m48-ch25 - Chapter 25 Value at Risk Question 25.1. Since...

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Chapter 25 Value at Risk Question 25.1. Since the price of stock A in h years ( S h ) is log-normal, P (S h <S 0 ) = P µ α 1 2 σ 2 h + σ hZ < 0 (1) = P ³ Z< ³ σ 2 α σ ´ h ´ = N ³ σ 2 α σ ´ h ´ . (2) Using the parameters and h = 1 / 365 this is N ( . 01832 ) = . 4927. If we use h = 1 / 252 the value would be N ( . 2205 ) = . 4912. Question 25.2. A95% VaR uses Z 1 =− 1 . 645 and the 99% VaR uses Z 2 2 . 326. Given the horizon h (in years), the value of 10 million will be 10 e ( α σ 2 / 2 ) h + σ hZ i million. Table One shows these values as well as the loss (VaR). Table One (Problem 25.2) 95% Values 1 day 10 day 20 day A 9,747,824 9,242,241 8,960,529 B 9,622,055 8,866,025 8,445,521 Loss (VaR) 1 day 10 day 20 day A 252,176 757,759 1,039,471 B 377,945 1,133,975 1,554,479 99% Values 1 day 10 day 20 day A 9,644,067 8,934,714 8,541,801 B 9,468,836 8,427,213 7,860,500 Loss (VaR) A 355,933 1,065,286 1,458,199 B 531,164 1,572,787 2,139,500 Question 25.3. Using the normal approximation, the portfolio will have a mean return of α p = 16 . 8% and standard deviation of σ p = 32 . 17%. Letting h be the holding period, there is a 95% (or 99%) chance the 315
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Part 5 Advanced Pricing Theory value of the portfolio will exceed $10m × h 1 + α p h + σ p hZ i i (3) where Z 1 =− 1 . 645 (95%) and Z 2 2 . 326. See Table Two for the numerical answers. Table Two (Problem 25.3) Values 1 day 10 day 20 day 95% 9,727,621 9,170,135 8,853,356 99% 9,612,863 8,807,239 8,340,143 Loss (VaR) 1 day 10 day 20 day 95% 272,379 829,865 1,146,644 99% 387,137 1,192,761 1,659,857 Question 25.4. The portfolio mean is α p = 16 . 3% and the standard deviation is σ p = 28 . 65%. Letting h be the holding period, there is a 95% (or 99%) chance the value of the portfolio will exceed $10m × h 1 + α p h + σ p hZ i i (4) where Z 1 1 . 645 (95%) and Z 2 2 . 326 (99%). See Table Three for the numerical answers. Table Three (Problem 25.4) Values 1 day 10 day 20 day 95% 9,757,785 9,264,584 8,986,124 99% 9,655,581 8,941,387 8,529,054 Loss (VaR) 1 day 10 day 20 day 95% 242,215 735,416 1,013,876 99% 344,419 1,058,613 1,470,946 Question 25.5. See Table Four for the numerical answers. Risk is not eliminated at ρ 1 for the portfolio volatility is σ p = 15%. If we had 60% in A and 40% in B, then σ p = q . 6 2 (. 09 ) + . 4 2 (. 45 ) 2 2 (. 6 )(. 4 3 45 ) = 0. (5) 316
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Chapter 25 Value at Risk Table Four (Problem 25.5) ρ =Ð1 w/ mean w/o mean Values 1 day 10 day 20 day Values 1 day 10 day 20 day 95% 9,875,459 9,637,640 9,514,508 95% 9,870,857 9,591,613 9,422,454 99% 9,821,953 9,468,439 9,275,222 99% 9,817,351 9,422,412 9,183,167 Loss (VaR) 1 day 10 day 20 day Loss (VaR) 1 day 10 day 20 day 95% 124,541 362,360 485,492 95% 129,143 408,387 577,546 99% 178,047 531,561 724,778 99% 182,649 577,588 816,833 ρ = Ð0.5 w/ mean w/o mean Values 1 day 10 day 20 day
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m48-ch25 - Chapter 25 Value at Risk Question 25.1. Since...

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