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FRM Exam Part II · Estimating Default Probabilities

Credit Ratings and Historical Default Rates for FRM Part II

Updated 11 October 2026 · Fact-checked

Agencies map borrowers to rating classes, and historical data show default rates rising as ratings fall. The cumulative default probability is the chance of default by year t. The marginal rate is the extra default in one year. The hazard rate is default in a year given survival to its start.

Understand Credit Ratings and Historical Default Rates

A credit rating is an agency's opinion on a borrower's creditworthiness. S&P and Fitch use AAA down to D. Moody's uses Aaa down to C. Ratings of BBB- / Baa3 and above are investment grade. Below that is speculative grade (high yield).

Agencies publish historical default rates for each rating class. The main table shows the cumulative default probability: the share of issuers in a class that defaulted by the end of year 1, 2, 3 and so on. Cumulative rates never fall as the horizon grows, because a default cannot be undone.

The usual pattern: default rates rise sharply as ratings fall. For investment grade, cumulative default rates rise faster than linearly with time, because good credits tend to be downgraded before they default. For poor ratings the cumulative curve is often concave. Weak issuers that survive the early years are the stronger ones, so yearly default rates fall. Also, ratings are not perfect: rates vary with the economic cycle, and defaults cluster in recessions.

Three related measures matter. The cumulative default probability Q(t) is the unconditional chance of default by t. The marginal (unconditional) default probability for year t is the chance of defaulting in that year as seen today: Q(t) − Q(t−1). The hazard rate (conditional default probability) is the chance of defaulting in year t given survival to the start of that year. Do not mix them up; exams test exactly this difference.

Historical rates are real-world probabilities. Default probabilities implied by bond yields or CDS spreads are risk-neutral and are usually higher, because they include risk premia and liquidity effects.

Key formulas to remember

Marginal (unconditional) default probability
Marginal(t) = Q(t) − Q(t−1)
Q(t) is the cumulative default probability by the end of year t. Q(0) = 0.
Survival probability
S(t) = 1 − Q(t)
Probability of no default through year t.
Hazard rate (annual conditional default probability)
h(t) = [Q(t) − Q(t−1)] ÷ [1 − Q(t−1)]
Default in year t given survival to the start of year t.
Cumulative from annual hazards
1 − Q(t) = Π (1 − h(i)) for i = 1 to t
Survival is the product of yearly conditional survival probabilities.
Constant continuous hazard rate λ
Q(t) = 1 − e^(−λt); λ = −ln(1 − Q(t)) ÷ t
Use when the question says a constant hazard (default intensity). Approximately λ ≈ spread ÷ (1 − recovery rate).

How to solve Credit Ratings and Historical Default Rates questions

Most questions give a cumulative table or a hazard rate and ask for another measure. Fix the definition first, then calculate.

  1. 1Identify what is given: cumulative Q(t), marginal, or hazard rate, and whether time is discrete (annual) or continuous.
  2. 2Identify what is asked, and write its definition in one line.
  3. 3If given cumulative rates, take differences for the marginal rate: Q(t) − Q(t−1).
  4. 4For the hazard rate, divide that marginal rate by the survival probability at the start of the year, 1 − Q(t−1).
  5. 5If given hazards, build survival by multiplying (1 − h) across years, then Q = 1 − survival.
  6. 6For constant continuous hazard, use Q(t) = 1 − e^(−λt) and solve with logarithms.
  7. 7Check sense: cumulative rises, hazard ≥ marginal, and probabilities lie between 0 and 1.
  8. 8Interpret the result: real-world, rating-class average, and sensitive to the cycle.

Quickest way: Difference, then divide by survival

When to use it: Any question that moves between cumulative, marginal and hazard rates in discrete years.

  1. Marginal = this year's cumulative minus last year's cumulative.
  2. Hazard = marginal ÷ (1 − last year's cumulative).
  3. Hazard is always at least the marginal, and the two are equal only in year 1.
  4. Cumulative from hazards: multiply the survival terms, then subtract from 1.
  5. Eliminate options where hazard is below marginal or cumulative decreases.

