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Explain the derivation of economic capital.

FRM® Part II · Credit Risk Measurement and Management · Capital Structure in Banks · FRM2.CR.4.g

The idea

Economic capital is derived from the portfolio loss distribution. First pick a confidence level tied to the target credit rating, say 99.9 percent over one year. Then read off the loss that is exceeded only with the remaining probability. Finally subtract expected loss, because EL is covered by pricing and provisions. Writing the loss quantile as Q, the capital left after subtracting expected loss is EC = Q − EL. In practice it is often expressed as a multiple of UL, where k is the capital multiplier. The relationship is EC = k × UL. For 99.9 percent, k is about 3.09 if losses were normal, but often substantially larger for the skewed, fat-tailed credit loss distribution. The multiplier is therefore found from the actual distribution, not from the normal table. Total capital is allocated to assets through their UL contributions.

Formulas

Economic capital from a loss distribution

EC = Q − EL

Q
the loss at the chosen confidence level, such as the 99.97th percentile over one year
EL
the expected loss

Expected loss is covered by pricing and provisions, so it is subtracted. Reporting the quantile itself is the usual error.

Economic capital as a multiple of UL

EC = k × UL

k
the capital multiplier for the chosen confidence level
UL
the portfolio unexpected loss

Take k from the actual skewed loss distribution; the normal value, about 3.09 at 99.9 percent, understates it.

A practice question

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

How is economic capital for credit risk derived from the credit loss distribution?

  1. It is the loss at the chosen confidence level minus the expected loss. - correct
  2. It is the expected loss plus one standard deviation.Why not: Confuses economic capital with a one-sigma loss and ignores the confidence level.
  3. It is the loss at the chosen confidence level, with no adjustment for expected loss.Why not: Leaves expected loss in, although it is covered by pricing and provisions.
  4. It is the average of all losses in the distribution.Why not: Describes the mean loss, which is EL, not capital.

Economic capital = quantile loss - EL, since EL is covered by pricing and reserves and capital absorbs the unexpected portion up to the confidence level.

A harder one

Credit losses are approximated as normal with expected loss USD 30 million and standard deviation (UL) USD 50 million. Economic capital, defined as the loss quantile less the expected loss, at 99.9 percent is closest to (z for 99.9 percent one-sided is 3.090):

  1. USD 184.5 million.Why not: Adds the expected loss, which gives the loss quantile rather than economic capital.
  2. USD 164.5 million.Why not: Uses a two-sided critical value of 3.291; the tail is one-sided.
  3. USD 154.5 million. - correct
  4. USD 116.3 million.Why not: Uses the 99 percent value of 2.326 instead of the 99.9 percent value.

Economic capital = z x UL = 3.090 x 50 = 154.5 million; the quantile would be EL + z x UL = 184.5, from which EL is subtracted.

The hardest tier

A bank sets economic capital at 8 times portfolio UL and allocates it to assets by their UL contributions. Asset A has a stand-alone UL of USD 3 million, asset B USD 4 million, and the correlation is 0.5. The economic capital allocated to asset A is closest to:

  1. USD 19.7 million. - correct
  2. USD 24.0 million.Why not: Multiplies the stand-alone UL by the capital multiple, ignoring diversification.
  3. USD 20.9 million.Why not: Allocates total capital in proportion to stand-alone ULs.
  4. USD 27.6 million.Why not: Treats the correlation as one in the contribution term.

UL(P) = sqrt(9+16+12) = 6.083; total EC = 8 x 6.083 = 48.7. ULC(A) = 3 x (3+0.5x4)/6.083 = 2.466; EC(A) = 19.7.

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