The handbook of portfolio mathematics ( PDFDrive.com )

CHAPTER 1 The Random Process and Gambling Theory 3 Independent versus Dependent Trials Processes 5 Exact Sequences, Possible Outcomes, and the Mathematical Expectation Less than Zero Spe

The Handbook of

Portfolio Mathematics

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The Handbook of

Portfolio Mathematics

Formulas for Optimal Allocation & Leverage

RALPH VINCE

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Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

Chapters 1–10 contain revised material from three of the author’s previous books, Portfolio Management Formulas: Mathematical Trading Methods for the Futures, Options, and Stock Markets (1990), The Mathematics of Money Management: Risk Analysis Techniques for Traders (1992), and The New Money Management: A Framework for Asset Allocation

(1995), all published by John Wiley & Sons, Inc.

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Library of Congress Cataloging-in-Publication Data:

1 Portfolio management–Mathematical models 2 Investments–Mathematical models.

“You must not be extending your empire while you are at war or runinto unnecessary dangers I am more afraid of our own mistakes

than our enemies’ designs.”

—Pericles, in a speech to the Athenians during the Peloponnesian

War, as represented by Thucydides

v

vi

CHAPTER 1 The Random Process and Gambling Theory 3

Independent versus Dependent Trials Processes 5

Exact Sequences, Possible Outcomes, and the

Mathematical Expectation Less than Zero Spells Disaster 18

The Runs Test, Z Scores, and Confidence Limits 27

vii

The Central Limit Theorem 52

CHAPTER 3 Reinvestment of Returns and Geometric

Measuring a Good System for Reinvestment—The

Finding the Optimal f by the Geometric Mean 122

How to Figure the Geometric Mean Using

A Simpler Method for Finding the Optimal f 128

Consequences of Straying Too Far

Finding Optimal f via Parabolic Interpolation 157

Optimal f for Small Traders Just Starting Out 175

One Combined Bankroll versus Separate Bankrolls 180

Efficiency Loss in Simultaneous Wagering or

Time Required to Reach a Specified Goal and the

The Estimated Geometric Mean (or How the Dispersion

Characteristics of Utility Preference Functions 218

Alternate Arguments to Classical Utility Theory 221

Solutions of Linear Systems Using Row-Equivalent Matrices 246

CHAPTER 8 The Geometry of Mean Variance Portfolios 261

Mathematical Optimization versus Root Finding 312

Upside Limit on Active Equity and the Margin Constraint 341

f Shift and Constructing a Robust Portfolio 342 Tailoring a Trading Program through Reallocation 343

Important Points to the Left of the Peak in the n + 1

Further Characteristics of Long-Term Trend Followers 369

CHAPTER 12 The Leverage Space Portfolio Model in

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It’s always back there, bubbling away It seems I cannot shut off my

mind from it Every conversation I ever have, with programmers andtraders, engineers and gamblers, Northfield Park Railbirds and War-rensville Workhouse jailbirds—those equations that describe these verythings are cast in this book

Let me say I am averse to gambling I am averse to the notion of creatingrisk where none need exist, averse to the idea of attempting to be rewarded

in the absence of creating or contributing something (or worse yet, taxing aman’s labor!) Additionally, I find amorality in charging or collecting interest,and the absence of this innate sense in others riles me

This book starts out as a compilation, cleanup, and in some cases,reformulation of the previous books I have written on this subject I’mstanding on big shoulders here The germ of the idea of those previous bookscan trace its lineage to my good friend and past employer, Larry Williams Inthe dust cloud of his voracious research, was the study of the Kelly Criterion,and how that might be applied to trading What followed over the comingyears then was something of an explosion in that vein, culminating in abetter portfolio model than the one which is still currently practiced.For years now I have been away from the markets—intentionally In apeculiar irony, it has sharpened my bird’s-eye view on the entire industry.People still constantly seek me out, bend my ears, try to pick my hollow,rancid pumpkin about the markets It has all given me a truly gigantic field

of view, a dizzying phantasmagoria, on who is doing what, and how.I’d like to share some of that with you here

We are not going to violate anyone’s secrets here, realizing that most ofthese folks work very hard to obtain what they know What I will speak of

is generalizations and commonalities in what people are doing, so that wecan analyze, distinguish, compare, and, I hope, arrive at some well-foundedconclusions

