162 THE HANDBOOK OF PORTFOLIO MATHEMATICS
or
G=
⎛
⎝
∏T
i= 1
⎛
⎝
(
1 +
(
Ai
W
f
))Pi⎞
⎠
⎞
⎠
1 /Pi
(4.18a)
where: T=Number of different scenarios.
TWR=Terminal wealth relative.
HPRi=Holding period return of theith scenario.
Ai=Outcome of theith scenario.
Pi=Probability of theith scenario.
W=Worst outcome of allnscenarios.
f=Value forfwhich we are testing.
Just as you could use Equation (4.04) to solve Equation (4.03), like-
wise you can use Equation (4.18a) to solveanyoptimalfproblem. It will
yield the same answers as the Kelly formulas when the data correctly has
a Bernoulli distribution. It will yield the same answers as previously men-
tioned formulas if you pump a distribution of trades through it (where the
probability of each trade is 1/T). This formula can be used to maximize the
expected value of the logarithm of any starting quantity of anything when
there is exponential growth involved. We will now see how to employ this
formula in the context ofscenario planning.
Scenario Planning
People who forecast for a living, be they economists, stock market fore-
casters, meteorologists, government agencies, or the like, have a notorious
history for incorrect forecasts. Most decisions anyone must make in life
usually require that the individual make a forecast about the future.
There are a couple of pitfalls that immediately crop up. To begin with,
people generally make more optimistic assumptions about the future than
the actual probabilities. Most people feel that they are far more likely to
win the lottery this month than they are to die in an auto accident, even
though the probabilities of the latter are greater. This is true not only on
the level of the individual; it is even more pronounced at the group level.
When people work together, they tend to see a favorable outcome as the
most likely result.
The second pitfall—and the more harmful—is that people make
straight-line forecasts into the future. People predict what the price of a
gallon of gas will be two years from now; they predict what will happen
with their jobs, who the next president will be, what the next styles will
