<< стр. 6(всего 11)СОДЕРЖАНИЕ >>
event of default by the reference entity.
The table in the bottom of Figure 16.1 has all the information we need
to conduct the valuation exercise just described. In particular, using the
168 16. The Credit Curve

TABLE 16.4
CDS-implied Coupons on
Principal-protected Notes

Maturity AZZ Bank XYZ Corp.
(years) (percent)

1 2.02 4.04
2 2.54 6.24
3 3.00 8.84
4 3.41 11.57
5 3.78 14.28

Note. Assumed recovery rates: AZZ Bank,
50 percent; XYZ Corp., 10 percent.

method described in Section 16.1, we can rely on the CDS premiums and
zero-coupon bond prices shown in that table to obtain the survival probabil-
ities of the reference entities (Table 16.1). Armed with these probabilities,
obtaining the coupons on the п¬Ѓve-year PPNs written on AZZ Bank and
XYZ Corp. is simply a matter of using the appropriate values on the right
side of equation (16.13). Table 16.4 shows the resulting term structures of
PPN coupons for AZZ Bank and XYZ Corp.

16.3.2 PPNs vs. Vanilla Notes
Just as we used CDS-implied survival probabilities to value PPNs of various
maturities, we can obtain a CDS-implied term structure of coupons on
vanilla п¬Ѓxed-rate notes issued by the AZZ Bank and XYZ Corp. directly.
According to equation (15.15), and assuming that these notes are sold at
par, the fair value of such coupons is given by
n
[1 в€’ Z(t, Tn )Q(t, Tn )] в€’ Z(t, Tj )[Q(t, Tjв€’1 ) в€’ Q(t, Tj )]X
j=1
C= n
Z(t, Tj )Q(t, Tj )Оґj
j=1
(16.14)

where, again, F is the par value of the notes.
Figure 16.2 shows the term structures of principal-protected and vanilla
п¬Ѓxed-rate notes that reference AZZ Bank and XYZ Corp. As one would
expect, investors have to give up some yield in order to obtain the principal-
protection feature. The п¬Ѓgure also shows that the spread between PPN and
vanilla yields is wider for lower-rated entities, consistent with the intuition
that principal protection is more highly valued for riskier entities than for
relatively safe п¬Ѓrms such as AZZ Bank.
16.4 Other Applications and Some Caveats 169

FIGURE 16.2. Implied Term Structure of Coupons on Principal-protected and
Vanilla Notes
Note. Assumed recovery rates: AZZ Bank, 50 percent; XYZ Corp.,
10 percent.

16.4 Other Applications and Some Caveats
The examples examined in this chapter illustrated several practical appli-
cations of credit curves, ranging from using CDS-implied survival prob-
abilities to determine the fair values of coupons on principal protected
notes to marking existing CDS positions to market. Other uses include
relative-value trading, such as taking long and short positions in two
related instruments based on the view that existing discrepancies in their
valuationsвЂ”as implied, say, by survival probabilities derived from CDS
quotesвЂ”will dissipate in the near future.
As we saw in Section 6.3, however, the real world rarely behaves exactly
as dictated by results reported in textbooks. Factors such as accrued
interest, liquidity, and market segmentation still very much drive wedges
between the prices of otherwise closely related credit instruments. As a
result, not only do market participants rely on more complex versions of
the simple illustrative framework described in this chapter, but they also
use their judgment and experience to assess current and prospective market
conditions when working with the credit curve of any given reference entity.
17
Main Credit Modeling Approaches

In this chapter we review and summarize the credit risk literature with
a special focus on the main modeling approaches for valuing instruments
subject to default risk. Section 17.1 summarizes the so-called п¬Ѓrm value
or structural approach to credit modeling, which traces its origins to the
work of Black and Scholes (1973) and Merton (1974). Credit models
in this tradition focus on the analysis of the capital structure of individ-
ual п¬Ѓrms in order to price their debt instruments. The discussion of the
structural approach relies on some basic results regarding vanilla call and
put options. Most of these results are discussed only at an intuitive level in
this chapter. Readers interested in additional detail on the pricing of such
options are encouraged to consult, for instance, the books by Baxter and
Rennie (2001) and Wilmott, Dewynne, and Howison (1999), which
provide excellent introductions to option pricing.
The reduced-form or default-intensity-based approach is addressed in
Section 17.2, where we discuss models based on the seminal work of Jar-
row and Turnbull (1995). The reduced-form approach does not directly
attempt to link defaults to the capital structure of the п¬Ѓrm. Instead, it
models defaults to be exogenous stochastic events. Work in this strand
of the credit risk literature is primarily interested in developing essentially
statistical models for the probability of default over diп¬Ђerent time horizons.
In Section 17.3 we brieп¬‚y compare the structural and reduced-form
approaches, both on methodological and empirical grounds. That sec-
tion also highlights the main thrust of a вЂњhybridвЂќ approachвЂ”motivated
by the work of Duп¬ѓe and Lando (2001)вЂ”that incorporates elements of
172 17. Main Credit Modeling Approaches

both the structural and reduced-form approaches. The chapter concludes
with Section 17.4, where we outline the basic tenets of the ratings-based
approach to credit modeling.

17.1 Structural Approach
To understand the essence of the structural approach to credit modeling we
will discuss the theoretical framework п¬Ѓrst proposed by Black and Scholes
(1973) and Merton (1974). Later on, we shall address some of the
most important extensions of this basic framework.

17.1.1 The Black-Scholes-Merton Model
Consider a hypothetical п¬Ѓrm with a very simple capital structure: one zero-
coupon bond with face value K and maturity T and one equity share. In
keeping with the notation introduced in Chapter 15, let Z d (t, T ) denote
the time-t price of a bond maturing at time T with a face value of \$1.
As a result, the price of the bond with face value K is Z d (t, T )K. (We
d
are dropping the 0 subscript on Z0 (t, T ) to emphasize that this will not
generally be a zero-recovery bond.) To represent the market value of the
equity share, we shall introduce a new variable, E(t), where limited liability
implies that the market value of equity cannot be negative.
The assumption of only one share and one bond is not restrictive. More
generally, one can think of K as being the total value of the п¬ЃrmвЂ™s debt,
where all debt is in the form of zero-coupon bonds maturing at time T , and
of E as corresponding to the total value of the shares issued by the п¬Ѓrm.
By way of the basic market value identity, which states that the market
value of the shareholderвЂ™s equity is equal to the diп¬Ђerence between the
market value of the assets and liabilities of the п¬Ѓrm, we can write:

A(t) = E(t) + Z d (t, T )K (17.1)

where A(t) stands for the market value of the assets of the п¬Ѓrm.
The basic idea behind the Black-Scholes-Merton (BSM) model is very
straightforward. Default is quite simply deп¬Ѓned as a situation where, at
time T , when the п¬ЃrmвЂ™s debt, K, becomes due, the value of the п¬ЃrmвЂ™s
assets, A(T ), falls short of K:

default в‡” A(T ) < K (17.2)

Figure 17.1 illustrates the main points of the model. The п¬Ѓgure shows the
evolution of the value of the п¬Ѓrm over time. For as long as A remains above
17.1 Structural Approach 173