But I am not in the markets’ trenches anymore My time has been spent

on software for parametric geometry generation of industrial componentryand “smart” robots that understand natural language and can go out and do

xiii

things like perform research for me, come back, draw inferences, and cuss their findings with me These are wonderful endeavors for me, allowing

dis-me to extend my litany of failures

Speaking of which, in the final section of this text, we step into the silent, blue-lit morgue of failure itself, dissecting it both in a mathematicaland abstract sense, as well as the real-world one In this final chapter, thetwo are indistinguishable

near-When we speak of the real world, some may get the mistaken impression

that the material is easy It is not That has not been a criterion of mine here.What has been a criterion is to address the real-world application of theprevious three books that this book incorporates That means looking at theprevious material with regard to failure, with regard to drawdown Moneymanagers and personal traders alike tend to have utility preference curvesthat are incongruent with maximizing their returns Further, I am aware of noone, nor have I ever encountered any trader, fund manager, or institution,who could even tell you what his or her utility preference function was.This is a prime example of the chasm—the disconnect—between theoryand real-world application

Historically, risk has been defined in theoretical terms as the variance(or semivariance) in returns This, too, is rarely (though in certain situations)

a desired proxy for risk Risk is the chance of getting your head handed toyou It is not, except in rare cases, variance in returns It is not semivariance

in returns; it is not determined by a utility preference function Risk isthe probability of being ruined Ruin is touching or penetrating a lowerbarrier on your equity So we can say to most traders, fund managers, andinstitutions that risk is the probability of touching a lower barrier on equity,such that it would constitute ruin to someone Even in the rare cases wherevariance in returns is a concern, risk is still primarily a drawdown to a lowerabsorbing barrier

So what has been needed, and something I have had bubbling away for

the past decade or so, is a way to apply the optimal f framework within the

real-world constraints of this universally regarded definition of risk That is,

how do we apply optimal f with regard to risk of ruin and its more familiar

and real-world-applicable-cousin, risk of drawdown?

Of course, the concepts are seemingly complicated—we’re seeking tomaximize return for a given level of drawdown, not merely juxtapose returnsand variance in returns Do you want to maximize growth for a given level

of drawdown, or do you want to do something easier?

So this book is more than just a repackaging of previous books onthis subject It incorporates new material, including a study of correlations

between pairwise components in a portfolio (and why that is such a bad

idea) Chapter 11 examines what portfolio managers have (not) been doingwith regards to the concepts presented in this book, and Chapter 12 takes

the new Leverage Space Portfolio Model and juxtaposes it to the probability

of a given drawdown to provide a now-superior portfolio model, based onthe previous chapters in this book, and applicable to the real world

I beg the reader to look at everything in this text—as merely my ticulation of something, and not an autocratic dictation Not only am I notinfallible, but also my real aim here is to engage you in the study of some-thing I find fascinating, and I want to share that very raw joy with you.Because, you see, as I started out saying, it’s always back there, bubblingaway—my attraction to those equations on the markets, pertaining to allo-cation and leverage It’s not a preoccupation with the markets, though—to

ar-me it could be the weather or any other dynamic system It is the allure ofnailing masses and motions and relationships with an equation

This book covers my thinking on these subjects for more than two and

a half decades There are a lot of people to thank I won’t mention them,either—they know who they are, and I feel uneasy mentioning the names

of others here in one way or another, or others in the industry who wish toremain nameless I don’t know how they might take it

There is one guilty party, however, whom I will mention—Rejeanne.

This one, finally, is for you

RALPHVINCE

Chagrin Falls, Ohio

August 2006

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This is a book in two distinct parts Originally, my task was to distill the

previous three books on this subject into one book In effect, Part Icomprises that text

It’s been reorganized, rehashed, and reworked to resemble the originaltexts while creating a contiguous path of reasoning, which takes us from thebasic gambling theory and statistics, through the introduction of the Kelly

criterion, optimal f , and finally onto the Leverage Space Portfolio Model

for multiple-simultaneous positions

The Leverage Space Portfolio Model addresses allocations and leverage.Often these are two distinct facets, but herein they refer to the same thing

Allocation is the relative leverage between multiple portfolio components.

