资金管理数学MathematicsofmoneymanagementVinceRalph

1、1 / 1427THE MATHEMATICS OF MONEY MANAGEMENT:RISK ANALYSIS TECHNIQUES FOR TRADERSby Ralph VincePublished by John Wiley & Sons, Inc.Library of Congress Cataloging-in-Publication DataVince. Ralph. 1958-The mathematics of money management: risk analysis techniques for traders / by Ralph Vince.Includ

2、es bibliographical references and index.ISBN 0-471-54738-71. Investment analysisMathematics. 2. Risk managementMathematics3. Program trading (Securities)HG4529N56 1992 332.60151-dc20 91-33547Preface and DedicationThe favorable reception of Portfolio Management Formulas exceeded even the greatest exp

3、ectation 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

4、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 contribute

5、d 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 Ba

6、rtlett 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

7、in one form or another.This book has left me utterly drained, and I intend it to be my last.Considering this, Id like to dedicate it to the three people who have influenced me the most. To Rejeanne, my mother, for teaching me to appreciate a vivid imagination; to Larry, my father, for showing me at

8、an early age how to squeeze numbers to make them jump; to Arlene, my wife, partner, and best friend. This book is for all three of you. Your influences resonate throughout it.Chagrin Falls, OhioR. V.March 1992IndexIntroduction.5Scope of this book.5Some prevalent misconceptions.6Worst-case scenarios

9、and stategy .6Mathematics notation.7Synthetic constructs in this text.7Optimal trading quantities and optimal f.8Chapter 1-The Empirical Techniques .9Deciding on quantity.9Basic concepts.9The runs test.10Serial correlation.11Common dependency errors.12Mathematical Expectation .13To reinvest trading

10、profits or not.14Measuring a good system for reinvestment the Geometric Mean.14How best to reinvest .15Optimal fixed fractional trading .15Kelly formulas .16Finding the optimal f by the Geometric Mean.16To summarize thus far .17Geometric Average Trade.17Why you must know your optimal f.18The severit

11、y of drawdown.18Modern portfolio theory.19The Markovitz model.19The Geometric Mean portfolio strategy.21Daily procedures for using optimal portfolios.21Allocations greater than 100%.22How the dispersion of outcomes affects geometric growth.23The Fundamental Equation of trading.24Chapter 2 - Characte

12、ristics of Fixed Fractional Trading and Salutary Techniques26Optimal f for small traders just starting out.26Threshold to geometric.26One combined bankroll versus separate bankrolls .27Threat each play as if infinitely repeated.28Efficiency loss in simultaneous wagering or portfolio trading.28Time r

13、equired to reach a specified goal and the trouble with fractional f29Comparing trading systems .30Too much sensivity to the biggest loss.30Equalizing optimal f.31Dollar averaging and share averaging ideas.32The Arc Sine Laws and random walks.33Time spent in a drawdown.34Chapter 3 - Parametric Optima

14、l f on the Normal Distribution.35The basics of probability distributions.35Descriptive measures of distributions.35Moments of a distribution.36The Normal Distribution.37The Central Limit Theorem.38Working with the Normal Distribution.38Normal Probabilities.39Further Derivatives of the Normal.41The L

15、ognormal Distribution.41The parametric optimal f.42The distribution of trade P&Ls .43Finding optimal f on the Normal Distribution.44The mechanics of the procedure.45Chapter 4 - Parametric Techniques on Other Distributions.49The Kolmogorov-Smirnov (K-S) Test.49Creating our own Characteristic Dist

16、ribution Function.50Fitting the Parameters of the distribution .52Using the Parameters to find optimal f.54Performing What Ifs.56Equalizing f.56Optimal f on other distributions and fitted curves.57Scenario planning .57Optimal f on binned data .60Which is the best optimal f? .60Chapter 5 - Introducti

17、on to Multiple Simultaneous Positions under the Parametric Approach62Estimating Volatility.62Ruin, Risk and Reality.63Option pricing models .63A European options pricing model for all distributions .66The single long option and optimal f.67The single short option .70The single position in The Underl

18、ying Instrument.71Multiple simultaneous positions with a causal relationship .71Multiple simultaneous positions with a random relationship.73Chapter 6 - Correlative Relationships and the Derivation of the Efficient Frontier74Definition of The Problem.74Solutions of Linear Systems using Row-Equivalen

19、t Matrices .77Interpreting The Results.78Chapter 7 - The Geometry of Portfolios.81The Capital Market Lines (CMLs) .81The Geometric Efficient Frontier .82Unconstrained portfolios.84How optimal f fits with optimal portfolios.85Threshold to The Geometric for Portfolios.86Completing The Loop.86Chapter 8

