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FRM Exam Part II · Credit Value at Risk

CreditRisk+ and Actuarial Models for Credit Portfolio Loss

Updated 11 October 2026 · Fact-checked

CreditRisk+ is an actuarial credit portfolio model that treats the number of defaults as a Poisson process with a random default rate. You group exposures into bands, find the expected defaults in each band, then build the portfolio loss distribution with a recursion. It needs no asset values and ignores rating migration.

Understand CreditRisk+ and Actuarial Models

Start with one loan. It either defaults or it does not. If you have many loans and each has a small default probability, the total number of defaults behaves like a Poisson count. Poisson needs only one input: the mean number of defaults, μ. That is the sum of the individual default probabilities.

CreditRisk+ was developed by Credit Suisse First Boston. It borrows from insurance, where you model claim counts and claim sizes. Here a default is a claim and the loss given default is the claim size. It is a default-mode model: a borrower either defaults or does not. There is no rating migration and no revaluation of the loan. Asset values and asset correlations are not used.

If default rates were fixed, the Poisson model would understate tail losses, because real defaults cluster in bad years. So CreditRisk+ makes the default rate random. Each obligor's default rate depends on one or more background factors (often sectors) that follow a gamma distribution. Obligors are independent given the factors. Correlation therefore comes only from sharing those factors. Mixing Poisson with gamma gives a negative binomial count, which has a fatter tail than Poisson.

To get losses, you round each exposure (net of recovery) into a multiple of a common exposure unit, then group loans into bands by that multiple. For each band you know the loss per default and the expected number of defaults. A probability generating function turns this into the full loss distribution. The Panjer recursion computes it step by step with no simulation. That is why CreditRisk+ is called an analytical model.

Against CreditMetrics: CreditMetrics is a mark-to-market model with rating migration, uses asset-value correlations and usually needs Monte Carlo simulation. CreditRisk+ is default-only, uses sector default-rate volatility and is analytical. CreditRisk+ needs less data, but it cannot capture losses from downgrades or spread changes.

Key formulas to remember

Poisson probability of n defaults
P(n) = e^(-μ) × μ^n ÷ n!
μ is the expected number of defaults. n = 0, 1, 2, ... Mean and variance are both μ.
Expected number of defaults
μ = Σ p_i
Sum of obligor default probabilities. Valid as an approximation when each p_i is small.
Expected loss
EL = Σ p_i × v_i
v_i is exposure net of recovery, i.e. loss given default in currency terms.
Count variance with random default rate
Var(N) = μ + Var(Λ)
Λ is the random mean default count with mean μ. Variance exceeds the mean, so the count is overdispersed. Gamma mixing gives a negative binomial.
Band expected loss in units
ε_j = v_j × μ_j
v_j is the loss per default in exposure units for band j. μ_j is the expected defaults in band j.
Panjer recursion (single factor, fixed rate)
A(0) = e^(-μ), A(n) = (1 ÷ n) × Σ ε_j × A(n − v_j), summed over bands with v_j ≤ n
A(n) is the probability that portfolio loss equals n exposure units. Here μ is the total across bands.

How to solve CreditRisk+ and Actuarial Models questions

Use this order for any CreditRisk+ question, whether it asks for a default count, a loss probability or a model comparison.

  1. 1Identify the model mode. CreditRisk+ means default only, no migration, no asset values, and an analytical solution.
  2. 2Compute μ as the sum of default probabilities, or per band as the number of loans times PD.
  3. 3Decide whether the default rate is fixed or random. Fixed gives Poisson with variance μ. Random gives a variance above μ and a fatter tail.
  4. 4For a count question, apply P(n) = e^(-μ) μ^n ÷ n!. Use complements for 'at least' or 'more than'.
  5. 5For a loss question, net out recovery to get loss per default. Round to exposure units and group into bands.
  6. 6Compute each band's ε_j = v_j × μ_j. Start the recursion at A(0) = e^(-μ), then compute A(1), A(2), and so on.
  7. 7Add the A(n) values for cumulative probabilities. Remember expected loss = Σ ε_j units.
  8. 8Interpret the result. Say what it means for tail loss and capital, and name the model's limits, such as no migration.

Quickest way: Poisson shortcut and band table

When to use it: Use when the question gives a small set of bands or a single μ and asks for a probability or an expected loss.

  1. Write μ first. Compute e^(-μ) once and reuse it, since A(0) and P(0) are both e^(-μ).
  2. For counts, build P(1) = P(0) × μ, P(2) = P(1) × μ ÷ 2, and so on. Each term is the previous one times μ ÷ n.
  3. For 'at least one default', use 1 − e^(-μ).
  4. For losses, list bands as (v_j, μ_j) and compute ε_j. Use the recursion only up to the loss level asked for.
  5. Cross-check: the expected loss must equal Σ v_j × μ_j.
  6. If an option says the variance equals the mean, check whether the default rate is random. If it is, variance exceeds the mean.

Common mistakes in CreditRisk+ and Actuarial Models

  • Saying CreditRisk+ models rating migration or uses mark-to-market values.

