the weakness of the Jarrow and

Turnbull [1995] model, Jarrow, Lando, and Turnbull

[1997] (JLT) present a more sophisticated reduced-form

model where default is modeled as the first time a con-

tinuous-time Markov chain with K states (where states 1

to K - 1 could be associated with credit ratings 1

being AAA; and the K-th state being default) hits the

absorbing state K (default state). JLT combine this Markov

chain specification with LGD characterized as a fractionof an otherwise similar default-risk-free claim. This spec-

ification nests the simpler model (i.e., default follows a

constant hazard rate or Poisson arrival process) and pro-

vides the opportunity to specify a more realistic evolution

of default intensity. The increased flexibility comes at a

cost of estimation difficulty. The Markov chain greatly

increases the number of parameters to be estimated as the

model requires specification of an entire generator matrix

to arrive at transition probabilities for each possible change

in state. JLT resolve this problem by suggesting the use of

historical transition probability matrices available from

companies like Standard and Poors. While these histori-cal matrices are easily obtained, the empirical validity of

this approach has yet to be demonstrated. The point to

remember is that mathematical tractability not eco-

nomics drives the choice of how to specify a reduced-

form model.

Duffie and Singleton [1998] (DS) use reduced-

form models to value risky debt as if it were default-risk-

free by replacing the usual short term default-risk-free rate

with the default-adjusted, short-rate process. Their model

is distinguished by the parameterization of LGD in terms

of the fractional reduction of the market value of the debt

that occurs upon default. They then present examples of

how to specify a reduced-form model in the context of

popular default-risk-free term structure models (e.g.,Heath, Jarrow, and Morton [1992]). Mathematically, they

write down the following:

D(F, r, T) = EQ[exp(-T0R

tdt)F]

R = rt+ h

tL

t+ l

t(22)

In this formulation we evaluate the expectation

under the equivalent martingale measure Q. In other

words, we take the expectation with respect to risk-neu-

tral probabilities. rtis still the default-risk-free rate. We

introduce ht

as the hazard-rate for default at time t. In

other words, htis the arrival intensity at time t (under

Q) of a Poisson process whose first jump occurs at

default. Note that taking the expectation under the

risk-neutral measure essentially13 transforms h into Q or

the risk-neutral probability of default. (Note that ht

will not equal the true instantaneous probability of

default as long as the market price of risk associated

with the Poisson process is non-zero.) Ltrepresents the

fractional loss given default. The advantage of this spec-

ification is that currently available term structure mod-

els for default-r isk-free debt can be applied to this

problem with little adjustment. Using available creditspread data, the implied risk-neutral mean loss rate

(htL

t) can be estimated. Notice in this description a new

variable, lt, has been added to the default-adjusted short

rate. This variable represents the fractional carrying

costs of the defaultable claim. By introducing lt, liquidity

effects or a liquidity premium can be included in the

model. Alternative specifications of this model focus on

different assumptions regarding the processes governing

the following:

1. Default process embodied in the hazard rate, ht.

2. Default-risk-free process embodied in the short-rate, r

t.

3. LGD process embodied in the fractional reduction (of

the debts market value), Lt.

4. Fractional carrying cost process embodied in lt.

These estimates can then be used to price other

debt and credit derivatives. A particularly thorny problem

in this framework involves disentangling htand L

t. With-





13/18

out a wide range of debt securities deriving value from the

same issuer (e.g., liquidly traded bond, credit derivatives),

the components of the mean loss rate cannot be estimated

separately. Given the paucity of data in this field, efficient

estimation of each individual parameter in this modeling

framework can be a daunting task.Duffie and Lando [1997] (DL) use the DS frame-

work with a twist. They demonstrate how to formulate

a structural model such that it can be estimated as a

reduced-form model presented in the second specifica-

tion. They begin with a diffusion process for the firms

asset value and a default barrier that marks the asset value

at which the firm defaults. They next derive a formula

for the hazard rate, ht, in terms of the asset value volatil-

ity, the default barrier, and the conditional distribution

of asset value given the history of information available

to investors. The mechanism creating the inaccessible

default stopping time is imperfect accounting informa-

tion. With imperfect accounting data, the current mar-

ket price of the firms debt relies not only on current

accounting data, but also on historical accounting data.

