THE JOURNAL OF BUSINESS Both for a theoretical development The optimal decision for the second nd for purposes of practical computa- period is obtained from (18)as tion, the solution method is very much complicated if statistical dependence 5.7A1-400 4.84 (23 among yields in different periods (i.e, serial correlation) is to be allowed for. This expression defines the best possible We shall therefore assume throughout decision, for any value of wealth at the that such dependence is absent, although beginning of the second period. When this decision rule is adopted, the ex- TABLE 1 pected value of final wealth will be, ac- rding to(19) AssEt 1 中(41)=aaxB2[U(4x2) Yield Yield 2.94 A 2.7675 +const E1=1 Vi=.01 Ea=5 v2=2.25 It is the expectation of this function hich is to be maximized by the first Initial wealth: Ao=200 period decision, which we achieve by Utility function: U=A2-(1/1,000)41 TABLE 2 this certainly means some loss of ger erality. The basic nature of the approach is the same however, and the conclusions we are to derive are certainly unaffected by this simplification Dependence among yields within any period would be rela ively easy to handle, but for a theoreti- cal development it does not seem worth maximizing the expectation of the func the extra trouble. Also, we shall ignore tion in parentheses transaction costs E1(A1 2.7675 B. A TWO-PERIOD EXAMPLE WITH QUADRATIC UTILITY Using formula ( 18)again (with a To illustrate the procedure, we shall 2.7675/2490), we then get the optimal an develop in some detail a numerical ex- as ample with two assets with random 57A0-360=1612.(24) yields. To simplify the notation as much 4.84 as possible, it is assumed that the yields Thus, the optimal decision to be effected X1and X2 are independent and that their immediately is to invest about 80.6 per distributions are the same in both peri cent in asset 1 and the remainder in asset ods. We take a1 and ay to be investment 2 in the first asset in periods 1 and 2, re- The possible outcomes of the spectively. The data of the example are period portfolio are then as show given in Table 1 Table 2(each with probability t) his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions222 THE JOURNAL OF BUSINESS Both for a theoretical development and for purposes of practical computa￾tion, the solution method is very much complicated if statistical dependence among yields in different periods (i.e., serial correlation) is to be allowed for. We shall therefore assume throughout that such dependence is absent, although this certainly means some loss of gen￾erality. The basic nature of the approach is the same, however, and the conclusions we are to derive are certainly unaffected by this simplification. Dependence among yields within any period would be rela￾tively easy to handle, but for a theoreti￾cal development it does not seem worth the extra trouble. Also, we shall ignore transaction costs. B. A TWO-PERIOD EXAMPLE WITH QUADRATIC UTILITY To illustrate the procedure, we shall develop in some detail a numerical ex￾ample with two assets with random yields. To simplify the notation as much as possible, it is assumed that the yields X1 and X2 are independent and that their distributions are the same in both peri￾ods. We take a, and a2 to be investment in the first asset in periods 1 and 2, re￾spectively. The data of the example are given in Table 1. TABLE 1 ASSET 1 ASSET 2 Yield P Yield P 0.0 .5 -1.0 .5 0.2 .5 2.0 .5 E1= . 1 V1=.01 E2=. 5 V2=2.25 Initial wealth: A o=200 Utility function: U=A2-(1/1,OOO)A2 The optimal decision for the second period is obtained from (18) as a2, -=.A 400, (23) 4.84 This expression defines the best possible decision/for any value of wealth at the beginning of the second period. When this decision rule is adopted, the ex￾pected value of final wealth will be, ac￾cording to (19): k1(Ai) =maxE2[U(A2) I a2 2.94 2.Al 27675 7 A2 + const. 2.42 2490 / It is the expectation of this function which is to be maximized by the first￾period decision, which we achieve by TABLE 2 Xi X2 Ai a 2 a2 in % Xi X: Ai ~~~~~~~ of Al 0.0 -1 161.2 107.2 66.5 0.2 -1 193.4 145.1 75.0 0.0 2 277.6 244.3 80.0 0.2 2 309.8 282.2 91.1 maximizing the expectation of the func￾tion in parentheses: 2.76 752 maxE1 Al- - A1l a, 2490 / Using formula (18) again (with a= 2.7675/2490), we then get the optimal a, as 5.7Ao-360 (2) a14.84 Thus, the optimal decision to be effected immediately is to invest about 80.6 per cent in asset 1 and the remainder in asset 2. The possible outcomes of the first￾period portfolio are then as shown in Table 2 (each with probability '). This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions