FRM Exam Part II · Credit Value at Risk

Credit VaR Fundamentals and the Credit Loss Distribution

Updated 11 October 2026 · Fact-checked

Credit VaR is the loss at a chosen confidence level on a credit portfolio's loss distribution over a horizon, usually one year. Expected loss is the mean loss. Credit VaR minus expected loss gives unexpected loss, which economic capital typically covers. To solve: find the percentile, subtract the mean, interpret.

Understand Credit VaR Fundamentals and Loss Distribution

A bank's credit portfolio has many loans. Most years, few borrowers default and losses are small. In a rare bad year, many borrowers default together and losses are very large. So the loss distribution is right-skewed with a fat right tail. It is not a bell curve.

This is why credit risk differs from market risk. Market returns are often modelled as roughly symmetric, and the horizon is short (a day or ten days). Credit loss distributions are strongly asymmetric, the horizon is usually one year, and the confidence level is very high (often 99.9% for capital). Loans are also usually held and not traded, so there is rarely a price history to build the distribution from. Models are needed instead.

Expected loss (EL) is the mean of the loss distribution. For one exposure, EL = PD × LGD × EAD. A bank treats it as a cost of doing business. It prices it into loan spreads and covers it with provisions or reserves.

Credit VaR is the loss at a given percentile of the distribution, such as the 99.9th. It says: losses should not exceed this amount with that confidence over the horizon. Unexpected loss (UL) in this context is the gap between credit VaR and EL. It is the loss above the average that the bank must hold capital against. Do not confuse it with the standard deviation of losses. Some texts also call that standard deviation unexpected loss, so read the question's definition.

Economic capital is the capital the bank's own model says it needs to absorb unexpected losses at its target confidence level. It is usually credit VaR minus EL. It differs from regulatory capital, which is set by Basel rules. Because the tail is fat, going from 99% to 99.9% raises capital by much more than it would under a normal distribution.

Key formulas to remember

Expected loss (single exposure): EL = PD × LGD × EAD; PD is probability of default over the horizon. LGD is loss given default as a fraction of EAD. EAD is exposure at default.
Expected loss (portfolio): EL(portfolio) = Σ EL(i); Expected losses add up exactly, with no diversification effect and no correlation needed.
Credit VaR: Credit VaR(α) = the α-percentile of the loss distribution; Measured as total loss over the horizon, for example 99.9% over one year.
Unexpected loss (VaR-based): UL = Credit VaR(α) − EL; The tail loss beyond the mean. Economic capital is usually set equal to this.
Economic capital: Economic capital = Credit VaR(α) − EL; Check the question. If it says capital covers the whole VaR, use the VaR. Standard practice is VaR minus EL.
Unexpected loss (standard deviation, single exposure, fixed EAD and LGD): UL = EAD × LGD × √(PD × (1 − PD)); Applies when LGD is a known constant. This is the standard deviation of loss, a different definition of UL.
Portfolio UL is not additive: UL(portfolio) ≤ Σ UL(i); Holds when default correlations are below 1. Diversification reduces portfolio UL.

How to solve Credit VaR Fundamentals and Loss Distribution questions

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

1Read the horizon, confidence level and the definition of UL the question uses (VaR minus EL, or standard deviation).
2List the inputs: PD, LGD, EAD for each exposure, or the percentile given in the loss table or distribution.
3Compute EL. Use PD × LGD × EAD for each exposure and add them.
4Find credit VaR: read the loss at the stated percentile of the distribution. For a table, accumulate probabilities until you reach the confidence level.
5Compute UL or economic capital as credit VaR minus EL, unless told otherwise.
6Check units and scale (₹ crore, USD million, per cent of portfolio).
7Interpret in a sentence: EL is covered by pricing and provisions, UL by capital, and the tail beyond VaR is not covered.

Quickest way: EL first, then percentile, then subtract

When to use it: Use it for any multiple-choice item that gives PD, LGD, EAD or a loss quantile and asks for EL, UL or capital.

Calculate EL = PD × LGD × EAD at once. Many options are traps built from missing one factor.
If a quantile is given, capital = quantile − EL. Do not stop at the quantile.
If only a standard deviation is given, check whether the question wants a multiple of it. Do not assume the distribution is normal.
For a concept question, remember: EL is mean, UL is tail beyond mean, VaR = EL + UL, skewed and fat-tailed.
Eliminate options that add ULs across loans, or that call EL a risk needing capital.

Common mistakes in Credit VaR Fundamentals and Loss Distribution

Treating the credit loss distribution as normal and using a normal multiple of the standard deviation.
Market VaR habits carry over.
Fix: Remember the distribution is skewed with a fat right tail. Use the stated percentile, not a z-score, unless the question says to assume normality.
Setting economic capital equal to credit VaR without subtracting EL.
Students forget EL is already covered by pricing and provisions.
Fix: Use economic capital = credit VaR − EL unless the question defines it otherwise.
Leaving out LGD or EAD when computing EL.
PD is the most familiar input.
Fix: Always write EL = PD × LGD × EAD first and check all three are used.
Adding unexpected losses across loans to get portfolio UL.
EL is additive, so students assume UL is too.
Fix: Only EL is additive. Portfolio UL is lower than the sum when default correlation is below 1.
Using a one-day or ten-day horizon for credit VaR.
Market risk VaR conventions are copied.
Fix: Credit VaR commonly uses a one-year horizon, because loans are held and credit events are slow-moving.
Mixing up the two meanings of unexpected loss.
Texts use UL for both the standard deviation of loss and VaR minus EL.
Fix: Read the question's wording. If it links UL to capital or VaR, use VaR − EL.

Worked examples

Example 1

A bank has a corporate loan with EAD of USD 20 million, PD of 2% over one year and LGD of 45%. The one-year 99.9% credit VaR of the bank's portfolio is USD 180 million and the portfolio's expected loss is USD 30 million. What is the loan's expected loss, and what economic capital does the portfolio need?

Show the solution

Loan EL = PD × LGD × EAD = 0.02 × 0.45 × 20 million.
0.02 × 0.45 = 0.009. 0.009 × 20 million = USD 0.18 million.
Portfolio economic capital = credit VaR − EL = 180 − 30 = USD 150 million.
The USD 30 million expected loss is covered by pricing and provisions, not by capital.

Answer: Loan EL is USD 0.18 million. Portfolio economic capital is USD 150 million.

Example 2

A small credit portfolio has the following one-year loss distribution: loss of ₹0 with probability 90%, ₹10 crore with probability 6%, ₹30 crore with probability 3%, and ₹80 crore with probability 1%. Find the expected loss, the 99% credit VaR, and the unexpected loss using VaR minus EL.

Show the solution

EL = 0.90 × 0 + 0.06 × 10 + 0.03 × 30 + 0.01 × 80.
= 0 + 0.6 + 0.9 + 0.8 = ₹2.3 crore.
Cumulative probability: ₹0 gives 90%, ₹10 crore gives 96%, ₹30 crore gives 99%, ₹80 crore gives 100%.
The 99th percentile is the smallest loss with cumulative probability of at least 99%. That is ₹30 crore.
UL = 30 − 2.3 = ₹27.7 crore.

Answer: EL is ₹2.3 crore, 99% credit VaR is ₹30 crore, and UL is ₹27.7 crore.

Exam tips

Questions often hide EL inside a loss table. Compute the probability-weighted mean before anything else.
Read the confidence level carefully. In a discrete table, VaR is the smallest loss whose cumulative probability reaches that level.
Expect concept items asking why credit VaR differs from market VaR: skew, fat tail, longer horizon, higher confidence and illiquid positions.
Know who covers what: pricing and provisions cover EL, economic capital covers UL, and losses beyond VaR fall on shareholders and creditors.
When a stem says economic capital, check whether it is internal (model-based) or regulatory (Basel rules).

Practice questions from Credit Value at Risk

Credit VaR Fundamentals and Loss Distribution 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 IPortfolio Credit Risk, Credit VaR and Correlation

Credit VaR Fundamentals and Loss Distribution: frequently asked questions

What is the difference between credit VaR and market VaR?

Market VaR usually uses a short horizon, such as one or ten days, on traded positions with price histories and a roughly symmetric return distribution. Credit VaR uses a horizon of about one year, a very high confidence level and a skewed, fat-tailed loss distribution. It usually needs a model, because loans rarely have price histories.

What is the difference between expected loss and unexpected loss?

Expected loss is the average loss you anticipate, equal to PD × LGD × EAD. Unexpected loss is the loss beyond that average, up to a tail level such as the 99.9th percentile. EL is priced and provisioned for. UL is what capital is held against.

How do you calculate credit VaR from a loss distribution?

Pick the confidence level and horizon, then find the loss at that percentile of the distribution. For a table, add probabilities from the smallest loss upward until the cumulative total reaches the confidence level. The loss at that point is credit VaR.

How is economic capital linked to credit VaR?

Economic capital is normally credit VaR minus expected loss at the bank's target confidence level. It is the buffer the bank's own model says it needs to survive unexpected credit losses. It is separate from the regulatory capital required under Basel rules.