bv_cvxbook_extra_exercises

# E it is a sure win betting scheme in economics we say

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Unformatted text preview: Risk-return trade-oﬀ in portfolio optimization. We consider the portfolio risk-return trade-oﬀ problem of page 185, with the following data: p= ¯ 0.12 0.10 0.07 0.03 , Σ= 0.0064 0.0008 −0.0011 0.0008 0.0025 0 −0.0011 0 0.0004 0 0 0 0 0 0 0 . (a) Solve the quadratic program minimize −pT x + µxT Σx ¯ subject to 1T x = 1, x 0 for a large number of positive values of µ (for example, 100 values logarithmically spaced between 1 and 107 ). Plot the optimal values of the expected return pT x versus the standard ¯ T Σx)1/2 . Also make an area plot of the optimal portfolios x versus the standard deviation (x deviation (as in ﬁgure 4.12). (b) Assume the price change vector p is a Gaussian random variable, with mean p and covariance ¯ Σ. Formulate the problem maximize pT x ¯ subject to prob(pT x ≤ 0) ≤ η 1T x = 1, x 0, as a convex optimization problem, where η < 1/2 is a parameter. In this problem we maximize the expected return subject to a constraint on the probability of a negative return. Solve the problem for a large number of values of η between 10−4 and...
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## This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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