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This section details the loss distribution relative to a single standalone banking exposure. The subsections address the same issues using a simple portfolio of two exposures only.

FULL VALUATION OF MIGRATION RISK

Under full valuation mode, there are as many values as there are migration states, plus the default state. A single facility has a distribution of values at a preset horizon, one for each migration, including one for the default state. Expected loss, loss volatility and loss percentiles are derived from this loss distribution. Migrations increase the number of values at the horizon. With several period horizons, the number of branches increases as K^N, K being the number of transitions, including the default state, for a single period, and N being the number of periods, since there are K transitions for each period.

The migration risk technique allows us to generate distributions of values at the horizon for single facilities as well as for portfolios of facilities and bonds. Many models use this technique. KMV Portfolio Manager models migrations through asset value changes but allows us to use matrix valuation given the final default probability. All other models value migration risk based on the migration matrix technique.

The following illustrates the mark-to-future process and the VaR for migration risk. The facility is a bullet facility of face value 1000, with maturity 2 years, a coupon of 5.30% corresponding to a rating B. The cash flows generated by the facility are 53 at end of year 1 (Eoy 1) and 1053 at Eoy 2. The risk-free rate is 5%, the credit spread for a B rating is 30 basis points, and the appropriate risky discount rate is 5.30%. The price at date 0 is identical to face value since the facility pays the market yield exactly.

There is a spectrum of values at the end of period 1 since the facility survives up to date 2. Revaluations at the horizon under no default depend on the final risk class and discount the last cash flow of date 2, 1053, at the risky yield corresponding to the risk class at the horizon. For valuation, we do not need the default probabilities at the final horizon. Rather, we need the credit spreads corresponding to each rating².

The 1-year transition matrix serves to determine the possible risk classes at the horizon. The credit spreads as end of year 1 correspond to each Eoy 1 credit state. A subset of the transition matrix, starting from rating class B, with final credit classes and corresponding credit spreads is given in Table 43.1. Note that migrations can generate gains as well as losses. The values, after payment of the cash flow of 53 at Eoy 1, discount the last cash flow at Eoy 2 with a discount rate equal to the risk-free rate of 5% plus the credit spread of the final rating at Eoy 1 (Figure 43.3).

In this example of a standalone facility, there are six migrations, with the probabilities from the transition matrix. For each migration, there is a value at date 1. Table 43.1 provides the calculations of all final values. The distribution of the facility values at the horizon, from which all loss statistics derive, follows. As usual for credit risk, the value distribution exhibits skewness and a larger downside tail than upside tail. The losses or gains are the algebraic differences between the final value at Eoy 1 and the current value.

Figure 43.4 provides the probability distribution of gains and losses. The loss percentile at 99% is —4.726, from Table 43.1, using the 1000 value as the origin for counting losses. The loss distribution is highly skewed to the left (losses are on the left-hand side). The various loss statistics deriving from the distribution are: the expected value at Eoy 1, the volatility of the value at Eoy 1 due to migrations plus the loss at a given percentile. Percentile losses are discrete because the number of states at Eoy 1 is small (Table 43.2).