Utility and Portfolio Choice

# One period model 2 quadratic utility function or the

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Unformatted text preview: . © P. KOLM. 14 Portfolio Choice • Portfolio choice is dynamic and multi-period in nature, max E 0 (u(C 0,C 1,...,CT −1,WT )) , or ⎛T −1 ⎞ ⎜ u(C , t ) + u( ,T )⎟ ⎟ max E 0 ⎜∑ WT t ⎟ ⎜ t =0 ⎟ ⎝ ⎠ where C t denotes consumption in period t • For most applications the “full” problem is too difficult to solve for realistic portfolios so a number of simplifying assumptions are typically applied: 1. One period model 2. Quadratic utility function or the return distribution is jointly normally distributed → mean-variance optimization RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 15 1. The One Period Model max E (u(W )) {wi } where ⎛ ⎞ ⎜W − ∑ w ⎟(1 + r ) + ∑ w (1 + r ) W =⎜ 0 i⎟ f i i ⎟ ⎜ ⎝ ⎠ i i = W0 (1 + rf ) + ∑ wi (ri − rf ) i Necessary and sufficient conditions (For you: derive these) E ⎡⎢u ′(W )(ri − rf )⎤⎥ = 0 ⎣ ⎦ for all i E ⎡⎢u ′′(W )(ri − rf )2 ⎤⎥ < 0 ⎣ ⎦ for all i (Here we see that u ′ > 0 and u′′ < 0 m...
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## This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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