The Leverage Space Trading Model: Reconciling Portfolio Management Strategies and Economic Theory

Preface

I can explain ...

This material began as a panoply of notes for a series of talks I gave in late 2007 and in 2008, after the publication of The Handbook of Portfolio Mathematics.

In those talks, I spoke of how there exists a plethora of market analysis, selection and timing techniques including charts and fundamental analysis, trading systems, Elliot waves, and on and on—all sorts of models and methods, technical and otherwise, to assist in timing and selection.

You wouldn’t initiate a trade without using the analysis you specialize in, but there is another world, a world of quantity, a world out there someplace, which has either been dark and unknown or, at best, fraught with heuristics. You will begin to understand this as I show you how those heuristics have evolved and are, very often, just plain wrong. Numerous Nobel Prizes have been awarded based on some of those widely accepted principles. I am referring specifically to the contemporary techniques of combining assets into a portfolio and determining their relative quantities. These widely accepted approaches, however, are wrong and will get you into trouble. I will show you how and why that is. They illuminate nothing, aside from providing the illusion of safety through diversification. In the netherworld of quantity, those flawed techniques still leave us in the dark.

There are still fund managers out there who use those atavistic techniques. They stumble blindly along the dim, twisted pathways of that netherworld. This is akin to trading without your charts, systems, or other analytical tools. Yet most of the world does just that. (See Figure P.1.)

And whether you acknowledge it or not, it is at work on you, just as gravity is at work on you.

Pursuing my passion for this material, I found there is an entire domain that I have sought to catalogue, regarding quantity, which is just as important as the discipline of timing and selection. This other area is shrouded in darkness and mystery, absent a framework or even a coordinate system. Once I could apply a viable framework, I found this dark netherworld alive with fascinating phenomena and bubbling with geometrical relationships. Most importantly, the effects of our actions regarding quantity decisions were illuminated.

FIGURE P.1 Market Analysis and Position Sizing (Both Equally Necessary)

I have encountered numerous frustrations while trying to make this point since the publication of Portfolio Management Formulas in 1990: People are lazy. They want a card they can put into a bank machine and get money. Very few want to put forth the mental effort to think, or to endure the necessary psychological pain to think outside of their comfortable, self-imposed boxes. They remain trapped within their suffocating, limited mental notions of how things should operate. Incidentally, I do not claim immunity from this.

When I alluded to quantity as the other, necessary half of trading, I was being overly generous, even apologetic about it. After all, some of the members of my audiences were revered market technicians and notable panjandrums. Indeed, I believe that quantity is nearly 100 percent of the matter, not merely half, and I submit that you are perhaps better off to disregard your old means of analysis, timing, and selection altogether.

Yes, I said 100 percent.

On Saturday, 26 January 2008, I was having lunch in the shadow of Tokyo Tower with Hiroyuki Narita, Takaaki Sera, and Masaki Nagasawa. Hiro stirred the conversation with something I had only marginally had bubbling in my head for the past few years.

He said something, almost in passing, about what he really needed as a trader. It knocked me over. I knew, suddenly, instantly, that what he was (seemingly rhetorically) asking for is what all traders need, that it is something that no one has really addressed, and that the answer has likely been floating in the ether all around us. I knew at that moment that if I thought about this, got my arms around it, it would fulminate into something that would change how I viewed everything in this discipline which I had been obsessed with for decades.

In the following days, I could not stop thinking about this. Those guys in Tokyo didn’t have to do a hard sell on me that day. I knew they were right, and that everything I had worked on and had compulsively stored in a corner of my mind for decades was (even more so) just a mere framework upon which to construct what was really needed.

I might have been content to stay holed up in my little fort along the Chagrin River, but an entirely new thread was beginning to reveal itself.

On the flight home, in the darkness of the plane, unable to sleep, in the margins of the book I was reading, I began working on exactly this.

That’s where this book is going.

