Portfolio selection problem

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In its elementary form, this is the same as the capital budgeting problem, except that the objective is to minimize the risk, rather than maximize expected return. Let LaTeX: x_j be the percent of capital invested in the j-th opportunity (e.g., stock or bond), so LaTeX: x must satisfy LaTeX: \textstyle x \ge 0 and LaTeX: \textstyle \sum_j x_j = 1. Let LaTeX: v_j be the expected return per unit of investment in the j-th opportunity, so that LaTeX: vx is the sum total rate of return per unit of capital invested. It is required to have a lower limit on this rate: LaTeX: \textstyle vx \ge b (where LaTeX: b is between LaTeX: \min v_j and LaTeX: \max v_j). Subject to this rate of return constraint, the objective is the quadratic form, LaTeX: x^TQx, where LaTeX: Q is the variance-covariance matrix associated with the investments (i.e., if the actual return rate is LaTeX: V_j, then LaTeX: \textstyle Q(i,j) = E[(V_i - v_i)(V_j - v_j)].

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