228 THE HANDBOOK OF PORTFOLIO MATHEMATICS
But what if a win of $2 were only one-and-a-half times as valuable to
us as a loss of $1? To determine how to maximize utility then, we assign a
util value of−1 to the losing scenario, the tails scenario, and a utils value
of 1.5 to the winning scenario, the heads scenario. Now, we determine the
optimalfbased upon these utils rather than dollars, and we find it to be
.166666, or to bet 16^2 / 3 % on each and every play to maximize our geometric
average utility. That means we divide our total cumulative utils to this point
by .166666 to determine the number of contracts.
We can then translate this into how many contracts we have per dollars
in our account, and, from there, figure what thefvalue (between zero and
one) is that we are really using (based on dollars, not utils).
If we do this, then the original two-to-one coin-toss curve of wealth
maximization, which peaks at .25 (Figure 9.2), still applies, and we are at
the .166666fabscissa.Thus, we pay the consequences of being suboptimal in
terms offon our wealth. However, there is a secondfcurve—one based on
our utility—which peaks at .166666, and we are at the optimalfon this curve.
Notice that, if we were to accept the .25 optimalfon this curve, we would
be way right of the peak and would pay the concomitant consequences of
being right of the peak with respect to our utility.
Now, suppose we were profitable in this holding period, and we go in
and update the outcomes of the scenarios based on utils, only this time,
since we have more wealth, the utility of a winning scenario in the next
holding period is only 1.4 utils. Again, we would find our optimalfbased
on utils. Again, once we determined how many units to trade in the next
holding period based on our cumulative utils, we could translate it into
what thef(between zero and one) is for dollars, and we would find it to be
nonuniform from the previous holding period.
The example shown is one in which we assume a sequence of more than
one play, where we are reusing the same money we started with. If there
was only one play, one holding period, or we received new money to play at
each holding period, maximizing the arithmetic expected utility would be
the optimal strategy. However, in most cases we must reuse the money on the
next play, the next holding period, that we have used on this last play, and,
therefore, we must strive to maximize geometric expected growth. To some,
this might mean maximizing the geometric expected growth of wealth; to
others, the geometric expected growth of utility. The mathematics is the
same for both. Both have two curves inn+1 space: a wealth maximization
curve and a utility maximization curve. For those maximizing the expected
growth of wealth, the two are the same.
If the reader has a different utility preference curve other than ln (wealth
maximization), he may apply the techniques herein, provided he substitutes
autilsquantity for the outcome of each scenario rather than a monetary