ńņš. 6 |

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 ļ¬ve-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 ļ¬xed-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 ā’ Z(t, Tn )Q(t, Tn )] ā’ Z(t, Tj )[Q(t, Tjā’1 ) ā’ Q(t, Tj )]X

j=1

C= n

Z(t, Tj )Q(t, Tj )Ī“j

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

ļ¬xed-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 ļ¬gure 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 ļ¬rms 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

valuationsā”as implied, say, by survival probabilities derived from CDS

quotesā”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 ļ¬rm value

or structural approach to credit modeling, which traces its origins to the

work of Black and Scholes (1973)[9] and Merton (1974)[59]. Credit models

in this tradition focus on the analysis of the capital structure of individ-

ual ļ¬rms 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)[6] and Wilmott, Dewynne, and Howison (1999)[74], 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)[46]. The reduced-form approach does not directly

attempt to link defaults to the capital structure of the ļ¬rm. 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 diļ¬erent time horizons.

In Section 17.3 we brieļ¬‚y compare the structural and reduced-form

approaches, both on methodological and empirical grounds. That sec-

tion also highlights the main thrust of a āhybridā approachā”motivated

by the work of Duļ¬e and Lando (2001)[23]ā”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 ļ¬rst proposed by Black and Scholes

(1973)[9] and Merton (1974)[59]. 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 ļ¬rm 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 ļ¬rmā™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 ļ¬rm.

By way of the basic market value identity, which states that the market

value of the shareholderā™s equity is equal to the diļ¬erence between the

market value of the assets and liabilities of the ļ¬rm, 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 ļ¬rm.

The basic idea behind the Black-Scholes-Merton (BSM) model is very

straightforward. Default is quite simply deļ¬ned as a situation where, at

time T , when the ļ¬rmā™s debt, K, becomes due, the value of the ļ¬rmā™s

assets, A(T ), falls short of K:

default ā” A(T ) < K (17.2)

Figure 17.1 illustrates the main points of the model. The ļ¬gure shows the

evolution of the value of the ļ¬rm 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 ļ¬rm does not default. Under such circumstances, the ļ¬rmā™s

creditors receive K and the shareholders get to keep the residual A(T )ā’K.

Should A(T ) fall short of K at time T , however, the ļ¬rm defaults, with

the creditors taking over the ļ¬rm, 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 debtholderā™s payout at time T can be written as

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

As for the shareholders, they either receive A(T ) ā’ K or nothing at time

T . Their payoļ¬ at time T can be more succinctly written as

E(T ) = Max(A(T ) ā’ K, 0) (17.4)

Equations (17.2) through (17.4) summarize some of the key implications

of the simple BSM framework. Examining (17.2) ļ¬rst, we can see that,

in the context of the model, the default probability associated with this

hypothetical ļ¬rm 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 ļ¬rm having survived through t.

174 17. Main Credit Modeling Approaches

FIGURE 17.2. Payoļ¬s 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 ā’ 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 ļ¬rmā™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 debtholderā™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 ā’ 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 ļ¬rmā™s debt and that of the embedded put option:

Z d (T, T )K = K ā’ 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 = eā’R(t,T )(T ā’t) K ā’ p(t, A(t); T, K)

= Z(t, T )K ā’ 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 debtholdersā™ position is equivalent to being

long a riskless bond and short a put option written on the value of the

ļ¬rm. But who bought this put option? The shareholders implicitly did.

To see this ļ¬rst 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 ) ā’ K, 0) = A(T ) ā’ K + Max(K ā’ A(T ), 0)

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

write:

E(t) = A(t) ā’ eā’R(t,T )(T ā’t) K + p(t, A(t); T, K) (17.11)

which establishes that the shareholdersā™ position can also be thought of as

including a long position on a put option written on the value of the ļ¬rm,

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 ļ¬rm is such that

Z d (t, T ) = eā’R (t,T )(T ā’t)

d

Thus:

1

Rd (t, T ) = ā’ log Z d (t, T ) (17.12)

T ā’t

and the credit spread is trivially given by Rd (t, T ) ā’ 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 ļ¬rm 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 ļ¬rms, and the value of that option is higher the lower is the

credit quality of the ļ¬rm.

