For investment managers through to the individual the task of solving their particular portfolio problems remains a principal objective. It has been shown that an efficient portfolio can be specified in terms of its expected return and profit variance, (risk), that is the first two moments of the investor's subjective yield distribution. Selection of an efficient portfolio can always be achieved by quadratic or homogeneous programming. An integral part of the efficient portfolio selection process by homogeneous programming lies in the use of the truncated minimax criterion which gives a measure of risk preference, m. Varying m will give the complete set of efficient portfolios from all possible ones. This is detailed in chapter one where it is also shown that an optimal portfolio which allocates the budget with maximal caution can be selected from among the efficient ones under the additional criterion that the marginal value of the investment dollar is not exceeded by its marginal cost. Using a specified algorithm an optimal portfolio is selected from stocks qualifying as Trustee Investments under the Trustee Amendment Act 1974 listed on tho New Zealand Stock Exchange. Chapter two details the manner in which a five year data base of weekly observations, 1979 to 1983 inclusive, was developed for this operation and gives the preliminary results of expected return and profit variance for the stocks selected. A printout of the complete data file is included in the appendix. Chapter three of this thesis shows in detail the manner in which the algorithm is applied and gives a final result using weekly data over the four year period, 1980 - 1983 inclusive. The characteristics of this optimal portfolio are shown together with details of its performance over the twelve month period Jan - Dec 1984. Finally consideration is given to the robust nature of the portfolio selection system by looking both at a range of efficient portfolios selected from the four year data and also an optimal result from the full five year data.