FIGURE 17.1. Simple Illustration of the Black-Scholes-Merton Model

K at T , the п¬Ѓrm does not default. Under such circumstances, the п¬ЃrmвЂ™s
creditors receive K and the shareholders get to keep the residual A(T )в€’K.
Should A(T ) fall short of K at time T , however, the п¬Ѓrm defaults, with
the creditors taking over the п¬Ѓrm, receiving its full value A(T ), and the
shareholders receiving nothing.
Thus, the debtholders either receive K or A(T ), whichever is lower. If we
let Z d (T, T )K denote the amount that the debtholders actually receive at
time T , the debtholderвЂ™s payout at time T can be written as

Z d (T, T )K = K в€’ Max(K в€’ A(T ), 0) (17.3)

As for the shareholders, they either receive A(T ) в€’ K or nothing at time
T . Their payoп¬Ђ at time T can be more succinctly written as

E(T ) = Max(A(T ) в€’ K, 0) (17.4)

Equations (17.2) through (17.4) summarize some of the key implications
of the simple BSM framework. Examining (17.2) п¬Ѓrst, we can see that,
in the context of the model, the default probability associated with this
hypothetical п¬Ѓrm is simply given by the probability that A will be lower
than K at time T :

default probability = Probt [A(T ) < K] (17.5)

where, as in previous chapters, we are interested in risk-neutral probabil-
ities. In particular, Probt [.] denotes a risk-neutral probability conditional
on all available information at t and on the п¬Ѓrm having survived through t.
174 17. Main Credit Modeling Approaches

FIGURE 17.2. Payoп¬Ђs of Long Positions in Vanilla Call and Put Options

In addition, from (17.2) we can also infer that the recovery value of the
bond upon default is simply A:

recovery value of defaulted bond = A(T ) (17.6)

which corresponds to a recovery rate in the event of default of A(T )/K.
Thus, we have just shown how we can use this simple structural model
to revisit two familiar credit concepts discussed in Chapter 15: default
probabilities and recovery rates.
Turning now to equations (17.3) and (17.4), it can be shown that they
lend themselves to an option-theoretic approach to the valuation of default-
able bonds and equities. This is a key insight of the BSM model. For
instance, upon closer inspection of Max(K в€’ A(T ), 0), the last term on the
right-hand side of (17.3), one can see that this corresponds to the payout of
a put option written on the value of the п¬ЃrmвЂ™s assets, where the strike price
of the option is K. This is shown in the lower panel of Figure 17.2. In par-
ticular, the holder of this option would stand to gain if A were to fall below
K by time T , which can be thought of as the expiration date of the option.
But this expression appears with a negative sign in (17.3), which implies
that, rather than being the holder of such a put, the debtholder wrote the
option. Thus, one can think of the debtholderвЂ™s position as being equivalent
to a portfolio composed of a long position in a riskless zero-coupon bond
with a face value of K and a short position in the just-described put option.
17.1 Structural Approach 175

To sum up:

defaultable bond = riskless bond в€’ put option on A (17.7)

Thus, if we let p(T, A(T ); T, K) denote the value of the put option at T ,
we can use equation (17.7) to write the following relationship between the
market value of the п¬ЃrmвЂ™s debt and that of the embedded put option:

Z d (T, T )K = K в€’ p(T, A(T ); T, K) (17.8)

and, expressing this relationship in terms of time-t prices, we obtain:

Z d (t, T )K = eв€’R(t,T )(T в€’t) K в€’ p(t, A(t); T, K)

= Z(t, T )K в€’ p(t, A(t); T, K) (17.9)

which is an expression for the price of a defaultable bond under the BSM
model.1
We have just shown that the debtholdersвЂ™ position is equivalent to being
long a riskless bond and short a put option written on the value of the
п¬Ѓrm. But who bought this put option? The shareholders implicitly did.
To see this п¬Ѓrst note that a quick look at (17.4) will reveal that what we
have on the right-hand side of that equation is essentially the payout of a
call option written on A, struck at K (see the upper panel of Figure 17.2).
Thus

equity share = call option on A (17.10)

But now note that the payout of that call can be rewritten as:

E(T ) = Max(A(T ) в€’ K, 0) = A(T ) в€’ K + Max(K в€’ A(T ), 0)

Thus, in the absence of arbitrage opportunities, for any t < T we can
write:

E(t) = A(t) в€’ eв€’R(t,T )(T в€’t) K + p(t, A(t); T, K) (17.11)

which establishes that the shareholdersвЂ™ position can also be thought of as
including a long position on a put option written on the value of the п¬Ѓrm,
a put that they implicitly bought from the bondholders.2

1
As discussed in Chapter 15, R(t, T ) is the time-t yield to maturity on a riskless
zero-coupon bond that will mature at time T with a face value of \$1.
2
Readers familiar with basic option pricing theory will recognize equation (17.11)
as the so-called put-call parity condition, which can be shown to be true regardless of
176 17. Main Credit Modeling Approaches

To derive model-implied risk spreads, recall that the yield to maturity,
d
R (t, T ), on the zero-coupon bond issued by the п¬Ѓrm is such that

Z d (t, T ) = eв€’R (t,T )(T в€’t)
d

Thus:
1
Rd (t, T ) = в€’ log Z d (t, T ) (17.12)
T в€’t

and the credit spread is trivially given by Rd (t, T ) в€’ R(t, T ).
Equation (17.9) implies that the higher the value of the put option implic-
itly sold by the bondholder to the shareholder, the wider the gap between
the prices of the defaultable and riskfree bonds and, equivalently, the wider
the corresponding credit spread. In terms of risk-neutral probabilities, a
high value of p(.) suggests that the put option is more likely to be exercised
than otherwise, which, in this context, amounts to saying that the п¬Ѓrm is
more likely to default. Thus, according to this model, issuers of defaultable
bonds pay yields that are higher than those on otherwise comparable risk-
less bonds because such issuers are implicitly buying a put option on the
value of their п¬Ѓrms, and the value of that option is higher the lower is the
credit quality of the п¬Ѓrm.

17.1.2 Solving the Black-Scholes-Merton Model
Thus far we have discussed a few key results based on the Black-Scholes-
Merton framework, but we have not actually solved the model. For instance,
we have used the model to argue that the bondholder is essentially short a
put option on the value of the п¬Ѓrm, but we have not derived the price of
this option or shown explicitly how that price, and thus the п¬ЃrmвЂ™s credit
spread, relates to the capital structure of the п¬Ѓrm. Indeed, up until now all
of the results we have derivedвЂ”default probabilities, recovery value, risky
bond prices and spreads, etc.вЂ”depend importantly on A, the value of the
п¬Ѓrm, but the evolution of A itself has not yet been addressed.
A central assumption in the BSM framework is that, based on actual
(not necessarily risk-neutral) probabilities, A(t) evolves continuously over
time following a geometric Brownian motion:

dA(t)
= Вµdt + ПѓdW (t) (17.13)
A(t)

how we model the evolution of AвЂ”see, for instance, Wilmott, Howison, and Dewynne
(1999).
17.1 Structural Approach 177

where dW (t) is an inп¬Ѓnitesimal increment in a standard Brownian motion,
and Вµ and Пѓ are constants that primarily determine the average (trend)
rate of growth and the volatility of A(t), respectively.3
It can be shown that (17.13) implies that the п¬Ѓrm value is lognormally
distributed, which greatly simpliп¬Ѓes the derivation of explicit formulas for
all the quantities discussed in the previous section. For instance, the well-
known Black-Scholes formula for the price of a put option, which is derived
under the assumption of lognormality of the price of the underlying, is
given by4

p(t, A(t); T, K, Пѓ) = Keв€’r(T в€’t) N (в€’d2 ) в€’ A(t)N (в€’d1 ) (17.14)

with

log(A(t)/K) + (r + Пѓ 2 /2)(T в€’ t)
в€љ
d1 в‰Ў (17.15)
Пѓ T в€’t
в€љ
d2 в‰Ў d1 в€’ Пѓ T в€’ t (17.16)

where, for simplicity, we have assumed that the term structure of riskless
interest rates is п¬‚at and constant, i.e.,

