STOCHASTIC

ON OPTIMAL MYOPIC PORTFOLIO POLICIES 325
As most portfolio models do, we assume, in addition to stochastically constant
returns to scale, perfect liquidity and divisibility of the assets at each (fixed) decision
point, absence of transaction costs, withdrawals, capital additions, and taxes,
and the opportunity to make short sales. Furthermore, we assume, until Section VI,
that the yields in the various periods are stochastically independent.
Since the end-of-period capital position is given by the proceeds from current
savings, or the negative of the repayment of current debt plus interest, plus the proceeds
from current risky investments, we have
where
Combining (1) and (2) we obtain
"i
xj+i = r,Zij + JjMtf > = 1, 2, ... , (1)
1-2
X>« = *y i = 1, 2,.... (2)
J=I
AT.
*y+i = H(0« - r,)za + rjXj j = 1, 2 (3)
l'e2
Let us now assume that fj(xj) is given for some horizon /. Then, as Mossin shows,
we may write, by the principle of optimality, 3
/,(*,) = max £[/y+l(zy+l)l j = 1, . . . , J - 1 (4)
.,.«*,.(*,.)
where Zj{x,) is the set of feasible investments at decision point j given that capital
is Xj. When there are two assets in each period (i.e., Afy = 2 for all7) and Zj(x,) =
\ZIJ: 0 < zn < Xj], that is, borrowing and short sales are ruled out, Mossin concludes
that/,•(*,•) = ajjix,) + bj (where at > 0 and b,- are constants), j = 1, . . . ,
/ — 1, that is, that the induced short-run utility functions at decision points 1, . . . ,
/ — 1 are completely myopic, if and only if (1) fj(x) is either logarithmic or a power
function, or (2) r, = r2 = . . . = o_i = 1 and/^z) is one of
fj(x) = -*-"*; (S)
fj(x) = log (x + M) ; (6)
//(*) = j^rj ( Xx + ")'-" x x ^ 0, x ^ 1, (7)
where n ?* 0 and X are constants. Note that X and n cannot both be negative.
While the first conclusion is beyond dispute, the second is incorrect, as are the
conclusions concerning partial myopia in general, except in a severely restricted
sense. We shall first demonstrate the assertion in the preceding case and then show
that it also holds when borrowing and short sales are not ruled out.
III. AN EXAMPLE
Assume that rx = n = . . . = rj^ = 1 and that there is only one risky opportunity
(i.e., Mj = 2) in each period. Moreover, assume that the proceeds ft; of
3 Richard Bellman, Dynamic Programming (Princeton, N.J.: Princeton University Press, 1957).
402 PART IV. DYNAMIC MODELS REDUCIBLE TO STATIC MODELS