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THE MATHEMATICS OF MONEY MANAGEMENT:
THE MATHEMATICS OF MONEY MANAGEMENT:
RISK ANALYSIS TECHNIQUES FOR TRADERS
by Ralph Vince
Published by John Wiley & Sons, Inc.
Library of Congress Cataloging-in-Publication Data
Vince. Ralph. 1958-The mathematics of money management: risk analysis techniques for traders / by Ralph Vince.
Includes bibliographical references and index.
ISBN 0-471-54738-7
1. Investment analysis—Mathematics.
2. Risk management—Mathematics
3. Program trading (Securities)
HG4529N56 1992 332.6'01'51-dc20 91-33547
Preface and Dedication
The favorable reception of Portfolio Management Formulas exceeded even the greatest expectation I ever had for the book. I had written it to
promote the concept of optimal f and begin to immerse readers in portfolio theory and its missing relationship with optimal f.
Besides finding friends out there, Portfolio Management Formulas was surprisingly met by quite an appetite for the math concerning money
management. Hence this book. I am indebted to Karl Weber, Wendy Grau, and others at John Wiley & Sons who allowed me the necessary latitude
this book required.
There are many others with whom I have corresponded in one sort or another, or who in one way or another have contributed to, helped me with,
or influenced the material in this book. Among them are Florence Bobeck, Hugo Rourdssa, Joe Bristor, Simon Davis, Richard Firestone, Fred Gehm
(whom I had the good fortune of working with for awhile), Monique Mason, Gordon Nichols, and Mike Pascaul. I also wish to thank Fran Bartlett of
G & H Soho, whose masterful work has once again transformed my little mountain of chaos, my little truckload of kindling, into the finished product
that you now hold in your hands.
This list is nowhere near complete as there are many others who, to varying degrees, influenced this book in one form or another.
This book has left me utterly drained, and I intend it to be my last.
Considering this, I'd like to dedicate it to the three people who have influenced me the most. To Rejeanne, my mother, for teaching me to appre-
ciate a vivid imagination; to Larry, my father, for showing me at an early age how to squeeze numbers to make them jump; to Arlene, my wife, part-
ner, and best friend. This book is for all three of you. Your influences resonate throughout it.
Chagrin Falls, Ohio R. V.
March 1992
- 2 -
Index
Introduction
..............................................................................
61
..............................................................................................
5
Estimating Volatility
..........................................................................
61
Scope of this book
................................................................................
5
Ruin, Risk and Reality
.......................................................................
62
Some prevalent misconceptions
...........................................................
6
Option pricing models
........................................................................
62
Worst-case scenarios and stategy
.........................................................
6
A European options pricing model for all distributions
.....................
65
Mathematics notation
...........................................................................
7
The single long option and optimal f
.................................................
66
Synthetic constructs in this text
...........................................................
7
The single short option
.......................................................................
69
Optimal trading quantities and optimal f
.............................................
8
The single position in The Underlying Instrument
............................
70
Chapter 1-The Empirical Techniques
.......................................................
9
Multiple simultaneous positions with a causal relationship
...............
70
Deciding on quantity
............................................................................
9
............
72
Basic concepts
......................................................................................
9
The runs test
.......................................................................................
10
...................................................................................................
73
Serial correlation
................................................................................
11
Definition of The Problem
.................................................................
73
Common dependency errors
..............................................................
12
Solutions of Linear Systems using Row-Equivalent Matrices
76
Mathematical Expectation
.................................................................
13
Interpreting The Results
.....................................................................
77
To reinvest trading profits or not
.......................................................
14
Chapter 7 - The Geometry of Portfolios
.................................................
80
Measuring a good system for reinvestment the Geometric Mean
14
The Capital Market Lines (CMLs)
.....................................................
80
How best to reinvest
...........................................................................
15
The Geometric Efficient Frontier
.......................................................
81
Optimal fixed fractional trading
.........................................................
15
Unconstrained portfolios
....................................................................
83
Kelly formulas
...................................................................................
16
How optimal f fits with optimal portfolios
........................................
84
Finding the optimal f by the Geometric Mean
...................................
16
Threshold to The Geometric for Portfolios
........................................
85
To summarize thus far
.......................................................................
17
Completing The Loop
........................................................................
85
Geometric Average Trade
..................................................................
17
Chapter 8 - Risk Management
................................................................
88
Why you must know your optimal f
..................................................
18
Asset Allocation
.................................................................................
88
The severity of drawdown
.................................................................
18
Reallocation: Four Methods
...............................................................
90
Modern portfolio theory
.....................................................................
19
Why reallocate?
..................................................................................
92
The Markovitz model
.........................................................................
19
Portfolio Insurance – The Fourth Reallocation Technique
................
92
The Geometric Mean portfolio strategy
.............................................
21
The Margin Constraint
.......................................................................
95
Daily procedures for using optimal portfolios
...................................
21
Rotating Markets
................................................................................
96
Allocations greater than 100%
...........................................................
22
To summarize
.....................................................................................
96
How the dispersion of outcomes affects geometric growth
...............
23
Application to Stock Trading
.............................................................
97
The Fundamental Equation of trading
...............................................
24
A Closing Comment
..........................................................................
97
APPENDIX A - The Chi-Square Test
....................................................
98
..............................................................................................
26
APPENDIX B - Other Common Distributions
......................................
99
Optimal f for small traders just starting out
.......................................
26
The Uniform Distribution
..................................................................
99
Threshold to geometric
......................................................................
26
The Bernouli Distribution
................................................................
100
One combined bankroll versus separate bankrolls
.............................
27
The Binomial Distribution
...............................................................
100
Threat each play as if infinitely repeated
...........................................
28
The Geometric Distribution
.............................................................
101
28
The Hypergeometric Distribution
....................................................
101
The Poisson Distribution
..................................................................
102
..........................................................................................................
29
The Exponential Distribution
...........................................................
102
Comparing trading systems
................................................................
30
The Chi-Square Distribution
............................................................
103
Too much sensivity to the biggest loss
..............................................
30
The Student's Distribution
................................................................
103
Equalizing optimal f
...........................................................................
31
The Multinomial Distribution
..........................................................
104
Dollar averaging and share averaging ideas
......................................
32
.......................................................
104
The Arc Sine Laws and random walks
..............................................
33
Time spent in a drawdown
.................................................................
34
.........................................................................................
106
Chapter 3 - Parametric Optimal f on the Normal Distribution
...............
35
The basics of probability distributions
...............................................
35
Descriptive measures of distributions
................................................
35
Moments of a distribution
..................................................................
36
The Normal Distribution
....................................................................
37
The Central Limit Theorem
...............................................................
38
Working with the Normal Distribution
..............................................
38
Normal Probabilities
..........................................................................
39
Further Derivatives of the Normal
.....................................................
41
The Lognormal Distribution
..............................................................
41
The parametric optimal f
....................................................................
42
The distribution of trade P&L's
..........................................................
43
Finding optimal f on the Normal Distribution
...................................
44
The mechanics of the procedure
........................................................
45
Chapter 4 - Parametric Techniques on Other Distributions
...................
49
The Kolmogorov-Smirnov (K-S) Test
...............................................
49
Creating our own Characteristic Distribution Function
.....................
50
Fitting the Parameters of the distribution
...........................................
52
Using the Parameters to find optimal f
..............................................
54
Performing "What Ifs"
.......................................................................
56
Equalizing f
........................................................................................
56
Optimal f on other distributions and fitted curves
.............................
56
Scenario planning
...............................................................................
57
Optimal f on binned data
....................................................................
60
Which is the best optimal f?
...............................................................
60
- 3 -
Multiple simultaneous positions with a random relationship
...........
.....
Efficiency loss in simultaneous wagering or portfolio trading
..........
The stable Paretian Distribution
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
- 4 -
Introduction
Readers will find this book to be more abstruse than its forerunner.
Hence, this is not a book for beginners. Many readers of this text will
have read Portfolio Management Formulas . For those who have not,
Chapter 1 of this book summarizes, in broad strokes, the basic concepts
from Portfolio Management Formulas . Including these basic concepts
allows this book to "stand alone" from Portfolio Management Formu-
las .
SCOPE OF THIS BOOK
I wrote in the first sentence of the Preface of Portfolio Manage-
ment Formulas , the forerunner to this book, that it was a book about
mathematical tools.
This is a book about machines.
Here, we will take tools and build bigger, more elaborate, more
powerful tools-machines, where the whole is greater than the sum of the
parts. We will try to dissect machines that would otherwise be black
boxes in such a way that we can understand them completely without
having to cover all of the related subjects (which would have made this
book impossible). For instance, a discourse on how to build a jet engine
can be very detailed without having to teach you chemistry so that you
know how jet fuel works. Likewise with this book, which relies quite
heavily on many areas, particularly statistics, and touches on calculus. I
am not trying to teach mathematics here, aside from that necessary to
understand the text. However, I have tried to write this book so that if
you understand calculus (or statistics) it will make sense and if you do
not there will be little, if any, loss of continuity, and you will still be
able to utilize and understand (for the most part) the material covered
without feeling lost.
Certain mathematical functions are called upon from time to time in
statistics. These functions-which include the gamma and incomplete
gamma functions, as well as the beta and incomplete beta functions-are
often called functions of mathematical physics and reside just beyond
the perimeter of the material in this text. To cover them in the depth nec-
essary to do the reader justice is beyond the scope, and away from the
direction of, this book. This is a book about account management for
traders, not mathematical physics, remember? For those truly interested
in knowing the "chemistry of the jet fuel" I suggest Numerical Recipes,
which is referred to in the Bibliography.
