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Unformatted text preview: Haas School of Business University of California at Berkeley UGBA 103 Avinash Verma P ORTFOLIO T HEORY : N-S ECURITY P ORTFOLIOS Recall that in the last note, we discussed two-security portfolios. In this note, we expand our analysis to all securities in the market. As before, we represent portfolios and securities on a graph by plotting their expected return on the vertical axis and the standard deviation of their returns on the horizontal axis. Correlation between the returns on two securities determines whether portfolios of the two lie on a straight line, on a curve, or on two line segments that meet on the vertical axis. 1. Because on average, returns on any two securities, say Security 1 and Security 2 , are unlikely to be either perfectly positively or perfectly negatively correlated [ 12 1 + ; 12 1 - . ], portfolios of the two securities will lie along a curve in the graph below rather than on a straight line between the two securities, as they do when , 1 12 + = or on two straight line segments that meet on the vertical axis, which occurs when 1 12- = . 2. When we introduce a third security, say Security 3 , we are likely to find that it is not perfectly correlated with the two existing securities by the same logic. Thus, in the graph below, all portfolios of Security 2 and Security 3 also lie on a curve between the two. 3. We can now construct portfolios of these three securities by forming portfolios of portfolios of two securities. The objective in combining the three securities in a portfolio, as always, is to get as far up or North [maximize expected return], and as far to the left or West [minimize risk as measured by the standard deviation of portfolio returns] as possible. Thus, we can take a portfolio of Security 2 and Security 3 , and combine it with a portfolio of Security 1 and Security 2 , to move further northwest as shown in the graph on the next page. Portfolio Theory 1 Security 1 Security 2 P E Security 1 Security 2 P E Security 3 Haas School of Business University of California at Berkeley UGBA 103 Avinash Verma 4. Now that we have a way in which we can deal with more than two securities, we can introduce all risky securities in the market in our graphical analysis. When we keep combining all available risky securities in portfolios of portfolios until we have gotten as far to the northwest as possible, we shall get a smooth curve as shown in the in the graph below. The curve will be smooth, which is to say, free of any dents or non-convexities , because whenever we see any such dent, we shall smooth it out by forming a portfolio of two portfolios on either side of the dent as a part of our expected-return-maximizing and risk-minimizing behavior....
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- Spring '08