STOCHASTIC

326 THE JOURNAL OF BUSINESS
this opportunity are
and that
where fj(xj) is given by (9).
It is easily verified that
Thus,
22 .J
_ (0 with probability 1/2
fty ~~ 13 with probability 1/2
(9) clearly belongs to the class (7). (4) and (3) now give
where
fdoes not exist Xj-l < 0
0 < *j_i < d .
+ d) xj-i > d
-I(ZJ--I) = \xj-i
ll/2(^_,
| = 1l/2(3s,_, + d)
fj-l(.Xj-C
IOJ_I(*J-_I + d) 1
>' 2 + l/2d» 2
/2
0 < Xj_! < d
xj-i > d
aU J (8)
fj{xj) = (xj + d)^ d>0. (9)
/y(xy) = max £{/y+i[(&y - l)z2y + *y]} i = 1, . . . , 7 - 1 , (10)
0£«jy£*y
(11)
(12)
o/_, = l/2[(l/2)" 2 + 2» 2 ] . (13)
We now observe that fj-i(x) is a positive linear transformation of fj(x) only for
x > d; for x < d,fj^(x) < aj-ifj(x).
Proceeding with the solution to (10), we obtain
h(% ' > jfly(*y + « X,->bs 3 l '-'-' J l ' U * ;
where ay is a positive constant, gy(xy) < O.J(XJ + d) in for 0 < Xy < 6„
and
J> = b, J - 1 , J- - 1 • (16)
It is easily determined that hj(x,) is highly irregular except for j = / — 1.
When / = 11 and d = 1,000, we obtain from (IS) that Ai = 1.023 million. Thus,
when the horizon is ten periods distant, the optimal amount to invest in opportunity
2 is, in this example, proportional to Xy + d only if initial wealth xt exceeds
$1 million by a substantial margin. Furthermore, while/y(x) is a positive linear transformation
of fj(x) for x > bj, j = 1, . . . , J — 1, it is not for x < bs, that is, for
Xi < 1.023 million, x2 < 511,000, etc., in the above example. Since the constant
bj > 0 depends on the distribution functions Fj, . . . , Fj-i, the short-run utility
functions induced by the terminal utility function (9) are clearly not myopic. In
other words, to make an optimal decision at decision point j, not only Fj but Fj+h
. . . , Fj-i must be known.
IV. BORROWING AND SOLVENCY
The nonmyopic nature of the induced utility functions/i(xi), /2(x2), • • • ,fj-i(xj-i)
in the preceding example is clearly attributable to the constraint
0 < zij < XJ j = 1, .. . , J - 1 , (17)
which precludes borrowing and short sales. We shall now relax this constraint.
3. MYOPIC PORTFOLIO POLICIES 403