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FRM Exam Part II · Introduction to Credit Risk Modeling and Assessment

Unexpected Loss and Credit VaR: Formulas and Economic Capital

Updated 11 October 2026 · Fact-checked

Expected loss (EL) is the average credit loss you price and provision for. Unexpected loss (UL) is the standard deviation of losses around that average. Credit VaR is the loss quantile at a confidence level. Economic capital is credit VaR minus EL, the buffer for losses above the average.

Understand Unexpected Loss and Credit VaR

A loan book loses money in two ways. Some default is normal and predictable. You expect a certain average loss every year. This is expected loss (EL). Banks cover it through loan pricing and provisions, so it is a cost of doing business, not a risk to capital.

The real risk is that actual losses differ from the average. Unexpected loss (UL) measures this. It is the standard deviation of the portfolio loss distribution. For a single loan, UL depends on PD, LGD, EAD and how uncertain recovery is. For a portfolio, UL depends on default correlation too. Higher correlation means less diversification and larger UL.

The credit loss distribution is very different from a market return distribution. It is highly skewed, with a long right tail. Most years losses are small. Rarely, many borrowers default together and losses are large. So you cannot use a normal-based multiple of UL with confidence. The tail must be read from the actual or simulated distribution.

Credit VaR is the loss at a chosen confidence level, such as 99.9%, over a horizon, usually one year. It is the quantile of the loss distribution. Economic capital is credit VaR minus EL. The logic is that EL is already covered by pricing and provisions, so capital only needs to absorb the loss above EL, out to the chosen quantile.

Credit VaR differs from market VaR in practice. Credit uses a longer horizon (typically one year versus days), a higher confidence level, a skewed distribution, and illiquid positions that are held, not traded. Data is scarcer, so models are used to build the distribution.

Key formulas to remember

Expected loss (single exposure)
EL = PD × LGD × EAD
LGD is a fraction of EAD. Use the same horizon for PD, usually one year.
Portfolio expected loss
EL_p = Σ EL_i
EL is additive across exposures, regardless of correlation.
Unexpected loss, single exposure (fixed LGD and EAD)
UL = EAD × LGD × √(PD × (1 − PD))
Default is a Bernoulli event, so its standard deviation is √(PD(1−PD)). Assumes LGD is known with certainty.
Unexpected loss with random LGD
UL = EAD × √(PD × σ_LGD² + LGD² × PD × (1 − PD))
Adds recovery uncertainty. σ_LGD is the standard deviation of LGD. Assumes LGD is independent of default.
Portfolio unexpected loss, two exposures
UL_p = √(UL₁² + UL₂² + 2ρ × UL₁ × UL₂)
ρ is the correlation between the two loss outcomes. UL is not additive unless ρ = 1.
Credit VaR
Credit VaR = quantile of the loss distribution at confidence level α
Measured as total loss, including EL, unless the question defines it relative to the mean.
Economic capital
EC = Credit VaR − EL
Also called the unexpected loss at the chosen confidence level. Some questions define EC as the full quantile, so read the wording.

How to solve Unexpected Loss and Credit VaR questions

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

  1. 1Identify what is asked: EL, UL, credit VaR, or economic capital. Note the confidence level and horizon.
  2. 2List the inputs per exposure: PD, LGD, EAD. Check that LGD is a fraction and PD matches the horizon.
  3. 3Compute EL = PD × LGD × EAD for each exposure, then add for the portfolio.
  4. 4Compute UL for each exposure using √(PD(1−PD)) and add the LGD variance term only if LGD volatility is given.
  5. 5Combine ULs for the portfolio with the correlation. Use the square root formula, never a simple sum, unless ρ = 1.
  6. 6Get credit VaR from the loss distribution quantile or the given multiple of UL. Then subtract EL to get economic capital.
  7. 7Check the definition: is the answer total loss or loss above EL? Match the wording in the question.
  8. 8Sanity check: EC should be positive, UL should fall with lower correlation, and EL should not depend on correlation.

Quickest way: Four-line shortcut for capital questions

When to use it: Use when the question gives a UL and a capital multiple, or a quantile and an EL, and asks for economic capital.

  1. Write EL first: PD × LGD × EAD.
  2. If a multiple k of UL is given, capital = k × UL. If a quantile is given, capital = quantile − EL.
  3. For two exposures, use √(a² + b² + 2ρab). Test the answers: with ρ = 0 the result is below a + b.
  4. Eliminate options that add ULs directly or that make EC negative.

