C ’ N ’1 (θ) 1 ’ β 2

±= (19.31)

β

which, together with (19.26), implies

C ’ N ’1 (θ) 1 ’ β 2

ω ’1 (θ) = (19.32)

β

17

By symmetry of the normal distribution we mean the fact that, for any normally

distributed random variable ± that has an expected value of zero,

Prob[± ≥ U ] = Prob[± ¤ ’U ]

where U is any arbitrary real number.

230 19. The Basics of Portfolio Credit Risk

and we arrive at our ¬nal expression for the (approximate) unconditional

loss distribution of a large homogeneous portfolio:

√

C’N ’1 (θ) 1’β 2

Prob[L ¤ θ] = N (’ ) (19.33)

β

19.3 Default Correlation and Loss Distribution

We are now ready to start tying together some of the di¬erent topics dis-

cussed in this chapter by examining how the probability distribution of

future losses is a¬ected by the degree of default correlation among the

issuers in the portfolio. This analysis also expands on an early exercise on

correlation and loss distribution, discussed in Chapter 10.

Our goal is to examine the crucial role that default correlation plays

in the determination of a portfolio™s loss distribution function and, as we

shall see in greater detail in Chapters 20 and 21, in the valuation of multi-

name credit derivatives. To highlight that role, we shall examine two large

homogeneous portfolios”which we shall call portfolios A and B”that are

identical in every respect, except for their extent of return, and, thus,

default correlation. In particular, for a given horizon, the individual default

probabilities of the entities represented in either portfolio are assumed to be

5 percent, which, following the spirit of equation (19.25), means that both

portfolios have expected default-related losses of 5 percent. (For simplicity,

we continue to assume zero recovery rates for all entities.) As for their

degree of correlation with the common factor, we shall assume that portfolio

A has a β of 0.2, and portfolio B™s β is 0.5.

Figure 19.5 shows the unconditional loss distribution functions for the

two portfolios described above. The loss distributions are quite di¬erent

despite the fact that both portfolios have the same expected loss. In par-

ticular, the loss distribution of portfolio A (low β) shows a virtually zero

probability of default-related losses amounting to more than 15 percent

of the portfolio. In contrast, for portfolio B (high β), that probability is

distinctively positive, although still relatively small. Portfolio B also has a

higher probability of very small losses than does portfolio A, re¬‚ecting the

fact that its higher correlation results in a greater tendency for its refer-

ence entities to either survive or default together. In statistical parlance,

the portfolio with the greater default correlation has a loss distribution

with “fatter tails,” i.e., the loss distribution of portfolio B assigns greater

odds to extreme events than does that of portfolio A.

From a practical perspective, the basic insights derived from the analysis

of portfolios A and B are very powerful. First, the degree of default cor-

relation in a portfolio can dramatically a¬ect its risk characteristics. For

instance, while holding a portfolio that includes a large number of issuers

19.4 Monte Carlo Simulation: Brief Overview 231

FIGURE 19.5. Unconditional Loss Distribution Functions for Two Large Homo-

geneous Portfolios

generally contributes to diversi¬cation across di¬erent types of risks, that

diversi¬cation may be signi¬cantly reduced if the degree of default corre-

lation among the issuers is high. Indeed, higher odds of extreme events

can be thought of as reduced diversi¬cation. For instance, as we discussed

in Chapter 10, in the extreme case of perfect correlation either all of the

issuers in the portfolio survive or default together, which is akin to holding

only one asset (no diversi¬cation).

A second basic insight from Figure 19.5 regards the valuation of credit

derivatives that involve some tranching of credit risk, such as baskets and

portfolio default swaps, which we will examine more fully in the next two

chapters. As we saw in Chapters 9 and 10, the valuation of these derivatives

can be importantly a¬ected by the probability distribution of losses in

the underlying portfolio, which, as we have just seen, depends crucially

on the degree of default correlation among the entities represented in the

portfolio.

19.4 Monte Carlo Simulation: Brief Overview

As an alternative to the large portfolio approximation, one can use Monte

Carlo simulation methods to compute the (approximate) loss distribution

232 19. The Basics of Portfolio Credit Risk

function of a portfolio. In their simplest form, described below, simulation

methods place less mathematical demands on the user, at the cost of longer

computer running times than the version of the large portfolio method

discussed in Section 19.2.18 An advantage of simulation-based methods is

that they are best applied to smaller portfolios, for which the large portfolio

approximation method is less suitable. In addition, they can be easily used

in the analysis of heterogeneous portfolios and in versions of the model that

allow for more than one common factor.

We can use equations (19.6) and (19.9)”repeated below for

convenience”to illustrate the basic principle of the Monte Carlo simulation

approach:

Default by ¬rm i <=> Ri,t ¤ Ci

1 ’ βi

2

Ri,t = βi ±t + i,t

where, as assumed before, ± and i are mutually independent random

variables that are normally distributed with zero mean and unit variance.

The basic thrust of simulation-based methods is very straightforward. It

consists of generating a large number of draws from the standard normal

distribution for ± and, for all i in the portfolio, for i . For instance, for

a portfolio with 20 reference entities, each draw will consist of 21 values

randomly selected according to the standard normal distribution, and, for

each draw, one compares the resulting return Ri for each entity to its

default boundary, Ci , to determine whether or not a default has occurred.

After a su¬ciently large number of draws, one can count the number of

defaults and estimate the probability distribution of losses as

Prob[j defaults] = Mj /M

where Mj is the number of draws where there were j defaults in the

portfolio, and M is the total number of draws.19

We illustrate the simple simulation-based method described above for

an equally weighted homogeneous portfolio with 40 reference entities, each

with a one-year individual default probability of 5 percent and a degree of

correlation (β) of 0.2. The solid line in Figure 19.6 shows the correspond-

ing loss distribution function for this portfolio, based on 500,000 draws

from the standard normal distribution. To keep the analysis comparable

18

Hull (2003)[41] provides an overview of simulation techniques in ¬nance.

19

If one is interested in the distribution of percentage losses in the portfolio, one can

modify these calculations accordingly. For instance, Mj could stand for the number of

draws where percentage losses in the portfolio amounted to j percent.

19.4 Monte Carlo Simulation: Brief Overview 233

40

beta=0.2

35 beta=0.5

30

probability (percent)

25

20

15

10

5

0

0 5 10 15 20 25 30 35 40 45 50

percentage loss

FIGURE 19.6. Unconditional Loss Distribution Functions for Smaller Portfolios

(based on 500,000 simulations of the model)

to that of large portfolios, we also computed the probability loss distribu-

tion for an otherwise identical portfolio with a β of 0.5. We arrive at some

of the same conclusions discussed in Section 19.3. Portfolios with a larger

degree of default correlation have more probability mass at tail events than

do portfolios with less correlation. Again, we see the bene¬ts of e¬ective

diversi¬cation (less default correlation) at work.

19.4.1 How Accurate is the Simulation-Based Method?

We have mentioned some desirable features of the simulation-based

method”such as ease of use and applicability to both small and het-

erogeneous portfolios, as well as to multifactor models”and a poten-

tial drawback”Monte Carlo simulations tend to be computer intensive.

Simulation-based methods would be of limited value, however, if the result-

ing loss distribution function were a poor representation of the true function

implied by the model”equation (19.24)”or if it took an unreasonably

large number of simulations”and thus substantial computing time”for

the simulation method to get it right.