R(t, T ) = r for all t and T

where r is the instantaneous riskless rate of interest.5
It can be shown that the price of the put option, and consequently the
п¬ЃrmвЂ™s credit spread, is increasing in

eв€’r(T в€’t) K
A(t)

3
Intuitively, the reader who is not familiar with continuous time processes can think
of (17.13) loosely as the continuous time analog of the following discrete process:
At+Оґt в€’ At
= ВµОґt + Пѓ t+Оґt
At
where is a zero-mean normally distributed variable with variance Оґt. This naive dis-
cretization of (17.13) is not exact, however, and is featured in this footnote only as
a reference point to the novice reader. Hull (2003) and Mikosch (1999) provide
accessible discussions of continuous-time results that are central to the pricing of options.
4
Textbook-like derivations of the Black-Scholes formula abound. See, for instance,
Baxter and Rennie (2001) and Wilmott et al. (1999). Appendix B provides a brief
summary of the lognormal distribution.
5
Appendix A contains a discussion of basic concepts in bond math.
178 17. Main Credit Modeling Approaches

which can be interpreted as the п¬ЃrmвЂ™s leverage ratio. This is consistent
with oneвЂ™s intuition, in that more highly leveraged п¬Ѓrms tend to face wider

17.1.3 Practical Implementation of the Model
Armed with the theoretical results discussed thus far, one might now be
eager to use the model with real data. It turns out, however, that taking
the model to the data is generally not a trivial matter. For instance, in the
real world, a п¬ЃrmвЂ™s liabilities are not just made up by zero-coupon bonds.
In addition, balance-sheet information can sometimes be noisy indicators
of the true state of the п¬Ѓrm, a phenomenon that became patently clear with
the events surrounding the Enron and WorldCom corporations in 2001 and
2002 in the United States.6
It should also be noted that the very same variable that plays a central
role in the BSM model, the value of the п¬Ѓrm (A), is not observed in practice.
Thus, even for a hypothetical п¬Ѓrm with the simple debt structure assumed
in the basic BSM model, one is still presented with data challenges. What
one does observe are daily п¬‚uctuations in share prices of the п¬Ѓrm and the
book value of the п¬ЃrmвЂ™s liabilities (typically at a quarterly frequency). As
the number of shares outstanding for a given п¬Ѓrm is commonly a known
quantity, one can then estimate the market capitalization of the п¬Ѓrm or
the value of its equity. Given estimates of the equity value, as well as the
model-implied result that this value should be equal to the price of a call
option on the value of the п¬Ѓrm, one can back out the implied values of A
and Пѓ for a given assumption for the stochastic process for the value of the
п¬Ѓrm.7

17.1.4 Extensions and Empirical Validation
Major contributions to the structural approach to credit risk modeling
include the treatment of coupon-bearing bondsвЂ”Geske (1977) and
Geske and Johnson (1984)вЂ”and the incorporation of stochastic riskless
rates into the frameworkвЂ”Shimko, Tejima, and van Deventer (1993).
The BSM framework has also been extended into the class of so-called
вЂњп¬Ѓrst-passageвЂќ models, which include the work of Black and Cox (1976),
Leland (1994), Longstaп¬Ђ and Schwartz (1995), and others. These
models address one limitation of the original BSM framework, which only

6
Even in the absence of accounting fraud, corporate balance sheets may not reveal
the entire state of a п¬Ѓrm. For instance, вЂњbalance-sheet noiseвЂќ may be introduced by
ambiguities in certain accounting deп¬Ѓnitions.
7
Backing out A(t) from equity prices involves some technical steps that go beyond
the scope of this introductory book. For more on this see, e.g., Duп¬ѓe and Singleton
(2003).
17.1 Structural Approach 179

FIGURE 17.3. A Simple First-passage Model

allows for defaults to occur at the maturity date of the underlying debt.
In addition, they allow for a more general treatment of the default bound-
ary, which no longer necessarily corresponds to the face value of the п¬ЃrmвЂ™s
debt. For instance, Leland (1994) models the default boundary as being
the outcome of equity holdersвЂ™ eп¬Ђorts to maximize the value of their stake
in the п¬Ѓrm.
Some of the basic features of п¬Ѓrst-passage models are illustrated in
Figure 17.3. Defaults are now assumed to occur at the п¬Ѓrst time that the
value of the п¬Ѓrm, A(t), crosses the вЂњdefault boundary,вЂќ B. In contrast, in
the basic characterization of the BSM framework, a default, if any, could
only occur at the maturity date of the п¬ЃrmвЂ™s debt. Figure 17.3 depicts two
possible paths for A(t), one where A(t) never falls to B before the end of the
time horizon, T вЂ”labeled a non-default pathвЂ”and one where A(t) crosses
the boundaryвЂ”labeled a default path. We will review a simple п¬Ѓrst-passage
model in the next section.
A well-known oп¬Ђshoot of the BSM framework is the commercially avail-
able KMV model, an analytical tool provided by MoodyвЂ™s KMV. This
model uses a large proprietary database of defaults to compute default
probabilities for individual issuersвЂ”see Crosbie (2002). Similar to the
simple BSM setup, the KMV model looks at equity market volatility and
prices to infer the volatility and level of the п¬ЃrmвЂ™s value, which provide
a measure of the distance to defaultвЂ”or how far, in terms of standard
deviations Пѓ of its value A, the п¬Ѓrm is from its default boundary K in
Figure 17.1. Using proprietary methods, MoodyвЂ™s KMV then translates
the distance from default into вЂњexpected default frequencies,вЂќ which is how
180 17. Main Credit Modeling Approaches

MoodyвЂ™s KMV calls its estimator of default probabilities over a one-year
horizon.
The empirical evidence on structural models is not conclusive and has
focused on a modelвЂ™s implications for both the shape of the credit spread
curve and its level. For instance, Jones, Mason, and Rosenfeld (1984)
reported that the BSM model tended to overprice corporate bonds (under-
predict credit spreads), but Delianedis and Geske (1998) found that
BSM-style models have predictive power for rating migrations and defaults.
The lack of consensus among empirical researchers reп¬‚ects both the fact
that there are diп¬Ђerent types of structural modelsвЂ”some have more plau-
sible empirical implications than othersвЂ”as well as data problems, such as
the relative illiquidity of many corporate bonds, which makes it harder to
obtain meaningful market quotes.
Intuitively, the empirical п¬Ѓnding that the BSM framework has a tendency
to generate credit spreads that are too low relative to observed spreads,
especially at the short end of the credit spread curve, can be thought of as
stemming from a feature that is at the very core of many structural models.
In particular, the traditional forms of both the basic BSM and п¬Ѓrst-passage
models assume that the value of the п¬Ѓrm evolves as in equation (17.13),
which, from a technical standpoint, implies that A(t) has continuous tra-
jectories and is thus not subject to jumps. As a result, if the value of the
п¬Ѓrm is suп¬ѓciently above the default barrier, the probability that it will
suddenly touch the barrier over the very near term is virtually zero and
very small for short maturities, regardless of the creditworthiness of the
п¬Ѓrm. Hence, the likelihood of a near-term default by a п¬Ѓrm that is not
in п¬Ѓnancial distress is virtually zero, and so is the п¬ЃrmвЂ™s near-term credit
spread implied by the model. Indeed, regarding the shape of the credit
spread curve for a п¬Ѓrm that is not in п¬Ѓnancial distress, traditional struc-
tural models tend to suggest that the curve starts at or near zero at the
very short end and then typically roughly follows a hump-shaped pattern
as the horizon under consideration is lengthened. In contrast, empirical
studies suggest that short-term spreads are generally a fair amount above
zero, and that it is not at all uncommon for credit spread curves to be п¬‚at
or even downward sloping.8
One extension to the basic BSM model that was largely motivated by
the desire to generate a better п¬Ѓt between model-implied and observed
credit spread curves is the work of Zhou (1997). Zhou assumed that
the dynamics of the value of the п¬Ѓrm has two components: a continuous
component that is similar to that assumed in the traditional BSM and
п¬Ѓrst-passage models, and a discontinuous вЂњjumpвЂќ component, which, as
the name suggests, allows the value of the п¬Ѓrm to change suddenly and

