The rise of the quants marschak, sharpe, black, scholes and merton repost

Contents Preface to the Great Minds in Finance series viii Part I Jacob Marschak Part II William Forsyth Sharpe, John Lintner, Jan Mossin, and Jack Treynor Part III Fischer Black and M

The Rise of the Quants

Great Minds in Finance

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This series explores the lives and times, theories and applications of those who have contributed most significantly to the formal study of finance It aims to bring to life the theories that are the foundation of modern finance, by examining them within the context of the historical backdrop and the life stories and characters of the “great minds” behind them

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The Rise of the Quants

Marschak, Sharpe, Black, Scholes, and Merton

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Contents

Preface to the Great Minds in Finance series viii

Part I Jacob Marschak

Part II William Forsyth Sharpe, John Lintner,

Jan Mossin, and Jack Treynor

Part III Fischer Black and Myron Scholes

Part IV Robert Merton

21 Applications 157

Part V What We Have Learned

List of Figures

10.2 Various choices of risk and return along

Preface to the Great Minds

in Finance series

This series covers the gamut of the study of finance – from the significance

of financial decisions over time and through the cycle of one’s life to the ways in which investors balance reward and risk; from how the price of a security is determined to whether these prices properly reflect all available information – we will look at the fundamental questions and answers in finance We delve into theories that govern personal decision- making, those that dictate the decisions of corporations and other similar entities, and the public finance of government This will

be done by looking at the lives and contributions of the key players upon whose shoulders the discipline rests

By focusing on the great minds in finance, we draw together the concepts that have stood the test of time and have proven themselves

to reveal something about the way humans make financial decisions These principles, which have flowed from individuals, many of whom have been awarded the Nobel Memorial Prize in Economics for their insights (or perhaps shall be awarded some day), allow us to see the financial forest for the trees

The insights of these contributors to finance arose because these great minds were uniquely able to glimpse a familiar problem through

a wider lens From the greater insights provided by a more expansive view, they were able to focus upon details that have eluded previous scholars Their unique perspectives provided new insights that are the measure of their genius The giants who have produced the theories and concepts that drive financial fundamentals share one important charac-teristic: they have developed insights that explain how markets can be used or tailored to create a more efficient economy

The approach taken is one taught in our finance programs and practiced by fundamentals analysts We present theories to enrich and motivate our financial understanding This approach is in contrast to the tools of technicians formulated solely on capitalizing on market inefficiencies without delving too deeply into the very meaning of efficiency in the first place From a strictly aesthetic perspective, one cannot entirely condemn the tug- of- war of profits sought by the technicians, even if they do little to enhance – and may even detract from – efficiency The mathematics and physics of price movements

and the sophistication of computer algorithms is fascinating in its own right Indeed, my appreciation for technical analysis came from my uni-versity studies toward a Bachelor of Science degree in physics, followed immediately by a PhD in economics.

However, as I began to teach economics and finance, I realized that the analytic tools of physics that so pervaded modern economics have strayed too far from explaining this important dimension of human financial decision- making To better understand the interplay between the scientific method, economics, human behavior, and public policy,

I continued with my studies toward a Master of Accountancy in taxation, an MBA, and a Juris Doctor of Law

As I taught the economics of intertemporal choice, the role of money and financial instruments, and the structure of the banking and financial intermediaries, I recognized that my students had become increasingly fascinated with investment banking and Wall Street Meanwhile, the developed world experienced the most significant breakdown of financial markets in almost eight decades I realized that this once- in- a- lifetime global financial meltdown arose because we had moved from an economy that produced things to one in which, by

2006, generated a third of all profits in financial markets, with little

to show but pieces of paper representing wealth that had value only if some stood ready to purchase them

I decided to shift my research from academic research in esoteric fields of economics and finance and toward the contribution to a better understanding of markets by the educated public I began to write a regular business column and a book that documented the unraveling of

the Great Recession The book, entitled Global Financial Meltdown: How

We Can Avoid the Next Economic Crisis, described the events that gave

rise to the most significant economic crisis in our lifetime I followed

that book with The Fear Factor, which explained the important role of

fear as a sometimes constructive and at other times destructive ence in our financial decision- making I then wrote a book on why

influ-many economies at first thrive and then struggle to survive in The Rise

and Fall of an Economic Empire Throughout, I try to impart to you,

the educated reader, the intuition and the understanding that would,

at least, help you to make informed decisions in increasingly volatile global economies and financial markets