Figure 19.7 takes an informal look at the accuracy of the Monte Carlo

simulation approach for two portfolios that are identical in every respect

234 19. The Basics of Portfolio Credit Risk

FIGURE 19.7. How Accurate is the Simulation-based Method? (Homogeneous

equally weighted portfolios with 100 assets; total face value =

$100,000)

except for their degree of asset correlation. In particular, each port-

folio is composed of 100 assets with individual default probabilities of

5 percent and zero recovery rates. For each portfolio, the ¬gure shows

“actual” (analytical) values of the loss distribution, as well as a set of

Monte-Carlo-based values. The results suggest that, even for computations

involving a number of simulations as low as 50,000”which take only a few

seconds to run in a well equipped laptop”the Monte Carlo method seems

to do a very good job capturing both the level and the shape of the true

loss distribution function.

The results in Figure 19.7 are only illustrative, however. A more formal

evaluation of the Monte Carlo approach would involve examining the results

of a large number of simulation exercises”for instance, running 100,000

simulations of the model 1,000 times and computing the mean and standard

deviation of all 1,000 results”so that one could look at the variability of

the ¬nal results. It should also be noted, that the simple simulation method

described in this chapter can be improved considerably through the use of

19.4 Monte Carlo Simulation: Brief Overview 235

“variance-reduction” techniques, which are designed to reduce the amount

of random noise that is inherent in simulation-based methods.20

19.4.2 Evaluating the Large-Portfolio Method

We can now compare results obtained through the large-portfolio approx-

imation method to those generated by the simulation-based approach,

which, as we have just seen, can be made very accurate. Similar to the

spirit of the last subsection, one can view this exercise as a very infor-

mal evaluation of the large-portfolio approximation approach™s ability to

capture the main features of the loss distribution function of progressively

smaller portfolios under di¬erent correlation assumptions.

Figure 19.8 summarizes the main ¬ndings. We consider several hypo-

thetical portfolios designed along the same lines as the ones examined thus

far in this chapter. The top panel shows the unconditional loss distribution

FIGURE 19.8. An Informal Evaluation of the Large-Portfolio Approximation

Method

20

See, for instance, Hull (2003)[41] for an overview of variance reduction techniques.

236 19. The Basics of Portfolio Credit Risk

functions for four equally weighted homogeneous portfolios that are identi-

cal in every aspect, except for the number of reference entities included in

the portfolio, which varies from 100 to 1,000. For all portfolios, the default

probabilities of the individual entities are assumed to be 5 percent and

β = 0.2. The Monte-Carlo-based results were derived based on 500,000

simulations of the model. The ¬gure also shows the loss distribution func-

tion based on the large-portfolio approximation. Not surprisingly, the

approximation works best for the largest portfolio, but it is noteworthy

that even for the 200-asset portfolio, the large-portfolio approximation

seems to have been able to capture the general shape and level of the

loss distribution.

The lower panel of Figure 19.8 depicts results for the same cases examined

in the upper panel, but we now examine scenarios with a bit more return

correlation among the entities in the portfolios, β = 0.5. The results suggest

that the approximation works relatively well, especially as we move away

from the left end of the distribution. On the whole, the ¬gure suggests

that the large-portfolio approximation seems to work “better” for portfolios

with higher correlation than for those with lower correlation. While we are

being vague here about what better means in this context, our intuition

would suggest that in the limit case of perfect correlation, the portfolio

essentially behaves as a single asset and the loss distributions of small and

large portfolios should essentially become the same.

Once again, we should emphasize that this section provides only an infor-

mal evaluation of the large-portfolio approximation approach. Our purpose

here was only to provide some of the ¬‚avor of the practical and methodolog-

ical issues that one is likely to face when dealing with real-world portfolios

and more advanced techniques. Issues not addressed here include the fact

that real-life portfolios are not strictly homogeneous, which would add

another source of approximation error to the large portfolio approach.

One ¬nal point about Figure 19.8. The loss distribution functions of

the various portfolios shown is remarkably similar, especially for the port-

folios with more than 100 reference names, although one can see a pattern

of slightly less dispersion of likely losses”more of the probability mass is

in the center of the distribution”for the larger portfolios. Yet, from an

investment and portfolio management perspective, it may be substantially

more costly to monitor, say, a portfolio with more than 500 names, as

opposed to one with 200. (Not to mention that, in most markets, one

may be hard pressed to ¬nd 500+ names that ¬t the pro¬le sought by the

investor/manager.) One small simple lesson learned from Figure 19.8 then

is that the diversi¬cation bene¬ts, in terms of less disperse loss distribu-

tions, achieved by adding more and more names to the portfolio eventually

are likely to be outweighed by the potentially higher costs of construct-

ing and managing portfolios in which an ever larger number of entities is

represented.

19.5 Conditional vs. Unconditional Loss Distributions 237

FIGURE 19.9. Conditional vs. Unconditional Loss Distribution Functions

19.5 Conditional vs. Unconditional Loss

Distributions

Suppose one has a view as to where the common factor will be by a

certain date and then computes the loss distribution function of a given

portfolio based on that view. The resulting function is none other than

the conditional loss distribution function that we discussed earlier in this

chapter. Suppose, in particular, that one makes the seemingly reasonable

forecast that ± will be at its expected value of zero at the end of the rele-

vant time horizon and then computes the corresponding (conditional) loss

distribution of the portfolio.

How much o¬ would one be by making risk assessments based on partic-

ular views regarding the common factor? Figure 19.9 answers this question

for the portfolio described in the previous section, with β set to 0.5. It

shows that conditioning the distribution of losses on the expected value

of ± would lead one to underestimate signi¬cantly the default risk in the

portfolio, exposing the danger of what would otherwise look like a sensible

assumption regarding the evolution of the common factor. More generally,

ignoring the uncertainty related with future values of the common factor

can lead one to understate substantially the likelihood of large losses in the

portfolio.

238 19. The Basics of Portfolio Credit Risk

19.6 Extensions and Alternative Approaches

We discussed a simple approach for modeling the credit risk in a portfolio

and described three basic methods for deriving a portfolio™s loss distribu-

tion function. The ¬rst method involved the analytical derivation of the

loss distribution function. The second technique was the large-portfolio

approximation motivated by the work of Vasicek (1987)[73] and others.

The third was a simple simulation-based method that can also be used for

small portfolios. We also highlighted the key role that default correlation

plays in determining the range of likely losses that can be experienced in a

portfolio.

There are several directions in which the basic model discussed in this

chapter can be extended. For instance, portfolios with heterogeneous assets

can be examined in a relatively straightforward manner with the help of

simulation-based methods, a topic that we examine further in Chapter 20.

Other extensions include allowing for defaults to take place at any time”

rather than only at the maturity date of the contract”the treatment of

accrued premiums, and the analysis of multi-period contracts”an issue

that we address in Chapter 23.

Additional modeling approaches for examining portfolio credit risk fall

along the lines of the more advanced versions of the single-default models

that we studied in Part III of the book. For instance, they include both

intensity- and ratings-based models. We brie¬‚y discuss examples of such

models in Chapter 22.

20

Valuing Basket Default Swaps

We now start applying some of the techniques introduced in Chapter 19 to

the valuation of basket default swaps, the mechanics of which we discussed

in some detail in Chapter 9. We start by reviewing some of the basic ideas

addressed in that chapter, which mostly centered on two-entity baskets,

and then move on to the valuation of more realistic cases involving baskets

with several reference entities.

