Optimal f

So you've got an idea and you want to know how good it is..

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The geometric mean

First, I'll start with a very simple but outstanding concept, it's called the Geometric Mean (GM), sounds scary, but essentially it's the "per trade return" (or growth) of your stake for the series of trades under evaluation.

For example, let's say that after 10 trades you've increased your stake from 100 to 150 units (I’ll call them contracts from now on, although, it could very well be 100 share lots). In this case we would calculate GM in the following way:

GM = (150/100) ^ 1/10 = 1.5 ^ 0.1 = 1.0414

Or, in other words, the growth per trade GM is 4.14%, for a total growth after 10 trades of 50%.

Now, let's step back a little, I specifically highlighted the outstanding significance of the GM concept based on the fact that GM has to be first and foremost greater than 0%. If your system produces negative growth, obviously, you're losing money; go back to the drawing board and look for something else! None of the math we'll be seeing will ever improve your system…

And, of course, the best system is the one with the greatest GM.

Hold it there. It seems too simple to be true...

Unfortunately, we have to throw in a couple of more nuances to really have an optimal system. But, look at it on the bright side, we'll be protecting our fannies, - a small price to pay...

Optimum allocation of funds
The first inescapable reality is that the portion of our account at risk affects our returns. Let’s go from one extreme to the other to clarify. If you place 100% of your account at risk on every trade, it will take very little time to wipe you out… On the other hand, if you put 1/1000 of your account at risk on every trade, you’ll probably never get wiped out, but your returns will be awful, better put your money in a savings account…

So, there’s obviously an optimum allocation somewhere in the middle of these two extremes. In order to find this optimum, we define f as the ratio of the biggest loss per contract tothe stake per contract involved. Or, in other words,

f = biggest loss per contract / stake per contract.

And, we will look for an optimal f where GM is maximized.

Let’s look at the following sequence of trade results, where our stake on each trade is $1,000:

-$250, $300, $500, -$100, $500

For this example, f = 250 / 1000 = 0.25.

And, we also need a relation between GM and f. As usual, I’ve oversimplified this process to get quickly to the bone. For those wanting further background of the following formula take a look at Empirical Techniques in Ralph Vince’s book: Mathematics of Money Management. I’ll only add that this relation not only considers fractional but also reinvestment of wins in subsequent trades:

GM = [{1 + (f * (-T1 / BL))} ^ (1/N)] * [{1 + (f * (-T2 / BL))} ^ (1/N)] * ...
...[{1 + (f * (-TN / BL))} ^ (1/N)] , where

T = Profit or loss for a trade with the sign reversed;
so a win ends with a – sign,
and a loss with a + sign.

BL = Biggest loss per contract of all N trades, always a negative number.

N = Total number of trades involved.

So, in order to find optimal f (and subsequently, the optimal number of contracts to trade), we must iterate for different f values. For our previous example, the calculations in a spread sheet give the following results:

GM (f =0.1) =
[{1 + (0.1 * (250 / -250))} ^ (1/5)] * [{1 + (0.1 * (-300 / -250))} ^ (1/5)]
*[{1 + (0.1 * (-500 / -250))} ^ (1/5)] * [{1 + (0.1 * (100 / -250))} ^ (1/5)]
*[{1 + (0.1 * (-500 / -250))} ^ (1/5)] = 1.068609059

GM (f = 0.1) = 1.068609059
GM (f = 0.2) = 1.123339717
GM (f = 0.3) = 1.164855131
GM (f = 0.4) = 1.193019253
GM (f = 0.5) = 1.206835267
GM (f = 0.6) = 1.204063741

So, optimal f = 0.5, because GM(f = 0.6) is declining. Hence, for a biggest loss of $250, we determine from our f relation the number of contracts by considering 1 contract per $500 of our equity account.

As a warning, I'd like to transcribe Ralph Vince's remarks from his Mathematical Methods book:

"We know that if we are using optimal f when we are fixed fractional trading, we can expect substantial drawdowns in terms of percentage equity retracements. Optimal f is like plutonium. It gives you a tremendous amount of power, yet it is dreadfully dangerous. These substantial drawdowns are the problem, particularly for novices, in that trading at the optimal f level gives them the chance to experience a cataclysmic loss sooner than they ordinarily might have."

And he goes on to state:

"You will have enormous difficulty in finding a portfolio with at least 5 years of historical data to it and all market systems employing the optimal f that has had any less than a 30% drawdown in terms of equity retracement! This is regardless of how many markets you employ. If you want to be in this and be mathematically correct, you better expect to be nailed for 30% to 95% equity retracements. This takes enormous discipline, and very few people can emotionally handle this".

In order to protect ourselves from a naked exposure, one should seek to cover the risk by spending some resources into buying some alternative derivative (LO options) of the underlying (CL futures); therefore, improving the method altogether.

We will continue the discussion...

Saturday, March 18, 2006 in Trader lessons, Trader methods