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FRM Exam Part II · Credit Value at Risk

Single-Factor Model and Vasicek Credit VaR Explained

Updated 11 October 2026 · Fact-checked

The single-factor model links each borrower's default to one common market factor plus its own shock. The Vasicek formula gives the default rate conditional on a bad factor outcome: WCDR = N[(N⁻¹(PD) + √ρ × N⁻¹(0.999)) ÷ √(1 − ρ)]. Basel IRB uses it to set capital.

Understand Single-Factor Model and Vasicek Credit VaR

Start with one borrower. In the Merton view, it defaults when a latent asset value falls below a threshold. If the unconditional default probability is PD, the threshold is N⁻¹(PD), where N is the standard normal CDF.

Now take many borrowers. Each one's latent variable is Xᵢ = √ρ × M + √(1 − ρ) × Zᵢ. M is the common factor (the state of the economy). Zᵢ is the idiosyncratic shock. Both are standard normal and independent. ρ is the asset correlation between any two borrowers. This setup is the Gaussian copula single-factor model: it ties all defaults together through one normal factor.

Fix the factor at a value M. Defaults are then independent, and the conditional default probability is PD(M) = N[(N⁻¹(PD) − √ρ × M) ÷ √(1 − ρ)]. A low M (bad economy) raises PD(M).

In a large homogeneous portfolio, idiosyncratic risk diversifies away. The default rate then equals PD(M). The worst case default rate at confidence X is found by setting M = N⁻¹(1 − X), the bad tail. This gives WCDR(X) = N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1 − ρ)]. Credit VaR is the loss from this rate: WCDR × EAD × LGD.

Basel IRB builds on this. Capital for unexpected loss is K = LGD × [WCDR(99.9%) − PD], then scaled by a maturity adjustment and a scaling factor in the formulas. Expected loss (PD × LGD) is covered by provisions, so it is subtracted. Higher ρ means fatter tail risk and more capital.

Key formulas to remember

Latent variable
Xᵢ = √ρ × M + √(1 − ρ) × Zᵢ
M, Zᵢ independent standard normals. Default if Xᵢ < N⁻¹(PD).
Conditional default probability
PD(M) = N[(N⁻¹(PD) − √ρ × M) ÷ √(1 − ρ)]
Valid for a given factor value M. Falls as M rises.
Worst case default rate (Vasicek)
WCDR(X) = N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1 − ρ)]
Large homogeneous portfolio. X is the confidence level, e.g. 99.9%.
Credit VaR
Credit VaR = WCDR × LGD × EAD
Total loss at the confidence level. Unexpected-loss capital subtracts PD × LGD × EAD.
Unexpected loss capital (Basel IRB core)
K = LGD × [WCDR(99.9%) − PD]
Before the maturity adjustment and other scaling in the Basel formula.

How to solve Single-Factor Model and Vasicek Credit VaR questions

Use this order for any Vasicek or IRB question.

  1. 1Write down PD, ρ, confidence level X, LGD and EAD from the question.
  2. 2Compute N⁻¹(PD) and N⁻¹(X) from the standard normal table.
  3. 3Compute √ρ and √(1 − ρ).
  4. 4Plug into the WCDR formula: numerator = N⁻¹(PD) + √ρ × N⁻¹(X), then divide by √(1 − ρ).
  5. 5Convert the result to a probability with N(·).
  6. 6Multiply WCDR by LGD and EAD for Credit VaR.
  7. 7If asked for capital or unexpected loss, subtract PD × LGD × EAD (or use K = LGD × (WCDR − PD)).
  8. 8Sanity check: WCDR must exceed PD, and rise with ρ and X.

Quickest way: Plug-in shortcut for WCDR

When to use it: Use when PD, ρ and the confidence level are given and the options are numerically spread.

  1. Memorise N⁻¹(0.99) = 2.326, N⁻¹(0.999) = 3.090.
  2. Compute the numerator and divide by √(1 − ρ) to get one z-score.
  3. Convert z to a probability using known anchors: N(−1.645) ≈ 5%, N(−2.326) ≈ 1%, N(0) = 50%.
  4. Eliminate options below PD or above 100% first, then pick the closest.

