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Financial Instruments, Markets and Conventions / Credit: Corporate Bonds, Credit Spreads, and the Default Option

The Risky Bond Identity: Default Probability, Recovery Rates, and the Credit-Spread Decomposition

The previous lesson established the institutional picture: credit ratings grade default risk, and credit spreads are the market's price for bearing it. This lesson builds the quantitative model that connects the spread to the probability and severity of default.

The central insight is a clean decomposition:

Risky bond⏟what you buy=Risk-free bond⏟risk-free cash flows−Default option⏟you wrote to the issuer\underbrace{\text{Risky bond}}_{\text{what you buy}} = \underbrace{\text{Risk-free bond}}_{\text{risk-free cash flows}} - \underbrace{\text{Default option}}_{\text{you wrote to the issuer}}

You, as bondholder, own a portfolio of risk-free cash flows — but you have written the issuer a put option: the right to walk away from those cash flows (by defaulting) and deliver the firm's assets at a distressed value. The spread is the fair premium for that option.

The hazard rate model gives the compact mathematics: a constant default intensity λ\lambda generates a survival probability e−λte^{-\lambda t} and links to the observed spread through the credit triangle: s≈λ(1−R)s \approx \lambda(1-R), where RR is the recovery rate.
Learning objectives. By the end of this note you should be able to:
  • state the risky bond identity and interpret it as a decomposition into a risk-free bond and a short default option;
  • define the hazard rate λ\lambda as the instantaneous conditional default probability and derive the survival probability and cumulative default probability;
  • use the credit triangle s≈λ(1−R)s \approx \lambda(1-R) to estimate λ\lambda from an observed spread and recovery rate;
  • extract term-structure-consistent hazard rates from a schedule of credit spreads across maturities;
  • describe historical recovery rates by seniority and explain why recovery rates are negatively correlated with default rates;
  • explain why the spread-implied default probability from the credit triangle exceeds the historically realised default probability.

In this lesson

  1. The risky bond identity
  2. The hazard rate and survival probability
  3. Recovery rates
  4. The credit triangle
  5. The spread-implied vs. historical default probability
  6. A more precise calculation: bootstrapping hazard rates from bond prices

Before you start

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