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Account for the difficulties of modeling diversification benefits, in particular summing up a firm's risk capital and spreading economic capital among its business lines.

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

The idea

Diversification makes the firm's economic capital less than the sum of the business lines' stand-alone capital, because losses rarely all arrive together. Adding stand-alone figures assumes perfect correlation and overstates capital. With two units A and B and correlation ρ, firm capital under normal-style scaling is written Capital. Capital = √(A² + B² + 2 × ρ × A × B). The diversification benefit, written Benefit, is the shortfall. Benefit = A + B − Capital. The harder problem is spreading the benefit back to the lines. Incremental capital (firm with the unit less firm without it) understates each line, and the pieces do not sum to the total. Pro rata scaling is simple but ignores which unit really diversifies. Covariance or marginal allocation sums to the total but moves when other units change. Correlations also rise in stress, so the benefit shrinks when it is needed most. In an exam, link each technique to its weakness.

Formulas

Firm capital for two units with correlated losses

Capital = √(A² + B² + 2 × ρ × A × B)

A
the stand-alone capital of unit A
B
the stand-alone capital of unit B
ρ
the correlation between the two units' losses

Holds when capital scales like a standard deviation. Equals A + B only when ρ is 1; correlations rising in stress push it back towards the simple sum.

Diversification benefit

Benefit = A + B − Capital

A
the stand-alone capital of unit A
B
the stand-alone capital of unit B
Capital
the diversified firm capital

The shortfall of firm capital below the sum of stand-alone figures. Reporting the firm capital instead of the benefit is the usual trap.

Covariance allocation of firm capital to unit A

Allocation(A) = A × (A + ρ × B) ÷ Capital

A
the stand-alone capital of unit A
B
the stand-alone capital of unit B
ρ
the correlation between the two units' losses
Capital
the diversified firm capital

The unit allocations sum to firm capital, unlike incremental charges, but A's share moves when B changes its positions.

A practice question

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

Why is the firm's economic capital normally less than the sum of its business lines' stand-alone capital?

  1. Because the lines' losses are not perfectly correlated, so they rarely peak together. - correct
  2. Because stand-alone capital is measured at a higher confidence level than firm capital.Why not: All figures use the same confidence level; the difference arises from correlation.
  3. Because expected losses are deducted from firm capital but not from stand-alone capital.Why not: Both measures are net of expected loss on the same basis.
  4. Because regulators allow banks to hold less than the sum of the line requirements.Why not: Describes a supervisory allowance, not the statistical reason behind the diversification benefit.

Summing stand-alone capital implicitly assumes a correlation of one. With less-than-perfect correlation the diversified firm total is lower and the difference is the diversification benefit.

A harder one

Two business lines have stand-alone economic capital of USD 60 million and USD 80 million. Their losses have a correlation of 0.3, and firm capital scales like a standard deviation. What is the diversification benefit?

  1. USD 113.5 millionWhy not: Reports the firm's diversified capital, not the benefit.
  2. USD 40.0 millionWhy not: Assumes zero correlation (square root of the sum of squares = 100).
  3. USD 26.5 million - correct
  4. USD 55.6 millionWhy not: Uses a correlation of -0.3 in the formula instead of +0.3.

Firm capital = sqrt(60² + 80² + 2 x 0.3 x 60 x 80) = 113.5. Benefit = 140 - 113.5 = 26.5 million.

The hardest tier

A bank's model uses a correlation of 0.2 between two units with stand-alone capital of USD 90 million and USD 50 million, giving firm capital of USD 111.4 million. In a crisis, the correlation rises to 0.8. Closest to what increase in firm capital would the crisis require?

  1. USD 6.6 millionWhy not: Reports the diversification benefit that remains at a correlation of 0.8.
  2. USD 22.1 million - correct
  3. USD 28.6 millionWhy not: Reports the whole benefit at 0.2, as if it all disappeared.
  4. USD 133.4 millionWhy not: Reports the new level of firm capital instead of the increase.

Capital at 0.8 = sqrt(90² + 50² + 2 x 0.8 x 90 x 50) = 133.4. Increase = 133.4 - 111.4 = 22.1 million. The benefit shrinks from 28.6 to 6.6 when it is needed most.

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