Finding Optimal-f

Optimal-f is a method described by Ralph Vince in the book "Portfolio Management Formulas : Mathematical Trading Methods for the Futures, Options, and Stock Markets". To understand it we will need a few definitions. Here are some terms that Vince uses and a bit of a translation.

Note from the definition of f that a fixed fraction of equity or %Risk strategy is assumed - each trade in the series risks the same fraction of equity. Optimal-f is found in a round about way.

HPRn = 1 - f * (Pn / WCS) {Remember that WCS is a negative number }

TWR = HPR1 * HPR2 * ... HPRn

These equations express TWR in terms of f. Optimal-f is the value of f that maximizes TWR. This is most easily done with a spreadsheet. (Excel Sheet)

f =
57.4%    
n Pn HPRn TWRn
1 155 2.98 2.98
2 245 4.12 12.27
3 -35 0.55 6.80
4 75 1.96 13.30
5 -45 0.43 5.67
6 25 1.32 7.47
WCS
-45
   

The values in the Pn column are the profits from each trade. The WCS is the lowest profit. The HPRn values are calculated from Pn, WCS and f - this is the gain on each trade as a fraction of equity before the trade. TWRn is the gain for the first n trades as a fraction of initial equity. The last TWR is the ratio of ending equity to starting equity - the profit ratio for the total series of trades.

The goal is to maximize the last TWR by adjusting f. For this example, f = 57.4% gives the maximum and thus Optimal value of f. This can be found by trying various values of f until the last TWR is as large as possible or with Excel use the solver.

Risk of 57.4% of equity gives a final equity that is 7.47 times the starting equity. A very nice return for 6 trades. The drawdown picture is not so nice. From 12.27 to 6.8 is a 44% drawdown and the next is 57%.

The optimal-f method says nothing about drawdown. The results are the same no matter what the sequence of the trades. If the two losses come one after the other then the worst drawdown is 76% but the TWS and Optimal-f are unchanged.

The goal is simply to maximize return and the calculation is strongly biased by the worst loss but unaffected by the order of the trades.

Analysing the same data using TradeSim gives these results for 57.4% risk and 6 trades.

The mean equity is 40% per trade which is the same as found with Optimal-f, but 15% of trials lose money.

Drawdown is likely to be 57% but has a 10% likelihood of rising as high as 87%.

The two problems with Optimal-f are that it is strongly dependent on the worst loss and it does not consider the effect of other losses or the order of the trades.

The example here is made with hypothetical data but the problems are very real as shown by the next article.


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The charts on this page are produced using the TradeSim software.

Copyright 2002, Larry Sanders

Last update 2002.04.17