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FRM Exam Part II · Portfolio Credit Risk

Credit Portfolio Risk and Diversification Explained

Updated 11 October 2026 · Fact-checked

Portfolio credit risk is the risk of losing money across many loans at once. Expected loss adds up linearly across names. Unexpected loss does not: it falls when default correlation is low and the portfolio is diversified, and rises with correlation and concentration. Solve questions by separating EL from UL, then checking correlation and concentration.

Understand Credit Portfolio Risk and Diversification

A single loan has two outcomes that matter: it pays or it defaults. Its risk is measured by its probability of default (PD), loss given default (LGD) and exposure at default (EAD). For one name, the loss is lumpy. You either lose a large amount or nothing.

A portfolio changes this. Expected loss (EL) is the average loss you plan for. It is additive: the portfolio EL is the sum of each loan's EL, whatever the correlation. Banks cover EL through pricing and provisions.

Unexpected loss (UL) is the standard deviation of portfolio loss. It measures how far actual losses can swing from EL. Capital is held against losses beyond EL, up to a chosen confidence level. This is the credit VaR minus EL, usually called economic capital.

UL is not additive. It depends on default correlation, the tendency of borrowers to default together because they share economic drivers such as the business cycle, sector, or region. If defaults were independent, many small loans would make losses very predictable. If defaults are perfectly correlated, diversification gives no benefit and the portfolio behaves like one big loan.

Concentration works against diversification. Name concentration means a few large exposures. Sector or geographic concentration means many names sharing the same risk driver. Even a portfolio of many names stays risky if they are highly correlated. Diversification removes idiosyncratic risk (name-specific) but not systematic risk (common factors). Credit loss distributions are skewed with a fat right tail, so diversification helps less in the tail than the standard deviation suggests.

Key formulas to remember

Single-name expected loss
EL = PD × LGD × EAD
Uses expected values. Assumes PD, LGD and EAD are treated as independent unless stated.
Portfolio expected loss
EL_P = Σ EL_i
Additive. Does not depend on default correlation.
Single-name unexpected loss (LGD and EAD fixed)
UL = EAD × LGD × √(PD × (1 − PD))
Bernoulli default standard deviation. With stochastic LGD, add the LGD variance term.
Two-name portfolio UL
UL_P = √(UL_1² + UL_2² + 2 × ρ × UL_1 × UL_2)
ρ is the correlation of default events (losses). UL_P is at most UL_1 + UL_2, with equality when ρ = 1.
Default correlation
ρ = [P(A ∩ B) − PD_A × PD_B] ÷ √(PD_A(1 − PD_A) × PD_B(1 − PD_B))
P(A ∩ B) is the joint default probability. Default correlation is typically small, often a few percent, even when asset correlation is higher.
Economic capital
Economic capital = Credit VaR (at confidence level) − EL
Capital covers unexpected loss only. EL is covered by provisions and pricing.
Independent-default limit
UL_P = √(Σ UL_i²) when ρ = 0
Diversification benefit is largest here.

How to solve Credit Portfolio Risk and Diversification questions

Use this order for any question on credit portfolio risk, diversification or concentration.

  1. 1Identify what is asked: expected loss, unexpected loss, capital, or a qualitative effect of correlation or concentration.
  2. 2List the inputs for each exposure: PD, LGD, EAD, and any correlation or joint default probability.
  3. 3Compute EL for each name as PD × LGD × EAD, then add them. Correlation plays no role here.
  4. 4If UL is needed, compute each stand-alone UL using √(PD × (1 − PD)) × LGD × EAD.
  5. 5Combine ULs using the correlation formula. Never add ULs unless ρ = 1.
  6. 6If asked for capital, subtract EL from the tail loss (credit VaR) at the stated confidence level.
  7. 7Check the direction: higher ρ or more concentration raises UL and tail loss; more independent names lowers UL.
  8. 8Match the answer to the option that respects these directions and the units given.

Quickest way: Direction-first elimination

When to use it: Use for conceptual or comparison questions where options differ in direction, not in calculation.

  1. EL asked? Sum it. Ignore correlation.
  2. UL or capital asked? Higher correlation or concentration means higher UL and fatter tail.
  3. Two-name UL with ρ = 0: take the square root of the sum of squares. With ρ = 1: add them.
  4. Any option saying diversification removes systematic risk, or that UL is additive, is wrong.
  5. For numbers, compute only what you need, then check the result lies between √(ΣUL²) and ΣUL.