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 ļ¬rm, but we have not derived the price of

this option or shown explicitly how that price, and thus the ļ¬rmā™s credit

spread, relates to the capital structure of the ļ¬rm. Indeed, up until now all

of the results we have derivedā”default probabilities, recovery value, risky

bond prices and spreads, etc.ā”depend importantly on A, the value of the

ļ¬rm, 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)

= Āµdt + ĻdW (t) (17.13)

A(t)

how we model the evolution of Aā”see, for instance, Wilmott, Howison, and Dewynne

(1999)[74].

17.1 Structural Approach 177

where dW (t) is an inļ¬nitesimal increment in a standard Brownian motion,

and Āµ and Ļ 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 ļ¬rm value is lognormally

distributed, which greatly simpliļ¬es 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, Ļ) = Keā’r(T ā’t) N (ā’d2 ) ā’ A(t)N (ā’d1 ) (17.14)

with

log(A(t)/K) + (r + Ļ 2 /2)(T ā’ t)

ā

d1 ā” (17.15)

Ļ T ā’t

ā

d2 ā” d1 ā’ Ļ T ā’ t (17.16)

where, for simplicity, we have assumed that the term structure of riskless

interest rates is ļ¬‚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

ļ¬rmā™s credit spread, is increasing in

eā’r(T ā’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+Ī“t ā’ At

= ĀµĪ“t + Ļ t+Ī“t

At

where is a zero-mean normally distributed variable with variance Ī“t. 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)[41] and Mikosch (1999)[60] 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)[6] and Wilmott et al. (1999)[74]. 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 ļ¬rmā™s leverage ratio. This is consistent

with oneā™s intuition, in that more highly leveraged ļ¬rms tend to face wider

credit spreads.

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 ļ¬rmā™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 ļ¬rm, 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 ļ¬rm (A), is not observed in practice.

Thus, even for a hypothetical ļ¬rm with the simple debt structure assumed

in the basic BSM model, one is still presented with data challenges. What

one does observe are daily ļ¬‚uctuations in share prices of the ļ¬rm and the

book value of the ļ¬rmā™s liabilities (typically at a quarterly frequency). As

the number of shares outstanding for a given ļ¬rm is commonly a known

quantity, one can then estimate the market capitalization of the ļ¬rm 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 ļ¬rm, one can back out the implied values of A

and Ļ for a given assumption for the stochastic process for the value of the

ļ¬rm.7

17.1.4 Extensions and Empirical Validation

Major contributions to the structural approach to credit risk modeling

include the treatment of coupon-bearing bondsā”Geske (1977)[30] and

Geske and Johnson (1984)[31]ā”and the incorporation of stochastic riskless

rates into the frameworkā”Shimko, Tejima, and van Deventer (1993)[72].

The BSM framework has also been extended into the class of so-called

āļ¬rst-passageā models, which include the work of Black and Cox (1976)[10],

Leland (1994)[53], Longstaļ¬ and Schwartz (1995)[56], 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 ļ¬rm. For instance, ābalance-sheet noiseā may be introduced by

ambiguities in certain accounting deļ¬nitions.

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., Duļ¬e and Singleton

(2003)[25].

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 ļ¬rmā™s

debt. For instance, Leland (1994)[53] models the default boundary as being

the outcome of equity holdersā™ eļ¬orts to maximize the value of their stake

in the ļ¬rm.