Thus, when we speak of leverage, we are also speaking of allocation, and

vice versa

Likewise, money management and portfolio construction, as

prac-ticed, don’t necessarily refer to the same exercise, yet in this text, they

do Collectively, whatever the endeavor of risk, be it a bond portfolio, acommodities fund, or a team of blackjack players invading a casino, the

collective exercise will be herein referred to as allocation.

I have tried to keep the geometric perspective on these concepts, andkeep those notions about them intact The first section is necessarily heavy

on math The first section is purely conceptual It is about allocation andleverage to maximize returns without respect to anything else

Everything in Part I was conjured up more than a decade or two ago Iwas younger then

Since that time, I have repeatedly been approached with the question,

“How do you apply it?” I used to be baffled by this; the obvious (to me)answer being, “As is.”

As used herein, a ln utility preference curve is one that is characteristic

of someone who acts so as to maximize the ratio of his or her returns to therisk assumed to do so

The notion that someone’s utility preference function could be

any-thing other than ln was evidence of both the person’s insanity and weakness

xvii

I saw it as a means for risk takers to enjoy the rush of their compulsive

gam-bling under the ruse of the academic justification of utility preference.

I’m older now (seemingly not tempered with age—you see, I still knowthe guy who wrote those previous books), but I have been able to at leastaccept the exercise—the rapture—of working to solve the dilemma of op-timal allocations and leverage under the constraint of a utility preference

curve that is not ln.

By the definition of a ln utility preference curve, given a few paragraphsago, a sane1 person is therefore one who is levered up to the optimal f

level in a game favorable to him or minimizes his number of plays in a gameunfavorable to him Anyone who goes to a casino and plunks down all he iswilling to lose on that trip in one play is not a compulsive gambler But whodoes that? Who has that self-control? Who has a utility preference curve

that is ln?

That takes us to Part II of the book, the part I call the real-world

applica-tion of the concepts illuminated in Part I, because people’s utility preference

curves are not ln

So Part II attempts to tackle the mathematical puzzle posed by ing to employ the concepts of Part I, given the weakness and insanity ofhuman beings What could be more fun?

attempt-* attempt-* attempt-*

Many of the people who have approached me with the question of “How doyou apply it?” over the years have been professionals in the industry Since,ultimately, their clients are the very individuals whose utility preferencecurves are not ln, I have found that these entities have utility preferencefunctions that mirror those of their clients (or they don’t have clients forlong)

Many of these entities have been successful for many years Naturally,their procedures pertaining to allocation, leverage, and trading implemen-tation were of great interest to me

Part II goes into this, into what these entities typically do The best ofthem, I find, have not employed the concepts of the last chapter except invery rudimentary and primitive ways There is a long way to go

Often, I have been criticized as being “all theory—no practice.” Well,

Part I is indeed all theory, but it is exhaustive in that sense—not on portfolio

construction in general and all the multitude of ways of performing that,but rather, on portfolio construction in terms of optimal position sizes (i.e.,

in the vein of an optimal f approach) Further, I did not want Part I to be

1Academics prefer the nomenclature “rational,” versus “sane.” The subtle differencebetween the two is germane to this discussion

a mere republishing, almost verbatim, of the previous books Therefore, Ihave incorporated some new material into Part I This is material that hasbecome evident to me in the years since the original material was published.Part II is entirely new I have been fortunate in that my first exposure tothe industry was as a margin clerk I had an opportunity to observe a sizableuniverse of ways people go about doing things in this business Later, thanks

to my programming abilities, from which the other books germinated, I hadexposure to many professionals in the industry, and was often privy to howthey practiced things, or was in a position where I could reverse-engineer it Ihave had the good fortune of being on a course that has afforded me a bird’s-eye view of the way people practice their allocation, leverage, and tradingimplementations in this business Part II is derived from that high-altitudebird’s-eye view, and the desire to provide a real-world implementation ofthe concepts of Part I—that is, to make them applicable to those peoplewhose utility preference functions are not ln

* * *

Things I have written of in the past have received a good deal of criticismover the years I welcome it, and a chance to address it To me, it says peopleare thinking about these ideas, trying to mold them further, or remold thoseareas where I may have been wrong (I’m not so much interested in being