8
See, for instance, Fons (1994) and Sarig and Warga (1989).
17.1 Structural Approach 181

unexpectedly by a sizable amount. Because of the possibility that a jump
may occur at any time, so-called jump-diп¬Ђusion models do not necessar-
ily have the property that near-term credit spreads are implausibly low.
Indeed, credit spread curves implied by such models can have a variety of
shapes, including upward-sloping, hump-shaped, and inverted.

17.1.5 Credit Default Swap Valuation
We will use a simple extension of the basic BSM model, a п¬Ѓrst-passage
model with a п¬Ѓxed default boundary, to illustrate the pricing of a credit
default swap in the context of the structural approach to credit modeling.
As noted above, some of the key features of the model can be seen in
Figure 17.3.
We continue to assume that, in terms of actual probabilities, the value
of the п¬Ѓrm, A(t), evolves according to (17.13) and that the riskless yield
curve is constant and п¬‚at at r. In the context of this simple model, the
п¬Ѓrm is assumed to default the moment that its value touches the default
boundary B.
Our interest is in computing the risk-neutral probability Q(t, T ), condi-
tional on all information available at time t and no default at that time,
that the п¬Ѓrm will not default by a given future date T , where T may denote,
for example, the horizon over which one is exposed to the п¬Ѓrm. Thus, we
can then write:

Q(t, T ) в‰Ў Probt {A(s) > B for all s в€€ (t, T )} (17.17)

where the notation s в€€ (t, T ] denotes all values of s that are greater than
t and less than or equal to T .
Given the assumed process for the evolution of A(t)вЂ”equation (17.13)вЂ”
there is a readily available formula for Q(t, T )вЂ”see, e.g., Musiela and
Rutkowski (1998):9

b(t) + (r в€’ Пѓ 2 /2)(T в€’ t)
в€љ
Q(t, T ) = N
Пѓ T в€’t
в€’b(t) + (r в€’ Пѓ 2 /2)(T в€’ t)
2(rв€’Пѓ 2 /2)
в€љ
в€’e b(t)
N (17.18)
Пѓ2
Пѓ T в€’t

9
Alternatively, a crude approach for computing Q(t, T ) would involve simulating a
large number of risk-neutral paths for A(t) and counting the number of paths where A
breached the default boundary. This so-called Monte Carlo simulation method can be
time consuming, however, and is best reserved for п¬Ѓrst-passage problems that do not
have an easily obtainable analytical solution.
182 17. Main Credit Modeling Approaches

where b(t) в‰Ў log(A(t)/B) and N (.) is the cumulative standard normal
distribution.10
We are now ready to use the model to price a vanilla credit default
swap written on the п¬Ѓrm depicted in Figure 17.3. As we saw in Chapter 6,
pricing a credit default swap means determining the premium S that will
be paid periodically by the protection buyer. Taking S initially as given,
we п¬Ѓrst compute the time-t value of the вЂњpremiumвЂќ and вЂњprotectionвЂќ legs
of a contract with maturity at T . For simplicity, assume that the premium
is paid continuously and that the contract has a notional amount of \$1. As
a result, the present value of the premium leg is

T
О·(t) = SZ(t, v)Q(t, v)dv (17.19)
t

which can be thought of as the continuous-time analog of (16.1).
To value the protection leg, we note that it is equivalent to a contingent
claim that pays (1 в€’ X) at the time of default, provided default happens
before T , where X is the recovery rate of the deliverable obligation(s) of the
reference entity. The value of such a claim is given by the continuous-time
analog of equation (16.3). Letting Оё(t) denote the value of the protection
leg, we can write

T
Оё(t) = (1 в€’ X) Z(t, v)[в€’dQ(t, v)] (17.20)
t

A default swap typically has zero market value at its inception, and thus
pricing such a contract is equivalent to п¬Ѓnding the value of S that makes
the two legs of the swap have equal value. This is given by

T
(1 в€’ X) Z(t, v)[в€’dQ(t, v)]
t
S= (17.21)
T
Z(t, v)Q(t, v)dv
t

Following Pan (2001), if we now deп¬Ѓne the annualized probability of
default as

в€’ log(Q(t, T ))
Л†
h(t, T ) =
T в€’t

10
The normal and lognormal distributions are brieп¬‚y discussed in Appendix B.
17.2 Reduced-form Approach 183

Л† ВЇ Л†
and if we assume that h(t, T ) в‰€ h for all t and T , i.e., h is approximately
constant as t and T vary, we obtain11
T ВЇ
Z(t, v)eв€’h(vв€’t) hdv
ВЇ
(1 в€’ X) ВЇ
Sв‰€ = (1 в€’ X)h
t
(17.22)
T ВЇ
Z(t, v)eв€’h(vв€’t) dv
t

which gives us the intuitive (and, by now, familiar) result that the credit
default swap premium is closely connected to the recovery rate and the
annualized probability of default associated with the reference entity. Thus,
starting from a model for the evolution of the asset value of the п¬Ѓrm, we
were able to price a credit default swap agreement written on that п¬Ѓrm.

17.2 Reduced-form Approach
Rather than tying defaults explicitly to the вЂњfundamentalsвЂќ of a п¬Ѓrm, such
as its stock market capitalization and leverage ratio, the reduced-form
approach takes defaults to be exogenous events that occur at unknown
times. Let П„ denote the time of default, which, of course, is a stochas-
tic variable. A central focus of the reduced-form approach is to propose
a model that assigns probabilities to diп¬Ђerent outcomes involving П„ . For
instance, conditional on all information available at time t and given no
default at that time, one might want to know the (risk-neutral) probability
that a given reference entity will not default within the next year. Contin-
uing to use the notation introduced earlier in this book, this probability
can be written as

Q(t, T ) в‰Ў Probt [П„ > T |П„ > t] (17.23)

where time T in this speciп¬Ѓc example is exactly one year from today, i.e.,
T в€’ 1 = 1 year.
As we saw in the previous section, structural models also allow for the
time of default to be stochastic, but in those models П„ is determined
endogenously by the evolution of the value of the п¬Ѓrm, A(t). In con-
trast, in reduced-form models, the stochastic properties of П„ are speciп¬Ѓed
as an exogenous process that is not directly related to the balance sheet of
the п¬Ѓrm.

