bv_cvxbook_extra_exercises

Once you 111 know this you must choose the quantities

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Unformatted text preview: i are λi to one, then the return matrix R ∈ Rm×m is given by rij = λi if j = i rij = −1 otherwise. 107 Show that there is no sure-win betting scheme (or arbitrage opportunity) if m i=1 1 = 1. 1 + λi In fact, you can verify that if this equality is not satisfied, then the betting strategy xi = 1/(1 + λi ) 1 − m 1/(1 + λi ) i=1 always results in a profit. The common situation in real life is m i=1 1 > 1, 1 + λi because the bookmakers take a cut on all bets. 13.8 Log-optimal investment. We consider an instance of the log-optimal investment problem described in exercise 4.60 of Convex Optimization. In this exercise, however, we allow x, the allocation vector, to have negative components. Investing a negative amount xi W (t) in an asset is called shorting the asset. It means you borrow the asset, sell it for |xi W (t)|, and have an obligation to purchase it back later and return it to the lender. (a) Let R be the n × m-matrix with columns rj : R= r 1 r 2 · · · rm . We assume that the elements rij of R are all pos...
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