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The Foundations of Money Management I

People have always wanted to win at the stock exchange. But the existing industry of attracting money to the market with promising-named books, metastocks and finams of all kinds exploits our common prejudices, making us seek wrong things at wrong places. We're busy looking for a "magic" indicator or trading system that will keep us winning 90% of the time.

I've found such a system. With numerous tests it almost never had under 90% profitable trades. The results of one such a test are given in Table 1 in Omega Research TradeStation format. The code for the system is in Appendix 1; you may copy it to Omega TradeStation or SuperCharts and go along winning (in the sense they usually mean winning, that is, having a profit on most trades). The system's main secret is a pseudo-random number generator (too "pseudo" in TradeStation, but doesn't matter much). Then it all goes as usual: if the position is profitable, close it. If the market goes against us, turn investors. Having enjoyed working and socializing with customers of two brokerages over a couple of years, I can insist that is just what most traders do - except the fact they formally replace the random number generator with analytic forecasts, indicator signals, the neighbor's opinion in the pit or just a momentary impulse. The problem is that winning at an exchange and earning money at an exchange are far from being the same.

Surely, the profit seen in the Table 1 example is casual, a result of a lucky dice roll, whereas it would not be profitable in most cases. But if one changes the system entry parameters to more reasonable levels, i.e. sets mmstp=1, pftlim =4, maxhold =10, this will make the system profitable in most tests.

So exploiting the principal idea of speculation - close losing trades fast and let profits grow - combined with money management allows to earn money even from random trades. Most people act just opposite to this principle; they let losses grow, hoping the market turns and proves how right have they been, and quickly close their profitable positions to prove how right they're at the moment. Most beginners and many self-styled pros, as our experience shows, are sure that the skill of market forecasting equals the ability to earn money at the market. Getting a profit on a given trade for them means proving their prognostic abilities and, consequently, their skill in making money.

A person unfamiliar with trading as a business could be puzzled by the fact that "successful investing and trading have nothing in common with forecasting"*. There is bad news and good news. The bad news is: markets cannot be prognosed. The good news is: one doesn't need to do that to have profit. We are concerned not with getting a profit on every trade, but on making large sums when we're right. The number of profitable trades may in this case be less than losing, that is, it is possible to use worse-than-random forecasting!

As a famous trader Paul Tudor Jones said: "I may be stopped four or five times per trade until it really start moving". That is, Paul may win only on a measly 20-25% times! Yet he'd had three-figure (percents) of income in five consecutive years with very low capital corrections1. Almost 100% of Steve Cohen's very large profits are taken off 5% of trades, and only 55% of his trades are profitable at all. Despite that in the last seven years he'd made 90% per year on the average, and had only three losing months (the worst losses were -2%)2.

The widely used by professional methods of trend following, as a rule, bring about 30-40% of profit. Profits or losses in any given trade do not matter - as long as the amount of money earned per average trade is positive. This value is called mathematical expectancy. The mathematical expectancy equals the sum of products of profit probabilities minus the sum of products of losses probabilities, multiplied by the losses' size

Simplified, the expectancy may be estimated as the probability of profits multiplied by the average profit minus probability of losses multiplied by the average loss. In terms of the Omega Research TradeStation this looks like:

Total Net Profit$562.70Open position P/L($75.60)
Gross Profit$1,269.40Gross Loss($706.70)
Total #of trades276Percent profitable92.75 %
Number winning trades256Number losing trades20
Largest winning trade$54.90Largest losing trade($126.50)
Average winning trade$4.96Average losing trade ($35.33)
Ratio avg win/avg loss.14Avg trade (win &loss)$2.04
Max consec.Winners39Max consec.losers2
Avg #bars in winners1Avg #bars in losers17
Account size required$177.30Return on account317.37%

In a newsgroup discussion one follower of Elliott's theory said: "Market is no gambling - we make no bets". Not being an Elliott adherent, for whom everything is pre-arranged, we do make bets. Since the result of any trade is unknown, any trade is a bet where we win or lose a certain sum. The principal difference between gambling (betting) and market trades (speculations) is first, that gambling creates its own risks and speculations re-distribute the risks already present on the market; second, the on a market a trader is able to provide himself with a statistical advantage, that is, a positive expectancy.

