answer the third part of homework:Risk Budgeting.

I.\tMatrix\tAlgebra\tand\tPortfolio\tMath\t(44\tpoints,\t4\tpoints\teach) Let Ri denote the simple return on asset i (i = 1, , N) with E[Ri] = i, var(Ri) = i2 and cov(Ri, Rj) = ij. Define the (N 1) vectors R R1 , , RN , (1 ,, N ) , m m1 , , mN , x x1 , , xN , y y1 , , y N , t t1 , , t N , 1 1, ,1 and the (N N) covariance matrix 12 12 1N 12 2 2 2N 1N 2N . 2 N The N 1 vectors m, x, y and t contain portfolio weights that sum to one. Using matrix algebra, answer the following questions. 1. For the portfolios defined by the vectors x and y give the expression for the portfolio returns, (Rp,x and Rp,y), the portfolio expected returns (p,x and p,y), the portfolio variances ( 2 2 p , x and p , y ), and the covariance between Rp,x and Rp,y ( xy ). 2. For portfolio x, derive the N 1 vector of marginal contributions to portfolio volatility defined by MCR p , x x p (x) x . 3. Write down the optimization problem and give the Lagrangian used to determine the global minimum variance portfolio assuming short sales are allowed. Let m denote the vector of portfolio weights in the global minimum variance portfolio. 4. Write down the optimization problem and give the Lagrangian used to determine an efficient portfolio with target return equal to 0 assuming short sales are allowed. Let x denote the vector of portfolio weights in the efficient portfolio. 5. Continuing with question 4, derive the first order conditions for determining the efficient portfolio x with target return 0 . Write these first order conditions as a system of linear equations in the form Az = b and show how the portfolio x can be determined from this system. 6. Briefly describe how you would compute the efficient frontier containing only risky assets (Markowitz bullet) when short sales are allowed. 7. Write down the optimization problem used to determine the tangency portfolio, assuming short sales are allowed and the risk free rate is given by rf . Let t denote the vector of portfolio weights in the tangency portfolio. 8. Continuing with question 7, is there an analytical solution (i.e., matrix algebra mathematical formula) for the tangency portfolio when short sales are allowed? If so, give the matrix algebra formula for this solution. 9. Write down the equations for the expected return ( e ) and standard deviation ( e ) of p p efficient portfolios consisting of the tangency portfolio and T-bills, where the T-bill rate (riskfree rate) is given by rf and t denotes the vector of portfolio weights in the tangency portfolio. 10. The previous computations for efficient portfolio allowed for short sales (i.e., negative values for asset shares). However, in many practical situations short sales are not allowed. Give three reasons why short sales may be prohibited. 11. Write down the optimization problem and give the Lagrangian used to determine an efficient portfolio with target return equal to 0 assuming short sales are not allowed. Let x denote the vector of portfolio weights in the efficient portfolio. II.\tEfficient\tPortfolios\t(36\tpoints,\t4\tpoints\teach) The graph below shows the efficient frontier (allowing short sales) computed from three Vanguard mutual funds: S&P 500 Index (vfinx), European Stock Index (veurx) and the US Long Term Bond Index (vbltx). Evfinx 0.010 Global Min vfinx Tangency veurx vbltx 0.000 0.005 Portfolio ER 0.015 Efficient Frontier T-Bills 0.00 0.01 0.02 0.03 0.04 0.05 Portfolio SD Figure 1 Markowitz Bullet Expected return and standard deviation estimates for specific assets are summarized in the table below. These estimates are based on monthly simple return data over the five year (60 month) period May 2009 - May 2014 (same data as in the class project but slightly different time period). Table 1 Portfolio Statistics Asset Mean (E[R]) VFINX VEURX VBLTX T-Bills Global Min Portfolio Efficient Portfolio with Mean=0.0148 (labeled Evfinx) Tangency Portfolio Standard deviation (SD(R)) 0.0148 0.0116 0.0083 0.0010 0.0114 0.0387 0.0574 0.0255 0.0000 0.0159 0.0148 0.58 -0.19 0.61 Weight in Efficient Portfolio with Mean = 1.48% 1.35 -0.69 0.34 Weight in Tangency portfolio 0.0232 0.0124 Weight in Global Min Portfolio 0.80 -0.33 0.53 0.0166 Using the above information, please answer the following questions. 1. Compute annualized means and standard deviations from the monthly statistics in Table 1 for the three portfolios vfinx, vbltx, and veurx using the square-root-of-time-rule (this is only an approximation because we have simple returns) and put these results in the Table below. Also, compute the annualized T-Bill rate. Asset vfinx vbltx veurx TBills Annualized Mean Annualized SD 2. Using the annualized information from part 1, compute the annualized Sharpe ratios for each of the three portfolios and put the results in the table below. Which portfolio is ranked best using the Sharpe ratio? Asset vfinx vbltx veurx Annualized Sharpe Ratio 3. The Sharpe ratios reported above are estimates. Briefly describe how you can compute standard errors and 95% confidence intervals for these estimated Sharpe ratios. 