130 THE HANDBOOK OF PORTFOLIO MATHEMATICS
loss of−$600 by .01 we will divide it by .02. Therefore, we will begin this
next pass through with a starting value of 600/.02=30000.
P&L WORKING VALUE
30000
+ 100 30100 ((30000/30000)*100)+ 30000
− 500 29598.33 ((30100/30000)*−500)+ 30100
+ 1500 31078.2465 ((29598.33/30000)*1500)+29598.33
− 600 30456.68157 ((31078.2465/30000)*−600)+31078.2465
Here the TWR=30456.68157/30000=1.015222719We keep on repeating the procedure until we obtain the value forfthat
results in the highest TWR. The answers we obtain for TWRs, as well as
for the optimalf, will be the same with this technique as with the previous
technique using the HPRs.
Definition of the Problem
It is a mathematical fact that when two people face the same sequence of fa-
vorable betting/trading opportunities, if one uses the optimalfand the other
uses any different money-management system, then the ratio of the optimal
fbettor’s stake to the other person’s stake will increase as time goes on,
with higher and higher probability. In the long run, the optimalfbettor will
have infinitely greater wealth than any other money-management-system
bettor with a probability approaching one.
Furthermore, if a bettor has the goal of reaching a prespecified for-
tune, and is facing a series of favorable betting/trading opportunities, the
expected time needed to reach the fortune will be less with optimalfthan
with any other betting system.
Obviously, the optimalfstrategy satisfies desirable property number
1 for money management, as it makes the most amount of money that is
mathematically possible using a fixed fractional betting strategy on a game
where you have the edge. Since optimalfincorporates the probability of
winning as well as the amounts won and lost, it also satisfies desirable
property numbers 3 and 4. Not much has been discussed about desirable
property number 2, the security aspect, but this will be treated in the closing
chapters.
Let’s go back and reconsider the following sequence of bets/trades: