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Full valuation mixes several effects. First, it combines the risk effect with the relative richness or poorness effect, or the influence of the excess spread of an asset over

the market required yields. Second, a longer maturity increases the length of exposure to credit risk. Third, the effect of a longer maturity on value depends on excess spread. Finally, migrations alter the risk. The effects of moving forward in time include:

• The drift of time, resulting in a lower number of remaining cash flows and related discount factors. This is the roll-down effect, keeping all other parameters constant, or the amortization effect.

• The usage of forward parameters rather than their current values such as the forward yields, both risk-free and risky, and the corresponding credit spreads and risk-neutral probabilities.

IMPLEMENTING THE VALUATION BUILDING BLOCK IN CREDIT RISK MODELS

Valuation is a key building block for determining the value distribution at a forward horizon. In risk models, valuation is forward, at the horizon, for all possible credit risk scenarios, in order to derive a distribution of future values. This section describes how matrix valuation, based on transition matrix and credit spreads, and risk-neutral valuation apply to find the distribution of values at a future horizon. It also compares the valuation building block of major models using full valuation of migration risk. The subsections summarize the valuation formulas using contractual flows and market risky yields, to show how matrix valuation operates and indicate which techniques credit risk models use.

Future Valuation

The valuation formulas for a forward date are similar to those providing valuation as of the current date, except that the stream of cash flows rolls down when getting closer to maturity. Formulas use yield to maturity as discount rate. The valuation of the debt as of date 0 is the discounted value of all contractual future flows F_t at the risky yield to maturity. RYV(t) is the value of risky debt at date t. The risk-neutral probability is d_t= d*(0,t) when calculations are as of current date 0, and d*(1, t)whenweare at a forward date 1. Risky yields to maturity y are risk-free yield yf plus credit spread cs: y = yf + cs, either spot or forward. For a current valuation, we use 0 as the current date. For a mark-to-future valuation at the horizon, we use 1 as the horizon date, and discounting applies to dates 2 and beyond. The formulas for forward valuations are as follows:

Matrix Valuation

When the facility defaults, the loss is net of recoveries, or loss under default. When looking at forward valuation at the horizon, there are as many possible credit states as there are migrations. The distribution of values is the foundation of the credit risk

VaR with full valuation models. There are as many values at the horizon as there are risk classes, including default. Mark-to-future designates the forward valuation for all possible migrations. The principles for valuing a risky debt at the current date and at a forward date are similar, except that the current value is certain while the forward valuation results in an entire distribution of values.

Forward valuation applies to debts that do not default between current and forward dates, and to facilities with maturities longer than the horizon. Under the migration matrix technique, there are a finite number of credit states at the horizon. Using the forward credit spreads to discount contractual flows beyond the horizon provides the values at the horizon for all credit states (Figure 42.1).