Risk-Return Analysis Volume 3

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[T]he race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.

—Ecclesiastes 9:11

Excuse me sir, but

How much longer would it take

If we did it right

The first time?

—Harry Max Markowitz

Life’s tragedy is that

We get old too soon

And wise too late.

—Benjamin Franklin

If not now, when?

If not me, who?

—Hillel, the Elder

CONTENTS

Preface

The Rational Decision Maker

Words of Wisdom

John von Neumann

Acknowledgments

13. Predecessors

Introduction

René Descartes

There Is No Is, Only Was and Will Be

Working Hypotheses

RDM Reasoning

David Hume

Eudaimonia

Financial Economic Discoveries

Economic Analyses That Have Stood the Test of Time

Constructive Skepticism

Isaac Newton, Philosopher

Fields Other Than Physics

Karl Popper

Mysticism

Caveats

Charles Peirce

Immanuel Kant

What an RDM Can Know A Priori

14. Deduction First Principles

Introduction

The Great Debate

One More Reason for Studying Cantor’s Set Theory

Very Few Understood It

Finite Cardinal Arithmetic

Relative Sizes of Finite Sets

Finite Ordinal Arithmetic

Standard Ordered Sets (SOSs)

Finite Cardinal and Ordinal Numbers

Cantor (101)

Theorem

Proof

Corollary

Proof

Transfinite Cardinal Numbers

The Continuum Hypothesis

Transfinite Cardinal Arithmetic

Lemma

Transfinite Ordinal Numbers

Examples of Well-Ordered and Not Well-Ordered Sets

Transfinite Ordinal Arithmetic

Extended SOSs

Lemma

Proof

The Paradoxes (a.k.a. Antimonies)

Three Directions

From Aristotle to Hume to Hilbert

British Empiricism versus Continental Rationalism

Who Created What?

Cantor Reconsidered

Brouwer’s Objections

Axiomatic Set Theory

Peano’s Axioms (PAs)

Hilbert’s Programs

Whitehead and Russell

Zermelo’s Axioms

The Axiom of Choice

The Trichotomy Equivalent to the Axiom of Choice

Kurt Gödel (1906–1978)

Thoralf Skolem (1887–1863)

15. Logic is Programming is Logic

Introduction

Terminology

Number Systems and the EAS Structures Built on Them

Deductive Systems as Programming Languages

A Variety of Deductive DSSs

Alternative Rules of Inference

Ladders and Fire Escapes

Organon 2000: From Ancient Greek to Symbolic Logic

So, What’s New?

Immediate Consequences

Two Types of Set Ownership

Modeling Modeling

EAS-E Deduction: Status

16. The Infinite and The Infinitesimal

Points and Lines

Fields

Constructing the Infinitesimals

Infinite-Dimensional Utility Analysis

The Algebraic Structure Called A Field

17. Induction Theory

Introduction

The Story Thus Far

Concepts

Basic Relationships

Examples

Objective Probability

The Formal M59 Model

Initial Consequences

Bayes’s Rule

A Bayesian View of MVA

Judgment, Approximation and Axiom III

(1) A Philosophical Difference between S54 and M59

Examples of Clearly Objective Probabilities"

Propositions about Propositions

A Problem with Axiom II

Are the πj Probabilities the Scaling of the πj?

The πj Mix on a Par with Objective Probabilities

18. Induction Practice

Introduction

R. A. Fisher and Neyman-Pearson Hypothesis Tests

The Likelihood Principle

Andrei Kolmogorov

A Model of Models

The R.A. Fisher Argument

Bayesian Conjugate Prior Procedures

19. Eudaimonia

Review

Eudaimonia for the Masses

Notes

References

Index

PREFACE

This third volume of a four-volume book, Risk-Return Analysis: The Theory and Practice of Rational Investing, begins with Chapter 13. The subject matter of the present chapter was a given; the size of the actual chapter needed to adequately cover the topic, was a variable. It turned out that the requisite size was that of a small monograph. Chapter 14 is even larger, and is still very much in process. It was therefore deemed best to first publish Chapter 13 on its own so that interested readers could begin the study of its contents.

As explained in the prefatory material of Volume I, the topics of the four volumes parallel the topics of the last four chapters of Markowitz (1959). Specifically, Volume I considers single-period decision analyses assuming known odds as did Markowitz (1959) Chapter 10. Volume II considers many-period analyses, still assuming known odds, as did Markowitz (1959) Chapter 11. The present volume is concerned with rational decision-making in the face of uncertainty, i.e., when odds are not known, as did Markowitz (1959) Chapter 12. The fourth volume will cover application matters not covered, or not adequately covered, in the prior three volumes. The chief difference between Chapter 13 of Markowitz (1959) and the planned Volume IV of this book, is that Markowitz (1959) Chapter 13 was written a priori, when the world had no experience with the practical application of MVA (Mean Variance Analysis), whereas Volume IV of the present book is based on over a half-century of MVA experience.

