required. The end result of such a cooperative effort might be a National Loan Mortality table that could be as useful in establishing banks’ loan loss reserves (based on expected losses) as the National Life Mortality tables are in pricing life insurance.3
126
FIGURE 8.1 Comparison of Credit Risk Plus and CreditMetrics.
BBB Loan
CreditMetrics Possible Path of Default Rate
Time Horizon
Default Rate
BBB Loan
Possible Path
of Default Rate D
BBB
AAA
Time Horizon
discrete $20,000 severity or loss bands). The smaller the bands, the less the degree of inaccuracy that is built into the model as a result of banding.
The two degrees of uncertainty—the frequency of defaults and the severity of losses—produce a distribution of losses for each exposure band.
Summing (or accumulating) these losses across exposure bands produces a distribution of losses for the portfolio of loans. Figure 8.2 shows the link between the two types of uncertainty and the distribution of default losses.
Although not labeled by CSFP as such, we call the model in Figure 8.2 Model 1.The computed loss function, assuming the Poisson distribution for individual default rates and the banding of losses, is shown in Figure 8.3.
The loss function is quite “symmetric” and is close to the normal distribu- tion, which it increasingly approximates as the number of loans in the port- folio increases. However, as discussed by CSFB (1997), default rates and loss rates tend to exhibit “fatter tails” than are implied by Figure 8.3.
Specifically, the Poisson distribution implies that the mean default rate of a portfolio of loans should equal its variance, that is,
or
(8.9) σ = mean
(8.8) σ2 =mean
FIGURE 8.2 The CSFP Credit Risk Plus model.
Frequency of Defaults
Distribution of Default Losses
Severity of Losses
Using figures from Carty and Lieberman (1996) on default rates, CSFP shows that, in general, equation (8.9) does not hold, especially for lower quality credits. For B-rated bonds, Carty and Lieberman find the mean de- fault rate is 7.62 percent and the square root of the mean is 2.76 percent, but the observed σis 5.1 percent, or almost twice as large as the square root of the mean (see Figure 8.3). Thus, the Poisson distribution appears to un- derestimate the actual probability of default.
The question is: What extra degree of uncertainty might explain the higher variance (fatter tails) in observed loss distributions? The additional uncertainty modeled by CSFP is that the mean default rate itself can vary over time (or over the business cycle). For example, in economic expansions, the mean default rate will be low; whereas in economic contractions, it may rise significantly.7 In the extended model (which we shall call Model 2), there are three types of uncertainty: (1) the uncertainty of the default rate around any given mean default rate, (2) the uncertainty about the severity of loss, and (3) the uncertainty about the mean default rate itself [modeled as a gamma distribution by CSFB (1997)]. Credit Risk Plus derives a closed- form solution for the loss distribution by assuming that these types of un- certainty are all independent.8
Appropriately modeled, a loss distribution can be generated along with expected losses and unexpected losses that exhibit observable fatter tails.
The latter can then be used to calculate a capital requirement, as shown in FIGURE 8.3 Distribution of losses with default rate uncertainty and severity uncertainty.
Probability
Model 1
Actual Distribution
of Losses
Losses
Figure 8.4. Note that this economic capital measure is not the same as the VARmeasured in Chapter 6 under CreditMetrics because CreditMetrics al- lows for upgrades and downgrades that affect a loan’s value. By contrast, there are no nondefault migrations in the CSFP model. Thus, the CSFP cap- ital measure is closer to a loss-of-earnings or book-value capital measure than a full market value of economic capital measure. Nevertheless, its great advantage is in its parsimonious data requirements. The key data inputs are mean loss rates and loss severities, for various bands in the loan portfolio, both of which are potentially amenable to collection, either internally or ex- ternally. A simple “discrete” example of the CSFP Model 1 will illustrate the minimal data input that is required.
An Example
Suppose a bank divides its loan portfolio into exposure bands (denoted as v by CSFP); that is, it has many different sizes of loans, and each potentially has a different loss exposure. At the lowest end of the exposure levels, it iden- tifies 100 loans, each of which has $20,000 of exposure.9We can think of this band (v=1) as containing all loans for which the exposures, when rounded up to “the nearest $20,000,” are $20,000. The next two exposure bands
FIGURE 8.4 Capital requirement under the CSFP Credit Risk Plus model.
Probability
Expected Loss
Economic Capital
99th Percentile Loss Level
Loss 0
would represent all loans with a “rounded” exposure of $40,000 (v=2) and
$60,000 (v=3), respectively.
As a first step, we want to compute the distribution of losses for the first band. In CSFP Credit Risk Plus, each band can be viewed as a separate portfolio, and the total loss distribution is then an aggregation of the (inde- pendent) individual loss distributions.
Suppose that, based on historic data, an average of 3 percent of loans with this level of loss exposure ($20,000) default. There are currently 100 loans in the portfolio of this type, so the expected mean default rate (m) is 3. However, the actual default rate is uncertain and is assumed to follow a Poisson distribution (see Figure 8.1). Given this assumption, we can com- pute the probability of 0 defaults . . .ndefaults, and so on, by using the for- mula, for the Poisson distribution:
where e=exponential=2.71828,m=mean number of defaults, !=facto- rial, n=number of defaults of interest, n=1. . . N.
