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IAI Actuarial Core Principles · Economic Modelling · Simple models for credit risk

In a two-state (default/no default) model with a constant hazard rate λ and zero recovery, a risky zero-coupon bond maturing at time t is priced under the risk-neutral measure with risk-free force of interest δ. Which expression gives the bond's price per unit face value?

The price is e^{-(δ+λ)t}. Survival to time t has probability e^{-λt}, the bond pays one unit only on survival, and discounting at the risk-free force gives the product, so the spread equals the hazard rate.

  1. Ae^{-δt} + λt
  2. Be^{-(δ+λ)t}Correct
  3. Ce^{-δt}(1 - λ)
  4. De^{-δt}/(1+λt)
  5. e^{-(δ-λ)t}

Explanation

Survival probability to t is e^{-λt}. With zero recovery the payoff is 1 only on survival, so price = e^{-δt}e^{-λt} = e^{-(δ+λ)t}. The credit spread is therefore λ exactly.

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