Each volume of this book distinguishes between the actions of a hypothetical Rational Decision Maker (RDM) and those of a Human Decision Maker (HDM). Our focus is on rational—i.e., RDM—action, and HDM approximations to it. The reason that the HDM makes an appearance in these volumes at all, is because (as the subtitle of this book promises) the book deals with practice as well as theory. I do not mean by this the questionable practices documented in behavioral studies such as that of Odean (1999) or McKay (2013, 1841). Rather, I mean existing and proposed HDM practices that seek to emulate the behavior, and approximate the results, of the RDM. Discussions of this sort are scattered throughout the volumes of this book, including, but not only, Chapter 11 in Volume II, on Judgment and Approximation.

THE RATIONAL DECISION MAKER

The RDM is a fictitious creation. Like other fictitious creations—such as the unicorn, or the fictional as opposed to the historical St. Nicholas—each storyteller can ascribe to it whatever attributes his or her audience can be persuaded are appropriate for the entity. The RDM of Mossin and Samuelson, discussed in Chapter 9, is a solitary investor. As Chapter 6 (in Volume II) spelled out in some detail, my RDM can be an individual (of either gender of course), a family, or an institution such as a university or symphony orchestra.

As a specific example, the frontice material of Volume II introduced a prototypical RDM as a family with husband, wife, four children, a dog, and a cat—all rational. But, as Markowitz (1959) emphasized, a rational entity is neither omniscient nor omnipotent. In particular, the rational cat cannot catch the irrational rat. This is perfectly plausible despite the fact that a hungry natural cat can eventually catch some natural rat. Presumably, our rational—but fussy, of course—cat would prefer to finish the tuna in its bowl than to eat a raw rat. Its moves toward catching the rat are perfunctory. Perhaps the Rational Cat (RC) does not really want to catch the Irrational Rat (IR) because it is a clean and sanitary white rat—saved by RC’s good-hearted family from its fate in a research lab. Perhaps playing cat and mouse with IR is RC’s only entertainment when the humanoid RDMs are out for the day, and either the RC or the Rational Dog (RD) is tired of playing hide and seek.

WORDS OF WISDOM

A friend of mine told me that when he got his copy of Volume II, he read my Ode to Rationality to his family, which they enjoyed, but they wondered why I chose a quote from the Rubàiyàt as my other offering. I explained why, and decided that it might be useful to the reader if I explained the reasons for my various frontice-page quotes.

The frontice page of Volume I contains an acknowledgment of the various members of Mrs. Markowitz’s (the other Dr. Markowitz in our house) and my extensive family. Since then, our family has become even more extensive. Specifically, we now have a twentieth great grandchild, namely a boy whose first name is Max, as is my middle name.

The frontice material of Volume II reflects its contents. As compared to Volume I, concerned with single-period decisions with known odds, Volume II adds a time dimension, as well as a risk dimension, to the analysis. It seemed to me that the quote from the Rubàiyàt perfectly captured the concepts of time and the dangers of the real world. Its somber tone seemed to be a nice balance to the light-hearted Ode. The implication of the pair of quotes is this: the world is a dangerous place, but try not to lose your sense of humor. As already noted, Volume III is concerned with rational action under uncertainty. Accordingly, it considers questions such as: what kinds of things can we know, how do we acquire such knowledge, and how should we act on the basis of this knowledge. In particular, how should one go about playing a successful game-of-life, and what’s in for us to play our game-of-life in what is generally considered a moral manner. Such philosophical topics call for some philosophical quotes.

The first quote is from Ecclesiastes. It starts with [T]he race is not to the swift, and ends with But time and chance happeneth to them all. That could be the slogan of this entire, four-volume book.

The second offering on the frontice page concerns doing things right the first time. Doing it right does not mean insisting on perfection. It means not achieving appreciably less than time and resource constraints permit because of impatience. In the case of Chapter 13, to a large extent doing it right meant doing it twice. Fortunately, I am blessed that Tony Batman (the CEO of 1st Global of Dallas, Texas, and sponsor of this book) and I share a common view of doing the present project right.

The Benjamin Franklin quote is sometimes paraphrased as

So soon old,

So late smart.

But the Franklin quote speaks of becoming wise rather than becoming smart. One must distinguish smartness from wiseness, just as one must distinguish wisdom from knowledge. One speaks of words of wisdom and volumes of knowledge, rather than words of knowledge and volumes of wisdom. Note, however, that there is no wisdom without knowledge. Wisdom is only the final summing up: the inner insight.

The Hillel quote reminds me of the story of the Jewish sociology professor who sought the reason for a common Jewish mannerism. He asked his rabbi, who sent the professor to an older and wiser rabbi, who sent him to a still older and wiser rabbi, etc., until the professor sat, in the old country, in the presence of the oldest and wisest rabbi. The professor asked, Wise old rabbi, why is it that when you ask a Jew a question, he answers with another question? The wise old rabbi transfixed the professor with a pointing finger and an intense gaze, and said, Why not?

