ProSizer TM 384
Unicorn Research Corporation
4621 N. Landmark Dr.
Orlando, FL 32817-1235
Overall Rating:1/2 (out of 4)
Available by download from
http://unicorn.us.com/trading/
Cost: $37.00
Requirements: The complete software package is contained in an Excel Workbook. The user must have installed Microsoft Excel 97 or later.
The software is designed to assist any trader of stocks, futures, forex or any other tradeable financial instrument by analyzing the effect of different “money management” or position sizing strategies used with the trader’s system. The idea is that by changing the amount risked on each trade, the trade-off between return and risk (the maximum drawdown) can be controlled. The vendor of ProSizer, Unicorn Trading, believes that a simple, low-cost Excel spreadsheet tool can determine the optimal position sizing strategy for a particular trading system.
Software design
The software is contained in an Excel workbook with four worksheets. They are labeled: Data, Equity Chart, Combo and Documentation. The user must carefully paste their trading profit and loss data (for a “single lot,” a term that is undefined) into the appropriate cells on the data sheets. Then, on a distant cell, the user must type the row number of the last line of the data. Any slip-up in this input can permanently damage the workbook, so the user must make a virgin copy of the entire workbook before embarking on this awkward data input process.
The user then selects any one of the five available position sizing strategies: 1) Fixed Lot, 2) Fixed Fraction, 3) Fixed Ratio, 4) Risk % of Equity, or 5) Volatility % of Equity. Any of the five strategies may be combined or simply used by itself for a total of 15 combinations. Once a combination is selected, the user must input 10 additional data points including initial equity dollars, margin cost per lot, fixed % of equity, and in a distant cell that is easily overlooked, the number of “Monte Carlo Iterations.” Finally the user presses a button in the data sheet and, in the words of the software, a “Monte Carlo simulation” is performed. More precisely the software either simply permutates the order of the input profit and loss trades or, optionally, re-samples the trades — with replacement — then reports some bare-bones statistics from the sampling.
The sampling statistics, such as the mean, median of the sampled return and maximum drawdown, are reported along with a quantity improperly termed a “standard deviation.” These sampling measures reveal the inadequacy of the analysis. The Help file asserts that the sampled maximum drawdown is distributed as a log-normal statistical distribution. This is not even approximately correct. A geometric Brownian model of security prices (used to derive many option formulae) was recently solved for the analytic form of the statistical distribution of the maximum drawdown. (I worked on the problem many years ago with a Nobel-laureate and another distinguished scientist without success.) The asymtotic form of the statistical distribution, when the (long-term return^2 * time horizon )/ bar-to-bar variance is large, does not converge to a log-normal. In this case the expected value of the max-drawdown asymptotically approaches 0.5*sig^2 / mu * ( log( x ) + 0.5 ) where x = mu ^ 2 * T /( 2*sig^2 ) . Here, mu is the Brownian “drift” or linear regression slope of the prices, sig^2 is the variance of the bar-to-bar log return, and T is the number of bars of horizon over which the max-drawdown is measured. When x is positive and very small, the expected value of the maximum drawdown approaches 1.7726* x ^ 0.5 * sig^2/ mu, and for short sellers, when mu is large and negative, the expected value is 2* sig ^ 2 * ( x + 0.5 )/Abs( mu )
The important point here is that most traders do not have more than 250 trades available, so the maximum drawdown statistics will assume a complicated (but now known) distribution. The distribution is highly skewed. If the number of trades is a small number, say below about 30, the re-sampling employed by ProSizer will be infused with a large sampling error, (regardless of the number of re-samples), and the statistics will be suspect. Above this number, re-sampling can converge to a reasonable estimate of the original – but unknown – empirical distribution. Still, summary statistics like a standard deviation to inform the trader about the possibly large maximum drawdowns that may be experienced in the future cannot be relied upon. A much better guide is to use the geometric Brownian formulae given above.
Considering the highly skewed nature of the maximum drawdown distribution and the skewness that may be introduced by sampling a small number of trades, the only acceptable way to describe the dispersion of the maximum drawdown is by use of confidence limits. The software should report the 90% or 95% quantile of the distribution but ProSizer does not report this important information.
The software reports and draws charts of the Calmar ratio of return/maximum drawdown, but these are unreliable and cannot be used due to software problems. Furthermore, the Calmar ratios are difficult to interpret because they divide a return percentage by a maximum drawdown percentage. The resulting percentage of a percentage ratio is conceptually difficult to understand. For example, if the trader starts with an initial equity of $5,000 and on a re-sample run immediately experiences a $2,000 drawdown, but later, after the system produces gains and the trader experiences a $10,000 drawdown with a $50,000 equity, the software would only record the small $2,000 drawdown for a 30% drawdown; not the “smaller” $10,000, 20% drawdown. The resulting statistical mean of the Calmar ratio of a percentage of a percentage would be an unreliable guide to the trade-off between return and risk.
Indeed the Calmar ratio is a crude scalar measure of the reward/risk trade-off. Much better would be a maximum drawdown-return chart of the cloud of re-samples, so that a proper two dimension portrayal of the trade-off could be appreciated. A better appreciation of the number and percentage of re-samples that dominate in the upper “North-West” portion of the chart would be much more useful.
Other features
One useful feature is contained on the EquityChart sheet. Here, for one re-sample, the software draws a chart of both the return and percentage drawdown versus time. It is instructive to press the re-sample button and experience the resulting curves that may be realized in future trading, assuming that the number of trades is large enough and that the future follows the empirical distribution of the past trades, which of course, it won’t. (Markets are not stationary, which is the source of their intrigue.)
Finally, ProSizer 384 contains a Combo sheet which reports the percentage returns and maximum drawdowns for a pair of position sizing strategies when their parameters are varied. However, I could not determine how the combination strategies are implemented and the documentation sheet does not illuminate the subject.
What is really needed is an optimization module that could find the sizing parameters that optimize a selected trading statistic, (e.g., average return or maximum drawdown or even the Calmar ratio). Use of modern Particle Swarm or Differential Evolution methods could be implemented to quickly accomplish this task. The ProSizer attempt at this task is laudable and the implementation is certainly visually beautiful, but its concept is un-explained.
The analysis presented is inadequate to properly measure many questions raised by the challenge. Further, the software suffers from bugs and on many runs with different inputs, the reported statistical measures appear to be out of bounds. Additionally the statistical measures reported in the spreadsheet are conceptually inadequate and misleading. For these reasons, the software cannot be recommended. It is not a bargain even at its low price.
Paul H. Lasky is the chief investment strategist for the P & B Fund and is a quantitative analyst and consultant to other funds.
