Question #1 (30 points)
You are a financial advisor and a portfolio manager. You are going to help your client, Anna, to find an optimal portfolio for her retirement (a complete portfolio). After due diligence, you have decided to invest in both stocks X and Y.
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You have made the following return forecasts for two stocks, X and Y:
|Bear market||Normal market||Bull market|
|T-bills (risk-free rate)||2%||2%||2%|
Assume that Anna is able to borrow and lend at the risk-free rate.
As the portfolio manager, you have to determine the optimal investment weight (%) of each stock to put in the optimal risky portfolio, P:
|Stock||% of Capital Allocated|
- Determine the optimal weights for stocks X and Y and then compute the expected return, standard deviation, and reward-to-variability ratio of the optimal risky portfolio, P.
- As your client’s financial advisor, you propose to allocate part of Anna’s money in the optimal risky portfolio (P), which is managed by you, and part of it in the T-bills (the proxy for risk-free asset.) Calculate the optimal proportion (%) of Anna’s complete portfolio (denote this as C) that should be invested in the optimal risky portfolio (P), assuming Anna’s utility function is E(rc) – 0.5Asc2 ,where A = 4.
- Calculate the expected return, standard deviation, and reward-to-variability ratio of Anna’s complete portfolio, C.
- Explain to Anna in details the asset allocation process that you have taken for her to arrive at her complete portfolio taking into consideration of her utility function. Draw a labeled graph to help illustrate your explanation on the process and the results (all steps above from (a) to (c)). That is, a graph showing how Anna’s complete portfolio (C) is determined given the capital allocation line and the utility function. You must label correctly the CAL, utility curves, portfolio P, portfolio C, and risk-free rate in the graph as well as the X- and Y-axis to receive full credit.
Question #2 (10 points)
A pension fund manager is considering investing in two funds to be included in the pension plan’s current portfolio, which is a well-diversified portfolio as proxied by the Russell 3000 index. The first fund under consideration is a growth stock fund, the second a value stock fund. The expected return, standard deviation, standard deviation of residuals, and beta of the funds are as follows:
|Expected return||Standard Deviation||Beta||Standard deviation of residuals|
|Growth fund (G)||10%||20%||1.1||12%|
|Value fund (V)||12%||12%||1.6||20%|
The correlation between the two fund returns is .80. The market risk premium is expected to be 6% and the risk-free rate is 2%.
Determine and explain which fund should be included in the pension fund’s current portfolio. Justify your answers with a risk-adjusted performance measure and show all your work.
Question #3 (15 points)
- The CAPM assumes that investors have homogeneous expectations. Explain what this assumption means and discuss its implications to investors.
- Explain what the separation/mutual fund theorem is, and discuss its importance to (i) investors and (ii) portfolio/asset managers.
Question #4 (10 points)
Jane’s company uses the single index model to estimate beta for stocks A and B. The risk-free rate over the period was 2%, and the market’s average return was 10%. Performance is measured using an index model regression on excess returns.
|Stock A||Stock B|
|Index model regression estimates||1% + 0.8(rM -rf)||-2% + 1.2(rM + rf)|
|Residual standard deviation, se||30%||40%|
|Standard deviation of market returns||22%|
Compute the nonsystematic risk (variance) of a portfolio with 40% in stock A,
40% in stock B and 20% in the risk-free asset.
Question #5 (25 points)
Kay, a portfolio manager at KD Asset Management, focuses on stock selection in her strategy. Her firm does not allow short-selling. She receives the following forecasts based on private information from her research associates:
- Given this private information and ability to pick mispriced stocks, determine the composition of the optimal risky portfolio (P) for Kay to hold using the Treynor-Black model, assuming her current portfolio mimics the market index. Show all computations.
- Calculate the Sharpe ratio of the optimal risky portfolio, P, managed by Kay and compare it to that of the market index portfolio. Show your computations.
- If Kay’s client is risk averse with a coefficient of risk aversion of 4 (i.e., A=4), compute the exact makeup of the complete portfolio of the client (C), i.e., specify the percentage of funds that should be invested in each stock, the market index portfolio, and the risk-free asset. Show all computations.
- Based on the results you obtained above, graph and label the old and new investment opportunity sets as well as the optimal risky portfolio (P), active portfolio (A), market index portfolio (M), Kay’s client portfolio (C), and the risk-free rate (rf) for full credit.
Question #6 (10 points)
Ando Kenichi, a portfolio manager for a Japanese small cap fund at Sakura Investment Company in Tokyo. His benchmark for performance of 2021 is the top quartile of fund managers who manage funds in Tokyo in 2021. In other words, the performance of Sakura for 2021 will be evaluated based on the average performance of the top quartile of fund managers in January 2022, the earliest.
Evaluate and discuss whether Mr. Ando has (or has not) a valid benchmark according to the properties identified by Richards (2001).
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