17.2.1 Overview of Some Important Concepts
Before we go on to describe the basic features of the reduced-form approach,
we should stop to introduce some additional notation and review a few key

11
Л†
If h(t, T ) is constant, the expression in (17.22) becomes an equality.
184 17. Main Credit Modeling Approaches

concepts. We shall focus primarily on extending the risk-neutral valuation
approach to the case of stochastic interest rates and on the notion of forward
default probabilities.

17.2.1.1 Stochastic Interest Rates
Up until now we have been taking the riskless interest rate r to be time
invariant. We will now start relaxing this assumption in order gradually to
bring the modeling framework a bit closer to the real world. In particular,
let us allow the riskless rate to vary deterministically from month to month,
but, for now, we will continue to insist on time-invariant rates within the
month. Consider an investor who puts \$1 in a riskless bank account that
pays interest on a continuously compounding basis. At the end of the п¬Ѓrst
month the investor will have er(1)(t1 в€’t0 ) in the bank, where r(1) is the
riskless rate prevailing that month, expressed on an annual basis, and t1 в€’t0
is the fraction of a year represented by the п¬Ѓrst month. Carrying on with
this exercise, and assuming no withdrawals, after two months the investorвЂ™s
bank account balance will be

er(1)(t1 в€’t0 ) er(2)(t2 в€’t1 ) = er(1)(t1 в€’t0 )+r(2)(t2 в€’t1 )

where r(2) is the rate applied to the second month and (t2 в€’ t1 ) is deп¬Ѓned
as above. Generalizing, after n months, the account balance will be

r(i)(ti в€’tiв€’1 )
n
e i=1

If instead of allowing the riskless rate to vary only from month to month,
we had allowed for perfectly predictable daily changes in r, the above
scheme would still work, with r(i) being redeп¬Ѓned to mean the interest
rate corresponding to day i and (ti в€’ tiв€’1 ) representing the fraction of the
year represented by the ith day. If we shorten the period over which r is
allowed to vary to the point where (ti в€’ tiв€’1 ) becomes inп¬Ѓnitesimally small,
the value of the bank account between the current instant in time t0 = t
and some future instant tn = T becomes
T
r(v)dv
e t

where dv is the length of the inп¬Ѓnitesimally small time interval.
Recall that the expression above is the time-T value of a \$1 deposit made
at time t. It is easy to see then that the time-t value of a dollar received at
time T would be given by

ОІ(T ) в‰Ў eв€’
T
r(v)dv
(17.24)
t
17.2 Reduced-form Approach 185

and we arrive at the well-known result that, in a world with deterministic
interest rates, the price of a riskless zero-coupon bond that matures at time
T is simply given by ОІ(T ).
With stochastic interest rates, the discount factor can no longer be
deп¬Ѓned as above because the future values of r are not known. Instead,
the price of a zero-coupon bond that pays out \$1 at T can be shown to be
the expected value of (17.24):

Z(t, T ) = Et eв€’
T
Лњ r(v)dv
(17.25)
t

Лњ
where Et [.] denotes the expected value of вЂњ.вЂќ based on information available
at time t, computed using risk-neutral probabilities.12

17.2.1.2 Forward Default Probabilities
Given (17.23), it is straightforward to see that the time-t probability of
default before time U , U > t, is given by

1 в€’ Q(t, U ) в‰Ў Probt [П„ в‰¤ U |П„ > t] (17.26)

We can also write down the probability of default between future times T
and U , as seen from time t, for t < T < U . As discussed in Chapter 15, this
is simply the probability of surviving through time T minus the survival
probability through time U :

Probt [T < П„ < U |П„ > t] = Probt [П„ > T |П„ > t] в€’ Probt [П„ > U |П„ > t]
= Q(t, T ) в€’ Q(t, U ) (17.27)

Equation (17.27) is an expression for the unconditional forward proba-
bility of default associated with this п¬Ѓrm. We say unconditional because
we are making no explicit particular stipulations about what will happen
between today, time t, and time T . We are still assuming, however, that
there is no default today and that we are using all information available at
time t.
One might be interested in the probability, conditional on all available
time-t information, that the п¬Ѓrm will survive through some future time
U , given that it has survived through an earlier future time T , but hav-
ing no other time-T information about either the state of the п¬Ѓrm or

12
See Neftci (2002) for additional non-technical insights into risk-neutral proba-
bilities and the derivation of an explicit expression for (17.25). Bjork (1998) provides
further details.
186 17. Main Credit Modeling Approaches

of the economy. This is the conditional forward probability of survival,
Probt [П„ > U |П„ > T ]. By the Bayes rule, we can write13

Probt [П„ > U |П„ > t] Q(t, U )
Probt [П„ > U |П„ > T ] = = (17.28)
Probt [П„ > T |П„ > t] Q(t, T )

and hence the forward conditional probability of default regarding the
future time period [T, U ] is given by

Probt [П„ в‰¤ U |П„ > T ] = 1 в€’ Probt [П„ > U |П„ > T ]

Q(t, U ) в€’ Q(t, T )
=в€’ (17.29)
Q(t, T )

17.2.1.3 Forward Default Rates
We can now introduce a key concept in the context of reduced-form models,
the default rate H(t, T ), deп¬Ѓned as the risk-neutral default probability
associated with a given horizon divided by the length of the horizon14

Probt [П„ в‰¤ T |П„ > t] 1 в€’ Q(t, T )
H(t, T ) в‰Ў = (17.30)
T в€’t T в€’t

The forward default rate, as seen at time t, that corresponds to the future
period [T, U ] is analogously deп¬Ѓned as

Probt [П„ в‰¤ U |П„ > T ] Q(t, U ) в€’ Q(t, T ) 1
H(t, T, U ) в‰Ў =в€’ (17.31)
U в€’T U в€’T Q(t, T )

If we now let U в‰Ў T +в€†T , we can deп¬Ѓne the time-t instantaneous forward
default rate as:

h(t, T ) в‰Ў lim H(t, T, T + в€†T ) (17.32)
в€†T в†’0

13
The Bayes rule simply says that, given two events A and B, and denoting
Prob[A&B] as the probability that both A and B occur, one can write
Prob[A&B] = Prob[A]Prob[B|A]
where Prob[A] is the unconditional probability that A will take place and Prob[B|A] is
the probability of B occurring given that A has taken place.
In the context of (17.28), we can think of event A as the event deп¬Ѓned as П„ > T ,
B as П„ > U , and, given U > T , A&B corresponds to П„ > U . (To be sure, all of the
corresponding probabilities would have to be deп¬Ѓned as being conditional on information
available at time t, as well as on survival through that time.)
14
Unless otherwise indicated, all probabilities discussed in this chapter are risk-neutral
probabilities.
17.2 Reduced-form Approach 187

and it is not diп¬ѓcult to see that

в€‚Q(t, T ) 1
h(t, T ) = в€’ (17.33)
в€‚T Q(t, T )

To see how we arrived at (17.33), recall that (17.31) implies that

[Q(t, T + в€†T ) в€’ Q(t, T )] 1
h(t, T ) в‰Ў lim в€’ (17.34)
в€†T в†’0 в€†T Q(t, T )

but, assuming that certain technical conditions on Q(.) are satisп¬Ѓed, the
expression within the square brackets is nothing more than the deп¬Ѓnition
of the negative of the derivative of Q(t, T ) with respect to T .
Equation (17.33) is probably the single most important preliminary
result derived so far in Section 17.2. In particular, integrating both sides
of (17.33) from T to U , we obtain

