bv_cvxbook_extra_exercises

The lorentz curve is a plot of the fraction fi of

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Unformatted text preview: itive, which implies that the log-optimal investment problem is feasible. Show the following property: if there exists a v ∈ Rn with 1T v = 0, RT v RT v = 0 0, (40) then the log-optimal investment problem is unbounded (assuming that the probabilities pj are all positive). (b) Derive a Lagrange dual of the log-optimal investment problem (or an equivalent problem of your choice). Use the Lagrange dual to show that the condition in part a is also necessary for unboundedness. In other words, the log-optimal investment problem is bounded if and only if there does not exist a v satisfying (40). (c) Consider the following small example. We have four scenarios and three investment options. The return vectors for the four scenarios are 2 r1 = 1.3 , 1 2 r2 = 0.5 , 1 The probabilities of the three scenarios are p1 = 1/3, p2 = 1/6, 108 0.5 r3 = 1.3 , 1 p3 = 1/3, 0.5 r4 = 0.5 . 1 p4 = 1/6. The interpretation is as follows. We can invest in two stocks. The first stock doubles in value in each period with a probabi...
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