# Daníelsson, J., B. N. Jorgensen, X. Yang, and C. G. de Vries (2007). Optimal portfolio allocation under a probabilistic risk constraint and the incentives for ﬁnancial innovation. Annals of Finance 4(3), 345–367.

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We characterize the investor's optimal portfolio allocation subject to a budget constraint and a probabilistic VaR constraint in complete markets environments with a finite number of states. The set of feasible portfolios might no longer be connected or convex, while the number of local optima increases exponentially with the number of states, implying computational complexity. The optimal constrained portfolio allocation may therefore not be monotonic in the state--price density. We propose a type of financial innovation, which splits states of nature, that is shown to weakly enhance welfare, restore monotonicity of the optimal portfolio allocation in the state--price density, and reduce computational complexity