sol9 - ACTSC 445 Asset-Liability Management Fall 2008...

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ACTSC 445: Asset-Liability Management – Fall 2008 Department of Statistics and Actuarial Science, University of Waterloo Solutions for Unit 9 Exercises 1. We have that the variance of the overall return on the portfolio, σ 2 V , is equal to 6 . 5 × 10 - 5 . Therefore VaR 0 . 99 , 5 = 5 × 200 000 × 6 . 5 × 10 - 5 × 2 . 33 = 8400 . 93. The sum of the VaR of the two portfolios is 2( 5 × 100 , 000 × 0 . 01) × 2 . 33 = 10420 . 08, so diversification reduces VaR by 2019.14. 2. Can write dV = ∂V ∂f 1 f 1 + ∂V ∂f 2 f 2 where f 1 and f 2 are not correlated since they come from a principal components analysis. The in- formation in the problem tells us that dV = 6 f 1 - 4 f 2 with a variance of 20 2 and 8 2 for f 1 and f 2 , respectively. Therefore, Var( dV ) = σ 2 V = (6 × 20) 2 + (4 × 8) 2 , and so σ V = 124 . 19. Therefore, VaR 0 . 90 , 5 = 5 × 124 . 19 × 1 . 282 = 356 . 02. 3. We have that L = V 0 - V = V 0 - V 0 (1 + R ) = - RV 0 . Therefore CTE = E( - RV 0 | - RV 0 > σV 0 z α ) = E( - RV 0 | - R > σz α ) = - V 0 1 P ( - R > σz α ) - σz α -∞ r e - r 2 / 2 σ 2 2 πσ dr = V 0 σ 2 π (1 - α ) - σz α -∞ - r σ 2 e - r 2 / 2 σ 2 dr = V 0 σ 2 π (1 - α ) e - z 2 α / 2 .
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