FRM Exam Part I · Measuring Credit Risk

Portfolio Credit Risk, Credit VaR and Default Correlation

Q: What is the difference between default correlation and asset correlation?

Asset correlation is the correlation between the normally distributed asset returns in the Vasicek or Merton framework. Default correlation is the correlation between the default events themselves. For the same pair of names, default correlation is usually much lower than asset correlation.

Q: How do you calculate economic capital for credit risk in FRM Part I?

Find the credit VaR at the stated confidence level, for example WCDR × LGD × EAD under Vasicek. Then subtract expected loss (PD × LGD × EAD). The remainder is unexpected loss, which is the economic capital.

Q: Why does the Vasicek model need a large portfolio?

The WCDR formula assumes idiosyncratic risk has been fully diversified away. This holds for a large, granular portfolio with similar exposures. With few names or large concentrations, actual tail losses can be higher than the formula suggests.

Q: What is the Gaussian copula in credit risk?

It is a way of joining each name's default probability into a joint distribution using normal asset returns and a correlation parameter. It is simple and widely used, but it has no tail dependence. It can understate the chance of many defaults clustering in a crisis.

Updated 11 October 2026 · Fact-checked

Portfolio credit risk is the risk of losses from many borrowers defaulting together. Default correlation and concentration fatten the loss distribution's tail. Credit VaR is the loss at a chosen confidence level. Under the Vasicek model, a large portfolio's worst-case default rate is N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1−ρ)]. Economic capital is credit VaR minus expected loss.

Understand Portfolio Credit Risk, Credit VaR and Correlation

A single loan has a simple loss story: it defaults with probability PD, and you lose EAD × LGD if it does. In a portfolio, expected loss just adds up. The risk does not. If defaults were independent, losses would average out and the loss distribution would be tight. In reality, borrowers share the same economy, so they tend to default together. That default correlation gives the loss distribution a long right tail.

Concentration works the same way. Large exposures to one name, one sector or one region mean a few events can cause big losses. A diversified portfolio has a thinner tail than a concentrated one with the same expected loss. Expected loss is the same, but unexpected loss and credit VaR are not.

The Vasicek single-factor model gives a tractable way to link these ideas. Each borrower's asset return has a systematic part (one common factor M, such as the state of the economy) and an idiosyncratic part (Z). A borrower defaults when its asset return falls below a threshold N⁻¹(PD). The asset correlation ρ sets how strongly borrowers move with the common factor. Given a bad value of M, defaults in a large portfolio become predictable, because the idiosyncratic noise diversifies away. This gives a worst-case default rate (WCDR) at confidence level X.

Credit VaR is the portfolio loss that will not be exceeded with probability X over the horizon, usually one year. Economic capital (unexpected loss at that confidence) is credit VaR minus expected loss. The Basel IRB capital formula is built on this same Vasicek logic, with a 99.9% confidence level.

The Gaussian copula is the machinery behind this. It links each name's marginal default probability to a joint distribution through a correlation parameter. Joint default probability for two names is the bivariate normal probability that both asset returns fall below their thresholds. Higher correlation raises the joint default probability and fattens the tail. A known weakness is that the Gaussian copula has no tail dependence, and a single correlation number can understate risk in crises.

Key formulas to remember

Expected loss (portfolio): EL = Σ (PDᵢ × LGDᵢ × EADᵢ); EL is additive across exposures. It does not depend on correlation.
Default correlation (two names): ρ_D = [P(A and B default) − PD_A × PD_B] ÷ √[PD_A(1−PD_A) × PD_B(1−PD_B)]; This is the correlation of the two 0/1 default indicators. It is usually much smaller than asset correlation.
Joint default probability: P(A and B default) = PD_A × PD_B + ρ_D × √[PD_A(1−PD_A) × PD_B(1−PD_B)]; The same formula rearranged. Independence means ρ_D = 0.
Vasicek asset return: Aᵢ = √ρ × M + √(1−ρ) × Zᵢ; M is the common factor and Zᵢ is the idiosyncratic factor. Both are standard normal and independent. Default occurs if Aᵢ < N⁻¹(PD).
Conditional PD given M: PD(M) = N[(N⁻¹(PD) − √ρ × M) ÷ √(1−ρ)]; A low (bad) M raises the conditional default rate.
Worst-case default rate: WCDR(X) = N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1−ρ)]; X is the confidence level, e.g. 0.999. N⁻¹(0.999) ≈ 3.09 and N⁻¹(0.99) ≈ 2.33. Applies to a large, granular portfolio with one common factor.
Credit VaR and economic capital: Credit VaR = WCDR × LGD × EAD; Economic capital = Credit VaR − EL; Assumes constant LGD and a large homogeneous portfolio.
Basel IRB capital charge (no maturity adjustment): K = LGD × [WCDR(99.9%) − PD]; Capital per unit of EAD. Multiply by EAD for the amount.
Variance of default count (n equal names): Var = n × PD × (1−PD) × [1 + (n−1) × ρ_D]; Assumes equal PD and equal pairwise default correlation ρ_D. It shows why correlation raises risk.

