Q2) (5 Points) Assume the correlations for the US, UK and Japan equity indices from the last 101 years are: Correlation coefficients US UK Japan US 1.00 0.55 0.21 UK 0.55 1.00 0.33 Japan 0.21 0.33 1.00 Annual Returns Standard Deviation 8.70% 20.20% 7.60% 20.00% 9.30% 30.30% What is the quarterly 5% VAR for a portfolio with an equal investment in each index? Step 1: Annual E(Rus) = 8.7%, 0 us = 20.20% E(Ruk) = 7.6%, 0 UK = 20.00% E(RJP) = 9.3%, JP = 30.30% T(US,UK) = 0.55, ((UK,JP) = 0.33, r(US.JP) = 0.21 and Weights are equally distributed 1/3 Step 2: Quarterly Expected Return Annual Expected Returns E(Rp) = WUS * E (RUS) + WUK) + WJP * E(RJP) E (Rp) = 1/3 * 8.7 + 1/3 * 7.6 + 1/3 * 9.3 E (RP) = 8.53 Quarterly Expected Returns E(RP)(Quarterly) = E(Rp)(annual) * 3/12 E(Rp) (quarterly) = 2.13 % Quarterly Standard Deviation Annual Standard Deviation op = wis *ois + Win + OK +W;p *03p +2 + Wus * WUK + TUSUK * OUS #QUK + 2 * WUK + Wsp#ruk,JP* JUK * 1 JP +2 + Wus* WjpurusJP * QUS + OJP Op= V 45.34 + 44.44 + 102.01 + 49.38 + 44.44 + 28.56 Op= V314.17 Op= 17.725 = 314.17 =17.725 % Quarterly Standard Deviation op(Quarterly) = 0p(Annual) * V3/V12 =17.725 * 1.7321 /3.4641 = 8.86% At 5% in the left tail z=1.64 Quarterly VAR (5%) = u 20 = 2.13 1.64 * 8.86 = -12.40% Quarterly VAR at 5% would be -12.40% In 5% of the worst circumstances minimum loss in a quarter would be 12.40% Q3) (10 Points) Using the data table from Q2), assume you want to construct a portfolio where you are fully invested in just US and Japan equities. You want to target a portfolio that has a standard deviation of 25.3974%. How should you allocate your investment between the US and Japan to achieve this risk target? There are actually two different allocations that will work, solve for both. Which allocation has the highest portfolio return? Hint and as a quick reminder, to solve for an unknown X value, where the equation is of the general form: ax? + bx+c = 0, the value of X can have two values, which can be computed as follows: -6 + 62 - 4ac 2a and x= -b - 62 4ac 2a Q2) (5 Points) Assume the correlations for the US, UK and Japan equity indices from the last 101 years are: Correlation coefficients US UK Japan US 1.00 0.55 0.21 UK 0.55 1.00 0.33 Japan 0.21 0.33 1.00 Annual Returns Standard Deviation 8.70% 20.20% 7.60% 20.00% 9.30% 30.30% What is the quarterly 5% VAR for a portfolio with an equal investment in each index? Step 1: Annual E(Rus) = 8.7%, 0 us = 20.20% E(Ruk) = 7.6%, 0 UK = 20.00% E(RJP) = 9.3%, JP = 30.30% T(US,UK) = 0.55, ((UK,JP) = 0.33, r(US.JP) = 0.21 and Weights are equally distributed 1/3 Step 2: Quarterly Expected Return Annual Expected Returns E(Rp) = WUS * E (RUS) + WUK) + WJP * E(RJP) E (Rp) = 1/3 * 8.7 + 1/3 * 7.6 + 1/3 * 9.3 E (RP) = 8.53 Quarterly Expected Returns E(RP)(Quarterly) = E(Rp)(annual) * 3/12 E(Rp) (quarterly) = 2.13 % Quarterly Standard Deviation Annual Standard Deviation op = wis *ois + Win + OK +W;p *03p +2 + Wus * WUK + TUSUK * OUS #QUK + 2 * WUK + Wsp#ruk,JP* JUK * 1 JP +2 + Wus* WjpurusJP * QUS + OJP Op= V 45.34 + 44.44 + 102.01 + 49.38 + 44.44 + 28.56 Op= V314.17 Op= 17.725 = 314.17 =17.725 % Quarterly Standard Deviation op(Quarterly) = 0p(Annual) * V3/V12 =17.725 * 1.7321 /3.4641 = 8.86% At 5% in the left tail z=1.64 Quarterly VAR (5%) = u 20 = 2.13 1.64 * 8.86 = -12.40% Quarterly VAR at 5% would be -12.40% In 5% of the worst circumstances minimum loss in a quarter would be 12.40% Q3) (10 Points) Using the data table from Q2), assume you want to construct a portfolio where you are fully invested in just US and Japan equities. You want to target a portfolio that has a standard deviation of 25.3974%. How should you allocate your investment between the US and Japan to achieve this risk target? There are actually two different allocations that will work, solve for both. Which allocation has the highest portfolio return? Hint and as a quick reminder, to solve for an unknown X value, where the equation is of the general form: ax? + bx+c = 0, the value of X can have two values, which can be computed as follows: -6 + 62 - 4ac 2a and x= -b - 62 4ac 2a