FRM Exam Part II · Credit Value at Risk

Portfolio Credit Risk and Default Correlation Explained

Updated 11 October 2026 · Fact-checked

Portfolio credit risk is the risk of loss from defaults across many borrowers at once. Default correlation is the tendency of defaults to cluster. Expected loss ignores correlation, but loss volatility and credit VaR rise with it. To solve questions, compute expected loss, then variance using PD, exposure, LGD and correlation, then interpret.

Understand Portfolio Credit Risk and Default Correlation

A credit portfolio is a set of exposures. Each one has a probability of default (PD), a loss given default (LGD) and an exposure at default (EAD). The expected loss of the portfolio is just the sum of the individual expected losses. Correlation does not change it.

Correlation changes the spread of losses. If defaults were independent, good and bad outcomes would offset each other and the portfolio loss would be fairly stable. If defaults are positively correlated, borrowers tend to default together, usually because they share the same economic drivers. Then you get many quiet years and a few very bad ones. Loss volatility is higher and the tail of the loss distribution is fatter.

Default correlation is the correlation between default indicators. For two obligors with default probabilities p1 and p2 and joint default probability p12, it is (p12 − p1p2) ÷ √[p1(1 − p1)p2(1 − p2)]. Default correlations are usually small, often a few percent, yet they matter a lot. Asset (or factor) correlation in a Merton or Vasicek model is a different and larger number. Do not mix them up.

Concentration risk means a large share of exposure sits with one name, sector or region. Single-name concentration is measured by exposure weights, and a common index is the Herfindahl index, the sum of squared weights. Higher values mean less diversification. Diversification cuts idiosyncratic risk but cannot remove the systematic part. As the number of names grows, portfolio loss volatility falls toward a floor set by correlation.

Finally, marginal risk contributions split portfolio risk among positions. The contribution of a position is its weight times its marginal effect on portfolio risk. Contributions add up to the total, so they work for capital allocation. A position with high exposure, high PD or high correlation with the rest of the book contributes more.

Key formulas to remember

Portfolio expected loss: EL_P = Σ EADi × PDi × LGDi; Additive. Correlation has no effect on it.
Default correlation (two obligors): ρD = (p12 − p1 × p2) ÷ √[p1(1 − p1) × p2(1 − p2)]; p12 is the joint default probability. Independent defaults give p12 = p1 × p2 and ρD = 0.
Joint default probability: p12 = p1 × p2 + ρD × √[p1(1 − p1) × p2(1 − p2)]; Rearranged from the correlation formula. Positive correlation raises joint default above p1 × p2.
Loss variance of a two-name portfolio (fixed LGD): σ² = Σ Σ Li × Lj × Cov(Di, Dj), where Li = EADi × LGDi; Cov(Di, Di) = pi(1 − pi). Cov(Di, Dj) = ρD × σDi × σDj, with σDi = √[pi(1 − pi)].
Homogeneous portfolio loss volatility: σ_P = L × √[p(1 − p)] × √[N × (1 + (N − 1)ρD)]; Total loss volatility for N equal exposures L, same p, same pairwise ρD. Dividing by N gives the average per-loan volatility, L × √[p(1 − p)] × √[(1 + (N − 1)ρD) ÷ N]. As N rises, the per-loan figure approaches L × √[p(1 − p)] × √ρD.
Herfindahl index: H = Σ wi²; Equal weights give H = 1/N. Higher H means more concentration.
Risk contribution: RCi = wi × ∂σ_P/∂wi = wi × Cov(Ri, R_P) ÷ σ_P; Contributions sum to total portfolio risk σ_P (Euler allocation).

How to solve Portfolio Credit Risk and Default Correlation questions

Use this order for any question on default correlation, concentration or risk contributions.

1Identify what is given: PD, LGD, EAD, weights, and whether the correlation is default correlation or asset correlation.
2Compute the expected loss of each position and add them. Remember this does not depend on correlation.
3Compute each position's loss standard deviation: Li × √[pi(1 − pi)], with Li = EAD × LGD (assume LGD fixed unless told otherwise).
4Combine them using the correlation: variance = Σ of variances + 2 × Σ covariances. For identical names use the homogeneous formula.
5State the direction: higher correlation gives higher volatility and a fatter tail, but the same expected loss.
6For risk contributions, compute each position's covariance with the portfolio, divide by portfolio volatility and multiply by weight. Check they sum to the total.
7Interpret: link the result to diversification limits, concentration or capital allocation.

Quickest way: Direction-first shortcut

When to use it: Use when options differ in direction or order of magnitude and you have little time.

Ask whether the item is expected loss or loss volatility. Correlation only moves the second.
If independence is assumed, p12 = p1 × p2 and correlation is 0. This is often the answer to a trap option.
For equal names, multiply single-name volatility by √[(1 + (N − 1)ρD) ÷ N]. This gives the average per-loan volatility, which is the volatility of the equal-weighted portfolio. Multiply by N to get total portfolio loss volatility in currency terms. Check the limits of the multiplier: ρD = 0 gives 1/√N, ρD = 1 gives 1.
For risk contributions, rank by weight × correlation with the portfolio. Remove any option where the contributions do not sum to the total.

