FRM Exam Part II · Fundamentals of Credit Risk

Expected vs Unexpected Loss and Credit VaR Explained

Updated 11 October 2026 · Fact-checked

Expected loss (EL) is the average credit loss, equal to PD × LGD × EAD. Unexpected loss (UL) is the standard deviation of losses around that average. Credit VaR is the loss at a chosen confidence level, and economic capital is credit VaR minus EL. To solve questions, find EL, then UL or the quantile, then subtract.

Understand Expected vs Unexpected Loss and Credit VaR

Every loan carries a chance of default. The average loss you expect over a year is the expected loss (EL). Banks treat EL as a cost of doing business. They price it into loan spreads and cover it with provisions. EL is not a risk in the sense of surprise, because it is anticipated.

The real risk is that actual losses differ from EL. This variability is unexpected loss (UL). In most exam questions, UL is the standard deviation of the loss distribution. For a single loan, losses are either zero (no default) or a large amount (default), so the variance depends on the default probability and on how uncertain the loss given default is.

Credit loss distributions are not normal. They are right-skewed with a fat right tail. Most years show small losses. Rarely, many defaults cluster and losses are very large. Because of this, a multiple of UL is not a reliable tail measure, and models use the actual quantile of the distribution.

Credit VaR is the loss at a chosen confidence level, for example 99.9% over one year. It is the quantile of the loss distribution. Some sources measure credit VaR from the mean, so it is the quantile minus EL. Read the question carefully for which definition it uses.

Economic capital is the buffer a bank holds against unexpected losses. It equals credit VaR at the target confidence level minus EL. EL is covered by provisions and pricing. Capital covers the gap between EL and the tail loss. The confidence level is usually linked to the bank's target credit rating. A higher target rating means a higher confidence level and more capital.

Key formulas to remember

Expected loss (single exposure): EL = PD × LGD × EAD; LGD = 1 − recovery rate. Use the same time horizon for PD, usually one year.
Unexpected loss, fixed LGD: UL = EAD × LGD × √[PD × (1 − PD)]; Standard deviation of loss for one loan when LGD is known with certainty.
Unexpected loss, random LGD: UL = EAD × √[PD × σ_LGD² + LGD² × PD × (1 − PD)]; Assumes default and LGD are independent. σ_LGD is the standard deviation of LGD.
Portfolio expected loss: EL_P = Σ EL_i; Expected losses add up directly. No diversification effect.
Portfolio unexpected loss: UL_P = √(Σ Σ ρ_ij × UL_i × UL_j); Diversification arises when default correlation ρ is below 1. For two loans: UL_P² = UL₁² + UL₂² + 2ρUL₁UL₂.
Credit VaR: Credit VaR = quantile of the loss distribution at confidence level α; Some definitions measure it as the quantile minus EL. Check the question.
Economic capital: EC = Credit VaR(α) − EL; Where credit VaR is the absolute loss quantile. Capital covers unexpected loss only.
Normal approximation of capital: EC ≈ k × UL_P; k is the multiplier, e.g. 2.33 at 99%. Only a rough guide, because credit losses are skewed.

How to solve Expected vs Unexpected Loss and Credit VaR questions

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

1Write down the inputs: EAD, PD, LGD or recovery rate, and the horizon. Convert recovery to LGD with LGD = 1 − recovery.
2Compute EL = PD × LGD × EAD.
3Compute UL. Use the fixed-LGD formula unless the question gives a standard deviation of LGD.
4For a portfolio, add ELs directly. Combine ULs with the correlation formula, never by simple addition unless ρ = 1.
5Find credit VaR. It is either given as a quantile, or approximated as EL plus a multiple of UL under a stated normal assumption.
6Compute economic capital = credit VaR − EL. Check whether the question already measures VaR from the mean.
7Interpret the result. State what is covered by pricing and provisions and what is covered by capital, and note the skewed distribution if relevant.

Quickest way: Fast path: EL, UL, then subtract

When to use it: Use when the question gives a single loan or a simple two-asset portfolio with all inputs stated.

Compute EL first. Often an option equals EL, so do not stop there.
For fixed LGD, compute √[PD(1 − PD)] and multiply by LGD × EAD.
For two loans, use the square-root rule UL_P² = UL₁² + UL₂² + 2ρUL₁UL₂. Check ρ = 0 and ρ = 1 as bounds.
Compute capital as multiplier × UL, then reconcile with the options. If the question gives VaR in absolute terms, subtract EL.