“right” about any of this as I am about “this”) Though I have not consciouslyintended that, this book, in many ways, answers some of those criticisms.The main criticism was that it was too theoretical with no real-worldapplication The criticism is well founded in the sense that drawdown wasall but ignored For better or worse, people and institutions never seem tohave utility functions that are ln Yet, nearly all utility functions of peopleand institutions are ln within a drawdown constraint That is, they seek tomaximize the ratio of returns to risk (drawdown) within a certain draw-down That disconnect between what I have written in the past has now,more than a decade later, been resolved

A second major criticism is that trading at optimal f is too wild for any

mere human I know of no professional funds that have traded at the optimal

f levels I have known people who have traded at optimal f , usually for short

periods of time, in only a single market, before panicking in a drawdown.There it is again: drawdown You see, it wasn’t so much this construct oftheir utility preference curve (talk about too theoretical!) as it was their

drawdown that was incongruent with their trading at the optimal f level.

If you are getting the notion that we will be looking into the nature ofdrawdown later on in this book, when we discuss what I have been doing

in terms of working on this material for the past decade-plus, you’re right.We’re going to look at drawdown herein beyond what anyone has

Which takes us to the third major criticism, being that optimal f or the

Leverage Space Model allocates without respect to drawdown This, too,has now been addressed directly in Chapter 12 However, as we will see

in that chapter, drawdown is, in a sequence of independent trials, but onepermutation of many permutations Thus, to address drawdown, one mustaddress it in those terms

The last major criticism has been that regarding the complexity of culation People desire a simple solution, a heuristic, something they couldperform by hand if need be

cal-Unfortunately, that was not the case, and that desire of others is nowsomething even more remote In the final chapter, we can see that one mustperform millions of calculations (as a sample to billions of calculations!) inorder to derive certain answers

However, such seemingly complex tasks can be made simple by aging them up as black-box computer applications Once someone under-stands what calculations are performed and why, the machine can do theheavy lifting Ultimately, that is even simpler than performing a simple cal-culation by hand

pack-If one can put in the scenarios, their outcomes, and probability ofoccurrence—their joint probabilities of occurrence with other scenarios

in other scenario spectrums—one can feed the machine and derive thatnumber which satisfies the ideal composition, the optimal allocations andleverage among portfolio components to satisfy that ln utility preferencefunction within a certain drawdown constraint

To be applicable to the real world, a book like this should, it would

seem, be about trading This is not a book on how to trade the markets.

(This makes the real-world application section difficult.) It is about howvery basic, mathematical laws are working on us—you and me—when weengage in a stream of risk-related outcomes wherein we don’t have controlover those outcomes Rather, we have control only over the relative impacts

on us In that sense, the mathematics applies to us in trading

I don’t want to pretend to know a thing about trading, really Just as I

am not an academic, I am also not a trader I’ve been around and workedfor some amazing traders—but that doesn’t mean I am one

That’s your domain—and why you are reading this book: To augment

the knowledge you have about trading vis- `a-vis cross-pollination with these

outside formulas And if they are too cumbersome, or too complicated,

please don’t blame me I wish they were simply along the lines of 2+ 2 Butthey are not

This is not by my design When you trade, you are somewhat trying tointuitively carve your way along the paths of these equations, yet you areoblivious to what the equations are You are, for instance, trying to maximize

your returns within a certain probability of a given drawdown over the nextperiod.

But you don’t really have the equations to do so Now you do Don’tblame me if you find them to be too cumbersome These formulas arewhat we seek to know—and somehow use—as they apply to us in trading,whether we acknowledge that or not I have heard ample criticism aboutthe difficulties in applications In this text, I will attempt to show you what

others are doing compared to using these formulas However, these

formu-las are at work on everyone when they trade It is in the disparity betweenthe two that your past criticisms of me lie; it is in that very disparity that

my criticisms of you lie

When you step up to the service line and line up to serve to my backhand,say, the fact that gravity operates with an acceleration of 9.8 meters persecond squared applies to you It applies to your serve landing in the box

or not (among other things), whether you acknowledge this or not It is anexplanation of how things work more so than how to work things You aretrying to operate within a world defined by certain formulas It does notmean you can implement them in your work, or that, because you cannot,they are therefore invalid Perhaps you can implement them in your work.Clearly, if you could, without expense to the other aspects of “your work,”wouldn’t it be safe to say, then, that you certainly wouldn’t be worse off?And so with the equations in the book Perhaps you can implementthem—and if you can, without expense to the other aspects of your game,then won’t you be better off? And if not, does it invalidate their truths anymore than a tennis pro who dishes up a first serve, oblivious to the 9.8 m/s2

at work?

