## A logarithmic measure of trading returns

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Written by Forex Automaton
Tuesday, 23 December 2008 15:32

This technical note concerns monitoring of returns generated by the automated trading systems in the course of the optimization runs. Typically, the distributions (see figures in the first note on the subject of optimization where actual results were shown) have very long tails, extending into high yield area. As every event (histogram count) in the distribution represents an independent trading history (a virtual trader), the tail events represent virtual traders with outstanding returns. In order to be able to compare results of various simulated trading runs visually without losing much information, I introduce a logarithmic measure of such returns.

If trader's total capital (equity) is E1 at the moment t1 and E2 at the moment t2, I calculate annualized return on investment as

R = (E2/E1)T/(t2-t1)-1,

where T is duration of the year, expressed in the same units as t1 and t2. So defined, R can not be below -1, which means one can not lose more than 100% of the principal. R is not limited in the positive direction. The Compact Measure Of Return (CMOR for short) useful exclusively for optimization studies is simply

CMOR = ln(R+2)

There is no particular sense in adding 2 to R and taking the natural logarithm of the sum, other than obtaining a convenient quantity with the following properties:

• CMOR is always positive
• R of -1 (the extreme of losing everything) corresponds to CMOR of 0;
• R of 0 (preserving the principal) corresponds to CMOR of 1;
• CMOR>ln(2) = 0.69... means making money;
• the tail of positive returns is compressed.

Fig.1:Distribution of returns for paper trading in real AUD/JPY and simulated reference markets, using the logarithmic measure CMOR. Vertical axis: normalized counts. Horizontal axis: CMOR measure. The vertical axis reflects the estimated likelihood of getting a particular investment return from the algorithm under the hypothetical scenario of a "know-nothing" application of the algorithm, or an application under the random choice of control parameters without any optimization. Both histograms are normalized to the unit sum of bin counts (not the unit area integral).

Fig.1 presents the result for the same computation runs as the figures in the original article, but compared to the original article, we get to see much more data as the tail is compactified. Needless to say, the returns have to be considered in the return vs risk context where higher returns generally accompany higher risk, as can be illustrated with the data in the original article. The usual caveats about validity and credibility of "paper trading" results as a measure of merit of any trading system apply.

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Last Updated ( Friday, 11 September 2009 16:27 )