Common mistakes in Portfolio Credit Risk and Default Correlation

Saying higher default correlation raises expected loss.
Students link more risk with more loss in every sense.
Fix: Expected loss is a sum of individual expected losses. Correlation changes only the volatility and tail, such as unexpected loss and credit VaR.
Confusing asset correlation with default correlation.
Both are called correlation and appear in the same models.
Fix: Asset correlation drives the latent variable in Merton or Vasicek models. Default correlation is the correlation of default indicators and is usually much smaller. Check which one the question gives.
Using the variance of a Bernoulli default as p instead of p(1 − p).
Students forget that a default indicator is a 0/1 variable.
Fix: Default variance is p(1 − p). Its standard deviation is √[p(1 − p)].
Believing that adding more names removes all risk.
Diversification is over-generalised from independent assets.
Fix: With positive correlation, volatility falls toward a floor based on correlation. Systematic risk stays. Only idiosyncratic risk diversifies away.
Treating standalone risk as a position's contribution to portfolio risk.
Students ignore the covariance with the rest of the book.
Fix: Use weight × covariance with portfolio ÷ portfolio volatility. Contributions add to the total, standalone risks do not.
Measuring concentration only by the number of names.
More names feels more diversified.
Fix: Look at weights too. One large exposure among many small ones leaves high concentration. Use the Herfindahl index and also check sector and region.

Worked examples

Example 1

Two obligors each have PD of 2%. Their default correlation is 0.10. What is the joint default probability, and how does it compare with the independent case?

Show the solution

Variance of each default indicator: 0.02 × 0.98 = 0.0196.
Standard deviation: √0.0196 = 0.14.
Covariance term: ρD × σ1 × σ2 = 0.10 × 0.14 × 0.14 = 0.00196.
Independent joint probability: 0.02 × 0.02 = 0.0004.
Joint default probability: 0.0004 + 0.00196 = 0.00236.

Answer: Joint default probability is 0.236%. This is about 5.9 times the independent value of 0.04%.

Example 2

A portfolio has 4 equal loans, each with exposure ₹10 crore, LGD 100%, PD 5%, and pairwise default correlation 0.20. Find expected loss and loss standard deviation.

Show the solution

Expected loss per loan: 10 × 0.05 = ₹0.5 crore. Portfolio EL = 4 × 0.5 = ₹2 crore.
Single-loan loss standard deviation: 10 × √(0.05 × 0.95) = 10 × √0.0475 = 10 × 0.2179 = ₹2.179 crore.
Correlation multiplier: √[(1 + 3 × 0.20) ÷ 4] = √(1.6 ÷ 4) = √0.4 = 0.6325. Average per-loan volatility = 2.179 × 0.6325 = ₹1.378 crore.
Total portfolio loss standard deviation = N × average per-loan volatility = 4 × 1.378 = ₹5.51 crore.
Check with the total variance: N × L² × p(1 − p) × [1 + (N − 1)ρD] = 4 × 4.75 × 1.6 = 30.4, and √30.4 = ₹5.514 crore.

Answer: Expected loss is ₹2 crore and total loss standard deviation is about ₹5.51 crore. With zero correlation the standard deviation would be √(4 × 4.75) = ₹4.36 crore, so correlation raises volatility but not expected loss.

Exam tips

Read the question for the word correlation and decide at once whether it is default, asset or loss correlation.
Expect direction-of-effect questions. Expected loss is unchanged by correlation, while unexpected loss and tail VaR rise.
Check limits to test an option: zero correlation and perfect correlation give known values.
For risk contribution questions, check that the numbers add to the portfolio total before choosing.
Name concentration types precisely: single-name, sector and geographic.

Practice questions from Credit Value at Risk

Portfolio Credit Risk and Default 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 and Default Correlation: frequently asked questions

How does default correlation affect credit VaR?

Higher default correlation makes defaults cluster, so the loss distribution gets a fatter right tail. Credit VaR and unexpected loss rise. Expected loss stays the same.

Is default correlation the same as asset correlation?

No. Asset correlation is the correlation between firms' asset values or latent factors in a structural model. Default correlation is the correlation between default events and is normally much lower. The two are linked through the model.

Why can diversification not remove all credit risk?

Borrowers share common drivers such as the economic cycle. This systematic part is correlated across names. Adding names removes idiosyncratic risk only, so volatility settles at a floor set by correlation.

What is a marginal risk contribution in a credit portfolio?

It shows how much a position adds to portfolio risk, based on its size and its covariance with the rest of the portfolio. Contributions sum to total portfolio risk, so banks use them to allocate economic capital.