This dependence on historical information remains

despite the Markovian nature of the underlying asset

value. In a model of this sort, perfect accounting data

would imply credit spreads go to zero as maturity goes

to zero. With imperfect accounting data, however, credit

spreads remain bounded away from zero even if maturity

approaches zero. In this sense, DL recast a structural

model in the reduced-form framework.Cathcart and El-Jahel [1998] present another

example of a reduced-form model that approaches the

structural approach. In their model, default occurs when

a signaling process instead of asset value hits some

lower barrier. They include a stochastic CIR process for

the default r isk-free rate and assume the signaling process

and default risk-free rate process are uncorrelated. They

argue anecdotally their model produces credit spread

term structures more consistent with observed credit

spreads than other formulations. Again, these claims are

yet to be tested rigorously.

These characterizations of reduced-form modelspresented above represent examples of approaches to

implementing this type of model. This discussion is by no

means comprehensive. For a more detailed overview of

these types of models see Lando [1997]. (See Exhibit 3 for

a brief summary of the characteristics of some of the

more popular reduced-from models published in the

finance literature.)

PREVIOUS EMPIRICAL RESEARCH

While the theoretical literature on contingent-

claims models for risky debt has mushroomed, the empir-

ical literature has been nearly non-existent. A number of

studies have looked at the relationships among relevantvariables in this framework. However, few articles focus

on testing properly specified contingent-claims models.

Lack of good bond data, noisiness in even the best bond

data, and the apparent inefficiency of the corporate bond

markets contribute to the dearth of good empirical evi-

dence in this area. Moreover, the complexity of many

bond indentures (all kinds of new options are dreamed up

by enterprising lawyers and investment bankers) makes fit-

ting parsimonious models a troublesome task. The other

major difficulty concerns estimation of the actual corpo-

rate risk-free rate. (Should we follow convention and use

the market for U.S. treasuries to find proxies for the

default risk-free rate?)

Testing Structural Models

While the theory has become increasingly sophis-

ticated, the empirical testing of structural models has

stagnated. Early work by Jones, Mason, and Rosenfeld

[1984] found the contingent-claims model produced

credit spreads significantly lower than actual credit spreads.

Moreover, they found the model did no better than a naive

model (discounting cash flows at the risk-free rate) in pric-ing investment grade debt. Later, Franks and Torous

[1989] confirmed the finding that actual credit spreads

were much greater than predicted credit spreads. These

studies essentially extinguished hope that a standard BSM

model would yield reasonable empirical estimates.

In that same year, Sarig and Warga [1989] estimated

the term structure of credit spreads using a small number

of zero coupon corporate bonds and zero coupon U.S.

treasury bonds. They demonstrated curve shapes (slightly

upwardly sloping for investment grade debt, humped

shaped for lower grade debt, and downward sloping for

very low grade debt) consistent with the contingent-claims model predictions. Unfortunately, their small sam-

ple size and lack of rigorous statistical testing prevented

them from drawing strong conclusions. Nonetheless, their

paper resuscitated hopes of empirically verifying the con-

tingent-claims modeling approach. More recently, Wei and

Guo [1997] use the contingent-claims framework to test

both the Merton [1974] model and the Longstaff and

Schwartz [1995] model. Their results favor Mertons





14/18

model and demonstrate the promise of the framework.

Unfortunately, their data cover only 1992 for the Eurodol-

lar market. Their small sample size and narrow focus

weaken the impact of their results. Another problem with

their study concerns the question of what the spread in

the Eurodollar market actually represents. While some

portion of that spread undoubtedly compensates for credit

risk, other non-credit characteristics likely explain the bulk

of this spread. The difficulty lies in identifying the appro-

priate corporate risk-free interest rate.

Another interesting empirical study recently com-

pleted by Delianedis and Geske [1998] uses the BSM

framework to estimate risk neutral default probabilities.

They find that rating migrations (using S&P credit ratings)

and defaults are detected months before in the equity mar-

kets. Their findings lend further support to the contin-

gent-claims framework for modeling default. More

importantly, their results demonstrate that default is likely

modeled better as a diffusion process than as a Poisson

event. The implication of this research is that most

reduced-form models relying on the Poisson characteri-zation of default will underperform structural models

focused on causal drivers of default.

Other non-BSM models, while successful in char-

acterizing qualitative aspects of the data predicted by

BSM-type structural models, have had difficulty fitting the

data. Fons [1987] uses a risk-neutral model to look at low-

grade bonds. He concludes:

Either that there is systematic mispricing of low-

rated corporate bonds by investors or that the risk-

neutral model derived herein cannot fully capture

the markets assessment of the probability of default

on these securities [p. 98].

