If the trader uses fixed fractional position sizing to take smaller positions as the account value falls, then he will reduce his bets as the account goes down (see “Constant vs. fixed fractional”). For example, a fixed fractional trader might risk 2% of the current account value on each losing trade. This will reduce the risk of losing half of the account because the losses will get smaller as the account gets smaller. More losses will be required to lose some large fraction of the account.
We can modify the Cox and Miller equation for fixed fractional position sizing. In their equation, the risk of losing one standard deviation is e^(-(2*a)/d) and the number of times d must be lost to lose z is given by z/d. In the equation below, the risk of losing one standard deviation remains e^(-(2*a)/d) and the number of times d must be lost to lose z is given by ln(1-z)/ln(1-d).
For fixed fractional position size, the risk of ruin equation is:
R = e^((-2*a/d)*(ln(1-z)/ln(1-d)))
risk of losing z fraction
of the account
2.71828, the base of the
ln(1-z) is the natural logarithm of (1-z)
The fraction of the account that might be lost, in this case 50%,
average, or mean return,
in this case 6%, or 0.06
standard deviation of returns,
in this case 13%, or 0.13
Plugging in the numbers, we find that:
R = e^(-0.923*(-0.693/-0.139))
R = e^-4.594
R = 0.010
There is a 1% chance of losing 50% of the account from any point on the equity curve. This trader will spend 1% of his time at least 50% below a prior maximum equity high. Using fixed fractional position size reduced the position size as the account value fell, and it reduced the risk of a 50% drawdown from 2.9% to 1%.
Mathematician Tim Tillson has used the software Mathematica to show that the fixed fractional risk of ruin equation can be expressed more simply as:
R = (1-z)^(-2*a/(d*ln(1-d)))