How to solve Portfolio Credit Risk, Credit VaR and Correlation questions

Use this order for any portfolio credit risk question. It keeps you from mixing up PD, asset correlation and default correlation.

1Identify what is asked: expected loss, unexpected loss, credit VaR, economic capital, default correlation or joint default probability.
2List the inputs: PD, LGD, EAD, confidence level X, and which correlation you have (asset correlation ρ or default correlation ρ_D).
3If the question is about expected loss, compute PD × LGD × EAD and sum over exposures. Stop there if that is all that is asked.
4For Vasicek, convert to z-scores: find N⁻¹(PD) and N⁻¹(X). Both come from the normal table or a calculator.
5Plug into WCDR = N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1−ρ)]. Compute the numerator first, then divide, then take N( ).
6Multiply WCDR by LGD and EAD to get credit VaR. Subtract EL to get economic capital if asked.
7For two-name questions, use the default correlation formula or its rearranged form to get the joint default probability, then build the loss distribution from the four outcomes.
8Sanity check: WCDR must be greater than PD, and higher ρ or higher X must raise it. Capital must be positive.

Quickest way: Vasicek WCDR in under two minutes

When to use it: Use for a large homogeneous portfolio when you are given PD, ρ, LGD, EAD and a confidence level, and you need credit VaR or capital.

Memorise N⁻¹(0.999) ≈ 3.09 and N⁻¹(0.99) ≈ 2.33.
Get N⁻¹(PD) from the table. For a PD below 50% it is negative (for example, N⁻¹(0.01) ≈ −2.33 and N⁻¹(0.02) ≈ −2.05).
Compute numerator = N⁻¹(PD) + √ρ × N⁻¹(X). Divide by √(1−ρ).
Read N( ) of the result. Round only at the end.
Credit VaR = WCDR × LGD × EAD. Capital = that minus PD × LGD × EAD.
Scan the answer options. WCDR must exceed PD, and capital must be below credit VaR. This usually rules out two options immediately.

Common mistakes in Portfolio Credit Risk, Credit VaR and Correlation

Treating asset correlation and default correlation as the same number.
Both are called correlation and both feed into joint default probability.
Fix: Asset correlation ρ is for the normal asset returns in Vasicek. Default correlation ρ_D is for the 0/1 default events and is usually much lower. Use the formula that matches the input you are given.
Reporting credit VaR as economic capital.
Both are tail losses, so the difference is easy to forget.
Fix: Economic capital is unexpected loss: credit VaR minus expected loss. Read the question wording and subtract EL when capital is asked.
Using the wrong sign for N⁻¹(PD) or for N⁻¹(X).
Candidates drop the negative sign on N⁻¹(PD) or use 1 − X by habit.
Fix: For small PD, N⁻¹(PD) is negative. In WCDR you add √ρ × N⁻¹(X), where N⁻¹(0.999) is positive 3.09. Check that WCDR comes out above PD.
Forgetting to multiply by LGD (or using LGD = 100%).
Candidates stop at WCDR, which is a default rate and not a loss rate.
Fix: Loss = default rate × LGD × EAD. Always apply LGD unless the question says zero recovery.
Believing diversification removes correlation risk in a Vasicek portfolio.
Granularity removes idiosyncratic risk, so it feels like all risk is gone.
Fix: Granularity removes only the Z part. The systematic factor M remains, and that is what WCDR captures. Capital stays positive even for infinitely many names.
Dividing by the wrong variance term when finding default correlation.
Candidates use PD × (1−PD) for one name only, or forget the square root.
Fix: The denominator is the square root of the product of both names' PD(1−PD) terms. Compute each term, multiply, then take the square root.

