Evidently, if we put too little at the stake, we won’t cover our expenditures of time, energy and beer, too. It is much less evident, yet so, that if we start betting too much, sooner or later we are going to lose the entire capital. Economical theories and common sense both keep telling us that the higher the risk, the more the profit. This statement is untrue: the dependece between risk and profit is non-linear.

Let us imagine there are only two outcomes in our treading: losing the bet wit ha probability 100 - PctWin, or winning WinToLoss * bet size with a probability PctWin. In this case the mathematical expectation will be:

Expectancy = PctWin * 0.01 * WinToLoss - (1 - PctWin * 0.01)

Suppose that the PctWin and WinToLoss parameters are set and we can only control the bet size. Let us then review the dependence between profit and bet size after 100 trades with different PctWin and WinToLoss values using Monte Carlo modeling. To do this we repeat over and over 100-trade series for every combination of the bet size, PctWin, WinToLoss parameters. The exact outcome (profit or loss) will be determined by a random number generator.

Here is an example of implementing Monte Carlo methods in TradeStation (the code for the corresponding TradeStation signal is shown in Appendix 1). Copy it to PowerEditor, create in StrategyBuilder a strategy with this signal, apply it to any plot and launch parameter optimization in TradeStation as shown below.

This strategy will save to a file the profit for all combinations of parameters and random trade outcomes. One should keep in mind that the number of bars multiplied by the number of combinations mustn’t exceed 65536 (the maximal number of lines in an Excel file). The Random (100) function will generate an uniformly distributed random value between 1 and 100. Then the PctWin-Random will define with a PctWin probability whether the given trade brings profit or loss, and the profit size will be equal to WinToLoss.

Then we can plot in Excel the plots indicating the profit for the given parameters. For example, let us recall the game played by scientists from the previous article, where the bet won in 60% of cases and lost in 40%. To plot the dependence between average profit and bet size in that game, we must:

• Launch in TradeStation an optimization of a strategy by the PctRisk parameter = 5, 10, …, 90 with constant PctWin = 60%, WinToLoss = 1;
• Open in Excel the file D:TS_ExportMTrading_MMII.csv;
• Enter the values of the parameters to be optimized in column F and the following formulas in column G:

=SUMIF (A$1:A$20860,"=5",E$1:E$20860)/COUNTIF (A$1:A$20860,"=5")

=SUMIF (A$1:A$20860,"=10",E$1:E$20860)/COUNTIF (A$1:A$20860,"=10")

etc.

We then will see a plot like shown in Ill. 2.

The shape and values of the curve may differ somewhat in different runs, since random values are random, but the profit will invariably first rise and then descend as the risk grows.

All the multitude of money management algorithms may be divided in two principal classes: martingale and antimartingale.

Martingale methods state that the risk should increase as the capital decreases. These methods are popular with traders trying to extract profit from a series of losses.

Let us review an application of martingale in roulette. We bet 1$ on a color and every time we lose, we double the bet. Next time after we win, we start at 1$ again. If we lose 10 times in a row, which may happen with a probability of (19/37)^10 or 0,13%, we’ll have to bet $1024 to win $1. Since in such a case the expected profit/risk ratio is disastrously low, it is often supposed that martingale methods may not be used in trading. But, one should keep in mind that in popular trend-following methods

But, one should be well aware that in popular trend-following methods

1) profits are usually 2-3 times larger than losses

2) series of small losses are typically interspersed with large profits

So martingale methods in our opinion deserve a serious study.

Antimartingale methods state the direct opposite: the risk size should be increased as the capital grows and decreased as the capital decreases.

The known antimartingale methods advise to risk a fixed fraction of the capital (fixed fractional):

• Trade a constant number of stocks – with some conditions this method can be considered an antimartingale;
• Use the whole accessible capital;
• Trade one lot per X dollars on account;
• Divide the account into equal shares corresponding to the assets traded;
• Risk a part of the capital;
• Take the risk in proportion to the traded assets’ volatility;
• Use the Kelly method, optimal f anf their variants.

The fixed ratio method by Ryan Jones can also be considered antimartingale. This method states that the relation of the number of stocks traded to the capital gain necessary to increase the number of stocks should remain constant. Ryan Jones was so sure of his method’s advantages that last year he resolved to break the World Trading Cup record of Larry Williams standing since 1987. Williams then increased this capital from $10,000 to $1,147 000 in a year of real S&P and T-Bonds trading. Ryan Jones didn’t make it to 2000 year winners, but at May 31, 2001 he was a sure leader with a +226% result.

A positive aspect of antimartingale methods is that they allow the account to grow in geometrical progression.

The most popular method of money management is no money management. There are three variants of it:

1. Money management for gamblers

This method includes betting on a single trade all the accessible capital wit the maximal allowable leverage. No matter what the result, close the account and leave either with 100% loss or with a profit equal to

Recommended for newbies wishing for quick profits. This method is especially good when using a leverage of 1:100and higher: in the absence of a strategy with a positive mathematical expectation this method is optimal. The most important in this method is understanding that the strategy is used once, as luck only is exploited, not statistical advantage, which according to the law of large numbers can come true only in a large series of profits and losses.

2. Fixed number of lots

This method states: independent of the account state, always enter the position with the same (usually an even) number of lots.

Let’s apply this method to the simplest model system known as the “dynamic channel”: Buy one lot if the average day price ((high + low)/2) grows over its minimum by X points;

The code for this system with those algorithms is shown in Appendix 2.

The results of trading a fixed number of lots with $100000 starting capital and 0.66 margin are shown in Table 1 (here and below the results are taken from TradeStation Strategy Performance Reports).

Table 1. Fixed number of lots, simplest system.

Let us remark that a further increase of lots activates an implicit antimartingale money management in one direction: we cannot open positions larger than our current capital, so if the capital dectreases, so will the position size. When capital grows, the position size will remain constant. So let us redefine the method as follows: independent of the account size, always enter the position with the same (usually even) number of lots, if the current capital allows that; otherwise enter the position with the maximal possible number of lots.

Although this method is fairy safe, it does not allow the account to grow in geometrical progression, so we do not recommend using it.

3. «Bet it all»

This method states: use all the available resources when opening a position.

In other words, w open the maximal possible position every time.

Let us review how results of this method depend on the leverage with the starting capital of $100000 (table 2).

As you can see, even losing just 4 cents per share in a trade when our strategy is profitable, with a 2:1 or larger leverage we eventually lose the entire capital!

This method increases risk without an adequate increase of profit, so we cannot recommend using it.

4. Number of lots per fixed sum of money3

Number of lots = Capital / Х_ dollars

For instance, if we’re trading one lot per$1000, then, if we have $100000 on account, then we can trade 100 lots.

The method’s advantage is that a trade will never be rejected as being too risky – but again, in some cases this may turn out to be a disadvantage.