FRM Exam Part II · Estimating Default Probabilities

Default Probabilities from CDS Spreads and Asset Swaps

Updated 11 October 2026 · Fact-checked

A CDS spread is the annual price of default protection. Under the credit triangle, spread ≈ hazard rate × (1 − recovery rate), so hazard rate = spread ÷ (1 − R). Survival to time t is e^(−λt). With several maturities, bootstrap the hazard rates. The CDS-bond basis compares CDS spread with bond spread.

Understand Default Probabilities from CDS Spreads and Asset Swaps

A credit default swap (CDS) is insurance against default of a reference entity. The buyer pays a spread each year, quoted in basis points. If the entity defaults, the seller pays the loss: face value minus the recovery value of the bond. So the spread must compensate the seller for the chance of default and the size of the loss.

That gives the credit triangle: spread ≈ hazard rate × loss given default. The hazard rate λ is the instantaneous default intensity, and loss given default is (1 − R), where R is the recovery rate. Rearranged, λ = s ÷ (1 − R). If the hazard rate is constant, the probability of surviving to time t is e^(−λt), and the cumulative default probability is 1 − e^(−λt).

One spread gives one average hazard rate. Real curves slope, so you use CDS spreads at 1, 3, 5, 7 and 10 years. Bootstrapping finds the hazard rate for the first period from the shortest CDS, then the next period so that the next CDS has zero value at inception (premium leg PV = protection leg PV). In the exam you will mostly use the average-hazard shortcut.

These probabilities are risk-neutral, not real-world. The spread also pays for risk premium, liquidity and counterparty effects. So implied default probabilities are usually higher than historical default rates. Use them for pricing and relative value, not as forecasts of actual defaults.

The CDS-bond basis is CDS spread minus the bond's spread over the risk-free rate. The asset swap spread is the usual way to measure the bond spread: you swap the bond's fixed coupons into floating, and the spread over the floating benchmark is the asset swap spread. In theory the basis is near zero, because a bond plus CDS protection should be roughly risk-free. In practice it is rarely zero. A negative basis (CDS cheaper than bond spread) often reflects bond illiquidity or funding stress. A positive basis can reflect bond scarcity, funding advantages of owning bonds, or the cheapest-to-deliver option in the CDS.

Key formulas to remember

Credit triangle: s ≈ λ × (1 − R); s is the annual CDS spread in decimal (120 bp = 0.012). It is an approximation that assumes a constant hazard rate.
Hazard rate from spread: λ = s ÷ (1 − R); Gives the average annual default intensity. Lower assumed recovery gives a lower implied hazard rate. A 40% recovery is a common standard assumption.
Survival probability: Q(t) = e^(−λt); Constant hazard rate. Probability of no default up to time t.
Cumulative default probability: PD(t) = 1 − e^(−λt); For small λt this is close to λt, but use the exponential when asked for precision.
Forward (period) hazard rate: λ(T1,T2) = (λ̄2 × T2 − λ̄1 × T1) ÷ (T2 − T1); λ̄ is the average hazard rate to each maturity. This is the quick bootstrap between two maturities.
Bootstrap condition: PV of premium payments = PV of expected protection payout; A par CDS has zero value at start. Premium leg includes the accrual paid on default.
CDS-bond basis: Basis = CDS spread − bond spread; Bond spread = bond yield minus the risk-free (or swap) rate, or the asset swap spread. Positive means CDS is the more expensive measure of credit risk.
Par asset swap spread: Asset swap spread ≈ bond yield − swap rate (par bond); Approximation for a bond at par. Non-par bonds need an adjustment for the premium or discount.

How to solve Default Probabilities from CDS Spreads and Asset Swaps questions

Use this order for any question on default probabilities from CDS spreads or the basis.

1Read the data and convert units. Turn basis points into decimals (150 bp = 0.015). Note the recovery rate and the horizon.
2Decide what is asked: a hazard rate, a cumulative or survival probability, a period (forward) probability, or the basis.
3For a single spread, apply λ = s ÷ (1 − R). Make sure you divide by (1 − R), not R.
4Convert the hazard rate to a probability: survival = e^(−λt), default = 1 − e^(−λt). Use the correct t.
5For several maturities, compute the average hazard λ̄ at each maturity. Then find period hazards using (λ̄2·T2 − λ̄1·T1) ÷ (T2 − T1).
6For basis questions, compute the bond spread over the risk-free or swap rate (or take the asset swap spread), then subtract from the CDS spread.
7Interpret the sign: positive basis, negative basis, and likely drivers such as liquidity, funding, delivery option or counterparty risk.
8State that the probabilities are risk-neutral and so overstate real-world default probabilities.

Quickest way: Credit triangle shortcut with exponential survival

When to use it: Use when the question gives a spread and a recovery rate and asks for default probability, or gives two maturities and asks for the probability in a later period.

