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The present paper studies a multi-stage portfolio optimization problem with bankruptcy and stage-by-stage value at risk (VaR) constraints that impose boundary for probability of the percentage of the investor's capital shortfall at each stage. The goal function is the mean value of final investor's capital. Making use of the stage-by-stage VaR constraints and a multivariate normal model for rates of return, the method of dynamic programming is applied. Due to peculiarities of this optimal control problem of the Markov chain with a set of absorbing states, an optimal investment policy turns out to be a relatively simple policy. More exactly, at each stage optimal portfolio depends only on the number of stage, but not on the value of current investor's capital. The initial problem is reduced to a sequence of one-stage portfolio optimization problems. Analysis of such one-stage problem is mainly based on known results, providing sufficient and necessary conditions for fulfilment of Slater's constraint qualification, as well as conditions for optimality. In addition, we extend the obtained results to a non-normality situation by use of elliptical distributions.

Optimal investment policy in a multi-stage problem with bankruptcy and stage-by-stage probability constraints