FRM Exam Part II · Fundamentals of Credit Risk

Credit Spreads and Risk-Neutral vs Real-World Default Probabilities

Q: How do I extract default probability from a bond spread?

Subtract the risk-free yield from the bond yield to get the spread. Divide by LGD, which is 1 minus the recovery rate, to get the hazard rate. Convert to a cumulative PD with 1 − e^(−λT) for horizon T.

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

Spreads include more than expected loss. Investors demand a premium for default risk that is systematic and clusters in bad times, plus compensation for illiquidity and tax effects. Risk-neutral PDs absorb all of this, so they come out higher.

Q: What is the credit spread puzzle?

It is the finding that observed spreads are much larger than what historical default and recovery experience would justify. The excess is attributed to risk premia, liquidity and other factors. It is why spread-implied PDs should not be read as forecasts.

Q: How are CDS spreads related to the hazard rate?

For a CDS, the spread is approximately the hazard rate times LGD, so λ ≈ CDS spread ÷ (1 − R). This gives a risk-neutral hazard rate, which is the right input for pricing other credit products.

Updated 11 October 2026 · Fact-checked

A credit spread is the extra yield a risky bond pays over a risk-free bond. As a first approximation, spread ≈ hazard rate × LGD, where LGD = 1 − recovery rate. Spreads imply risk-neutral default probabilities, which are usually higher than real-world (historical) ones because investors demand a risk premium.

Understand Credit Spreads and Risk-Neutral vs Real-World Default Probabilities

A corporate bond pays a higher yield than a risk-free bond of the same maturity. The difference is the credit spread. It pays you for expected default losses and, beyond that, for bearing risk.

Start with expected loss. If a bond defaults with annual probability λ (the hazard rate, or default intensity) and loses a fraction LGD when it does, the expected annual loss is λ × LGD. In a simple model, the spread must cover this loss, so spread ≈ λ × LGD. Recovery rate R and LGD are linked: LGD = 1 − R. This gives the common extraction: λ ≈ spread ÷ (1 − R).

Probabilities taken from market prices are risk-neutral (also called Q-measure or market-implied). They are the probabilities that, when used to discount expected cash flows at the risk-free rate, reproduce observed prices. Real-world (P-measure, physical or historical) probabilities come from observed default frequencies, such as rating agency tables.

The two differ. Risk-neutral default probabilities are usually much higher than historical ones. Reasons: investors are risk averse and demand compensation for default risk that is hard to diversify, since defaults cluster in bad times; bonds are less liquid than government bonds; and there are tax and other effects on yields. Part of the spread is therefore not expected loss. The gap between observed spreads and the spread explained by historical losses is often called the credit spread puzzle.

Use the right measure for the right job. For pricing and valuing derivatives such as CDS, or for CVA, use risk-neutral probabilities. For scenario analysis, capital, expected loss, and risk management, use real-world probabilities. Mixing them is a classic exam error.

Key formulas to remember

Credit spread: s = y(risky) − y(risk-free); Use bonds of the same maturity, currency and coupon structure where possible.
Spread approximation: s ≈ λ × (1 − R) = λ × LGD; λ is the annual risk-neutral hazard rate. Approximation for small values and a flat hazard rate.
Hazard rate from spread: λ ≈ s ÷ (1 − R); Gives the risk-neutral default intensity. Also holds approximately for CDS spreads.
Cumulative default probability: PD(T) = 1 − e^(−λT); Assumes a constant hazard rate. Survival probability is e^(−λT).
Annual PD (approximate): PD ≈ 1 − e^(−λ) ≈ λ for small λ; Hazard rate and annual probability are close only when λ is small.
Risky bond price (one-year zero): P = [(1 − PD) × 100 + PD × R × 100] ÷ (1 + r); Expected payoff discounted at the risk-free rate gives the risk-neutral PD.
Risk-neutral to real-world gap: λ(Q) = λ(P) × multiplier, usually multiplier > 1; Multiplier is a descriptive ratio, not a fixed rule. Not always the same across ratings.

How to solve Credit Spreads and Risk-Neutral vs Real-World Default Probabilities questions

Use this method for any question that links spreads, default probability, recovery and the two probability measures.

1Identify what is asked: a spread, a hazard rate, a PD, or a conceptual comparison of risk-neutral and real-world measures.
2Note the recovery rate R. Compute LGD = 1 − R. If R is not given, check for an assumed value in the question.
3Check whether the rates are continuously compounded or annual, and whether the spread is in basis points. Convert bps to decimals (100 bps = 0.01).
4Apply s ≈ λ × LGD to find λ or s. If the question gives bond prices, use the expected-payoff pricing equation instead.
5Convert hazard rate to cumulative PD with PD(T) = 1 − e^(−λT) if a multi-year probability is needed.
6State which measure you have: spreads imply risk-neutral PD. Do not call it historical PD.
7If comparing with historical PD, expect the risk-neutral value to be higher and name the reasons: risk premium, liquidity, taxes.
8Check the answer for reasonableness: λ should be above s (since LGD < 1) and PD should lie between 0 and 1.