Common mistakes in Expected vs Unexpected Loss and Credit VaR

Treating EL as a risk that needs capital.
EL looks like a large loss number, so it feels like something to hold capital against.
Fix: EL is covered by pricing and provisions. Capital covers only the unexpected part, credit VaR minus EL.
Using UL = EAD × LGD × PD.
It is confused with the EL formula.
Fix: UL needs the square root of PD × (1 − PD). Check that PD appears under a root.
Adding the ULs of two loans to get portfolio UL.
ELs add, so students assume ULs do too.
Fix: ULs add only when ρ = 1. Use the correlation formula. Lower correlation gives a lower portfolio UL.
Assuming credit losses are normal and using 2.33 × UL as the 99% loss.
Market VaR often uses the normal distribution.
Fix: Credit losses are right-skewed with a fat tail, so the normal multiple usually understates the true quantile. Use it only when the question tells you to.
Forgetting to convert recovery rate to LGD.
The question gives recovery, not LGD, and the formula needs LGD.
Fix: Always compute LGD = 1 − recovery before anything else.
Subtracting EL when VaR is already measured from the mean, or failing to subtract when it is absolute.
Definitions of credit VaR differ between sources.
Fix: Check whether the quantile is an absolute loss or a deviation from EL, then apply EC = absolute VaR − EL once.

Worked examples

Example 1

A bank lends USD 10 million to a firm. One-year PD is 2%, recovery rate is 60% and LGD is treated as certain. Compute EL and UL.

Show the solution

LGD = 1 − 0.60 = 0.40.
EL = 0.02 × 0.40 × 10,000,000 = USD 80,000.
√[PD × (1 − PD)] = √(0.02 × 0.98) = √0.0196 = 0.14.
UL = 10,000,000 × 0.40 × 0.14 = USD 560,000.

Answer: EL = USD 80,000 and UL = USD 560,000.

Example 2

A portfolio has an expected loss of USD 4 million and a loss standard deviation (UL) of USD 9 million. A bank approximates the 99.9% loss quantile as EL plus 3.09 × UL. Compute the credit VaR and economic capital.

Show the solution

Credit VaR (absolute) = 4 + 3.09 × 9 = 4 + 27.81 = USD 31.81 million.
Economic capital = credit VaR − EL = 31.81 − 4 = USD 27.81 million.
Check: this equals 3.09 × UL = 3.09 × 9 = 27.81 million.
Note the weakness: actual credit losses are skewed with a fat tail, so the true 99.9% quantile is likely above this normal approximation.

Answer: Credit VaR is USD 31.81 million and economic capital is USD 27.81 million.

Exam tips

Always compute EL even if the question asks for UL. Options often include EL as the trap answer.
Read the definition of credit VaR in the stem. Whether it is measured from zero or from the mean changes the capital answer.
When a question mentions skewness or fat tails, the expected conclusion is that a normal multiple of UL understates tail loss.
For two-loan portfolios, test the bounds: ρ = 1 gives the sum of ULs, ρ = 0 gives the root of the sum of squares.
Link the confidence level to the bank's target rating when the question asks why capital is set at 99.9% or similar.

Practice questions from Fundamentals of Credit Risk

Expected vs Unexpected Loss and Credit VaR 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 vs Unexpected Loss and Credit VaR: frequently asked questions

What is the unexpected loss formula for a single loan?

With a fixed LGD, UL = EAD × LGD × √[PD × (1 − PD)]. If LGD is random and independent of default, add the LGD variance term: UL = EAD × √[PD × σ_LGD² + LGD² × PD × (1 − PD)].

What is the difference between credit VaR and economic capital?

Credit VaR is the loss quantile at a chosen confidence level. Economic capital is that quantile minus expected loss, because EL is covered by pricing and provisions. Always check whether the question measures VaR from zero or from the mean.

Why is the credit loss distribution skewed?

Most years bring few defaults and small losses. Occasionally defaults cluster, producing very large losses. This gives a long right tail, so the mean is below the tail quantiles and normal approximations understate extreme losses.

Does diversification reduce expected loss?

No. Portfolio EL is the sum of individual ELs. Diversification lowers unexpected loss and economic capital when default correlations are below 1.