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FRM Part 2 · Worked study notes

Credit Exposure and Credit VaR

Credit exposure measures what can be lost if a counterparty fails; credit VaR is a quantile of a specified credit-loss distribution. Define whether the reported measure is total quantile loss or quantile loss minus expected loss before comparing figures.

The method at a glance

Credit loss: average versus tail

Illustrative comparison · thousand · bars start at zero

  • Expected loss12
  • 99% total loss VaR600
  • 99% VaR minus EL588

One-year exposure of 1 million, PD 2%, LGD 60%. The distribution has only two outcomes: zero or 600,000.

  • 01Exposure

    How it works: Specify drawn amounts, potential drawings, collateral and netting.

    Watch for: Using notional as exposure without examining the contract.

  • 02Loss distribution

    How it works: Model default, recovery and dependence over the chosen horizon.

    Watch for: Applying a symmetric normal shape to a highly skewed loan loss.

  • 03Convention

    How it works: Distinguish VaR(loss) from VaR(loss) - E[loss].

    Watch for: Comparing a total loss quantile with unexpected loss as though identical.

Original worked example

A two-outcome credit distribution

A one-year exposure of 1 million has PD 2% and fixed LGD 60%. Loss is zero with probability 98% and 600,000 with probability 2%. Use the smallest loss whose cumulative probability reaches the confidence level.

  1. 1

    Expected loss = 0.02 × 600,000 = 12,000.

  2. 2

    At 95%, cumulative probability at zero is already 98%, so VaR is zero.

  3. 3

    At 99%, the required quantile is 600,000; subtracting EL gives 588,000.

99% total loss VaR is 600,000; the corresponding VaR-minus-EL figure is 588,000. The definition changes the reported number.

Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.

Apply the method and check its limits

Before you start: Discrete loss distributions and cumulative probabilities.

Build the cumulative distribution before reading VaR

For the single loan, cumulative probability is 98% at loss zero and 100% at loss 600,000. The smallest loss reaching 95% confidence is zero; the smallest reaching 99% is 600,000. Quantiles can jump abruptly as confidence changes when the distribution has probability masses. Do not interpolate a smooth normal curve into this stated two-outcome model.

Distinguish loss dispersion from a chosen capital convention

Expected loss is the probability-weighted mean. Unexpected loss is sometimes used for a standard deviation and sometimes for a tail quantile minus expected loss, depending on the convention. A reported “credit VaR” must identify the definition, horizon and confidence level. Subtracting EL changes the number but does not remove concentration or default dependence.

Why concentration matters

Pooling many small independent loans can smooth losses, but common industry or macroeconomic factors create correlated defaults. Recovery can also deteriorate when defaults increase. Fixed-LGD examples are useful for understanding mechanics; a realistic portfolio analysis must consider whether default, exposure and recovery worsen together.

Work through the topic

Build a discrete credit-loss distribution, calculate its mean and quantile, and explain how dependence changes the capital interpretation without changing every input.

Define the loss unit and default states

A credit-loss model needs a horizon, exposure definition, default event and loss severity. For a default-only model, a surviving loan may have zero modeled credit loss even if its market spread changes. That is a modeling boundary, not a claim that its market value cannot change. Decide whether migration and mark-to-market effects belong in the question before constructing the distribution.

Take two loans, each with exposure 200, loss given default 50% and one-year default probability 10%. Each default therefore produces a loss of 100. Assume independence, fixed exposures and deterministic recoveries. The portfolio has three possible loss totals: zero, 100 or 200. The units could be thousands of currency units, but the same unit must be used throughout.

No default has probability 0.9² = 81%. Exactly one default has probability 2 × 0.1 × 0.9 = 18%. Both default with probability 0.1² = 1%. Check that the probabilities sum to one before calculating a risk measure. A missing state or double-counted single-default outcome will distort both the mean and the tail.

Original two-loan default-only distribution

0
Probability: 81%
Cumulative probability: 81%
100
Probability: 18%
Cumulative probability: 99%
200
Probability: 1%
Cumulative probability: 100%

Calculate expected loss and state the VaR convention

Expected loss is 0 × 0.81 + 100 × 0.18 + 200 × 0.01 = 20. It also equals the sum of the two standalone expected losses, each 0.10 × 200 × 0.50 = 10. This reconciliation checks the arithmetic and clarifies that expectation adds even though tail aggregation requires the joint distribution.

Using the lower-quantile definition, 95% VaR is 100 because cumulative probability first reaches at least 95% at that loss. Some credit-risk discussions call the full quantile credit VaR; others report unexpected loss as the quantile minus expected loss. Under the latter convention, the figure is 100 - 20 = 80. Label the result explicitly instead of treating the two figures as competing answers.

Expected shortfall asks a different question. The worst 5% contains the 1% probability of losing 200 and 4 percentage points of the probability of losing 100. Thus 95% ES is (0.01 × 200 + 0.04 × 100)/0.05 = 120. This is a useful check that a loss quantile does not describe the average severity beyond its cutoff.

Change dependence while preserving marginal default probabilities

Now replace independence with perfect common default: both loans default together with probability 10% and otherwise both survive. Each loan still has the same 10% PD, exposure and LGD. Portfolio expected loss remains 20. However, the portfolio now loses either zero with probability 90% or 200 with probability 10%, so its 95% VaR becomes 200.

Under a quantile-minus-mean convention, unexpected loss becomes 180 instead of 80. The change comes from joint default behavior, not from a higher sum of expected losses. This deliberately simple case demonstrates why a portfolio model needs dependence assumptions and why a borrower-level PD report cannot by itself establish diversification.

Real models also face uncertainty about exposure at default, recoveries, common factors and the estimation sample. Borrowers can draw facilities before default, and collateral values can decline during the same downturn. Stress the joint mechanisms rather than perturbing each input independently and assuming that the combined result captures all interactions. Any economic-capital use needs an explicit horizon, confidence level and treatment of expected loss.

Why can expected loss stay constant when tail risk rises?

Expectation depends on the average loss of each position, while portfolio tail risk depends on which losses occur together. Changing dependence can concentrate losses into fewer, more severe states without changing each position’s marginal expected loss.

Is the quantile-minus-mean figure automatically regulatory capital?

No. It is a defined statistical quantity in this example. Regulatory capital uses an applicable framework with its own exposure, parameter and aggregation rules. Do not substitute a simplified portfolio quantile for those requirements.

Try a changed assumption

If PD rises from 2% to 6% with fixed exposure and LGD, what happens to 95% VaR?

Show answer and reasoning

The no-loss probability falls to 94%, so 95% VaR becomes 600,000. EL rises to 36,000. The quantile jumps even though PD changes continuously.

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Why can 95% VaR be zero for a risky loan?

If the probability of no loss exceeds 95%, that quantile can be zero despite a severe default loss. Expected loss, higher quantiles and tail measures provide complementary information.

Sources and scope

Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.

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