FRM Exam Part II · Portfolio Credit Risk

Expected Loss, Unexpected Loss and Economic Capital Explained

Q: What is the difference between expected loss and unexpected loss?

Expected loss is the average credit loss, calculated as PD × LGD × EAD. Unexpected loss is the standard deviation of losses around that average. Expected loss is priced and provisioned. Unexpected loss is what capital protects against.

Q: How is economic capital different from regulatory capital?

Regulatory capital is the minimum set by supervisors under Basel rules. Economic capital is the bank's internal estimate of capital needed to cover losses at a chosen confidence level, often linked to its target credit rating. The two can differ in size and method.

Q: How do I calculate economic capital from credit VaR?

Take the portfolio loss at the chosen confidence level, such as 99.9%, and subtract expected loss. The result is the capital held for unexpected losses. Check the question's definition, as some sources use credit VaR directly.

Q: Why does diversification reduce unexpected loss but not expected loss?

Expected losses add across exposures regardless of correlation. Unexpected loss depends on how defaults move together. When default correlation is below 1, some losses offset in timing, so portfolio UL is below the sum of individual ULs.

Updated 11 October 2026 · Fact-checked

Expected loss (EL) is the average credit loss: EL = PD × LGD × EAD. Unexpected loss (UL) is the standard deviation of loss around that average. Credit VaR is the loss quantile at a confidence level. Economic capital is credit VaR minus EL, the buffer for unexpected losses.

Understand Expected Loss, Unexpected Loss and Economic Capital

Every loan can default. Three inputs describe it: PD (probability of default over a horizon, usually one year), LGD (share of exposure lost if default happens, equal to 1 minus the recovery rate) and EAD (exposure at default, the amount owed when the borrower fails).

Expected loss is the mean of the loss distribution: EL = PD × LGD × EAD. It is a cost of doing business. Banks cover it through loan pricing (credit spreads) and provisions, not capital.

Actual losses vary from year to year. Unexpected loss measures that variation and is the standard deviation of portfolio loss. For one exposure with fixed LGD and EAD, UL = EAD × LGD × √(PD × (1 − PD)). Credit loss distributions are skewed with a long right tail, so the standard deviation alone does not describe the worst cases.

Credit VaR is the loss at a chosen confidence level, such as 99.9%, over one year. Economic capital is the bank's own estimate of capital needed to absorb losses to that confidence level. It is credit VaR minus EL, because EL is already covered by pricing and provisions. Regulatory capital is set by supervisors under Basel rules. Economic capital reflects the bank's internal view of risk and its target rating, and the two can differ.

At portfolio level, losses do not simply add up. Default correlation reduces diversification, so portfolio UL is less than the sum of the individual ULs unless correlation is perfect. Capital is then allocated back to exposures using risk contributions, which sum to the portfolio total.

Key formulas to remember

Expected loss (single exposure): EL = PD × LGD × EAD; PD, LGD and EAD must use the same horizon. LGD = 1 − recovery rate.
Expected loss (portfolio): EL(portfolio) = Σ EL(i); Expected losses always add, whatever the correlation.
Unexpected loss (single exposure, fixed LGD and EAD): UL = EAD × LGD × √(PD × (1 − PD)); Standard deviation of the default loss. Assumes LGD is known and constant. If LGD is random, add its variance.
Unexpected loss with random LGD: UL = EAD × √(PD × σ(LGD)² + LGD² × PD × (1 − PD)); σ(LGD) is the standard deviation of LGD, assumed independent of default.
Portfolio UL (two exposures): UL(p) = √(UL₁² + UL₂² + 2 × ρ × UL₁ × UL₂); ρ is the default correlation. UL(p) ≤ UL₁ + UL₂, with equality only when ρ = 1.
Economic capital: EC = Credit VaR(α) − EL; Credit VaR here is the loss quantile at confidence α. Some texts call this the unexpected loss at that confidence.
Risk contribution: RC(i) = UL(i) × ∂UL(p)/∂UL(i), and Σ RC(i) = UL(p); For the two-exposure case RC₁ = UL₁ × (UL₁ + ρ × UL₂) ÷ UL(p). Contributions add to the portfolio UL.

How to solve Expected Loss, Unexpected Loss and Economic Capital questions

Use this order for any question on EL, UL, credit VaR or economic capital.

1Identify what is asked: EL, UL, credit VaR, economic capital or a capital allocation.
2List PD, LGD, EAD and the horizon. Convert recovery rate to LGD if needed (LGD = 1 − recovery).
3Compute EL for each exposure as PD × LGD × EAD, in currency, not as a percentage.
4Compute UL for each exposure. Check whether LGD is fixed or random, then pick the correct formula.
5For a portfolio, add ELs directly. Combine ULs using the correlation formula, never by simple addition unless ρ = 1.
6If asked for capital, take the loss quantile (credit VaR) and subtract EL. Check whether the question gives credit VaR or already gives the unexpected part.
7For allocation, compute each exposure's risk contribution and confirm the contributions sum to portfolio UL.
8State the interpretation: EL is priced and provisioned, capital covers the unexpected tail.

