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# CREDIT VAR AND MATRIX VALUATION

Credit risk losses (or gains) depend on events, default and migrations between the current date and horizon that affect the value of facilities. Future valuation at horizon is required to assess all credit loss statistics, expected loss, unexpected loss and loss distribution for credit risk VaR.

The "matrix valuation" methodology, introduced originally by Credit Metrics[1], considers a discrete number of credit states at horizon, matching rating classes, for assigning a final value to a facility. The probabilities of reaching each of the final states, including the default state, result from transition matrices. For each credit state, different credit spreads apply, which allows calculating the value of any facility or bond at this future time point from the contractual cash flows up to maturity. From the distribution, we derive the expected values at the future date, value volatility and VaR as percentiles of value changes. The methodology relies on migrations and ratings, plus a credit spread attached to each rating. Because the credit spreads vary with rating and residual maturity, they are tabulated in a matrix cross-tabulating ratings and maturities. This revaluation technique is sometimes referred as "matrix valuation" and is a common building block for several credit portfolio models.

When looking at forward valuation at horizon, there are as many possible credit states as there are migrations, inclusive of the default state. When the facility defaults, the value of the facility is the loss under default. When the facility migrates to another credit state, it is revalued at horizon using the credit spreads mapped to each credit state or rating. There are as many values at horizon as there are risk classes, including default. The distribution of values allows deriving credit risk measures and credit risk VaR.

Credit Metrics uses the migration matrix technique. Applied to a single facility, as illustrated in Figure 46.1, the technique provides only a small number of final values. When applied to a portfolio of facilities, the migrations results in a much higher number of values. Using the forward credit spreads to discount contractual flows beyond horizon provides the values at horizon for all credit states.

FIGURE 46.1 Migration risk and distribution of future values

The valuation formulas of an asset for a forward date are similar to those providing valuation as of the current date. The current date is zero and the horizon for modeling credit risk is one year (date 1).

Forward valuation should use the forward rates and credit spreads for the horizon as seen from today. Since we isolate credit risk, there is no need to consider changes of interest rates. The valuation discounts future payoffs prior to the future date 1. Discount factors applying to the payoffs after the future date 1 are based on forward rates. The forward risky rates are y(0, 1, t), with 0 being the current date, 1 the horizon for future valuation and t the dates of future payoffs prior to date 1 and up to maturity T. Forward credit spreads are derived from forward rates between two future dates. Forward default probabilities from the horizon of the future valuation up to the maturity of a facility are the marginal default probabilities applicable between two future dates, conditional upon no prior default.

The migrations are considered between now and horizon, using, for example, published transition matrices matching that period. The formula for forward valuations as of date 1 is as follows:

The discount rate is the forward risky yield to maturity T starting at one, as seen from today and up to the maturity of the asset.

A maturity of the asset, T, larger than 1 year is explicit in the above formula. In practice, for shorter maturities than horizon, we consider the final payoff under the two states, default or no default, discounted to today. Gains and losses due to credit risk result from comparing forward values at horizon with the current value. The following example illustrates the technique.

• [1] The Credit Metrics technique is developed in the J.P. Morgan technical document [37].

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