* * *

This is, in its totality, what I know about allocations and leverage in trading

It is the sum of all I have written of it in the past, and what I have savoredover the past decade-plus As with many things, I truly love this stuff I hope

my passion for it rings contagiously herein However, it sits as dead and cold

as any inanimate abstraction It is only your working with these concepts,your application and your critiques of them, your volley back over the net,that give them life

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P A R T I

Theory

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2

C H A P T E R 1

The Random Process and Gambling Theory

We will start with the simple coin-toss case When you toss a coin in

the air there is no way to tell for certain whether it will land heads

or tails Yet over many tosses the outcome can be reasonably dicted

pre-This, then, is where we begin our discussion

Certain axioms will be developed as we discuss the random process

The first of these is that the outcome of an individual event in a

ran-dom process cannot be predicted However, we can reduce the possible outcomes to a probability statement

Pierre Simon Laplace (1749–1827) defined the probability of an event

as the ratio of the number of ways in which the event can happen to thetotal possible number of events Therefore, when a coin is tossed, the prob-ability of getting tails is 1 (the number of tails on a coin) divided by 2 (thenumber of possible events), for a probability of 5 In our coin-toss example,

we do not know whether the result will be heads or tails, but we do knowthat the probability that it will be heads is 5 and the probability it will be

tails is 5 So, a probability statement is a number between 0 (there is no

chance of the event in question occurring) and 1 (the occurrence of the event is certain)

Often you will have to convert from a probability statement to odds andvice versa The two are interchangeable, as the odds imply a probability,and a probability likewise implies the odds These conversions are givennow The formula to convert to a probability statement, when you knowthe given odds is:

Probability= odds for/(odds for + odds against) (1.01)

3

If the odds on a horse, for example, are 4 to 1 (4:1), then the probability

of that horse winning, as implied by the odds, is:

Probability= 1/(1 + 4)

= 1/5

= 2

So a horse that is 4:1 can also be said to have a probability of winning

of 2 What if the odds were 5 to 2 (5:2)? In such a case the probability is:

Probability= 2/(2 + 5)

= 2/7

= 2857142857

The formula to convert from probability to odds is:

Odds (against, to one)= 1/probability − 1 (1.02)

So, for our coin-toss example, when there is a 5 probability of thecoin’s coming up heads, the odds on its coming up heads are given as:

Odds= 1/.5 − 1

= 2 − 1

= 1This formula always gives you the odds “to one.” In this example, we wouldsay the odds on a coin’s coming up heads are 1 to 1

How about our previous example, where we converted from odds of5:2 to a probability of 2857142857? Let’s work the probability statementback to the odds and see if it works out

Most people can’t handle the uncertainty of a probability statement; itjust doesn’t sit well with them We live in a world of exact sciences, andhuman beings have an innate tendency to believe they do not understand

an event if it can only be reduced to a probability statement The domain

of physics seemed to be a solid one prior to the emergence of quantum

physics We had equations to account for most processes we had observed.These equations were real and provable They repeated themselves overand over and the outcome could be exactly calculated before the eventtook place With the emergence of quantum physics, suddenly a theretoforeexact science could only reduce a physical phenomenon to a probabilitystatement Understandably, this disturbed many people.

I am not espousing the random walk concept of price action nor am Iasking you to accept anything about the markets as random Not yet, any-way Like quantum physics, the idea that there is or is not randomness inthe markets is an emotional one At this stage, let us simply concentrate

on the random process as it pertains to something we are certain is dom, such as coin tossing or most casino gambling In so doing, we canunderstand the process first, and later look at its applications Whether therandom process is applicable to other areas such as the markets is an issuethat can be developed later

ran-Logically, the question must arise, “When does a random sequence gin and when does it end?” It really doesn’t end The blackjack table con-tinues running even after you leave it As you move from table to table in

be-a cbe-asino, the rbe-andom process cbe-an be sbe-aid to follow you be-around If you tbe-ake

a day off from the tables, the random process may be interrupted, but itcontinues upon your return So, when we speak of a random process of Xevents in length we are arbitrarily choosing some finite length in order tostudy the process

