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FRM Exam Part I · Measuring Credit Risk

Reduced-Form Models and Credit Spreads for FRM Part I

Updated 11 October 2026 · Fact-checked

A reduced-form model treats default as a random event that arrives at a hazard rate (intensity), without modelling the firm's assets. The credit spread is approximately hazard rate × LGD. Solve by finding the spread over the risk-free rate, dividing by LGD, then using survival probability = e^(−λt).

Understand Reduced-Form Models and Credit Spreads

A reduced-form model does not ask why a firm defaults. It simply says default is a surprise event that can happen at any moment, with some probability per unit of time. That probability rate is the hazard rate (or default intensity), written λ. This differs from a structural model such as Merton, which links default to the firm's asset value falling below its debt.

If λ is constant, the time to default follows an exponential distribution. The probability of surviving to time t is e^(−λt). The probability of default by time t is 1 − e^(−λt). For small λt this is close to λt. The hazard rate is a conditional measure: it is the default rate in the next instant, given survival so far. The unconditional default probability in a given year is different, because the firm must first survive to reach that year.

A risky bond pays less than a risk-free bond because investors face default loss. The extra yield is the credit spread. A useful approximation is spread ≈ λ × LGD, where LGD = 1 − recovery rate. Equivalently, λ ≈ spread ÷ (1 − R). This is the average annual loss rate that the spread must compensate for.

The default probability you extract from spreads is risk-neutral (Q-measure). It is usually higher than the real-world (P-measure) probability estimated from historical default data such as rating agency studies. The gap exists because spreads also pay for risk premia, such as bearing default risk that rises in bad times, plus liquidity and tax effects. So never use a spread-implied PD as a forecast of actual defaults. Use it for pricing and valuation, and use historical PDs for expected loss and risk measurement.

Key formulas to remember

Survival probability (constant hazard)
Q(t) = e^(−λt)
Probability of no default up to time t. λ is the hazard rate per year.
Cumulative default probability
PD(t) = 1 − e^(−λt)
Close to λt when λt is small.
Credit spread approximation
s ≈ λ × (1 − R) = λ × LGD
s is the spread over the risk-free rate. R is the recovery rate. Approximation, best for short maturities and small spreads.
Hazard rate from spread
λ ≈ s ÷ (1 − R)
Gives the risk-neutral hazard rate when the spread is used.
Average hazard rate from cumulative PD
λ̄ = −ln(1 − PD(t)) ÷ t
Use when you are given cumulative default probability.
Unconditional default probability in year t
Q(t−1) − Q(t) = e^(−λ(t−1)) − e^(−λt)
Default occurs in year t only if the firm survived to its start.
Risky bond price (zero-coupon)
P = e^(−(r + s)T) where s = λ × LGD
Uses continuous compounding. r is the risk-free rate.

How to solve Reduced-Form Models and Credit Spreads questions

Use this sequence for most questions on hazard rates, spreads and risk-neutral default probabilities.

  1. 1Identify what is given: spread, risk-free rate, recovery rate, hazard rate, or cumulative PD, and the time horizon.
  2. 2Check the measure. Spread-implied numbers are risk-neutral. Historical or rating-based numbers are real-world.
  3. 3Convert the spread into a hazard rate using λ = s ÷ (1 − R). Make sure the spread is in decimals (100 bp = 0.01).
  4. 4Decide what is asked: survival, cumulative PD, or default in a specific year.
  5. 5Apply Q(t) = e^(−λt) and PD(t) = 1 − Q(t). For a specific year, take the difference of survival probabilities.
  6. 6If asked to price a bond, discount at r + s or compare with a risk-free bond to find the loss.
  7. 7Sanity check: PD must lie between 0 and 1, and longer horizons give higher cumulative PD.

Quickest way: Spread ÷ LGD shortcut

When to use it: Use when the question asks for a hazard rate or one-year PD from a spread and a recovery rate, and options are well separated.

  1. Compute LGD = 1 − R.
  2. Compute λ = spread ÷ LGD.
  3. For one year, PD ≈ λ (more exactly 1 − e^(−λ)).
  4. For longer horizons, use 1 − e^(−λt) on your calculator with the e^x key.
  5. Eliminate options with PD above 1 or with the wrong direction for the horizon.

