Kelly capital allocations seem to favor "animal spirits"

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Written by Forex Automaton   
Monday, 19 April 2010 07:42

If you are following Danica trading system updates, you must have seen the table of Kelly capital allocations. These are estimates of the capital allocations to trades which, according to the theory of J.Kelly, maximize the speed of capital growth and are tuned to the historical success rate of the insight (forecasts in our case) on which the trading is based. According to Kelly, 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. The implementation of Kelly approach therefore is based on data base requests with specific selection conditions. Kelly allocations would, with enough data, converge to zero if darts throwing is used to make decisions. Same outcome would ensue if the real markets were replaced by hypothetic efficient markets of the academic finance. In this post I present and discuss a more complete version of the table, including negative allocations, some of which seem to apply consistently to a particular class of trades.

Table 1. Table 1 gives trading capital allocation according to Kelly Criterion. The allocation depends on the strength of the forecast move, reported as next-current in the system output. The distribution of this quantity for each forex pair is split into four quartiles, and their boundaries are given on the quartile boundaries line. The allocation is the fraction of capital to risk on the protective stop, assuming that this is the only trade submitted.  Negative allocations are cases where staying  on the sidelines is indicated. These allocations are calculated applying the selective condition of L0 trigger (predicted moves for low, high and close all in the same direction).


Q1
Q2
Q3
Q4
AUD/USD quartile boundaries   -0.00364   -0.00068   0.00193  
allocation, % -11.   2.4   7.0   9.9
EUR/USD quartile boundaries   -0.00517   -0.00093   0.00280  
allocation, % -5.2   -4.0   2.0   9.9
GBP/USD quartile boundaries   -0.00661   -0.00099   0.00415  
allocation, % -0.43   7.3   16   1.2
USD/CAD quartile boundaries   -0.00525   -0.00140   0.00231  
allocation, % 11   2.8   -4.0   -1.7
USD/CHF quartile boundaries   -0.00543   -0.00148   0.00299  
allocation, % 4.9   4.4   3.7   -12.
USD/JPY quartile boundaries   -0.5289   -0.1435   0.2512  
allocation, % 3.3   -2.8   6.5   -1.1
AUD/JPY quartile boundaries   -0.4453   -0.1047   0.2086  
allocation, % -1.7   -3.1   24   21
CHF/JPY quartile boundaries   -0.3680   -0.0535   0.2282  
allocation, % -21.   -4.2   5.0   1.8
EUR/AUD quartile boundaries   -0.0087   -0.0037   0.0012  
allocation, % 1.7   0.0   -12.   2.9
EUR/CHF quartile boundaries   -0.0026   -0.0013   0.0014  
allocation, % -1.6   11.   2.1   -1.5
EUR/GBP quartile boundaries   -0.00206   -0.00071   0.00098  
allocation, % 0.4   0.5   -5.3   18.5
EUR/JPY quartile boundaries   -0.5657   -0.0547   0.3574  
allocation, % -11.   -1.6   11.   8.0
GBP/CHF quartile boundaries   -0.00749   -0.00181   0.00411  
allocation, % 14.   6.8   -9.2   1.8
GBP/JPY quartile boundaries   -0.8555   -0.1742   0.4810  
allocation, % -13.   7.5   5.6   6.4

Thus, the Kelly allocation coefficients are a figure of merit for both the system and the forex markets of today from the trader's point of view.

I know of no other example of a quantitative study on how much capital to risk on a trade, tied to past performance of a specific system. Most gurus give you numbers which vary from 0.5% to 10%. For example, the legendary commodity trader W.D. Gann ("How to Make Profits in Commodities") recommended risking about 2-5% per trade and never more than 10%.

As you see, some allocations are negative which indicates a mistrust of the Kelly analysis towards certain classes of predictions.

Initially I was expecting to see smaller allocations for the predicted moves of smaller magnitude and larger allocations for the predicted moves of larger magnitude. In reality however, if there is a common pattern, it is to put more faith into "risk on" forecasts, and less faith into "risk off" ones.

The concept of risk on and risk off days in the markets is not new and is based on an observation that risky assets (those having properties of interest-bearing assets) rise and fall as a group or in a correlated fashion. They rise on so-called "risk on" days and fall on the "risk off" days, which are defined by this very phenomenon. On the risk aversion days market participants flee into instruments providing the minimal return.

Forex trades, each of which consists in selling (borrowing) one currency and buying (depositing) another, can be easily classified along the "risk on" and "risk off" lines, since the interest earned or paid depends on the currency.

Currently, being long AUD/JPY or AUD/USD are the two most aggressive "risk-on" trade ideas, due to high interest differential on these trades.

In the table, you see that the Kelly allocation component of the system mistrusts "risk off" ideas. "Risk off" is somewhat of a misnomer since following these bearish "risk averse" forecasts, in historic perspective, turns out to be dangerous from the point of view of Kelly's algorithm. (Back-testing trading for Danica begins in 2006). Most of the negative allocations in the table belong to the risk-off situations. Namely, shorting AUD/USD on strongly bearish forecast (Q1) is not recommended; shorting EUR/USD on any bearish forecast (either Q1 or Q2) is not recommended; shorting AUD/JPY EUR/JPY  on any bearish forecast (either Q1 or Q2) is not recommended;  shorting GBP/JPY on a strongly bearish forecast (Q1) is not recommended.

CAD, not exactly a high-yielder, is nevertheless a commodity currency (as is AUD) and commodity currency rallies are often animated by the same animal spirits that drive carry trade. Thus, Kelly allocator's mistrust towards CAD bearishness is not surprising given its mistrust towards Aussie bearishness.

A plausible explanation  may be that the chart patterns of exchange rates with high interest differential are typically sharply asymmetric, in such a way that the risk averse movements happen more rapidly. The forecasting component itself may very well lack full corrective adjustment for this asymmetry. The forecasting component may underestimate the abruptness of the correction and take it for the beginning of a sustained trend -- which it too often isn't. Use of Kelly allocation corrects this deficiency.

relative profit distribution, logarithmic scale 1.1

Fig.1. Distribution of Kelly allocation coefficients for all 14 markets and all 4 predicted move magnitudes. A normal distribution fit is shown.1=100%.

If one choses to treat both positive and negative Kelly coefficients as independent measurements of a certain  single hypothetic quantity, then standard statistical analysis yields 1.8% as an estimate of that quantity, with two-sigma statistical significance of its positiveness (and the Gaussian fit in Fig.1 yields a somewhat larger mean and larger uncertainty of the mean). Mind you, this would imply betting on as well as against carry trade, thus disregarding what was just said about "risk on and off" days and peculiarities of carry trade unwinding.

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Last Updated ( Monday, 19 July 2010 16:33 )