Technical Note Introducing the Loan Pool Specific Factor in CreditManager A New Tool for Decorrelating Loan Pools Driven by the Same Market Factor Attila Agod, András Bohák, Tamás Mátrai Attila.Agod@ Andras.Bohak@ Tamas.Matrai@ Introduction In the CreditMetrics framework, the value of a pool of loans at the risk horizon is determined by the state of its driving market factors and the idiosyncratic factors of the individual loans. However, if the loan pool consists of hundreds of loans, most of the risk originating from the idiosyncratic factors is diversified away. This leaves the horizon values driven mostly by the market factors. While this behavior is intuitive for standalone pools, it has an undesirable side effect for portfolios containing multiple large loan pools, which differ in some systematic dimension, but which are mapped to the same market factor. As an example, consider two pools of mortgages issued in two different cities. If no city specific mortgage factors are available, the two mortgage pools get mapped to the same country level factor (e.g., a National Home Price Index), leading to the challenges outlined above. Since the horizon values of the loan pools are determined by the same index, the simulated horizon values for the pools are highly correlated. This extra correlation makes the portfolio less diversified and drives up the risk numbers. In this Technical Note, we present an enhancement to the Loan Pool and Mortgage Pool models in CreditManager that allows decorrelating the horizon values of distinct pools driven by the same market factor. We introduce a Loan Pool Specific Factor that applies only to a single pool of assets and is not correlated to any market factor. Since only the loans of the particular pool are exposed to its Pool Specific Factor, the factor can be associated with the feature that distinguishes the pool from the rest of the investment universe, e.g., in the example of mortgage pools, the Pool Specific Factor can be interprested as the city specific component of the Home Price Index. Please refer to the disclaimer at the end of this document 1 of 9
Unlike the MSCI index time series that are stored and updated regularly, Loan Pool Specific Factors are pure random numbers generated on the fly for the given loan pool. In the next section, The Extended Model,, we show how the Pool Specific Factor can be incorporated into the existing CreditMetrics framework. In the section after, we give an example for estimating the variance of Pool Specific Factors based on historical writedown data of two loan pools. The Extended Model Loan Pools in CreditManager are described with the following information: Number of exposures (N): number of loans in the pool. Probability of default (p): individual default probabilities for a given horizon. Recovery rate (R): recovery rate of the segment. Asset correlation (): pairwise correlation between obligor asset returns. Driving index (S): the systematic factor driving the gains or losses of the pool. In CreditMetrics, the default indicator of the i th loan (Z i ) in a pool is proxied by the linear combination of two standard normal variables, a systematic factor (S) and a loan specific variable ( i ): Z S 1 (1) i i The extended model assumes that the common factor (S) is further decomposed into an observable market factor (M) and a Loan Pool Specific Factor (P): S P 1 M (2) Thus, we introduce the following two new concepts: Loan Pool Specific Factor (P): standard normal unobservable variable that is orthogonal to every other factor and drives only the modeled pool. Loan Pool Specific Factor variance (): The ratio of the variance explained by the Pool Specific Factor to that explained by the whole systematic factor. Its value can range from 0 to 1. In the extended model, the asset correlation () keeps its original meaning; i.e., it is the correlation of loan values within the same pool. However, the correlation of assets in two distinct (but identically parameterized) pools decreases to 1. Thus the other interpretation of is the percentage by which the correlation of assets decreases in two separate pools. To illustrate the effect of the Pool Specific Factor on the horizon values of two distinct pools, we will consider two identical portfolios of 1,000 loans with a default probability of 1 percent, an average recovery rate of 50 percent, and an average asset correlation of 50 percent. In Figure 1, the cloud corresponds to the horizon values of the two pools calculated by a Monte Carlo simulation using 1,000 scenarios. In this setting the pools are driven by the same market factor and Pool Specific Factors do not Please refer to the disclaimer at the end of this document 2 of 9
play any role, so the majority of the idiosyncratic risk is diversified away and the horizon values are highly correlated. On the other hand, if the Pool Specific Factors account for 10 percent, i.e., a small fraction of the variance of the systematic factors of the pools, the horizon values become far less correlated, as shown in Figure 2. Figure 1. Horizon values of two large identical pools driven by the same single market factor. Figure 2. Horizon values of two large identical pools, where 10 percent of the variance of the systematic factor is associated with Loan Pool Specific Factors. 1 1 0.95 0.95 Pool 2 Horizon Value 0.9 0.85 Pool 2 Horizon Value 0.9 0.85 0.8 0.8 0.75 0.75 0.8 0.85 0.9 0.95 1 Pool 1 Horizon Value 0.75 0.75 0.8 0.85 0.9 0.95 1 Pool 1 Horizon Value Introducing the Loan Pool Specific Factor increases the diversification potential of the loan pools. To illustrate this effect, we calculated the 99% Value at Risk of the portfolio consisting of two identically parameterized pools using the same parameters as in the above example. Figure 3 shows that increasing the Pool Specific Factor variance in both pools together from 0 to 1 (that is, shifting the pools from asymptotically perfectly correlated to independent) decreases the 99% VaR by about 30%. Much of the decrease in the Value at Risk is obtained from small values of the Pool Specific Factor variance. Figure 3. 99 percent VaR of the two pools together as a function of Loan Pool Specific Factor variance. 99% Value at Risk 0.09 0.085 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0 0.2 0.4 0.6 0.8 1 Variance of the Pool Specific Factor Please refer to the disclaimer at the end of this document 3 of 9
