<<

. 8
( 11)



>>


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,

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