The Journal of finance point above the line(1- X1-X2=0) is not attainable because it violates the condition that X3=1-X1-X220 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 the isomean curves are a system of parallel straight lines; the cally% isomean and isovariance curves. Specifically they tell us ance curves are a system of concentric ellipses(see Fig. 2). For example if un us equation 1 can be written in the familiar form X2=a+ 6X1; specifically(1) E X. Thus the slope of the isomean line associated with E= Eo is-(1 u3)/(u2-u3)its intercept is(E0-43)/(42-43). 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 ge 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 and nother, C?, then C1 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 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. ts with a 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 4=4= 43. In the As to the assumptions implicit in our description of the isovariance curves see footnote