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FRM Exam Part II · Credit Derivatives

CDS Valuation and Pricing: Premium Leg, Protection Leg and Par Spread

Updated 11 October 2026 · Fact-checked

A CDS is priced so that the present value of the premium leg equals the present value of the protection leg at inception. The par spread is the premium rate that makes this true. A quick estimate is the credit triangle: spread ≈ hazard rate × loss given default (1 − recovery rate).

Understand CDS Valuation and Pricing

A credit default swap (CDS) is insurance on a reference entity's debt. The protection buyer pays a regular premium (the spread) until default or maturity. If default occurs, the protection seller pays the loss: notional × (1 − recovery rate).

To value it, you split it into two legs. The premium leg is the expected present value of the spread payments. Each payment is weighted by the probability the entity survives to that date, and then discounted. The protection leg is the expected present value of the payout, weighted by the probability of default in each period. The par spread is the spread that makes the two legs equal, so the swap has zero value at inception.

Default probabilities come from a hazard rate λ, the instantaneous default intensity. If λ is constant, the survival probability to time t is Q(t) = e^(−λt). The probability of default between 0 and t is 1 − e^(−λt). Hazard rates are usually bootstrapped from quoted CDS spreads across maturities.

For a quick estimate, the credit triangle says spread s ≈ λ × LGD, where LGD = 1 − R. It holds best for a flat hazard rate and a flat curve, ignoring accrued premium and discounting detail. So λ ≈ s ÷ (1 − R). The probabilities you get from CDS spreads are risk-neutral. They include a risk premium, so they are usually higher than real-world (historical) default probabilities. Use risk-neutral probabilities for pricing, and real-world ones for expected loss and capital.

The CDS-bond basis = CDS spread − bond spread (usually the bond's asset-swap or z-spread over the risk-free benchmark). A negative basis means the CDS is cheap relative to the bond. A negative-basis trade buys the bond and buys CDS protection to capture the gap. Basis drivers include funding costs, the cheapest-to-deliver option, counterparty risk, liquidity and bond prices far from par.

Key formulas to remember

Credit triangle
s ≈ λ × (1 − R)
s is the annual CDS spread, λ the constant hazard rate, R the recovery rate. An approximation.
Hazard rate from spread
λ ≈ s ÷ (1 − R)
Gives a risk-neutral intensity. Convert s from basis points to decimals first.
Survival probability (constant hazard rate)
Q(t) = e^(−λt)
Cumulative default probability is 1 − e^(−λt).
Premium leg PV
PV(premium) = s × Σ [Δt × Q(tᵢ) × D(tᵢ)] (plus accrual-on-default term)
Δt is the accrual period, D the discount factor. The sum is often called the risky annuity or RPV01.
Protection leg PV
PV(protection) = (1 − R) × Σ [(Q(tᵢ₋₁) − Q(tᵢ)) × D(tᵢ)]
Q(tᵢ₋₁) − Q(tᵢ) is the probability of default in period i. Multiply by notional for money amounts.
Par spread
s* = PV(protection) ÷ risky annuity
The spread that sets the CDS value to zero at inception.
Value of an existing CDS (to protection buyer)
V ≈ (s_market − s_contract) × risky annuity × notional
Positive if spreads have widened since the contract was struck.
CDS-bond basis
Basis = CDS spread − bond spread
Negative basis: CDS spread below bond spread.

How to solve CDS Valuation and Pricing questions

Use this order for most CDS pricing questions, and check the question for whether it wants an approximation or a full leg calculation.

  1. 1Identify what is asked: par spread, hazard rate, survival probability, a leg's PV, or the mark-to-market of an existing CDS.
  2. 2Convert inputs: spreads from basis points to decimals, and note the recovery rate R and LGD = 1 − R.
  3. 3If the question gives a flat spread and recovery rate and asks for a default intensity, use λ ≈ s ÷ (1 − R).
  4. 4Build survival probabilities Q(t) = e^(−λt) for each payment date, and default probabilities as differences of Q.
  5. 5Compute the premium leg as spread × Σ(Δt × Q × D), and the protection leg as (1 − R) × Σ(default probability × D).
  6. 6Set the legs equal for the par spread, or compare the market spread with the contract spread times the risky annuity for mark-to-market.
  7. 7For basis questions, subtract the bond spread from the CDS spread and read the sign, then name the likely driver.
  8. 8Interpret: say whether probabilities are risk-neutral or real-world, and sanity-check the sign and size of your answer.

Quickest way: Credit triangle shortcut

When to use it: Use it when the question gives a spread and a recovery rate and asks for a hazard rate, default probability or approximate spread, with no discount factors.