Worked examples

Example 1

A bank has a large, granular corporate loan portfolio with total EAD of USD 500 million. Each loan has PD = 2% and LGD = 45%. Asset correlation is ρ = 0.20 and the confidence level is 99.9%. Calculate the one-year credit VaR and the economic capital. Use N⁻¹(0.02) = −2.0537 and N⁻¹(0.999) = 3.0902.

Show the solution

Expected loss = 500 × 0.02 × 0.45 = USD 4.5 million.
√ρ = √0.20 = 0.4472. √(1−ρ) = √0.80 = 0.8944.
√ρ × N⁻¹(0.999) = 0.4472 × 3.0902 = 1.3820.
Numerator = −2.0537 + 1.3820 = −0.6717.
Divide: −0.6717 ÷ 0.8944 = −0.7510.
WCDR = N(−0.7510) ≈ 0.2263, or about 22.6%.
Credit VaR = 0.2263 × 0.45 × 500 = USD 50.9 million.
Economic capital = 50.9 − 4.5 = USD 46.4 million.

Answer: Credit VaR ≈ USD 50.9 million. Economic capital (unexpected loss) ≈ USD 46.4 million.

Example 2

Two borrowers have PDs of 2% and 5%. The probability that both default within a year is 0.4%. (a) What is the default correlation? Choose the closest: A) 0.4% B) 9.8% C) 19.6% D) 25.0%. (b) What is the probability that exactly one of them defaults?

Show the solution

Independent joint probability = 0.02 × 0.05 = 0.001. Covariance of the default indicators = 0.004 − 0.001 = 0.003.
Standard deviation product = √[0.02 × 0.98 × 0.05 × 0.95] = √[0.0196 × 0.0475] = √0.000931 = 0.03051.
ρ_D = 0.003 ÷ 0.03051 = 0.0983, or about 9.8%. This is option B.
For (b): P(only A defaults) = 0.02 − 0.004 = 0.016.
P(only B defaults) = 0.05 − 0.004 = 0.046.
P(exactly one defaults) = 0.016 + 0.046 = 0.062.

Answer: (a) Default correlation ≈ 9.8%, option B. (b) The probability that exactly one defaults is 6.2%.

Exam tips

Questions often give N⁻¹ values or table values. Check which one is supplied before you start, so you do not waste time.
Expect conceptual questions on what raises credit VaR: higher PD, higher ρ, higher confidence level, higher LGD or concentration. Know the direction of each effect.
Know the difference between expected loss (provisions, priced into spreads) and unexpected loss (covered by capital). Many questions test this split.
Learn the limits of the Gaussian copula: no tail dependence and a single correlation parameter. These come up in crisis-related questions.
Use a financial calculator or spreadsheet logic only for N( ) and N⁻¹( ). Do all other steps by hand to avoid entry errors.

Practice questions from Measuring Credit Risk

Portfolio Credit Risk, Credit VaR and Correlation in other exams

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

Portfolio Credit Risk, Credit VaR and Correlation: frequently asked questions

What is the difference between default correlation and asset correlation?

Asset correlation is the correlation between the normally distributed asset returns in the Vasicek or Merton framework. Default correlation is the correlation between the default events themselves. For the same pair of names, default correlation is usually much lower than asset correlation.

How do you calculate economic capital for credit risk in FRM Part I?

Find the credit VaR at the stated confidence level, for example WCDR × LGD × EAD under Vasicek. Then subtract expected loss (PD × LGD × EAD). The remainder is unexpected loss, which is the economic capital.

Why does the Vasicek model need a large portfolio?

The WCDR formula assumes idiosyncratic risk has been fully diversified away. This holds for a large, granular portfolio with similar exposures. With few names or large concentrations, actual tail losses can be higher than the formula suggests.

What is the Gaussian copula in credit risk?

It is a way of joining each name's default probability into a joint distribution using normal asset returns and a correlation parameter. It is simple and widely used, but it has no tail dependence. It can understate the chance of many defaults clustering in a crisis.