RALPH VINCE

Selby Library, Sarasota

August 2008

Introduction

This is a storybook, not a textbook. It is the story of ideas that go back roughly three centuries, and how they have, and continue, to change. It is the story of how resources should be combined, when confronted with one or more prospects of uncertain outcome, where the amount of such resources you will have for the next prospect of uncertain outcome is dependent on what occurs with this outcome. In other words, your resources are not replenished from outside.

It is a story that ultimately must answer the question, "What are you looking to accomplish at the end of this process, and how do you plan to implement it?" The answer to this question is vitally important because it dictates what kinds of actions we should take. Given the complex and seemingly pathological character of human desires, we are presented with a fascinating puzzle within a story that has some interesting twists and turns, one of which is about to take place.

There are some who might protest, Some of this was published earlier! They would certainly be correct. A lot of the material herein presents concepts that I have previously discussed. However, they are necessary parts of the thread of this story and are provided not only for that reason but also in consideration of those readers who are not familiar with these concepts. Those who are familiar with the past concepts, peppered throughout Parts I and II of this story, can gloss over them as we build with them in Part III.

As mentioned in the Preface, this material began as a panoply of notes for a series of talks I gave in late 2007 and in 2008 after the publication of The Handbook of Portfolio Mathematics (which, in turn, built upon previous things I had written of, along with notes of things pertaining to drawdowns, which I had begun working on in my spare time while at the Abu Dhabi Investment Authority, a first-rate institution consisting of first-rate and generous individuals whom I had the extreme good fortune to be employed by some years ago). I designed those talks to illustrate the concepts in the book, in a manner that made them simpler, more intuitive, essentially devoid of mathematics and, therefore, more easily digestible. I have drawn from those talks and fleshed them out further for this book, with their mathematical derivations and explanations of how to perform them. This comprises a good portion of the early sections of this text. This background, at least conceptually, is necessary to understand the new material.

One idea, discussed at length in the past, needs to be discussed before we begin the story. It is the concept of Mathematical Expectation. This is called Expected Value, by some, and it represents what we would expect to make, on average, per play, when faced with a given prospect—an outcome we cannot yet know, which may be favorable or not. The concept is introduced in 1657 in a treatise by the Dutch Mathematician and Physicist Christian Huygens, at the prompting of Blaise Pascal.

This value, the Mathematical Expectation (ME), is simply the sum of the products of the probabilities and payoffs of all the ways something might turn out:

where:

Pi = the probability associated with the ith outcome

Ai = the result of the ith outcome

n = the total number of possible outcomes

For example, assume we toss a coin and if it’s heads we win two units and if it’s tails we lose one unit. There are two possible outcomes, +2 and -1, each with a probability of 0.5.

An ME of 0 is said to be a fair gamble. If ME is positive, it is said to be a favorable gamble, and if negative, a losing gamble. Note that in a game with a negative ME (that is, most gambling games), the probability of going broke approaches certainty as you continue to play.

The equation for Mathematical Expectation, or expected value, is quite foundational to studying this discipline.

Mathematical Expectation is a cornerstone to our story here. Not only is it a cornerstone to gambling theory, it is also a cornerstone to principles in Game Theory, wherein payoff matrices are often assessed based on Mathematical Expectation, as well as the discipline known as Economic Theory. Repeatedly in Economic Theory we see the notion of Mathematical Expectation transformed by the theory posited. We shall see this in Chapter 6.

However prevalent and persistent the notion of Mathematical Expectation, it must be looked at and used with the lens of a given horizon, a given lifespan. Frequently, viable methods are disregarded by otherwise-intelligent men because they show a negative Mathematical Expectation (and vice versa). This indicates a misunderstanding of the basic concept of Mathematical Expectation.

By way of example, let us assume a given lottery that is played once a week. We will further assume you are going to bet $1 on this lottery. Let us further assume you are a young man, and you plan to play this for 50 years. Thus, you expect 52 * 50 = 2,600 plays you will be able to make.