The use by Jews of questions for purposes other than interrogation can have serious objectives. The quote form Hillel asked when and who. The answer is not always me, right now. For example, the first Monday of December, my colleagues (Mary Midge McDonald and Lilli Alexander) and I meet to discuss which tasks must be done before December 15 (when Christmas cards should be in the mail), which before December 25, which before December 31, and which may be done the following year. We also make sure that everyone knows who will do what. In addition to who and when other important questions include what, how, and why. The use by Jews of questions for noninterrogatory purposes is ancient. In particular, Hillel, the Elder, died in 10 A.D., and is said to have lived 120 years, just like Moses. Conceivably, the noninterrogatory use of questions had been passed down from biblical times.

Benjamin Franklin did not offer wisdom in the form of questions, but he knew the value of questions of various sorts. He had a saying, which I will amplify a bit, to the effect that

He who knows how,

Will be supervised by

He who knows what and when;

Who will be employed by

He who knows why.

JOHN VON NEUMANN

How has this book stood on von Neumann’s shoulders? Let me count the ways:

Volume I centered on mean-variance approximations to expected utility. The expected utility max was proposed by Daniel Bernoulli (1954, 1738) as an alternative to the clearly faulty expected return, or expected win or loss, rule. But it remained only a plausible heuristic until von Neumann and Morgenstern (1944) presented an axiomatic justification for it.

This set of axioms opened a floodgate of applications, extensions and reexaminations. Volume I, as just noted, seeks to establish the range of applicability of MV analysis by examining theoretically and empirically the ability for such analysis to approximate the expected values of various utility functions, including Bernoulli’s logarithmic utility.

The natural extension of von Neumann’s axiomatic justification of the utility analysis of action in the face of risk (known odds) is the L. J. Savage (1954) axiomatic defense of the EU/PB (expected return/personal probability) maximum for choice under uncertainty.

This, in turn, produced the Bayesian revolution which has completely turned around most scholar’s view of statistical inference, both formal—the way people work with numbers—and informal—the way many (including me) view any learning process. The other consequence of von Neumann’s defense of expected utility has been the re-creation of alternative axiom systems deemed superior as prescriptive of desirable behavior, or descriptive of risk-facing behavior, such as Kahneman and Tversky’s (1979) prospect theory, from which Behavior Finance" sprang. Also, Allais’ (1953) was the first to question the conclusions and therefore the premises, of von Neumann’s analysis. This resulted in hierarchical utility, to be discussed at length in Chapter 15.

Volume II is centered on Decision Support Systems (DSSs) which help the family—or other enterprise—play its Game of Life. Of course "Game here refers to von Neumann’s Game Theory," as published in von Neumann (1928), and more completely in von Neumann and Morgenstern (1944). I consider the latter to be the fourth modern economic paradigm, including those of Adam Smith (2003, 1776), Karl Marx (2010, 1867), John Maynard Keynes (2011, 1936). Von Neumann’s contribution to economics is not confined to the game theory. It also includes optimum growth paths (von Neumann 1945–1946). Von Neumann made frequent visits to the Cowles Commission for Research in Economics, unfortunately for me, long before I arrived at Cowles. In particular, at the 50th Anniversary of the Cowles Commission, Lawrence Klein reviewed recent work on models of the U. S. and other economics. He was asked how his models differed from his Nobel-Prize winning work reported in Klein (1950). Klein described how the modern computer made all the difference. Rather than having equations to characterize the entire U. S. economy, he now had hundreds of equations and variables to trace out micro-relationships.

At the point Tjalling Koopmans signaled, in his quiet way, that he would like to say a few words. Koopmans reminisced that during WWII, von Neumann would frequently travel by train from Washington, D.C. to Los Alamos—with a stop-over in Chicago, so that von Neumann could visit the Cowles Commission. The latter would expect von Neumann to share with them his latest insights into game theory or growth optimal paths. Instead, von Neumann told them about a machine he was working on that could do thousands of calculations per minute. According to Koopmans, no one at the Cowles Commission could see any way that a machine that could do thousands of calculations per minute could ever be of any use to economics.

ACKNOWLEDGMENTS

I would very much like to thank those who sponsored this work and/or read it cover-to-cover and advised on its contents. These include Tony Batman, Ken Blay, Roger Brown, John Guerard and Alasdar Mullarney.

I also want to express my thanks to and love for my wife Barbara. Barbara is suffering from the after-effects of a stroke, but otherwise we are both in our nineties and in good health. She still recognizes me. In particular, after dinner each night (she needs to be fed of course), I take her hand and ask her if she recognizes me, her husband Harry. She nods yes and we reminisce. She cannot speak because of the stroke but nods her head and acknowledges with her eyes.

13

PREDECESSORS

INTRODUCTION

As explained in the Preface, this third volume of a four-volume book presents the theory of rational decision-making when odds are not known. Its central concern is how to go rationally from information to action. Here I use information in the broadest sense, including, e.g., not only computerized data (with errors of course), but also news reports of varying credibility and relevant third-party analyses whose methodology may be well-documented or