assign03soln

assign03soln - (2.11) The optimality conditions for this...

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Unformatted text preview: (2.11) The optimality conditions for this problem are tμ- Σ x = ul and l x = 0 , which implies that x =- u Σ- 1 l + t Σ- 1 μ and 0 = l x =- ul Σ- 1 l + tl Σ- 1 μ. Therefore, u = t l Σ- 1 μ l Σ- 1 l and x =- t l Σ- 1 μ l Σ- 1 l Σ- 1 l + t Σ- 1 μ = th 1 , which has μ p = μ x = tμ h 1 = tα 1 , σ 2 p = x Σ x = t 2 h 1 Σ h 1 = t 2 β 2 = t 2 α 1 , so t = ± σ p / √ α 1 . This gives us two straight lines in ( σ p ,μ p ) space: μ p = σ p √ α 1 and μ p =- σ p √ α 1 . .. . . . . . . . . . . . σ p ....................................................................................................................................... ....................................................................................................................................... . . . . . . . . . . . . . μ p 1 (2.13) Recall the two optimization problems: (2 . 1) min x ′ Σ x s.t. μ ′ x = μ p , l ′ x = 1 (2 . 3) min- tμ ′ x + 1 2 x ′ Σ x s.t. l ′ x = 1 (where t is allowed to be negative). Since both problems have convex quadratic objective functions and linear equality constraints, the necessary and sufficient conditions for the two problems are respectively given by (2 . 1 * ) braceleftBigg μ ′ x = μ p , l ′ x = 1- 2Σ x = u 1 μ + u 2 l for some u 1 , u 2 ∈ R (2 . 3 * ) braceleftBigg l ′ x = 1 tμ- Σ x = wl for some w ∈ R If x ∗ is optimal for (2.1), x ∗ must satisfy (2.1*); taking t =- u 1 / 2 and w = u 2 / 2, we have that x ∗ satisfies (2.3*). Thus x ∗ is optimal for (2.3) with parameter t =- u 1 / 2. Conversely, if x ∗ is optimal for (2.3), x ∗ must satisfy (2.3*); taking μ p = μ ′ x ∗ , u 1 =- 2 t and u 2 = 2 w , we have that x ∗ satisfies (2.1*). Thus x ∗ is optimal for (2.1) with parameter μ p = μ ′ x ∗ . (2.14a) From equation (2.8), the parametric-efficient portfolio for t ≥ 0 is x ( t ) = h + th 1 , where h = Σ − 1 l l ′ Σ − 1 l and h 1 = Σ − 1 μ- l ′ Σ − 1 μ l ′ Σ − 1 l Σ − 1 l . Consider the linear inequality system e ′ 1 x = e ′ 1 h + te ′ 1 h 1 ≥ e ′ 2 x = e ′ 2 h + te ′ 2 h 1 ≥ ....
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This note was uploaded on 03/29/2010 for the course CO CO372 taught by Professor Michael during the Winter '09 term at Waterloo.

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assign03soln - (2.11) The optimality conditions for this...

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