Money Management 9

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Money Management 8

Money Management


Dr. Van K. Tharp on Money Management & Position Sizing

Drawdowns
Manage Other's Money
Definition
Models



An Excerpt from Dr. Tharp's Report on Money Management.

John was a little shell-shocked over what had happened in the market over the last three days. He'd lost 70% of his account value. He was shaken, but still convinced that he could make the money back! After all, he had been up almost 200% before the market withered him down. He still had $4,500 left in his account. What advice would you give John?

Perhaps your answer is, "I don't know. I don't have enough information to know what John is doing." But you do have enough information. You know he only has $4,500 in his account and you know the kind of fluctuations his account has been going through. As a result, youhave enough information to understand his money management -- the most important part of his trading. And your advice should be, "Get out of the market immediately. You don't have enough money to trade." However, the average person is usually trying to make a big killing in the market, thinking that he or she can turn a $5,000 to $10,000 account into a million dollars in less than a year. While this sort of feat is possible, the chances of ruin for anyone who attempts it is almost 100%.

Ralph Vince did an experiment with forty Ph.D.s. He ruled out doctorates with a background in statistics or trading. All others were qualified. The forty doctorates were given a computer game to trade. They started with $1,000 and were given 100 trials in a game in which they would win 60% of the time. When they won, they won the amount of money they risked in that trial. When they lost, they lost the amount of money they risked for that trial.

Guess how many of the Ph.Ds had made money at the end of 100 trials? When the results were tabulated, only two of them made money. The other 38 lost money. Imagine that! 95% of them lost money playing a game in which the odds of winning were better than any game in Las Vegas. Why? The reason they lost was their adoption of the gambler's fallacy and the resulting poor money management.

Let's say you started the game risking $1000. In fact, you do that three times in a row and you lose all three times -- a distinct possibility in this game. Now you are down to $7,000 and you think, "I've had three losses in a row, so I'm really due to win now." That's the gambler's fallacy because your chances of winning are still just 60%. Anyway, you decide to bet $3,000 because you are so sure you will win. However, you again lose and now you only have $4,000. Your chances of making money in the game are slim now, because you must make 150%just to break even. Although the chances of four consecutive losses are slim -- .0256 -- it still is quite likely to occur in a 100 trial game.

Here's another way they could have gone broke. Let's say they started out betting $2,500. They have three losses in a row and are now down to $2,500. They now must make 300% just to get back to even and they probably won't do that before they go broke.

In either case, the failure to profit in this easy game occurred because the person risked too much money. The excessive risk occurred for psychological reasons -- greed, the judgmental heuristic of not understanding the odds, or in some cases, the desire to fail. However, mathematically their losses occurred because they were risking too much money.

What typically happens is that the average person comes into most speculative markets with too little money. An account under $50,000 is small, but the average account is only $5,000 to $10,000. As a result, these people are practicing poor money management just because their account is too small. Their mathematical odds of failure are very high just because they open an account that is too small.

Hundreds of thousands of hopefuls open up their speculative accounts yearly, only to be lead to the slaughter by others who are happy to take their money. Many brokers know these people don't have a chance, but they are happy to take their money in the form of fees and commissions. In addition, it takes many $5,000 accounts to feed a single multi-million dollar account that consistently gets a healthy rate of return.

Look at the table below. Notice how much your account has to recover from various sized drawdowns in order to get back to even. For example, losses as large as 20% don't require that much larger of a corresponding gain to get back to even. But a 40% drawdown requires a 66.7% gain to breakeven

Drawdown Gain to Recover
5 Percent 5.3% Gain
10 Percent 11.1% Gain
15 Percent 17.6% Gain
20 Percent 25% Gain
25 Percent 33% Gain
30 Percent 42.9% Gain
40 Percent 66.7% Gain
50 Percent 100% Gain
60 Percent 150% Gain
75 Percent 300% Gain
90 Percent 900% Gain

and a 50% drawdown requires a 100% gain. Losses beyond 50% require huge, improbable gains in order to get back to even. As a result, when you risk too much and lose, your chances of a full recovery are very slim.