Some of the basic features of ļ¬rst-passage models are illustrated in

Figure 17.3. Defaults are now assumed to occur at the ļ¬rst time that the

value of the ļ¬rm, A(t), crosses the ādefault boundary,ā B. In contrast, in

the basic characterization of the BSM framework, a default, if any, could

only occur at the maturity date of the ļ¬rmā™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 ā”labeled a non-default pathā”and one where A(t) crosses

the boundaryā”labeled a default path. We will review a simple ļ¬rst-passage

model in the next section.

A well-known oļ¬shoot of the BSM framework is the commercially avail-

able KMV model, an analytical tool provided by Moodyā™s KMV. This

model uses a large proprietary database of defaults to compute default

probabilities for individual issuersā”see Crosbie (2002)[16]. Similar to the

simple BSM setup, the KMV model looks at equity market volatility and

prices to infer the volatility and level of the ļ¬rmā™s value, which provide

a measure of the distance to defaultā”or how far, in terms of standard

deviations Ļ of its value A, the ļ¬rm is from its default boundary K in

Figure 17.1. Using proprietary methods, Moodyā™s KMV then translates

the distance from default into āexpected default frequencies,ā which is how

180 17. Main Credit Modeling Approaches

Moodyā™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 modelā™s implications for both the shape of the credit spread

curve and its level. For instance, Jones, Mason, and Rosenfeld (1984)[48]

reported that the BSM model tended to overprice corporate bonds (under-

predict credit spreads), but Delianedis and Geske (1998)[20] found that

BSM-style models have predictive power for rating migrations and defaults.

The lack of consensus among empirical researchers reļ¬‚ects both the fact

that there are diļ¬erent types of structural modelsā”some have more plau-

sible empirical implications than othersā”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 ļ¬nding 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 ļ¬rst-passage

models assume that the value of the ļ¬rm 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

ļ¬rm is suļ¬ciently 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

ļ¬rm. Hence, the likelihood of a near-term default by a ļ¬rm that is not

in ļ¬nancial distress is virtually zero, and so is the ļ¬rmā™s near-term credit

spread implied by the model. Indeed, regarding the shape of the credit

spread curve for a ļ¬rm that is not in ļ¬nancial 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 ļ¬‚at

or even downward sloping.8

One extension to the basic BSM model that was largely motivated by

the desire to generate a better ļ¬t between model-implied and observed

credit spread curves is the work of Zhou (1997)[76]. Zhou assumed that

the dynamics of the value of the ļ¬rm has two components: a continuous

component that is similar to that assumed in the traditional BSM and

ļ¬rst-passage models, and a discontinuous ājumpā component, which, as

the name suggests, allows the value of the ļ¬rm to change suddenly and

8

See, for instance, Fons (1994)[29] and Sarig and Warga (1989)[69].

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-diļ¬usion 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 ļ¬rst-passage

model with a ļ¬xed 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 ļ¬rm, A(t), evolves according to (17.13) and that the riskless yield

curve is constant and ļ¬‚at at r. In the context of this simple model, the

ļ¬rm 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 ļ¬rm will not default by a given future date T , where T may denote,

for example, the horizon over which one is exposed to the ļ¬rm. Thus, we

can then write:

Q(t, T ) ā” Probt {A(s) > B for all s ā (t, T )} (17.17)

where the notation s ā (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)ā”equation (17.13)ā”

there is a readily available formula for Q(t, T )ā”see, e.g., Musiela and

Rutkowski (1998)[61]:9

b(t) + (r ā’ Ļ 2 /2)(T ā’ t)

ā

Q(t, T ) = N

Ļ T ā’t

ā’b(t) + (r ā’ Ļ 2 /2)(T ā’ t)

2(rā’Ļ 2 /2)

ā

ā’e b(t)

N (17.18)

Ļ2

Ļ T ā’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 ļ¬rst-passage problems that do not

have an easily obtainable analytical solution.

