winning trade (e.g., $400)
AvgLoss is the average losing trade (€.g., $200)
investment is the amount invested (e.g., $10,000)
ProbWin is the probability (percentage) of a winning trade (e.g., 40)
ProbLoss is the probability (percentage) of a losing trade (€.g., .60)
MaxRisk is the maximum part of the investment that can be lost, in percent (€.g., .25)AvgWin% is @ABS(AvgWin/investment) -AvgLoss% is @ ABS(AvgLoss/investment)
Z is the sum of possible events, ProbWin*AvgWin% — ProbLoss*AvgLoss%
Ais the square root of the sum of the squares of possible events, (ProbWin*AvgWin%/2+ ProbLoss-AvgLoss%2)\(1/2) .Pis 5°(1 + (Z/A)) -
This can be written as the following spreadsheet example:
 
 
 
COMPOUNDING A PO. 1'lON 617
 
 
 
Row ColA Description of value or calculation in column A1 AvgWin 400.00 Enter average winning trade in $
2 AvgLoss 200.00 Enter average losing trade in $
3 Investment 40000.00 Enter initial capital invested in $
4 ProbWin 40 Enter probability of a winning trade
5 ProbLoss 60 (1 -A4)
6 MaxRisk 25 Enter maximum part of investment that can be lost (in %)7 AvgWin% .040 @abs(A1/A3)
8 AvgLoss% .020 @abs(A2/A3)
9 Zz .0040 A4*A7 + AS*AB
10 A 0297 (A4*A7A2 + AS*AB*2)4(1/2)
1" P 5674 0.9444
12 Risk of Ruin .1016 (1 = A11/A11)AB/A10)
 
In the above example, the risk of ruin is slightly greater than 10% based on historic data.The change in risk can be seen by varying the investment amount. Note that in row 11,when P = 1, the risk of ruin is 100%; therefore, P cannot be greater than 1.
COMPOUNDING A POSITION
At some point, all speculators find themselves adding to, or compounding, their position.Many traders view this as a means of concentrating on those commodities that have morepotential. There are two lines of thinking among these traders. When a trade becomesmore profitable, it is confirming its move and is thought to deserve more of a commitmentthan a trade that has not become profitable. On the other hand, by adding positions to atrade at preset intervals, the effect of a single poor entry point is reduced and a better aver-age entry price is created. This latter technique is called scaled-down buying in the securi-ties industry.
The following sections will assume that positions are added based on profitability as ameans of increasing leverage. There are a number of techniques used by experiencedtraders, but the time to add must be carefully selected. The situation chosen must havepotential for a long move with limited risk; the sustained consolidation period of a marketthat is priced near its historic lows would be a candidate. No matter how well chosen, eachmethod will result in the largest holdings at the highest (or lowest) price; when the mar-ket reverses, losses occur on a larger base and profits will disappear quickly. Compound-ing a position is very fragile, hard work and must be watched cautiously for a changingmarket; there are enough stories of speculators who leveraged small capital into a largefortune in less than a year’s time and then lost it ail in a week. As in all investments, the risk”balances the opportunities.
The Scaled-Down Pyramid
‘The patterns used to compound positions can be represented geometrically as pyramids.The standard pyramid, or upright pyramid, has a larger base than its top. The largest por-tion of profits are developed early and an adverse price move is not as likely to be disas-trous. The profit-compounding effect of this technique is comparably reduced. A favoritescaling method of this type adds one-half of the prior position at each opportunity (Figure23-12a). The maximum number of contracts to be held must be planned in advance. Thetotal position, if followed to completion, will be about twice the number of contracts thatwere initially entered; starting with 20 lots, 10, 5, 2, and 1 would be added, respectively. An 
618 RISK CONTROL
 
FIGURE 23-12 Pyramid stricture. (a) Scaled-down (upright) pyramid offers a small amountof compounding. (b) Adding equal amounts (inverted pyramid) gives maximum leverage.(©) Reflecting pyramid combines leverage and profit taking.
 
 
 
 
Additions Total position(a)10 | 40 J10 3010 2010 10Additions Total position
 
~ Additions and reductions . Total positions
advantage of this or any other pyramiding method is that an initial loss will be based on asmaller number of contracts than the final commitment.
Adding Equal Positions
As larger commitments are added, an inverted pyramid is created (Figure 23-126) in whichthe risk of an immediate loss due to a small reversal becomes greater.
‘Adding equal numbers of contracts reduces the proportion of the original commit-ment in the event the trend does not materialize but increases later exposure (when, hope-fully, there are large profits). The shift of risk can only be“determined by the trader. Withthis approach, as well as the other inverted pyramid formations, the speculator should follow the rule that no unsuccessful secondary position should offset the entire profits of theprior positions.
Adding Equal Amounts
To offset the effects of disproportionate risk, subsequent positions can be added based onthe new value of the market being traded. Comparing this to the previous method of equalcontracts, this would reduce the number of purchases in a rising market and increase them
EQUITY CYCLES : 619
 
in a declining market. The effect of increased volatility at higher levels would substantiatethis approach to adding as a means of maintaining the same relative effect of each newposition to the prior ones.
Maximum-Leverage Compounding
The greatest risk and the most potential is in adding positions using all profits accumu-lated to date. To do this necessitates a close working arrangement with the margin depart-ment of a brokerage firm. They might even want to install a direct telephone line.Whenever possible, new positions are added, based on the assumption that profits areconsidered a confirmation of the trend. Being completely leveraged is a tenuous position,requiring constant monitoring of the market; there must be a well-defined stop-loss at alltimes in anticipation of a premature reversal.
Reflecting Pyramid
‘One way to reduce the extreme risks of compounding is to remove positions once a specificprofit level has been reached. For example, if there is a reasonable expectation of an averageprofit of 60 points, a reflecting pyramid (Figure 23-12c) will add positions until the maxi-mum commitment is reached at one-half the expected profit and then reduce positions untilit is entirely closed out at 60. This technique can be modified to apply to any of the methodsof adding positions by requiring that the full commitment be achieved at a level below theexpected profit; full liquidation may be targeted for a point above the average profit.
Comparison of Compounding Methods
Table 23-6 and Figures 23-13@ and 23-136 show the risks and rewards of three methods ofcompounding compared with holding a single position from trade entry to exit. All forms ofpyramiding increase both profits and risk. Holding an unchanged position shows consis-tently lower returns with a mostly higher reward/risk ratio, especially at early equity levels.Scaled-down additions show an improved return and less attractive ratio; equal additionsshow much higher returns with a much lower reward/risk profile. The reflecting pyramidhas both higher returns than the single position and an improved ratio. At one-half theexpected return, it is clearly a better choice than the other methods. As expected profits areneared, the investor must choose between the extremely high leverage and profits of addingequal positions and the excellent reward/risk ratio of the reflecting pyramid.
EQUITY CYCLES
Every system has profit-loss cycles that can be seen clearly by plotting its daily or weeklyequity. Trending techniques show that once or twice each year, there are major increasesin profits corresponding to a trending market; at other times, there is a steady decline anda stabilizing pattern to the total equity. For a system to operate as it is expected, the posi-tions must be kept constant. Increasing positions as equity increases in a trending cycleresults in always being fully invested at the top of the cycle, when losses begin. Losses willbe on a larger base than profits, and equity will drop much faster than it increased.
The same effect can occur at the bottom of an’equity cycle after a nontrending marketperiod, A sustained losing streak may cause a speculator to reduce the investment in pro-portion to dwindling capital. If this happens, the result will be entering into a profitableperiod with a smaller investment than the prior losing period. The system must have dis-proportionately larger profits to recover the losses and achieve a net gain, An example of atypical system is shown, using a 100% gain for each profitable cycle, followed by a 50% losscycle. Assume this cycle is repeated twice each year; in one year, the following equity pat-tern will occur:ceevs:
ii 620 RISK CONTROL EQUITY CYCLES : 621
a
 
 
 
 
                   
 
 
 
 
 
 
 
 
 
 
i TABLE 23-6 Comparison of Compounding Methods* FIGURE 23-13 Returns and risks from compounding. (a) Equity and liquidated value4 i ‘Scaled to 10 (10-point reversal). (b) Reward to risk ratio.4 Price Add =Total «= quity = LiqVal_~—Risk—E-quity/Risk Equity LiqVal__—_——Risk 3000i No Pyramiding4 ‘ 0 10 10 0 100 100 0z 10 0 10 100 0 100 1.00i 20 0 10 200 100 400 2.00 Equal pestions (equity. 30 ° 10 300 200 100 3.00 2000# 40 0 10 400 300 100 4.00t 50 0 10 500 400 100 5.00i 60 0 10 600 500 100 6.00 3 «3 7 Eaual positions (lia val‘Scaled-Down (Upright) Pyramid scotedlown pyramid (equiti 0 32. 32 0 ~320 320 0 0 100 100 1000 etl oremtr 10 16 48 320-160-480 67 100 -50 150 electing pyramici(equty)4 20 3 56 800 240 560 1.42 250 75 175 Scaled-down pyramid cia vai)a 30 4 60 1360 760 600 227 425 237 187 Reflecting pyramid (lia val)40 2 62 1960 1340 620 37 612 418 193 No pyramiding (equity)‘ 50 1 63 2580 1950 630 4. 806 609 196 No pyramiding (liq ep: 60 0 63 3210 2580 1003 806g - =zt é Equal Positions 0 10 20 30 40 50 60° 10 (0 6 100 100 9 Price level increase7” 10 10 20 100 -100 200 50 (a) ~20 to 30 300 0 ~~ 300 1.00 :E- 30 10 40 600 200 © 400 150£ 40 Jo 50 1000 500 S00 2.00 -E 50 10 60 1500 900 600 2.50 Reflecting pyramid# 60 ° 60 2100 1500é. ° Reflecting Pyramidi° 10 10 0 100 100 ° .10 5 1s 100 -50 150 67 . -20 2 7 250 80 470 14730 0 17 420 250 170 247 Nopyramiding40 -2 15 590 440 150 3.93 350 5 10 740 640 100 740 3$0. -10 0 840 740 z Scaled-down pyramiding— —— I (upright)+1 poine = $100.————Change Total - Adding equal positionsin Equity Equity : aOriginal margin $10,000 ~Gain of 100% +10,060 20,000 | First 6 month: - ExpectedLoss of 50% =5,000 1 ‘rst & months ae TetumGain of 100% +10,000 25.000 |g. ong 6 monch- Loss of 50% -5,000 20,000 recond 6 monthsee ee 1 4 160
This leaves a net gain of 100% for the year. Each 100% profit was $10,000, and losses were$5,000-—the rate of return is always based on the original margin. Had the cycle started
 
  
 
622 RISK CONTROL.
with the losing phase, there would be a profit of 50% for the first year and 100% for eachsubsequent year. :
‘Many traders would not net 100% each year from a system that performs as the one inthe example. As their profits increased, positions would be added so that at the time thetotal equity was worth $20,000, the margin requirements would also be $20,000. The 50%loss is then applied to the total equity:
ee
Change Total
in Equity EquityOriginal margin $10,000Gain of 100% +10,000 20,000 First 6 monthsLoss of 50% —10,000 10,000Gain of 100% +10,000 20,000 Second 6 monthsLoss of 50% 10,000 10,000
—
‘Trading futures would be a great deal of effort for no return. :
Holding the investment constant as shown can also be viewed by studying the growthand decline of the account excess, called the reserve. The size of the reserve relative to totalequity is the key to sticcessful management. Starting with margin and reserves equal,reserves increase during profitable periods and decrease during losing ones. Proportion-ately more of the total equity is-traded during losing phases. This pattern can be used toimprove results safely, as follows:
Se
 
Change Total Reserve!
in Equity Morgin Reserve Equity EquityOriginal investment 10,000 10,000 20,000 50%Gain of 100% +10,000 10,000 20,000 30,000 67%Loss of 50% 5,000 10,000 15,000 25,000 60%Gain of 100% +10,000 10,000 25,000 35,000 71%Loss of 50% 5,000 10,000 20,000 30,000 67%
paisa
Using the natural equity cycles, hold the number of positions the same and allow thereserve to increase during profitable periods; maintain the same position size through thebeginning of the next losing period. When the equity drop has slowed or stabilized,the total equity can be redistributed into margin and reserve according to the original 50%formula. In the next example, the total equity of $25,000 is distributed 40% to margin and60% to reserve at the end of the first cycle. It is redistributed so that the next profit phase-will be entered with a larger base than the previous losing cycle. The result is a gradualincrease in profits:
re EEE EEN
 
Change Jota! Reservel
- in Equity Margin —_—_—Reserve Equity EquityOriginal investment 10,000 10,000 20,000 50%Gain of 100% +10,060 10,000 20,000 30,000 67%Loss of 50% 5,000 10,000 15,000 25,000 60%Redistribute . 412,500 12,500 25,000 50%Gain of 100% +10,000 12,500 ° - 25,000 37,500 67%Loss of 50% —-5,000 12,500 18,750 31,250 60%Redistriouce 15,625 15,625 31,250 50%
 
INVESTING AND REINVESTING: OPTIMAL f 623
 
Trading on Equity Cycles
Ifa moving average technique is traded, the equity resulting from this strategy will fluctu-ate with the trending nature of the market. By applying a moving average analysis to theequity itself, the trending and nontrending periods are identified by buy and sell signalsjust as though the equity series was a price series.
An equity buy means that the market has begun trending; the length of this perioddepends on the calculation period of the trend. A sell signal means that the market is nolonger trending. These signals can be taken as “buy the system” or “short the system”; thatis, enter all positions that the system currently holds or liquidate the entire portfolio andhold cash. An equity buy could also be taken as the point to redistribute the equity into theoriginal ratio of margin to reserves. If the normal profile of price movement is to spend alarge proportion in a sideways pattern, then there should be sustained periods of down-ward equity trends; trading equity cycles can improve performance.
INVESTING AND REINVESTING: OPTIMAL f
Optimal f is the optimal fixed fraction of an account that should be invested at any onetime, or the size of the bet to place on any one trade. The amount to be risked is measuredas a percentage of the portfolio size. The objective is to maximize the amount invested (putat risk), yet avoid the possibility of total loss. Trading a very small part of assets can be apoor use of capital, while trading too much guarantees bankruptcy or ruin. Optimal fis theideal portion of an investment that should be placed at risk at any one time. If you risk lessthan the optimal f then you are not generating the peak profits; however, if you trade morethan optimal f you assure eventual ruin.
Investing generally has a two-level optimal f: the part of the total portfolio put at riskcompared with that part held in cash equivalents, and the individual size of the commit-ment to each stock or futures market within that portfolio. This is particularly importantfor futures, where the high leverage of individual markets makes it very easy to risk toomuch on each trade.
  
Fixed Initial Investments -
Then, how much should be invested? The optimal amount is difficult to pinpoint becauseyou would have to know what risks lie ahead, and of course, that’s not possible. However,based on a very small likelihood of losing 50% of the portfolio when 50% is invested, onemight say that a portion under a 50% investment is best. To account for greater uncertain-ties in the future, you could increase your confidence by investing only 25% of the initialportfolio. This approach has negative effects, because a smaller relative investment reducesboth risk and returns. When the investment becomes too small, the returns are no longerattractive.
The chance of a catastrophic risk is an important concern for any investor or portfoliomanager. For practical purposes this is always figured on the historic profile of the data ortrading results. This still leaves uncertainty in the final values. Nevertheless, the mostextreme situation is often found by using the calculation for risk of ruin (in a previous sec-tion of this chapter), most often applied to gambling situations in which the bet sizes, pay-out, and odds are well defined. When there are enough test data and trades, this techniquehas been applied to trading systems (see the section “Wins Not Equal to Losses”).
Some analysts have tried to deal with the uncertainties of- price movements by using aMonte Carlo technique in testing, which shifts the sequence of blocks of data, or profit andloss results, so that they occur in random order, The worst results are considered the great-est risk. This approach may be unrealistically severe, yet even the real performance is notlikely to reflect the size of the risk in the future. For those traders applying a long-term 
624 RISK CONTROL
trend-following technique td capture moves that are based on economic or governmentpolicy, sustained profits are most often followed by a reversal before the trade is ended. Infact, the ending change of direction is directly related to the sustained move. To move thisdata around so that the loss comes at a different time may create a large loss without theoffsetting profit, which is a situation that is unfair to the trading strategy. Before applying aMonte Carlo analysis, it is first necessary to identify the dominant period of sequential cor-relation to avoid segmenting the series incorrectly.
For an initial investment, the optimal f is simply the maximum part of that portfoliothat can safely be traded without any significant risk of ruin. For those investors who takeout profits when they occur and continue to trade based on the same assumed initialinvestment, nothing need be changed unless exceptionally high risk causes a reassessmentand decrease in the amount of leverage. However, it is more common and more compli-‘cated for the investor to vary the amount committed to the market by either increasing ordecreasing leverage. This involves (1) determining che right time to change the leverage,(2) calculating the amount to increase the investment when there are profits, and (3) fig-uring the size of the reduction when there are losses exceeding some designated amount.These are issues that are addressed by optimal f
Finding Optimal f
Ralph Vince, in his popular book, Portfolio Management Formulas," focuses on optimal frisk of ruin, and other practical items. Optimal fis the ideal amount of an investment thatshould be put at risk at any one time. As background for this, we need to formulate therelationship where the percentage gain needed to recover a loss is larger than the per-centage lost.
i ~
Required gain = 5 — ‘percent loss
‘That is, a 50% loss requires a 100% gain to restore the original value. Because theamount risked on each trade depends on our expectations of loss, the results obtainedfrom the optimal f calculation will be the size of the bet, invested amount, or the numberof contracts to be traded, as a percentage of the maximum loss. As we already know, thevalue used as a maximum loss will be an estimate, because losses can always be greaterthan only those experienced in the market, whether theoretical or real. In addition, theoptimal f will be different for each system, depending upon its performance profile.
‘The mathematics needed to determine optimal fis based on the Kelly Betting System.”Kelly states that the optimum bet is the one that maximizes the growth function G(f):
Gf) =P «ind +B*f)+a-P)*iIna-f/)
where fis the optimum fixed fractionP is the probability of a winning bet or tradeBis the ratio of the average winning return to the average losing return .In is the natural log function .
The solution for finding the optimal fixed fraction to invest uses the geometric productand geometric mean, which represent the way in which profits and losses accrue.
Optimal f = max([] @ fx CFL)
+ LEEjet largest loss
 
us}) |, forf=.01t0 1.0
Ralph Vince, Port/olio Management Formuias (John Wiley & Sons, 1990, pp. 79-86). Jobn L. Kelly, Jr., “Kelly Betting System,” Bell System Technical Journal (July 1956).
INVESTING AND REINVESTING: OPTIMAL f 625
where max is the function that selects the maximum valueTl is the productlargest loss is the largest loss of any PL;PI, is the profit or loss for trade iNis the number of trades
By testing values of f between .01 and 1.0, and finding the geometric mean of all trades(each percentage profit or loss applied to the account value before the current trade), wefind the value of f that gives the best return. That value is the optimal f the amount of thetotal account that should be invested for each trade.
‘Asimpler way of expressing optimal f the amount invested on a single trade, is given as”
F=(PxR+V-1/R
where R= the ratio of average profit to average lossp= the probability of a winning trade
‘Therefore, if p =..50, there is an equal chance of a profit or a loss, and the average profit is$400 while the average loss is $200 (giving R = 2.0), then f= (.50(2 + 1) ~ 1)/2 = .5/2 =.25or 25% of the available capital. Given an equal chance of a profit or a loss, it is not likelythat there would be four losses in a row, each of 25%; however, the theory of runs showsthat, out of every 100 trades, there should be one’run of six. Eventually, there will be a runof four or five Idsses in a row. Optimal £ however, invests a fraction of the current equity;therefore, after 4 loss of 25%, the next investment is 25% of the balance, or 18.75% of theinitial equity. If there are further losses, that amount drops to 14.06%. After three losses ina row, instead of having lost 75% of the initial equity, the investment has only dropped by57.81%. Over time, with profits twice as large as losses, and winning trades normally alter-nating with losing trades, the losses will be recovered.
Observations of Optimal f
According to Alex Elder," there are some difficulties in using optimal f Because the valueis based on every historic trade, the ideal amount to invest on the next trade will keepchanging. In addition, if you trade a position larger than the optimal f, with average results,you can expect to go broke eventually because you are overinvesting. On the other hand,if you invest less than the optimal amount, then your risk decreases arithmetically, but yourprofits decrease geometrically, which is another bad scenario. Because this process canbecome very complicated, the simple solution for most investors is to keep trading thesame amount, leaving a large reserve so you are not caught short.
‘On the positive side, Dr. Elder concludes that the most useful results of optimal f isthat it shows the important principles: _
1. Never average down.2. Never meet margin calls. -3. Liquidate your worst position first.
Practically Speaking
Evaluating the risk and reward ofa trading program, even using a number of years of actualperformance, rarely gives results that are statistically accurate. Markets change and the per-formance profile of any trading system can vary significantly over long time periods. Deter-mining the amount to invest on a single trade based on the average risk and average return
» Robert B Rotella, The Elements of Successful Trading (The New York Institute of Finance, 1992, pp. $49-550).21 Dr. Alexander Elder, Trading for a Living John Wiley & Sons, 1993) 
 
626 RISK CONTROL,
 
of past performance can lead to a tragic end. When investing 25% in each trade, a loss thatis twice what is expected, followed by a smaller profit and another large loss (in a morevolatile period), could exhaust your capital, regardless of the statistics.
In general, the less you risk, the safer you are. From time to time there are price shocksthat produce profits and losses far greater than the averages show. Although the theory ofan optimal fixed fraction may be correct, the numbers used in the calculation are not oftenreliable.
‘COMPARING EXPECTED AND ACTUAL RESULTS
In the development of an economic model or trading system, the final selection is usually theresult of a performance comparison of the completed models. Often the results are given interms of profivloss ratios, annualized percentage profits, expected reliability (percentage ofprofitable trades to total trades), and potential risk. Although these statistics are common,their predictive qualities and sometimes their accuracy are not known. On occasion, theseresults are generated by a sample that is too small, and usually they are not the results of apredictive but an historic test. This does not mean that the model will be unsuccessful, butthat the pattern of success might vary far from the expected profit/loss ratio, reliability, andrisk. In actual trading, every speculator experiences a series of losses far exceeding anythingthat was expected; at that point, it is best to know whether this situation could occur withinthe realm of the system’s profile or whether the system has failed. For example, a movingaverage system is expected to have 1 out of 3 profitable trades with a profit/loss ratio of 4 : 1.But the first 10 trades of the system are losers. Should trading be stopped?
Binomial Probability
Consider the application of a random-number sequence to the trading, model. What is theprobability of / losses in 7 tries when the probability of a loss is p? Most of the work in thisarea of probability is credited to Bernoulli, whose study of a random walk is called aBernoulli process. A clear representation of a random walk is shown by the Pascal triangle(Figure 23-14), where each box represents the probability of being in a particular posi
at a specific time in a forward random walk. The result of this process is called a binomialdistribution. .
‘The forward random walk has an analogy to price movement, with the far edges of Pas-cal’s triangle showing the probability of a continuous sequence of wins or losses using ran-dom numbers