The Journal of finance t above the line(1-Xi-X2=0) is not attainable because it violates the condition that X3=1-X1-X2>0. We define an isomean curve to be the set of all points (portfolios) with a given expected return. Similarly an isovariance line is defined to be the set of all points (portfolios)with a given variance of return. An examination of the formulae for e and v tells us the shapes of the isomean and isovariance curves. Specifically they tell us that typically the isomean curves are a system of parallel straight lines; the isovari- ance curves are a system of concentric ellipses(see Fig. 2). For example if u2* u3 equation 1 can be written in the familiar form X2=a+ bX1; specifically(1) E Thus the slope of the isomean line associated with e Eo is-(u1 u3)/(u2-43)its intercept is(Eo-43/(u2 -u3). If we change E we change the intercept but not the slope of the isomean line. This con firms the contention that the isomean lines form a system of parallel Similarly, by a somewhat less simple application of analytic geome try, we can confirm the contention that the isovariance lines form a family of concentric ellipses. The "center"of the system is the point which minimizes V. We will label this point X. Its expected return and variance we will label E and V Variance increases as you move away from X. More precisely, if one isovariance curve, Cl, lies closer to X than another, C2, then Cu is associated with a smaller variance than C2 With the aid of the foregoing geometric apparatus let us seek the efficient sets. X, the center of the system of isovariance ellipses, may fall either inside or outside the attainable set Figure 4 illustrates a case in which X falls inside the attainable set In this case: X is efficient For no other portfolio has a V as low as X; therefore no portfolio can have either ller V(with the same or greater E)or greater E with the same or smaller V. No point (portfolio) with expected return E less than E is efficient. For we have e> e and v< v Consider all points with a given expected return E; i.e., all points on the isomean line associated with E. The point of the isomean line at which V takes on its least value is the point at which the isomean line 9. The isomean“curv e as described above except when u u= ua. In the latter case all portfolios have the same expected return and the invest l s to the assumptions implicit in our description of the isovariance curves see footnote84 The Journal of Finance point above the line (1 -X1 - Xz = 0) is not attainable because it violates the condition that X3 = 1 -XI -Xz > 0. We define an isomean curve to be the set of all points (portfolios) with a given expected return. Similarly an isovariance line is defined to be the set of all points (portfolios) with a given variance of return. An examination of the formulae for E and V tells us the shapes of the isomean and isovariance curves. Specifically they tell us that typicallyg the isomean curves are a system of parallel straight lines; the isovari￾ance curves are a system of concentric ellipses (see Fig. 2). For example, if ~2 p3 equation 1' can be written in the familiar form X2 = a + bX1; specifically (1) Thus the slope of the isomean line associated with E = Eois -(pl - j~3)/(.~2- p3) its intercept is (Eo - p3)/(p2 - p3). If we change E we change the intercept but not the slope of the isomean line. This con￾firms the contention that the isomean lines form a system of parallel lines. Similarly, by a somewhat less simple application of analytic geome￾try, we can confirm the contention that the isovariance lines form a family of concentric ellipses. The "center" of the system is the point which minimizes V. We will label this point X. Its expected return and variance we will label E and V. Variance increases as you move away from X. More precisely, if one isovariance curve, C1, lies closer to X than another, Cz, then C1 is associated with a smaller variance than Cz. With the aid of the foregoing geometric apparatus let us seek the efficient sets. X, the center of the system of isovariance ellipses, may fall either inside or outside the attainable set. Figure 4 illustrates a case in which Xfalls inside the attainable set. In this case: Xis efficient. For no other portfolio has a V as low as X; therefore no portfolio can have either smaller V (with the same or greater E) or greater E with the same or smaller V. No point (portfolio) with expected return E less than E is efficient. For we have E > E and V < V. Consider all points with a given expected return E; i.e., all points on the isomean line associated with E. The point of the isomean line at which V takes on its least value is the point at which the isomean line 9. The isomean "curves" are as described above except when = pz = pa In the latter case all portfolios have the same expected return and the investor chooses the one with minimum variance. As to the assumptions implicit in our description of the isovariance curves see footnote 12