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Marsha:Actually only about 500,000, dear. The co variances above the diagonal are the same asthe co variances below. But you are right, most of the estimates would be out-of-date or justgarbage.John:To say nothing about the expected returns: Garbage in, garbage out.3
Marsha:But John, you don’t need to solve for 1,000 portfolio weights. You only need a handful.Here’s the trick: Take your benchmark, the S&P 500, as security 1. That’s what you would end upwith as an indexer. Then consider a few securities you really know something about. Pioneer couldbe security 2, for example. Global, security 3. And so on. Then you could put your wonderfulfinancial mind to work.John:I get it. Active management means selling off some of the benchmark portfolio andinvesting the proceeds in specific stocks like Pioneer. But how do I decide whether Pioneer reallyimproves the portfolio? Even if it does, how mush should I buy?Marsha:Just maximize 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.The following table reproduces John’s notes on Pioneer Gypsum and Global Mining.Calculate the expected return, risk premium, and standard deviation of a portfolio investedpartly in the market and partly in Pioneer. (You can calculate the necessary inputs from thebeats and standard deviations given in the table). Does adding Pioneer to the marketbenchmark improve the Sharpe ratio? How mush should Jon invest in Pioneer and howmuch in the market?Pioneer GypsumGlobal MiningExpected return11.0%12.9%Standard deviation32%20%Beta0.651.22Stock price$87.50$105.00Repeat the analysis for Global Mining. What should John do in this case? Assume thatGlobal accounts for .75 of the S&P index.SolutionJohn neglected to mention the standard deviation of the S&P 500. We will assume 16%. Recallthat stock i’s beta is just the ratio of its covariance with the market (σim) to the market varianceσm2, where σm2= .162= .0256. For Pioneer Gypsum, β = .65 = σim/.0256, which gives acovariance of σim= .01664. The covariance also equals the correlation coefficient ρ times the product of the stock’s and market’s standard deviations σiandσm. For Pioneer, σim = ρσiσm= .01664 = ρ×.32×.16, which implies ρ = .325.