FRM Exam Part II · Estimating Default Probabilities

Estimating Default Probability from Bond Yields and Credit Spreads

Updated 11 October 2026 · Fact-checked

The credit spread is the corporate bond yield minus the risk-free yield for the same maturity. It pays for expected default loss, so spread ≈ hazard rate × (1 − recovery rate). Divide the spread by (1 − R) to get the risk-neutral hazard rate, then use 1 − e^(−λT) for cumulative default probability.

Understand Default Probabilities from Bond Yields and Credit Spreads

A corporate bond pays a higher yield than a risk-free bond of the same maturity. The extra yield is the credit spread. Part of it compensates you for the chance that the issuer defaults and you recover only part of the face value.

The hazard rate (default intensity) λ is the probability of default per year, given survival so far. If λ is constant, the probability of surviving T years is e^(−λT). The cumulative default probability is 1 − e^(−λT). The recovery rate R is the share of value you get back after default. Loss given default is 1 − R.

The key link is: spread ≈ λ × (1 − R). Read it as annual expected loss per unit of bond value. A 200 bp spread with 40% recovery implies λ of about 3.33% a year, not 2%. The spread alone is not the default probability.

The probabilities you get this way are risk-neutral. They come from market prices, so they price in investors' risk aversion. The spread also contains compensation for liquidity and other premia. So risk-neutral default probabilities are usually higher than real-world (historical) ones. Use risk-neutral probabilities to price and value. Do not use them as forecasts of actual default frequency or for expected loss and capital.

The exact route uses bond prices. The risk-free bond is worth more than the corporate bond. The gap is the present value of expected default loss. For a zero-coupon bond with recovery paid as a fraction of face value, you can solve directly for the default probability.

Key formulas to remember

Credit spread: s = y − y_f; Corporate yield minus risk-free yield, same maturity and same compounding. Convert bp to decimals: 150 bp = 0.015.
Spread, hazard rate and recovery (approximation): s ≈ λ × (1 − R); λ is the average annual risk-neutral hazard rate. It is an approximation, and works best for modest spreads.
Hazard rate from spread: λ ≈ s ÷ (1 − R); Higher assumed recovery means a higher implied hazard rate for the same spread.
Cumulative default probability (constant hazard): Q(T) = 1 − e^(−λT); Survival probability is e^(−λT). For small λT, Q(T) ≈ λT.
Default probability from zero-coupon prices: Q(T) = [1 − B_corp ÷ B_rf] ÷ (1 − R); Assumes recovery is a fraction R of the face value paid at maturity. With continuous compounding, B_corp ÷ B_rf = e^(−sT).
Average hazard rate from cumulative probability: λ = −ln(1 − Q(T)) ÷ T; Use this to turn a cumulative default probability back into an annual hazard rate.

How to solve Default Probabilities from Bond Yields and Credit Spreads questions

Use this order for any question that asks for a default probability, hazard rate or spread from market yields.

1Write down the corporate yield, the risk-free yield for the same maturity, and the recovery rate R. Check the compounding convention.
2Compute the spread s = y − y_f. Convert basis points to a decimal.
3Decide whether the question wants the approximation or the price-based exact method. If bond prices or zero-coupon yields are given, lean towards the price-based method.
4Approximation: λ = s ÷ (1 − R). Exact zero-coupon: Q = (1 − B_corp ÷ B_rf) ÷ (1 − R).
5Convert between annual hazard rate and cumulative probability: Q(T) = 1 − e^(−λT), or λ = −ln(1 − Q) ÷ T.
6State the result as risk-neutral. If asked for a comparison with historical default rates, note that the risk-neutral figure is usually higher.
7Sanity check: Q must lie between 0 and 1, and λ should be larger than s when R > 0.

Quickest way: Spread divided by loss given default

When to use it: Multiple-choice questions that give a spread (or two yields) and a recovery rate and ask for the annual default probability or hazard rate.

Spread = corporate yield − risk-free yield.
Divide by (1 − R). That is your annual hazard rate.
For multi-year cumulative probability, compute 1 − e^(−λT). If λT is small, λT is a quick estimate, but it slightly overstates the exact value.
Eliminate options that equal the raw spread. That is the usual trap.

Common mistakes in Default Probabilities from Bond Yields and Credit Spreads

Treating the credit spread as the default probability.
Spread and default probability are both small percentages, so they look interchangeable.
Fix: Spread is expected loss, not probability. Always divide by (1 − R) to get λ.
Multiplying by (1 − R) instead of dividing.
The formula s ≈ λ(1 − R) is remembered, but the rearrangement is rushed.
Fix: Solve for the unknown first. For λ, divide the spread by (1 − R). A λ smaller than the spread is a red flag.
Using the bond yield rather than the spread.
The question gives a 6.2% yield and the number gets used directly.
Fix: Subtract the risk-free yield for the same maturity first. Only the excess yield pays for default risk.
Confusing the annual hazard rate with the cumulative default probability.
Both are called default probabilities.
Fix: λ is per year. Cumulative Q(T) = 1 − e^(−λT). Check which one the question asks for and the horizon T.
Presenting risk-neutral probabilities as real-world forecasts.
The same words, default probability, cover two different measures.
Fix: Spread-implied probabilities include risk premia and liquidity effects. Use them for valuation and pricing. Real-world probabilities, from ratings and history, suit expected loss and capital.
Mixing units and maturities, such as basis points with percentages or a 5-year yield with a 10-year risk-free rate.
Data is quoted in different forms under time pressure.
Fix: Convert everything to decimals and match the maturity of the risk-free rate to the bond.

Worked examples

Example 1

A 5-year zero-coupon corporate bond yields 6.2% and the 5-year risk-free rate is 4.2%. Recovery is 40%. Estimate the average annual risk-neutral hazard rate and the 5-year cumulative risk-neutral default probability. Assume a constant hazard rate.

Show the solution

Spread = 6.2% − 4.2% = 2.0% = 0.02.
Hazard rate λ ≈ 0.02 ÷ (1 − 0.40) = 0.02 ÷ 0.60 = 0.03333, or 3.33% a year.
Cumulative probability Q(5) = 1 − e^(−0.03333 × 5) = 1 − e^(−0.16667).
e^(−0.16667) ≈ 0.8465, so Q(5) ≈ 1 − 0.8465 = 0.1535.

Answer: λ ≈ 3.33% a year; 5-year cumulative risk-neutral default probability ≈ 15.35%.

Example 2

A 3-year zero-coupon corporate bond yields 5.5% and the 3-year risk-free zero rate is 4.0%, both continuously compounded. Recovery is 30% of face value, paid at maturity. Find the 3-year default probability and the implied average annual hazard rate.

Show the solution

Spread = 5.5% − 4.0% = 1.5% = 0.015.
Price ratio B_corp ÷ B_rf = e^(−0.015 × 3) = e^(−0.045) ≈ 0.9560.
Q(3) = (1 − 0.9560) ÷ (1 − 0.30) = 0.0440 ÷ 0.70 ≈ 0.0629.
Average hazard rate λ = −ln(1 − 0.0629) ÷ 3 = 0.0649 ÷ 3 ≈ 0.0216.
Check with the approximation: 0.015 ÷ 0.70 = 2.14%, which is close.

Answer: 3-year default probability ≈ 6.29%; average annual hazard rate ≈ 2.16%.

Exam tips

Look for the giveaway wrong option that equals the raw spread or the spread times (1 − R). Compute by dividing.
Check whether the question wants annual hazard rate or cumulative probability, and over what horizon.
Questions often ask why spread-implied probabilities exceed historical ones. The answer is risk premium, plus liquidity and other non-default components in spreads.
Keep the exact price-based formula and the s/(1 − R) shortcut both ready. Use the price-based one when bond prices are given.
If recovery rises and the spread is fixed, the implied default probability rises. Be ready to explain that direction.

Practice questions from Estimating Default Probabilities

Default Probabilities from Bond Yields and Credit Spreads in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Default Probabilities from Bond Yields and Credit Spreads: frequently asked questions

What is the difference between credit spread and default probability?

The credit spread is a yield difference. It pays for expected loss from default, so it depends on both the default probability and the loss given default. Default probability is a likelihood. To go from one to the other, use λ ≈ s ÷ (1 − R).

How do I calculate risk-neutral default probability from bond prices?

Compare the price of a risk-free zero-coupon bond with a corporate zero of the same maturity. If recovery is a fraction R of face value, Q = (1 − B_corp ÷ B_rf) ÷ (1 − R). Then convert to a hazard rate with λ = −ln(1 − Q) ÷ T if needed.

Why are risk-neutral default probabilities higher than real-world ones?

Market spreads include compensation for bearing default risk, and often for illiquidity and other effects. The implied probability therefore overstates actual expected default frequency. Use risk-neutral figures for pricing, not for forecasting losses.

Is the formula spread = hazard rate × (1 − recovery) exact?

No. It is an approximation, best when spreads are modest. Exact treatment uses bond prices and the timing of recovery assumptions. In the exam, use the approximation unless you are given prices and told to use them.