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
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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).
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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
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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
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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.)
62 A SURVEY OF CONTINGENT-CLAIMS APPROACHES TO RISKY DEBT VALUATION SPRING 2000
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
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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
SPRING 2000 THEJOURNAL OF RISK FINANCE 63
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.
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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-
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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
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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
66 A SURVEY OF CONTINGENT-CLAIMS APPROACHES TO RISKY DEBT VALUATION SPRING 2000
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
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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
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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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