Common mistakes in Credit Portfolio Risk and Diversification

  • Adding unexpected losses across names

    EL is additive, so students assume UL is too.

    Fix: Only EL adds. UL needs the correlation formula, and the sum is an upper bound reached only when ρ = 1.

  • Saying correlation changes portfolio expected loss

    Correlation raises risk, so it feels like it should raise loss.

    Fix: Correlation changes the spread and tail of the loss distribution, not its mean.

  • Confusing default correlation with asset correlation

    Both are called correlation and both drive joint defaults.

    Fix: Default correlation is of default events and is usually lower than asset correlation. Use the one the question gives.

  • Believing many names always means diversified

    Number of loans is easy to count.

    Fix: Check for sector, region or common-factor overlap. Concentrated systematic risk survives a large name count.

  • Treating economic capital as total loss at the VaR level

    Credit VaR and capital are used loosely together.

    Fix: Capital = credit VaR − EL, because EL is already provisioned and priced.

  • Using PD instead of √(PD × (1 − PD)) for UL

    Mixing up expected value and standard deviation of a default event.

    Fix: EL uses PD. UL uses the Bernoulli standard deviation.

Worked examples

Example 1

A bank has two loans. Loan A: EAD $10 million, PD 2%, LGD 50%. Loan B: EAD $10 million, PD 2%, LGD 50%. Defaults are independent. Find the portfolio EL and UL (LGD fixed).

Show the solution
  1. EL of each loan = 0.02 × 0.50 × 10,000,000 = $100,000.
  2. Portfolio EL = 100,000 + 100,000 = $200,000.
  3. UL of each loan = 10,000,000 × 0.50 × √(0.02 × 0.98) = 5,000,000 × √0.0196 = 5,000,000 × 0.14 = $700,000.
  4. With ρ = 0: UL_P = √(700,000² + 700,000²) = 700,000 × √2 ≈ $989,950.
  5. Compare with the sum of ULs: $1,400,000. Diversification reduces UL.

Answer: EL = $200,000; UL ≈ $989,950, versus $1,400,000 if the ULs were simply added.

Example 2

Using the same two loans, defaults now have correlation ρ = 0.30. Find the portfolio UL and the diversification benefit versus the sum of stand-alone ULs.

Show the solution
  1. Stand-alone UL of each loan = $700,000.
  2. UL_P² = 700,000² + 700,000² + 2 × 0.30 × 700,000 × 700,000.
  3. Factor out 700,000²: UL_P² = 700,000² × (1 + 1 + 0.60) = 700,000² × 2.60.
  4. UL_P = 700,000 × √2.60 = 700,000 × 1.6125 ≈ $1,128,700.
  5. Sum of stand-alone ULs = $1,400,000.
  6. Diversification benefit = 1,400,000 − 1,128,700 ≈ $271,300.
  7. Portfolio EL is unchanged at $200,000, since EL does not depend on correlation.

Answer: UL ≈ $1,128,700, a diversification benefit of about $271,300. EL stays $200,000. Higher correlation raises UL from $989,950 to $1,128,700.

Exam tips

  • Expect a question testing that EL is additive and UL is not. Read the options for this trap first.
  • When a vignette mentions one sector or region dominating the book, think concentration and systematic risk, not name count.
  • Check whether the question gives default correlation or asset correlation, and use it accordingly.
  • For capital questions, subtract EL from the stated percentile loss.
  • Sanity check numerical UL: it must sit between the independent case and the fully correlated sum.

Practice questions from Portfolio Credit Risk

Credit Portfolio Risk and Diversification in other exams

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

Credit Portfolio Risk and Diversification: frequently asked questions

What is the difference between expected loss and unexpected loss in a credit portfolio?

Expected loss is the average loss, PD × LGD × EAD, and it adds across loans. Unexpected loss is the standard deviation of loss and it depends on correlation. Banks cover EL with pricing and provisions, and UL with capital.

How does default correlation affect portfolio credit risk?

Higher default correlation means defaults cluster, so the loss distribution gets a fatter right tail and higher UL. It does not change expected loss. At correlation of one, diversification gives no benefit.

How does concentration affect credit portfolio risk?

Concentration raises UL and tail losses. Name concentration means a few large exposures. Sector or geographic concentration means exposures share risk drivers and so are more correlated.

Can diversification remove all credit risk?

No. Diversification removes idiosyncratic risk but not systematic risk from common factors such as the economic cycle. In a downturn, correlations tend to rise and the benefit shrinks.