Common mistakes in Unexpected Loss and Credit VaR

  • Adding unexpected losses across exposures to get portfolio UL.

    EL is additive, so students assume UL is too.

    Fix: Only EL adds. Combine UL with the correlation formula. Simple addition applies only when ρ = 1.

  • Using a normal multiple such as 2.33 × UL for every credit VaR.

    Habit from parametric market VaR.

    Fix: Credit losses are skewed with a fat right tail. Use the quantile given, or the multiple the question states.

  • Forgetting to subtract EL when computing economic capital.

    Students treat credit VaR as the capital figure.

    Fix: EC = credit VaR − EL when VaR is the total loss quantile. Read how the question defines each term.

  • Using √PD instead of √(PD × (1 − PD)) in single-loan UL.

    Forgetting that default is a Bernoulli variable.

    Fix: Variance of a default indicator is PD(1−PD). Take the square root and multiply by LGD × EAD.

  • Mixing PD horizons, such as a monthly PD with a one-year capital horizon.

    Inputs come from different tables.

    Fix: Convert everything to the credit VaR horizon, normally one year, before computing.

  • Saying credit VaR and market VaR differ only by confidence level.

    Both are called VaR.

    Fix: List the differences: horizon, skewed distribution, illiquid holding, and heavy reliance on models and default correlation.

Worked examples

Example 1

A bank has a USD 10 million loan with PD of 2%, LGD of 40% and no LGD volatility. Compute EL and UL.

Show the solution
  1. EL = 0.02 × 0.40 × 10,000,000 = USD 80,000.
  2. √(PD × (1 − PD)) = √(0.02 × 0.98) = √0.0196 = 0.14.
  3. UL = 10,000,000 × 0.40 × 0.14 = USD 560,000.

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

Example 2

A portfolio has two loans. Loan A has UL of USD 300,000 and loan B has UL of USD 400,000. Loss correlation is 0.5. Portfolio EL is USD 150,000 and the 99.9% credit VaR is USD 2,150,000. Find the portfolio UL and the economic capital.

Show the solution
  1. UL_p² = 300,000² + 400,000² + 2 × 0.5 × 300,000 × 400,000.
  2. = 9.0 × 10¹⁰ + 16.0 × 10¹⁰ + 12.0 × 10¹⁰ = 37.0 × 10¹⁰.
  3. UL_p = √(3.7 × 10¹¹) ≈ USD 608,276.
  4. Economic capital = credit VaR − EL = 2,150,000 − 150,000 = USD 2,000,000.

Answer: Portfolio UL is about USD 608,276 and economic capital at 99.9% is USD 2,000,000.

Exam tips

  • Read whether 'credit VaR' is total loss or loss relative to the mean. This decides whether you subtract EL.
  • Expect conceptual options on why credit VaR differs from market VaR: longer horizon, skewed tail, illiquidity, model dependence.
  • When correlation changes, track UL only. EL does not move with correlation.
  • Do the arithmetic with variances first, then take one square root at the end.
  • Link the answer to meaning: EL is covered by pricing and provisions, EC by capital.

Practice questions from Introduction to Credit Risk Modeling and Assessment

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.

Unexpected Loss and Credit VaR: frequently asked questions

What is the difference between expected loss and unexpected loss?

Expected loss is the average credit loss, equal to PD × LGD × EAD. Unexpected loss is the standard deviation of losses around that average. EL is covered by pricing and provisions, while UL drives the capital requirement.

How do you calculate economic capital for credit risk?

Find the credit VaR at the chosen confidence level from the loss distribution. Subtract expected loss. The result is the capital needed to absorb losses above the average. Always check how the question defines credit VaR.

How is credit VaR different from market VaR?

Credit VaR usually uses a one-year horizon and a very high confidence level such as 99.9%. The loss distribution is skewed with a long right tail, and positions are illiquid. Market VaR typically uses short horizons and is often approximated with a more symmetric distribution.

Why is portfolio unexpected loss less than the sum of individual ULs?

Defaults are not perfectly correlated, so losses on different loans partly offset each other. This is diversification. Only when correlation is 1 does portfolio UL equal the sum of individual ULs.