lecture17 - Portfolio Optimization Expected returns over...

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e62: lecture 17 15/11/10 Portfolio Optimization Expected returns over the next year Invest all funds in aapl? “Don’t put all your eggs in one basket” Optimize risk adjusted profit 1 stock expected return goog 5% yhoo 4% msft 3% aapl 6% max expected profit - ρ × variance of profit e62: lecture 17 15/11/10 Positive definite: h > 0 Convex function: Unique minimizer: Negative definite: h < 0 Concave function: Unique maximizer: Quadratic Functions of a Scalar 2 f ( x ) = hx 2 + cx + d x * = - c/ 2 h x * = - c/ 2 h f ( α x + (1 - α ) y ) α f ( x ) + (1 - α ) f ( y ) x, y N , α (0 , 1) f ( α x + (1 - α ) y ) α f ( x ) + (1 - α ) f ( y ) x, y N , α (0 , 1) e62: lecture 17 15/11/10 Quadratic Functions of a Vector Positive definite matrix: H > 0 Convex function: Unique minimizer: Negative definite matrix: H < 0 Concave function: Unique maximizer: 3 x T Hx > 0 x N , x = 0 f ( x ) = x T Hx + cx + d x T Hx < 0 x N , x = 0 f ( α x + (1 - α ) y ) α f ( x ) + (1 - α ) f ( y ) x, y N , α (0 , 1) f ( α x + (1 - α ) y ) α f ( x ) + (1 - α ) f ( y ) x, y N ,
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