Common mistakes in Credit Ratings and Historical Default Rates

  • Treating the cumulative rate for year 3 as the probability of defaulting in year 3.

    The table column looks like a yearly rate.

    Fix: Cumulative is default by year 3. For year 3 alone, take Q(3) − Q(2).

  • Computing the hazard rate as marginal ÷ 1, ignoring survival.

    Students stop after the subtraction.

    Fix: Divide by 1 − Q(t−1). Hazard is conditional on survival.

  • Adding annual hazard rates to get the cumulative probability.

    It is a quick shortcut that looks plausible.

    Fix: Multiply survival terms (1 − h). Adding overstates the cumulative probability.

  • Dividing by 1 − Q(t) instead of 1 − Q(t−1) for the hazard.

    Confusing the end of the year with the start.

    Fix: The condition is survival to the start of the year, so use the previous year's cumulative.

  • Assuming default rates rise with maturity for every rating.

    Cumulative rates rise, so students assume yearly rates do too.

    Fix: For investment grade, hazard tends to rise with time. For speculative grade, it often falls as weak issuers drop out.

  • Treating historical default rates as risk-neutral probabilities for pricing.

    Both are called default probabilities.

    Fix: Historical rates are real-world. Spread-implied ones are risk-neutral and typically higher.

Worked examples

Example 1

A B-rated class has cumulative default probabilities of 4.0% after year 1, 9.0% after year 2 and 14.0% after year 3. What is the hazard rate (conditional default probability) in year 3?

Show the solution
  1. Marginal in year 3 = 14.0% − 9.0% = 5.0%.
  2. Survival to the start of year 3 = 1 − 0.09 = 0.91.
  3. Hazard = 0.05 ÷ 0.91 = 0.05495.

Answer: About 5.5%. The unconditional (marginal) probability is 5.0%, and the hazard rate is higher because it is conditional on survival.

Example 2

A bond has a constant annual hazard rate of 2% (discrete, annual). What is the probability it defaults within 3 years?

Show the solution
  1. Annual survival = 1 − 0.02 = 0.98.
  2. Three-year survival = 0.98 × 0.98 × 0.98 = 0.941192.
  3. Cumulative default = 1 − 0.941192 = 0.058808.

Answer: About 5.88%. Adding 3 × 2% = 6% would overstate it.

Exam tips

  • Read whether the question says unconditional, marginal or conditional. These words decide the formula.
  • Annual versus continuous hazard: if the question gives λ, use e^(−λt). If it gives yearly percentages, multiply survival terms.
  • Expect interpretation items: speculative grade hazards often decline with time, investment grade hazards often rise.
  • Remember that historical rates are real-world, and spread-implied rates are risk-neutral and higher.
  • Do the subtraction first, then the division. Writing both steps stops slips.

Practice questions from Estimating Default Probabilities

Credit Ratings and Historical Default Rates in other exams

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

Credit Ratings and Historical Default Rates: frequently asked questions

How do I calculate marginal default probability from cumulative?

Subtract the previous year's cumulative probability from the current year's. For year 4, it is Q(4) − Q(3). This is the unconditional chance of defaulting in that single year, seen from today.

What is the difference between hazard rate and unconditional default probability?

The unconditional probability is measured from today and ignores whether the firm survives. The hazard rate is conditional on surviving to the start of the year. It equals the unconditional probability divided by the survival probability at the start of that year.

Do default rates always rise with lower ratings?

Broadly yes, and historical tables show much higher default rates for speculative grade than investment grade. Individual years can be noisy, and rates swing with the economic cycle.

Are historical default rates the same as default probabilities from CDS spreads?

No. Historical rates are real-world estimates. CDS or bond spreads give risk-neutral probabilities, which are usually higher because they include risk premia and liquidity effects.