 ## Kelly capital allocation improves system profitability while reducing risk

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Written by Forex Automaton
Wednesday, 14 April 2010 12:42

Currently there is little doubt that the next upgrade of our trading system will introduce some form of trade idea selection into the system's output. This article reports our recent progress towards implementing Kelly Criterion for capital allocation to trades. This is our first quantitative attempt to see what Kelly does to the expected profit per trade and the shape of its distribution.

The histogramming approach used in this study is similar to the one used before, when what will be called for the sake of definitiveness Day Close strategy was introduced.

The quantity being histogrammed is the day's profit expressed in the units of (as a fraction of) price. The trading strategy under study consists in taking Danica's forecast for the day, selecting currency pairs where the predicted directions of change, labeled as "next-current", have the same sign for day's high, low and close (the condition known as trigger word 3 of the L0 trigger), and placing a trade having the previous day's extreme (day's low for a long trade, day's high for a short trade) as a protective stop. The day is defined to last 24 hours.

The essence of Kelly's insight into the problem capital allocation is, in brief, in establishing the fact that the rate of capital growth is maximized by having the allocation of capital to outcomes, given the forecast, match the probability distribution of those outcomes, given the forecast.

In other words, when being torn between going long and going short, what matters is the number of times you were successful in going long under present circumstances in the past vs the number of times you were successful in going short under present circumstances in the past. If these numbers are equal or roughly equal, your capital allocation to trade will be zero or near zero. That way, Kelly capital allocation component will keep you out of the market if the system is of little value or the market is nearly efficient. The recommended allocation will be the larger, the greater are the odds of success.

Speaking of "circumstances" which, mathematically, define the condition associated with conditional probability in Kelly's formalism, the forecast at hand is the most important (but not necessarily the only important) component of what constitutes these "circumstances". The distribution of forecasts is split into four quartiles, and the identifier of the quartile is currently the only condition imposed when making a request to the database of past outcomes. Thus it is the magnitude of the predicted price move that constitutes the condition of "relevance" for the history precedents on file, and a rich subject of study is whether this is indeed the best choice.

Needless to say, on each step of the system's evolution through time, the Kelly algorithm can only use the data that are available to the system at that step in time -- there is no causality-violating feed-back from the future.

Effect of Kelly allocation weight on profitability is modeled by making the weight with which each profit/loss quantity eneters the histogram equal the Kelly allocation. (The weight is an increment in the bin magnitude recorded after the profit/loss is histogrammed, or "falls" into its proper bin). That way, the simple average profit becomes a weighted average, reflecting the effect of capital allocation on the system profitability in a natural way. In this approach, after the histograms are normalized (to the unit total probability), the effect of weights shows up as a subtle difference in shapes. 1.1 1.2

Fig.1. Distribution of day-to-day profit or loss. Gray symbols show results for the strategy where L0 trigger word=3 is required but positive Kelly allocation is not, and the Kelly allocation is ignored (case A in table). Red symbols correspond to the strategy where L0 trigger word=3 and positive Kelly allocation are required, and Kelly allocation enters as a histogram weight for the profit/loss (case C in table). Both histograms are normalized to the unit intergal.

Kelly allocation used as a weight in histogram:

ignored (A) positive, calculated regardless of L0 (B) positive, L0=3 is required in Kelly algorithm (C)
# of events, thousands 11.8 6.9 7.1
relative profit per day, average 3.1 ×10-4 8.0 ×10-4 8.7 ×10-4
relative profit per day, RMS 7.1 ×10-3 6.5 ×10-3 6.7 ×10-3

Table 1. Effect of Kelly allocation on the trading system figures of merit.

In case C, L0=3 is required in Kelly algorithm only when determining quartile boundaries for the forecast distribution, needed to classify forecasts. A more logical approach would impose a requirement of L0=3, in addition to the forecast move of a given strength, as part the condition logic when forming the conditional distribution of price moves. This remains to be done, and it remains to be seen whether this approach will result in quantitative improvements.

The effect of Kelly allocation weight in figures and table is nothing short of amazing. For the first time in a long history of studies I am seeing an increase in an average profit accompanied by a decrease in the RMS. In other words, Kelly allocation achieves an increase in profit accompanied by a decrease in risk. The figures show that while Kelly shifts the mean by redistributing the center of gravity on both slopes of the distribution in the positive direction (of greater profits), see Fig.1.2, it also suppresses the tails, as is seen very well on the logarithmic scale in Fig.1.1.

Last Updated ( Tuesday, 31 August 2010 13:30 )