20.1 Basic Features of Basket Swaps

As we saw in Chapter 9, a basket default swap is a credit derivative contract

where the protection seller agrees to take on some of the credit risk in a

basket (portfolio) of reference entities. For instance, in the ¬rst-to-default

(FTD) basket the protection seller commits to making a payment to the

protection buyer to cover the ¬rst default among the entities referenced by

the basket. As with a standard credit default swap, the default-contingent

payment made by the protection seller is typically equal to the face value

of the defaulted asset minus its recovery value. In exchange for this kind of

protection, the protection buyer makes periodic payments to the seller”the

FTD premium”until the maturity date of the contract.

After payments associated with the ¬rst default in the FTD basket are

settled, both parties are relieved of further obligations under the contract.

Thus, assuming the protection buyer bought no additional default pro-

tection on the basket, any losses related to further defaults among the

240 20. Valuing Basket Default Swaps

remaining reference entities in the basket will be borne out entirely by the

protection buyer. Similar to other credit derivative contracts, if no default

takes place during the life of the contract, the protection seller keeps on

collecting the FTD premium from the buyer until the expiration of the

contract.

Second- and third-to-default baskets are similarly de¬ned. We will take a

quick look at the second-to-default basket towards the end of this chapter.

20.2 Reexamining the Two-Asset FTD Basket

Continuing with a brief review of Chapter 9, we reexamine the valua-

tion principles surrounding the simplest basket swap, one that references

only two debt issuers, but we now start to incorporate some of the results

discussed in Chapter 19.

For a given time horizon, say one year, let ωA and ωX denote the risk-

neutral default probabilities associated with XYZ Corp. and AZZ Bank,

the two reference entities included in the basket. We further assume that

the default correlation between the two entities is ρA,X , which, as we saw

in Chapter 19”equation (19.5)”can be written as

ωA&X ’ ωA ωX

ρA,X = (20.1)

ωA (1 ’ ωA ) ωX (1 ’ ωX )

where ωA&X is the probability that both XYZ and AZZ will default within

a speci¬c horizon.

Let us consider a one-year FTD basket involving only one premium pay-

ment date, at the end of the contract. The probability that the protection

seller will have to make a payment under the contract is simply equal to

the probability ωA or X of at least one default in one year™s time. From

Chapter 9, we know:

ωA or X = ωA + ωX ’ ωA&X (20.2)

For simplicity, we shall assume that both reference entities have a zero

recovery rate and that defaults, if any, will only take place in one-year™s

time. We further assume that the total notional amount of the basket is

$10 million, with each reference entity accounting for exactly half of the

total. Thus, the expected value of the payment made by the protection

˜

seller, denoted below as E[protection payment], is

E[protection payment] = ωA or X — $5 million + (1 ’ ωA or X ) — 0 (20.3)

˜

i.e., with probability ωA or X the protection seller will have to pay for one

default”$5 million given our assumptions”and with probability 1’ωA or X

20.3 FTD Basket with Several Reference Entities 241

there will be no default in the basket, in which case the protection seller

will not have to make any payment.

Given that the contract typically has zero market value at its inception

(no money changes hands at the inception of the basket swap), and, again,

assuming a single premium payment date, at the end of the contract, it

must be the case that the premium paid by the protection buyer is equal

to the expected payment made by the protection seller.1 Thus, the value

of the FTD premium payment is:

Sbasket = ωA or X — $5 million (20.4)

which is customarily quoted in terms of basis points (see Section 6.1). For

instance, if ωA or X is equal to .01”a 1 percent probability that either one

of the reference entities will default in one-year™s time”then the premium

for this one-year FTD basket would be 100 basis points, and the total

premium paid by the protection buyer would be $50,000.

A few additional points are worth remembering about the two-asset FTD

basket. First, as we have just seen, the key ingredient in the determination

of the FTD premium is the probability that any one of the reference entities

in the basket will default during the life of the contract. Second, that prob-

ability depends importantly on the default correlation of the two entities.

Indeed, as we saw in Chapters 9 and 19 (Section 19.1.1), as we approach

the polar cases of (i) perfect correlation, ρA,X = 1, and (ii) no correlation,

ρA,X = 0, coupled with a su¬ciently low value for the probability that

both reference entities will default while the contract is in force, the FTD

basket premium tends to the default probability of the weaker asset and to

the sum of the default probabilities of each asset, respectively.2

20.3 FTD Basket with Several Reference Entities

In practice, basket swaps are generally written on more than two reference

entities. The basic results derived from the two-asset case do carry over to

baskets with several entities, but the valuation exercise does become a bit

more involved.

20.3.1 A Simple Numerical Example

We start with a simple example that can be used to ¬x some important

points regarding more realistic cases involving baskets with more than two

1

Unless otherwise indicated, throughout most of Part IV, we shall assume that future

premiums are discounted at the same rate as expected future protection payments.

2

The reader is invited to verify this using the diagrams discussed in Section 19.1.

242 20. Valuing Basket Default Swaps

TABLE 20.1

Basic Valuation of Five-Asset Basket Swapsa

(Total notional = $50 million; individual recovery rates = 0)

Possible Outcome Payment by FTD Payment by STD

protection sellerb protection sellerb

outcomes probabilities

(No. of defaults) (percent) ($ millions) ($ millions)

(1) (2) (3) (4)

0 90.0 0 0

1 6.0 10 0

2 3.0 10 10

3 0.5 10 10

4 0.3 10 10

5 0.2 10 10

a Each reference entity represents an equal portion of the portfolio ($10 million).

b Values shown in columns 3 and 4 are the total payments made by FTD and STD

protection sellers under each possible default outcome.

reference entities. Table 20.1 shows all possible default outcomes for a one-

year basket swap that references ¬ve entities, along with the risk-neutral

probabilities, shown in column 2, associated with each outcome.3 There

are six such possible outcomes, ranging from no defaults to defaults by

all entities referenced by the basket. For instance, there is a 90 percent

probability that none of the entities referenced by the basket will default

during the life of the basket and a 6 percent probability that only one of

the entities will default.

We now consider a reference portfolio with the characteristics featured

in Table 20.1. The total face value of the portfolio is $50 million, and each

of the reference entities accounts for an equal piece, $10 million, of the

portfolio. We continue to assume a zero recovery rate for all the entities

referenced by the basket.

The third column in Table 20.1 shows, for each possible default out-

come, the payment made by a protection seller in a ¬rst-to-default basket

swap written on the portfolio. Naturally, if there are no defaults, the

FTD protection seller owes nothing to the protection buyer, but if at

least one of the reference entities defaults, she is liable for the entire loss

associated with that asset, $10 million in this example. Note that, pro-

vided there is at least one default, the total payment made by the FTD

3

Note that columns 1 and 2 essentially correspond to the loss distribution function

of the portfolio.

20.3 FTD Basket with Several Reference Entities 243

protection seller is always $10 million as she is only liable to cover the ¬rst

default.

What would be the FTD premium for this ¬ve-asset FTD basket?

