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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 = R G- R f G To find the x i s we take derivatives w.r.t. each x i set them equal to zero and solve d dx i = z i 2 i + N X j 6 = i z j ij = 0 , i = 1 , N or R i- R f = z i 2 i + N X j 6 = i z j 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 d dx i < 0. But if the maximum occurs at a positive x i then d dx i = 0 (see figure below). To summarize d dx i which can me written as equality d dx i + u i = 0...
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- Spring '11