I have been reading Vince, R. (1992), The Mathematics of Money Management: Risk Analysis Techniques for Traders, John Wiley & Sons, Inc. Interesting book. A little confusing, but interesting. The central theme of the book is determining what the author terms "optimal f". Now just what this quantity is remains somewhat mysterious, but I have gathered that it is what is normally thought of as percentage risk. That is, what percentage of your trading capital you risk on any one trade. I am going to denote this as r (and assume that it is equivalent f, but expressed as a percentage).
Now the conventional wisdom is that you never have r > 1. A string of losses with larger r can leave your trading account severely depleted. At the same time, using r < 1 means that you are risking very little per trade, and consequently stand to gain commensurately little.
I have always just assumed that your winnings would scale linearly with r. For example, if you make $100 in a month using r = 1, then in the same month you would have made $200 with r = 2. Seems reasonable, right? Apparently not. The author of the book illustrates this point very nicely with an example of a simple coing tossing game in which you stand to win $2 (heads) or lose $1 (tails). In this case, it turns out that the optimal f is 0.25 (equivalent to r = 25). In other words, each time you play you should risk 25% of your capital.
So I wondered whether something similar applied with one of my trading strategies. So I chose a reasonably good set of parameters and then varied the percentage risk.
Wow, isn't that interesting? As r increases the payoff (average nett winnings per trade) increases until it reaches a peak (at r = 2.75) then declines. For r > 5.75 the payoff becomes negative and then gets progressively worse. The percentage drawdown also increases with r. Well, that stands to reason: the more you risk, the more you are likely to lose at any given time!
So, why is the simple linear assumption wrong? Well, evidently trading is a little non-linear. Consider the following two plots which show the evolution in account balance with (top) r = 2.75 and (bottom) r = 8.00.
Initially the curves look rather similar. However, whereas r = 2.75 incurs a relatively deep depression during the losing streak and is able to recover, r = 8.00 is almost completely depleted and really doesn't bounce back.
I am not suggesting that we should trade with r > 1, but it is interesting to know that there is an optimal percentage risk. Perhaps this might in some instances be at r < 1? More research required.