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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 σ 2 R = w 2 ( . 25) 2 + (1 w ) 2 (0) 2 = w 2 ( . 25) 2 . and the standard deviation of the return is σ 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" = σ 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 σ R • Once either E ( R ) or σ 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 σ 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 μ 1 and μ f – and the standard deviation of the risky asset is σ 1 , • then – the expected return on the portfolio is wμ 1 + (1 w ) μ f – and the standard deviation of the portfolio’s return is w σ 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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This note was uploaded on 02/05/2012 for the course ECON 4140 taught by Professor A during the Spring '11 term at York University.
 Spring '11
 A

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