The Leverage Space Trading Model

Reconciling Portfolio Management Strategies and Economic Theory

Ralph Vince

• Publisher: Wiley
• Year: 2009
• Print ISBN:
  9780470455951
• Edition: 1
• eBook ISBN:
  9780470496022
• Topic: Business
• Subtopic: Investments & Securities
• Index: Business

The cornerstone of money management and portfolio optimization techniques has remained the same throughout history: maximize gains and minimize risk. Yet, asserts Ralph Vince, the widely accepted approaches of combining assets into a portfolio and determining their relative quantities are wrong—and will cost you. They illuminate nothing, he says, aside from providing the illusion of safety through diversification. Although numerous Nobel Prizes have been awarded based on some of those widely accepted principles, their popular acceptance does not constitute real-world validation. What has been needed is a viable alternative to directly address these real-world dictates.

In The Leverage Space Trading Model, Vince offers a groundbreaking contribution to the literature that builds on a lifetime of expert analysis to deliver not only a superior new portfolio model, but takes the entire discipline of portfolio management to a new level.

In this book, Vince—who has made many important intellectual contributions to the field for over two decades—departs radically from informed orthodoxy to present an entirely new approach to portfolio management. At its core, The Leverage Space Trading Model basically tells how resources should be combined to maximize safety and profitability given the dictates of the real world. But, as the author points out, given the complex and seemingly pathological character of human desires, we are presented with a fascinating puzzle. Research has found that human beings do not primarily want to maximize gains—our psychological makeup is such that we instead tend to possess seemingly more complex desires. If the models don't work, if we are ultimately unable to satisfy our more complex desires, what's the alternative? As Vince shows, the answer is to utilize the Leverage Space Model as a "framework" to achieve the specific ends a trader or portfolio manager seeks.

The author's new allocation paradigm avoids the troubles that come with mean variance models—which most models are—and quantifies drawdowns to achieve a growth-optimal portfolio within a given drawdown constraint, in a manner that satisfies these seemingly pathological human desires. And for those who don't wish to get involved with the mathematics, Vince has presented the text in a manner of two congruent, simultaneous channels, with math and without.

Most simply put, this book will change how you think about money management and portfolio allocations.

PART I

The Single Component Case: Optimal f

CHAPTER 1

The General History of Geometric Mean Maximization

Geometric mean maximization, or “growth-optimality,” is the idea of maximizing the size of a stake when the amount you have to wager is a function of the outcome of the wagers up to that point. You are trying to maximize the value of a stake over a series of plays wherein you do not add or remove amounts from the stake.

The lineage of reasoning of geometric mean maximization is crucial, for it is important to know how we got here. I will illustrate, in broad strokes, the history of geometric mean maximization because this story is about to take a very sharp turn in Part III, in the reasoning of how we utilize geometric mean maximization. To this point in time, the notion of geometric mean maximization has been a criterion (just as being at the growth-optimal point, maximizing growth, has been the criterion before we examine the nature of the curve itself).

We will see later in this text that it is, instead, a framework (something much greater than the antiquated notion of “portfolio models”). This is an unavoidable perspective that gives context to our actions, but our criterion is rarely growth optimality. Yet growth optimality is the criterion that is solved mathematically. Mathematics, devoid of human propensities, proclivities, and predilections, can readily arrive at a clean, “optimal” point. As such, it provides a framework for us to satisfy our seemingly labyrinthine appetites.

On the ninth of September 1713, Swiss mathematician Nicolaus Bernoulli, whose fascination with difference equations had him corresponding with French mathematician Pierre Raymond de Montmort, whose fascination was finite differences, wrote to Montmort about a paradox that involved the intersection of their interests.

Bernoulli described a game of heads and tails, a coin toss in which you essentially pay a cover charge to play. A coin is tossed. If the coin comes up heads, it is tossed again repeatedly until it comes up tails. The pot starts at one unit and doubles every time the coin comes up heads. You win whatever is in the pot when the game ends. So, if you win on the first toss, you get your one unit back. If tails doesn’t appear until the second toss, you get two units back. On the third toss, a tails will get you four units back, ad infinitum.

Thus, you win 2^q-1 units if tails appears on the qth toss.

