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The risk-neutral default probability is such that the risky debt expected value at the horizon is equal to that of the risk-free debt. The risk-neutral probability from 0 to n is d*(0, n), instead of d(0, n), the natural default probability. It is such that:

This risk-neutral default probability is higher than the observed default probability of 1.5%. The risk-neutral probability is the probability that would apply in a risk-neutral universe by definition since it makes equal the expected value with its certainty equivalent.

Natural probabilities are real world probabilities, as opposed to the risk-neutral probabilities. Nevertheless, risk-neutral probabilities are also real world probabilities because they are those effectively used by investors to value the risk premium attached to risky flows! Our convention is as follows:

• Risk-neutral default probabilities are the default probabilities adjusted for the risk premium. They serve for valuation using the expected flows and the risk-free rate as discount rate.

• Real world default probabilities, or natural default probabilities, are those effectively observed. These natural probabilities do not serve for valuation. Valuation discounts contractual flows using the risky rate.

Risk aversion implies that investors behave as if they used default probabilities higher than natural default probabilities. In order to find predictors of actual probabilities, we need a correction for risk aversion. The value of risk aversion is the difference between the expected value of the future flow at actual default probabilities and the certain equivalent flow using risk-neutral probabilities. We know the certain equivalent flow, which is 1.06, and the expected flow with actual probabilities, 1.06488. The ratio of the certain equivalent flow to the expected value of the flow is lower than 1. It is:

The risk premium, as a future value at the horizon, is 0.0081 per unit invested.

Similar calculations could use the credit spreads since they quantify risk aversion. Therefore, in order to derive implied natural default probabilities from the credit spreads, we need to equate the value of the expected flows under risk-neutral probabilities with the value of the contractual flows at the risky yields. However, credit spreads value risk aversion and recovery rate. Unless we know the embedded recovery rate in market spreads, we cannot value risk aversion.

General Formulas with Multiple Period Horizons

This section makes explicit the relationship between credit spreads and risk-neutral default probabilities, given recovery. It shows the consequences of equating valuation using risky market yields and valuation using risk-neutral probabilities. The example extends to maturities longer than 1 year by splitting them down into zero-coupons of all contractual maturities. Any coupon bond can be stripped into a series of zero-coupons, whose terminal flows are identical to all interest and principal flows, so that the analysis extends to all bonds.

Alternatively, for multiple periods, an equivalent valuation technique uses yields to maturity instead of zero-coupon yields. This formulation extends to multiple periods up to maturity T. The yield to maturity Ytm is such that the present value of a risky debt is equal to its price. This yield to maturity embeds an averaged credit spread. RYV is the value of the risky debt and RFV is the value of the same risk-free debt. In what follows, we use d*(0,t) = d_t*.