Investment Decisions In economics and Finance
Appendix 2
Case study: John and Marsha on a Portfolio Selection
The scene:
John and Marsha hold hands in a cozy French restaurant in downtown Manhattan. Marsha is a futures-market trader. John manages a $125 million common-stock portfolio for a large pension fund. They have just ordered tournedos financiere for the main course and flan financiere for dessert. John reads the financial pages of The Wall Street Journal by candlelight.
John: Wow! Potato futures hit their daily limit. Let¡¦s add an order of gratin Dauphinoise. Did you manage to hedge the forward interest rate on that euro loan?
Marsha: John, please fold up that paper. (He does so reluctantly.) John, I love you. Will you marry me?
John: Oh, Marsha, I love you too, but¡Kthere¡¦s something you must know about me ¡V something I¡¦ve never told anyone.
Marsha(concerned): John, what is it?
John: I think I am a closet indexer.
Marsha: What? Why?
John: My portfolio returns always seem to track the S&P 500 market index. Sometimes I do a little better, occasionally a little worse. But the correlation between my returns and the market returns is over 90%.
Marsha: What¡¦s wrong with that? Your client wants a diversified portfolio of large-cap stocks. Of course your portfolio will follow the market.
John: Why doesn¡¦t my client just buy an index fund? Why are they paying me? Am I really adding value by active management? I try, but I guess I¡¦m just an¡Kindexer.
Marsha: Oh, John, I know you¡¦re adding value. You were a star security analyst.
John: It¡¦s not easy to find stocks that are truly over- or undervalued. I have firm opinions about a few, of course.
Marsha: You were explaining why Pioneer Gypsum is a good buy.
John: Right, Pioneer. (Pulls handwritten notes form his coat pocket.) Stock price $87.50. I estimate the expected return as 10% with an annual standard deviation of 30%.
Marsha: Only 10%? You¡¦re forecasting a market return of 12%.
John: Yes, I¡¦m using a market risk premium of 7% and the risk-free interest rate is about 5%. That gives 12%. But Pioneer¡¦s beta is only 0.3. I was going to buy 30,000 shares this morning, but I lost my nerve. I¡¦ve got to stay diversified.
Marsha: Have you tried modern portfolio theory?
John: MPT? Not practical. Looks great in textbooks, where they show efficient frontiers with 5 or 10 stocks. But I choose from hundreds, maybe thousands, of stocks. Where do I get the inputs for 1,000 stocks? That¡¦s a million variances and covariances!
Marsha: Actually only about 500,000, dear. The covariances above the diagonal are the same as the covariances below. But you¡¦re right, most of the estimates would be out-of-date or just rubbish.
John: To say nothing about the expected returns. Rubbish in, rubbish out.
Marsha: But John, you don¡¦t need to solve for 1,000 portfolio weights. You only need a handful. Here¡¦s the trick: Take you benchmark, the S&P 500, as
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security 1. That¡¦s what you would end up with as an indexer. Then consider a few securities you really know something about. Pioneer could be security 2, for example. And so on. Then you could put your wonderful financial mind to work.
John: I get it: active management means selling off some of the benchmark portfolio and investing the proceeds in specific stocks like Pioneer. But how do I decide whether Pioneer really improves the portfolio? Even if it does, how much should I buy?
Marsha: Just maximise the Sharpe ratio, dear.
John: I¡¦ve got it. The answer is yes!
Marsha: What¡¦s the question?
John: You asked me to marry you. The answer is yes. Where should we go on our honeymoon?
Marsha: How about Australia? I¡¦d love to visit the Melbourne Stock Exchange.
QUESTION
€žh Table 1 reproduces John¡¦s notes on Pioneer Gypsum. Calculate the expected return, risk premium, and standard deviation of a portfolio invested partly in the market and partly in Pioneer. (You can calculate the necessary inputs from the betas and standard deviations given in the table.) Does adding Pioneer to the market benchmark improve the Sharpe ratio? How much should John invest in Pioneer and how much in the market? (John neglected to mention the standard deviation of the S&P 500. We will assume 12%. )