Following the same logic outlined in the valuation of the two-asset basket,

if, as customary, the contract has zero market value at its inception, then

the FTD premium owed by the protection buyer to the protection seller

should be equal to the (risk-neutral) expected value of the payment made

by the protection seller. Armed with the probabilities shown in Table 20.1,

this expected value is $1 million:

FTD premium = .9 — 0 + .1 — $10 million = $1 million

i.e., with probability 90 percent there is no default among the entities

referenced by the basket, and the protection seller makes no payment

under the contract; but with probability 10 percent there is at least one

default, in which case the protection seller owes the buyer the $10 million

loss associated with the ¬rst default. Thus the premium is $1 million for

$10 million worth of protection, which amounts to 1,000 basis points.

The last column of Table 20.1 illustrates possible payments made by a

protection seller in a second-to-default (STD) basket swap written on the

portfolio. By de¬nition, such a seller would owe nothing under the terms

of the STD agreement if less than two defaults take place, but would be

liable for $10 in the event of a second default, and nothing more for any

additional defaults. Proceeding as with the FTD basket, one would ¬nd

that the STD premium amounts to 400 basis points, considerably less than

that of the FTD basket, given a much smaller probability of at least two

defaults than that associated with one default.

20.3.2 A More Realistic Valuation Exercise

Table 20.1 was an illustration used to provide some intuition about the

valuation of basket default swaps. In real-world situations, however, one

will not be presented with the risk-neutral probabilities associated with

every possible default outcome regarding the entities referenced in a basket

swap. Indeed, arriving at a table such as Table 20.1 is perhaps the most

crucial step in the basket swap valuation exercise. We outline below a

simple method for estimating the needed default probabilities.

Consider a one-year FTD basket that references ¬ve entities, each with

di¬erent, but unknown, default probabilities. As in the previous section, the

total notional amount of the basket is $50 million, equally divided among

the ¬ve reference entities, and all entities have a zero recovery rate. To

avoid the need to model explicitly the joint probability density of default

times, we follow the traditional BSM framework”see Chapter 17”and

assume that defaults, if any, only occur at the maturity date of the contract,

244 20. Valuing Basket Default Swaps

TABLE 20.2

Reference Entities in a Hypothetical Five-Asset Basket Swap

(Total notional = $50 million; individual recovery rates = 0)

Reference entity CDS premium Notional amount

(basis points) ($ million)

(1) (2) (3)

Entity #1 25 10

Entity #2 45 10

Entity #3 50 10

Entity #4 60 10

Entity #5 100 10

an assumption that also simpli¬es the treatment of accrued premiums.4

We also assume that one-year CDS contracts written on each reference

are negotiated in a liquid market, and Table 20.2 lists the corresponding

premiums.

Given the information in Table 20.2, we are then asked to value a ¬rst-

to-default basket swap written on this portfolio. To do so we shall rely

on the one-factor credit risk model discussed in Chapter 19, with a slight

modi¬cation to allow us to leave the homogeneous portfolio case that was

examined in that chapter. Namely, for each of the ¬ve entities in the basket,

we assume that their returns vary in response to a common factor, ±, and

an entity-speci¬c factor i , where the subscript i = 1, . . . 5 identi¬es each

reference entity. Thus, as in equation (19.9):

1 ’ βi

2

Ri,t = βi ±t + (20.5)

i,t

where Ri,t , ±, and i have been standardized as discussed in Chapter 19,

and Ri is the return on owning reference entity i.

Recall that the model de¬nes default as a situation where the return Ri

reaches or falls below a given threshold Ci :

Default by entity i <=> Ri ¤ Ci (20.6)

and thus the probability ωi of default by entity i by the maturity date of

the contract is given by

ωi = Prob[Ri ¤ Ci ] = N (Ci ) (20.7)

4

Chapter 17 discussed the notion of the probability density of default time in a

single-issuer context.

20.3 FTD Basket with Several Reference Entities 245

where N (.) is the cumulative density function of the standard normal

distribution.

Given that our interest lies in risk-neutral probabilities, ωi can be

obtained from the observed CDS premiums for each entity”see Chapters 6

and 16”and Ci can be computed by inverting (20.7), as discussed in

Chapter 19. Lastly, to parameterize βi , one can regress stock returns for

each of the entities on the common factor ±, where ± could correspond, for

instance, to standardized stock market returns.5 To keep things simple, we

shall assume that βi = .5 for all companies.

We now have all ingredients necessary to run the valuation exercise.

Using the simulation-based method described in Chapter 19, we obtain

the results presented in Table 20.3. Such results are analogous to the

information presented in Table 20.1, except that we have now derived

the probabilities associated with each default outcome on the basis of

market data”CDS premiums and equity returns”and of an explicit port-

folio credit risk model. Carrying on with the same calculations done for

Table 20.1, we arrive at the result that the FTD basket premium for this

portfolio is about 265 basis points, which is well in excess of the CDS

premiums for any one of the reference entities included in the basket.

TABLE 20.3

Valuation of a Five-Asset FTD Basket Swapa

(β = .5; total notional = $50 million; individual recovery rates = 0)

Possible Outcome Payment by FTD Loss to unhedged

protection sellerb

outcomes probabilities investor

(No. of defaults) (percent) ($ millions) ($ millions)

(1) (2) (3) (4)

0 97.35 0 0

1 2.51 10 10

2 0.13 10 20

3 0.01 10 30

4 0.00 10 40

5 0.00 10 50

a The referenced portfolio is the one described in Table 20.2.

bValues shown in column 3 are the total payments made by the FTD protection seller

under each possible default outcome.

5

This regression-based approach is only a rough approximation, however, because

the resulting correlations are computed based on estimates of actual probabilities, rather

than risk-neutral probabilities.

246 20. Valuing Basket Default Swaps

As we discussed in Chapter 9, from the standpoint of investors (protec-

tion sellers), the FTD basket swap represents an opportunity to leverage

one™s credit exposure: While the protection seller is exposed to the credit

risk in debt instruments that total $50 million in terms of notional amount

in the example just described, the actual potential loss is limited to

$10 million, all while earning a premium that corresponds to a credit qual-

ity that is well inferior to that of each individual entity referenced in the

portfolio.

To conclude this subsection, we compare the expected loss of the FTD

protection seller to that of an unhedged investor who owns the entire portfo-

lio.6 The total default-related losses experienced by such an investor under

each possible default outcome are shown in the last column of Table 20.3.

If we multiply each element of column 2 by its counterpart in column 4,

and sum the resulting numbers across all possible default outcomes, we

¬nd that the unhedged investor faces an expected loss of about 0.56 per-

cent of the entire portfolio, compared to an expected loss of 2.65 percent

for the FTD protection seller”column 2 times column 3. Note, however,

that the maximum loss of the unhedged investor is ¬ve times as much as

that of the FTD protection seller. Moreover, for this particular portfo-

lio, the expected (dollar) losses of the FTD protection seller”who is long

only the ¬rst loss”and of the unhedged investor”who is long the entire

portfolio”are $265,000 and $280,000, respectively.

20.4 The Second-to-Default Basket

Let us brie¬‚y revisit the second-to-default (STD) basket, which is a con-

tract where default protection is bought and sold for the second default,

and the second default only, among all the entities in a given portfolio.

Column 2 in Table 20.3 has all that we need to value a STD basket writ-

ten on the portfolio described in Table 20.2, once we note the fact that

the potential losses of the STD protection seller are zero if the number

of defaults falls below 2 and $10 million thereafter (see Table 20.1). The

reader is then invited to verify that the STD premium for our hypothetical

portfolio amounts to about 14 basis points. The fact that the STD pre-

mium is so low re¬‚ects the very small probabilities associated with more

than one default in the overall portfolio.

