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=
0.03 , Σ= 0.0064 0.0008 −0.0011
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 (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.
- Fall '13
- The Aeneid