In real investment situations, one desires to only minimize downside risk or portfolio loss without affecting the upside potentials. This can be accomplished by mean semi-variance optimization but not by mean variance. In the Black-Scholes setting, this paper proposes for the very practical yet intractable dynamic mean semi-variance portfolio optimization problem, an almost analytical solution. It proceeds by reducing the multi-dimensional portfolio selection problem to a one-dimensional optimization problem, which is then expressed in terms of the normal density, leading to a very simple and efficient numerical algorithm. A numerical comparison of the efficient frontier for the mean variance and semi-variance portfolio optimization problem is presented.
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