Let us review betting on a color when playing roulette. There are 18 red sectors, 18 black and the zero. The expectancy of winning for a single bet on a color is 18/37 - (18+1/37) = - 1/37. On the average the house wins from a single gambler this amount multiplied by the bet size. Despite the fact some gamblers may win a lot, it is the house that wins always - because of the biased expectancy, not because the dealer knows where the ball stops.

To be just we should mention that it is possible to create a "gambler's advantage" - so a mathematician Edward Thorp has developed strategies with a positive expectancy for playing blackjack, which he'd successfully used in Las Vegas gambling houses. When they stopped letting him in, he published his methods1, after which blackjack rules had to be altered to remove the gambler advantage. In late sixties Thorp took interest in shares market and became a manager for a private investing partnership: " Our significant rival then was a Harry Markowitz, a future Nobel prize winner. After 20 months we had +39,9% profit compared to Dow Jones' +4,2%. Markowitz went negative in a couple of years, and we're satisfied with our stable results- about 20% yearly (standard deviation around 6%0 and zero correlation with the market".

The market allows to play games with a positive expectancy. This is a necessary condition for successful stock trading. Actually, as Ralph Vince says, "it doesn't matter how negative or how positive; only positive or negative matters". A doubtful claim from our point of view; a larger positive expectancy is superior to a smaller one.

Besides expectancy, most traders have problems understanding risk. For instance, a historian by education, (former) head of a regional investing company with assets over a million dollars by summer 1997 was sure that "risk doesn't exist so it cannot be measured" and also sure that "one shouldn't sell shares at a loss". What can one say about amateurs then- Risk does exist and it can be measured. It is considered that risk is a volatility measured as the standard deviation of the changes of actives traded. This holds true for investing risk, speculative risk is more adequately defined as standard deviation of capital changes. By both those definitions risk is heavily underestimated. According to Murphy's laws, the worst is yet to come; We shall employ the following definition: risk is the amount of money we are ready to lose before withdrawing from a losing trade.

Before opening a position it is necessary to define the point where we close the position wit a loss to save capital - the so-called stop loss1, or where we open an opposite position, having made sure of our mistake concerning the market direction - the so-called stop-and-reverse. The difference between the entry point and the stop loss point multiplied by the number of lots is the starting risk or 1 R2, independent of how and in which units we measure the stop level, be it dollars, percents, volatility units or six-packs. This definition of risk is not equal to the first definition - the risk may be many times the 1 R if the stops are not executed due to lack of discipline3, gaps against the position or unexpectedly high slippage. The profit, then, can be defined in units of risk per share or in multiples of R. In terms of multiples the basis rule of speculation will be formulated as: keep losses at the level of 1 R as long as possible and let profits reach many times R.

The expectancy in multiples of R will mean how much can we win or lose per unit of risk in an average trade. To calculate expectancy in terms of multiples of R we must place the results of our trades in a table with the following columns: Number of lots Profit or Loss Starting risk Multiple of R

The Profit or Loss must take into account broker commissions and slippage. Multiple of R is calculated by dividing the second column by the third. Then to calculate expectance it is enough to add up the values of the fourth column and divide by the number of trades. This method is also works with "intuitive" trading.

So, we do have a winning strategy - what next?

We can open a brokerage account and bet all our capital with the maximal leverage.

