STOCHASTIC

(h) Show that the optimal policy for the logarithmic utility is
/,(*) = A, log* + B,; z*0 = lffa';
zfi solves maxZ(eZE\ogR(i= \, ...,n); A, = YJ=o«'; and
B, = -/I, log/1, + loga X a'-'/l, + £ a'-'/f^logtffe*..,) for t = 0,...,T,
i=t+i i=t+i
except that zf, = 0(/= 1, ...,n) and 5T = 0.
The solutions given above show that the optimal portfolio selection problem in each period
requires the expected utility maximization of a single-period utility function with constant
relative risk aversion r, in period t.
(i) Show that rt = 1 - A, +1 = 1 - y U= (+1 a', with y = 0 for the logarithmic case.
Note that risk tolerance increases or decreases with age according as risk tolerance is greater
or less than that of the logarithm.
(j) Show that for the power-law utility function, the optimal policy as T-* oo is
ft{W) = /1(A) w\ A = ya'/(l -a); z*0 = 1 - a;
z,*i solves maxz £ z SEIR"-"] (i = 1,..., n); and
/1(A) =