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Unformatted text preview: 0 ) − w0 · u (w0 ) − µx − (σx + µx ) . 2 w0 u￿￿ (w0 ) K.S. Tan/Actsc 372 F11 u(w0 ) − 12 w 20 ￿￿ · u (w0) ￿ Modern Portfolio Theory & CAPM – p. 28 ￿ ( 0) − w20u ￿￿ww0) µx u( − 2 (σx + ￿ µ2 ) x 2 Further assuming µx ￿ σx , then ￿ ￿ 1 2 ￿￿ 2u￿ (w0 ) 2 E[u(w0 (1+Rx ))] = u(w0 )− w0 ·u (w0 ) − µx − σx . 2 w0 u￿￿ (w0 ) For risk averse agent: ￿ ￿ 2 max E[u(w0 (1 + Rx ))] ⇔ max 2τ µx − σx ) x x where τ =− u￿ (w0 ) w0 u￿￿ (w0 ) which reduces to “formulation III" of the optimization. 1 Aside: τ is known as the Arrow-Pratt measure of relative risk-aversion. K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 29 Investor Preferences & Distributional Moments ˜ ˜ Let W be the terminal wealth with E[W ] = µ ˜ Applying Taylor expansion (around µ) to u(W ): 1 ˜ ˜ ˜ u(W ) ≈ u(µ) + u￿ (µ)(W − µ) + u￿￿ (µ)(W − µ)2 2 1 1 ˜ ˜ + u￿￿￿ (µ)(W − µ)3 + u￿￿￿￿ (µ)(W − µ)4 + · · · 3! 4! ˜ ⇒ E[u(W )] ≈ u(...
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This note was uploaded on 01/04/2012 for the course ACTSC 372 taught by Professor Maryhardy during the Fall '09 term at Waterloo.

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