This is the second and final part of the article on Money Management. Last week we introduced the idea and defined ‘fixed dollar’ and the ‘percent at risk’ methods. This week it gets interesting. We look at the ‘Optimal f’ method and compare all methods with examples.

The mathematics of this system has its roots in solving the problem of interference in data transmission over telephone lines – truly. However the system was adopted for gambling and then again for trading futures.

Optimal f refers to the mathematically optimal fixed fraction of total equity that is allocated to any one trade. The optimal fraction is defined at the one that offers the maximum long term growth of equity.

This is really no different than the previous version except we have mathematically determined the figure with will produce the best long term profits.

For trading systems, the mathematics of Optimal f simply takes into account system parameters as defined by your past results or hypothetical results and returns a figure that represent the dollars in your account that should represent one contract.

Now the formula itself is just a little to detailed to list here. For those wanting to head down this path I strongly recommend Ralph Vince’s first book Portfolio Management Formulas. It is heavy on the maths, but the proof is very convincing!

Playing with coins
The Optimal f formula will return a decimal that represent the fraction of the total equity. To then come to the amount to trade, you divide the largest loss by this fraction.

In our coin example, the optimal f is 0.25. The largest loss as we know is \$1.00:

\$1.00 / 0.25 = 4

Therefore we would be one unit (dollars) every \$4.00. This is the figure that without question offers the best long term growth given the payoff levels (\$2 profit and \$1 loss) and probability (50% for each).

The same goes for more complex futures trading systems where you have a measurable string of profit and losses. You can use the Optimal f calculation to determine the fixed fraction position size that would have resulted in maximum equity growth given those parameters.

The Proof Is In The Tossing
So how much advantage does Optimal f give you? Is it really worth learning the maths? Would it just be OK to make a rough guess at the optimal faction and trade that way?

Well, the numbers are nothing short of astounding. Let’s say we give three traders a bankroll of \$100,000 each and have them trade a fixed fraction of equity.

For some reason the first three names that have popped into my head are Peter, Paul and Mary – so we will use those. Peter is conservative and bets 10%. Paul has done some homework and knows to bet 25%. Mary arbitrarily chooses 40% – thinking “it’s only risking less than half” of my money and that’s not too bad”.

If we simulate using alternating returns (profit, loss, profit, loss…etc), then after 50 trades we would have the following equity curves:

Peter’s (10%) and Mary’s (40%) have the same outcome. Starting with \$100,000, they both end up with just less than \$685,000 – a return on 585%! Not too bad.Paul on the other hand has just over \$1,900,000 – a return of over 1,800%! It is very interesting to note that for such seemingly small changes in bet size, the optimal fraction showed a return of more than three times the others.

Another interesting point about Peter (10%) and Mary’s (40%) results are that drawdowns for the 40% allocation are up to six times that of the 10% allocation. So you make the same amount of money for six times the worry! This suggests it is better to err on the lower side of the Option f figure.

Now ask yourself this question, if you were presented with this game (the 2:1 coin toss), and you wanted to make the most money, what would be your approach?

What if you decided to be very aggressive and bet say 55% of your stake on each toss? Well you will go broke with a probability that approaches 100% the more you play. In the 50 toss example, you would end up with a loss of 75%! That is a loss of 75% from a game that has such advantageous odds.

Hopefully those figures will convince you there is something to this ‘money management‘ stuff.

Mathematical expectation
Now you may have got to this stage of the article and be thinking “Hey, remember that sorry little system I designed that never made any money? Maybe with a clever money management strategy that thing would work!”.

Well unfortunately, it’s not that easy. Any system will still have to make money under the simplest of money management strategies to then make more money under the Optimal f method.

In statistical terminology, the system will have to have a positive mathematical expectation. A mathematical expectation is what you would on average make per trade. This figure has to be positive. That is you have to be able to make money on each trade on average.

The figure is quite simple to calculate for simple examples. Back to the coin toss. If you could play the coin toss game where you win \$1 for heads and lose \$1 for tails, then you can easily figure out that on average you would breakeven, right? In technical terms it is calculated as such:

= P(W)*W + P(L)*L

Where
W = profit per winning bet
P(W) = probability of that win
L = Loss per losing bet
P(L) = probability of that loss

For the coin toss the mathematical expectation is:

50% * \$1.00 + 50% * -\$1.00 = \$0.00

That one is pretty straight forward. What about our \$2.00 win and \$1.00 loss game? The mathematical expectation for that one is:

50% * \$2.00 + 50% * -\$1.00 = +\$0.50

That is on average, each game will win 50 cents. That is why you would play it under virtually any money management method.

