Marsha Actually only about 500000 dear The co variances above the diagonal are

Marsha actually only about 500000 dear the co

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Marsha: Actually only about 500,000, dear. The co variances above the diagonal are the same as the co variances below. But you are right, most of the estimates would be out-of-date or just garbage. 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 up with as an indexer. Then consider a few securities you really know something about. Pioneer could be security 2, for example. Global, security 3. 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 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 invested partly in the market and partly in Pioneer. (You can calculate the necessary inputs from the beats and standard deviations given in the table). Does adding Pioneer to the market benchmark improve the Sharpe ratio? How mush should Jon invest in Pioneer and how much in the market? Pioneer Gypsum Global Mining Expected return 11.0% 12.9% Standard deviation 32% 20% Beta 0.65 1.22 Stock price \$87.50 \$105.00 Repeat the analysis for Global Mining. What should John do in this case? Assume that Global accounts for .75 of the S&P index. Solution John neglected to mention the standard deviation of the S&P 500. We will assume 16%. Recall that stock i’s beta is just the ratio of its covariance with the market (σ im ) to the market variance σ m 2 , where σ m 2 = .16 2 = .0256. For Pioneer Gypsum, β = .65 = σ im /.0256, which gives a covariance of σ im = .01664. The covariance also equals the correlation coefficient ρ times the product of the stock’s and market’s standard deviations σ i and σ m . For Pioneer, σ im = ρσ i σ m = .01664 = ρ×.32×.16, which implies ρ = .325.