Managing Other People's Money

In the futures industry, when an account goes down in value, it's called a drawdown. Suppose you open an account for $50,000 on August15th. For a month and a half, the account goes straight up and on September 30th, it closes at a high of $80,000 for a gain of 60%. At this point, you may still be in all of the same trading positions. But as a professional, your account is "marked to the market" at the end of the month and statements go out to your clients indicating what their respective accounts are worth.

Now, lets say that your positions start to go down in value around the 6th of October. Eventually, you close them out around the 14th of October and your account is now worth about $60,000. And let's say, for the sake of discussion, that your account at the end of October is worth $60,000. Essentially, you've had a peak-to-trough drawdown (peak = $80,000, trough = $60,000) of $20,000 or 25%. This may have occurred despite the fact that all of your trades were winners. It doesn't really matter as far as clients are concerned. They still believe that you just lost $20,000 (or 25%) of their money.

Let's say that you now make some losing trades. Winners and losers, in fact, come and go so that by August 30th of the following year, the account is now worth $52,000. It has never gone above $80,000, the previous peak, so you now have a peak-to-trough drawdown of$28,000 -- or 35%. As far as the industry is concerned, you have an annual rate of return of 4% (i.e., the account is only up by $2,000) and you are now labeled as having 35% peak-to-trough drawdown. And the ironic thing is that most of the drawdown occurred at a time in which you didn't have a losing trade -- you just managed to give back some of your profits. Nevertheless, you are still considered to be a terrible money manager. Money managers typically have to wear the label of the worst peak-to-trough drawdown that they produce for their clients for the rest of their lives.

Think about it from the client's viewpoint. You watched $28,000 of your money disappear. To you it's a real loss. You could have asked for your money on the first of October and been $28,000 richer.

Trading performance, as a result, typically is best measured by one's reward-to-risk ratio. The reward is usually the compounded annual rate of return. In our example, it was 4% for the first year. The risk is considered to be the peak-to-trough drawdown which in our example was35%. Thus, this traders reward-to-risk ratio was 4/35 or 0.114 -- a terrible ratio.

Typically, you want to see ratios of 2 better in a money manager. For example, if you had put $50,000 in the account and watched it rise to$58,000 you would have an annual rate of return of 16%. Let's say that when your account has reached $53,000, it had drawn down to$52,000 and then gone straight up to $58,000. That means that your peak to trough drawdown was only 0.0189 ($1,000 drawdown divided by the peak equity of $53,000). Thus the reward-to-risk ratio would have been a very respectable 8.5. People would flock to give you money with that kind of ratio.

Let's take another viewpoint and assume that the $50,000 account is your own. How would you feel about your performance in the two scenarios? In the first scenario you made $2,000 and gave back $28,000. In the second scenario, you made $7,000 and only gave back$1,000.

Let's say that you are not interested in 16% gains. You want 40-50% gains. In the first, scenario you had a 60% gain in a month and a half. You think you can do that several times at year. And you're willing to take the chance of giving all or most of it back in order to do that. You wouldn't make a very good money manager, but you might be able to grow your own account at the fastest possible rate of return if you could "stomach" the drawdowns.

Both winning scenarios, plus numerous losing scenario, are possible using the same trading system. You could aim for the highest reward to risk ratio. You could aim for the highest return. Or you could be very wild, like the Ph.D.s in the Ralph Vince game and lose much of your money by risking too much on any given trade.

Interestingly enough, a research study (Brinson, Singer, and Beebower, 1991) has shown that money management (called asset allocation in this case) explained 91.5% of the returns earned by 82 large pension plans over a ten year period. The study also showed that investment decisions by the plan sponsors pertaining to both the selection of investments and their timing, accounted for less than 10% of the returns. The obvious conclusion is that money management is a critical factor in trading and investment decision making. (Determinants of Portfolio Performance II: An Update, Financial Analysts Journal, 47, May-June, 1991, p 40-49.)