Estimating the Pool Specific Factor Variance Technical Note This section shows how we estimate the variance of the Loan Pool Specific Factors. We begin with a model generated time series of writedowns of two identically parameterized loan pools, using the maximum likelihood method for the estimation. The parameters of the pools are set to the same values we used in the previous section, and we assume that we observed five years of monthly loss data (i.e., 60 observations) for both pools. The demonstrated calculation can be generalized for two pools having differing parameters for default probability and recovery. In the first chart of Figure 4, one time series of the normalized writedowns and the cumulative writedowns for both pools are plotted as a function of time. The cumulative loss functions tend to grow unevenly; for example, most of the observed writedowns are relatively small and the majority of the loss is accumulated in a few large steps. The writedown histories of the two pools are not perfectly correlated, which is caused mainly by the Pool Specific Factors which account for 10 percent of the variance of the systematic factors. Our goal is to estimate the Pool Specific Factor variance based on the observed history of the pools. The estimation consists of three major steps (the detailed derivation of each step is given in the Appendix). 1. Estimate the monthly default probability (p) individually for the loan pools based on the observed rate of defaults. (See equation A1 in the Appendix.) 2. Estimate the asset correlation () for both pools separately. (See equation A5 in the Appendix). 3. Estimate the Pool Specific Factor variance using the above two estimates. (See equation A9 in the Appendix.) Please refer to the disclaimer at the end of this document 4 of 9
Figure 4. Top left: Simulated monthly writedowns of two loan pools (left axis) and the corresponding cumulative writedowns (right axis). Top right: Histogram of the estimates of monthly default probabilities. Bottom left: asset correlation histogram. Bottom right: Pool Specific Factor variance histogram. To test the level of error in the estimation, we carried out 1,000 experiments with the same model parameters; we then estimated the monthly default probability, the asset correlation and the Pool Specific Factor variance from the model generated writedowns. In Figure 4, the histograms show the frequency of the estimated values. The red line indicates the predefined value of the model parameter, and the green line shows the average of the estimates in the 1,000 experiments. The red and green lines are very close, so the estimation appears unbiased. On the other hand, as the horizontal blue lines show, the standard error of the estimation is pretty significant. The relative error of the default probability and the Pool Specific Factor variance estimates are about 50 percent, whereas the asset correlation can be estimated with a relative error of about 20 percent in the investigated case. In general, the estimation error can be reduced if we rely on longer history or more loans in the asset pools. In practice, the estimation can be further improved by using the writedown history of more pairs of pools, if it can be safely assumed that the variance of the Pool Specific Factor is the same in all pools. Please refer to the disclaimer at the end of this document 5 of 9
Conclusion We presented an enhancement to the pooled instrument models in CreditManager that enables users to decorrelate the horizon values of distinct pools driven by the same market factor. We introduced the concept of Pool Specific Factor, and showed how it fits into the CreditMetrics framework, and discussed how its variance can be estimated using historical writedown data. Our numerical experiments confirm that the Pool Specific Factor allows for modeling risk better for portfolios of loan pools for markets where no detailed index structure is available. Please refer to the disclaimer at the end of this document 6 of 9
Appendix: Estimation of Model Parameters Technical Note The first step is estimating the default probability based on the time average of the number of defaulted loans (D(t)), or equivalently, based on the time average of the loss (L(t)): 1 1 1 pˆ D t L t T 1 R T (A1) t t In the second step, we estimate the asset correlations separately for the loan pools. The conditional probability of default for given value (s) of the systematic factor: p s p s 1 1 (A2) The conditional probability that we observe D(t) number of defaults out of N loans for given value (s) of the systematic factor is proportional to s : Dt p 1 p s s s N Dt (A3) We get the log likelihood function by integrating s with the density of the normally distributed systematic factor ((s)) and summing up for every observation date: l s s ds (A4) t log The maximum likelihood estimation for the asset correlation is the value at which () takes its maximum: ˆ arg max l (A5) In the third part of the process, we estimate the Pool Specific Factor variance. The probability of default conditional on the market factor value (m) and the Pool Specific Factor value (h) can be derived by substituting Eq. 2 of the main section into A2: 1 p 1 m h pmh, (A6) 1 The probability that we observe D j (t) number of defaults in the j th pool is proportional to: Dj t N Dj t p 1 p (A7) mh,, j mh, mh, Please refer to the disclaimer at the end of this document 7 of 9
So, the log likelihood function can be calculated by integrating the above proxy with the density of the two Pool Specific Factors and the density of the market factor: l log hdh mdm t mh,, j (A8) j1,2 The Pool Specific Factor variance can be estimated by finding the maximum of the log likelihood function after entering the already estimated default probability ( ˆp ) and the asset correlation ( ˆ ) into A6 A8: ˆ arg max l (A9) Please refer to the disclaimer at the end of this document 8 of 9
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