The question is “What should you pay to enter this game, in order for it to be a ‘fair’ game based on Mathematical Expectation?”

Suppose you win one unit with probability .5, two units with probability .25, four units with probability .125, ad infinitum. The Mathematical Expectation is therefore:

(1.01)

The expected result for a player in such a game is to win an infinite amount. So just what is a fair cover charge, then, to enter such a game?¹ This is quite the paradox indeed, and one that shall rendezvous with us in the sequel in Part III.

The cognates of geometric mean maximization begin with Nicolaus Bernoulli’s cousin, Daniel Bernoulli.²,³ In 1738, 18 years before the birth of Mozart, Daniel made the first known reference to what is known as “geometric mean maximization.” Arguably, his paper drew upon the thoughts and intellectual backdrop of his era, the Enlightenment, the Age of Reason. Although we may credit Daniel Bernoulli here as the first cognate of geometric mean maximization (as he is similarly credited as the father of utility theory by the very same work), he, too, was a product of his time. The incubator for his ideas began in the 1600s in the belching mathematical cauldron of the era.

Prior to that time, there is no known mention in any language of even generalized optimal reinvestment strategies. Merchants and traders, in any of the developing parts of the earth, evidently never formally codified the concept. If it was contemplated by anyone, it was not recorded.

As for what we know of Bernoulli’s 1738 paper (originally published in Latin), according to Bernstein (1996), we find a German translation appearing in 1896, and we find a reference to it in John Maynard Keynes’ 1921 Treatise on Probability.

In 1936, we find an article in The Quarterly Journal of Economics called “Speculation and the carryover” by John Burr Williams that pertained to trading in cotton. Williams posited that one should bet on a representative price and that if profits and losses are reinvested, the method of calculating this price is to select the geometric mean of all of the possible prices.

Interesting stuff.

By 1954, we find Daniel Bernoulli’s 1738 paper finally translated into English in Econometrica.

When so-called game theory came along in the 1950s, concepts were being widely examined by numerous economists, mathematicians, and academicians, and this fecund backdrop is where we find, in 1956, John L. Kelly Jr.’s paper, “A new interpretation of information rate.” Kelly demonstrated therein that to achieve maximum wealth, a gambler should maximize the expected value of the logarithm of his capital. This is so because the logarithm is additive in repeated bets and to which the law of large numbers applies. (Maximizing the sum of the logs is akin to maximizing the product of holding period returns, that is, the “Terminal Wealth Relative.”)

In his 1956 paper in the Bell System Technical Journal, Kelly showed how Shannon’s “Information Theory” (Shannon 1948) could be applied to the problem of a gambler who has inside information in determining his growth-optimal bet size.

When one seeks to maximize the expected value of the stake after n trials, one is said to be employing “The Kelly criterion.”

The Kelly criterion states that we should bet that fixed fraction of our stake (f) that maximizes the growth function G( f ):

(1.02)

where:

f = the optimal fixed fraction

P = the probability of a winning bet/trade

B =the ratio of amount won on a winning bet to amount lost on a losing bet

ln( ) = the natural logarithm function

Betting on a fixed fractional basis such as that which satisfies the Kelly criterion is a type of wagering known as a Markov betting system. These are types of betting systems wherein the quantity wagered is not a function of the previous history, but rather, depends only upon the parameters of the wager at hand.

If we satisfy the Kelly criterion, we will be growth optimal in the long-run sense. That is, we will have found an optimal value for f (as the optimal f is the value for f that satisfies the Kelly criterion).

In the following decades, there was an efflorescence of papers that pertained to this concept, and the idea began to embed itself into the world of capital markets, at least in terms of academic discourse, and these ideas were put forth by numerous researchers, notably Bellman and Kalaba (1957), Breiman (1961), Latane (1959), Latane and Tuttle (1967), and many others.

Edward O. Thorp, a colleague of Claude Shannon, and whose work deserves particular mention in this discussion, is perhaps best known for his 1962 book, Beat the Dealer (prov...

Table of contents

• Title Page
• Copyright Page
• Praise
• Preface
• Introduction
• PART I - The Single Component Case: Optimal f
• PART II - The Multiple Component Case: The Leverage Space Portfolio Model
• PART III - The Leverage Space Praxis
• Bibliography
• Index