You now understand the importance of money management. Let's now look at various money management models, so that you can see how money management works.



Money Management Defined

Money management is that portion of one's trading system that tells you "how many" or "how much?" How many units of your investment should you put on at a given time? How much risk should you be willing to take? Aside from your personal psychological issues, this is the most critical concept you need to tackle as a trader or investor.

The concept is critical because the question of "how much" determines your risk and your profit potential. In addition, you need to spread your opportunity around into a number of different investments or products. Equalizing your exposure over the various trades or investments in your portfolio gives each one an equal chance of making you money.

I was intrigued when I read Jack Schwager's Market Wizards in which he interviews some of the world's top traders and investors. Practically all of them talked about the importance of money management. Here are a few sample quotes:

"Risk management is the most important thing to be well understood. Undertrade, undertrade, undertrade is my second piece of advice. Whatever you think your position ought to be, cut it at least in half." -- Bruce Kovner

"Never risk more than 1% of your total equity in any one trade. By risking 1%, I am indifferent to any individual trade. Keeping your risk small and constant is absolutely critical." -- Larry Hite

"You have to minimize your losses and try to preserve capital for those very few instances where you can make a lot in a very short period of time. What you can=t afford to do is throw away your capital on suboptimal trades." -- Richard Dennis

Professional gamblers play low expectancy or even negative expectancy games. They simply use skill and/or knowledge to get a slight edge. These people understand very clearly that money management is the key to their success. Money management for gamblers tends to fall into two types of systems -- martingale and anti-martingale systems. And investors and traders should know about these models.

Martingale systems increase winnings during a losing streak. For example, suppose you were playing red and black at the roulette wheel. Here you are paid a dollar for every dollar you risk, but your odds of winning are less than 50% on each trial. However, with the martingale system you think you have a chance of making money through money management. The assumption is that after a string of losses you will eventually win. And the assumption is true -- you will win eventually. Consequently, you start with a bet of one dollar and double the bet after every loss. When the ball falls on the color you bet, you will make a dollar from the entire sequence of wagers.

The logic is sound. Eventually, you will win and make a dollar. But two factors work against you when you use a martingale system. First, long losing streaks are possible, especially since the odds are less than 50% in your favor. For example, one is likely to have a streak of 10 losses in a row in a 1,000 trials. In fact, a streak of 15 or 16 losses in a row is quite probable. By the time you reached ten in a row, you would be betting $2,048 in order to come out a dollar ahead. If you lose on the eleventh throw, you would have lost $4,095. Your reward-to-risk ratio is now 1 to 4095.

Second, the casinos place betting limits. At a table where the minimum bet was a dollar, they would never allow you to bet much over $50 or$100. As a result, martingale betting systems, where you risk more when you lose, just do not work.

Anti-martingale systems, where you increase your risk when you win, do work. And smart gamblers know to increase their bets, within certain limits, when they are winning. And the same is true for trading or investing. Money management systems that work call for you to increase your risk size when you make money. That holds for gambling and for trading and investing.

The purpose of money management is to tell you how many units (shares or contracts) you are going to put on, given the size or your account. For example, a money management decision might be that you don't have enough money to put on any positions because the risk is too big. It allows you to determine your reward and risk characteristics by determining how many units you risk on a given trade and in each trade in a portfolio. It also helps you equalize your trade exposure in a portfolio.

Some people believe that they are "managing their money" by having a "money management stop." Such a stop would be one in which you get out of your position when you lose a predetermined amount of money -- say $1000. However, this kind of stop does not tell you "how much" or "how many", so it really has nothing to do with money management.

Nevertheless, there are numerous money management strategies that you can use. In the remainder of this report, I'm going to present different money management strategies that work. Some are probably much more suited to your style of trading than others. Some workbest with stock accounts, while others are designed for futures account. All of them are Anti-martingale strategies.

The rest of this article is continued in Dr. Tharp's special report on money management.