Quickest way: Spread ÷ LGD shortcut

When to use it: Use when a question gives a spread and recovery rate and asks for a hazard rate or approximate PD, and the answer options are well separated.

Convert the spread to a decimal: 240 bps = 0.024.
Divide by (1 − R): with R = 40%, λ = 0.024 ÷ 0.6 = 0.04.
For a one-year PD use λ directly if small, or 1 − e^(−λ) if the options are close.
For T years use 1 − e^(−λT).
For conceptual options, choose the one that says risk-neutral PD exceeds real-world PD and cites risk premium.

Common mistakes in Credit Spreads and Risk-Neutral vs Real-World Default Probabilities

Using the spread as the default probability.
Students forget that the spread compensates for loss given default, not just the chance of default.
Fix: Always divide by LGD: λ ≈ s ÷ (1 − R).
Using R instead of 1 − R in the denominator.
Recovery and loss given default sound alike under time pressure.
Fix: Write LGD = 1 − R first. A higher recovery rate means a higher implied hazard rate for the same spread.
Treating a spread-implied PD as a real-world forecast.
The number looks like a probability, so it is assumed to be physical.
Fix: Label it risk-neutral. Use it for pricing. Use historical or model-based PDs for expected loss and capital.
Assuming the whole spread is expected default loss.
The simple formula hides other components.
Fix: Remember that spreads include a default risk premium and compensation for liquidity and tax effects. This is why historical expected loss explains only part of the spread.
Forgetting to convert basis points.
Spreads are quoted in bps, formulas need decimals.
Fix: Divide bps by 10,000 before any calculation.
Applying cumulative PD formula with annual probability instead of hazard rate, or skipping the exponent for long horizons.
Linear approximation works for one year and is carried over blindly.
Fix: For multi-year horizons, use PD(T) = 1 − e^(−λT).

Worked examples

Example 1

A five-year corporate bond yields 5.20% and the five-year risk-free yield is 3.00% (both continuously compounded). Assume a recovery rate of 40%. Estimate the average annual risk-neutral hazard rate and the five-year cumulative risk-neutral default probability. Options: A) 3.67% and 16.8%; B) 2.20% and 10.4%; C) 5.50% and 24.0%; D) 1.32% and 6.4%.

Show the solution

Spread = 5.20% − 3.00% = 2.20% = 0.022.
LGD = 1 − 0.40 = 0.60.
λ ≈ 0.022 ÷ 0.60 = 0.036667, about 3.67%.
Cumulative PD = 1 − e^(−0.036667 × 5) = 1 − e^(−0.18333).
e^(−0.18333) ≈ 0.8325, so PD ≈ 0.1675, about 16.8%.

Answer: A) Hazard rate about 3.67% a year and five-year risk-neutral PD about 16.8%.

Example 2

A bank sees a BBB-rated issuer with a one-year bond spread of 150 bps and an assumed recovery rate of 40%. Historical data show a one-year default rate of 0.20% for BBB issuers. What is the risk-neutral hazard rate, and about how many times larger is it than the historical default rate? Options: A) 2.50%, about 12.5 times; B) 1.50%, about 7.5 times; C) 0.90%, about 4.5 times; D) 3.75%, about 18.8 times.

Show the solution

Spread = 150 bps = 0.015.
LGD = 1 − 0.40 = 0.60.
Risk-neutral λ ≈ 0.015 ÷ 0.60 = 0.025, or 2.50%.
Ratio = 2.50% ÷ 0.20% = 12.5.
The gap reflects risk premium, liquidity and other non-default components of the spread. The 2.50% is a risk-neutral figure and should be used for pricing, not as a forecast of actual defaults.

Answer: A) Risk-neutral hazard rate is 2.50%, about 12.5 times the historical default rate.

Exam tips

Questions usually give a spread in bps and a recovery rate. Convert and divide by LGD before looking at options.
Expect conceptual items asking which measure to use: risk-neutral for pricing and CVA, real-world for expected loss, stress testing and capital.
Know the reasons risk-neutral PDs exceed real-world PDs: risk premium for clustered defaults, liquidity, and tax effects. Name at least two.
Check if the question wants a hazard rate or a cumulative PD. They are different numbers over multi-year horizons.
The same s ≈ λ × LGD logic applies to CDS spreads, so a CDS question may be this topic in disguise.

Practice questions from Fundamentals of Credit Risk