Quickest way: EL first, then ρ shortcut

When to use it: Use for numerical MCQs with two or three exposures and limited time.

Write EL = PD × LGD × EAD. Multiply in currency units, and watch the decimal places.
If correlation is 0, portfolio UL = √(UL₁² + UL₂²). If 1, add the ULs. Any option above the sum of ULs is wrong.
If economic capital is asked and credit VaR is given, subtract total EL and stop.
Eliminate options that add ELs wrongly, ignore LGD, or use PD instead of √(PD(1 − PD)).

Common mistakes in Expected Loss, Unexpected Loss and Economic Capital

Treating economic capital as the full credit VaR.
Students forget that EL is already covered by pricing and provisions.
Fix: Use EC = Credit VaR − EL unless the question states a different definition.
Adding individual ULs to get portfolio UL.
ELs add, so students assume ULs do too.
Fix: Use the correlation formula. Simple addition holds only when ρ = 1.
Using PD instead of √(PD × (1 − PD)) in UL.
Confusing the mean of the default indicator with its standard deviation.
Fix: UL is a standard deviation. The default indicator has variance PD × (1 − PD).
Using the recovery rate as LGD.
Questions often give recovery, and students plug it straight in.
Fix: Compute LGD = 1 − recovery rate before any calculation.
Assuming economic capital equals regulatory capital.
Both are called capital for credit risk.
Fix: Regulatory capital follows Basel rules set by supervisors. Economic capital is the bank's internal estimate at its chosen confidence level and may use different assumptions.
Expecting risk contributions to sum to the sum of stand-alone ULs.
Mixing stand-alone UL with contribution to portfolio UL.
Fix: Contributions sum to portfolio UL, which is lower than stand-alone ULs when ρ < 1.

Worked examples

Example 1

A bank lends $20 million to a borrower with a one-year PD of 2%. Recovery rate is 40%. Find the expected loss and the unexpected loss, assuming LGD is fixed.

Show the solution

LGD = 1 − 0.40 = 0.60.
EL = 0.02 × 0.60 × 20,000,000 = $240,000.
√(PD × (1 − PD)) = √(0.02 × 0.98) = √0.0196 = 0.14.
UL = 20,000,000 × 0.60 × 0.14 = $1,680,000.

Answer: EL = $240,000 and UL = $1,680,000.

Example 2

A portfolio has two loans. Loan A has UL of $3 million and Loan B has UL of $4 million. Default correlation is 0.5. The portfolio has 99.9% credit VaR of $22 million and total EL of $2 million. Find the portfolio UL, the economic capital, and the risk contribution of Loan A.

Show the solution

UL(p) = √(3² + 4² + 2 × 0.5 × 3 × 4) = √(9 + 16 + 12) = √37 ≈ $6.08 million.
Economic capital = 22 − 2 = $20 million.
RC(A) = UL(A) × (UL(A) + ρ × UL(B)) ÷ UL(p) = 3 × (3 + 0.5 × 4) ÷ 6.083 = 3 × 5 ÷ 6.083 ≈ $2.47 million.
Check: RC(B) = 4 × (4 + 0.5 × 3) ÷ 6.083 = 4 × 5.5 ÷ 6.083 ≈ 3.62. Sum = 2.47 + 3.62 = 6.09, equal to UL(p) apart from rounding.

Answer: Portfolio UL ≈ $6.08 million, economic capital = $20 million, and Loan A's risk contribution ≈ $2.47 million.

Exam tips

Read whether the question gives LGD or recovery rate. This is the most common trap.
Check whether the question wants credit VaR or economic capital. They differ by EL.
When correlation is below 1, expect the portfolio UL to be lower than the sum of stand-alone ULs. Use this to eliminate options.
For allocation questions, confirm the contributions add to the portfolio total.
Be ready to explain in words why EL is covered by pricing and provisions, while capital covers UL.

Practice questions from Portfolio Credit Risk

Expected Loss, Unexpected Loss and Economic Capital in other exams

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

FRM Part IExpected and Unexpected Loss

Expected Loss, Unexpected Loss and Economic Capital: frequently asked questions

What is the difference between expected loss and unexpected loss?

Expected loss is the average credit loss, calculated as PD × LGD × EAD. Unexpected loss is the standard deviation of losses around that average. Expected loss is priced and provisioned. Unexpected loss is what capital protects against.

How is economic capital different from regulatory capital?

Regulatory capital is the minimum set by supervisors under Basel rules. Economic capital is the bank's internal estimate of capital needed to cover losses at a chosen confidence level, often linked to its target credit rating. The two can differ in size and method.

How do I calculate economic capital from credit VaR?

Take the portfolio loss at the chosen confidence level, such as 99.9%, and subtract expected loss. The result is the capital held for unexpected losses. Check the question's definition, as some sources use credit VaR directly.

Why does diversification reduce unexpected loss but not expected loss?

Expected losses add across exposures regardless of correlation. Unexpected loss depends on how defaults move together. When default correlation is below 1, some losses offset in timing, so portfolio UL is below the sum of individual ULs.