Key facts
- With fixed exposure and loss severity, expected credit loss equals PD × LGD × EAD.
- PD and LGD are fractions; EAD and the resulting expected loss are currency amounts.
- Loss standard deviation and a loss quantile minus expected loss are different measures of unexpected loss.
- With individual loss distributions fixed, default dependence can change portfolio dispersion but not total expected loss.
- Basel IRB risk-weight functions address unexpected loss; expected losses are compared with eligible provisions separately.
Expected credit loss is the probability-weighted average loss. With fixed exposure and loss severity, . Unexpected loss concerns variation around that average. Distinguish loss standard deviation from a tail-capital amount: a chosen loss quantile minus expected loss. Basel’s internal ratings-based (IRB) approach treats expected and unexpected losses separately. (BCBS, IRB explanatory note) (BCBS, CRE30.1–30.2)
This guide supports study for the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP®). (GARP, FRM Exam information) Our questions complement GARP’s own study materials; they do not replace them. See the course page for context. All numerical examples below use our own illustrative assumptions, not observed default rates or regulatory parameter values.
What do PD, LGD and EAD mean?
| Input | Meaning | Unit |
|---|---|---|
| PD | Probability of default over the stated horizon | Probability |
| LGD | Loss given default: the fraction of exposure lost if default occurs | Fraction |
| EAD | Exposure at default: the amount exposed when default occurs | Currency |
Enter PD and LGD as decimals; multiplying them by EAD gives a currency amount. Keep the horizon consistent. Basel’s PD estimation requirements use one-year default rates. (BCBS, CRE31) (BCBS, CRE36)
In a simple recovery model, LGD equals one minus the recovery rate. The recovery basis matters: Basel measures economic loss, including material discount effects and collection costs. A nominal recovery received later is not necessarily the same as an immediate net recovery. EAD can exceed today’s drawn balance because borrowers may draw further funds around default. (BCBS, CRE36)
Why does the expected-loss formula work?
Consider a default-only model: loss is zero without default. Let equal one on default and zero otherwise. With fixed LGD and EAD, the loss on default is a fixed amount, so the Bernoulli mean gives: (NIST, binomial moments (secondary source))
This is an average across possible outcomes, not the loss predicted for an individual loan. If severity or exposure is random and PD is positive, the law of total expectation gives the more general identity: (Chen, probability moments (secondary source))
Do not automatically multiply unconditional averages: dependence can matter. Multiplying the conditional averages of LGD and EAD gives the conditional expected product only when their conditional covariance is zero. (Chen, probability moments (secondary source)) Basel explicitly requires attention to links between default frequency, recovery and exposure. Its regulatory estimates follow specified conventions, including downturn considerations. (BCBS, CRE36)
Worked example: a credit line’s expected loss
Assume a one-year PD of 2.4%, net recovery of 60%, £500,000 already drawn and £300,000 still available. For this model, fix additional drawings at default at 50% of the undrawn amount. This assumed conversion factor is not a Basel prescription.
The expected loss rate is of EAD. But the actual default loss in this model is £260,000; otherwise it is zero. The £6,240 average is neither possible realised outcome. Using recovery instead of LGD, or using only the current balance, would answer a different calculation.
How do you calculate unexpected loss as standard deviation?
In the dispersion convention, define UL as the standard deviation of loss. A Bernoulli indicator has variance , so our fixed-severity model gives: (NIST, binomial moments (secondary source))
EL locates the mean; this UL measures dispersion around it. Standard deviation is not a worst-case loss, a loss quantile or a capital requirement by itself. With random recovery or exposure, calculate variance from their joint distribution with default. The fixed-severity formula need not apply. (Chen, probability moments (secondary source))
Our single loan has only two possible loss outcomes. It is not normally distributed, so applying a normal multiplier to its standard deviation would not produce a justified credit VaR. The loss distribution, not standard deviation alone, determines the quantile.
Why does portfolio dependence matter?
For two loans, expected losses add whether defaults are independent or dependent, provided each loan’s loss distribution stays fixed. Expanding the variance of their combined loss using the definitions of variance and covariance gives: (Chen, probability moments (secondary source))
Take two loans identical to our example. Total EL is £12,480. With independent defaults, covariance is zero and portfolio UL is √2 times the unrounded single-loan UL: about £56,275. If both loans always default together, portfolio loss doubles in every outcome, making UL about £79,585. EL remains £12,480.
These are different assumptions about the same individual loans. The comparison explains why summing expected losses is straightforward but summing standalone standard deviations generally misstates portfolio dispersion. Dependence can change portfolio variance through covariance; dependence structures with the same covariance give the same variance when the individual loss distributions stay fixed. Independence allows diversification here; identical default outcomes remove that benefit.
What does credit VaR measure?