A Special Report on Money Management By Van K. Tharp Ph.D.

In this special report, written by Van K. Tharp, Ph.D., you'll learn dozens of different models of money management-one of which could make a big difference for you. The biggest secret that most people neglect in their quest for big profits is proper money management. Research has proven that about 90% of the variance in performance between portfolio managers is due to money management. For the average trader, it makes the difference between losing, and winning big-depending upon your objectives. Probably the most valuable book any trader could own.


What you will learn from this report:

How to meet your objectives using money management

The definition of money management

27 models of money management with three ways to measure equity

How to design high reward risk systems for managing money

Four techniques to produce maximum profits

Dr. Tharp calls this type of money management a "secret" because few people seem to understand it, including many people who've written books on the topic. Some people call itrisk control, others call it diversification, and still others call it how to "wisely" invest your money. However, the money management that is the key to top trading and investing simply refers to the algorithm that tells you "how much" with respect to any particular position in the market.

As if the issue of money management weren't confusing enough on its own, there are also many psychological biases that keep people from practicing sound money management. And, there are practical considerations, such as not understanding money management or not having sufficient funds to practice sound money management.

This report is written to give you an overall understanding of the topic and to show you various models of money management. To order this exclusive report call IITM at800-385-IITM (4486) or at 919-852-3664. The report is $79.95 plus shipping and handling.

Last revised: July 08, 1999

Five Tips To Give You More Discipline

They Will Help You Earn Bigger, More Consistent Profits (Without the Stress) by Changing Your Thinking -- GUARANTEED!

Responsibility
Mistakes
Mental State Control
Change What You're Doing
Scan Your Body

Tip #1: Take Responsibility for Everything That Happens to you. One of the keys to peak performance is to make the assumption that you create everything that happens to you. For example, if you give your money to someone and then run off with it, you must still take responsibility. You made the decision to give that person money. You made a decision about how much information you needed from that person before you gave them money. Thus, even though they committed an illegal act, you are still ultimately responsible.

When this sounds like I'm asking you to feel guilty for you mistakes, the truth is exactly the opposite. What I am asking you to do is set yourself free by taking control over the rest of your life instead of being a victim.

Tip #2: There are only two types of mistakes. You might say that you biggest problem is your spouse. If you believe that, then it is probably true. In addition, you are also doomed to repeat that mistake for the rest of you life. No, I'm not saying that you can't divorce your spouse. You probably can and may do so. But you will probably just find another spouse that will give you the same problems. However, if you look at the problem different, such as noticing that I get angry when my spouse does X, then you have taken a step toward controlling the problem. The reason is that you have now traced the problem to a mental state that you can elect to own -- your anger. You don't have to get angry at your spouse when that person does X. You can elect to have another response. That is the essence of discipline.

Now you can also apply this to the process of trading or investing. The mistakes you make are in some way related to negative mental states -- whether it's an inability to pull the trigger or compulsiveness, you can trace it to a negative mental state.

Tip #3: Discipline Involves Controlling Your Mental State. My home study course has over 15 ways to control your mental state. This means that you control your life and not your mental state. You don't have to be the victim of fear. Instead, you can just notice "Oh, I'm starting to do that fear thing and I need to take control."

Tip #4: If you don't like the results you are getting, respond differently! How are you producing fear or the mental state you don't want? Notice what you are doing and do something else or do it differently. For example, are you producing fear by something you tell yourself? Then tell yourself something else. Or change the nature of the voice. Try saying the same thing in a voice like Mickey Mouse. If you are producing fear by something your are seeing in your mind, then picture something else. Or you might change the nature of the picture. Make it black and white or move it further away. This amounts to taking charge of the way you run your brain.

Tip #5: Change what you are doing with your body to change the reaction of your mind. When your mental state is inappropriate, then scan your body. Notice what you are doing with your body. If there is tension, relax that area. If your posture is bad, straighten up. Also notice your breathing. Take regular deep breaths and you can literally change your state. As an exercise, try imitating people's walking. Notice what it feels like to walk in another person's shoes. This will convince you how important it is to change what you do with your body to change your mental states.