Q(t, U )
eв€’
U
= Probt [П„ > U |П„ > T ]
h(t,v)dv
= (17.35)
T
Q(t, T )

where the last equality results from (17.28).
Equation (17.35) is the forward survival probability associated with the
future period [T, U ], given survival through time T and conditional on
all available time-t information. Indeed, we can now express default and
survival probabilities over any horizon for a given entity simply on the basis
of the instantaneous forward default rate h(t, s).15 For instance, the time-t
probability that the п¬Ѓrm will survive through time U is

Probt [П„ > U |П„ > t] = Q(t, U ) = eв€’
U
h(t,v)dv
(17.36)
t

For t = T , we can rewrite (17.34) as

[Q(t, t + в€†t) в€’ Q(t, t)] 1
h(t, t) в‰Ў lim в€’ (17.37)
в€†t Q(t, t)
в€†tв†’0

but, by deп¬Ѓnition, Q(t, t) = 1, and thus, for inп¬Ѓnitesimally small dt,

Probt [П„ в‰¤ t + dt|П„ > t] = h(t, t)dt (17.38)

15
Readers familiar with yield curve models will notice the analogy between the concept
of instantaneous forward interest rates and that of instantaneous forward default rates,
especially regarding their relationship to zero-coupon bond prices and survival proba-
bilities, respectively. (Appendix A provides a brief overview of instantaneous forward
interest rates.)
188 17. Main Credit Modeling Approaches

which, in words, is the time-t instantaneous probability of default, assuming
no default at t. Thus, for small time intervals в€†t, the probability of default
between t and t + в€†t, given no default by t, is approximately equal to
h(t, t)в€†t.16

17.2.2 Default Intensity
In the context of most reduced-form models, the random nature of defaults
is typically characterized in terms of the п¬Ѓrst вЂњarrivalвЂќ of a Poisson process.
In particular, for a given reference entity, if we assume that defaults arrive
(occur) randomly at the mean risk-neutral rate of О» per year, the time-t
risk-neutral probability of no default by time T can be written as17

Q(t, T ) = eв€’О»(T в€’t) (17.39)

provided О» is time invariant and, of course, given no default by time t. In
the credit risk literature О» is commonly called the intensity of default or
the hazard rate.
If О» is not constant over time, but varies deterministically, we can fol-
low the same logic discussed in Section 17.2.1 to show that if О» changes
continuously over inп¬Ѓnitesimally small time intervals we can write

Q(t, T ) = eв€’
T
О»(v)dv
(17.40)
t

and thus, as suggested by a comparison between (17.40) and (17.36), with
deterministic О», the terms default intensity and forward default rate can
essentially be used interchangeably.

16
Given the deп¬Ѓnition of derivative in classic calculus:
[Q(t, T + dt) в€’ Q(t, T )]
в€‚Q(t, T )
в‰Ў lim в€’
в€‚T dt
dtв†’0

we can write
в€‚Q(t, T )
Q(t, T + dt) в‰€ Q(t, T ) + dt = Q(t, T )[1 в€’ h(t, T )dt]
в€‚T
For T = t, and given that Q(t, t) = 1, we can write:
1 в€’ Q(t, t + dt) в‰€ h(t, t)dt
which becomes an equality if Q(t, T ) has a derivative at T = t and we let dt в†’ 0.
Thus, given that 1 в€’ Q(t, t + dt) = Probt [П„ в‰¤ t + dt|П„ > t], we arrive at (17.38)вЂ”Neftci
(2002) discusses basic notions from classic calculus that are relevant for п¬Ѓnance.
17
For a Poisson process with a constant mean arrival rate of О», a basic result from
statistics states that the time until the вЂњп¬Ѓrst arrival,вЂќ П„ , is a random variable with
probability density function О»eв€’О»П„ , which characterizes the exponential distribution.
From this, equation (17.39) easily follows. See Appendix B for further details on the
Poisson and exponential distributions.
17.2 Reduced-form Approach 189

But default intensity can vary stochastically over time in response to,
say, unanticipated developments regarding the economy or the п¬Ѓnancial
condition of the п¬Ѓrm. With stochastic default intensity, it can be shown
that, under some technical conditions that go beyond the scope of this
bookвЂ”see, for instance, Lando (1998)вЂ”the expression for the time-t
survival probability of a given п¬Ѓrm can be written in a way that is entirely
analogous to (17.25):

Q(t, T ) = Et eв€’
T
Лњ О»(v)dv
(17.41)
t

Models where (17.41) holds are typically called doubly stochastic models of
default because they assume that not only is the time of default a random
variable, but so is the mean arrival rate of default at any given point in time.
A comparison between equations (17.41) and (17.36) suggests that the
equality between time-t forward default rates and future values of intensity
generally breaks down when О» is assumed to be stochastic. For instance,
consider the value of default intensity at some future time v. That value,
О»(v), incorporates all information available at time v, including the values
of any stochastic factors that may aп¬Ђect default intensity at time v, such as
the prevailing states of the economy and of the п¬Ѓrm at time v. In contrast,
the forward default rate h(t, v) can be thought of as the intensity rate
for the future time v, as seen on the basis of currently available (time-t)
information, and conditional on the п¬Ѓrm surviving through time v.
Thus, whereas О»(v) is based on all information available at time v, the
only time-v information on which h(t, v) is conditional is the survival of
the п¬Ѓrm through time v. Still, from the perspective of time t, it can be
shown the following relationship between forward default rates and default
intensity holds:

h(t, t) = О»(t) (17.42)

which says that when defaults occur according to a Poisson process, todayвЂ™s
(time-t) instantaneous default rate associated with the inп¬Ѓnitesimally small
time interval [t, t + dt] is simply the default intensity of the reference entity,
which is assumed to be known at time t. Moreover, we can write the
conditional instantaneous time-t probability of default as:

Probt [П„ в‰¤ t + dt|П„ > t] = О»(t)dt (17.43)

Reduced-form models can essentially be characterized in terms of the
particular assumptions that they make regarding how О» changes over time.
Indeed, the case of constant default intensity constitutes the simplest of all
reduced-form models.
190 17. Main Credit Modeling Approaches

A simple model that admits random variation in default intensity
involves the following speciп¬Ѓcation for the evolution of О»:

ВЇ
dО»(t) = k[О» в€’ О»(t)]dt + Пѓ О»(t)dW (t) (17.44)

where dW (t) is an inп¬Ѓnitesimal increment in a standard Brownian motion.18
Equation (17.44) closely parallels the yield curve model proposed by Cox,
Ingersoll, and Ross (1985). In words, it says that, although the default
intensity associated with a given reference entity varies stochastically over
ВЇ
time, it has a tendency to revert to its long-run mean of О», where k indicates
the degree of mean reversion.