INDEPENDENT VERSUS DEPENDENT

TRIALS PROCESSES

We can subdivide the random process into two categories First are thoseevents for which the probability statement is constant from one event tothe next These we will call independent trials processes or sampling withreplacement A coin toss is an example of just such a process Each tosshas a 50/50 probability regardless of the outcome of the prior toss Even

if the last five flips of a coin were heads, the probability of this flip beingheads is unaffected, and remains 5

Naturally, the other type of random process is one where the outcome

of prior events does affect the probability statement and, naturally, the

probability statement is not constant from one event to the next Thesetypes of events are called dependent trials processes or sampling withoutreplacement Blackjack is an example of just such a process Once a card isplayed, the composition of the deck for the next draw of a card is differentfrom what it was for the previous draw Suppose a new deck is shuffled

and a card removed Say it was the ace of diamonds Prior to removing thiscard the probability of drawing an ace was 4/52 or 07692307692 Now that

an ace has been drawn from the deck, and not replaced, the probability ofdrawing an ace on the next draw is 3/51 or 05882352941

Some people argue that dependent trials processes such as this are ally not random events For the purposes of our discussion, though, wewill assume they are—since the outcome still cannot be known before-hand The best that can be done is to reduce the outcome to a probabilitystatement Try to think of the difference between independent and depen-

re-dent trials processes as simply whether the probability statement is fixed (independent trials) or variable (dependent trials) from one event to the

next based on prior outcomes This is in fact the only difference

Everything can be reduced to a probability statement Events wherethe outcomes can be known prior to the fact differ from random eventsmathematically only in that their probability statements equal 1 For ex-ample, suppose that 51 cards have been removed from a deck of 52 cardsand you know what the cards are Therefore, you know what the one re-maining card is with a probability of 1 (certainty) For the time being, wewill deal with the independent trials process, particularly the simple cointoss

A= Amount you can win/Amount you can lose

So, if you are going to flip a coin and you will win$2 if it comes up heads,but you will lose$1 if it comes up tails, the mathematical expectation perflip is:

This formula just described will give us the mathematical tion for an event that can have two possible outcomes What about situa-tions where there are more than two possible outcomes? The next formulawill give us the mathematical expectation for an unlimited number of out-comes It will also give us the mathematical expectation for an event withonly two possible outcomes such as the 2 for 1 coin toss just described.Hence, it is the preferred formula.

expecta-Mathematical Expectation=

N



i =1(Pi∗ Ai) (1.03a)

where: P= Probability of winning or losing

A= Amount won or lost

N= Number of possible outcomes

The mathematical expectation is computed by multiplying each possiblegain or loss by the probability of that gain or loss, and then summing thoseproducts together

Now look at the mathematical expectation for our 2 for 1 coin tossunder the newer, more complete formula:

ME= 33 ∗ (−1) + 33 ∗ (−2) + 33 ∗ 3

= −.33 − 66 + 99

= 0Consider betting on one number in roulette, where your mathematicalexpectation is:

ME= 1/38 ∗ 35 + 37/38 ∗ (−1)

= 02631578947 ∗ 35 + 9736842105 ∗ (−1)

= 9210526315 + (−.9736842105)

= −.05263157903

If you bet $1 on one number in roulette (American double-zero), youwould expect to lose, on average, 5.26 cents per roll If you bet $5, you

would expect to lose, on average, 26.3 cents per roll Notice how

differ-ent amounts bet have differdiffer-ent mathematical expectations in terms of amounts, but the expectation as a percent of the amount bet is always the same

The player’s expectation for a series of bets is the total of the pectations for the individual bets So if you play $1 on a number inroulette, then$10 on a number, then $5 on a number, your total expectationis:

ex-ME= (−.0526) ∗ 1 + (−.0526) ∗ 10 + (−.0526) ∗ 5

= −.0526 − 526 − 263

= −.8416

You would therefore expect to lose on average 84.16 cents

This principle explains why systems that try to change the size oftheir bets relative to how many wins or losses have been seen (assuming

an independent trials process) are doomed to fail The sum of expectation bets is always a negative expectation!

negative-EXACT SEQUENCES, POSSIBLE OUTCOMES,

AND THE NORMAL DISTRIBUTION

We have seen how flipping one coin gives us a probability statement withtwo possible outcomes—heads or tails Our mathematical expectationwould be the sum of these possible outcomes Now let’s flip two coins.Here the possible outcomes are:

combi-If we were to flip three coins, we would have:

Notice here that if we were to plot the asterisks vertically we would

be developing into the familiar bell-shaped curve, also called the Normal

or Gaussian Distribution (see Figure 1.1).1

FIGURE 1.1 Normal probability function

1Actually, the coin toss does not conform to the Normal Probability Function in apure statistical sense, but rather belongs to a class of distributions called the Bi-nomial Distribution (a.k.a Bernoulli or Coin-Toss Distributions) However, as Nbecomes large, the Binomial approaches the Normal Distribution as a limit (pro-vided the probabilities involved are not close to 0 or 1) This is so because theNormal Distribution is continuous from left to right, whereas the Binomial is not,and the Normal is always symmetrical whereas the Binomial needn’t be Since weare treating a finite number of coin tosses and trying to make them representative

of the universe of coin tosses, and since the probabilities are always equal to 5,

we will treat the distributions of tosses as though they were Normal As a furthernote, the Normal Distribution can be used as an approximation of the Binomial ifboth N times the probability of an event occurring and N times the complement ofthe probability occurring are both greater than 5 In our coin-toss example, sincethe probability of the event is 5 (for either heads or tails) and the complement is.5, then so long as we are dealing with N of 11 or more we can use the NormalDistribution as an approximation for the Binomial

Finally, for 10 coins:

Notice that as the number of coins increases, the probability of

get-ting all heads or all tails decreases When we were using two coins, theprobability of getting all heads or all tails was 25 For three coins it was.125, for four coins 0625; for six coins 0156, and for 10 coins it was 001

POSSIBLE OUTCOMES AND

The term “exact sequence” here means the exact outcome of a randomprocess The set of all possible exact sequences for a given situation is

called the sample space Note that the four-coin flip just depicted can be

four coins all flipped at once, or it can be one coin flipped four times (i.e.,

it can be a chronological sequence)

If we examine the exact sequence T H H T and the sequence H H T T,the outcome would be the same for a person flat-betting (i.e., betting 1 unit

on each instance) However, to a person not flat-betting, the end result ofthese two exact sequences can be far different To a flat bettor there areonly five possible outcomes to a four-flip sequence:

4 Heads

3 Heads and 1 Tail

2 Heads and 2 Tails

1 Head and 3 Tails

4 Tails

As we have seen, there are 16 possible exact sequences for a coin flip This fact would concern a person who is not flat-betting We willrefer to people who are not flat-betting as “system” players, since that ismost likely what they are doing—betting variable amounts based on somescheme they think they have worked out

four-If you flip a coin four times, you will of course see only one of the

16 possible exact sequences If you flip the coin another four times, youwill see another exact sequence (although you could, with a probability

of 1/16 = 0625, see the exact same sequence) If you go up to a gamingtable and watch a series of four plays, you will see only one of the 16 exact

sequences You will also see one of the five possible end results Each exact

sequence (permutation) has the same probability of occurring, that being

.0625 But each end result (combination) does not have equal probability

of occurring:

3 Heads and 1 Tail 25

2 Heads and 2 Tails 375

1 Head and 3 Tails 25

Most people do not understand the difference between exact sequences (permutation) and end results (combination) and as a result falsely con- clude that exact sequences and end results are the same thing This is a

common misconception that can lead to a great deal of trouble It is the end results (not the exact sequences) that conform to the bell curve—the Normal Distribution, which is a particular type of probability distribution.

An interesting characteristic of all probability distributions is a statistic

known as the standard deviation.

For the Normal Probability Distribution on a simple binomial game,such as the one being used here for the end results of coin flips, the stan-dard deviation (SD) is:

be+ or − 1 standard deviation from the center line, 95.45% between + and

− 2 standard deviations from the center line, and 99.73% between + and

− 3 standard deviations from the center line (see Figure 1.2) Continuingwith our 10-flip coin toss, 1 standard deviation equals approximately 1.58

We can therefore say of our 10-coin flip that 68% of the time we can expect

to have our end result be composed of 3.42 (5− 1.58) to 6.58 (5 + 1.58)being heads (or tails) So if we have 7 heads (or tails), we would be beyond