Next time you are having a problem trading, ask yourself how am I doing this? Change what you are doing with your body, and if you do it right, you should notice a big change in your behavior.

For more information about discipline see Dr. Tharp's Home Study Course or the Peak Performance Trading Seminar.

Last revised: July 08, 1999

BASIC TECHNIQUES

Exploiting Positions With Money Management by Daryl Guppy

Here's a trading technique adding positions to successful trades.

Even in a bull market, there is a feeling of triumph when a trade goes our way. When this happens, the novice trader feels that getting the trade right is enough in itself and that profits will automatically follow. He is invariably disappointed when profits turn out to be smaller than expected, and often, he will redirect his attention to derivative markets to try to leverage winning trades into larger profits. This in turn exposes him to a higher level of risk than he anticipates. A better solution would be to apply money management techniques to equity markets to grow profits more effectively. This is also an important way of reducing risk.

The entry is just after the 10- and 30-day moving average crossover signal, and it was made at $5.52. The exit at $6.37 is also triggered by the crossover of the 10- and 30-day exponential moving average. The return on trading capital depends on how the initial position taken at $5.52 is added to in October 1998. We buy 6,000 shares for a total cost of $33,120. If this same parcel were sold at $6.37 following a simple buy-and-hold strategy, then we collect $5,100 profit for a 15.3% return on capital employed in the trade.

Here's how risk changes, even with comparable positions and stop-loss exits. Risk, here, is trade risk, measured by the dollar loss incurred with adverse price movements. This includes both capital reduction and reduction in open profit. Risk is directly related to the money management technique selected. By comparing outcomes, the trader can exploit his winning positions more effectively. Let's examine a method to increase profits while reducing risk that I call the grow-up strategy. I use the name to distinguish it from adding to winning positions using constant dollar or constant position size.

When traders first approach the market, they concentrate on choosing the right analytical tools. They believe that if they get this right, profits will automatically follow. As time goes on, however, they realize that success is more closely tied to the way they trade and to trading discipline. They understand analysis tools are a starting point, not an end point. Truly successful traders take the next step by using money management to control risk.

Fund managers and institutional traders have a selection of well-defined money management formulas. Texts by Ralph Vince and Fred Gehm serve well as a good introduction to this area, but their solutions are less applicable to private traders. The private trader finds generally less information available, and even less of it applicable to portfolios at his level of capital. Take a closer look at the strategies for loading up the winners. Many trading books suggest that loading up winners is a good strategy, and because it is so self-evident, traders spend no time exploring the outcomes of their advice.

My objective is to increase the total position size as the trade continues to move in my favor. The ultimate outcome is to have most of my trading capital tied up in positions making money rather than in positions losing money.

Despite intensive analysis and research of potential and actual trading positions, many traders approach money management armed with a collection of old wives' tales. High on this list of adages is the assumption that traders improve trading outcomes by adding to winning positions. The concept is sound. The way the concept is implemented, however, is often less successful.

Most traders reach for one of two common strategies. The first is to add new positions to a winning trade of the same parcel size (that is, the same number of shares in each new position) as the initial position; the second is to add new positions that are the same dollar size. Both strategies appear successful while the trend continues, but they expose the trader to unexpected risk when the trend reverses.

Daryl Guppy is an author and a private equity and derivatives trader. He speaks regularly on trading in Australia and Asia. He can be contacted via www.ozemail.com.au/~guppy.

Excerpted from an article originally published in the September 1999 issue of Technical Analysis of STOCKS & COMMODITIES magazine. All rights reserved. © Copyright 1999, Technical Analysis, Inc.

CLASSIC TECHNIQUES

Matching Money Management With Trade Risk

by Daryl Guppy

Manage your trades using technical analysis by identifying risk points as well as setting profit objectives. This Australia-based author shares some of his favorite techniques.