6

By an unhedged investor, we mean one who is holding the portfolio, but who has

not hedged his or her credit risk exposure via baskets, credit default swaps, or any other

hedging vehicle.

20.5 Basket Valuation and Asset Correlation 247

20.5 Basket Valuation and Asset Correlation

In Chapter 9, and also in Section 20.2, we used the simple two-asset basket

example to suggest that default correlation can have a signi¬cant e¬ect

on the valuation of basket swaps. We can now verify this result using the

more realistic basket described in Section 20.3.2. To do this we propose

the following exercise: compute the one-year FTD premiums for baskets

referencing 11 portfolios. The portfolios are identical to the one described

in Table 20.2, except for the degree of correlation, which we vary from zero

to one. We continue to assume that the premium is paid annually.

Figure 20.1 summarizes the results, which are consistent with the main

conclusions drawn from the two-asset case. In particular, the ¬gure plots

one-year FTD premiums for di¬erent values of β. It shows that when β = 1,

the FTD premium is equal to the default probability of the reference entity

with the lowest credit quality (highest default probability), which is 1 per-

cent or 100 basis points in this case. As for the case of no correlation, β = 0,

the FTD premium is essentially equal to the sum of the CDS premiums of

each individual entity referenced in the basket, or 280 basis points. As the

¬gure shows, for intermediate values of β the FTD premium varies between

these two polar cases, but the relationship between asset correlation and

the FTD premium is very nonlinear.

FIGURE 20.1. Asset Correlation and the FTD Premium

248 20. Valuing Basket Default Swaps

One practical lesson from Figure 20.1 regards the importance of trying

to obtain good estimates for the degree of asset correlation in the portfolio.

A large error in the estimation of β could result in a signi¬cant mispricing

of the basket and, consequently, lead to a credit risk exposure that could

di¬er substantially from the intended one. For instance, a FTD protection

seller who overestimated the degree of asset correlation in the portfolio

could end up charging a much lower premium than would be warranted by

the risk pro¬le of the portfolio.

20.6 Extensions and Alternative Approaches

We have mentioned already, at the end of Chapter 19, the several directions

in which the simple model used here for the valuation of basket default

swaps has been extended, as well as alternative modeling approaches for

examining portfolio credit risk. For instance, the work of Li (2000)[55] is of

particular interest in that he proposes a method that explicitly addresses

issues related to the joint probability density of default times. He also

illustrates an application of his method to the valuation of a basket swap.

For convenience, we have been limiting ourselves essentially to the anal-

ysis of contracts involving only one premium payment date and assets with

zero recovery rates. We should note, however, that the methods described

here can be extended to deal with multi-year contracts with several pre-

mium payment dates and with assets with nonzero recovery rates. Such

extensions would be analogous to what we did in Part III when we went

from examining zero-coupon bonds to coupon-paying bonds.

21

Valuing Portfolio Swaps and CDOs

As we discussed in Chapter 10, portfolio default swaps share many of the

characteristics of basket default swaps. Some salient di¬erences include the

fact that they tend to reference a larger number of entities than do baskets

and that default protection is bought and sold in terms of percentage losses

in the portfolio, as opposed to with reference to the number of individual

defaults. For instance, an investor might enter into a portfolio default swap

where it agrees to cover the ¬rst ten percent in default-related losses in

the portfolio. In exchange, the investor receives periodic payments from

the protection buyer. Given these basic similarities, many methods used

for the valuation of portfolio default swaps are variations of those used to

value baskets.

Portfolio default swaps are important in their own right, but they are also

of interest because they can be thought of as building blocks for the increas-

ingly common synthetic CDO structure, which we discussed in Chapter 14.

We conclude this chapter with a brief discussion of additional valuation

issues that are relevant for the pricing of these structures.

21.1 A Simple Numerical Example

We start the discussion of valuation principles for portfolio default swaps

with a simple numerical example. Similar to the analysis surrounding

Table 20.1, when we examined basket swaps, the goal of this discussion

is to provide some basic intuition into the valuation exercise.

250 21. Valuing Portfolio Swaps and CDOs

TABLE 21.1

Valuation of Portfolio Default Swaps: A Simple Numerical Example

(Total notional = $400 million; individual recovery rates = 0)

Portfolio swap lossesa

Possible Outcome Portfolio

¬rst-loss second-loss

outcomes probabilities losses

($ millions) ($millions)

(No. of defaults) (percent) ($ millions)

(1) (2) (3) (4) (5)

0 52.73 0 0 0

1 23.28 20 20 0

2 11.21 40 40 0

3 5.80 60 40 20

4 3.03 80 40 40

5 1.80 100 40 60

6 0.92 120 40 80

7 0.56 140 40 80

8 0.29 160 40 80

9 0.19 180 40 80

10 0.10 200 40 80

11 0.05 220 40 80

12 0.03 240 40 80

13 0.01 260 40 80

14 0 280 40 80

15 0 300 40 80

16 0 320 40 80

17 0 340 40 80

18 0 360 40 80

19 0 380 40 80

20 0 400 40 80

a Valuesshown in columns 4 and 5 are the total losses of FTD and STD protection sellers

under each possible default outcome.

Consider a portfolio composed of bonds issued by 20 entities, all with a

zero recovery rate. The ¬rst column of Table 21.1 lists all possible default

outcomes associated with this portfolio over the next year, ranging from

no defaults to all 20 reference entities in the portfolio defaulting together.

The second column shows the corresponding risk-neutral probabilities asso-

ciated with each of the 21 default outcomes.1 As in previous chapters

in this part of the book, we continue to assume that defaults, if any,

1

Columns 1 and 2 together essentially correspond to the loss distribution function of

this portfolio.

21.1 A Simple Numerical Example 251

only occur at the maturity date of the contract, which is assumed to be

one year. For instance, the second row of column 2 tells us that there is a

23.3 percent probability of the reference portfolio experiencing one default

at the maturity date of the swap.

We assume that each of the entities represented in the portfolio accounts

for an equal portion of the total value of the portfolio. Thus, for a port-

folio that corresponds to a total notional amount of $400 million, each

default results in a loss of $20 million, or 5 percent of the total. Default-

related losses for each possible default outcome are shown in column 3.

As a result, as we did in Chapter 20, we can multiply columns 2 and 3

on an element-by-element basis and sum the resulting numbers across all

possible default outcomes to arrive at the expected (risk-adjusted) loss in

the overall portfolio over the life of the contract:

20

Expected loss = ωi li (21.1)

i=0

which is roughly $20 million, or 5 percent of the portfolio™s face value. (In

the above expression, ωi and li correspond to the ith elements of the second

and third columns of Table 21.1, respectively.)

Consider now a portfolio default swap written on the portfolio detailed in

Table 21.1. The protection seller in this swap commits to absorb all default-

related losses up to 10 percent of the notional amount of the contract,

which corresponds to a maximum loss of $40 million. The possible outcomes

facing such a protection seller, in terms of his losses under di¬erent default

scenarios, are listed in column 4. Multiplying columns 2 and 4 and summing

across all outcomes as in equation (21.1) leads to an expected loss for

this investor in the order of $14.3 million, or 35.6 percent of the ¬rst-loss

piece.

