## Synthetic Constructs In This Text

As you proceed through the text, you will see that there is a certain geometry to this material. However, in order to get to this geometry we will have to create certain synthetic constructs. For one, we will convert trade profits and losses over to what will be referred to as holding period returns or HPRs for short. An HPR is simply 1 plus what you made or lost on the trade as a percentage. Therefore, a trade that made a 10% profit would be converted to an HPR of 1 + .10 = 1.10. Srmilarly, a trade that lost 10% would have an HPR of 1 + (-.10) = ,90. Most texts, when referring to a holding period return, do not add 1 to the percentage gain or loss. However, throughout this text, whenever we refer to an HPR, it will always be 1 plus the gain or loss as a percentage.

Another synthetic construct we must use is that of a market system. A market system is any given trading approach on any given market (the approach need not be a mechanical trading system, but often is). For example, say we are using two separate approaches to trading two separate markets, and say that one of our approaches is a simple moving average crossover system. The other approach takes trades based upon our Elliott Wave interpretation. Further, say we are trading two separate markets, say Treasury Bonds and heating oil. We therefore have a total of four different market systems. We have the moving average system on bonds, the Elliott Wave trades on bonds, the moving average system on healing oil, and the Elliott Wave trades on heating oil.

A market system can be further differentiated by other factors, one of which is dependency. For example, say that in our moving average system we discern (through methods discussed in this text) that winning trades beget losing trades and vice versa. We would, therefore, break our moving average system on any given market into two distinct market systems. One of the market systems would take trades only after a loss (because of the nature of this dependency, this is a more advantageous system), the other market system only after a profit. Referring back to our example of trading this moving average system in conjunction with Treasury Bonds and healing oil and using the Elliott Wave trades also, we now have six market systems: the moving average system after a loss on bonds, the moving average system after a win on bonds, the Elliott Wave trades on bonds, the moving average system after a win on heating oil, the moving average system after a loss on heating oil, and the Elliott Wave trades on heating oil.

Pyramiding (adding on contracts throughout the course of a trade) is viewed in a money management sense as separate, distinct market systems rather than as the original entry. For example, if you are using a trading technique that pyramids, you should treat the initial entry as one market system. Each add-on, each time you pyramid further, constitutes another market system. Suppose your trading technique calls for you to add on each time you have a $1,000 profit in a trade. If you catch a really big trade, you will be adding on more and more contracts as the trade progresses through these $1,000 levels of profit. Each separate add-on should be treated as a separate market system. There is a big benefit in doing this. The benefit is that the techniques discussed in this book will yield the optimal quantities to have on for a given market system as a function of the level of equity in your account. By treating each add-on as a separate market system, you will be able to use the techniques discussed in this book to know the optimal amount to add on for your current level of equity.

Another very important synthetic construct we will use is the concept of a unit. The HPRs that you will be calculating for the separate market systems must be calculated on a "1 unit" basis. In other words, if they are futures or options contracts, each trade should be for 1 contract. If it is stocks you are trading, you must decide how big 1 unit is. It can be 100 shares or it can be 1 share. If you are trading cash markets or foreign exchange (forex), you must decide how big 1 unit is. By using results based upon trading 1 unit as input to the methods in this book, you will be able to get output results based upon 1 unit. That is, you will know how many units you should have on for a given trade. It doesn't matter what size you decide 1 unit to be, because it's just an hypothetical construct necessary in order to make the calculations. For each market system you must figure how big 1 unit is going to be. For example, if you are a forex trader, you may decide that 1 unit will be one million U.S. dollars. If you are a stock trader, you may opt for a size of 100 shares.

Finally, you must determine whether you can trade fractional units or not. For instance, if you are trading commodities and you define 1 unit as being 1 contract, then you cannot trade fractional units (i.e., a unit size less than 1), because the smallest denomination in which you can trade futures contracts in is 1 unit (you can possibly trade quasifractional units if you also trade minicontracts). If you are a stock trader and you define 1 unit as 1 share, then you cannot trade the fractional unit. However, if you define 1 unit as 100 shares, then you can trade the fractional unit, if you're willing to trade the odd lot.

If you are trading futures you may decide to have 1 unit be 1 minicon-tract, and not allow the fractional unit. Now, assuming that 2 minicontracts equal 1 regular contract, if you get an answer from the techniques in this book to trade 9 units, that would mean you should trade 9 minicontracts. Since 9 divided by 2 equals 4.5, you would optimally trade 4 regular contracts and 1 minicontract here.

Generally, it is very advantageous from a money management perspective to be able to trade the fractional unit, but this isn't always true. Consider two stock traders. One defines 1 unit as 1 share and cannot trade the fractional unit; the other defines 1 unit as 100 shares and can trade the fractional unit. Suppose the optimal quantity to trade in today for the first trader is to trade 61 units (i.e., 61 shares) and for the second trader for the same day it is to trade 0.61 units (again 61 shares).

I have been told by others that, in order to be a better teacher, I must bring the material to a level which the reader can understand. Often these other people's suggestions have to do with creating analogies between the concept I am trying to convey and something they already are familiar with. Therefore, for the sake of instruction you will find numerous analogies in this text. But I abhor analogies. Whereas analogies may be an effective tool for instruction as well as arguments, I don't like them because they take something foreign to people and (often quite deceptively) force fit it to a template of logic of something people already know is true. Here is an example:

The square root of 6 is 3 because the square root of 4 is 2 and 2 + 2 =4.

