Credit spreads are related to default probabilities and loss given default. The relationship results from the two alternate methods for valuing a risky bond:

• discounting at the risk-free rate the expected payoff

• discounting at the risky rate the contractual payoffs.

The discount factor, in continuous time, applying to a future date t for calculating a present value as of today is:

where y is the discount rate. The bond value discounts the promised payoffs at the risky yield to maturity y as:

In this summation, t is any intermediate date, when a payoff occurs, between now and maturity T.

For using risk-neutral valuation, it is necessary to derive the probabilities of default and survival over each time interval, and to discount at the risk-free rate such expected payoffs. The risk-neutral intensity of default is X(t) and the recovery rate is R in percentage of amount due. For simplification purposes, we assume that both are constant. In practice, we can also use discrete time intervals over which they can be assumed constant.

Rather than using integrals for valuing the bond, we consider valuation over a small time interval in a continuous time setting[1]. The small period of time At is defined by two time points, t and t + At, such that At tends towards zero. The default probability from t to t + At is XAt. The cumulative survival probability until t viewed from today is the exponential function exp(-Xf). The expected payoff at any intermediate date t is F under no default, with probability exp(-Xt). The payoff under default is R with a probability equal to the probability of default conditional on survival until t. This probability is equal to the product of the probability of survival until t multiplied by the probability of default in the interval t to t + At, or exp(-Xt)XAt.

Over a small interval of time At, using the risk-neutral valuation, the value of the bond at the beginning of the interval [t, t + At] discounts the payoffs under survival and recoveries under default:

Since the argument of the exponential is small, its first-order approximation is exp(-w) ~ 1 -u:

The value of the bond at the beginning of the interval, t, is the discounted value of the payoff at the end of the interval, or Ft+At. The value using the risky yield y, equal to the risk-free rate plus the credit spread s, is:

The value of the bond using risk neutral valuation discounts its expected payoff at the risk-free rate:

From this formula we can factor out the payoff because the recovery rate is a percentage of the payoff. Expanding the bracket terms, we obtain:

We ignore the second-order terms (At2) when the small interval At tends towards zero, and simplify the second expression to:

We obtain two alternative valuations of the bond value:

risk-neutral valuation: F(bond) ~ Ff+Ai {1 - At [(y + (l - R)]} risky yield valuation: Ff bond) ~ Fi+Ai {1 - At (yf+ s)} Eliminating the contractual payoff and equating those two values, we extract the credit spread s:

The credit spread is the product of the risk-neutral default intensity and loss given default (LGD). This shows that the credit spread equals expected loss under risk-neutral probabilities.

  • [1] This is inspired by the simple presentation by Duffie and Singleton [28].
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