182 17. Main Credit Modeling Approaches

where b(t) ā” 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 ļ¬rm 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 ļ¬rst compute the time-t value of the āpremiumā and āprotectionā 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

Ī·(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 ā’ 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 Īø(t) denote the value of the protection

leg, we can write

T

Īø(t) = (1 ā’ X) Z(t, v)[ā’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 ļ¬nding the value of S that makes

the two legs of the swap have equal value. This is given by

T

(1 ā’ X) Z(t, v)[ā’dQ(t, v)]

t

S= (17.21)

T

Z(t, v)Q(t, v)dv

t

Following Pan (2001)[65], if we now deļ¬ne the annualized probability of

default as

ā’ log(Q(t, T ))

Ė

h(t, T ) =

T ā’t

10

The normal and lognormal distributions are brieļ¬‚y discussed in Appendix B.

17.2 Reduced-form Approach 183

Ė ĀÆ Ė

and if we assume that h(t, T ) ā h for all t and T , i.e., h is approximately

constant as t and T vary, we obtain11

T ĀÆ

Z(t, v)eā’h(vā’t) hdv

ĀÆ

(1 ā’ X) ĀÆ

Sā = (1 ā’ X)h

t

(17.22)

T ĀÆ

Z(t, v)eā’h(vā’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 ļ¬rm, we

were able to price a credit default swap agreement written on that ļ¬rm.

17.2 Reduced-form Approach

Rather than tying defaults explicitly to the āfundamentalsā of a ļ¬rm, 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 Ļ„ 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 diļ¬erent outcomes involving Ļ„ . 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 ) ā” Probt [Ļ„ > T |Ļ„ > t] (17.23)

where time T in this speciļ¬c example is exactly one year from today, i.e.,

T ā’ 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 Ļ„ is determined

endogenously by the evolution of the value of the ļ¬rm, A(t). In con-

trast, in reduced-form models, the stochastic properties of Ļ„ are speciļ¬ed

as an exogenous process that is not directly related to the balance sheet of

the ļ¬rm.

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

Ė

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 ļ¬rst

month the investor will have er(1)(t1 ā’t0 ) in the bank, where r(1) is the

riskless rate prevailing that month, expressed on an annual basis, and t1 ā’t0

is the fraction of a year represented by the ļ¬rst month. Carrying on with

this exercise, and assuming no withdrawals, after two months the investorā™s

bank account balance will be

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

where r(2) is the rate applied to the second month and (t2 ā’ t1 ) is deļ¬ned

as above. Generalizing, after n months, the account balance will be

r(i)(ti ā’tiā’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 redeļ¬ned to mean the interest

rate corresponding to day i and (ti ā’ tiā’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 ā’ tiā’1 ) becomes inļ¬nitesimally 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 inļ¬nitesimally 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

Ī²(T ) ā” eā’

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 Ī²(T ).

With stochastic interest rates, the discount factor can no longer be

deļ¬ned 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 eā’

T

Ė r(v)dv

(17.25)

t

Ė

where Et [.] denotes the expected value of ā.ā 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 ā’ Q(t, U ) ā” Probt [Ļ„ ā¤ U |Ļ„ > 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 < Ļ„ < U |Ļ„ > t] = Probt [Ļ„ > T |Ļ„ > t] ā’ Probt [Ļ„ > U |Ļ„ > t]

= Q(t, T ) ā’ Q(t, U ) (17.27)

Equation (17.27) is an expression for the unconditional forward proba-

bility of default associated with this ļ¬rm. 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 ļ¬rm 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 ļ¬rm or

12

See Neftci (2002)[62] for additional non-technical insights into risk-neutral proba-

bilities and the derivation of an explicit expression for (17.25). Bjork (1998)[7] provides

further details.

