Portfolio selection Two conditions--at least--must be satisfied before it would be prac- tical to use efficient surfaces in the manner described above First, the investor must desire to act according to the e-v maxim. Second, we must be able to arrive at reasonable ui and oi. we will return to these matters later Let us consider the case of three securities. In the three security case our model reduces to 1)E x X;=1 4)X;≥0fo et 3′)X3=1-X If we substitute(3) in equation(1)and(2)we get E and V as functions of X1 and X2. For example we find 1)E=H3+X1(41-43)+X2(2-3) The exact formulas are not too important here(that of V is given be- low).We can simply write a) E=E(X1, X2) b) V=V(X1, X2) c)X1>0,X2≥0,1-X1-X2≥0 By using relations (a),(b),(c), we can work with two dimensional geometry The attainable set of portfolios consists of all portfolios which satisfy constraints(c)and(3)(or equivalently(3)and (4)). The at tainable combinations of X1, X2 are represented by the triangle abc in Figure 2. Any point to the left of the X2 axis is not attainable because it violates the condition that X1> 0. Any point below the X1 axis is not attainable because it violates the condition that X2>0. Any 2x on xi o+2x ti a 2 *a x+ 2o3a+ oa)+2x1 xa(0ua-0na-oa t oaPortfolio Selection 83 Two conditions-at least-must be satisfied before it would be prac￾tical to use efficient surfaces in the manner described above. First, the investor must desire to act according to the E-V maxim. Second, we must be able to arrive at reasonable pi and uij. We will return to these matters later. Let us consider the case of three securities. In the three security case our model reduces to 4) Xi>O for i=l,2,3. From (3) we get 3') Xs= 1-XI--Xz Ifwe substitute (3') in equation (1)and (2) we get E and V as functions of X1 and Xz. For example we find 1') E' =~3 +x1(111 -~ 3 +) x2 (112 - 113) The exact formulas are not too important here (that of V is given be￾low).8 We can simply write a) E =E (XI, Xd b) V = V (Xi, Xz) By using relations (a), (b), (c), we can work with two dimensional geometry. The attainable set of portfolios consists of all portfolios which satisfy constraints (c) and (3') (or equivalently (3) and (4)). The at￾tainable combinations of XI, X2 are represented by the triangle abc in Figure 2. Any point to the left of the Xz axis is not attainable because it violates the condition that X1 3 0. Any point below the X1 axis is not attainable because it violates the condition that Xz 3 0. Any