[1+A)/(1-A)f-1 " [(1+ )/(1- ) -

where all terms are the same as above, and G is the goal in units of trading capital. Wins Not Equal to Losses

The basic equations just presented are generally applied to gambling situations, where the size of profits and losses are the same. This requires that the percentage of winning events exceed the losing events to avoid ruin. Futures trading, however, often results in more losing trades than profitable ones and must, therefore, retum much larger profits than losses. Such a shncture is common to all conservation of capital, or trend-following, sjstems. This situation can be applied to the more complex form where

Cr= the total cital available for trading (in units)

Cg = the cutoff point where level of ruin is reached (C, <

= - Ck, coital available to be risked

E = the expected mean return per trade, the probability-weighted sum of values that a trade might take

e Trading & Money (Revised Edition (Irwn, li*95»

FIGURE23-11 Risk of ruin based on capital.

where ., is the possible profit or loss value, and p, is the probabiUty of PL, occurring <0<p,<l).

bl is the expected squared mean retum per trade, the probability-weighted sum of all Ihe squared values of a trade,

where PI, and p, are defined above.

=0.5+£/(2\) and the risk of njin is

Introducing an oiDfective and a desired level of capital L, the risk of ruin R becomes [<l-f)/Pl°. [0.~P)/Pf-l

where

c = i/Vii

As in the first situation, using equal profits and losses, the rid; increases as the objective L increases. Ralph Vince, in Portfolio Management Formulas (Wiley, NewYork, 1990) derived similar results from P. Griffins woik. The Theory of Blackjack (Gamblers Press, Las Vegas, 1981), which claims to provide a "fair approximation" of rid; Vinces approach has been modified for convenience and given in a way that allows apreaddieet formulas:

Rid; of Ruin = ((1 - P)/P)A(Ma.xRisk/A)

where the following terms are defined in the order needed for calculation:

AvgWin is the average winning trade (e.g., $400)

AvgLoss is the average losing trade (e.g., $200)

Inveshnent is the amount invested (e.g., $10,000)

ProbWin is the probability (percentage) of awinning trade (e.g., .40)

ProbLoss is the probability (percentage) of a losing frade (e.g., .60)

MasRisk is the ma.ximum parto of the investment that can be lost, in percent (,e.g., .25)

AvgWin<io is [aABS(AvgWin/invedment)

AvgLosSo is [aABS(AvgLossinvestment)

Z is the sum of possible events, ProbWin* AvgWin" - ProbLoss* AvgLosSSo

A is the square root of the sum of the squares of possible events, (ProbWin-AvgWin" 2

+ ProbLoss.AvgLoss"oA2)A(I/2)

P is .5*(-I (Z/A))

This can be written as the following apreadsheet example

in the above example, the risk of ruin is slightly greater than 10°o based on historic data The change in risk can be seen by varjing the investment amount. Note that in row 11, when P= 1, the rid; of ruin is 100° o; therefore, P cannot be greater than 1.

COMPOUNDING A POSITION

At some point, all apeculators find themselves adding to, or compounding, their position. Many traders view this as a means of concentrating on those commodities that have more potential. There are two lines of thinking among these traders. When a trade becomes more profitable, it is confirming its move and is thought to deserve more of a commitment than a trade that has not become profitable. On the other hand, by adding positions to a trade at preset intervals, the effect of a single poor entry point is reduced and a better average entry price is created. This latter technique is called scaled-down bujing in the securities industrj.

The following sections will assume that positions are added based on profitability as a means of increasing leverage. There are a number of techniques used by experienced traders, but the time to add must be carefiilly selected The situation chosen must have potential for a long move with limited risk; the sustained consolidation period of a maiket that is priced near its historic lows would be a candidate. No matter how well chosen, each method will result in the laigest holdings at the highest (or lowest) price; when the maiket reverses, losses occur on a laiger base and profits will disappear quickly. Compounding a position is very fragile, hard work and must be watched cautiously for a changing maiket; there are enough stories of speculators who leveraged small capital into a laige fortune in less than a years time and then lost it all in aweek. As in all investments, the rid; balances the opportunities.

The Scaled-Down Pyramid

The pattems used to compound positions can be represented geomeU-ically as pjramids. The standard pjramid or upright pjramid, has a larger base than its top. The laigest portion of profits are developed early and an adverse price move is not as likely to be disastrous. The profit-compounding effect of this technique is comparably reduced. A favorite scaling method of this tjpe adds one-half of the prior position at each opportunity (Figure 23-12a). The maximum number of contracts to be held must be planned in advance. The total position, if followed to completion, will be about twice the number of confracts that were initially entered; starting with 20 lots, 10, 5, 2, and 1 would be added, respectively. An

FIGURE 23-12 Pjramid stmcture. (a) Scaled-down (upright) pjramid offers a small amount of compounding, (b) Adding equal amounts (inverted pjramid) gives maximum leverage, (c) Reflecting pjramid combines leverage and profit taking.