186 17. Main Credit Modeling Approaches

of the economy. This is the conditional forward probability of survival,

Probt [Ļ„ > U |Ļ„ > T ]. By the Bayes rule, we can write13

Probt [Ļ„ > U |Ļ„ > t] Q(t, U )

Probt [Ļ„ > U |Ļ„ > T ] = = (17.28)

Probt [Ļ„ > T |Ļ„ > t] Q(t, T )

and hence the forward conditional probability of default regarding the

future time period [T, U ] is given by

Probt [Ļ„ ā¤ U |Ļ„ > T ] = 1 ā’ Probt [Ļ„ > U |Ļ„ > T ]

Q(t, U ) ā’ Q(t, T )

=ā’ (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 ), deļ¬ned as the risk-neutral default probability

associated with a given horizon divided by the length of the horizon14

Probt [Ļ„ ā¤ T |Ļ„ > t] 1 ā’ Q(t, T )

H(t, T ) ā” = (17.30)

T ā’t T ā’t

The forward default rate, as seen at time t, that corresponds to the future

period [T, U ] is analogously deļ¬ned as

Probt [Ļ„ ā¤ U |Ļ„ > T ] Q(t, U ) ā’ Q(t, T ) 1

H(t, T, U ) ā” =ā’ (17.31)

U ā’T U ā’T Q(t, T )

If we now let U ā” T +āT , we can deļ¬ne the time-t instantaneous forward

default rate as:

h(t, T ) ā” lim H(t, T, T + āT ) (17.32)

āT ā’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 deļ¬ned as Ļ„ > T ,

B as Ļ„ > U , and, given U > T , A&B corresponds to Ļ„ > U . (To be sure, all of the

corresponding probabilities would have to be deļ¬ned 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 diļ¬cult to see that

ā‚Q(t, T ) 1

h(t, T ) = ā’ (17.33)

ā‚T Q(t, T )

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

[Q(t, T + āT ) ā’ Q(t, T )] 1

h(t, T ) ā” lim ā’ (17.34)

āT ā’0 āT Q(t, T )

but, assuming that certain technical conditions on Q(.) are satisļ¬ed, the

expression within the square brackets is nothing more than the deļ¬nition

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 )

eā’

U

= Probt [Ļ„ > U |Ļ„ > 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 ļ¬rm will survive through time U is

Probt [Ļ„ > U |Ļ„ > t] = Q(t, U ) = eā’

U

h(t,v)dv

(17.36)

t

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

[Q(t, t + āt) ā’ Q(t, t)] 1

h(t, t) ā” lim ā’ (17.37)

āt Q(t, t)

ātā’0

but, by deļ¬nition, Q(t, t) = 1, and thus, for inļ¬nitesimally small dt,

Probt [Ļ„ ā¤ t + dt|Ļ„ > 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 āt, the probability of default

between t and t + āt, given no default by t, is approximately equal to

h(t, t)āt.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 ļ¬rst āarrivalā 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 Ī» per year, the time-t

risk-neutral probability of no default by time T can be written as17

Q(t, T ) = eā’Ī»(T ā’t) (17.39)

provided Ī» is time invariant and, of course, given no default by time t. In

the credit risk literature Ī» is commonly called the intensity of default or

the hazard rate.

If Ī» 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 Ī» changes

continuously over inļ¬nitesimally small time intervals we can write

Q(t, T ) = eā’

T

Ī»(v)dv

(17.40)

t

and thus, as suggested by a comparison between (17.40) and (17.36), with

deterministic Ī», the terms default intensity and forward default rate can

essentially be used interchangeably.

16

Given the deļ¬nition of derivative in classic calculus:

[Q(t, T + dt) ā’ Q(t, T )]

ā‚Q(t, T )

ā” lim ā’

ā‚T dt

dtā’0

we can write

ā‚Q(t, T )

Q(t, T + dt) ā Q(t, T ) + dt = Q(t, T )[1 ā’ h(t, T )dt]

ā‚T

For T = t, and given that Q(t, t) = 1, we can write:

1 ā’ Q(t, t + dt) ā h(t, t)dt

which becomes an equality if Q(t, T ) has a derivative at T = t and we let dt ā’ 0.

