Reduced Form Models and Hazard Rates in Credit Risk

Understand Reduced Form Models and Hazard Rates

Key rules to remember

Survival probability (constant hazard)

P(τ > t) = e^(−λt)

τ is the default time. Time to default is exponential with mean 1/λ.

Survival probability (time-varying hazard)

P(τ > t) = exp(−∫₀ᵗ λ(s) ds)

For piecewise constant λ, add λ × length over each period inside the exponent.

Default probability by time t

P(τ ≤ t) = 1 − e^(−λt)

Use the risk-neutral λ for pricing and the real-world λ for real-world risk measurement.

Hazard rate definition

λ(t) = f(t) ÷ S(t)

f is the density of τ and S is the survival function. This is the same idea as force of mortality.

Defaultable zero-coupon bond, recovery of face value at maturity

Price = e^(−rT) × [R + (1 − R) e^(−λT)]

Assumes constant r and λ, independent, with recovery R of face value paid at T if default occurs before T.

Defaultable zero-coupon bond, zero recovery

Price = e^(−(r + λ)T)

The spread equals λ when R = 0.

Approximate credit spread

s ≈ (1 − R) × λ

Approximation for small λ. Rearrange to get λ ≈ s ÷ (1 − R).

How to solve Reduced Form Models and Hazard Rates questions

1. 1Identify what is given: hazard rate λ (real-world or risk-neutral), risk-free rate r, recovery rate R, maturity T, and whether recovery is paid at default or at maturity.
2. 2State your assumptions: constant or piecewise hazard, constant interest, independence of default and interest rates, and the recovery convention.
3. 3Find the survival probability using e^(−∫λ) and the default probability as one minus it.
4. 4Check which probability measure the question needs. Use risk-neutral λ for pricing. Use real-world λ for expected loss or real-world probabilities.
5. 5Write the payoff in each state: no default pays face value, default pays the recovery amount.
6. 6Discount the expected payoff at the risk-free rate under the risk-neutral measure, to get the price.
7. 7If asked for the spread or the implied hazard rate, compare with the risk-free bond or use s ≈ (1 − R)λ, and show it is an approximation.
8. 8Check reasonableness: the price should be below the risk-free price, and the spread should be positive.

Quickest way: Price by splitting into survival and default parts

Common mistakes in Reduced Form Models and Hazard Rates

• Using the real-world default probability to price a bond.

  Students see a historical default rate given in the question and assume it is the pricing input.

  Fix: Price only with risk-neutral probabilities. Use real-world ones only when the question asks about actual expected losses.

• Writing default probability as λT for any T.

  The approximation λ dt works for very small intervals and gets over-applied.

  Fix: Use 1 − e^(−λT). Use λT only if the question says the period is small and approximation is acceptable.

• Treating the hazard rate as the unconditional probability of default in a year.

  The word rate hides that it is conditional on survival so far.

  Fix: Remember λ(t) is the instantaneous rate given no default before t. The one-year default probability is 1 − e^(−λ) for constant λ.

• Forgetting the recovery term, or applying recovery to the wrong amount.

  Students rush and use e^(−(r+λ)T), which is correct only for zero recovery.

  Fix: Write the payoff in both states first. Check whether recovery is a fraction of face value and when it is paid.

• Using s = λ when recovery is positive.

  The zero-recovery result is remembered and applied everywhere.

  Fix: Use s ≈ (1 − R)λ. Only with R = 0 does the spread equal λ.

• Adding hazard rates of different periods without multiplying by period length.

  Piecewise hazard questions look like a list of rates.

  Fix: Integrate: add λ × years for each period inside the exponent, then exponentiate once.

Worked examples