Outline CDS spreads and show how they can serve to infer hazard rates.
The idea
A CDS spread is the annual premium, as a fraction of notional, that makes the swap worth zero at inception. It reflects the expected annual loss from default. Let λ be the average hazard rate and R the recovery rate. The credit triangle approximation says the spread is the hazard rate times the loss given default: Spread = λ × (1 − R). Rearranged, the average hazard rate over the life of the swap is: Hazard = Spread ÷ (1 − R). A 120 basis point spread with 40 percent recovery implies a hazard rate of 0.012 ÷ 0.6, or 2 percent. For a horizon of T years and a constant hazard rate, the cumulative default probability is: PD = 1 − e^(−λ × T). Spreads of different maturities give average hazard rates that bootstrap into forward hazard rates. Let λ1 and λ2 be the average rates to T1 and T2. The forward hazard between them is: Forward = (λ2 × T2 − λ1 × T1) ÷ (T2 − T1). Mistakes include dividing by R instead of 1 − R, and treating the spread as the default probability.
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
The credit triangle. A higher assumed recovery raises the implied hazard rate for the same spread.
Cumulative default probability with a constant hazard rate
cumulative PD = 1 − e^(−λ × T)
- λ
- the constant hazard rate per year
- T
- the horizon, in years
Use the hazard implied by the spread, not the spread itself.
Forward hazard rate between two CDS maturities
forward hazard = (λ₂ × T₂ − λ₁ × T₁) ÷ (T₂ − T₁)
- λ₁, λ₂
- the average hazard rates implied by the shorter and longer CDS spreads
- T₁, T₂
- the shorter and longer maturities, in years
Each λ × T is a cumulative hazard. Using the longer average hazard as the forward rate understates it on a rising curve.
A practice question
Which expression is the standard approximation linking the CDS spread s, the hazard rate lambda and the recovery rate R?
- s = lambda x RWhy not: Uses the recovery rate instead of loss given default.
- s = lambda / (1 - R)Why not: Divides by loss given default instead of multiplying.
- s = lambda x (1 + R)Why not: Adds recovery to one; the loss is one minus recovery.
- s = lambda x (1 - R) - correct
The spread compensates for the expected annual loss, the hazard rate times loss given default, so lambda = s / (1 - R).
A harder one
A 5-year CDS spread is 240 basis points and the recovery rate is assumed to be 40%. What is the implied 5-year cumulative probability of default, assuming a constant hazard rate?
- 11.31%Why not: Uses the spread itself as the hazard rate, ignoring the 60% loss given default.
- 18.13% - correct
- 20.00%Why not: Multiplies the hazard rate by five instead of using 1 - exp(-lambda T).
- 25.92%Why not: Divides the spread by the recovery rate instead of by one minus the recovery rate.
lambda = 0.0240 / 0.60 = 4.0%; cumulative PD = 1 - exp(-0.04 x 5) = 18.13%.
The hardest tier
Using a 50% recovery rate, the 1-year CDS spread is 80 basis points and the 2-year spread is 120 basis points. Treating the average hazard rate to each maturity as the cumulative hazard divided by time, what is the probability, as seen today, that the firm defaults during year two?
- 3.15%Why not: Gives the conditional probability in year two, given survival through year one.
- 4.69%Why not: Gives the two-year cumulative probability of default.
- 2.40%Why not: Reports the two-year average hazard rate as the year-two probability.
- 3.10% - correct
Average hazards are 1.6% and 2.4%. Cumulative hazard is 1.6% at one year and 4.8% at two, so year-two unconditional PD = exp(-0.016) - exp(-0.048) = 3.10%. The year-two forward hazard is 3.2%.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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