Assuming that the swap is fairly valued and that no money changes hands

at its inception, and relying on arguments entirely analogous to the ones

used in previous chapters, we then arrive at a premium of 35.63 percent,

or 3,563 basis points, for this ¬rst-loss contract, i.e., the protection buyer

promises to pay $14.3 million (3,563 bps — $40 million) in exchange for the

protection provided by the contract.

Column 5 lists, for every possible default outcome, the total payment

made by a second-loss protection seller who committed to cover all default-

related losses falling between 10 and 30 percent of the portfolio. Again,

multiplication and addition of columns 2 and 5 show that, on a risk adjusted

basis, such a protection seller can expect to make a payment of $5.2 million

under the terms of this contract. Given a second-loss piece of $80 million”

(30’10) percent — $400 million”the corresponding protection premium is

647 basis points, or 6.47 percent.

252 21. Valuing Portfolio Swaps and CDOs

21.2 Model-based Valuation Exercise

As with other multi-name credit derivatives, obtaining a loss distribution

function for the underlying portfolio is perhaps the most important part

of the valuation exercise. In the previous section we skipped this problem

altogether as our goal was simply to build some intuition on the valuation

process.

Consider, as an example, an equally weighted reference portfolio with 100

companies and corresponding to a total notional amount of $400 million.

The portfolio is homogeneous with each of its constituent companies having

a risk-neutral default probability of 10 percent over the life of the one-year

portfolio default swaps that we will examine. For simplicity, we continue

to assume a zero recovery rate. It turns out that we can value any default

swap written on this portfolio by relying on essentially the same model-

and simulation-based approach that we discussed in Chapter 19, much as

we did in Chapter 20 in the valuation of basket default swaps. In partic-

ular, to model the credit risk in this portfolio, we assume that individual

company returns and defaults behave as in equations (20.5) through (20.7).

To examine the likelihood of di¬erent default scenarios associated with this

portfolio, one could take the following steps:

• Step 1: Assign values to the key model parameters, β and C. For

instance, as we noted in Chapter 20, an estimated value for β could

be obtained by regressing standardized stock market returns for the

companies included in the portfolio on the modeler™s choice for the

common factor, such as standardized changes in a marketwide stock

index, and C can be obtained by inverting equation (20.7);

• Step 2: Based on the standard normal distribution, draw a large

number of random values for the common factor ± and for each of the

company-speci¬c factors i , i = 1, . . . , 100;

• Step 3: For each reference entity i, use equation (20.5) to compute

its return, Ri , for each value of ± and i and record whether or not

that value of Ri constituted a default by entity i, as speci¬ed by

equation (20.6);

• Step 4: Once individual defaults have been counted and recorded for

all reference entities under all values of ± and i , the probabilities

associated with di¬erent default outcomes can be computed by taking

the ratio of the number of occurrences of that outcome over the total

number of random draws in Step 1.2

2

Each outcome corresponds to a given number of defaults in the portfolio,

as in Table 21.1, and each random draw corresponds to a value for the vector

[±, 1 , 2 , . . . , 100 ].

21.2 Model-based Valuation Exercise 253

9

8

7

probability (percent )

6

5

4

3

2

1

0

0 10 20 30 40 50 60 70 80 90 100

percentage loss in the portfolio

FIGURE 21.1. Loss Distribution Function for a Portfolio with 100 Assets

(one-year horizon)

Once the above steps are followed, one has essentially obtained the

loss distribution function associated with the portfolio, which is shown

in Figure 21.1 for β = .5. Given the loss distribution function, which e¬ec-

tively corresponds to columns 1 and 2 in Table 21.1, the rest of the valuation

exercise proceeds in exactly the same fashion as in the numerical example

illustrated in that table.

Take, for instance, the valuation of two one-year portfolio default swaps

written on the portfolio underlying the loss distribution function shown in

Figure 21.1. The ¬rst swap is a ¬rst-loss contract covering losses of up to

20 percent of the portfolio, and the second is a second-loss contract covering

losses between 20 and 50 percent. We will discuss the valuation results

by examining the maximum and expected default-related losses of three

investors, one who is long the entire portfolio without having bought any

default protection, one who has sold protection via the ¬rst-loss contract,

and another who has sold protection via the second-loss contract. We shall

call each of these the unhedged investor, the ¬rst-loss investor, and the

second-loss investor, respectively.

Table 21.2 displays key statistics for each investor. Column 2”rows

1 through 3”shows the corresponding maximum losses, which are $400

million for the unhedged investor, $80 million”20 percent of $400

million”for the ¬rst-loss investor, and $120 million”(50 ’ 20 = 30) percent

of $400 million”for the second-loss investor. Thus, similar to basket

swaps, portfolio swaps give investors an opportunity to lever up their

254 21. Valuing Portfolio Swaps and CDOs

TABLE 21.2

Valuation of Portfolio Default Swapsa

(β = .5; total notional = $400 million; individual recovery rates = 0)

Investor type Maximum loss Expected loss

(percentb )

($ millions) ($ millions)

(1) (2) (3) (4)

1. unhedged investor 400 40 10

2. ¬rst-loss investor 80 34 43

3. second-loss investor 120 5 4.4

Memo:

Prob[portfolio loss ≥ 20%] = 14.7 percent

Prob[portfolio loss ≥ 50%] = 0.6 percent

a Results

based on 500,000 simulations of the credit risk model.

b Percentage

losses for ¬rst- and second-loss investors are reported relative to their

maximum losses”see column 2.

credit exposure. For instance, while the ¬rst-loss investor is exposed to

all reference entities in this $400 million portfolio, he or she is liable only

for the ¬rst $80 million of default-related losses, all while enjoying, as we

shall see below, a substantial premium. Of course this “substantial pre-

mium” is there for a reason. For instance, as the memo lines in Table 21.2

show, there is a 14.7 percent chance that the ¬rst-loss investor will lose

the entire ¬rst-loss piece, whereas the probabilities of complete losses for

the second-loss and unhedged investors are, respectively, 0.6 percent and

virtually zero.

To compute the fair value of the premiums owed to the ¬rst- and second-

loss investors in Table 21.2, we use the loss distribution function associated

with the reference portfolio”Figure 21.1”and carry out the same calcula-

tions described for Table 21.1. The results are shown in columns 3 and 4.

For instance, the expected losses of the ¬rst-loss and unhedged investors are

relatively close, at $34.4 million and $40 million, respectively. However, rel-

ative to their maximum losses, expected percentage losses are much higher

for the ¬rst-loss investor (about 43 percent) than for the unhedged investor

(10 percent).

Thus assuming that no money changed hands at the inception of the

¬rst-loss contract and that market forces acted to rule out any arbitrage

opportunities, the ¬rst-loss investor receives a premium from the protec-

tion buyer in that contract that amounts to 43 percent of the ¬rst-loss

piece, or 4,300 basis points. Likewise, the fair value of the premium for

the second-loss portfolio default swap is about 440 basis points as the

21.3 The E¬ects of Asset Correlation 255

risk-adjusted expected loss of the second-loss investor is approximately

4.4 percent.3

21.3 The E¬ects of Asset Correlation

As with the default basket, the degree of correlation among the entities

included in the reference portfolio is an important determinant of the port-

folio swap premium. We illustrate this point in Figure 21.2, which shows

the premiums for three one-year portfolio default swaps written on the

portfolio described in Section 21.2 for values of β varying from 0 to 1. The

¬rst two contracts”labeled ¬rst- and second-loss”are the ones examined

in Section 21.2. The third one”labeled third-loss”is essentially a con-

tract that would absorb any residual losses after the protection provided

by both the ¬rst- and second-loss contracts is exhausted. One can think

of a protection-selling position in the third contract as equivalent to the

residual risk exposure of an investor who owns the entire portfolio, but who

has bought protection through the ¬rst two contracts.

