SEC08 - UNIVERSITY OF PENNSYLVANIA THE WHARTON SCHOOL FNCE...

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UNIVERSITY OF PENNSYLVANIA THE WHARTON SCHOOL FNCE 601 FINANCIAL ANALYSIS LECTURE NOTES Franklin Allen Fall 2003 QUARTER 1 - WEEK 6 Tu: 10/7/03 and Th: 10/9/03 Copyright 2003 by Franklin Allen
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FNCE 601 - Section 8 - Page 1 Section 8: The Capital Asset Pricing Model Read Chapters 7 and 8 BM The Capital Asset Pricing Model (CAPM) We next turn to the derivation of the CAPM. We are going to assume that capital markets are well functioning and that everybody has the same beliefs about the means and standard deviations of all stocks. Everybody’s perception of the efficiency locus of all the stocks in the market is therefore the same. In other words, every investor’s perception of the north-west boundary of the attainable portfolios is the same. It is helpful to think of each portfolio on this boundary as corresponding to a mutual fund. If this is the situation investors face, then the portfolio or mutual fund each chooses will depend on his or her preferences. People such as Person 1 who are fairly risk averse will choose
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FNCE 601 - Section 8 - Page 2 π π mutual funds such as X with low return and low risk; people such as Person 2 who are not very risk averse will choose mutual funds such as Y with high return and high risk. Now suppose that there is a risk-free asset, something like a T-bill, and this yields r f with certainty. Can Person 1, for example, do better than X now that the risk-free asset is there?
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FNCE 601 - Section 8 - Page 3 Suppose we hold a portfolio with a proportion π in the mutual fund X which has a mean return of r X and variance Var X and 1 - π invested in the risk-free asset which has a mean return of r f and a variance of 0. The covariance of the risk-free asset and X is also 0 since r f -Er f = 0. What are the mean and standard deviation of this portfolio? Using the formulas for two-security portfolios from Section 7(iv) with π 1 = π and π 2 = 1 - π : Er port = π r X + (1 - π )r f Var port = π 2 Var X + (1 - π ) 2 × 0 + 2 π (1 - π ) × 0 = π 2 Var X SD port = π SD X The expected return is just a weighted average of r f and r X as you’d expect. The formula SD = π SD X simply says that if only one of the stocks in a portfolio varies the standard deviation of the portfolio is the proportion of that stock in the portfolio times the stock’s standard deviation. As we vary π what do we trace out? To see this suppose r f = 0.05, r X = 0.15, and SD X = 2. Now consider the following. π = 0.25 r port = 0.075 SD port = 0.5 π = 0.50 r port = 0.100 SD port = 1.0 π = 0.75 r port = 0.125 SD port = 1.5 Hence we trace out a straight line.
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Is Person 1 better off investing in a combination of the risk-free asset and X? Yes, he can reach a higher indifference curve if he buys a combination of the risk-free asset and X than if he just invests in X. X
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SEC08 - UNIVERSITY OF PENNSYLVANIA THE WHARTON SCHOOL FNCE...

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