Here the most important thing - the money management begins. To clear the situation here is a pair of facts. Ralph Vince invented a game, where bet size was the only moveable parameter. He chose forty doctors of sciences (i.e. not the dumbest people at least) as players, none of which were professional traders or studied statistics. The doctors played a game where 100 random trades were generated, one by one. Every one began at $1000, and before every trade one had to make a single decision - how much (up to 50% of the capital) to bet. 60% of the time the players won their bet, and 40% of the time they lost their bet. This game has an expectancy of 20 cents per dollar risked, i.e. in the long run the player can receive 1 dollar 20 cents per dollar. The academicals made their 100 bets, enough to resolve the expectancy. Making the same trades, they finished the game with different results. Guess how much of them increased their starting capital? Two of forty. 95% of doctors lost money playing a game with a positive expectation!1

Van Tharp made an even more striking example. In an Asian Tour for Dow Jones Telerate TAG (Technical Analysis Group) he gave lectures in 8 cities before 50-100 listeners each time, most of them professional traders for large companies or banks that traded shares, bonds or exchange rates on Forex. In an analogous game over a half of highly professional traders lost!2 Another personal example - a trader offered a similar game to a friend employed by Charles Schwab as a leading analyst. At the first level the distribution of multiples of R with an expectancy of 0,45 and 60% profitable trades. To get to the second level one had to make 50% profit in 100 trades. The result was "I cannot get to level 2 in a day!"3. In 1991 Brinson, Singer and Beebower published a research of the efficiency of 82 portfolio managers in a 10-year period, which showed that 91,5% of all profit was generated by asset distribution3. The asset distribution meant the division of capital between cash, shares and bonds. Only 8,5% of profit was due to buying and selling the right stocks and bonds at the right time.

Let us play the game described by Vince. If there was no risk, i.e. we knew the result of each trade beforehand, it would make sense to bet all the capital each time. So every player would have gained $1000 ..(1.2 ^100)=$82,817,974,522.01 .

In reality, if we bet all $1000 on the first trade, we have a 40% risk to lose all at the first attempt. Even if we win and have $2000, betting all on the next trade would be exactly as insane.

Now suppose we bet $200 at a time. So if five first trades are losing, we again lose all. The probability of such an event is small, just over 1%. But are we ready for such a "small" risk, if we can lose all the money? Suppose we lose in the first two trades (16% probability), so we'd lose 40% of the capital. Beginning from the next trade we must gather 67% of profit just ot restore the starting capital. This effect is called "asymmetric leverage"5.

Table 2 shows that loses of over 50% need improbably large profits just to recover; so if we risk relatively large sums and lose our chances to end up wit a profit are negligible.

% loss102030405060708090
% profit required to recover11,125,042,966,7100150223,3400900

The result in the doctors' case is explained not only by oversized bets. A widely spread pitfall is so-called "gambler's error": People tend to suppose that after a series of losses the probability of a profit increases, so we raise our bets. But in this game the probability is not affected by previous results and always remains at 60%.

Suppose that we bet a certain percent of our capital and record the current capital after each trade. Repeat the 100-trades sequence again and again, and after a lot (1000 or so) series we'll be able to estimate the distribution of results. Evidently, we'll have different end profits, since the game is random-based. This is called Monte Carlo modeling.

Let us arrange the 1000 profit performances from 1000 series from smaller to larger. Then let us divide this range into 100 parts with equal number of variants in each - so every such a percentile will have 10 variants of performance. The first percentile will contain 10 worst results, and its top limit (number 10) will correspond to what they usually formulate as: "In 1% of cases the results will be inferior to- value". Statistically this percentile is called k-1. The border of the 50 percentile (k-50) would correspond to: "In 505 of the cases the result will be inferior to-"

Table 3 displays the outcomes of the 1000 series with different bet sizes in percents of the capital.

Bet sizek-50 DD, %k-99 DD, %Max DD, %Worst profit casek-1 profitk-50 profitk-95 profit

With 10% bet for each trade the minimal capital after 100 trades was 181,1$. In 1% of all trades our capital was under $405 (Profit k1). In 50% the trading yielded $4501 and less (Profit k-50). In 95% of cases the end capital was below $22411 (Profit k-95), and, corerespondingly, in 5% of cases the end capital was above $22411.

