Calculation of investment portfolios with risk free borrowing and lending

We colisider the .problem of portfolio selection for a risk averse investor wishing to allocate his resources among several investment opportunities in order to maximize the expected utility of final wealth. The calculation of the optimal investment proportions generally requires the solution of a stochastic program whose dimension is the number of risky investments. The computations are simplified dramatically when there is a risk free asset and the investment returns are jointly normally distributed. In this case Tobin has shown that the investment proportions in the risky assets are independent of the utility function and Lintner has shown that these proportions may be obtained from the solution of a fractional program. It is shown under mild hypotheses that the fractional program has a pseudo-concave objective and that the program always has a unique solution. The solution may be sought in several ways, perhaps most efficiently via Lemke's algorithm applied to a linear complementarity problem. The optimal investment proportions in all assets may be found by solving a stochastic program having one random variable and one decision variable via a search technique. Data on the major pooled Canadian equity pension funds were used to provide an empirical test of the suggested solution approach. Five common classes of utility functions were utilized with varying parameter values. For each class there are smooth curves that related the investment in the risk free asset to the parameters of the utility function. The investor is more risk averse when faced with quarterly data.