Now let’s say that this particular lottery has a one-in-two-million chance of winning a $1 million jackpot (this is for mere illustrative purposes, most lotteries having much lower probabilities of winning. For example, Powerball, as presently played in the United States, has less than a 1-in-195,000,000 chance of winning its jackpot). Thus we see a negative expectation in our example lottery of:

Thus, we expect to lose -$0.4999995 per week, on average, playing this lottery (and based on this, we would expect to lose over the course of the 50 years we were to play this, 2,600 * -0.4999995 = -$1,300).

Mathematical Expectation, however, is simply the average, outcome (i.e., it is the mean of this distribution of the ways the future plays might turn out). In the instant case, we are discussing the outcome of 2,600 plays taken from a pool of two million possible plays, allowing for sample and replacement. Thus, the probability of seeing the single winning play in any randomly chosen 2,600 is:

From this, we can say that (1 - .0013 = .9987) 99.87 percent of the people who play this lottery every week for the next 50 years will lose $2,600. About 1/8 of 1 percent (.0013) will win $1 million (thus netting 1,000,000 - 2,600 = $997,400). Clearly the mode of the distribution of outcomes for these 2,600 plays is to lose $2,600, even though the mean, as given by the Mathematical Expectation, is to lose $1,300.

Now, let’s reverse things. Suppose now we have one chance in a million of winning $2 million. Now our Mathematical Expectation is:

A positive expectation. If our sole criteria was to find a positive expectation, you would think we should accept this gamble. However, now the probability of seeing the single winning play in any randomly chosen 2,600 is:

In this positive expectation game, we can expect 99.74 percent of the people who play this over the next 50 years to lose $2,600. So is this positive expectation game a good bet? Is it a bet you would play expecting to make $1.000001 every week?

To drive home this important point we shall reverse the parameters of this game one more time. Assume a lottery wherein you are given $1 every week, with a one-in-one-million chance of losing $2 million. The Mathematical Expectation then is:

Thus, we expect to lose -1.000001 per week, on average, playing this lottery (and based on this, we would expect to lose over the course of the 50 years we were to play this, 2,600 * -1.000001 = -$2,600).

Do we thus play this game, accept this proposition, given its negative Mathematical Expectation? Consider the probability that the 2,600 weeks we play this for will see the two million loss:

Thus, we would expect that 99.74 percent (1 - .0026) of the people who play this game will never see the $2 million loss. Instead, they will be given a dollar every week for 2,600 weeks. Thus, about 399 out of every 400 people who play this game will not see the one-in-a-million chance of losing $2 million over the course of 2,600 plays.

I trace out a path through 3D space not only of the places I go, but on a planet that revolves roughly once every 24 hours, about a heliocentric orbit of a period of roughly 365 1/4 days, in a solar system that is migrating through a galaxy, in a galaxy that is migrating through a universe, which itself is expanding.

Within this universe is an arbitrary-sized chunk of matter, making its own tangled path through 3D space. There is a point in time where my head and this object will attempt to occupy the same location in 3D space. The longer I live, the more certain I will see that moment.

Will I live long enough to see that moment? Likely not. That bet is a sure thing; however, its expectation approaches 1.0 as the length of my life approaches infinity. Do you want to accept that bet?

Clearly, Mathematical Expectation, a cornerstone of gambling theory, of money management as well as Economic Theory, must be utilized with the lens of a given horizon, a given lifespan. Hence the often-overlooked caveat in the definition provided earlier for Mathematical Expectation, as you continue to play.

Often you will see the variable N throughout. This refers to the number of components in a portfolio, or the number of games played simultaneously. This is not to be confused with the lowercase n, which typically herein will refer to the total number of ways something might turn out.

Readers of the previous books will recognize the term market system, which I have used with ubiquity. This is simply a given approach applied to a given market. Thus, I can be trading the same market with two different approaches, and therefore have two separate market systems. On the other hand, I can be trading a basket of markets with the same system and then have a basket of market systems. Typically, a market system is one component in a portfolio (and the distribution of outcomes of a market system is the same as the distribution