week_11_section - Department of Economics University of...

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Department of Economics Financial Economics University of California, Berkeley Economics 136 November 9, 2003 Fall 2006 Economics 136: Financial Economics Section Notes for Week 11 1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset This material is in BKM chapter 7. Assume for the moment that an investor must decide how to invest all of her wealth and has only two options: a risk-free asset such as Treasury Bills (T-Bills) and a risky portfolio of stocks (such as a mutual fund). Since all of her wealth must be invested, the decision that she makes can be summarized by one parameter, the fraction of her wealth that she invests in the risky portfolio, w . Since she must allocate all of her wealth to either the mutual fund or T-Bill, the fraction of her wealth invested in T-Bill must be, 1 - w . If we assume a functional form for the investors objective (or utility) function, then we can determine the optimal fraction of wealth for the investor to put into the risky portfolio, w * . Let R P denote the returns of the combined T-Bill and mutual fund portfolio and R MF denote the returns of the mutual fund. Assuming that people like high mean return and dislike high return variance then they would like to solve the following optimization, max w ± E [ R P ] - 1 2 A · V ar ( R P ) ² = max w ± R f + w ( E [ R MF ] - R f ) - 1 2 w 2 · A · V ar ( R MF ) ² Then the first order condition (foc) is 0 = E [ R MF ] - R f - w · A · V ar ( R MF ) w * = E [ R MF ] - R f A · V ar ( R MF ) Note: w * gives the optimal fraction of wealth invested in the risky portfolio. The total portfolio which invests w * in the risky portfolio and 1 - w * in the risk-free asset will result in the optimal portfolio which has an expected return and standard deviation such that the investor’s indifference curve is tangent to the CAL at this point. This is why it is optimal. It gives the investor the highest possible utiity subject to the mean and standard deviation of investment possibilities (the CAL is the frontier of these possibilities). 1
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2 Finding the Optimal Risky Prtfolio This material is in BKM chapter 8. In the last section, we took the expected return and standard deviation of the risky portfolio
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week_11_section - Department of Economics University of...

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