Most of us think trading is a rational process, but many private traders approach trading the same way that other people approach a wishing well. Those people throw money into the well, make a wish and wait for their wish to come true. In the same vein, some private traders throw money into the market and all they wish for is a profit. Sometimes the wish comes true, but most times, just like a wishing well, it is a waste of money and time. For traders using the market as a substitute for a wishing well, trading is a very emotional experience.

With that in mind, let's take a closer look at the way emotion interferes with good trading and at some of the ways that chart analysis can help us establish trading objectives more effectively.

Most readers will protest that trading is not like a wishing well; we don't just toss money in and hope for fat profits. By and large, we prefer to believe that we are too sophisticated for that. Instead, we analyze the tradable instrument in question, applying sound technical analysis to charts and price data, and then, and only then, make a trading decision. This is a rational process, and if our analysis is correct, the tradable's price will increase when we go long.

Unfortunately, describing the entry decision in pretty words and trading jargon does not alter the trader's intent nor the outcome. Trading based on the wishing-well approach is characterized by poor trade management illustrated by the lack of a plan. Just hoping to make a profit is not a plan. It is an objective, but it tells us nothing about how we are going to get there.


Assume that all the sound analysis has been done, and Woodside appears to be irresistible at$8.60. As prices tumble toward the chosen entry level, the trader must concentrate on the possibility that his or her decision might be wrong.

Daryl Guppy is a full-time private position trader. He is the author of several books, including Share Trading: An Approach to Buying and Selling (with editorial assistance from Alexander Elder) and Trading Tactics: An Introduction to Finding, Exploiting and Managing Profitable Share Trading Opportunities and Trading Asian Shares: Buying and Selling Asian Shares for Profit. He is a regular contributor to the Sydney Futures Exchange magazine. He can be contacted via E-mail at www.ozemail.com.au/~guppy.

Excerpted from an article originally published in the May 1998 issue of Technical Analysis of STOCKS & COMMODITIES magazine. All rights reserved. © Copyright 1998, Technical Analysis, Inc.

October 1998 Letters to the Editor

SECURING OPTIMAL F

Editor,

In your July 1998 issue, you published an article titled "Secure fractional money management." The technique uses Ralph Vince's optimal f, and purports to limit the drawdowns that occur when optimal f is applied. But the authors do not consider the biggest problem with optimal f applications.

Optimal f is a random variable. It depends on the largest loss experienced to date and all the trade results to date. Because these variables are subject to large statistical fluctuations, optimal f is subject to random variations. The method set forth by the authors does not address the drawdowns that can occur when future trades are subject to an optimal f different from the calculated value.

A partial solution is to just replace the largest loss to date in Vince's formula with a conservative choice of a larger loss. Still, the resulting optimal f will be exposed to large variations from the trade history employed. The best way to address these problems is to simply rerun the optimal f calculation over several different time frames containing the same number of trades and then average the results for optimal f. That way, the result will be much more "secure."

PAUL H. LASKY via E-mail

------------------------------------------------
INSECURE ABOUT SECURE F

Editor,

The article "Secure fractional money management" by Leo J. Zamansky and David C. Stendahl that appeared in the July 1998 issue of STOCKS & COMMODITIES contains a major error. Optimal f is not the percentage of equity to trade, as stated in the article. Optimal f is used to figure the number of contracts to trade:

number of contracts to trade = (equity) (opt f)/-profit of worst trade

The equations that the authors give are correct, but because of their error in thinking, they misapply the equations.

In the example of the three series given in the article, the authors correctly come up with an optimal f of 1/3. But then to interpret this as being able to buy only three contracts is wrong. Using their interpretation, you could buy only three contracts in each of the series and end up with $101,500, a terminal wealth relative (TWR) of 101,500/100,000 = 1.015. This is far from the TWR of 1.185 that is seen in their Figure 3. Using the correct interpretation, you don't buy three but 66 contracts!