Thus, the probability of 3 defaults is:10
and, the probability of 8 defaults is:
The probability that a different number of defaults will occur and the cu- mulative probabilities are listed in Table 8.2. The distribution of defaults for band 1 is shown in Figure 8.5. Calculation of the distribution of losses in band 1 is straightforward because, by assumption (and rounding), the loss severity is constant in the v=1 band at $20,000 per loan. Figure 8.6
Prob. defaults
%
8 2 71828 3
8 008 0 8
3 8
( )
=( )
×= =
. −
. .
! Prob. defaults
%
3 2 71828 3
3 224 22 4
3 3
( )
=( )
×= =
. −
. .
!
(8.10) Prob. ndefaults e m
n!
m n
( )
= −shows the distribution of losses where the mean number of defaults is 3.
The expected loss is then $60,000 (=3×$20,000) in band 1 of the loan portfolio. The 99th percentile (unexpected) loss rate shows slightly less than 8 loans out of 100 defaulting, which puts the probability of 8 loans default- ing equal to 0.8 percent. Using 8 loans as an approximation,11the 99 percent unexpected loss rate is $160,000 (=8×$20,000) on portfolio v=1. Viewed in isolation from the rest of the loan portfolio, the capital requirement would
FIGURE 8.5 Distribution of defaults: Band 1.
Probability
.008 .05 .168 .224
Defaults
8 4
3 2
1 0
TABLE 8.2 Calculation of the Probability of Default Using the Poisson Distribution
N Probability Cumulative Probability
0 0.049787 0.049789
1 0.149361 0.199148
2 0.224042 0.423190
3 0.224042 0.647232
.. .
8 0.008102 0.996197
be $100,000 (the unexpected loss minus the expected loss, or, $160,000−
$60,000).12This type of analysis would be repeated for each loss severity band—$40,000, $60,000, and so on—taking into account the mean default rates for these higher exposure bands and then aggregating the band expo- sures into a total loan loss distribution.
Continuing the discrete example of a CSFP-type model, suppose, for simplicity, that the band 2 portfolio (v=2), with average loss exposure of
$40,000, also contained 100 loans with a historic average default rate of FIGURE 8.6 Loss Distribution for single loan portfolio. Severity
rate=$20,000 per $100,000 loan.
0
Probability
0.25
0.15
0.05 0.1 0.2
0
Amount of Loss in $ Expected
Loss
Economic Capital
Unexpected Loss
350,000 400,000 250,000 300,000
160,000 200,000 60,000 100,000
FIGURE 8.7 Single loan portfolio. Severity rate=$40,000 per $100,000 loan.
0
Probability
0.25
0.15
0.05 0.1 0.2
0
Amount of Loss in $
350,000 400,000 250,000 300,000
150,000 200,000 50,000 100,000
3 percent (m=3). Figure 8.7 shows the loss distribution for the $40,000 band (v=2) portfolio alone. Figure 8.8 shows the aggregation of losses across the two portfolio bands, v=1 and v=2. If these were the only types of loans made, this would be the loss distribution for the entire loan port- folio. Notice that, in adding the loan distributions for the two bands, the total loss distribution in Figure 8.8 looks more “normal” than the individ- ual loss distributions for v=1 and v=2.13
Finally, this calculation is likely to underestimate the true capital re- quirement because we assumed that the mean default rate was constant in each band. To the extent that mean default rates themselves are variable (e.g., they increase systematically in each band as the “national” default rate increases), the loss distribution will have fatter tails than are implied in this example (and shown in Figure 8.8). Moreover, when the mean default rate in the economy varies and the default rates in each band are linked to economywide default rates, then the default rates in each band can no longer be viewed as independent. (There is a systematic default correlation element among loans; see Chapter 11.)14Indeed, exposure amounts them- selves may even be affected by systemic risk factors; something not incorpo- rated into even advanced versions of Credit Risk Plus.
SUMMARY
We have reviewed two insurance-based approaches to credit risk analysis.
Mortality analysis offers an actuarial approach to predicting default rates, FIGURE 8.8 Loss distribution for two loan portfolios with severity rates of
$20,000 and $40,000.
0
Probability
0.120
0.060
0.020 0.040 0.080 0.100
0.000
Amount of Loss in $
350,000 400,000 250,000 300,000
150,000 200,000 50,000 100,000
which might be thought of as an alternative to some of the traditional ac- counting-based models for calculating expected losses and loan loss re- serves. However, the predictive usefulness of mortality rates very much depends on the size of the sample of loans/bonds from which they are cal- culated. Credit Risk Plus, an alternative to CreditMetrics, calculates capital requirements based on actuarial approaches found in the property insur- ance literature. Its major advantage is the rather minimal data input re- quired (e.g., no data on credit spreads are required). Its major limitation is that it is not a full VARmodel because it concentrates on loss rates rather than loan value changes. It is a default model (DM) rather than a mark-to- market (MTM) model.
135
9
A Summary and Comparison of New Internal Model Approaches
I
n Chapters 4 through 8, we described key features of some of the more prominent new models of credit risk measurement. At first sight, these ap- proaches appear to be very different and likely to produce considerably dif- ferent loan loss exposures and VARfigures. This chapter summarizes these new models and discusses key differences and similarities among them.Empirical evidence on predictive differences among these models is also discussed.