Therefore, since 3 + 3 = 6, then the square root of 6 must be 3.

Analogies explain, but they do not solve. Rather, an analogy makes the a priori assumption that something is true, and this "explanation" then masquerades as the proof. You have my apologies in advance for the use of the analogies in this text. I have opted for them only for the purpose of instruction.

### OPTIMAL TRADING QUANTITIES AND OPTIMAL f

Modem portfolio theory, perhaps the pinnacle of money management concepts from the stock trading arena, has not been embraced by the rest of the trading world. Futures traders, whose technical trading ideas are usually adopted by their stock trading cousins, have been reluctant to accept ideas from the stock trading world. As a consequence, modem portfolio theory has never really been embraced by futures traders.

Whereas modem portfolio theory will determine optimal weightings of the components within a portfolio (so as to give the least variance to a pre-specified return or vice versa), it does not address the notion of optimal quantities. That is, for a given market system, there is an optimal amount to trade in for a given level of account equity so as to maximize geometric growth. This we will refer to as the optimal f. This book proposes that modem portfolio theory can and should be used by traders in any markets, not just the stock markets. However, we must marry modem portfolio theory (which gives us optimal weights) with the notion of optimal quantity (opti-

mal f) to arrive at a truly optimal portfolio. It is this truly optimal portfolio that can and should be used by traders in'any markets, including the stock markets.

In a nonleveraged situation, such as a portfolio of stocks that are not on margin, weighting and quantity are synonymous, but in a leveraged situation, such as a portfolio of futures market systems, weighting and quantity are different indeed. In this book you will see an idea first roughly introduced in Portfolio Management Formulas, that optimal quantities are what we seek to know, and that this is a function of optimal weightings.

Once we amend modern portfolio theory to separate the notions of weight and quantity, we can return to the stock trading arena with this now reworked tool. We will see how almost any nonleveraged portfolio of stocks can be improved dramatically by making it a leveraged portfolio, and marrying the portfolio with the risk-free asset. This will become intuitively obvious to you. The degree of risk (or conservativeness) is then dictated by the trader as a function of how much or how little leverage the trader wishes to apply to this portfolio. This implies that where a trader is on the spectrum of risk aversion is a function of the leverage used and not a function of the type of trading vehicle used.

In short, this book will teach you about risk management. Very few traders have an inkling as to what constitutes risk management. It is not simply a matter of eliminating risk altogether. To do so is to eliminate return altogether. It isn't simply a matter of maximizing potential reward to potential risk either. Rather, risk management is about decision-making strategies that seek to maximize the ratio of potential reward to potential risk within a given acceptable level of risk.

To learn this, we must first learn about optimal f, the optimal quantity component of the equation. Then we must learn about combining optimal f with the optimal portfolio weighting. Such a portfolio will maximize potential reward to potential risk. We will first cover these concepts from an empirical standpoint (as was introduced in Portfolio Management Formulas), then study them from a more powerful standpoint, the parametric standpoint. In contrast to an empirical approach, which utilizes past data to come up with answers directly, a parametric approach utilizes past data to come up with parameters. These are certain measurements about something. These parameters are then used in a model to come up with essentially the same answers that were derived from an empirical approach. The strong point about the parametric approach is that you can alter the values of the parameters to see the effect on the outcome from the model. This is something you cannot do with an empirical technique. However, empirical techniques have their strong points, too. The empirical techniques are generally more straightforward and less math intensive. Therefore they are eas-

ier to use and comprehend. For this reason, the empirical techniques are covered first.

Finally, we will see how to implement the concepts within a user-speci-fied acceptable level of risk, and learn strategies to maximize this situation further.

There is a lot of material to be covered here. I have tried to make this text as concise as possible. Some of the material may not sit well with you, the reader, and perhaps may raise more questions than it answers. If that is the case, than I have succeeded in one facet of what I have attempted to do.

Most books have a single "heart," a central concept that the entire text flows toward. This book is a little different in that it has many hearts. Thus, some people may find this book difficult when they go to read it if they are subconsciously searching for a single heart. I make no apologies for this; this does not weaken the logic of the text; rather, it enriches it. This book may take you more than one reading to discover many of its hearts, or just to be comfortable with it.

One of the many hearts of this book is the broader concept of decision making in environments characterized by geometric consequences. An environment of geometric consequence is an environment where a quantity that you have to work with today is a function of prior outcomes. I think this covers most environments we live in! Optimal f is the regulator of growth in such environments, and the by-products of optimal f tell us a great deal of information about the growth rate of a given environment. In this text you will learn how to determine the optimal f and its by-products for any distributional form. This is a statistical tool that is directly applicable to many real-world environments in business and science. I hope that you will seek to apply the tools for finding the optimal f parametrically in other fields where there are such environments, for numerous different distributions, not just for trading the markets.

For years the trading community has discussed the broad concept of "money management." Yet by and large, money management has been characterized by a loose collection of rules of thumb, many of which were incorrect. Ultimately, I hope that this book will have provided traders with exactitude under the heading of money management.

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