Reinvestment of Returns and Geometric Growth Concepts 115
third bet, being a loss of $1, lowers your total winnings to $1. Since you
encountered a loss, however, you recapitalize your bet size to the base bet
($1) plus 50% of your winnings (.5 * $1) and hence bet $1.50 on the fourth
bet. The fourth bet was a winner, paying 1 to 1, so you made $1.50 on the
fourth bet, bringing your total winnings to $2.50. Since this last bet was a
winner, you will not recapitalize (step up or down) your bet size into the
fifth bet; instead, you stay with a bet size of $1.50 into the fifth bet.
On the surface, the reserve strategy seems like an ideal staking system.
However, like all staking systems, its long-term performance falls short of
the simple fixed fraction (small antimartingale) approach. Another popular
idea of gamblers/traders has been the base bet plus square root strategy,
whereby you essentially are always betting the same amount you started
with plus the square root of any winnings. As you can see, the possibilities
of staking systems are endless.
Many people seem to be partial, for whatever reason, to adding con-
tracts after a losing trade, a streak of losing trades, or a drawdown. Over
and over again in computer simulations (by myself and others) this turns
out to be a very poor method of money management. It is akin to the martin-
gale and small martingale. Since we have determined that trading is largely
an independent trials process, the past trades have no effect on the present
trade. It doesn’t matter whether the last 20 trades were all winners or all
losers.
It is interesting to note that those computer tests that have been per-
formed all bear out the same thing. In an independent trials process where
you have an edge, you are better off to increase your bet size as your stake
increases, and the bet size optimally is a fixed fraction of your entire stake.
Time and again authors have performed studies that take a very long stream
of independent outcomes with a net positive result, and have applied vari-
ous staking systems to betting/trading on these outcomes. In a nutshell, the
outcomes of every study of this type reach the same conclusion: that you
are better off using a staking system that increases the size of the bet in
direct proportion to the size of the total stake.
In another study, William T. Ziemba demonstrated in the June 1987 issue
ofGambling Timesmagazine that proportional betting was superior to any
other staking strategy.^4 Furthermore, Ziemba’s article demonstrated how
the optimal proportion (determined correctly by the Kelly formula in this
study) far outperforms any other proportion. The study simulated 1,000
seasons of betting on 700 horse races, starting you out with an initial stake
of $1,000. The test looked at such outcomes as how many seasons would
(^4) Ziemba, William T., “A Betting Simulation, The Mathematics of Gambling and In-
vestment,”Gambling Times, pp. 46–47, 80, June 1987.