The matrix approach uses discrete final credit states. The structural model of default provides a continuous distribution of credit states at horizon, which depends on the distance to default. Migration matrices group historical frequencies of transitions across risk classes over a specified horizon, for example one year. The final states include all risk classes plus the default state. The default option model also applies to migrations since it models the default probabilities that map with risk classes. The difference is that the distribution of migrations is continuous rather than discrete.

Each final asset value corresponds to a distance to default and a default probability. The distribution of asset values is a continuous distribution of migrations. For mapping asset values, or final distance to default, to discrete ratings, the continuous distribution has to be discretized. The process consists of defining ranges of asset values and distances to default that map to the default probabilities of ratings.

The lognormal distribution was used for deriving the default probability, given the debt obligation at horizon. But the same distribution can be used as above for deriving a standalone VaR. The VaR is the negative variation of value matching a percentile. It is defined as the negative difference between ^(a) and A at confidence level a. For obtaining such VaR under the structural model, it is sufficient to substitute to the debt value a value percentile (Figure 46.4).

FIGURE 46.3 Asset values and migrations

FIGURE 46.4 VaR of asset value

In Chapter 42 describing the structural model, value percentiles were defined for the ratio V(a)/V0 of final to initial value. The formula for the a percentile of this ratio was:

FIGURE 46.5 Percentiles of the ratio of final asset value to initial asset value (expected return 10%, volatility 30%, horizon I year)

Replicating the example, when using an expected return of 10%, a volatility of asset return of 30% and a horizon Tof one year, we find that the 1% percentile of the ratio V(a)/V0 is 40.7%. This result is summarized by:

The downside deviation of value matching the 1% percentile is the complement to 1, or 59.3%. This value, 59.3% x V0 is the VaR at 1% confidence level. The graph showing the ratio value, with the same data as the above example, and the percentiles is shown in Figure 46.5. It depends on the inputs used above. Figure 46.5 is a summary of migrations. Higher than 1 values of the ratio correspond to upward migrations, and lower than 1 values corresponds to adverse deviations. The graph is dependent on all inputs.

Note that with the standardized model using a standard normal distribution, the VaR at 1% confidence level matches a decline of value of-2.3263a, or 0.7. The decline of value is 70%, much higher than with the lognormal distribution because the downside tail of the normal distribution is more extended than the truncated at 0 left-hand tail of the lognormal distribution (Figure 46.5).

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