N = 0.5 *
[1 + (1 + 8 *
Profit/delta)^0.5]

where Profit = total closed trade
profit in dollars, delta = profit/contract to increase by one contract, and "^0.5" means that the expression in parentheses is raised
to the power of 0.5.

It's interesting to compare this equation to the corresponding equation
for fixed fractional trading:

N = ff * Equity/| trade risk |

where "trade risk" is the possible loss in dollars for the trade,
and the vertical bars (|) represent absolute value. Notice that the
relationship between the number of contracts and the profit is linear with fixed
fractional trading. As the profits accrue, the number of contracts increases
linearly. The rate of change of N with respect to account equity is constant
with the fixed fractional method; e.g., a $10,000 increase in profits results in
the same increase in the number of contracts regardless of whether that profit
occurred with a $15,000 account or a $150,000 account.

With fixed ratio trading, on the other hand, as you accrue more profits,
the number of contracts increases more slowly. A $10,000 profit with a
$20,000 account will increase the number of contracts more than if a $10,000
profit is made on a $200,000 account. For small account sizes, you'll
increase the number of contracts more quickly with fixed ratio position sizing.
However, when the account equity becomes larger, the number of contracts will
increase more slowly than with fixed fractional position sizing. This is why
fixed ratio position sizing is sometimes preferred for small
accounts.

Because the fixed ratio method depends on account size, how it performs
compared to the fixed fractional method over a series of trades depends on where
the drawdowns occur. If the biggest drawdown occurs late in the sequence of
trades, the fixed ratio method will do well because the fastest increase in the
number of contracts will have occurred during the most profitable period. On the
other hand, if the biggest runup in equity occurs late in the sequence of
trades, the fixed fractional method will do better because it will increase the
number of contracts more quickly at that point, whereas the rate of increase in
the number of contracts with the fixed ratio method will have already
slowed.

As an example, consider the following equity curve from a real sequence
of trades. I adjusted the delta for the fixed ratio method and the fixed
fraction for the fixed fractional method so that the worst-case
percentage drawdown was the same in each case. Fig. 1 shows the equity
curves for both methods when the trades occur in their historical sequence. The
fixed ratio method clearly delivers superior performance. The net profit is much
higher for the same maximum drawdown. Note that the primary run up in equity
occurred early in the sequence of trades. The fixed ratio method was more
aggressive early on when it mattered the most.

**Figure 1. Equity curve for historical sequence of trades using
fixed fractional ("Fix Fract") and fixed ratio ("Fix Ratio") position sizing.
The maximum peak-to-valley drawdown is the same (in percentage terms) in each
case.**

However, if we randomize the trade sequence, as in Fig. 2, the opposite
result is possible. These are the same trades as in Fig. 1, just in a different
order. Again, the parameters for the two methods have been adjusted to produce
the same maximum peak-to-valley percentage drawdown. In this case, the fixed
fractional method generates a much higher return for the same drawdown. In this
sequence of trades, the run up in equity occurred late in the sequence. As a
result, the fixed fractional method was more aggressive than the fixed ratio
method in increasing the number of contracts late in the sequence when it
counted most.

**Figure 2. Equity curve for randomized sequence of the same trades
as in Fig. 1 using fixed fractional ("Fix Fract") and fixed ratio ("Fix Ratio")
position sizing. The maximum peak-to-valley drawdown is the same (in percentage
terms) in each case.**

For any sequence of trades, one method will be better than the other.
However, even if we have a good idea of the *distribution* of our trades,
the sequence is always unknown. As I discussed in last month's newsletter, one
way to deal with the consequences of not knowing the sequence of a series of
trades is to use the Monte Carlo method. With the Monte Carlo method, we can
perform an analysis over many different, randomly chosen trade sequences and
evaluate the results in terms of statistics. In effect, this is a way to convert
the uncertainty of the trade sequence into a quantified (if probabilistic)
result. This method might be able to tell us whether the fixed
fractional or fixed ratio method is better for a given series of
trades.

We can take this analysis one step further by reconsidering the
equation presented above for the fixed ratio method. Notice the 0.5 exponent in
the equation for the number of contracts, N, in the fixed ratio method.
Consider what we would get if the 0.5 was replaced with 0. In that case, we get
N = 1. In other words, an exponent of zero represents fixed contract trading
with one contract per trade. What if the exponent has the value 1? In this case,
we find that the number of contracts, N, is proportional to the profit. This is
the basis of fixed fractional trading. In other words, an exponent of 1
represents fixed fractional position sizing.

There's nothing preventing us from choosing other exponent values as
well. With this in mind, we can write a more generalized form of the position
sizing equation as:

N = 0.5 * [1 + (1 + 8 *
Profit/delta)^m]

where the exponent m can vary from 0 to any positive number we like. With
m = 0, we get fixed contract trading. With m = 1, we have the equivalent of
fixed fractional trading. m = 0.5 gives us fixed ratio
trading.

Any value of m less than 1 (e.g., 0.5 or 0.10) will increase the number
of contracts more slowly for larger account equity values. Values of m larger
than 1 will increase the number of contracts more quickly as the account equity
increases. At m = 1 (i.e., fixed fractional), the rate of change in the number
of contracts is independent of account size.

We might expect that for any sequence of trades, there's an "optimal"
value of m. By optimal, I mean there's one value of m that produces the greatest
return for a given maximum drawdown. As noted above, since we don't know the
sequence of trades to expect in the future, calculating this optimal m for a
historical sequence of trades is probably a pointless exercise. However, it
might be interesting to use the Monte Carlo method to see what this optimal m
would be based on the statistical results of the Monte Carlo simulation. I've
yet to do that analysis, however.