Ralph Vince - Portfolio Mathematics

Laws of Growth, Utility, and Finite Streams 211

quit after one play to .5 for both the first play and the second. That is, if we
were to quit after the second play, we should optimally bet .5 on both the first
and second plays to maximize growth. (Remember, we allowed for the first
play to be anfvalue different from the second, yet they both came out the
same: .5 in this case. It is a fact that if you are looking to maximize growth,
thefthat is optimal—for finite as well as infinite streams—is uniform.)
We can see this if we take the first two possible combinations of tosses:

Toss#

12

Heads

Heads

Tails

Heads

Tails

Tails

Which can be represented by the following outcomes:

Toss#

12

2

2

− 1

2

− 1

− 1

These outcomes can be expressed as holding period returns for variousf
values. In the following, it is shown for anfof .5 for the first toss, as well as
for anfof .5 for the second:

Toss#

12

2

2

.5

2

.5

.5