Compute λ = s ÷ (1 − R) in decimal.
Multiply by t and take 1 − e^(−λt). For λt below about 0.05, λt alone is a fair check.
For a later period, find cumulative hazard at each maturity (λ̄ × T), subtract, and divide by the period length for the forward hazard.
For the period default probability from T1 to T2, take the survival to T1 times (1 − e^(−λ_forward × (T2 − T1))), or the difference in cumulative default probabilities.
For the basis, subtract bond spread from CDS spread. Check the sign before choosing an answer.

Common mistakes in Default Probabilities from CDS Spreads and Asset Swaps

Dividing the spread by R instead of (1 − R).
Students remember that recovery matters and plug in the wrong number.
Fix: Loss given default is (1 − R). With R = 40%, divide by 0.6, not 0.4.
Forgetting to convert basis points to decimals.
Spreads are quoted as 120 or 250, which feels like a plain number.
Fix: Divide by 10,000 first. 120 bp = 0.012. A result above 1 for a hazard rate is a warning sign.
Treating the hazard rate as the cumulative default probability.
The hazard rate looks like a probability, and the exam gives a multi-year horizon.
Fix: The hazard rate is annual intensity. Use 1 − e^(−λt) for the probability over t years.
Treating CDS-implied probabilities as real-world probabilities.
The formula looks like a statistical estimate of default.
Fix: CDS-implied probabilities are risk-neutral. They include risk premia and liquidity effects, so they are usually higher than historical default frequencies.
Getting the sign or definition of the basis backwards.
Students mix up which spread is subtracted.
Fix: Basis = CDS spread − bond spread. Negative basis means protection is cheap relative to the bond spread.
Using the average hazard rate for a specific later year.
The 5-year spread gives the average to year 5, and students apply it to year 3 alone.
Fix: When the curve slopes, compute the forward hazard between maturities with the bootstrap formula.

Worked examples

Example 1

A 5-year CDS on a corporate issuer trades at 120 bp. Assume a 40% recovery rate and a constant hazard rate. Find the hazard rate, the 1-year default probability and the 5-year cumulative default probability.

Show the solution

Convert the spread: s = 120 bp = 0.012.
Loss given default = 1 − 0.40 = 0.60.
Hazard rate λ = 0.012 ÷ 0.60 = 0.02, or 2% a year.
1-year default probability = 1 − e^(−0.02) = 1 − 0.980199 = 0.019801, about 1.98%.
5-year cumulative default probability = 1 − e^(−0.02 × 5) = 1 − e^(−0.10) = 1 − 0.904837 = 0.095163.
Note: these are risk-neutral probabilities.

Answer: λ = 2% a year; 1-year default probability ≈ 1.98%; 5-year cumulative default probability ≈ 9.52% (risk-neutral).

Example 2

A 5-year bond issued by the same company yields 4.60%. The 5-year swap rate is 3.30%. The 5-year CDS spread is 150 bp. Calculate the CDS-bond basis and interpret it.

Show the solution

Bond spread over the swap rate = 4.60% − 3.30% = 1.30% = 130 bp.
Basis = CDS spread − bond spread = 150 bp − 130 bp = +20 bp.
The sign is positive, so the CDS market prices more credit risk than the bond spread implies.
Possible reasons: the bond is in high demand or hard to borrow, funding advantages of holding the bond, the cheapest-to-deliver option inside the CDS, or counterparty risk priced into protection.
A trader who believes the basis will narrow could sell protection and short the bond, subject to funding and shorting costs.

Answer: The basis is +20 bp. The CDS is expensive relative to the bond spread.

Exam tips

Write down R and the unit of the spread before any calculation. Many wrong answers come from a missed conversion.
If two answer choices differ only by using R instead of (1 − R), the exam is testing that exact point. Check the denominator.
For basis questions, expect an interpretation: name the cause (liquidity, funding, delivery option, counterparty risk) and the direction of the trade.
Remember that CDS-implied probabilities are risk-neutral. If a question compares them with historical default rates, expect the implied figure to be higher.
If a curve is upward sloping, the forward hazard in later years is higher than the average hazard to that maturity.

Practice questions from Estimating Default Probabilities

Default Probabilities from CDS Spreads and Asset Swaps 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 CDS Spreads and Asset Swaps: frequently asked questions

What is the credit triangle formula?

The credit triangle says spread ≈ hazard rate × (1 − recovery rate). It links the CDS spread, the default intensity and the expected loss if default happens. You can solve for any one if you know the other two.

How do I convert a CDS spread into a default probability?

Divide the spread (in decimal) by (1 − R) to get the hazard rate λ. Then the default probability over t years is 1 − e^(−λt). For example, 120 bp with 40% recovery gives λ = 2% and a 5-year default probability of about 9.52%.

How do I bootstrap default probabilities from CDS spreads?

Start with the shortest CDS and find the hazard rate that sets the premium leg equal to the protection leg. Then move to the next maturity and solve for the next period's hazard rate, keeping the earlier ones fixed. In exam shortcuts, compute average hazards and use the forward formula between maturities.

What does a negative CDS-bond basis mean?

The CDS spread is lower than the bond spread over the risk-free rate. This often reflects bond illiquidity or funding stress. A trader may buy the bond and buy CDS protection to capture the difference, subject to funding costs and counterparty risk.