Ralph Vince - Portfolio Mathematics

Optimalf 171

Scenario Probability Result

A.3 − 20

B.4 0

C.3 30

Mathematical expectation=$3. 00

Optimal f=. 17

Geometric mean= 1. 0123

It doesn’t matter what these scenarios are, they can be anything. To
further illustrate this, they will simply be assigned letters, A, B, C in this
discussion. Further, it doesn’t matter what the result is; it can be just about
anything.
Our analysis determines that the black decision will present the follow-
ing scenarios:

Scenario Probability Result

A.3 − 10

B.4 5

C .15 6

D .15 20

Mathematical expectation=$2. 90

Optimalf=. 31

Geometric mean= 1. 0453

Many people would opt for the white decision, since it is the decision
with the higher mathematical expectation. With the white decision, you can
expect,on average, a $3.00 gain versus black’s $2.90 gain. Yet the black deci-
sion is actually the correct decision because it results in a greater geometric
mean. With the black decision, you would expect to make 4.53% (1.0453 – 1)
on averageas opposed to white’s 1.23% gain. When you consider the effects
of reinvestment, the black decision makes more than three times as much,
on average, as does the white decision!
The reader may protest at this point that, “We’re not doing this thing
over again; we’re only doing it once. We’re not reinvesting back into the
same future scenarios here. Won’t we come out ahead if we always select
the highest arithmetic mathematical expectation for each set of decisions
that present themselves to us?”
The only time we want to be making decisions based on greatest arith-
metic mathematical expectation is if we are planning on not reinvesting