In a later article (Fons [1994]), Fons tests his risk-

neutral model again and finds that it seriously underestimates

the spreads he obtains from fitting linear regressions through

data within different credit classes. His model specification

has particular difficulty with investment grade bonds. His

analysis provides general support in terms of apparent down-

ward sloping term structures of credit risk for lower rated

bonds; however, his r isk-neutral model abstracts too much

from the real world restricting the models ability to gen-

erate an accurate term structure of credit spreads.

The conventional wisdom, while praising the the-

oretical insights gained from structural models, dismisses

them as impractical for actual bond valuation. However,

small sample sizes, doubts about the quality of bond pric-

ing data, and the lack of focus on the appropriate default-risk-free rate leave us without conclusive evidence

regarding the power of structural models. The resolution

of these empirical issues awaits further research.

Testing Reduced-Form Models

Similar to the circumstances surrounding structural

models, few empirical articles have been written on


E X H I B I T 3Categorization of Reduced-Form Models

Reference Default Process Default Risk-Free Process

Jarrow, Lando, and Turnbull [1997] Markov chain in credit ratings Any desired term structure model

Duffie and Singleton [1998] Hazard rate Included in default process

Cathcart and El-Jahel [1998] Signaling process CIR

Reference Correlation LGD Process

Jarrow, Lando, and Turnbull [1997] Independent Constant fraction of default risk-free claim

Duffie and Singleton [1998] Unrestricted Fractional reduction in market value

Cathcart and El-Jahel [1998] Independent Constant fraction




15/18

reduced-form models. These models require that credit

spread data accurately reflect market expectations about

credit risk, recovery, and liquidity. Given the noisiness in

the data and the difficulty in finding actual bond prices,

this assumption is heroic. The question remains whether

the more complicated reduced-form models fit noise oruncover systematic relationships. We can only hope for

more and better data to answer this question.

An earlier article to follow the publication of JLTs

model was written by Duffee [1996a]. He sums three

independent square-root processes (similar to the CIR

[1985] model for the default risk-free term structure) to

arrive at the default-adjusted discount rate. He follows JLT

and assumes recovery (1 - LGD) is constant. His dataset

covers monthly prices from January 1985 to December

1994 for corporate bonds that make up the Lehman

Brothers Bond indexes. As an aside, this dataset likely con-

stitutes the best available bond pricing dataset given that

matrix prices are flagged. (See Warga [1991] and Duffee

[1996b] for details on these data.) He finds strong evidence

of misspecification with the model having particular dif-

ficulty simultaneously producing both flat term struc-

tures of credit spreads for investment-grade bonds with less

credit risk and steeper term structures of credit spreads for

investment-grade bonds with relatively more credit r isk.

Including non-investment grade bonds would likely mag-

nify the evidence of misspecification due to the humped

and downward sloping term structures that are com-

monly observed (see Sarig and Warga [1989]). On aver-age, however, the model appears to fit investment-grade,

corporate bond prices reasonably well. He makes sig-

nificant strides in implementing this modeling frame-

work; but concludes that

single-factor models of instantaneous default probabilities

. . .face a substantial challenge in matching the dynamic

behavior of corporate bond term structures (Duffee

[1996a, p. 26]).

While not focused on corporate bond spreads, the

paper written by Duffie and Singleton [1997] tests a DSversion of reduced-form models on defaultable swap

yields. They focus on the yields directly in order to avoid

questions concerning the true default risk-free rate. They

express the default-adjusted discount rate as the sum of

two independent square-root diffusions. One factor drives

credit risk and the other drives liquidity risk. They use

weekly Telerate data (which represent average bid and ask

rates quoted by large dealers) from January 4, 1988

through October 28, 1994. After using maximum likeli-

hood to estimate the model, they compute implied risky

zero-coupon bond yields. They then subtract the corre-

sponding U.S. treasury zero-coupon yields to arrive at

implied, defaultable swap spreads. They study these spreads

in the context of a multivariate vector autoregressionwith proxy variables for credit risk (spread between BAA-

and AAA-rated commercial paper) and liquidity (spread

between the generic three-month repo rate for the ten-

year treasury note and the repo rate of the current on-the-

run treasury note) and find that liquidity shocks are

short-lived while credit shocks have little short-term

impact followed by significant long-term impact. Their

model does a reasonable job of fitting the swaps yields with

the exception of the short-end of the term structure.

Questions still remain concerning the interpretation of

some of their parameter estimates.

