Use credit spreads to calculate default probability.

FRM® Part II · Credit Risk Measurement and Management · Credit Risk · FRM2.CR.12.c

The idea

A risky bond's yield spread over the risk-free rate compensates for expected default loss. The standard approximation is Spread = λ × (1 − R). Here λ is the average hazard rate, which is the annual default intensity, and R is the recovery rate. Rearranged, this gives the average hazard rate implied by an observed bond spread. Hazard rate = Spread ÷ (1 − R). Over a horizon of T years, the cumulative probability of default is then PD = 1 − exp(−λT). Survival over the same period is exp(−λT), so the two add to 1. With a term structure of spreads, compute λT for each maturity and take differences to get forward hazard rates and conditional default probabilities. Probabilities backed out of spreads are risk-neutral and exceed real-world historical ones, because spreads also contain risk premium, liquidity and tax effects. Exam point: never read the spread as the default probability, and strip out non-credit components before converting.

Formulas

Hazard rate implied by a credit spread

λ = spread ÷ (1 − R)

spread: the CDS spread or bond yield spread over the risk-free rate
R: the expected recovery rate

An approximation giving a risk-neutral, average annual intensity. Dividing by R instead of 1 − R is the classic slip.

Cumulative default probability from a constant hazard rate

PD = 1 − e^(−λT)

λ: the constant annual hazard rate
T: the horizon in years

Survival is e^(−λT), so the two add to 1. Using λT itself overstates the probability.

Default probability in a later period, given survival to its start

PD(T1 to T2) = 1 − e^(−(λ2 × T2 − λ1 × T1))

λ1: the average hazard rate to T1
T1: the start of the period, in years
λ2: the average hazard rate to T2
T2: the end of the period, in years

Each λ comes from that maturity's spread ÷ (1 − R). Without the conditioning on survival the probability is the smaller unconditional figure.

A practice question

From the question bank, with the reason each wrong option is wrong.

A one-year bond issued by a company trades at a yield spread of 150 basis points over the risk-free rate. Assuming a recovery rate of 40%, the average annual hazard rate implied by the spread is closest to:

0.90%.Why not: Multiplies the spread by (1 - R) instead of dividing by it.
1.50%.Why not: Reads the spread as the default probability and ignores recovery.
2.50%. - correct
3.75%.Why not: Divides the spread by the recovery rate R rather than by (1 - R).

Hazard rate = spread / (1 - R) = 1.50% / 0.60 = 2.50% per year.

A harder one

Bond spreads are 150 basis points at three years and 250 basis points at five years, and the recovery rate is 40%. Using average hazard rates, the probability of default during years four and five, conditional on survival to the end of year three, is closest to:

11.6%.Why not: Computes the unconditional probability of default in years four and five instead of conditioning on survival to year three.
8.0%.Why not: Applies the five-year average hazard rate to the two-year window rather than the forward hazard rate.
7.7%.Why not: Uses the 400 basis point forward spread as the hazard rate without dividing by (1 - R).
12.5%. - correct

Cumulative lambda x T: 3y = 0.015 x 3 / 0.6 = 0.075; 5y = 0.025 x 5 / 0.6 = 0.20833. The difference is 0.13333 over two years, so conditional probability = 1 - exp(-0.13333) = 12.5%.

The hardest tier

A one-year zero-coupon corporate bond yields 8.00% and a one-year risk-free zero yields 4.00%, both with annual compounding. The bond pays face value at maturity if no default occurs and 40% of face value on default. The default probability implied exactly by the bond prices is closest to:

6.67%.Why not: Uses the approximation spread / (1 - R) = 4.00% / 0.60 instead of the exact price comparison.
4.00%.Why not: Reads the yield spread directly as the default probability.
6.17%. - correct
10.00%.Why not: Divides the spread by the recovery rate (0.40) instead of using the loss given default.

Price equality: 1/1.08 = [(1 - PD) + PD x 0.4]/1.04, so 1 - 0.6 PD = 1.04/1.08 = 0.96296 and PD = 0.03704/0.6 = 6.17%.

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