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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Single index model - Short sales not allowed Risk free asset exists Kuhn-Tucker conditions If we assume short sales then we can simply maximize the slope and find the tangent to the efficient frontier subject to the constraint ∑ N i =1 x 1 = 1 max θ = ¯ R G- R f σ G To find the x i s we take derivatives w.r.t. each x i set them equal to zero and solve ··· dθ dx i = z i σ 2 i + N X j 6 = i z j σ ij = 0 , i = 1 , ··· N or ¯ R i- R f = z i σ 2 i + N X j 6 = i z j σ ij , i = 1 , ··· N If short sales are not allowed we have an extra set of constraints x i ≥ 0. We still take the derivative w.r.t. each x i but now if the maximum occurs at x i < 0 then it is not feasible for our problem. Then dθ dx i < 0. But if the maximum occurs at a positive x i then dθ dx i = 0 (see figure below). To summarize dθ dx i ≤ which can me written as equality dθ dx i + u i = 0...
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This note was uploaded on 06/02/2011 for the course STATS 183 taught by Professor Nicolas during the Spring '11 term at UCLA.
- Spring '11