Common mistakes in Reduced-Form Models and Credit Spreads

  • Treating spread-implied PD as the real-world probability of default

    Both are called default probabilities, so the measure is overlooked.

    Fix: Remember spreads include risk premia and liquidity effects, so risk-neutral PD is generally higher than historical PD.

  • Using λ = spread without dividing by LGD

    Students forget that only the loss portion is compensated by the spread.

    Fix: Always use λ = s ÷ (1 − R). With a 40% recovery, λ is spread ÷ 0.6.

  • Confusing hazard rate with unconditional annual PD

    Both are per-year numbers.

    Fix: The hazard rate is conditional on survival. Year-t default probability is e^(−λ(t−1)) − e^(−λt).

  • Forgetting to convert basis points to decimals

    Spreads are quoted in bp under time pressure.

    Fix: Write 150 bp as 0.0150 before any calculation.

  • Mixing up reduced-form and structural models

    Both give default probabilities.

    Fix: Structural models (Merton) tie default to asset value versus debt. Reduced-form models use an exogenous hazard rate and need no balance sheet.

  • Adding probabilities across years instead of using survival

    Treating yearly PDs as independent additions.

    Fix: Use cumulative PD = 1 − survival probability.

Worked examples

Example 1

A bond yields 250 bp over the risk-free rate. The recovery rate is 40%. Estimate the risk-neutral hazard rate and the probability of default within 5 years, assuming a constant hazard rate.

Show the solution
  1. LGD = 1 − 0.40 = 0.60.
  2. λ = 0.0250 ÷ 0.60 = 0.041667.
  3. λt = 0.041667 × 5 = 0.208333.
  4. Survival Q(5) = e^(−0.208333) ≈ 0.8119.
  5. PD(5) = 1 − 0.8119 = 0.1881.

Answer: Hazard rate ≈ 4.17% per year; 5-year risk-neutral default probability ≈ 18.8%.

Example 2

A firm has a constant hazard rate of 3% per year. What is the probability that it defaults during year 3 (between t = 2 and t = 3)?

Show the solution
  1. Survival to the start of year 3: Q(2) = e^(−0.06) ≈ 0.94176.
  2. Survival to the end of year 3: Q(3) = e^(−0.09) ≈ 0.91393.
  3. Default in year 3 = Q(2) − Q(3) = 0.94176 − 0.91393 = 0.02783.
  4. Check: this equals Q(2) × (1 − e^(−0.03)) = 0.94176 × 0.02955 ≈ 0.02783.

Answer: About 2.78%, which is below the 3% hazard rate because the firm must first survive two years.

Exam tips

  • Check the first word of the question: risk-neutral means spread-implied; real-world or historical means rating or default data.
  • Expect to be asked why risk-neutral PD exceeds real-world PD. Answer with risk premia, liquidity and tax effects.
  • Know the structural versus reduced-form contrast: asset-based and endogenous default versus intensity-based and exogenous default.
  • Practise e^x and ln on your calculator so each exponential step takes seconds.
  • For year-specific default questions, always go through survival probabilities.

Practice questions from Measuring Credit Risk

Reduced-Form Models 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.

Reduced-Form Models and Credit Spreads: frequently asked questions

What is the difference between reduced-form and structural credit models?

Structural models, such as Merton, explain default through the firm's asset value falling below its debt. Reduced-form models treat default as an unexpected event with a hazard rate and do not model the balance sheet. Reduced-form models are easier to calibrate to market spreads.

How do you get default probability from a credit spread?

Divide the spread by LGD to get the hazard rate, λ ≈ s ÷ (1 − R). Then PD over t years is 1 − e^(−λt). The result is a risk-neutral probability.

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

Spreads compensate investors not only for expected default loss but also for bearing default risk, which is worse in bad times, and for liquidity and tax effects. This pushes implied PDs above historical default rates.

Which PD should I use for expected loss and which for pricing?

Use real-world PDs when estimating expected and actual losses and for risk measurement. Use risk-neutral PDs when valuing credit instruments, such as bonds or credit derivatives, from market prices.