Set probabilities of default estimated from past experience beside those inferred from credit yield spreads, and weigh them against each other.
The idea
There are two ways to estimate default probabilities. Historical estimates count how often issuers of a given rating defaulted, which gives real-world probabilities. Spread-based estimates infer a default intensity from bond yield spreads or CDS spreads, which gives risk-neutral probabilities. The hazard rate comes from the spread s and the recovery rate R as follows: Hazard = s ÷ (1 − R). Spread-implied probabilities are generally well above historical ones, often by a factor of two or more. The gap arises because spreads also compensate for systematic default risk, which cannot be diversified and so earns a risk premium. Spreads also compensate for bond illiquidity and tax differences, while a historical frequency measures only average realised defaults. So only part of a spread, roughly the historical default intensity × (1 − R), is expected loss. Historical numbers suit forecasting actual losses, whereas spread-implied numbers reflect what the market charges, so neither is a free substitute for the other.
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
A risk-neutral intensity. It is generally well above the historical default intensity for the same rating.
Expected loss part of a credit spread
expected loss spread = historical intensity × (1 − R)
- historical intensity
- the real-world default intensity for the rating
- R
- the recovery rate
The rest of the spread is a premium for systematic risk, illiquidity and tax. Leaving out the (1 − R) overstates the expected loss part.
A practice question
Which is the main reason spread-implied default probabilities exceed historical ones?
- Historical frequencies include recoveries, while spreads do not, so spreads imply more defaults.Why not: Recovery enters the spread-to-hazard conversion and does not explain the gap on its own.
- Historical frequencies are always measured over horizons that are too short.Why not: Horizon length does not produce a systematic upward bias in spread-implied estimates.
- Investors require compensation for the systematic, non-diversifiable component of default risk, which a historical frequency does not include. - correct
- Spreads are computed from default probabilities that agencies deliberately overstate.Why not: Agencies do not overstate probabilities to set spreads; the difference is mostly a risk premium.
Default risk is correlated with the economy and with other risks, so it earns a premium; that premium makes risk-neutral probabilities higher than real-world ones.
A harder one
A bond has a yield spread of 210 basis points over the risk-free rate. The historical default intensity for its rating is 1.2% per year and the recovery rate is 40%. How many basis points of the spread are not explained by historical expected loss?
- 72Why not: Reports the expected loss part of the spread, not the unexplained part.
- 90Why not: Uses the default intensity as the loss rate and omits the 60% loss given default.
- 210Why not: Treats the whole spread as unexplained by expected loss.
- 138 - correct
Expected loss = 1.2% x 0.60 = 72 bp; unexplained spread = 210 - 72 = 138 bp, which is the premium for systematic risk, illiquidity and tax effects.
The hardest tier
A bond's yield spread is 180 basis points. Estimates suggest 40 basis points of it is compensation for illiquidity and tax effects. The historical default intensity is 1.5% a year and the recovery rate is 40%. How many basis points of the spread are left as a credit risk premium beyond historical expected loss?
- 90Why not: Subtracts expected loss but does not remove the illiquidity and tax component first.
- 50 - correct
- 140Why not: Removes the illiquidity and tax component but not the expected loss.
- -10Why not: Treats the 1.5% default intensity as the loss rate, forgetting the 60% loss given default.
Expected loss = 1.5% x 0.60 = 90 bp. Premium = 180 - 40 - 90 = 50 bp.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Estimating Default Probabilities
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