Thus, given that 1 ā’ Q(t, t + dt) = Probt [Ļ„ ā¤ t + dt|Ļ„ > t], we arrive at (17.38)ā”Neftci

(2002)[62] discusses basic notions from classic calculus that are relevant for ļ¬nance.

17

For a Poisson process with a constant mean arrival rate of Ī», a basic result from

statistics states that the time until the āļ¬rst arrival,ā Ļ„ , is a random variable with

probability density function Ī»eā’Ī»Ļ„ , 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 ļ¬nancial

condition of the ļ¬rm. With stochastic default intensity, it can be shown

that, under some technical conditions that go beyond the scope of this

bookā”see, for instance, Lando (1998)[51]ā”the expression for the time-t

survival probability of a given ļ¬rm can be written in a way that is entirely

analogous to (17.25):

Q(t, T ) = Et eā’

T

Ė Ī»(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 Ī» is assumed to be stochastic. For instance,

consider the value of default intensity at some future time v. That value,

Ī»(v), incorporates all information available at time v, including the values

of any stochastic factors that may aļ¬ect default intensity at time v, such as

the prevailing states of the economy and of the ļ¬rm 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 ļ¬rm surviving through time v.

Thus, whereas Ī»(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 ļ¬rm 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) = Ī»(t) (17.42)

which says that when defaults occur according to a Poisson process, todayā™s

(time-t) instantaneous default rate associated with the inļ¬nitesimally 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 [Ļ„ ā¤ t + dt|Ļ„ > t] = Ī»(t)dt (17.43)

Reduced-form models can essentially be characterized in terms of the

particular assumptions that they make regarding how Ī» 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 speciļ¬cation for the evolution of Ī»:

ĀÆ

dĪ»(t) = k[Ī» ā’ Ī»(t)]dt + Ļ Ī»(t)dW (t) (17.44)

where dW (t) is an inļ¬nitesimal increment in a standard Brownian motion.18

Equation (17.44) closely parallels the yield curve model proposed by Cox,

Ingersoll, and Ross (1985)[15]. In words, it says that, although the default

intensity associated with a given reference entity varies stochastically over

ĀÆ

time, it has a tendency to revert to its long-run mean of Ī», 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 Ļ„ , the time of default. This is the function G(s)

such that

Prob[s < Ļ„ ā¤ s + ds] = G(s + ds) ā’ G(s) ā g(s)ds

where g(s) is the probability density function (p.d.f.) of Ļ„ and the approx-

imation error in the above equation is negligible for very small values of

dsā”see Grimmett and Stirzaker (1998)[36], 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 < Ļ„ ā¤ s + ds] = Prob[Ļ„ > s] Prob[Ļ„ ā¤ s + ds|Ļ„ > 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):

ĀÆ

Ī»t+Ī“t ā Ī»t + k[Ī» ā’ Ī»t ]Ī“t + Ļ Ī»t t+Ī“t

where is a normally distributed random variable with mean zero and variance Ī“t. Thus,

we can think of Ī»t+Ī“t as a mean-reverting random variable with conditional variance

var[Ī»t+Ī“t |Ī»t ] equal to Ļ 2 Ī»t Ī“t .

19

G(.) is the so-called cumulative distribution function of Ļ„ . Appendix B reviews key

concepts surrounding cumulative distribution and probability density functions.

17.2 Reduced-form Approach 191

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

gt (s)ds = Probt [s < Ļ„ ā¤ s + ds] = Et eā’

s

Ė Ī»(u)du

Ī»(s)ds (17.45)

t

where we are assuming that ds is inļ¬nitesimally small and that s ā„ t.