1 standard deviation of the expected outcome (the expected outcome ing 5 heads and 5 tails)

be-Here is another interesting phenomenon Notice in our coin-toss ples that as the number of coins tossed increases, the probability of getting

exam-an even number of heads exam-and tails decreases With two coins the ity of getting H1T1 was 5 At four coins the probability of getting 50% headsand 50% tails dropped to 375 At six coins it was 3125, and at 10 coins 246

probabil-Therefore, we can state that as the number of events increases, the

prob-ability of the end result exactly equaling the expected value decreases

FIGURE 1.2 Normal probability function: Center line and 1 standard deviation in either direction

The mathematical expectation is what we expect to gain or lose, onaverage, each bet However, it does not explain the fluctuations from bet tobet In our coin-toss example we know that there is a 50/50 probability of atoss’s coming up heads or tails We expect that after N trials approximately

1/2 * N of the tosses will be heads, and1/2 * N of the tosses will be tails.Assuming that we lose the same amount when we lose as we make when

we win, we can say we have a mathematical expectation of 0, regardless ofhow large N is

We also know that approximately 68% of the time we will be+ or −

1 standard deviation away from our expected value For 10 trials (N= 10)this means our standard deviation is 1.58 For 100 trials (N = 100) thismeans we have a standard deviation size of 5 At 1,000 (N= 1,000) trials thestandard deviation is approximately 15.81 For 10,000 trials (N = 10,000)the standard deviation is 50

Notice that as N increases, the standard deviation increases as well

This means that contrary to popular belief, the longer you play, the

FIGURE 1.3 The random process: Results of 60 coin tosses, with 1 and 2 dard deviations in either direction

stan-further you will be from your expected value (in terms of units won or lost) However, as N increases, the standard deviation as a percent of N de-

creases This means that the longer you play, the closer to your expected

value you will be as a percent of the total action (N) This is the “Law ofAverages” presented in its mathematically correct form In other words, ifyou make a long series of bets, N, where T equals your total profit or lossand E equals your expected profit or loss, then T/N tends towards E/N as Nincreases Also, the difference between E and T increases as N increases

In Figure 1.3 we observe the random process in action with a toss game Also on this chart you will see the lines for+ and − 1 and 2standard deviations Notice how they bend in, yet continue outward for-ever This conforms with what was just said about the Law of Averages

60-coin-THE HOUSE ADVANTAGE

Now let us examine what happens when there is a house advantage volved Again, refer to our coin-toss example We last saw 60 trials at aneven or “fair” game Let’s now see what happens if the house has a 5%advantage An example of such a game would be a coin toss where if wewin, we win$1, but if we lose, we lose $1.10

in-Figure 1.4 shows the same 60-coin-toss game as we previously saw,only this time there is the 5% house advantage involved Notice how, in

FIGURE 1.4 Results of 60 coin tosses with a 5% house advantage

this scenario, ruin is inevitable—as the upper standard deviations begin tobend down (to eventually cross below zero)

Let’s examine what happens when we continue to play a game with anegative mathematical expectation

to make one million$1 bets at a 5% house advantage, it would be equallyunlikely for you to make money Many casino games have more than a5% house advantage, as does most sports betting Trading the markets

is a zero-sum game However, there is a small drain involved in the way

of commissions, fees, and slippage Often these costs can run in excess

or lose within+ or −10 in a fair game At a 5% house advantage this is+5 and −15 units At 1 standard deviation, where we can expect the finaloutcome to be with 68% probability, we win or lose up to 5 units in a fairgame Yet in the game where the house has the 5% advantage we can ex-pect the final outcome to be between winning nothing and losing 10 units!Note that at a 5% house advantage it is not impossible to win money after

100 trials, but you would have to do better than 1 whole standard tion to do so In the Normal Distribution, the probability of doing betterthan 1 whole standard deviation, you will be surprised to learn, is only.1587!

devia-Notice in the previous example that at 0 standard deviations from thecenter line (that is, at the center line itself), the amount lost is equal tothe house advantage For the fair 50/50 game, this is equal to 0 You wouldexpect neither to win nor to lose anything In the game where the househas the 5% edge, you would expect to lose 5%, 5 units for every 100 trials,

at 0 standard deviations from the center line So you can say that in

flat-betting situations involving an independent process, you will lose at the rate of the house advantage