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Account for the problems of using RAROC to measure performance, among them selecting a time horizon, measuring default probability and picking a confidence level.

FRM® Part II · Operational Risk and Resilience · Risk Capital Attribution and Risk-Adjusted Performance Measurement · FRM2.ORR.18.d

The idea

RAROC is only as good as its inputs, and three of them are judgement calls. The horizon, usually one year, may be too short for illiquid positions that cannot be sold or hedged quickly, so capital looks too low and RAROC looks better than reality. Default probabilities are hard to measure. Through-the-cycle averages understate losses in a downturn, point-in-time estimates are procyclical, and defaults cluster, so the tail is fatter than averages suggest. The confidence level, such as 99.9 percent or 99.97 percent to match an AA rating, is a policy choice. Raising it raises economic capital and lowers RAROC, so a project can pass at one level and fail at another. Further problems are that RAROC ignores systematic risk and that diversification is hard to allocate. In an exam, expect to explain how each choice moves capital and flips the accept or reject decision.

Formulas

Expected loss deducted in the RAROC numerator

EL = PD × LGD × EAD

PD
the probability of default over the horizon
LGD
loss given default, as a share of the exposure
EAD
the exposure at default

The PD chosen matters: a point-in-time PD in a downturn gives a larger EL and a lower RAROC than a through-the-cycle PD. Forgetting LGD or the exposure is the usual slip.

Confidence level matching a target rating

CL = 1 − PD

PD
the one-year default probability of the target rating

Used to calibrate economic capital to a rating. A higher confidence level raises capital and lowers RAROC; shifting the decimal place of PD is the usual trap.

A practice question

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

Which of the following is a recognised difficulty in applying RAROC?

  1. RAROC ignores expected loss, so risky loans always look attractive.Why not: False: expected loss is deducted in the numerator.
  2. RAROC can only be calculated for market-risk positions held in the trading book of the bank.Why not: False: RAROC is used for loans, projects and whole units across credit, market and operational risk.
  3. The result depends on judgemental choices such as the horizon and the confidence level. - correct
  4. RAROC cannot be compared with any shareholder required return, because it is a book measure.Why not: False: the comparison with the hurdle rate is exactly how the measure is used.

Horizon, default probability estimates and confidence level are subjective inputs; changing them changes economic capital and therefore RAROC and the accept or reject decision.

A harder one

A bank measures capital over a one-year horizon for a position that it could not sell or hedge for about two years in stressed markets. What is the likely effect on the RAROC?

  1. Capital is understated and RAROC overstated, as the risk persists beyond the horizon. - correct
  2. Economic capital is overstated and RAROC is understated, because the horizon is too long.Why not: Reverses the effect: a horizon shorter than the liquidation period understates risk.
  3. There is no effect, because a constant default probability makes capital independent of the horizon.Why not: Capital is the loss over the horizon, so it changes with the horizon even if annual PD is constant.
  4. RAROC is unchanged because expected loss and capital both scale with the horizon.Why not: Expected loss scales, but unexpected loss does not scale in the same way, and the problem is risk left uncovered.

For illiquid positions the horizon should reflect the time needed to exit or hedge. A one-year window omits losses that can occur during a longer holding period, so capital looks too low and RAROC too high.

The hardest tier

A bank sets each rating's default probability equal to its long-run average default rate and sizes economic capital at 99.9% with these inputs. Defaults in fact cluster in recessions. What is the likely consequence for its RAROC figures?

  1. Capital is overstated and RAROC is understated, because averages include recessions.Why not: Confuses the average with the tail: in a recession the default rate is well above the average.
  2. Only the numerator is affected, so capital and the denominator are unchanged.Why not: Default probabilities drive the loss distribution, so both expected loss and economic capital change.
  3. There is no bias, because expected loss is the same whichever way the average is formed.Why not: True of expected loss but not of the tail: clustering changes unexpected loss and capital.
  4. Tail capital is understated and RAROC overstated, because average rates miss clustering of defaults. - correct

Default rates vary through the cycle and are correlated across borrowers. Average PDs understate the 99.9% tail loss, so economic capital is too low and RAROC too high.

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