STOCHASTIC

known as partial myopia, and it states that the "induced utility function" U{w) can be computed
by pretending that all of the investor's wealth is invested in a risk-free asset with rate
of return r. The equivalence of U(w) and u{w) is known as complete myopia, and states that
in a multiperiod investment problem, the "induced utility function" remains unchanged.
In both cases, the multiperiod investment problem is simplified to a sequence of one-period
problems. See, however, the Hakansson paper for difficulties arising from endpoint maxima.
6. Referring to Exercise 5, suppose that the investor's utility for terminal wealth is
u(w) = log(w + A:), where A: > 0 is constant. Assume that v & r(l + (a 2 In 2 )), and assume that
borrowing and short sales are not allowed.
(a) Show that the optimal policy for a 1-period problem is to invest totally in the risky
asset. [Hint: If w0 is initial wealth and '(w0) £ (j>'(w0 + k/r).]
We now demonstrate that the optimal policy for an n-period problem is to invest totally
in the risky asset in each period. If A" is a normal random variate with mean m and variance
s 2 , let N(x;m,s 2 ) = PrpCg x\. Let
Ut(w) = max Eu[(w — a)r+ap].
a 6 [0, w]
(b) Show that U/(w) = (llw)F(w/k), where
F(x) = 1 - jll+xe"]- 1 dN{y,m,s 2 )
for appropriate m and s 2 .
For notational simplicity, let N(x) = N(x;m,s 2 ) in the following. Let ].
(c) Show that
0>2'()v) = - j(l-re-')Flje x \ dN(x).
(d) Show that F((wjk)e") is positive and strictly increasing in x.
(e) Using (d), show that /(w) g 0. This establishes the result for n -•
(f) Show by induction that the result is true for all n.
EU„ [(w — a) r + ap], show that
where
«>; + i(M')= l - j{l-re-')F„+Jje*\ dN(x),
f,«W = j -• j F(y exp(*i + ••• + x„)) dN(Xl) ••• dN{x„).
Hint: If