Assigment 2 free essay sample

1. Suppose that you can trade a riskless asset that yields 5% and two risky assets A and B. The expected return of asset A is 8% and that of asset B is 11%, while the standard deviation of asset A is 14% and that of asset B is 23%. The covariance between assets A and B is 0:0322. Solution . rA,B= CovAR(A,B) / [(? A)(? B)] = -0. 0322 / (14%)(23%) rA,B = -1 But when rA,B = -1, (? p)^2 = [wA(? A) – wB(? B)]^2, ? p = wA(? A) – wB(? B) Is there is no risk fo the portafolio, then ? p = 0 So this means that: 0 = 14%(wA) – 23%(1 – wA), solve this for wA, wA = 0. 6216 wB = 1 – wA, wB = 0. 3784 E(Rp) = 0. 6216(8%) + 0. 3784(11%) = 9. 1352% Suppose that each one of the securities has a value of $100, Cash Flow Today Cash Flow 1 year from today Buy 0. 6216 units of A -$62. 16 $62. 16(1. 08) = +$67. 13 Buy 0. 3784 units of B -$37. 84 $37. 84(1. 11) = +$42. 00 Short 1 unit of risk-free +$100 -$100(1. 05) = -$105 Net Cash Flow 0 +$4. 13 What today will be $4. 13/(1. 05) = $3. 93 2. You are the risk manager in a major investment bank. We will write a custom essay sample on Assigment 2 or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page The banks current portfolio consists of U. S. stocks (50%), bonds (20%), and derivatives (30%). The expected returns and standard deviations of these investments are Expected Return 13% 7% 25% Standard Deviation 25% 9% 50% A trader comes with a idea about investing in some new emerging markets: the markets of Polynesia, Micronesia, and New Caledonia. These markets have the following characteristics: Polynesia Micronesia New Caledonia Expected Return 18% 20% 22% Standard Deviation 30% 35% 28% Correlation with Stocks 0. 4 0. 2 0. 6 Correlation with Bonds 0. 3 0. 1 0. 2 Correlation with Derivatives 0. 2 0. 3 0. 4 Your job as risk manager is to determine how this investment would project the overall risk of the banks portfolio. Based on risk considerations alone, which of the three emerging markets is the best investment? Assume that the investment in the new market is based by borrowing on the riskless asset, and that it is a very small part of the banks overall investment. The market with the smallest risk contribution can be figured by determining the CovAR of each market to the portfolio. CovAR(RA, RM) = (? A)var(RM) Thus, it eliminates the need to determine Var(RM). If the CovAR of Polynesia to the bank’s portfolio can be measured by first determining the covariance between stocks, bonds, and derivatives. Then, these values can be summed to compute the CovAR of Polynesia to the bank’s portfolio. CovAR(Polynesia, Stocks) = xstocks(? Polynesia)(? stocks)(weight of stocks) = 0. 4 * 0. 3 * 0. 25 * 0. 5 = 0. 015 With similar calculations, CovAR(Polynesia, Bonds) = 0. 00162 and CovAR(Polynesia, Derivatives) = 0. 009 CovAR(Polynesia, portfolio) can be computed by summing CovAR(Polynesia, Stocks), CovAR(Polynesia, Bonds), and CovAR(Polynesia, Derivatives) CovAR(Polynesia, portfolio) = 0. 015 + 0. 00162 + 0. 009 = 0. 2562 With similar calculations, CovAR(Micronesia, portfolio) = 0. 02513, CovAR(NewCaledonia, portfolio) = 0. 038808 The best option of investment is the bank is Micronesia. 3. Stocks X, Y, and Z have the same expected return 8% and the same standard deviation 19% (a)Compute the standard deviation of the equally weighted portfolio if the correlation between all pairs of stocks is 1:0. Explain the intuition behind this result. r = 1, ? p = 19% With a perfect positive correlation, there is no idiosyncratic risk, thus there are no benefits of diversification. (b) Compute the standard deviation of the equally weighted portfolio if the correlation between all pairs of stocks is 0:5. Using excel, when r = 0. 5, ? p = 15. 51% (c) Compute the standard deviation of the equally weighted portfolio if the correlation between all pairs of stocks is 0:0. Using excel, when r = 0, ? p = 10. 