Fixed ratio MM
Fixed ratio MM was first introduced by Ryan Jones in his book The Trading Game in 1999 [13]. In the fixed ratio position sizing the key parameter is the so called "delta". This delta is the dollar amount of profit per traded unit to increase the number of units by one. A delta of $10,000 means that if you're currently trading one lot you would need to increase your account equity by $10,000 to start trading two lots. Once you get to two lots, you would need an additional profit of $20,000 to start trading three lots. Then, trading with three lots, you would need an additional profit of $30,000 to start trading four lots and so on. The base to calculate the number of traded lots in fixed ratio position sizing is the following equation:
N is the traded position size, NO is the starting position size, P is the total closed trade profit, and delta is the parameter discussed above. As the mathematicians know x05 means "square root of x".
A few points are worth mentioning. The profit P is the accumulated profit over all trades leading up to the one for which you want to calculate the number of lots. As a consequence the position size for the first trade is always N0 because you start with zero profits (P = 0). Please note that the account equity is not a factor in this equation, so changing the starting account size will not change the number of traded lots. Neither is the trade risk a factor in this equation  if trade risks are defined for the current sequence of trades, they will be ignored when fixed ratio position sizing is in effect. All that matters is the accumulated profit and the delta. The delta determines how quickly the lots are added or subtracted.
Ryan Jones' position sizing rule leads to a MM scheme which is a bit more conservative than the fixed fractional MM by Ralph Vince which was shown above. You can simply compare the two MM schemes as following (Figure 7.5):
Ralph Vince, fixed fractional lots = constant * accountsize Ryan Jones, fixed ratio lots = constant * squareroot(accountsize)
So in contrast to the above situation with fixed fractional MM where the number of traded lots is linearly increased with the account size, with the fixed ratio MM the number of lots is increased like a square root function. This MM type is a bit more agressive in the beginning, but becomes more conservative and slower with more gained trading capital.
Figure 7.5: Fixed Ratio vs. Fixed Fractional MM. As trading capital increases, the fixed ratio MM increases the number of lots more slowly, with a square root function, than the fixed fractional MM, which works linearly.
To see how fixed ratio MM works let's start with a delta of $100,000. This means that you can add the second lot only after your account value has increased by another $100,000 to $200,000 (Figure 7.6A). Obviously this MM is very slow and not much different from just trading one lot all the time so let's make it a bit more aggressive. (Figures 7.6B and C).
Figure 7.6: Applying fixed ratio MM. The aggressiveness of the MM is increased from A to C. Upper blue line: equity curve. Lower black area: number of traded lots. Starting account size = $100,000. LUXOR system on British pound/US dollar (FOREX), 30 minute bars, 21/10/20024/7/2008, with entry time filter and exits in place. Including $30 S+C per RT. An area with a recent drawdown is encircled. Chart created with Market System Analyzer.
Figure 7.6: Applying fixed ratio MM. The aggressiveness of the MM is increased from A to C. Upper blue line: equity curve. Lower black area: number of traded lots. Starting account size = $100,000. LUXOR system on British pound/US dollar (FOREX), 30 minute bars, 21/10/20024/7/2008, with entry time filter and exits in place. Including $30 S+C per RT. An area with a recent drawdown is encircled. Chart created with Market System Analyzer.
200 400 600 300 1,000
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200 400 600 300 1,000
Trade
As with the fixed fractional MM the more aggressively you increase the lot size, the higher equity peaks you can reach, but also the bigger drawdowns you get in phases where the trading system shows some weakness. Compared with the fixed fractional MM the fixed ratio MM shows a better return/risk ratio as you can see when you have a closer look on the delta=$500 fixed ratio MM (Figure 7.6C). There you see that the equity high is $11.5 million and the drawdown is about $1.8 million. Although the drawdown is still excessive, the return/drawdown ratio is, with 6.4 in this example, better than the comparable 5% fixed fractional MM where you have an equity peak of $22 million but then a drawdown of $8 million leading to a return/drawdown ratio of 2.8.
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