In fixed ratio position sizing the key parameter is the delta. This is the amount of profit per share/contract/unit to increase the number of shares/contracts/units by one. A delta of $3,000, for example, means that if you're currently trading one contract, you would need to increase
your account equity by $3,000 to start trading two contracts. Once you get to two contracts, you would need an additional profit of $6,000 to start trading three contracts. At three contracts, you would need an additional profit of $9,000 to start trading four contracts, and so on.
Fixed ratio position sizing was developed by Ryan Jones in his book "The Trading Game," John Wiley & Sons, New York, 1999. Based on an equation presented by Jones, it's possible to derive the following equation for the number of contracts in fixed ratio position sizing:
N = 0.5 * [((2 * N0 – 1)^2 + 8 * P/delta)^0.5 + 1]
where N is the position size, N0 is the starting position size, P is the total closed trade profit, and delta is the parameter discussed above. The carat symbol (^) represents exponentiation; that is, the quantity in parentheses is raised to the power following the carat (x^2 is “x squared; x^0.5 is “square root of x”).
A few points are worth noting. The profit, P, is the accumulated profit over all trades leading up to the one for which you want to calculate the number of shares/contracts/units.
Consequently, the position size for the first trade is always N0 because you always start with zero profits (P = 0). The account equity is not a factor in this equation. Changing the starting account size, for example, will not change the number of shares/contracts/units, provided there is enough equity to avoid dropping below zero.
The trade risk is also not 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 shares/contracts/units are added or subtracted. If the “trade in units” option is selected, delta will be the profit per unit to increase the number of units by one. For stock trading, without the trade-in-units option, the delta will be the profit per share to increase the number of shares by one.
Note that the equation above is applicable only to the case where a single delta is used for both increasing and decreasing the number of shares/contracts. MSA allows you to decrease at a different rate. This is accomplished by specifying a delta fraction for decreasing. The delta fraction is multiplied by the original delta to determine the delta for decreasing the position size. The delta fraction is set to the value 1.0 by default, which implies the two deltas will be the same.
If a delta fraction less than 1.0 is entered, the levels of increase and decrease will be closer together, which will cause the position size to decrease more frequently in drawdowns as the equity drops. This will provide more protection from drawdowns but less ability to recover. If a delta fraction greater than 1.0 is entered, the level of decrease will be farther from the level of increase, so the position size will not drop as quickly or change as often during a
drawdown. This will provide less protection during drawdowns but make is easier to recover. Note that if the optimization feature of MSA is used with fixed ratio position sizing, only the delta parameter will be optimized. The delta fraction will be unchanged by the optimization.
Generalized Ratio
As the name suggests, generalized ratio position sizing is a generalized version of fixed ratio position sizing. Generalized ratio position sizing uses an additional parameter – the exponent – to alter the characteristics of the fixed ratio equation. Generalized ratio position sizing is unique to MSA.
The number of shares/contracts/units in generalized ratio position sizing is given by the following equation:
N = 0.5 * [(1 + 8 * P/delta)^m + 1]
where N is the number of shares/contracts/units, P is the total closed trade profit, m is the exponent, which must be greater than or equal to zero, and delta is as described for the fixed ratio method. The carat symbol (^) represents exponentiation; that is, the quantity in
parentheses is raised to the power of m.
With an exponent of ½, the generalized ratio method is the same as the fixed ratio method with an initial position size of one. With exponent values less than ½ the position size increases more slowly with increasing profits than with the fixed ratio method. With exponents greater than ½, the position size increases more quickly than with the fixed ratio method. An exponent value of one provides the same proportionality as the fixed fractional method. An exponent value of zero gives one share/contract/unit for each trade.
The number of shares/contracts/units always starts at one with generalized ratio position sizing. This can be seen in the equation above. If the profit, P, is zero, N is equal to 1. For this reason, generalized ratio position sizing is more suitable for smaller accounts. Also, for stock trading, the “trade in units” option should be selected so that the initial number of shares will be equal to one unit. If the unit size is 100 shares, for example, the generalized ratio method will start with 100 shares.
Note that if the optimization feature of MSA is used with generalized ratio position sizing, only the delta parameter will be optimized. The exponent, m, will be unchanged by the optimization.