17.2.3 Uncertain Time of Default
The use of reduced-form models in the valuation of defaultable securities
and related credit derivatives often requires the derivation of a probability
distribution function for П„ , the time of default. This is the function G(s)
such that

Prob[s < П„ в‰¤ s + ds] = G(s + ds) в€’ G(s) в‰€ g(s)ds

where g(s) is the probability density function (p.d.f.) of П„ and the approx-
imation error in the above equation is negligible for very small values of
dsвЂ”see Grimmett and Stirzaker (1998), p. 90.19
The probability density of default time can be thought of as the product
involving the probability of no default by time s and the probability of
default in the interval from s to s + ds conditional on no default by time s:

Prob[s < П„ в‰¤ s + ds] = Prob[П„ > s] Prob[П„ в‰¤ s + ds|П„ > s]

which is an implication of the Bayes rule.
Conditional on all information available at time t, and given no default
by time t, we can use equations (17.41) and (17.43) to write the conditional

18
A discrete-time approximation helps convey some of the intuition behind equation
(17.44):

ВЇ
О»t+Оґt в‰€ О»t + k[О» в€’ О»t ]Оґt + Пѓ О»t t+Оґt

where is a normally distributed random variable with mean zero and variance Оґt. Thus,
we can think of О»t+Оґt as a mean-reverting random variable with conditional variance
var[О»t+Оґt |О»t ] equal to Пѓ 2 О»t Оґt .
19
G(.) is the so-called cumulative distribution function of П„ . Appendix B reviews key
concepts surrounding cumulative distribution and probability density functions.
17.2 Reduced-form Approach 191

p.d.f. of П„ as the function gt (s) such that:

gt (s)ds = Probt [s < П„ в‰¤ s + ds] = Et eв€’
s
Лњ О»(u)du
О»(s)ds (17.45)
t

where we are assuming that ds is inп¬Ѓnitesimally small and that s в‰Ґ t.
For nonrandom intensity, we can make use of (17.40), and equation
(17.45) simpliп¬Ѓes to

gt (s)ds = Probt [s < П„ в‰¤ s + ds] = Q(t, s)О»(s)ds

17.2.4 Valuing Defaultable Bonds
Suppose we want to value a zero-coupon bond that pays \$1 at time T if its
issuer has not defaulted by then. Otherwise the bond becomes worthless, a
d
zero recovery rate. As in previous chapters, we let Z0 (t, T ) denote todayвЂ™s
(time-t) price of this bond.
In Section 15.2.1, where survival probabilities and future values of the
riskless rate were assumed to be known with certainty, we argued that one
can write the price of a defaultable bond as the product of the bond issuerвЂ™s
risk-neutral survival probability and the price of a riskless zero-coupon
bond with the same maturity date and face value as the defaultable bond.
That result still holds when interest rates are stochastic but independent
from the default intensity process, an assumption that we will maintain
throughout this chapter.20 Thus, given (17.25) and (17.41), we can write:

Z0 (t, T ) = Et eв€’
T
Лњ
d [r(v)+О»(v)]dv
= Z(t, T )Q(t, T ) (17.46)
t

which gives the logical result that, in the absence of default risk, the
d
expression for Z0 (t, T ) reduces to Z(t, T ).
Given the above equation, it is relatively straightforward to derive the
yield spread associated with this risky bond. In particular, recall from
Chapter 15 that the yield to maturity of this bond is the annualized rate
Rd (t, T ) that discounts the face value of the bond back to its market price.
Thus, Rd (t, T ) is such that Z0 (t, T ) = eв€’R (t,T )(T в€’t) for a bond with a face
d
d

20
Two events are said to be independent if the knowledge that one has occurred does
not aп¬Ђect oneвЂ™s assessment of the probability that the other will occur. Whenever we
refer to independent random variables in this book, we mean independence with respect
to risk-neutral probabilities. Independence is discussed further in Appendix B. We brieп¬‚y
discussed some empirical results regarding the relationship between defaults and riskless
interest rates in Chapter 15.
192 17. Main Credit Modeling Approaches

value of \$1. As a result,
T
в€’ log(Z0 (t, T ))
d [f (t, v) + h(t, v)]dv
d t
R (t, T ) = = (17.47)
T в€’t T в€’t

where the second equality stems from equation (17.36) and from the basic
result, reviewed in Appendix A, that the price of a riskless zero coupon
bond can be expressed in terms of the time-t instantaneous forward riskless
interest rates, f (t, v), that span the remaining maturity of the bond:

Z(t, T ) = eв€’
T
f (t,v)dv
t

The risky bond spread S(t, T ), deп¬Ѓned as Rd (t, T ) в€’ R(t, T ), is given by

T
h(t, v)dv
t
S(t, T ) =
T в€’t

In light of (17.36), we can write

в€’ log(Q(t, T ))
S(t, T ) =
T в€’t

which has the intuitive implication that the higher the risk-neutral survival
probability Q(t, T ) of the bond issuer, the lower the corresponding risk
If we assume that both r and О» are deterministic and time invariant,
equation (17.46) reduces to

Z0 (t, T ) = eв€’(r+О»)(T в€’t)
d
(17.48)

and it is easy to see that, with zero recovery, the spread between the yield
to maturity on a defaultable bond, r + О», over that on a riskless bond, r,
is equal to the default intensity of the issuer of the defaultable bond:

S(t, T ) = О»

17.2.4.1 Non zero Recovery
Armed with the probability distribution of default times, we can now
discuss the valuation of defaultable bonds with a nonzero recovery rate.
In particular, we will take advantage of the fact that such bonds can be
thought of as a portfolio involving two simpler securities: an otherwise
comparable zero-recovery defaultable bond and a contingent claim that
pays X at the time of default, if a default occurs before the maturity date
17.2 Reduced-form Approach 193

of the bond, and zero otherwise, where X is the recovery value of the
original bond.
Let О¦(t, T ) be the time-t price of the contingent claim just described.
In order to value such a claim we rely on the continuous-time analog of
the argument developed in Section 15.4 in the valuation of defaultable
bonds with a nonzero recovery value. In particular О¦(t, T ) is equal to
the probability-weighted average of all possible recovery payment scenar-
ios involving the bond, where the weights are given by the risk-neutral
probability density function of the time of default. Thus,

T
О¦(t, T ) = X Z(t, v)gt (v)dv (17.49)
t

where gt (v) в‰Ў Et [eв€’ t О»(u)du О»(v)] is the probability density function of the
v
Лњ
default time, which we discussed in Section 17.2.3.
We are now ready to derive the valuation formula for a zero-coupon bond
that has a value of X in the event of default. Given the above discussion,
we can write

Z d (t, T ) = Z0 (t, T ) + О¦(t, T )
d
(17.50)

where the two right-hand side terms of (17.50) are given by (17.46) and
(17.49), respectively.
For the simple reduced-form model that assumes that both r and О» are
time invariant, the valuation formula for the defaultable zero-coupon bond
with recovery value X can be shown to be:

XО»
Z d (t, T ) = eв€’(r+О»)(T в€’t) + (1 в€’ eв€’(r+О»)(T в€’t) ) (17.51)
r+О»

17.2.4.2 Alternative Recovery Assumptions
Thus far in the context of reduced-form models we have essentially assumed
that investors recover, at the time of default, a fraction of the defaulted
instrumentвЂ™s original face value. It should be noted, however, that this is
only an assumption, and that there are alternative ways for reduced-form
models to handle the valuation of bonds with nonzero recovery values.
One alternative to the framework based on immediate recovery of face
value is the so-called equivalent recovery assumption. This was actually
the recovery assumption made in the seminal work of Jarrow and Turnbull
(1995). In the original Jarrow-Turnbull model, a defaulted security with
face value \$1 is immediately replaced by X otherwise equivalent riskfree
zero-coupon bonds, with 0 в‰¤ X в‰¤ 1. (Obviously, X = 1 would constitute
the case of a riskfree bond to begin with.) By вЂњotherwise equivalent,вЂќ we
194 17. Main Credit Modeling Approaches

mean that the newly issued riskfree zero-coupon bonds will have the same
maturity date and face value as the defaulted bond.
How does the valuation of defaultable bonds under the equivalent recov-
ery assumption diп¬Ђer from (17.50)? The assumption that, in the event of
default, the recovery payment only takes place at the original maturity of
the risky bond makes this valuation exercise simpler because we no longer
need to explicitly derive a p.d.f. for the time of default. To see this, note
that, at the maturity date T , the bond holder will either receive the full face
value of the bond (\$1) or the recovery value X, and the risk-neutral proba-
bilities associated with these events are Q(t, T ) and 1в€’Q(t, T ), respectively.
Thus the risk-adjusted expected value of the bondholderвЂ™s payout, based
on information available at time t, is

