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Outline how economic capital for operational risk can be sized, covering where each method is applied and its shortcomings.

FRM® Part II · Operational Risk and Resilience · Risk Measurement and Assessment · FRM2.ORR.4.f

The idea

Economic capital for operational risk can be sized in several ways. The loss distribution approach combines a frequency distribution, often Poisson, with a severity distribution, often lognormal, to simulate annual losses, and reads capital from a high quantile such as 99.9%. It suits areas with plentiful data, but its tail is extrapolated far beyond the sample and is unstable. Scenario-based methods use expert judgement for severe events and fit data-poor areas, but they are subjective and open to anchoring bias. Scorecards rate risk drivers and controls and are forward-looking, but they are subjective and weakly calibrated to losses. Let a and b be the stand-alone capital of two units, and ρ the correlation between their losses. Combined capital for the two then follows the square-root rule, in which Capital = √(a² + b² + 2ρab). Simply adding the two figures assumes perfect correlation, so the square-root result is lower whenever ρ is below 1. Expect to match a method with its data situation.

Formulas

Expected annual loss with Poisson frequency and lognormal severity

EL = λ × e^(μ + σ² ÷ 2)

λ
the mean number of loss events a year
μ
the mean of the log of the loss
σ
the standard deviation of the log of the loss

The lognormal mean is above its median e^(μ) because of the right skew. Using the median, dropping the halving of σ², or using σ instead of σ² are the traps.

Aggregating two capital charges with a correlation

Total capital = √(C₁² + C₂² + 2 × ρ × C₁ × C₂)

C₁
the stand-alone capital charge for the first risk
C₂
the stand-alone capital charge for the second risk
ρ
the correlation between the two risks

Below the simple sum unless the correlation is 1; adding the unit figures assumes perfect correlation. Subtracting the cross term or dropping it altogether are the slips.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the loss distribution approach (LDA) to operational risk, the aggregate annual loss distribution is built by combining:

  1. a distribution of risk scores with a distribution of control ratings.Why not: Mixes up the LDA with scorecards, which work with scores rather than loss distributions.
  2. a market risk VaR with a credit risk VaR, scaled by a fixed operational add-on.Why not: Combines unrelated risk types; the LDA uses operational loss frequency and severity.
  3. a frequency distribution for the number of events with a severity distribution for each loss. - correct
  4. the expected loss with the firm's total equity, divided by the number of business lines in the group.Why not: Invents a ratio that is not a distribution of losses.

LDA models how often losses occur (often Poisson) and how large each is (often lognormal or heavy-tailed), then simulates the annual sum. Capital is read from a high quantile, such as 99.9%, of that aggregate distribution.

A harder one

A bank models operational losses in one business line with Poisson frequency with mean 12 events a year and lognormal severity with mu = 10 and sigma = 1.5 (parameters of the log of the loss, in USD). What is the expected annual loss, closest to?

  1. USD 264 thousandWhy not: Uses the median severity exp(mu) instead of the lognormal mean exp(mu + sigma² / 2).
  2. USD 2,508 thousandWhy not: Uses exp(mu + sigma²) and forgets to halve the variance term.
  3. USD 560 thousandWhy not: Uses sigma / 2 instead of sigma² / 2 in the exponent.
  4. USD 814 thousand - correct

Mean severity = exp(10 + 1.5² / 2) = exp(11.125) = 67,846. Expected annual loss = frequency x mean severity = 12 x 67,846 = 814,155. The lognormal mean is well above its median because of the right skew.

The hardest tier

A bank fits a lognormal severity to five years of internal data (about 120 loss events) and sets 99.9% capital from the simulated aggregate distribution. One additional large loss in the next year raises the capital estimate by 35%. Which shortcoming does this best illustrate?

  1. The method ignores the frequency of losses, which should be the main input.Why not: Wrongly says frequency is ignored; the LDA models frequency explicitly.
  2. The method is purely qualitative, so any change in judgement alters the result.Why not: Describes scorecards or scenario judgement; the LDA is quantitative.
  3. The method cannot be applied to operational risk because loss data are normally distributed and symmetric.Why not: Mischaracterises the problem; operational losses are heavy-tailed, not normal, and the LDA does not assume normality.
  4. The 99.9% quantile extrapolates far beyond the sample, so a few tail points dominate the estimate. - correct

A 99.9% annual quantile is a one-in-1,000-year event, estimated from five years of data. A single large loss changes the fitted tail sharply, so estimates are unstable. Remedies are external data, scenarios and tail methods.

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