Let us review drawdowns (DD in the table). The drawdown is the difference between the maximal capital and its subsequent minimum before the new maximum is reached. With 10% bets in 50% of the cases the DD was over 48%, in 1% over 78% and the maximal DD was almost 90% of the capital. With bets over 30% of the capital we ape practically doomed to ruin. Once again we remind that this game has a positive expectancy - at win/loss probability 60% to 40% the win size relates to loss size as 1 to 1.

Steve Cohen says that: "the traders' general mistake is taking too large positions in relation to their portfolios. The, when the shares move against them, they are hurt too much to remain in control, they finally either panic or freeze in shock"1.

These examples described the importance of bet size in games with an undetermined outcome. So what is money management? An Internet search with those keywords yielded links to services for personal financial control, advices on handling others' money, how to control risk, on Turtle Trading, etc. According to Van Tharp, money management is NOT:
T a part of system that dictates how much you will lose in a given trade
T a way to exit a profitable trade
T is not diversification
T is not risk control
T is not avoiding risks
T is not a part of a system that maximizes performance
T is not a part of the system that tells where to invest

Money management is a part of a trading system that tells "how much". How many units of investitions should be held at a time? How much risk may be taken?

So, money management is controlling the bet size. Te most radical definition known to us is given by Ryan Jones3: money management is limited to defining what sum from your account should be risked on the next trade. Pay attention that this definition does not list as money management controlling the size of an already open position, which Van tharp allows.

Appendix 1. A system giving over 90% profitable trades.
Random System ¦1.
Copyright (C)2001 DT
Parameter values by default: mmstp =1,pflim =4,maxhold =10
Inputs: Bias(.025), {Random entry parameter}
mmstp(100), {Stop loss parameter}
pflim(.1), {Profit target limit}
maxhold(50); {maximum holding period};
trigger =random(1);
if trigger < bias then signal = -1;
if trigger >1 - bias then signal =1;
ATR =XAverage(Truerange,50);
{ Random Entry}
If signal =1 then Buy("Random_Mkt.LE")num contracts next bar at open;
If signal =1 then Sell("Random_Mkt.SE")num contracts next bar at open;
{ Standartized Exits}
if marketposition >0 then begin
Exitlong ("MM.LX")Next Bar at Entryprice -mmstp*ATR stop;
Exitlong ("Pt.LX")Next Bar at Entryprice +pflim*ATR limit;
if barssinceentry >=maxhold then
Exitlong ("Hold.LX")at close;
if marketposition <0 then begin
Exitshort ("MM.SX")Next Bar at Entryprice +mmstp*ATR stop;
Exitshort ("Pt.SX")Next Bar at Entryprice -pflim*ATR limit;
if barssinceentry >=maxhold then
Exitshort ("Hold.SX")at close;

Appendix 2. The simplest system number 2.
The Simplest System ¦2.
Copyright (C)2001 DT
if Marketposition <=0 then begin
if Price < LL then LL =Price;
if Price cross above LL +PtUp *.001 then begin
buy("Simpl.LE ")num contracts next bar at market;
HH =Price;
if Marketposition >=0 then begin
if Price >HH then HH =Price;
if Price cross below HH -PtDn *.001 then begin
Sell("Simpl.SE ")num contracts next bar at market;
LL =Price;

Appendix 3. Data output to a file to compute mathematical expectancy.
Expectancy Output
Copyright (C)2001 DT
Trades =TotalTrades;
R1 =PctUp *.001 *Bigpointvalue;
RMult =Positionprofit(1)/R1;
If barnumber =1 then
print(file("D:\TS_Export \M trading.csv"),"Qty",",","Profit",",","Initial Risk",",","R multiple");
If Trades <>Trades [1 ]then
print(file("D:\TS_Export \M trading.csv"),Num:10:0,",",Positionprofit(1):10:4,",",R1:10:4,",",RMult:10:4);