Section 9 - ECON 136: Financial Economics Section 9 (Oct...

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Section 9 (Oct 30th) Xing Huang 1 Economics Department, UC Berkeley 1 Portfolio Return and Portfolio Risk A mean-variance optimizing investor chooses her portfolio to maximize the value of E [ R ] 1 2 A ± V ar ( R ) , where A is a coe¢ cient of risk aversion. Consider a portfolio of two assets with weights w 1 and w 2 , expected returns E [ R 1 ] and E [ R 2 ] , and return variances 2 1 and 2 2 : Portfolio expected return = E [ R p ] = w 1 ± E [ R 1 ] + w 2 ± E [ R 2 ] Portfolio variance (risk) = 2 p = w 2 1 2 1 + w 2 2 2 2 + 2 w 1 w 2 12 Portfolio standard deviation = q 2 p = p If one of these assets (say asset 2) is risk free, then E [ R 2 ] = R f and 22 = 0 . Because a risk-free 12 = 0 . Thus, with one risky asset and one risk free asset: Portfolio expected return = E [ R p ] = w 1 ± E [ R 1 ] + w 2 ± R f Portfolio variance = 2 p = w 2 1 2 1 Portfolio standard deviation = q 2 p = w 1 1 ( w 1 ² 0 ) 2 Capital Allocation Line (CAL) and Optimal Portfolio Choice 2.1 Review Again consider a world with two assets. Asset 1 is a risky asset with E [ R 1 ] = R 1 and standard deviation 1 , and asset two is a risk-free asset with return R f : 1. Capital allocation line (CAL) ±a graph of all possible expected returns and standard devia- tions of a portfolio formed by combining the risky asset with the risk-free asset 2. Sharpe ratio (S) of the risky asset = R 1 R f 1 = the expected excess return of the risky asset per unit of its standard deviation; the reward-to-variability ratio of investing in the risky asset (investors like high Sharpe ratios); the slope of the CAL 3. Optimal Portfolio Choice: In a world with one risky asset and one risk free asset, a mean-variance optimizing investor with risk aversion A would invest the proportion of her wealth into the risky asset w ± 1 = R 1 R f 1 = S 1 1 1 These notes are enormously bene²ted from previous GSIs: Keith Jacks Gamble, Dan Hartley and Congyan Tan. Thank you all very much !!
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This note was uploaded on 01/25/2010 for the course ECON 136 taught by Professor Szeidl during the Fall '08 term at University of California, Berkeley.

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Section 9 - ECON 136: Financial Economics Section 9 (Oct...

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