FIGURE 21.2. Portfolio Swap Premiums and Asset Correlation

3

Recall, as noted in Chapter 19, that we are assuming that all parties to the contract

discount expected future payments at the same rate and that the premium is paid at

the maturity date of the contract.

256 21. Valuing Portfolio Swaps and CDOs

FIGURE 21.3. Probability of Portfolio Losses Equal to or Greater than

50 Percent

Figure 21.2 shows that as the degree of correlation in the portfolio

increases, the premium owed to ¬rst-loss investors decreases. Although this

may seem counterintuitive at ¬rst, a closer look at the nature of the contract

proves otherwise. As the degree of correlation in the portfolio increases”

see Figure 21.3”so does the probability of large losses. This happens

because higher correlation increases the chances that several reference

entities default together. Thus greater correlation increases the risk that

second- and third-loss investors will be called upon to cover default-related

losses, i.e., that they will have to pay up for losses beyond those covered

by the ¬rst-loss investors. Thus, the second- and third-loss investors need

to be compensated with larger premiums as more of the total risk in the

portfolio is being borne by them as β increases. Indeed, for very high corre-

lation, even the premium owed to second-loss investors starts declining in

this example as a substantial portion of the total risk in the portfolio is now

also shared with the third-loss investors. (The premium that would be owed

to a third-loss investor is shown as the dash-dotted line in Figure 21.2.)

One ¬nal note: When β = 1, the portfolio essentially behaves like a single

asset because either all reference entities survive or all default together.

In this extreme case, Figure 21.2 shows that all investors earn the same

premium, 1,000 basis points, which corresponds to the 10 percent default

probability of the individual entities included in the portfolio.4

4

As noted in Chapter 10, with nonzero recovery rates, the equality of premiums

across di¬erent classes of investors when β = 1 generally does not hold. The positive

21.4 The Large-Portfolio Approximation 257

21.4 The Large-Portfolio Approximation

In Chapter 19 we discussed the large-portfolio approximation method for

computing loss distribution functions for a given portfolio. In particular,

see equation (19.33), we found that the following expression approximates

the probability that default-related losses in a large homogeneous portfolio

will not exceed θ percent over a given time period:

√

C’N ’1 (θ) 1’β 2

Prob[L ¤ θ] = N (’ ) (21.2)

β

where L is the percentage default-related loss in the portfolio; N (.) is the

standard normal cumulative distribution function; N ’1 () is its inverse; and

C and β have their usual de¬nitions.

We can then use (21.2) to compute an approximate value for the prob-

ability that the ¬rst-loss investor described in Section 21.2 will lose the

entire ¬rst-loss piece as a result of defaults in the underlying portfolio.

That probability is approximately

√

C’N ’1 (.2) 1’β 2

1’ N (’ )

β

or 13.5 percent, which is close to the 14.7 percent probability reported

in Table 21.2. The analogous ¬gures for the second-loss contract are 0.5

percent and 0.6 percent, respectively.

One can also compute approximate expressions for the premiums for the

¬rst- and second-loss portfolio swaps described in Section 21.2, but these

calculations involve some mathematical manipulations that go beyond the

scope of this book. Interested readers will ¬nd the following formula in

O™Kane and Schlogl (2001)[64] for the premium owed under an arbitrary

portfolio swap written on a large homogeneous portfolio:

N2 ’N ’1 (Llb ), C, ’ 1 ’ β 2 ’ N2 ’N ’1 (Lub ), C, ’ 1 ’ β 2

Lub ’ Llb

(21.3)

where N2 () is the bivariate normal cumulative distribution function, and

Llb and Lub are, respectively, the lower and upper bounds of the range

of losses covered by the contract. For instance, for the ¬rst-loss contract

featured in Section 21.2, Llb = 0 and Lub = .2, whereas, for the second-loss

contract, Llb = .2 and Lub = .5.

recovery value of the assets in the reference portfolio provides an additional cushion to

higher-order investors, potentially reducing their expected losses.

258 21. Valuing Portfolio Swaps and CDOs

Using the above formula, we obtain (approximate) premiums of 4,356 and

417 basis points for the ¬rst- and second-loss portfolio swaps, respectively,

which are indeed close to the results reported in Table 21.2.5

21.5 Valuing CDOs: Some Basic Insights

The basic framework laid out in this chapter for the valuation of portfolio

default swaps easily translates into the foundation for a simple method for

valuing the tranches of CDOs (synthetic or cash ¬‚ow). Indeed, as we shall

see in Chapter 22, some of the methods used in practice in the valuation

of CDO structures can be cast as extensions of this simple framework.

For the sake of illustration, let us take a closer look at the synthetic CDO

structure. From our discussion in Chapter 14 we know that we can think

of a synthetic CDO approximately as a structure made up of one or more

portfolio default swaps combined, in the case of a funded synthetic CDO,

with an outright position in highly rated assets (the SPV collateral).

It is relatively straightforward to see that equity investors in a CDO

structure have a position that is akin to that of ¬rst-loss investors in a

portfolio default swap (or of second-loss investors if the institution that

originated the CDO retained the ¬rst-loss piece). Likewise, mezzanine and

senior tranche investors are long positions that are analogous to that of

second- and third-loss investors in portfolio swaps.

Indeed, Figure 21.2 would be very informative for someone considering a

CDO where equity tranche investors absorb the ¬rst 20 percent in default-

related losses, mezzanine tranche investors absorb losses between 20 and

50 percent, and senior investors absorb any remaining losses. Essentially,

one can reinterpret Figure 21.2 as showing the expected default-related

losses for each class of investors in this CDO under varying degrees of asset

correlation, assuming that the individual one-year default probability of

the underlying entities is 10 percent.6

21.5.1 Special Considerations for CDO Valuation

Crucial as they are, default correlation and the credit quality of the under-

lying reference entities are not the only factors determining the valuation of

CDO structures. For instance, an important di¬erence between CDOs and

5

Not surprisingly, the large-portfolio approximation works better for larger portfolios.

For instance for a 500-asset portfolio, the second-loss premium based on the simulation

method is 420 basis points.

6

A reading of, say, 2,500 basis points in Figure 21.2 can be thought of as a (risk-

neutral) expected loss of 25 percent of the notional amount represented by each CDO

tranche.

21.6 Concluding Remarks 259

portfolio default swaps is that the former typically incorporate so-called

“coverage tests” provisions, but the latter do not. These tests are part of

the legal structure of CDOs and are intended to protect investors in more

senior tranches against a deterioration in the credit quality of the pool of

collateral assets. In particular, a CDO structure may specify that its senior

tranches will be provided a certain cushion such that the ratio of the struc-

ture™s total par value to that of that tranche™s will not fall below a certain

“overcollateralization” level.

