F4160-4 - 4 Mean-Variance Analysis and Portfolio Separation...

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4. Mean-Variance Analysis and Portfolio Separation 4.1. Motivation Accepting the expected utility maximization framework, we now discuss the conditions for mean-variance analysis to be valid. An individual’s utility function may be expanded as a Taylor series around his expected end-of-period wealth (assume the Taylor series converges): u ( f W ) = u ( E ( f W )) + u 0 ( E ( f W ))( f W E ( f W )) + 1 2 u 00 ( E ( f W ))( f W E ( f W )) 2 + R 3 where R 3 = P n =3 1 n ! u ( n ) ( E ( f W ))( f W E ( f W )) n and u ( n ) ( · ) is the n th derivative of u ( · ) . Since the expectation and summation operators are interchangeable, the individual’s expected utility can be expressed as E ( u ( f W )) = u ( E ( f W )) + 1 2 u 00 ( E ( f W )) σ 2 ( f W ) + E ( R 3 ) (4.1) where E ( R 3 ) = X n =3 1 n ! u ( n ) ( E ( f W )) m n ( f W ) (4.2) and m n ( f W ) = E (( f W E ( f W )) n ) is the n th central moment of f W . Note that equation (4.1) indicates a preference for expected wealth and an aversion to variance of wealth for an individ- ual having an increasing and strictly concave utility function. However, equation (4.2) shows that expected utility cannot be de fi ned solely over the expected value and variance of wealth for arbitrary distributions and preferences, as indicated by the remainder term which involves higher order terms. Therefore, to motivate the mean-variance model, we need at least one of the following: 1. Quadratic preferences (for arbitrary distributions): u ( f W ) = f W b 2 f W 2 Under quadratic preferences, the third and higher order derivatives of u ( · ) are zero. That is, E ( R 3 ) = 0 and E ( u ( f W )) = E ( f W ) b 2 E ( f W 2 ) = E ( f W ) b 2 ³ ( E ( f W )) 2 + σ 2 ( f W ) ´ Therefore, when E ( f W ) and σ 2 ( f W ) are fi nite, quadratic utility is su cient (in fact, necessary for arbitrary distributions) for mean-variance analysis. 1
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Remark 1. Some undesirable properties are: a. Satiation: An increase in wealth beyond the satiation point decreases utility. b. Increasing absolute risk aversion in wealth: R A ( f W ) u 00 ( f W ) u 0 ( f W ) = b 1 b f W d R A ( f W ) d f W = b 2 (1 b f W ) 2 > 0 This implies that risky assets are inferior goods. 2. Rates of return follow a multivariate normal distribution (for arbitrary preferences): In general, a su cient condition for mean-variance framework is that ( µ, σ ) can com- pletely characterize the feasible choices, and such two-parameter distributions are closed ( stable ) under linear combinations, i.e., X, Y F aX + bY F Remark 2. A disadvantage of using normal distribution is that it is unbounded from below. This is inconsistent with limited liability and with economic theory, which at- tributes no meaning to negative consumption. De fi ne the relationship between wealth and return as e r f W W 0 W 0 = f W W 0 1 with µ E ( e r ) = E ( i W W 0 1) and σ 2 σ 2 ( e r ) = σ 2 ( i W ) W 2 0 . To show the representation of going from u ( f W ) to v ( µ, σ 2 ) , we can write the utility function as u ( · ) = u ( e r ; µ, σ 2 ) E ( u ( · )) = Z −∞ u ( e r ; µ, σ 2 ) f ( e r ; µ, σ 2 ) d e r Standardize e r by de fi ning e z = h r µ σ . Then e r = µ + σ e z , which implies that e z = ± when e r = ± , and d e r = σd e z . By change of variable, f ( e r ; µ, σ 2 ) = 1
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