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FRM Exam Part II · Portfolio Credit Risk

Vasicek Single-Factor Model and Large Homogeneous Portfolios

Updated 11 October 2026 · Fact-checked

The Vasicek single-factor model says each borrower's default depends on one common market factor and one borrower-specific factor. In a large homogeneous portfolio, the idiosyncratic part diversifies away. The default rate then depends only on the market factor, giving a conditional default rate and a worst-case default rate at a chosen confidence level.

Understand Vasicek Single-Factor Model and Large Homogeneous Portfolios

Start with one borrower. A Merton-style idea says the borrower defaults when its asset value falls below a threshold. In the Vasicek model, the standardized asset return is Aᵢ = √ρ · M + √(1 − ρ) · Zᵢ. M is the common market factor. Zᵢ is the borrower's own shock. Both are independent standard normal variables.

Default happens when Aᵢ falls below a threshold. The threshold is N⁻¹(PD), where PD is the unconditional (through-the-cycle) default probability. The parameter ρ is the asset correlation between any two borrowers. A higher ρ means borrowers move more together.

Now fix the market factor at a value M. Defaults become independent, because only the Zᵢ terms remain random. The conditional default probability is N[(N⁻¹(PD) − √ρ · M) ÷ √(1 − ρ)]. A bad economy means a low M, which raises this probability.

Now take a large homogeneous portfolio: many loans, equal size, same PD, same ρ, same LGD. By the law of large numbers, the portfolio default rate equals the conditional default probability. All remaining randomness comes from M. So the loss distribution is driven by one normal variable.

To get the worst-case default rate at confidence level X, use the M value that is exceeded only with probability 1 − X, which is M = −N⁻¹(X). This gives WCDR = N[(N⁻¹(PD) + √ρ · N⁻¹(X)) ÷ √(1 − ρ)]. Basel IRB uses this logic with X = 99.9%. Capital for credit VaR is about (WCDR − PD) × LGD × EAD, which is the unexpected loss. The full Basel formula also adds a maturity adjustment.

Key formulas to remember

Asset return in the one-factor model
Aᵢ = √ρ · M + √(1 − ρ) · Zᵢ
M and Zᵢ are independent standard normal. Default if Aᵢ < N⁻¹(PD).
Default threshold
K = N⁻¹(PD)
N is the standard normal CDF. N⁻¹ is its inverse.
Conditional default probability
PD(M) = N[(N⁻¹(PD) − √ρ · M) ÷ √(1 − ρ)]
Given the market factor M. Lower M means higher default rate.
Worst-case default rate (WCDR)
WCDR(X) = N[(N⁻¹(PD) + √ρ · N⁻¹(X)) ÷ √(1 − ρ)]
Default rate not exceeded with confidence X. Basel uses X = 99.9%, so N⁻¹(X) ≈ 3.09.
Credit VaR and unexpected loss
Credit VaR = WCDR × LGD × EAD; UL = (WCDR − PD) × LGD × EAD
Expected loss is PD × LGD × EAD. Basel IRB capital is based on UL.
Useful normal values
N⁻¹(95%) ≈ 1.645; N⁻¹(99%) ≈ 2.326; N⁻¹(99.9%) ≈ 3.090
Quote these when the question gives no tables.

How to solve Vasicek Single-Factor Model and Large Homogeneous Portfolios questions

Use the same sequence for any Vasicek or WCDR question. Keep PD and ρ in decimals.

  1. 1Identify PD, asset correlation ρ, confidence level X, and LGD and EAD if loss is asked.
  2. 2Compute N⁻¹(PD) and N⁻¹(X) from the given values or standard table.
  3. 3Compute √ρ and √(1 − ρ).
  4. 4If a market factor value M is given, use the conditional PD formula. If a confidence level is given, use the WCDR formula with + √ρ · N⁻¹(X).
  5. 5Evaluate the numerator, divide by √(1 − ρ), then apply N(·) to convert back to a probability.
  6. 6If loss is asked, multiply by LGD and EAD. Subtract PD × LGD × EAD if unexpected loss is asked.
  7. 7Check the answer: WCDR must exceed PD, and a higher ρ must give a higher WCDR.

Quickest way: Sign and sanity shortcut

When to use it: Use when options are spread apart and you need to avoid heavy normal-table work.

  1. Decide the direction: WCDR is always above PD, and rises with ρ and with X.
  2. Compute the argument of N(·) roughly: (N⁻¹(PD) + √ρ · N⁻¹(X)) ÷ √(1 − ρ).
  3. Convert with benchmark values: N(−1) ≈ 15.9%, N(−1.645) ≈ 5%, N(−2) ≈ 2.3%, N(0) = 50%, N(1) ≈ 84.1%.
  4. Eliminate options that are below PD or that fall the wrong way when ρ rises.