For nonrandom intensity, we can make use of (17.40), and equation

(17.45) simpliļ¬es to

gt (s)ds = Probt [s < Ļ„ ā¤ s + ds] = Q(t, s)Ī»(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 todayā™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 issuerā™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 eā’

T

Ė

d [r(v)+Ī»(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 ) = eā’R (t,T )(T ā’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 aļ¬ect oneā™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 brieļ¬‚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

ā’ log(Z0 (t, T ))

d [f (t, v) + h(t, v)]dv

d t

R (t, T ) = = (17.47)

T ā’t T ā’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 ) = eā’

T

f (t,v)dv

t

The risky bond spread S(t, T ), deļ¬ned as Rd (t, T ) ā’ R(t, T ), is given by

T

h(t, v)dv

t

S(t, T ) =

T ā’t

In light of (17.36), we can write

ā’ log(Q(t, T ))

S(t, T ) =

T ā’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

spread.

If we assume that both r and Ī» are deterministic and time invariant,

equation (17.46) reduces to

Z0 (t, T ) = eā’(r+Ī»)(T ā’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 + Ī», over that on a riskless bond, r,

is equal to the default intensity of the issuer of the defaultable bond:

S(t, T ) = Ī»

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 Ī¦(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 Ī¦(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

Ī¦(t, T ) = X Z(t, v)gt (v)dv (17.49)

t

where gt (v) ā” Et [eā’ t Ī»(u)du Ī»(v)] is the probability density function of the

v

Ė

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 ) + Ī¦(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 Ī» are

time invariant, the valuation formula for the defaultable zero-coupon bond

with recovery value X can be shown to be:

XĪ»

Z d (t, T ) = eā’(r+Ī»)(T ā’t) + (1 ā’ eā’(r+Ī»)(T ā’t) ) (17.51)

r+Ī»

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

instrumentā™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)[46]. 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 ā¤ X ā¤ 1. (Obviously, X = 1 would constitute

the case of a riskfree bond to begin with.) By āotherwise equivalent,ā 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 diļ¬er 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 1ā’Q(t, T ), respectively.

Thus the risk-adjusted expected value of the bondholderā™s payout, based

on information available at time t, is

Q(t, T ) + [1 ā’ 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 ā’ 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 ā’ X) + Z(t, T )X

d

(17.52)

and a comparison of (17.50) and (17.52) makes it clear that one obtains

diļ¬erent 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 Duļ¬e and Singleton (1999)[24]. Duļ¬e

and Singleton essentially assume that upon a default at time Ļ„ the bond

loses a fraction L of its market value. (This is equivalent to saying that

upon default the bondholder recovers a fraction 1 ā’ L of the no-default

market value of the bond.) Duļ¬e and Singleton show that, under such

circumstances, the value of a defaultable zero-coupon bond would satisfy

Z d (t, T ) = Et eā’

T

Ė [r(s)+LĪ»(s)]ds

(17.53)

t

which simpliļ¬es to

Z d (t, T ) = Z(t, T )Et eā’

T

Ė LĪ»(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

Eļ¬ect 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 ļ¬ve-year bond .779 5.00

a Based on the assumption of ļ¬‚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 eļ¬ect of the recovery assumption on the valuation of a

zero-coupon bond, we compare the prices of a ļ¬ve-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

eļ¬ect 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

eļ¬ectively received in full only at the bondā™s original maturity date so that

the investor eļ¬ectively has to be compensated for this ādelayā 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 ā’ 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 diļ¬erences in model-implied bond prices can be signiļ¬cantly 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 tableā™s footnote. Diļ¬erences in bond prices across diļ¬erent recovery

assumptions can also be much smaller than those in the table for coupon-

bearing defaultable bonds, as shown by Duļ¬e and Singleton (1999)[24]

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 Oā™Kane and Schlogl (2001)[64], 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-speciļ¬c results derived thus far, such as the valua-

tion formulae for defaultable zero-coupon bonds with zero recovery and

with fractional recovery of face valueā”equations (17.48) and (17.51),

respectivelyā”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 speciļ¬cations 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)[51],

Duļ¬e and Singleton (1999)[24], Jarrow and Yu (2001)[47], Madan and

Unal (2000)[58], Schonbucher (1998)[70], 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

ńņš. 6 |