S tanactsc 372 f11 modern portfolio theory capm p 26

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Unformatted text preview: odel (N Risky Assets): A Summary Optimal Portfolio Weights: x∗ = x∗ (τ ) = xMIN + τ z ∗ for τ ≥ 0, where xMIN = 1 e T Σ− 1 µ − 1 Σ−1 e and z ∗ = Σ−1 µ − Σe eT Σ−1 e e T Σ− 1 e Optimal risk-reward tradeoffs: 2 2 µx∗ = µxMIN + τ µz ∗ and σx∗ = σxMIN + τ 2 µz ∗ Efficient Frontier: 2 σx∗ − (µx∗ − µxMIN )2 2 = σxMIN µz ∗ K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 25 Motivation for Mean-Variance Optimization A mean-variance optimizer can be justified based on either of the following : (i) Investor has quadratic utility: (ii) Asset returns follow a multivariate normal distribution (R1 , R2 , · · · , RN ) ∼ M V N (µ, Σ) (i) Quadratic utility: u(x) = x − αx2 , x < (2α)−1 , α > 0 Suppose an initial wealth is invested in a portfolio x with random return Rx terminal wealth = w0 (1 + Rx ) utility of the terminal wealth: u(w0 (1 + Rx )) = w0 (1 + Rx ) − α[w0 (1 + Rx )]2 22 = w0 (1 − αw0 ) + (1 − 2αw0 )...
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