I have tried to cover my material as deeply as possible considering
that you do not have to know calculus or functions of mathematical
physics to be a good trader or money manager. It is my opinion that
there isn't much correlation between intelligence and making money in
the markets. By this I do not mean that the dumber you are the better I
think your chances of success in the markets are. I mean that intelli-
gence alone is but a very small input to the equation of what makes a
good trader. In terms of what input makes a good trader, I think that
mental toughness and discipline far outweigh intelligence. Every suc-
cessful trader I have ever met or heard about has had at least one experi-
ence of a cataclysmic loss. The common denominator, it seems, the
characteristic that separates a good trader from the others, is that the
good trader picks up the phone and puts in the order when things are at
their bleakest. This requires a lot more from an individual than calculus
or statistics can teach a person.
In short, I have written this as a book to be utilized by traders in the
real-world marketplace. I am not an academic. My interest is in real-
world utility before academic pureness.
Furthermore, I have tried to supply the reader with more basic infor-
mation than the text requires in hopes that the reader will pursue con-
cepts farther than I have here.
One thing I have always been intrigued by is the architecture of mu-
sic -music theory. I enjoy reading and learning about it. Yet I am not a
musician. To be a musician requires a certain discipline that simply un-
derstanding the rudiments of music theory cannot bestow. Likewise with
trading. Money management may be the core of a sound trading pro-
gram, but simply understanding money management will not make you
a successful trader.
This is a book about music theory, not a how-to book about playing
an instrument. Likewise, this is not a book about beating the markets,
and you won't find a single price chart in this book. Rather it is a book
about mathematical concepts, taking that important step from theory to
application, that you can employ. It will not bestow on you the ability to
tolerate the emotional pain that trading inevitably has in store for you,
win or lose.
This book is not a sequel to Portfolio Management Formulas .
Rather, Portfolio Management Formulas laid the foundations for what
will be covered here.
Many of the ideas covered in this book are already in practice by
professional money managers. However, the ideas that are widespread
among professional money managers are not usually readily available to
the investing public. Because money is involved, everyone seems to be
very secretive about portfolio techniques. Finding out information in
this regard is like trying to find out information about atom bombs. I am
indebted to numerous librarians who helped me through many mazes of
professional journals to fill in many of the gaps in putting this book to-
gether.
This book does not require that you utilize a mechanical, objective
trading system in order to employ the tools to be described herein. In
other words, someone who uses Elliott Wave for making trading deci-
sions, for example, can now employ optimal f.
However, the techniques described in this book, like those in Port-
folio Management Formulas , require that the sum of your bets be a
positive result. In other words, these techniques will do a lot for you, but
they will not perform miracles. Shuffling money cannot turn losses into
profits. You must have a winning approach to start with.
Most of the techniques advocated in this text are techniques that are
advantageous to you in the long run. Throughout the text you will en-
counter the term "an asymptotic sense" to mean the eventual outcome of
something performed an infinite number of times, whose probability ap-
proaches certainty as the number of trials continues. In other words,
something we can be nearly certain of in the long run. The root of this
expression is the mathematical term "asymptote," which is a straight line
considered as a limit to a curved line in the sense that the distance be-
tween a moving point on the curved line and the straight line approaches
zero as the point moves an infinite distance from the origin.
Trading is never an easy game. When people study these concepts,
they often get a false feeling of power. I say false because people tend to
get the impression that something very difficult to do is easy when they
understand the mechanics of what they must do. As you go through this
text, bear in mind that there is nothing in this text that will make you a
better trader, nothing that will improve your timing of entry and exit
from a given market, nothing that will improve your trade selection.
These difficult exercises will still be difficult exercises even after you
have finished and comprehended this book.
Since the publication of Portfolio Management Formulas I have
been asked by some people why I chose to write a book in the first
place. The argument usually has something to do with the marketplace
being a competitive arena, and writing a book, in their view, is analo-
gous to educating your adversaries.
The markets are vast. Very few people seem to realize how huge to-
day's markets are. True, the markets are a zero sum game (at best), but
as a result of their enormity you, the reader, are not my adversary.
Like most traders, I myself am most often my own biggest enemy.
This is not only true in my endeavors in and around the markets, but in
life in general. Other traders do not pose anywhere near the threat to me
that I myself do. I do not think that I am alone in this. I think most
traders, like myself, are their own worst enemies.
In the mid 1980s, as the microcomputer was fast becoming the pri-
mary tool for traders, there was an abundance of trading programs that
entered a position on a stop order, and the placement of these entry stops
was often a function of the current volatility in a given market. These
systems worked beautifully for a time. Then, near the end of the decade,
these types of systems seemed to collapse. At best, they were able to
carve out only a small fraction of the profits that these systems had just
a few years earlier. Most traders of such systems would later abandon
them, claiming that if "everyone was trading them, how could they work
anymore?"
Most of these systems traded the Treasury Bond futures market.
Consider now the size of the cash market underlying this futures market.
Arbitrageurs in these markets will come in when the prices of the cash
and futures diverge by an appropriate amount (usually not more than a
few ticks), buying the less expensive of the two instruments and selling
- 5 -
 
 

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