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Unformatted text preview: Portfolio Selection 1 PORTFOLIO SELECTION Trading off expected return and risk How should we invest our wealth? Two principles: we want to maximize expected return we want to minimize risk = variance Portfolio Selection 2 goals somewhat at odds riskier assets generally have higher expected return investors demand a reward for bearing risk called risk premium there are optimal compromises between expected return and risk Portfolio Selection 3 In this chapter maximize expected return with upper bound on the risk or minimize risk with lower bound on expected return. Portfolio Selection 4 Key concept: reduction of risk by diversification Diversification was not always considered favorably in the past Portfolio Selection 8 One risky asset and one riskfree asset Start with a simple example: one risky asset, which could be a portfolio, e.g., a mutual fund expected return is .15 standard deviation of the return is .25 one riskfree asset, e.g., a 30day Tbill expected value of the return is .06 standard deviation of the return is 0 by definition of riskfree. Portfolio Selection 9 Problem: construct an investment portfolio a fraction w of our wealth is invested in the risky asset the remaining fraction 1 w is invested in the riskfree asset then the expected return is E ( R ) = w ( . 15) + (1 w )( . 06) = . 06 + . 09 w . the variance of the return is 2 R = w 2 ( . 25) 2 + (1 w ) 2 (0) 2 = w 2 ( . 25) 2 . and the standard deviation of the return is R = . 25 w . Would w > 1 make any sense? Portfolio Selection 10 0.05 0.1 0.15 0.2 0.25 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 "Risk" = R = w/4 "Reward" = E(R) = .06+.09w Plotting the portfolios in rewardrisk space Portfolio Selection 11 need to choose w : choose the expected return E ( R ) , or the amount of risk R Once either E ( R ) or R is chosen, w can be determined. Portfolio Selection 12 Question: Suppose you want an expected return of .10? What should w be? Answer: .10 = .06 + .09 w w = 4/9 Question: Suppose you want R = . 05. What should w be? Answer: . 05 = w ( . 25) w = 0.2 Portfolio Selection 13 More generally, if the expected returns on the risky and riskfree assets are 1 and f and the standard deviation of the risky asset is 1 , then the expected return on the portfolio is w 1 + (1 w ) f and the standard deviation of the portfolios return is w 1 . Portfolio Selection 14 This model of one riskfree asset and one risky asset is simple but not useless finding an optimal portfolio can be achieved in two steps....
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 Spring '11
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