4. Find the efficient portfolio of risky assets only (i.e., a portfolio on the Markowitz bullet) that has an expected monthly return equal to 0.014 (1.4%). In this portfolio, how much is invested in vfinx, vbltx, and veurx? Show this portfolio on Figure 1. 5. How much should be invested in T-bills and the tangency portfolio to create an efficient portfolio with expected return equal to the average return on vfinx - 0.0148 (1.48%) ? What is the standard deviation of this efficient portfolio? Indicate the location of this efficient portfolio on Figure 1. 6. In the efficient portfolio you found in part 5, what are the shares of wealth invested in TBills, vfinx, vbltx, and veurx? 7. Assuming an initial $100,000 investment for one month, compute the 5% value-at-risk based on the normal distribution for the global minimum variance portfolio. 8. The efficient frontier of risky assets shown in Figure 1 allows for short sales (see the weights in the portfolios listed in Table 1). Using the graph below indicate the location of the efficient frontier of risky assets that does not allow short sales. vfinx veurx 0.010 Global Min 0.005 vbltx 0.000 Portfolio ER 0.015 Efficient Frontier T-Bills 0.00 0.01 0.02 0.03 Portfolio SD 0.04 0.05 9. Suppose you want to find the efficient portfolio of risky assets only that has an expected monthly return equal to 0.02 (2%) but that you are prevented from short selling. Is it possible to find such an efficient portfolio? Briefly explain why or why not. III. Risk\tBudgeting\t(20\tpoints,\t5\tpoints\teach) Table 2 below shows a risk budget report for the global minimum variance portfolio given in Table 1. The global minimum variance portfolio has some special properties that are revealed in the risk budget report below. Table 2 Risk Budget Report for Global Minimum Variance Portfolio p ,m 0.0159 Asset xi i vfinx veurx vbltx 0.585 -0.195 0.611 0.0387 0.0574 0.0255 MCRi 0.0159 0.0159 0.0159 CRi 0.00931 -0.00311 0.00972 PCRi 0.585 -0.195 0.611 i , p i , p 1. The risk budget report for a portfolio shows the additive decomposition of portfolio volatility into contributions from the assets in the portfolio. In the global minimum variance portfolio which asset has the highest contribution to portfolio volatility? Which asset has the lowest contribution? What is the relationship between an asset's percent contribution to risk, PCRi , and its allocation weight, xi ? Verify that the sum of the asset contributions to risk, CRi , add to portfolio volatility, p ,m 0.0159 . 2. Suppose the risk manager wants to reduce the portfolio volatility. For which assets should allocations be reduced, and for which assets should allocations be increased to achieve this goal? 3. Give the definition of the \"beta\" (denoted i , p ) of an asset's return with respect to the portfolio return. In lecture, we showed that i , p PCRi / xi . Using the information in the risk budget report compute i , p for each asset and put these values under the i , p column in the table. What relationship do you see? 4. Give the definition of the correlation (denoted i , p ) of an asset's return with respect to the portfolio return. In lecture, we showed that i , p MCRi / i . Using the information in the risk budget report compute i , p for each asset and put these values under the i , p column in the table. What relationship do you see? IV.\tStatistical\tAnalysis\tof\tEfficient\tPortfolios\t(20\tpoints\ttotal) The figures below show the simple returns of vfinx, veurx and vbltx along with 24-month rolling estimates of their average returns, standard deviations and pairwise correlations. 1. Briefly comment on the 24-month rolling estimates of the means, standard deviations and pairwise correlations. Which estimates appear to be constant and which do not? (8 points) 2. The CER model assumes that i and i and ij are constant over time. Given your response to 1, is this a reasonable assumption for vfinx, veurx and vbltx. (4 points) 3. The following figure shows 24-month rolling weights in the global minimum variance portfolio (allowing short sales) constructed from vfinx, veurx and vbltx. Which CER model estimates (i.e., i , i , ij ) determine the global minimum variance weights? Do you see evidence of substantial time variation in these weights? Can you explain any of this time variation by the time variation in the CER model estimates? (4 points) 4. The following figure shows 24-month rolling means and standard deviation of the global minimum variance portfolio (allowing short sales) constructed from vfinx, veurx and vbltx. What has been the impact on these values of the time variation in the 24-month estimates of i and i and ij? (4 points)