 Further analysis of the day-range strategy. Selecting the forecasts to trade. - Distribution of selected returns and its implications

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Written by Forex Automaton
Tuesday, 09 February 2010 12:45
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Further analysis of the day-range strategy. Selecting the forecasts to trade.
Distribution of selected returns and its implications
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To summarize what has been done so far: one seems to be able to bias an outcome of a day-range trade (a short or long position, depending on the direction of the forecast, with a profit target and a protective loss set at the previous day's high and low (or low and high respectively depending on whether the trade is long or short), on the basis of information available before the trade is placed.

Fig.4 takes a look at the distribution of profit/loss outcomes of such selected trades. The profits and losses are expressed as a fraction of price and data from all currency pairs are summed. 4.1 4.2 4.3

Fig.4. Distribution of profit relative to previous day's close for the 2365 entries (trades) forming the selection described in the previous figures. Chi-squared fits to the tails are exponential, as described in the text.

The distribution is clearly non-Gaussian and asymmetric. Curiously, the most likely outcome is a small loss. However, the distribution has positive mean, which implies that the longer tail of the positive outcomes (profits) outweighs the impact of the losses on average.

Left and right tails of the distribution are well described by exponents of the form

f(p) = exp(Constant+Slope×p),

where p is the relative profit being histogrammed. The larger is the slope, the steeper is the rise or fall. The slope on the right (profitable) side of the peak is clearly more gradual, which is what creates the positive mean.

To tell whether the mean is significant or not would require special analysis, since the simple Gaussian formula for the uncertainty of the mean, standard deviation (approximately equals RMS) over square root of the number of entries, would not work for this distribution.

One can toy-model the performance of a system by assuming that the capital evolves with number of trades n as

C = C0(1+(p-s)×L)n

where C0 is the starting capital, and n is the number of trades executed, p is a random number distributed according to Fig.4, separate for each of the n trades, s is spread expressed in the same units as p (fraction of price) and L is leverage. Such a toy model could be used to study the probability of ruin depending on the leverage applied and on the fraction of capital at risk, and to find the values of these parameters leading to the largest expected wealth after n trades, in spirit of Kelly Criterion, but on the basis of the actual distribution of trade outcomes.

The cost of leveraging (loan interest) can not be realistically modeled without knowing the duration of these trades and the interest rates used at each point in time.

For back-of-an-envelope estimates of the expected profit after n trades, the formula above can be simplifed to

E[C] = C0(1+E[p-s]×L×n),

where E[C] is the sought expectation and E[p] can be taken to be the mean from Fig.4, that is 0.00022. Terms containing higher order powers of p would be neglected.

The importance of having low spreads can be judged from the fact that a typical spread that retail forex trader pays is non-negligible compared to the expected profit. For example, a 1 pip EUR/USD spread today, as a fraction of price, is about 7×10-5 which is about 1/3 of the expected profit, averaged over currency pairs, expressed in the same unit! And EUR/USD has one of the lowest spreads. The expected profit for EUR/USD would be 5.7×10-4 which is higher than average and about 8 times the spread but that's getting too detailed -- the point is, the risk of surrendering the entire profit to the broker is very real unless one shops around. If your broker charges a commission rather than spread (zero-spread brokers), the calculation will be different.

Selectivity of the system remains fairly low with this set of criteria; with the trading history beginning in late April 2006, the average number of trades selected per day (from the 14 pairs) is around 2.4. Further research into how to improve the system is underway.

Last Updated ( Wednesday, 08 September 2010 16:00 )