97% (d) Compute the standard deviation of the equally weighted portfolio if the correlation between all pairs of stocks is 0:5. Using excel, when r = -0. 5, ? p = 0% (e) Explain intuitively in which case above (a) to (d) (if any) is the equally weighted portfolio the minimum variance portfolio? (No computation is needed. ) The minimum variance portfolio is seen when r = -0. 5. Here, ? p = 0% (f) How does your answer to part (e) change if stocks X, Y, and Z have the same expected return 11% instead of 8% and nothing else is changed? No change (g) How does your answer to part (e) change if stocks X, Y, and Z have the same standard deviation 15% instead of 19% and nothing else is changed? No change 4. Your rich uncle asks you financial advice. He is currently holding a portfolio of 30% T-bills and 70% Microsoft stock. The beta of Microsoft is 1. 2 and the standard deviation is 37. 95%. You decide to base your advice on the CAPM. The T-bill rate is 5%. The market portfolio has expected return 15% and standard deviation 20%. (a)What is the expected return of your uncles portfolio? E(RM) = 5% + 1. 2(15% 5%) = 17% E(Rp) = (3/10)5% + (7/10)17% = 13. 4% (b) What is the standard deviation of your uncles portfolio? (? p)^2 = (7/10)^2(0. 3795)^2 + (3/10)^2(0)^2 + 0 = 0. 0705699 ? p = 0. 0705699^0. 5 = 0. 26565 = 26. 565% (c) You decide to recommend to your uncle a portfolio that has the same expected return as his portfolio but the lowest possible standard deviation. Which is this portfolio, and what is its standard deviation? 13. 4% = wTP(15%) + (1 – wTP)(5%) solving for wTP, wTP = 0. 84 wT = 1 – wTP, wT = 0. 16 (? p)^2 = 0. 84^2(0. 2)^2 = 0. 028224 ?p = 0. 168 = 16. 8% 5. Consider market portfolio and three risky assets: A, B, and C. Over the next year, only three scenarios of how the economy will develop can happen with equal probability. The table below describes, in each scenario, returns predicted by analysts for the market portfolio and for the three risky assets. Economy Market A B C Boom 17% 11% 3% 2% Mediocre 6% 11% 3% 2% Recession -2% 2% 7% 4% (a) What are the expected returns and the standard deviations of returns from in- vesting into the market portfolio and into each of the three risky assets? E(RM) = (1/3)17% + (1/3)6% + (1/3)-2% = 7%E(RM) = 7% With similar calculations, E(RA) = 8%, E(RB) = 4. 33%, E(RC) =2. 67% (? M)^2 = (1/3)(17% 7%)^2 + (1/3)(6% 7%)^2 + (1/3)(-2% 7%)^2 = 0. 0061 ? M = 0. 0061^0. 5 = 0. 0779 = 7. 79% With similar calculations, ? A = 4. 24%, ? B = 1. 89%, ? C = 0. 94% (b) Covariance of returns of the market portfolio with asset A is where pBoom, pMediocre, and pRecession are the probabilities of the three scenarios to occur. The correlation of returns of the market portfolio with the returns of asset A is _(RM;RA) = Cov(RM;RA) _ (RM) _ (RA) Use the formulas above tend the covariance’s and correlations of returns of assets A, B, and C with the returns of the market portfolio. Using formula, Cov(RM,RA) = 0. 0027, Cov(RM,RB) = -0. 0012, Cov(RM,RC) = -0. 0006 Using formula, ? (RM,RA) = 0. 8171, ? (RM,RB) = -0. 8171, ? (RM,RC) = -0. 8171 (c)What are the betas of assets A, B, and C? ?A = CovAR(RM,RA) / Var(RM) = 0. 0027 / 0. 0061 = 0. 4451? A = 0. 4451 With similar calculations, ? B = -0. 1978, ? C = -0. 0989 (d)If the riskless rate is 3. 5%, what are the expected returns of A, B, and C as predicted by the CAPM? CAPM, E(Ri) = rf + ? i [ E(RM) – rf ] E(RA) = 3. 5% + 0. 4451( 7% 3. 5% ) = 5. 06%E(RA) = 5. 06% With similar calculations, E(RB) = 2. 81%, E(RC) = 3. 15% (e) Draw a graph that contains the riskless asset, the market portfolio, and the three risky assets A, B, and C. Draw the SML in this graph. (f) Find alphas of risky assets A, B, and C. Indicate alphas of each risky asset in the alpha(A) = 8% 5. 06% = 2. 94%, alpha(B) = 4. 33% 2. 81% = 1. 52%, alpha(C) = 2. 67% 3. 15% = -0. 48%