20、 - Risk Management.89Asset Allocation.89Reallocation: Four Methods.91Why reallocate? .93Portfolio Insurance The Fourth Reallocation Technique.93The Margin Constraint.96Rotating Markets.97To summarize .97Application to Stock Trading.98A Closing Comment .98APPENDIX A - The Chi-Square Test.100APPENDIX

21、B - Other Common Distributions.101The Uniform Distribution.101The Bernouli Distribution.101The Binomial Distribution.102The Geometric Distribution.103The Hypergeometric Distribution.104The Poisson Distribution .104The Exponential Distribution.105The Chi-Square Distribution.105The Students Distributi

22、on .106The Multinomial Distribution.107The stable Paretian Distribution .107APPENDIX C - Further on Dependency: The Turning Points and Phase Length Tests109IntroductionSCOPE OF THIS BOOKI wrote in the first sentence of the Preface of Portfolio Management Formulas, the forerunner to this book, that i

23、t 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

24、 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

25、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 sens

26、e 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 g

27、amma 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 necessary to do the reader justice is beyond the scope, and away fro

28、m 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 deepl

29、y 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 isnt much correlation between intelligence and making money in the markets. By this I do not mean that the dumber you are the better

30、I think your chances of success in the markets are. I mean that intelligence 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 successful trader I have

31、 ever met or heard about has had at least one experience 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 m

32、ore 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 wi

33、th more basic information than the text requires in hopes that the reader will pursue concepts farther than I have here.One thing I have always been intrigued by is the architecture of music -music theory. I enjoy reading and learning about it. Yet I am not a musician. To be a musician requires a ce

34、rtain discipline that simply understanding the rudiments of music theory cannot bestow. Likewise with trading. Money management may be the core of a sound trading program, but simply understanding money management will not make you a successful trader.This is a book about music theory, not a how-to

35、book about playing an instrument. Likewise, this is not a book about beating the markets, and you wont 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 ab

36、ility 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.Readers will find this book to be more abstruse than its

37、 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

38、 book to stand alone from Portfolio Management Formulas.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

39、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 putt

40、ing this book together.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 decisions, for example, can now employ optimal f.However, the techniques des

41、cribed in this book, like those in Portfolio 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 s

42、tart 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 encounter the term an asymptotic sense to mean the eventual outcome of something performed an infinite number of times, whose probability approaches certa

43、inty 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 between a moving point on the

44、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 d

45、o 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 select

46、ion. 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 wi

47、th the marketplace being a competitive arena, and writing a book, in their view, is analogous to educating your adversaries.The markets are vast. Very few people seem to realize how huge todays markets are. True, the markets are a zero sum game (at best), but as a result of their enormity you, the r

48、eader, 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 th

49、ink most traders, like myself, are their own worst enemies.In the mid 1980s, as the microcomputer was fast becoming the primary 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 cur

50、rent 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

51、 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 price

52、s 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 the more expensive. As a result, the divergence between the price of cash and futures will dissipate in short order. The only time that the rel

53、ationship between cash and futures can really get out of line is when an exogenous shock, such as some sort of news event, drives prices to diverge farther than the arbitrage process ordinarily would allow for. Such disruptions are usually very short-lived and rather rare. An arbitrageur capitalizes

54、 on price discrepancies, one type of which is the relationship of a futures contract to its underlying cash instrument. As a result of this process, the Treasury Bond futures market is intrinsically tied to the enormous cash Treasury market. The futures market reflects, at least to within a few tick

55、s, whats going on in the gigantic cash market. The cash market is not, and never has been, dominated by systems traders. Quite the contrary.Returning now to our argument, it is rather inconceivable that the traders in the cash market all started trading the same types of systems as those who were ma

56、king money in the futures market at that time! Nor is it any more conceivable that these cash participants decided to all gang up on those who were profiteering in the futures market, There is no valid reason why these systems should have stopped working, or stopped working as well as they had, simp

57、ly because many futures traders were trading them. That argument would also suggest that a large participant in a very thin market be doomed to the same failure as traders of these systems in the bonds were. Likewise, it is silly to believe that all of the fat will be cut out of the markets just bec

58、ause I write a book on account management concepts.Cutting the fat out of the market requires more than an understanding of money management concepts. It requires discipline to tolerate and endure emotional pain to a level that 19 out of 20 people cannot bear. This you will not learn in this book or

59、 any other. Anyone who claims to be intrigued by the intellectual challenge of the markets is not a trader. The markets are as intellectually challenging as a fistfight. In that light, the best advice I know of is to always cover your chin and jab on the run. Whether you win or lose, there are signi

60、ficant beatings along the way. But there is really very little to the markets in the way of an intellectual challenge. Ultimately, trading is an exercise in self-mastery and endurance. This book attempts to detail the strategy of the fistfight. As such, this book is of use only to someone who already possesses the