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This section addresses the equivalence between credit spreads and default probabilities under risk-neutrality. Risk-neutrality means that investors are indifferent between the expected value of a random flow and the certain flow having a value equal to the expected value of the random one. Since this is unrealistic, natural or real probabilities differ from risk-neutral probabilities. This provides an intermediate step to model credit spreads in relation to the default probabilities and the recovery rate. After introducing notations, we proceed to the calculation of the value of the risky debt.

There are two ways to value risky debt. Either we discount the risk-adjusted flows at the risk-free rate, in practice the expected future flows subject to default risk, or we discount the contractual flows at the risky market rate. Under arbitrage, these two values should be identical. This imposes a condition on the probabilities used to value the expected flows. The corresponding probabilities are risk-neutral probabilities. They differ from natural probabilities because of the market risk aversion. The above equivalence makes it easy to derive implied risk-neutral probabilities from risky debt prices.

The section uses a simple example of a 1-year debt to illustrate the formula. General formulas applying to several periods are given in the last subsection.

Notation

We use the following notation for spot yields and default probabilities.

Notation for spot transactions, the current date being 0:

Risky rate between any two dates: y(0, t) Risk-free rate between any two dates: y_f(0, t) Default probability between any two dates: d(0, t)

Recovery rate: equal to 1 — Lgd(%). This parameter is assumed constant over time Uncertain future flow at date t: F_t

Notation for forward transactions starting at the future date t up to the future date t + n :

Risky rate between any two dates: y(t, t + n) Risk-free rate between any two dates: y_f(t, t + n) Default probability between any two dates: d(t, t + n) The other notation is the same as above

Linking market spreads with the underlying credit risk parameters requires making explicit the market valuation as a function of these credit risk parameters. The simplest example would be an investor facing two investment opportunities for 1 year:

• A zero-coupon risky bond, with a 1-year maturity, yielding the risky rate y = 8%.

• A zero-coupon risk-free bond, with a 1-year maturity, yielding the risk-free rate y_f = 6%.

In both cases, the investor invests a value of 1 up to the horizon 1. The investment future value depends on the credit risk of the risky debt, and is the certain value (1 + y_f) with the risk-free debt. The risky debt has a default probability d(0,1) = 1.5% between the dates 0 and 1. In the event of default, the recovery rate R is 30%. From the above data, we show that the market requires a credit spread of 2% for the risky debt above the risk-free rate.