eco331-09-MeanVariance

# eco331-09-MeanVariance - Northwestern University Fall 2009...

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Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 MEAN-VARIANCE ANALYSIS AND PORTFOLIO CHOICE WITH MANY ASSETS 1. Mean is Good, Variance is Bad (or is it?) Portfolio choice with a single risky asset reduces to a simple trade-oﬀ: stocks (generally) yield a higher expected return than bonds, but they are risky. A risk-averse investor will put some money in stocks, but her optimal portfolio will have to balance these two considerations. A common measure of risk is the variance of returns. Recall that, if X is a random variable with probability distribution ( x 1 ,p 1 ; ... ; x n ,p n ), its variance is Var[ X ] = n X i =1 p i ( x i - E[ X ]) 2 = E[ X 2 ] - (E[ X ]) 2 (if the ﬁrst equality is the deﬁnition of variance; the second is a well-known and useful identity; make sure you see why it is true). Also recall that the standard deviation of a r.v. is the square root of its variance: thus, in particular, StDev [ X ] = p Var[ X ]. Similar deﬁnitions hold for continuous random variables. The idea is that, if returns vary a lot around their expectation, they are intuitively “risky”. Note however that one might argue that upside variability are not so bad, so one should really only take downside variability into account; it is indeed possible to do so, but the math is ugly, so this consideration is typically disregarded. In general, the DM’s preferences among risky assets, or random variables in general, are not solely a function of their means and variances. For instance, if the probability distribution of X is (100 , 9 32 ; 200 , 14 32 ; 300 , 9 32 ) and that of Y is (50 , 1 8 ; 200 , 3 4 ; 350 , 1 8 ), then E[ X ] = E[ Y ] = 200 and Var[ X ] = Var [ Y ] = 75, but a DM with utility u ( x ) = x will strictly prefer X to Y , as will a DM with power utility u ( x ) = x 1 - γ 1 - γ with e.g. γ = 3. However, the idea that “expected return is good, variance is bad” is so appealing (and, it turns out, simple) that people have tried to provide justiﬁcations for analyzing portfolio choice solely in terms of these two statistics. A “local” rationale for this is provided by the Arrow approximation, which we considered last time. Recall that the (insurance) risk premium π ( W, ˜ ± ) for a small risk ˜ ± with mean 0 is approximately 1 2 A ( W )Var[˜ ± ], where A ( w ) is the Arrow-Pratt measure of risk aversion. Also remember that the certainty equivalent and risk premium are related by the equation W - π ( W, ˜ ± ) = c W ± . If we now reinterpret W as the expectation of a random variable ˜ W representing the ﬁnal wealth of an investor who purchases a risky portfolio with Var[ ˜ W ] = Var[˜ ± ], we can write c ˜ W = E[ ˜ W ] - 1 2 A ( W )Var[ ˜ W ] : that is, if the variance of ˜ W is small, the certainty equivalent of terminal wealth is approximately equal to its expectation minus a constant times its variance. Since ranking random variables by their certainty equivalent is by deﬁnition the same as ranking them according to their expected 1

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## This note was uploaded on 11/17/2009 for the course ECONOMICS 331 taught by Professor Marciano during the Spring '09 term at Northwestern.

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eco331-09-MeanVariance - Northwestern University Fall 2009...

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