Common mistakes in Single-Factor Model and Vasicek Credit VaR

  • Using a minus sign on N⁻¹(X) in the WCDR formula.

    Students mix it up with the conditional PD formula, which uses −√ρ × M.

    Fix: In WCDR, M = N⁻¹(1 − X) = −N⁻¹(X), so the sign becomes plus. WCDR must exceed PD.

  • Forgetting to divide by √(1 − ρ).

    The numerator looks complete after adding the two terms.

    Fix: Always finish with the division, then apply N(·).

  • Treating Credit VaR as WCDR × EAD only.

    LGD is dropped when the question gives it late.

    Fix: Loss = WCDR × LGD × EAD. Check every input is used.

  • Reporting capital as the full loss rather than unexpected loss.

    Expected loss is covered by provisions, but candidates ignore this.

    Fix: Capital K = LGD × (WCDR − PD). Subtract expected loss.

  • Applying the formula to a small, concentrated portfolio.

    The formula looks general.

    Fix: It assumes a large homogeneous portfolio with one factor, so idiosyncratic risk is fully diversified.

  • Thinking higher correlation lowers risk.

    Confusing it with diversification in market risk.

    Fix: Higher ρ makes defaults cluster, so WCDR and capital rise.

Worked examples

Example 1

A large homogeneous loan portfolio has PD = 1%, asset correlation ρ = 0.20. Find the 99.9% worst case default rate. Use N⁻¹(0.01) = −2.326, N⁻¹(0.999) = 3.090, and N(0.30) ≈ 0.618 where needed.

Show the solution
  1. √ρ = √0.20 = 0.4472; √(1 − ρ) = √0.80 = 0.8944.
  2. Numerator = −2.326 + 0.4472 × 3.090 = −2.326 + 1.382 = −0.944.
  3. Divide: −0.944 ÷ 0.8944 = −1.055.
  4. WCDR = N(−1.055) ≈ 1 − 0.854 = 0.146 (about 14.6%).

Answer: WCDR is about 14.6%, far above the 1% PD, because correlated defaults cluster in a severe downturn.

Example 2

A bank has EAD of $200 million, LGD of 40%, PD of 1%, and a 99.9% WCDR of 14.6%. Find the Credit VaR and the unexpected loss capital.

Show the solution
  1. Credit VaR = 0.146 × 0.40 × $200 million = $11.68 million.
  2. Expected loss = 0.01 × 0.40 × $200 million = $0.8 million.
  3. Unexpected loss capital = 11.68 − 0.8 = $10.88 million.
  4. Check with K: 0.40 × (0.146 − 0.01) = 0.0544; × $200 million = $10.88 million.

Answer: Credit VaR is $11.68 million and unexpected loss capital is $10.88 million.

Exam tips

  • Know both sign cases: conditional PD uses −√ρ × M, WCDR uses +√ρ × N⁻¹(X).
  • Expect a sanity check question: how WCDR changes when ρ, PD or X rises. All raise it.
  • Remember Basel IRB subtracts expected loss and applies a 99.9% confidence level.
  • Learn the key assumptions: one factor, large homogeneous portfolio, normal distributions, constant LGD.
  • Round z-scores sensibly. Options are usually spaced widely.

Practice questions from Credit Value at Risk

Single-Factor Model and Vasicek Credit VaR: frequently asked questions

What is the Vasicek worst case default rate?

It is the default rate on a large homogeneous portfolio that will not be exceeded at a chosen confidence level, such as 99.9%. It comes from setting the common factor at its bad-tail value in the conditional PD formula.

How does the Gaussian copula single-factor model work?

Each borrower's latent asset value is a mix of one common normal factor and its own normal shock. Default occurs when the latent value falls below N⁻¹(PD). Correlation among borrowers comes only from the common factor.

How does Basel IRB use the Vasicek formula?

Basel IRB sets capital for unexpected loss using the WCDR at 99.9%. Capital per unit of exposure is LGD × (WCDR − PD), with further adjustments such as maturity in the regulatory formula.

What happens to WCDR when correlation increases?

It rises. Higher asset correlation makes defaults more likely to occur together in a bad economy, so the tail default rate is larger and capital increases.