  1. Write s as a decimal and LGD = 1 − R.
  2. Get λ = s ÷ LGD (or s = λ × LGD).
  3. Get the survival probability e^(−λt) and the default probability 1 − e^(−λt).
  4. For short horizons, default probability ≈ λt as a quick check.
  5. If options show a risk-neutral versus historical comparison, pick risk-neutral as the higher figure for CDS-implied values.

Common mistakes in CDS Valuation and Pricing

  • Using spread as the hazard rate, ignoring recovery

    Students remember that spread reflects default risk and skip the LGD adjustment.

    Fix: Always divide by (1 − R). With a 40% recovery, λ is s ÷ 0.60, which is larger than s.

  • Mixing basis points and decimals

    Spreads are quoted as 120 bp but formulas need 0.0120.

    Fix: Convert first. Write 120 bp = 0.0120 before any calculation.

  • Treating CDS-implied default probabilities as real-world

    The probabilities look like ordinary default odds.

    Fix: CDS-implied values are risk-neutral and contain a risk premium. Use them for pricing, not as a forecast of actual defaults.

  • Weighting the premium leg by default probability

    Both legs use probabilities, so students swap them.

    Fix: Premium payments are made only if the entity survives, so weight by survival Q. The payout is weighted by default probability in that period.

  • Getting the sign of the basis or its trade wrong

    Students memorise the trade without the definition.

    Fix: Basis = CDS − bond spread. Negative means protection is cheap, so buy the bond and buy protection.

  • Treating the credit triangle as exact

    It is a neat one-line formula.

    Fix: It is an approximation that assumes a flat hazard rate and ignores discounting and accrual details. Use legs for exact answers.

Worked examples

Example 1

A 5-year CDS on a corporate has a spread of 180 bp and the recovery rate is 40%. Using the credit triangle, estimate the constant hazard rate and the probability that the entity survives 5 years.

Show the solution
  1. Spread s = 180 bp = 0.0180. LGD = 1 − 0.40 = 0.60.
  2. λ = 0.0180 ÷ 0.60 = 0.03, or 3% per year.
  3. Survival to 5 years: Q(5) = e^(−0.03 × 5) = e^(−0.15).
  4. e^(−0.15) ≈ 0.8607.

Answer: Hazard rate ≈ 3% per year; 5-year survival probability ≈ 86.1% (risk-neutral), so default probability ≈ 13.9%.

Example 2

A bank bought 5-year protection on USD 10 million notional at a contract spread of 100 bp. The market spread for the same CDS is now 160 bp and the risky annuity (RPV01) is 4.3. What is the approximate mark-to-market value to the protection buyer?

Show the solution
  1. Spread change = 160 − 100 = 60 bp = 0.0060.
  2. Value per unit notional ≈ 0.0060 × 4.3 = 0.0258.
  3. Multiply by notional: 0.0258 × USD 10,000,000 = USD 258,000.
  4. Spreads widened, so protection bought at the lower spread has gained value.

Answer: The position is worth about USD 258,000 to the protection buyer (a gain).

Exam tips

  • Check whether the question wants risk-neutral or real-world probabilities. CDS-implied means risk-neutral.
  • Read the answer options for the sign of the CDS-bond basis before calculating, since many items are interpretation only.
  • Do the credit triangle first. Only build the legs if discount factors and payment dates are given.
  • Watch units: bp versus decimals, and per-year versus per-quarter premium payments.
  • Know the basis drivers: funding costs, cheapest-to-deliver option, counterparty risk and liquidity.

Practice questions from Credit Derivatives

CDS Valuation and Pricing: frequently asked questions

How do I calculate a CDS spread from a hazard rate?

Multiply the hazard rate by loss given default: s ≈ λ × (1 − R). For λ = 2% and R = 40%, s ≈ 0.02 × 0.60 = 1.2%, or 120 bp. This is the credit triangle approximation.

What is the difference between the premium leg and the protection leg?

The premium leg is the present value of spread payments, paid while the entity survives. The protection leg is the present value of the expected payout of (1 − R) × notional on default. The par spread equates them.

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

Risk-neutral probabilities are backed out of market prices, so they include compensation for default risk, liquidity and uncertainty. Real-world probabilities come from historical default experience. The gap is the risk premium.

What does a negative CDS-bond basis mean?

The CDS spread is lower than the bond spread, so protection is cheap relative to the bond. A trader may buy the bond and buy CDS protection to capture the difference, subject to funding and counterparty costs.