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Capital Under Default Mode

For a default-only model, ignoring migrations, and using book values only, the issue is simple. The value under no prior default is the book value. Hence, the zero-loss point is the aggregated book value as of horizon, conditional on no prior default, which is the current book value minus amortization until horizon. Capital should be net of expected loss EL, which includes both principal and accrued revenues. For a default model, the expected loss is the expectation of all book value losses at horizon. It is equal to book value multiplied by the default probability and multiplied by the loss given default. Considering the accrued portfolio net revenues, i?p, as an additional cushion for absorbing losses is another option. Only accrued portfolio revenues conditional on no prior default provide a cushion against losses. The most comprehensive value of capital is net of expected loss, and, eventually, net from accrued revenues conditional on no prior default.

Capital Under Full Migration Mode

For a full valuation model, the expected value at horizon is random and depends on excess spread of facilities over market spread and the related "roll-down" effect, migrations, the forward risk-free rates plus the forward credit spreads matching the final credit state.

For a single facility, the expected value results from all migrations, including migrations to the default state. For the default state, the expected value depends on value if there is no default plus the recoveries under default. Without prior default, the value at horizon is an expectation over all possible credit states other than default. The expected value is the weighted average of expected values under no prior default, multiplied by the survival probability, plus the value given default (recoveries) multiplied by the default probability. The equation is:

It is simpler to assume that (V default) is certain, as recoveries, which is restrictive. Otherwise, we should consider the expected value under default, which is the expectation of the recovery distribution. Such expected value embeds expected loss. The zero point for potential loss is the expected value conditional on no prior default, or E(V | no default) [1 - d(0, H)]. At this point, we can simply use the following notations:

E(V no default) =E(V no default) [ 1 - d(0, H)]

E(V default) =£(7| default) d(0, H)

The expected loss EL is the difference between the value under no default and the value under default.

ELp = E(V no default) -E(V default)

The capital is in excess of EL, or equal to the loss percentile minus expected loss, ignoring excess spread.

K(a) = L(a) -[£(7| no default) - E(V | default)]

If we choose to consider excess spread as an additional cushion against future losses, we deduct it from the above:

K(a) = L(a) -[£(7| no default) -E(V default)] - (ES | no default)

For a portfolio, the expected value also includes expected loss and expected excess spread. Capital should be determined with the same principles than above for a default mode model. The most general capital formula is similar to above equation:

£p(a) = Lp(a) - £(7P | no default) - (ESp | no default) - ELp

The zero point for measuring potential losses, or capital, is £(Fp | no default) + (ESp | no default) - ELp. The capital is valued as of horizon.

Moody's-KMV Portfolio Manager follows these principles. Credit Metrics uses the current value, and does not calculate the expected value under no prior default. Both are feasible choices for calculating the capital.

 
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