Problem-set-3

MGMT 411 - Problem set 3 Due on April 8th 1) Short questions 1. What are the problems of using unconstrained mean-variance optimization in practice? Explain the optimizer’s mirage . 2. Discuss the Black-Litterman model and the problem it is trying to solve. 3. Explain how to test the CAPM using time-series and cross-sectional tests. 4. Discuss the evidence for the following predictions of the CAPM: i) expected returns are increasing in market betas; ii) the slope of the SML is given by the market risk premium 5. Does the data support the prediction that a single factor, market beta, explains the cross-section of returns? Discuss the evidence relevant to this question. 2) CAPM and optimal portfolio formation Suppose the CAPM holds, company A has beta β A = 0 . 5 and volatility σ A = 12% , company B has beta β B = 1 . 5 and volatility σ B = 25% . The market return is 6% , market volatility is 15% , and the risk-free rate is equal to zero. All assets are uncorrelated. Reminder: formulas may require variance while the question is providing a standard deviation. 1. What would be your estimate of the aggregate risk aversion in this economy? 2. What is the expected return of company A and B? 3. Suppose an investor with risk aversion γ = 4 has to choose how much to invest in the risk-free asset and companies A and B. What is the optimal portfolio for this investor? 4. What is the optimal portfolio in the case γ = 2 ? Compute the leverage of the investor. 5. Compute the beta of the portfolio of an investor with risk aversion equals to γ = 2 and γ = 4 . Hint: the beta of a portfolio equals the average of the individual betas (weighted by the portfolio share). The beta of the risk-free asset is equal to zero. 3) The in-sample performance of the optimal mean-variance portfolio In this question, you will compare the in-sample performance of the optimal mean-variance portfolio against a few benchmarks. The portfolios will be combinations of the following five assets: SPY (S&P 500 ETF), EFA (a non-US equities ETF), IJS (a small-cap value ETF), EEM (an emerging-markets ETF), and AGG (a bond ETF). Download data starting in January 2014. 1. Create a variable called returns containing the monthly log returns for all assets. Report the an- nualized Return, annualized volatility, and annualized Sharpe ratio (assuming a risk-free rate of 1% a year), using the function table.AnnualizedReturns from the PerformanceAnalytics package. Hint: monthlyReturn(XYZ, method = “log”) computes the log monthly return for asset XYZ. 2. Using the function Return.portfolio , compute the portfolio returns with quarterly rebalancing for two portfolios: i) the equal-weight portfolio; ii) the (no-leverage) risk-parity portfolio, where the portfolio share is inversely related to volatility. Hint 1: you can use the command sds <- sapply(returns, sd) to compute a vector with the standard-deviation of returns for all assets. Hint 2: no leverage means that the portfolio share for the five assets adds up to one. 1