Problem Set #3

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ECN 410: Topics in Financial Economics Jeff Kubik Fall 2023 Problem Set #3 Due September 26th before class Question 1 You manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The t-bill rate is 7%. a) One of your clients chooses to invest 70% of a portfolio in your risky fund and 30% in t-bills. What is the expected return and standard deviation of your client's portfolio? b) What is the Sharpe Ratio of the risky portfolio you offer? What is the Sharpe Ratio of your client's portfolio? c) If another client wants a 15% expected return on a portfolio that invests in t-bills and the risky portfolio, what percentage of her assets will she have to put in the risky portfolio? What will be the standard deviation of the rate of return of this complete portfolio? d) A third client wants the highest expected return on a complete portfolio that does not have a standard deviation of returns greater than 20%. What would be this expected return? e) A fourth client is thinking about investing in a passive risky portfolio, the S&P 500 stock index, instead of your risky portfolio. Assume that the S&P 500 stock index has an expected return of 13% and a standard deviation of 25%. If the client is thinking about investing 70% of his portfolio in the passive index, then what is his expected return and standard deviation? Can you show that this client can do better using your risky portfolio instead of the passive index? f) What fee (as a percentage of assets invested) could you charge clients to invest in your risky portfolio so that they would be indifferent between using your portfolio or the S&P 500 stock index described above? (Hint: Think about how the fee would change the Sharpe Ratio of your risky portfolio.) Question 2 Let's use the Portfolio Visualizer website ( https://www.portfoliovisualizer.com/ ) we discussed in class. Like we did in class, scroll down to the part of the first page that lists links in the Portfolio Optimization section. Click on the "Efficient Frontier" link. a) Let's look at the efficient frontier of a portfolio consisting of two assets: the U.S. Stock Market and Global Ex-U.S. Stock Market (non-U.S. stocks). Data on the historical returns of the Global Ex-U.S. Stock Market are only available from 1986 onwards, so 1
make the start year 1986 and the end year 2023. Over that time period, what is the expected return and the standard deviation of returns of the U.S. Stock Market? (After you click View, select the Assets and Correlations tab). Over that same period, what is the expected return and the standard deviation of returns of the Global Ex-U.S. Stock Market? For the tangency portfolio, what is the weight on the two assets? b) Collect the same information for a portfolio that consists of the U.S. Stock Market and Gold. For these two assets, we can collect the information over the period 1972 to 2023. What is the expected return and the standard deviation of returns of the U.S. Stock Market over the period 1972 onward? Over that same period, what is the expected return and the standard deviation of returns of Gold? For the tangency portfolio, what is the weight on the two assets? c) Given that over this period, both the Global Ex-U.S. Stock Market and Gold have lower expected returns and higher standard deviations of returns than the U.S. Stock Market, why are the tangency portfolios so much different? Question 3 Let's look at another portfolio using Portfolio Visualizer. In this case, let's have five as- sets: the U.S. Stock Market, the Global Ex-U.S. Stock Market, 10-year Treasuries, REITs (a way to invest in real estate) and gold. Historical data on REITs only begins in 1994, so make the start year 1994 and the end year 2023. If you click the Data Points tab (after hitting View), you will see the holdings of all five of these assets at every percentile of the Efficient Frontier. In this circumstance, are any of the portfolios on the Efficient Frontier where you hold Foreign Stocks? Question 4 An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 14% and a standard deviation of return of 20%. Stock B has an expected return of 21% and a standard deviation of return of 39%. The correlation coefficient between the returns of A and B is .4. The risk-free rate of return is 5%. Would the proportion of the optimal risky portfolio that should be invested in stock A be .71 or .29? Question 5 You are considering investing $1000 in a (riskless) T-bill that returns 5% and a risky port- folio, P, constructed with 2 risky securities, X and Y. The weights of X and Y are 0.60 and 0.40 respectively. X has an expected rate of return of 0.14 and variance 0.01 and Y has an expected rate of return of 0.10 and a variance of 0.0081. You want to form a portfolio with a standard deviation of return of 0.11, can you figure out with the information provided what percentages of your money must you invest in the T-bill and P, respectively? 2
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