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FRM Part II · FRM Exam Part II · Risk Capital Attribution and Risk-Adjusted Performance Measurement

A bank allocates its USD 600 million firm-level economic capital to three units using the contribution of each unit to total portfolio standard deviation (risk contribution method). Standalone standard deviations of loss are: A 40, B 50, C 30 (USD million). The correlations between units are all 0.5. Unit A's covariance-based contribution is its weight in portfolio variance: contribution_A = sigma_A x (sum over j of sigma_j x rho_Aj) / sigma_P. What capital is allocated to A?

Unit A is allocated about USD 200 million. Its covariance with the portfolio is 3,200 out of total variance of 9,700, a 33 percent share, applied to USD 600 million of capital. Variance-based risk contributions sum exactly to the firm total, unlike standalone allocations.

  1. AUSD 150 million
  2. BUSD 200 million
  3. CUSD 216 millionCorrect
  4. DUSD 240 million

Explanation

Variance = 40²+50²+30² + 2(0.5)(40·50+40·30+50·30) = 1600+2500+900 + (2000+1200+1500) = 5000+4700 = 9700; sigma_P = 98.49. Cov of A with portfolio = 1600 + 0.5(2000+1200) = 3200. Contribution A = 3200/98.49 = 32.49. Contributions: B = (2500+0.5(2000+1500))/98.49 = 4250/98.49 = 43.15; C = (900+0.5(1200+1500))/98.49 = 2250/98.49 = 22.85. Sum = 98.49. A share = 3200/9700 = 32.99%. Capital = 0.3299 x 600 = USD 197.9 million, so closest is 200.

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