66.666 = 100,000 * 1/3/(500)

You always round down the number of contracts. Then, after winning $33,000 in the first, the second series gets 88 contracts: 88 = 133,000/(1,500). The third series gets 118contracts: 118 = 177,000/(1,500). You end up with $118,000, giving TWR = 1.18.

The authors' calculation of maximum drawdown of $7,500 is dwarfed by the actual maximum drawdown, which is 5/3 of equity, or $220,000, as it occurs in series 2. The correct value of the secure f is 0.01.

Perhaps the authors became confused with equalized optimal f (see page 83 of Ralph Vince's The Mathematics of Money Management). In this method, you can come up with a number that is a fraction of equity to trade with, but this number is neither f nor optimal f. (As far as I know, Vince does not identify this fraction, but it is evident from his equations.) In this method, you use the percentage loss or gain for each trade. In the authors' example, the trades become ($500/$10,000 = $0.05, 0.05, -0.05) instead of ($500, 500, -500). In this case, the equalized optimal f = 1/3; it is the same as optimal f because the buy price was always $10,000. Now use this equation: fraction of equity = equalized optimal f/ - return of worst losing trade

For the equalized optimal f of 1/3, the fraction of equity is 666.7% = (1/3)/(0.05). Yes, this does mean you are buying on margin. For the equalized secure f of 0.01, the fraction of equity is 20%.

After getting every other interpretation wrong, the authors do come up with the correct final answer of 20%. This leads me to believe that they do have access to a program that correctly generates these numbers for them.

Despite my criticism, Zamansky and Stendahl are to be congratulated for the idea of secure f. Fixed fractional money management is a wonderful and complex subject that deserves some attention. Too bad this article got the fundamentals wrong.

BRADEN A. BROOKS via E-mail

------------------------------------------------
Leo Zamansky and David Stendahl reply:

Thank you for the feedback about our July 1998 article, "Secure fractional money management."

Mr. Brooks is correct in saying that according to Ralph Vince, the formula is number of contracts =(equity) * (optimal f) / (- profit of worst trade). He is also correct that according to this formula, we buy66 contracts, not three. However, as we assumed in the article, one contract price is $10,000. To buy 66contracts, we need to have enough money to buy 66 contracts at $10,000 each. That makes66*$10,000 = $660,000. And that amount should be only one-third of the total capital available, which, as stated, is $100,000. The question we are answering is: How many contracts can we buy following optimal f? The answer is $33,000 / $10,000 = 3. If the contract price were $500, then the number of contracts to purchase would be exactly 66. If the contract price were less than $500, then the number of contracts to purchase also would be exactly 66.

Vince, in his book Portfolio Management Formulas, states on page 80, "Margin has nothing to do whatsoever with what is the mathematically optimal number of contracts to have on." We emphasize the word mathematically. In real trading, if you need to buy one contract, you need to have a certain amount of money in your account, say x, and if you buy n contracts, you need to have the amount of money equal to n*x. In the article's example we call it price, but in reality it is a margin requirement. Inother words, the formula for the number of contracts should be adjusted to the price of the contract and be modified to look as follows:

number of contracts to trade = (equity) (opt f)/max [price of contract, -profit of worst trade]

We agree that we should have specified that. Of course, if you follow this number of contracts purchased, the TWR is not going to be 1.18, because the contract price is too high and does not allow you to buy the number of contracts that would maximize it. However, the lower the price of a contract, the closer you will come to the calculated maximum of TWR.

We develop tools to use in trading futures. We cannot introduce a calculation that is not based on real trading rules. In real futures trading, the margin requirement per contract always exists and has a very similar meaning to the contract price in the game introduced.

As to the software we use to obtain our results, we use only the software written by us. In fact, readers can download the secure f calculator from our Web site, as mentioned in the article, and run it to obtain values for both secure f and optimal f. This should demonstrate the validity of this approach.

We appreciate the feedback from Mr. Brooks, who makes the important point regarding the difference between optimal f as a mathematical concept and trading using optimal f given the constraints imposed by the reality of futures trading.

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