Should defaults occur among the assets in the collateral pool and

bring overcollateralization ratios for senior tranches below the prescribed

minimum (“trigger”) levels, the CDO is said to have failed its overcollater-

alization tests. As a result, the CDO structure may require the diversion of

principal and interest cash ¬‚ows from lower tranches to pay down enough

of the principal of more senior tranches to bring the structure back into

compliance with its overcollateralization requirements. In the context of

this book, it su¬ces to say that coverage tests bring an additional level of

complexity to the valuation of CDOs, one that was not captured by the

portfolio default swap valuation exercise discussed in this chapter.

Some CDO structures call for the diversion of cash ¬‚ows away from lower

tranches even in the absence of default, if, for instance, the credit quality

of the underlying assets is deemed to have deteriorated signi¬cantly. In

contrast, if no defaults have occurred, a deterioration in the credit quality

of the reference portfolio has no cash-¬‚ow implications for protection sellers

in portfolio default swaps.

We mentioned other salient di¬erences between CDOs and portfolio

default swaps in Chapter 14. These include the credit quality of the SPV

collateral in a synthetic CDO, reinvestment and “manager” risk associated

with structures that include ramp-ups, removals, and replenishments, as

well as other aspects of the CDO™s legal structure. Along with coverage

tests, these factors should not be ignored in real-world attempts to value

CDO structures.

21.6 Concluding Remarks

The techniques described in this chapter are intended to serve as introduc-

tory illustrations of some of the key factors that in¬‚uence the valuation

of multi-name credit derivatives. Before using these or any other methods

to value portfolio products in the real world, the reader should consider

several key questions:

• Is the model a good description of return and default dynamics of the

underlying reference entities?

260 21. Valuing Portfolio Swaps and CDOs

• Should I allow for more than one common factor to determine the extent

of correlation among the reference entities?

• To which extent can I rely solely on market prices, such as CDS premi-

ums, as a proxy for the risk-neutral default probabilities that are fed to

the model?

Other issues such as non-normal shocks, uncertain recovery rates, and

time-varying correlations (e.g., greater default correlation during economic

downturns), which were not addressed by the simple modeling framework

described in this chapter, should also be taken into account and serve as

a reality check to would-be portfolio credit risk modelers. Still, imperfect

as the basic modeling framework described in this chapter may be, it con-

stitutes the basis for understanding more complex models that are used in

commercial applications. For instance, as we shall see in Chapter 22, the

CreditMetrics model, developed by the RiskMetrics Group, is essentially

a more elaborate version of the modeling framework discussed here and in

Chapters 19 and 20.7

Lastly, as we noted elsewhere in this part of the book, our basic model

represents only one of several approaches to assessing the extent of default-

related losses in a portfolio of credit-related instruments. For instance, a

well-known alternative method for valuing CDOs is the intensity-based

model of Du¬e and Garleanu (2001)[22].

7

CreditMetrics is a trademark of J.P. Morgan.

22

A Quick Tour of Commercial Models

As the credit markets have grown in both size and sophistication so have

the technical skills required to assess the risk-reward characteristics of an

ever-expanding array of new products and structures, such as multi-name

credit derivatives. Rather than developing in-house the analytical tools

and databases required to fully understand and examine these new prod-

ucts, many investors have turned to outside experts for technical assistance.

Indeed, several ¬rms have come to be known as leading providers of analyti-

cal services regarding portfolio credit risk. In this chapter we brie¬‚y discuss

some of the better-known models developed and marketed by these ¬rms

and compare them to the basic credit risk model discussed in Chapters 19

through 21.

We shall focus on four commercially available modeling approaches to

the analysis of portfolio credit risk: Moody™s Investors Service™s Binomial

Expansion Technique (BET), J.P. Morgan/RiskMetrics Group™s Credit-

Metrics model, Moody™s KMV™s KMV model, and Credit Suisse Financial

Products™ CreditRisk+ model.1 Given the number of approaches just men-

tioned, however, as well as the length and scope of this introductory book,

our discussion of each modeling framework will be brief and, for the most

part, non-technical. Basic sources for each approach are cited throughout

the chapter. In addition, Crouhy, Galai, and Mark (2000)[17] provide a

1

CreditMetrics is a trademark of J.P. Morgan; KMV is a trademark of Moody™s KMV

Corp.; CreditRisk+ is a trademark of Credit Suisse Financial Products.

262 22. A Quick Tour of Commercial Models

comprehensive comparison involving most of the models summarized in

this chapter.

22.1 CreditMetrics

Of the modeling approaches discussed in this chapter, this is the one that

is most closely related to the basic model described in Chapter 19 and

used in Chapters 20 and 21 for the valuation of basket and portfolio swaps.

Indeed, one can think of our basic model, described by equations (19.7)

through (19.9), as a simpli¬ed version of the CreditMetrics model. The

CreditMetrics approach is described in detail by Gupton et al. (1997)[37].

Similar to the basic portfolio credit risk model, the CreditMetrics model

is a Merton-style model that speci¬es defaults as situations where a variable

Ri , which is assumed to measure the creditworthiness of a given entity i,

falls below some threshold Ci . Another similarity regards the evolution of

Ri , which CreditMetrics also assumes to be a function of both common

(marketwide) and idiosyncratic random factors, where the former are the

main determinants of the extent of default correlation in the portfolio.

An important feature of CreditMetrics that was not captured by the basic

model discussed in Chapters 19“21 is that it is also designed to examine the

likelihood of “ratings transitions,” or the probability, for instance, that an

A-rated corporate borrower will be downgraded to, say, a BBB rating over a

given time horizon. Thus, whereas the basic model only allowed for a refer-

ence entity to be in one of two states”solvency and default”CreditMetrics

allows for as many states as the number of credit ratings under considera-

tion. The way ratings transitions are modeled in CreditMetrics is similar to

the manner in which we described the passage from solvency into default

in the basic model. In particular, continuing with the same example just

mentioned, the probability of ¬rm i being downgraded from A to BBB is

modeled as the probability that Ri falls below the threshold Ci,BBB . More

generally, one can write

Prob[downgrade to J-rating]=Prob[Ri ¤ Ci,J ] (22.1)

where, as always, the probability is de¬ned with respect to a given time

horizon.

22.2 The KMV Framework

We described the basic features of Moody™s KMV™s single-default model in

Part III in our introduction to structural models of credit risk. Moody™s

22.3 CreditRisk+ 263

KMV also o¬ers a related tool for analyzing portfolio credit risk, called

Portfolio Manager. The main output of Portfolio Manager is the loss dis-

tribution of the portfolio under consideration, from which, as we saw in

Chapters 20 and 21, one can value a wide array of multi-name credit

derivatives.

Portfolio Manager is similar to CreditMetrics in that it incorporates a

model where default correlations are captured through the dependence

of individual entities™ returns on common factors. In addition, as in the

CreditMetrics model, individual returns depend on a ¬rm-speci¬c fac-

tor, and defaults and credit migrations are characterized as situations

where individual returns fall below certain prescribed thresholds. The basic

Moody™s KMV framework is described by Crosbie (2002)[16].

Di¬erences between the CreditMetrics and Moody™s KMV approaches

include the fact that the latter uses an empirical distribution for returns

based on proprietary data, as opposed to the normal distribution used in

the CreditMetrics framework. Another di¬erence regards the fact that the

RiskMetrics Group makes the details of its model speci¬cation publicly

available, where Moody™s KMV does not.

22.3 CreditRisk+

Unlike the full versions of the CreditMetrics and Moody™s KMV models,