If however, the win payout was \$0.75 for a heads and the loss for a tails was still \$1.00, if would not make sense to play. Your mathematical expectation is:

50% * \$0.75 + 50% * -\$1.00 = -\$0.125

On average you would lose 12.5 cents per bet. Nothing but luck would help you make money in the long term on this one. This interestingly enough is how a casino will make their money. All casino games have a probability of winning and a probability of losing. A casino will set their payouts such that any game never has a positive mathematical expectation. When you think about it this way, you wonder why anyone would ever go to a casino!

So how can you calculate your mathematical expectation on your futures system? Unless you can work out a probability distribution on your trades, then you can simple use past results. Let’s say we have two systems, System A and System B. They have the following results after 20 single contract trades:

Now if we assume that these 20 trades are indicative of the systems overall, then we can simply take the average to work out the mathematical expectation. System A shows a positive expectation despite have a lower proportion of winners. System B wins more often but has a negative expectation give the size of a couple of the losses.

System A would be suitable for an Optimal f calculation. System B would have no Optimal f since it cannot make money in the first place!

Using the spreadsheet provided, the Optimal f works out to be 0.13. Dividing the largest loss by this figure we get:

100/0.13 = \$769

This means you would trade one contract per \$769 to achieve maximum long term growth from this system. A lower dollar figure would mean more risk resulting in lower  long term growth (on average). A higher dollar figure would mean you would not be using the system to it most potential and your returns will suffer.

In other words a figure greater than or less than the Optimal f number will result in lower returns over the long run.

Pitfalls in using Optimal f
First and foremost is the assumption that future results will match that of the past. This all comes down to your testing methods. The most important figure here is the estimate of the largest loss. You probably have not bee trading long enough if you have never been surprised by the extent of a loss in unusual circumstances (eg Sep 11th).

There is a real problem then of picking what the exact loss should be. Should you assume your system will run into another September 11th scenario? Should you take that month out of your figures? Each side of the argument has its pros and cons.

Another negative is the volatility of returns. Optimal f will show you the number of contracts to trade that maximises profits. Along the way however, this also means that drawdowns can be significant, particularly if you experience sequential losses. Anyone using Optimal f should be prepared for this.

The way around this is to trade multiple un-correlated systems at any one time. In English, that means to diversify.

If you on the other hand were to design a great system for one market and invest all your money in that, then you would have to be prepared for significant drawdown solely thanks to Optimal f.

Varying bet size based on previous result(s)
Now in all of the above, we have assumed that any one profit or loss from your system has no bearing on the next profit or loss. That is there is no sequential relationship or correlation. This is something a lot of people do not think about.

Head on down to the roulette table at pretty much any casino and you will see an electronic screen showing some statistics of past spins – things like ‘% black vs red’; ‘% odd versus even’ and the out comes of the last few spins. The idea is the gambler will use this information to work out where to place the next bet. You would expect the number of blacks and number of reds to both be close to 50% right? So if blacks drop and reds rise, you might think “hmmm, chances are black will come up next“.

If you are laughing at this suggestion, don’t. Some people do actually think this will work. If you are one of them ask “how?” How could it possibly work? Is there any way that the last spin has anything to do with the next spin? How could it? How then can you make a betting decision based of the past spins? It’s ridiculous.

Some people do make these decisions and in the markets it’s even more common. After a string of losses, a trader might say “chances are my luck will turn around and the next trade will be a winner.” Based on this type of thinking, a trader may increase or decrease position sizes according.

In roulette this is a stupid strategy (in fact playing roulette in the first place is stupid). You see it has everything to do with dependency. One spin has no bearing on another. In the markets, most mechanical trading systems are the same. One trade will have no bearing on another. I have seen systems out there that do have some form of correlation, but these are rare. If then, we were to make one assumption, it would be that a time series of trade results are not correlated.

Therefore, any betting strategy that increases or decreases the best size based on one of a string of past profits of losses, does not make sense – just like the roulette example.

Conclusion
The Optimal f method of position sizing shows incredible results over and above any other method of sizing. Ultimately however the benefits rest entirely on the accuracy of the parameters used in the calculation. This of course comes down to your testing and your homework. Unfortunately, nothing comes for free.

References
Balsara, NauzerMoney Management Strategies for Futures Traders
Schwager, Jack. Market Wizards and The New Market Wizards
Vince, Ralph. Portfolio Management Formulas
Vince, Ralph. The Mathematics of Money Management
Vince, Ralph. The New Money Management