    It is mixed up with CreditMetrics, which is also a portfolio credit model.

    Fix: Remember that CreditRisk+ is default-only. Losses arise only when a borrower defaults. CreditMetrics is the migration and revaluation model.

  • Claiming that CreditRisk+ uses asset correlations.

    Structural models such as Merton and CreditMetrics use asset values, and students assume all portfolio models do.

    Fix: In CreditRisk+, correlation comes from shared background factors that drive default rates. Obligors are independent given those factors.

  • Using a fixed default rate and stopping at Poisson.

    The Poisson formula is simple, so students apply it to every question.

    Fix: Check whether the question gives default-rate volatility. If it does, the variance is above the mean and the tail is fatter than plain Poisson.

  • Using the gross exposure instead of the loss given default in the bands.

    Students forget that the band loss is exposure net of recovery.

    Fix: Multiply exposure by (1 − recovery rate) first, then round into exposure units.

  • Treating μ as the probability of a default instead of the expected number of defaults.

    Both are described as 'default' numbers and look similar.

    Fix: μ can exceed 1. It is the portfolio's average default count. P(n) is the probability of exactly n defaults.

  • Calling the recursion a simulation.

    Loss distributions are usually built by Monte Carlo in other models.

    Fix: The Panjer recursion is exact arithmetic on the band inputs. No random draws are needed, which is a key advantage of the model.

Worked examples

Example 1

A loan portfolio has 200 independent borrowers, each with a one-year default probability of 1.5%. Using the Poisson approximation, find the probability of more than 2 defaults. Use e^(-3) = 0.049787.

Show the solution
  1. μ = 200 × 0.015 = 3 expected defaults.
  2. P(0) = e^(-3) = 0.049787.
  3. P(1) = P(0) × 3 = 0.149361.
  4. P(2) = P(1) × 3 ÷ 2 = 0.224042.
  5. P(at most 2) = 0.049787 + 0.149361 + 0.224042 = 0.423190.
  6. P(more than 2) = 1 − 0.423190 = 0.576810.

Answer: About 57.7%. The mean is 3 defaults, so more than 2 is more likely than not.

Example 2

A portfolio has two exposure bands, with a unit of $1 million of loss given default. Band 1 has loss 1 unit per default and 2 expected defaults. Band 2 has loss 2 units per default and 1 expected default. Assume a fixed Poisson default rate. Find the probability that portfolio loss is at most $2 million, and the expected loss. Use e^(-3) = 0.049787.

Show the solution
  1. Total μ = 2 + 1 = 3, so A(0) = e^(-3) = 0.049787.
  2. ε1 = 1 × 2 = 2 units. ε2 = 2 × 1 = 2 units.
  3. A(1) = (1 ÷ 1) × ε1 × A(0) = 2 × 0.049787 = 0.099574.
  4. A(2) = (1 ÷ 2) × [ε1 × A(1) + ε2 × A(0)] = 0.5 × (0.199148 + 0.099574) = 0.149361.
  5. Check A(2) directly: two band 1 defaults and no band 2 default, or no band 1 default and one band 2 default. That gives e^(-3) × (2 + 1) = 0.149361. It matches.
  6. P(loss ≤ 2 units) = 0.049787 + 0.099574 + 0.149361 = 0.298722.
  7. Expected loss = ε1 + ε2 = 4 units = $4 million.

Answer: P(loss ≤ $2 million) is about 29.9%. Expected loss is $4 million.

Exam tips

  • Questions often test the contrast with CreditMetrics. Memorise: CreditRisk+ is default-only, actuarial and analytical. CreditMetrics is mark-to-market, with migration and asset-value correlation.
  • If an option says CreditRisk+ needs asset values or equity prices, reject it.
  • For overdispersion questions, the answer is that random default rates make variance exceed the mean, giving a fatter tail than Poisson.
  • Do the arithmetic with a few decimals. Options are usually well apart, but e^(-μ) errors carry through the whole recursion.
  • When asked about limitations, name these: no migration or spread risk, exposures treated as fixed, and results that depend on the Poisson assumption of small default probabilities.

Practice questions from Credit Value at Risk

CreditRisk+ and Actuarial Models: frequently asked questions

Why does CreditRisk+ use the Poisson distribution?

For many borrowers with small default probabilities, the count of defaults is well approximated by a Poisson distribution. It needs only one parameter, the mean number of defaults. This keeps the model simple and analytical.

How does CreditRisk+ calculate the loss distribution?

It rounds net exposures into units and groups loans into bands. It finds the expected number of defaults in each band. Then the Panjer recursion computes the probability of each loss level without simulation.

What is the difference between CreditRisk+ and CreditMetrics?

CreditMetrics values the portfolio at a horizon, including rating migration, and uses asset-value correlations, often with simulation. CreditRisk+ considers only default, drives correlation through background factors, and solves analytically.

How are defaults correlated in CreditRisk+?

Default rates depend on common background factors, often sectors, modelled with gamma distributions. Given the factors, obligors default independently. Shared factors raise the chance of many defaults at once, which fattens the tail.