Define credit VaR here as a quantile of total credit loss over a stated horizon: the smallest loss threshold whose cumulative probability reaches confidence level . (Acerbi and Tasche, expected shortfall)
For a separate illustrative portfolio, assume the following complete one-year loss distribution:
| Loss | Probability | Cumulative probability |
|---|---|---|
| £0 | 90% | 90% |
| £80,000 | 7% | 97% |
| £200,000 | 2% | 99% |
| £600,000 | 1% | 100% |
The one-year 99% VaR is £200,000. If EL is already provided for, the additional amount needed to reach that threshold is: (BCBS, IRB explanatory note)
This tail-capital interpretation of unexpected loss differs from standard deviation. State whether “credit VaR” means the total-loss quantile or the excess over EL before subtracting anything. The £600,000 outcome still exceeds the chosen threshold: a quantile does not describe losses beyond it. For more on quantiles, see our guide to value-at-risk methods. (Acerbi and Tasche, expected shortfall)
Expected loss also differs from expected shortfall. EL averages all outcomes, while expected shortfall averages the worst fraction specified by the confidence level. (Acerbi and Tasche, expected shortfall) In this example, the worst 1% consists entirely of the £600,000 outcome, so 99% expected shortfall is £600,000. The three figures answer different questions: £15,600 is the overall average, £200,000 is the 99% threshold and £600,000 is the average within the worst 1%.
How does this connect to the Basel IRB approach?
The conceptual split carries into IRB: its risk-weight functions address unexpected loss, while expected loss is treated separately. The inputs include PD, LGD, EAD and, where relevant, effective maturity. Depending on the applicable approach, some inputs are bank estimates and others are supervisory values. (BCBS, CRE30.1–30.2) (BCBS, CRE31)
For corporate, sovereign, bank and retail exposures not in default, CRE35 specifies the EL rate as ; multiplying by EAD gives the EL amount. Banks compare total IRB EL with eligible provisions, with the difference subject to capital-treatment rules. This is more precise than saying that capital simply equals . (BCBS, CRE35)
The analogy is to a buffer above average loss. It does not make our assumed 99% quantile or Bernoulli standard deviation a Basel capital calculation. The actual risk-weight functions include correlation parameters and, for relevant exposures, maturity adjustments. (BCBS, CRE31) For revision, identify the horizon, exposure, recovery convention and requested risk measure before calculating. That keeps expected loss, dispersion and tail capital distinct.
Frequently asked questions
What is the expected credit loss formula?
With fixed exposure and loss severity, EL = PD × LGD × EAD. If severity or exposure is random and PD is positive, use PD times the conditional expected product of LGD and EAD given default.
Is LGD the same as the recovery rate?
No. In the simple recovery model, LGD is one minus the recovery rate. Basel’s economic-loss measurement also considers material discount effects and collection costs.
How is unexpected loss calculated for a single loan?
When UL means standard deviation in a default-only model with fixed LGD and EAD, it equals LGD × EAD × √[PD × (1 − PD)]. This is different from a loss quantile minus EL.
Does default dependence change portfolio expected loss?
Not when each loan’s loss distribution stays fixed: expected losses still add. Dependence can change portfolio loss dispersion through covariance, but dependence structures with the same covariance give the same variance.
How does Basel IRB distinguish expected and unexpected losses?
IRB risk-weight functions address unexpected loss. Expected losses are calculated separately and compared with eligible provisions under the capital-treatment rules.
Related tools and pages
Sources
- Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE35.1–35.9 — IRB expected losses, eligible provisions and capital treatment. Accessed 29 October 2026.
- Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE30.1–30.2 — IRB risk components and the separation of expected and unexpected losses. Accessed 29 October 2026.
- Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE31 — IRB risk-weight functions, input units, correlation and maturity. Accessed 29 October 2026.
- Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE36.63, 36.76–36.93 — PD horizon, economic loss, downturn LGD and additional drawings in EAD. Accessed 29 October 2026.
- Basel Committee on Banking Supervision, Bank for International Settlements: An Explanatory Note on the Basel II IRB Risk Weight Functions — §§2 and 4.4, expected loss and capital above the mean (5 July 2005). Accessed 29 October 2026.
- National Institute of Standards and Technology (statistical handbook, secondary source): NIST/SEMATECH e-Handbook §1.3.6.6.18 — Binomial distribution: mean and standard deviation. Accessed 29 October 2026.
- Yen-Chi Chen, University of Washington (lecture notes, secondary source): STAT 535, Lecture 1: Review on Probability and Statistics — expectation, covariance and total expectation. Accessed 29 October 2026.
- Carlo Acerbi and Dirk Tasche, arXiv: On the coherence of Expected Shortfall — arXiv v5; definitions 2.1, 2.2 and 2.6; proposition 3.2 (2 May 2002). Accessed 29 October 2026.
- Global Association of Risk Professionals (GARP): FRM Exam information: format and exam topics. Accessed 26 October 2026.
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