Q(t, T ) + [1 в€’ Q(t, T )]X

and, following the logic set out in Chapter 15, the time-t value of this
expected payout is:

Z d (t, T ) = Z(t, T ){Q(t, T ) + [1 в€’ Q(t, T )]X}

d
Recall that Q(t, T )Z(t, T ) is simply Z0 (t, T ). Thus, the bond valuation
formula becomes:

Z d (t, T ) = Z0 (t, T )(1 в€’ X) + Z(t, T )X
d
(17.52)

and a comparison of (17.50) and (17.52) makes it clear that one obtains
diп¬Ђerent values for a defaultable bond depending on the assumed recovery
scheme.
A third common recovery assumption is the fractional recovery of mar-
ket value framework, proposed by Duп¬ѓe and Singleton (1999). Duп¬ѓe
and Singleton essentially assume that upon a default at time П„ the bond
loses a fraction L of its market value. (This is equivalent to saying that
upon default the bondholder recovers a fraction 1 в€’ L of the no-default
market value of the bond.) Duп¬ѓe and Singleton show that, under such
circumstances, the value of a defaultable zero-coupon bond would satisfy

Z d (t, T ) = Et eв€’
T
Лњ [r(s)+LО»(s)]ds
(17.53)
t

which simpliп¬Ѓes to

Z d (t, T ) = Z(t, T )Et eв€’
T
Лњ LО»(s)ds
(17.54)
t

when the riskless rate process is independent of the default process.
17.2 Reduced-form Approach 195

TABLE 17.1
Eп¬Ђect of Alternative Recovery Assumptions on the Valuation of a Five-year
Zero-coupon Bonda

Recovery Assumption Price Yield Risk Spread
(\$) (percent) (basis points)

Fractional recovery of face value .699 7.18 218
Equivalent recovery .676 7.83 283
Fractional recovery of market value .664 8.20 320
No recovery .552 13.00 800
Memo: Riskless п¬Ѓve-year bond .779 5.00
a Based on the assumption of п¬‚at riskless curve. The riskless rate r is set at 5 percent,
and the default intensity of the bond issuer is assumed to be constant at 8 percent. For
the nonzero recovery rate cases, X is set at 60 percent (L = 40 percent). Face value of
the bonds = \$1.

To illustrate the eп¬Ђect of the recovery assumption on the valuation of a
zero-coupon bond, we compare the prices of a п¬Ѓve-year zero-coupon bond
implied by equations (17.50), (17.52), and (17.54) derived from an other-
wise identical reduced-form model and based on the same parameter values.
Suppose, for instance, that both the riskless interest rate and default inten-
sity are constant at 5 percent and 8 percent, respectively. Let the relevant
recovery rates be 60 percent under each of the recovery rate assumptions
examined.21
Table 17.1 shows that the recovery assumption can have a non-trivial
eп¬Ђect on the model-implied prices and yield-to-maturities of this bond,
with the latter ranging from 7.18 percent when the bond is priced under
the assumption of fractional recovery of the face value of the bond to 8.20
percent under the assumption of partial recovery of the market value of
the bond. The case of recovery of face value has the lowest yield given
that it involves an immediate payment to the bond holder upon default,
as well as a payment that represents 60 percent of the full par value of
the bond. Under the equivalent recovery case, even though the recovery
value still corresponds to 60 percent of the par value, that payment is
eп¬Ђectively received in full only at the bondвЂ™s original maturity date so that
the investor eп¬Ђectively has to be compensated for this вЂњdelayвЂќ by receiving
a higher yield.
The lowest bond price (highest yield) corresponds to the case involving
the fractional recovery of market value. This occurs because the market

21
For the recovery-of-market-value framework, we assume that L is equal to 1 в€’ X or
40 percent.
196 17. Main Credit Modeling Approaches

value of the bond just before default can be substantially below par for a
zero-coupon bond. The table also shows the price of the defaultable bond
under the assumption of a zero recovery value, which corresponds to a yield
of 13 percent (a spread of 800 basis points over the yield on a comparable
riskfree bond, shown as a memo item).
The diп¬Ђerences in model-implied bond prices can be signiп¬Ѓcantly less
dramatic than those shown in Table 17.1 for zero-coupon bonds of shorter
maturity. For instance, for a one-year bond, the theoretical yields would
range from 8 percent to 8.2 percent given the assumptions listed in
the tableвЂ™s footnote. Diп¬Ђerences in bond prices across diп¬Ђerent recovery
assumptions can also be much smaller than those in the table for coupon-
bearing defaultable bonds, as shown by Duп¬ѓe and Singleton (1999)
in the context of a reduced-form model with stochastic riskfree rates and
intensity.
Which recovery assumption should one favor? Each has its pros and
cons, and neither has gained complete acceptance among either academics
or practitioners. The assumption of fractional recovery of face value is clos-
est to the market convention for defaulted bonds, where the obligations of
a liquidated debtor tend to have the same value, assuming the same level
of seniority, regardless of their maturity date. Nonetheless, as discussed
by OвЂ™Kane and Schlogl (2001), the recovery of face value assumption
imposes upper bounds on the yields of defaultable bonds, as does the equiv-
alent recovery assumption. The latter has the advantage of being easier to
deal with analytically than the recovery of face value assumption. (This can
be seen by examining how much simpler it was to arrive at (17.52) than at
(17.50), which involved, for instance, the computation of the probability
distribution of default times.)
From an analytical perspective, the most tractable of the recovery
assumptions is likely the formulation involving fractional recovery of the
market value of the bond. Indeed, this assumption has the advantage of
making the valuation of defaultable bonds almost entirely analogous to
that of default-free bonds. One drawback of the recovery-of-market-value
assumption, however, is that one can no longer separately infer default
probabilities from observed market quotes using the simple steps outlined
in Chapter 16.

17.2.5 Extensions and Uses of Reduced-form Models
Most of the model-speciп¬Ѓc results derived thus far, such as the valua-
tion formulae for defaultable zero-coupon bonds with zero recovery and
with fractional recovery of face valueвЂ”equations (17.48) and (17.51),
respectivelyвЂ”came from the simple reduced-form model based on constant
intensity and riskless rate of interest and nonrandom recovery rates. To be
17.2 Reduced-form Approach 197

sure, these are over-simplistic assumptions, which were made solely for the
sake of analytical tractability and pedagogical convenience.
Richer models do exist with various degrees of complexity, ranging
from speciп¬Ѓcations with stochastic riskless rates but deterministically
time-varying intensity to fully stochastic models with uncertain recovery.
Examples of work in this strand of the literature include Lando (1998),
Duп¬ѓe and Singleton (1999), Jarrow and Yu (2001), Madan and
Unal (2000), Schonbucher (1998), and many others.
Reduced-form models are commonly used in practice to extract default
probability information from the prices of actively traded instruments and
use those probabilities to value, for instance, less liquid or more com-
plex credit derivatives. Nonetheless, given that we essentially did this
in Chapter 16 without ever having explicitly to resort to a model, one
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