Default Risk-Free Rate and Credit Spreads

One unresolved issue in this framework concerns

the relationship between default risk-free rates and credit

spreads. The one-factor models (which specify a stochas-

tic process only for the asset value) assume independence.

Separating the credit spreads from the time value of

money makes for much simpler models. More compli-

cated two-factor models introduce a relationship between

these variables. Unfortunately, we do not have much

guidance regarding the nature of this relationship. Longstaffand Schwartz [1995] report a negative relationship

between credit spreads and interest rates. (Their data sam-

ple is monthly from 1977 to 1992.) In their two-factor

model, the driver behind this relationship is the fact that

an increase in the interest rate increases the drift of the asset

value process. Consequently, the risk-neutral probability

of default decreases leading to lower credit spreads.

Other researchers have also found time periods

where this negative correlation appears. Duffee [1996b]

uses Lehman Brothers data to look at monthly spreads for

investment grade bonds from January 1985 through March

1995. He finds that for the highest credit quality bonds,changes in credit spreads are mostly unrelated to changes

in interest rates. For lower credit quality bonds that are still

investment grade, credit spreads appear to be negatively

correlated with interest rates. These tests were done on

noncallable bonds. He points out that other research

results based on samples of callable bonds will overstate the

negative correlation given the impact of interest rates on

the embedded call option (increased interest rates lead to





16/18

a less valuable call option which lowers the credit spread).

Having properly controlled for callability, Duffee finds

some evidence that lower quality investment grade bond

spreads are negatively correlated with interest rates. He

argues also that testing this relationship with refreshed

indexes (i.e., credit ratings are held fixed over time) willlikely understate the relationship between treasury yields

and credit spreads. Interestingly enough, he finds evi-

dence rejecting the hypothesis that the relationship

between treasury yields and credit spreads is driven by vari-

ations in credit quality. The difficulty with correlation

studies lies in the sensitivity to the chosen sampling

period. We still do not have enough data to measure

correlation accurately over a long-period of time. Even

if the data existed, the shifts in the underlying nature of

the economic processes may introduce non-stationarity

into the data rendering inferences on long-period sam-

ples meaningless. At this stage, we still need more explo-

ration into the data to paint a complete picture.

CONCLUSION

The valuation of risky debt remains a fertile field

for financial researchers. Currently, we can choose from

many theoretical models. Our focus at this stage should

be on assembling and analyzing quality pricing data for

risky debt. Without empirical results, choosing the best

model remains a difficult task.

ENDNOTES

1The term structure of credit risk is also called the

term structure of credit spreads, the risky term structure, or

the risk structure of interest rates. I will use these descriptions

interchangeably.2See Exhibits 1 and 2 for a categorization of several rep-

resentative structural models published in the finance literature.3Most of this section derives from the approach explained

in Black and Scholes [1973], Merton [1974], Vasicek [1996], and

the approach explained in Longstaff and Schwartz [1995].4I will refer to the value of the firm and the value of the

firms assets (meaning total assets of the business which equals

liabilities plus equity) interchangeably.5Generally speaking, the time argument will be sup-

pressed in this and subsequent equations: dz implies dz(t); VA

implies VA(t)

. In cases where the appropriate time argument can

be misunderstood it will be represented as follows: VA(T).

6The debt is assumed to mature at time T. We are

determining the value of the debt at time 0. Consequently, the

time to maturity also equals T.

7In this equation all subscripts refer to partial derivatives

with respect to the variable in the subscript.8These types of probabilities are also known as pseudo-

probabilities and quasi-probabilities. Essentially, they are adjust-

ments to the actual probabilities that maintain the two key

characteristics of probabilities the probabilities are greaterthan or equal to zero and have a sum of one.9 = cov(r

V, r

M)/

M10Note that R2 = regression sum of squares/total sum

of squares. In other words, this statistic measures how much of

an assets return is explained by the risk factors. In the case

of the CAPM or any other single factor model, R2 = 2.11Readers interested in the more complicated derivation

should consult Ingersoll [1987].12Jarrow and Turnbull [1995] actually use the symbol

to denote the hazard rate or default intensity. In order to avoid

confusion in the context of this survey and to maintain con-

sistency, I will refer to this parameter as h.

13More rigorously, if h is right continuous, then in thelimit as t 0, h

tis the risk-neutral conditional probability

given the information available at time t and given no default

by time t, that the firm will default before t + 1. This charac-

terization of htas the default probability is approximately true

for small time intervals. (See Duffie and Singleton [1998] for a

more extensive discussion of these properties.)

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