Optimal <I>f</I>

It is commonly assumed that risk is primarily a personal choice and that you will be linearly rewarded for the sort of risk you are willing to take on. This has been shown not to be true. In fact there seems to be an optimal amount of risk that should be taken on for the highest gains compared to risk of ruin, and this is called Optimal f , where you make bet sizes equal to a fixed fraction, f.

In particular when dealing with leveraged trading instruments, there comes a point where aggressively trading large positions does not lead to greater profits. As trade sizes become larger and larger, the trader becomes increasingly vulnerable to asymmetric leverage, a losing streak combined with aggressive trading can lead to a large drawdown that is not adequately compensated by gains on the winning trades.

Ralph Vince, in his book Portfolio Management Formulas introduced the concept of Optimal f with an example using coin flips, but attached I have made an Excel Spreadsheet that demonstrates the concept well enough.

The usual profile of gains versus fraction of the account traded is some sort of bell curve. I have set up this spreadsheet to use random numbers to represent the success of individual trades, every time the spreadsheet updates so does the equity curve.(it does this automatically when you make a change to the spreadsheet, including pressing delete in an empty cell).

The spreadsheet is pretty big, so I've zipped it up for faster downloading:optimalf.zip

Take a minute or two to see how alternative trade sequences effect the "equity curve". Vince uses a parameter "Terminal Wealth Ratio", which is a really fancy term to that means multiplication of wealth, ie TWR=2 means doubling your account.

The bright green cells represent the best and the worst possible results from your "system", enter a negative number and a positive number and the spreadsheet uses the randbetween() function (you'll need to install the analysis toolpack) to generate a random sequence of trades in between?

What do you notice about the Optimal f fraction? There are a couple of things that are easily apparent after playing with this spreadsheet for a while:

If the smallest possible loss is quite small, most series of trades lead to the Optimal fraction to trade being 100%, your profits always improve by trading more of your account in any one trade. This states that often for stock traders who are skilled at cutting losses on any trade it usually makes sense to trade the full account. While futures and option traders, or any stock trader who makes use of margin knows that this is a suicidal practice for leveraged traders, there is probably a reason why money management techniques never really took off among stock traders, the "reckless" approach of reinvesting all profits usually works just fine, and all this highfalutin money management stuff doesn't help much.
The "optimal" fraction to trade changes completely for every sequence of trades! This may surprise some people who are heavily into trading and have a collection of all the Vince books, or maybe it won't. No two series of trades ever seem to bring up the same Optimal f , and I'm not even talking about changing any parameters here. If you get a longer than average drawdown, the "optimal" fraction to trade, with perfect hindsight, would be a small one. If you get a particularly good lucky streak in your trade sequence, you'll get a more aggressive fraction.

Here really is the problem with Vince's wonderful Optimal f technique. It works only with perfect hindsight, and is just another curve-fitted parameter. Unless your system is going to behave in pretty much the same way as it did in the backtest period, the fraction you determine to be optimal will probably not be optimal in the next trade.

Once you get tired of playing with randomly generated trade sequences you can enter your own trades into the yellow box, the numbers are percentage points, so if you made 15% on your last trade enter 15 in the last cell. It would be interesting to see if your own trading style does lead to an optimal fraction to use, or if such a concept is useless.

Like most curve fitted parameters the Optimal f probably should not be taken too seriously. A single large drawdown or lucky streak in the backtest period will throw the optimal fraction way out of whack. While a lot of futures traders seem to swear by the technique, I can only assume that these are people with extremely consistent systems that work equally well in all market conditions. While I don't know of anyone who can make a boast to have such a robust trading system, Vince certainly does have his admirers.