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Under risk-neutral valuation, risk-free rates apply to the expected flows calculated with risk-neutral probabilities. KMV Portfolio Manager provides credit spread valuation and is the only model also providing risk-neutral valuation by modelling risk-neutral probabilities directly.

For each expected default frequency Edf, there is a risk-neutral probability embedding the effect of risk aversion. The risk-neutral probability is such that the asset provides the risk-free yield. It is the probability that the asset falls below the default point when it yields the risk-free rate r_f rather than the expected risky return. We know the default point at the horizon. When the asset provides a risk-free return lower than its actual risky yield, its horizon value is lower than if it provides the risky return. Therefore, the distribution of asset values at the horizon is below the distribution of the risky asset return. With the same default point, this lower distribution results in a higher probability of hitting the default point. This probability, higher than the natural default probability, is the risk-neutral probability. The areas under the two distributions, below the default point DP, are the default probabilities, actual or risk-neutral. The risk-neutral probability exceeds the natural probability.

The process needs to price risk aversion. Risk aversion is the difference between the certain equivalent of an uncertain flow and its expected value using natural probabilities. There are several indications of risk aversion. KMV Portfolio Manager uses the risk aversion applicable to equity returns in the Capital Asset Pricing Model (CAPM). In addition, it calibrates this risk aversion on bond data of varying maturities, which is volatile. The risk-neutral probabilities are consistent with this risk aversion. This valuation technique does not require credit spreads (Table 42.3).

APPENDIX: RESIDUAL MATURITY, VALUATION AND RISK MEASURE

In this technical section, we focus on value changes due to a pure maturity roll-down effect. The change in value through time is the basis of loss measures. By controlling all market parameters, we see the pure effect of the difference in valuation dates, as of today and at the horizon, which drives the value-based measure of credit risk loss. It makes it clear that under specific conditions, a loss occurs simply due to time decay.

For this purpose, we use a bullet loan. To isolate the revenue effect, we change some notations. The facility provides a return r. The market risky required yield is y as above, and includes a credit spread. It remains constant through time. The facility yield is a coupon C = r % x K ,where K is the capital repaid once at maturity T .Atthe endof each period, there is a coupon payment.

The facility value discounts the coupons until maturity, plus the principal at maturity, at the market required rate y. The value discounts the coupons. The capital amortizes once at maturity. The formula for a recurring coupon, plus the terminal discounted term (the capital equal to K) at maturity T is: