oothing,
(65-Day Eff Ratio and 16-Day Trend)
 
 
 
 
 
400 ABM nn aoe on ~| ED iz 300 we - oe - i2 Trend profits ‘Bm WO -- + -ee --......Withless noise ..5 wv NCP£ | .BNWT gy gag [ PB cra- 0 ———$ 98sSpe Ft=100 $4} 4} tt tt008 0.1 0.12 0.14 0.16 0.18 02 0.22 0.2465-Day Efficiency Ratio
 
When the efficiency ratio is high, then a trend system is a better strategy.When the efficiency ratio is low, then a countertrend system is best.
in the second case, a countertrend approach does not mean that you must set a short posi-tion when the market is moving up. It can also mean taking profits on a long position whenthere is a countertrend sell signal, or building a long position when a countertrend buy sig-nal occurs on a price drop. While the efficiency ratio is not specifically a directional indi-cator, it does seem to classify markets as trending or nontrending.
PROBABILITY OF SUCCESS AND RUIN ©
The relative size of trading profits and losses, the frequency of the losses, and the sequencein which they occur comprise the equity profile of traders and systems. This profile can beused to determine the capitalization necessary to maintain trading during the losing peri-ods and allow the system to return to its full potential. In investment terminology andprobability theory, the level at which you no longer have enough money to continue trad-ing is called the point of ruin, and the chance of getting there is the risk of ruin. The prob-ability of the risk of ruin is normally expressed as .
1-A/R=|——— -1+A :where 0<R<'1, 0 indicates no risk, and 1 certain ruinA=P~(1~P), P is the proportion of winning trades, also considered the trader's
advantagec= the beginning units of trading capitalPROBABILITY OF SUCCESS AND RUIN 61S
 
A system of trading that has 60% profitable trades and trading capital in $10,000 unitswill have a risk of ruin calculated as follows:
A= 0,60 ~ (1 ~ .060) = 0.20Re 1820) 0.80\"_/1\~\1+0.20) ~\120) \3When c= 1 ($10,000), R = 0.33, and when c = 2 ($20,000), R = 0.11. Therefore, the greaterthe trader’s advantage or the greater the capital, the smaller the risk of ruin (Figure 23-11).
When using profit goals, the chance of ruin should decrease as the goal becomessmaller. The relationship can be expressed as:
_ [1+a/G-A)]f-1
~ [+A -A)]*t 1" Osks1
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 thesize of profits and losses are the same. This requires that the percentage of winning eventsexceed the losing events to avoid ruin. Futures trading, however, often results in more los-ing trades than profitable ones and must, therefore, return much /arger profits than losses.Such a structure is common to all conservation of capital, or trend-following, systems. Thissituation can be applied to the more complex form'” where
C;= the total capital available for trading (in units)Ce = the cutoff point, where level of ruin is reached (C, < C7)C4 = Cr— Ca, Capital available to be riskedE = the expected mean return per trade, the probability-weighted sum of values that atrade might take
E=) @Lp)
faa
17 Fred Gehm, Quantitative Trading & Money Management (Revised Edition (Irwin, 1995)).
 
 
FIGURE 23-11 Risk of ruin based on capital.
Risk
 
Capital616 RISK CONTROLTS
where PL, is the possible profit or loss value, and p, is the probability of PL, occurring(@ <p,<1).
E, is the expected squared mean return per trade, the probability-weighted sum of allthe squared values of a trade,
N
E,= >, lip)
istwhere PL, and p; are defined above.
D=G,/VE;
P=0.5+E/(2VE;)and the risk of ruin is
e(F)
P
Introducing an objective and a desired level of capital Z, the risk of ruin R becomes
_ia -Py/PP-1
Rel Gc pptwhereG=VVEz ~
As in the first situation, using equal profits and losses, the risk increases as the objective Zincreases. - .
Ralph Vince, in Portfolio Management Formulas (Wiley, New York, 1990) derived sim-ilar results from P Griffin’s work, 7e Theory of Blackjack (Gamblers Press, Las Vegas,1981), which claims to provide a “fair approximation” of risk. Vince’s approach has beenmodified for convenience and given in a way that allows spreadsheet formulas:
Risk of Ruin = ((1 — P)/P)4(MaxRisk/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)
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 (e.g., .60)
MaxRisk is the maximum part of the investment that can be lost, in percent (e.g., .25)AvgWin% is @ ABS(AvgWin/Investment) : -
AvgLoss% is @ ABS(AvgLoss/investment) “
Z is the sum of possible events, ProbWin*AvgWin% — ProbLoss*AvgLoss%
A is the square root of the sum of the squares of possible events, (ProbWinsAvgWin%2+ ProbLoss*AvgLoss%*2)( 1/2) : -
P is .5*(1 + (Z/A)) ~
This can be written as the following spreadsheet example:COMPOUNDING A POs: 1'lON 617
 
 
 
Row CofA 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 10000.00 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 + A5*A8
10 A 0297 (AS*A7A2 + AS*A842)4(1/2)
W P 5674 0.9444
12 Risk of Ruin 1016 (1 —AT1VA11)*(A6/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 afl 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 (Figure23-124). 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. An618 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.{c) Reflecting pyramid combines leverage and profit taking.
 
 
Additions . . Total position
-10
 
~ Additions and reductions . Total positions(9 ‘
 
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 23-12) 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 fol-low 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 themEQUITY 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 untilitis 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:620
TABLE 23-6 Comparison of Compounding Methods*
RISK CONTROL
 
 
 
 
 
 
 
 
 
Scaled to 10Price Add Total Equity Liq Vat Risk Equity/Risk Equity Lig Val RiskNo Pyromiding0 10 10 0 -100 100 Q10 0 10 100 0 100 1.0020 0 10 200 100 100 2.0030 0 10 300 200 100 3.0040 0 10 400 300 100 4.0050 0 lo 500 400 100 3.0060 0 10 600 500 100 6.00Scoled-Down (Upright) Pyramid0 32 32 0 -320 320 0 0 -100 100lo 16 48 320 —160 480 67 100 —50 15020 8 56 800 240 560 142 250 75 7530 4 60 1360 760 600 2.27 425 237 (8740 2 62 1960 1340 620 3.17 612 418 19350 | 63 ”. 2580 1950 630 4.1 806 609 19660 0 63 3210 ~ 2580 1003 806Equal Positions0 10 10 0 -100 100 0lo 10 20 100 100 200 5020 10 30 300 0 300 1,0030 0 40 600 200 400 1.5040 10 50 1000 500 500 2.0050 10 60 1500 900 600 2.5060 0 60 2100 1500: Reflecting Pyramid° 40 10 0 100 4100 0lo 5 15 100 -50 (50 67 x20 2 (7 250 80 170 1.4730 0 17 420 250 170 2.4740 2 15 590 440 150 3.9350 a) 10 740 640 100 7.4060 ~10 0 840 740
SN
*1 point = $100.
i
 
 
Change Tota! -in Equity EquityOriginal margin : $10,000 onGain of 100% +10,000 20,000 | os & monthLoss of 50% 5,000 15,000 ‘est 6 monthsGain of 100% +10,000 25,000- Loss of 50% -5.000 20,000 { Second & months
 
This leaves a net gain of 100% for the year, Each 100% profit was $10,000, and losses were$5,600-—~the rate of return is always based on the original margin, Had the cycle startedEQUITY CYCLES ~ 628
 
 
FIGURE 23-13 Returns and risks from compounding. (a) Equity and liquidated vaiue(10-point reversal). (b) Reward to risk ratio.
3000
                 
Equal positions (equity)
Value
7? Equal positions (liq val)
f Scaled-down pyramid (equity)
Reflecting pyramid;(equity)
‘Scaled-down pyramid (iq val)Reflecting pyramid (liq va!)
1000
No pyramiding (equity)No pyramiding (liq val)
-0 10 20 30 40 50 60Price level increase(a)
         
Reflecting pyramid70
6.0
5.0 No pyramiding
Scaled-down pyramiding(upright)
40
Reward/risk ratio
3.0
Adding equal positions2.0
10
0 10 20 30 40 50 60~ Prics leve! increase wee
(6)
 622 RISK CONTROLSS EROwith 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:
 
 
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% +40,000 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 successful 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:
 
 
Change Total Reservel
in Equity Margin 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%
nce
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 gradualintrease in profits:
 
 
Change Jotal Reserve/
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 (0,000 15,000 25,000 60%Redistribute . 12,500 (2,500 25,000 50%Gain of 100% +10,000 {2,500 ° - 25,000 37,500 67%Loss of 50% 5,000 (2,500 18,750 31,250 60%
Redistribute 15,625 15,625 31,250 50%
 
 INVESTING AND REINVESTING: OPTIMAL f 623
 
Trading on Equity Cycles
If a 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 f is theideal portion of an investment that should be placed at risk at any one time. If you risk lessthan the optimal § 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-term624 RISK CONTROL
 
trend-following technique to 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 the 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 f is 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.
1
Requii in =——_—_—_—_---1quired gain 1- 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(/):
G(f) =P «In(Ql+B*f)+(~P)*Ind-f
where fis the optimum fixed fractionP is the probability of a winning bet or tradeB is the ratio of the average winning return to the average losing returnIn is the natural log function
ea
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.
Nn. >, Sfx (FL) y"' f= max OO for f= .01 to 1.Optinial f: anas([] (a largest li 3) ’ for f= .0 0
‘© Ralph Vince, Fort/olio Management Formuias (John Wiley & Sons, 1990, pp. 79-86).¥ Jobn L, Kelly, Jr, “Kelly Berting System,” Bel System Technical Journal (July 1956).INVESTING AND REINVESTING: OPTIMAL f 625
 
where max is the function that selects the maximum valueIT is the productlargest loss is the largest loss of any PL;PL, 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