The journal of finance For fixed probability beliefs(μ,σ动 the investor has a choice of vari ous combinations of E and V depending on his choice of portfolio X XN. Suppose that the set of all obtainable(E, v) combina tions were as in Figure 1. The e-v rule states that the investor would or should )want to select one of those portfolios which give rise to the (e, V) combinations indicated as efficient in the figure; i.e., those with minimum V for given E or more and maximum E for given V or less. There are techniques by which we can compute the set of efficient portfolios and efficient(E, v) combinations associated with given u E,V combinations officient FIG. 1 and oi. We will not present these techniques here We will,however, illustrate geometrically the nature of the efficient surfaces for cases in which N(the number of available securities) is small. The calculation of efficient surfaces might possibly be of practi se. Perhaps there are ways, by combining statistical techniques and the judgment of experts, to form reasonable probability beliefs a. We could use these beliefs to compute the attainable efficient combinations of(E, v). The investor, being informed of what(E, n) combinations were attainable, could state which he desired. We could hen find the portfolio which gave this desired combination82 The Journal of Finance For fixed probability beliefs (pi, oij) the investor has a choice of vari￾ous combinations of E and V depending on his choice of portfolio XI, . . . ,XN.Suppose that the set of all obtainable (E, V) combina￾tions were as in Figure 1.The E-V rule states that the investor would (or should) want to select one of those portfolios which give rise to the (E, V) combinations indicated as efficient in the figure; i.e., those with minimum V for given E or more and maximum E for given V or less. There are techniques by which we can compute the set of efficient portfolios and efficient (E, V) combinations associated with given pi attainable E, V combinations and oij. We will not present these techniques here. We will, however, illustrate geometrically the nature of the efficient surfaces for cases in which N (the number of available securities) is small. The calculation of efficient surfaces might possibly be of practical use. Perhaps there are ways, by combining statistical techniques and the judgment of experts, to form reasonable probability beliefs (pi, aij).We could use these beliefs to compute the attainable efficient combinations of (E, V). The investor, being informed of what (E, V) combinations were attainable, could state which he desired. We could then find the portfolio which gave this desired combination