Common mistakes in Vasicek Single-Factor Model and Large Homogeneous Portfolios

  • Subtracting √ρ · N⁻¹(X) in the WCDR formula.

    The conditional PD formula has a minus sign with M, and students copy it directly.

    Fix: The worst market factor is M = −N⁻¹(X), so the sign becomes plus. WCDR must be above PD.

  • Using ρ instead of √ρ.

    The factor loading is the square root of the correlation, which is easy to forget under time pressure.

    Fix: Always take √ρ and √(1 − ρ) before substituting.

  • Calling the capital charge WCDR × LGD × EAD.

    Students forget that expected loss is covered by provisions and pricing.

    Fix: Basel capital covers unexpected loss: (WCDR − PD) × LGD × EAD, before the maturity adjustment.

  • Using the 99% value 2.326 when the question says 99.9%.

    Students remember a single familiar VaR quantile.

    Fix: Match the confidence level in the question. Basel IRB is 99.9%, so use 3.090.

  • Treating the conditional default probability as the unconditional PD.

    Both are called default probabilities.

    Fix: Unconditional PD is the average of the conditional PD over all values of M. The two coincide for every M only when ρ = 0. At M = 0 the conditional PD is generally not equal to PD.

  • Applying the large-portfolio result to a small or concentrated portfolio.

    The approximation looks like a general formula.

    Fix: It needs many small, equal exposures with the same PD, ρ and LGD. Concentration or name risk adds loss variance that the model ignores.

Worked examples

Example 1

A large homogeneous portfolio has PD = 2%, asset correlation ρ = 0.20. Using N⁻¹(2%) = −2.054 and N⁻¹(99.9%) = 3.090, what is the 99.9% worst-case default rate? Use N(−1.07) ≈ 14.2%, N(−1.03) ≈ 15.2%, N(−1.0) ≈ 15.9%.

Show the solution
  1. √ρ = √0.20 = 0.4472.
  2. √(1 − ρ) = √0.80 = 0.8944.
  3. Numerator: −2.054 + 0.4472 × 3.090 = −2.054 + 1.382 = −0.672.
  4. Divide: −0.672 ÷ 0.8944 = −0.751.
  5. N(−0.751) ≈ 0.226 (since N(−0.75) ≈ 0.2266).

Answer: WCDR ≈ 22.6%, about eleven times the PD of 2%.

Example 2

A portfolio has EAD = $500 million, LGD = 40%, PD = 1%, and the 99.9% WCDR is 10%. Find expected loss, credit VaR and unexpected loss.

Show the solution
  1. Expected loss = 1% × 40% × $500m = $2.0 million.
  2. Credit VaR = 10% × 40% × $500m = $20.0 million.
  3. Unexpected loss = (10% − 1%) × 40% × $500m = 9% × 0.40 × $500m = $18.0 million.
  4. Check: credit VaR − expected loss = $20m − $2m = $18m.

Answer: Expected loss $2.0 million; credit VaR $20.0 million; unexpected loss $18.0 million.

Exam tips

  • Memorize N⁻¹(99%) = 2.326 and N⁻¹(99.9%) = 3.090. Most questions use one of them.
  • Check the sign first. WCDR below PD means you made an error.
  • Know the interpretation: ρ drives the gap between WCDR and PD, and ρ = 0 gives WCDR = PD.
  • Be ready to state the assumptions: one factor, normal shocks, large, equal, homogeneous exposures, constant LGD.
  • Distinguish expected loss, unexpected loss and credit VaR in case-style questions on Basel IRB capital.

Practice questions from Portfolio Credit Risk

Vasicek Single-Factor Model and Large Homogeneous Portfolios in other exams

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

Vasicek Single-Factor Model and Large Homogeneous Portfolios: frequently asked questions

What is the Vasicek model used for in Basel?

It underlies the Basel IRB capital formula. The formula uses a conditional default rate at the 99.9% confidence level, with asset correlation set by supervisors. Capital is based on the excess of that rate over the expected PD.

Why does a higher asset correlation raise the worst-case default rate?

Higher ρ means borrowers depend more on the common factor, so a bad market state hits many of them at once. Diversification removes less risk. The tail default rate therefore rises, while the average default rate stays at PD.

What does large homogeneous portfolio mean?

It means a very large number of small exposures with the same PD, LGD and asset correlation. Idiosyncratic risk then diversifies away. The portfolio default rate equals the conditional default probability given the market factor.

Is the worst-case default rate the same as the maximum possible default rate?

No. It is a quantile, the default rate not exceeded with the chosen confidence, such as 99.9%. Worse outcomes are possible, with probability 0.1%.