As I describe the theories that form the foundations of modern finance, I show how individuals born without great fanfare can come

to be regarded as geniuses within their own lifetime The lives of each

Preface to the Great Minds in Finance series ix

of the individuals examined in this series became extraordinary, not because they made an unfathomable leap in our understanding, but rather because they looked at something in a different way and caused

us all thereafter to look at the problem in this new way That is the test

of genius

x Preface to the Great Minds in Finance series

1

Introduction

This book is the third in a series of discussions about the great minds

in the history and theory of finance Each volume addresses the tributions of brilliant individuals to our understanding of financial decisions and markets

con-The first in the series began by establishing a framework upon which all subsequent discussions rest It discussed how individuals make decisions over time and why these decisions change as we age and our circumstances change The expansion of traditional economic models

to decision- making across time created the foundations of finance The early financial theorists, which included Irving Fischer, Frank Ramsey, John Maynard Keynes, Franco Modigliani, Milton Friedman, and others, recognized that the static time- independent models of classical economics were ill- equipped to describe how households balance the present and the future This topic of the first volume is variously called intertemporal choice by microeconomists and the Life Cycle Model by macroeconomists and financial theorists To you and me, it explains why we expect to earn interest on our investment even if we take little risk It also predicts why some of us conclude that the prevailing interest rate in a financial market presents a good opportunity to save, while others of a different ilk take the same interest rate as a good opportunity

to borrow

The inclusion of risk and uncertainty

Yet, however valuable were the contributions of the Life Cyclists to our understanding of savings over our lifetime, the inclusion of risk and uncertainty in our models remained elusive Then, in a dramatic explo-sion of theory over 15 years between 1937 and 1952, finance theory

2 The Rise of the Quants

went from a stage of infancy to a tool that allowed us to establish and understand equilibrium Finance began to incorporate the insight

of Frank Knight that there can be an unknown but often calculable estimate of risk that pervades an entire economic system, or systematic risk, and risk that affects a given security, labeled idiosyncratic or unsys-tematic risk The great mind John Maynard Keynes lamented:

[Under uncertainty] there is no scientific basis on which to form any calculable probability whatever We simply do not know Nevertheless, the necessity for action and for decision compels us as practical men

to do our best to overlook this awkward fact and to behave exactly

as we should if we had behind us a good Benthamite calculation of

a series of prospective advantages and disadvantages, each multiplied

by its appropriate probability waiting to be summed.1

The finance literature further clarified that there are calculable risks and that there are uncertainties that cannot be quantified In the 1930s, John von Neumann set about producing a model of expected utility that permitted the inclusion of risk Then, Leonard Jimmie Savage described how our individual perceptions affect the probability of uncertainty, and Kenneth Arrow was able to include these probabilities

of uncertainty in a model that established the existence of equilibrium

in a market for financial securities

With the existence of equilibrium and a better understanding of the meaning and significance of probability at hand, Harry Markowitz then packaged up these intuitions into a tidy set of insights we now call Modern Portfolio Theory In doing so, he demonstrated that an efficient portfolio minimizes and diversifies market risk through the choice of securities that take best advantage of the ways in which their returns are correlated with each other His notion of an efficient frontier of securities, the securities market line, provided new insights into how

an optimal finance portfolio could be developed Subsequently, James Tobin demonstrated how any individual’s preferred trade- off between risk and return could be met by a unique combination of a risk- free asset and a diversified mutual fund

However, while these insights were produced during the 1950s, suffi cient computing power necessary to apply them would not exist for another decade Despite the inability of the transition of these significant theoretical developments into practice, the great minds

of von Neumann, Savage, Arrow, and Markowitz created the sound theoretical framework from which finance could develop and establish

Introduction 3

itself on firm foundations as a discipline distinct from the questions most often posed in economics But, while von Neumann considered himself a physicist and mathematician, Savage published as a statis-tician, and Arrow at first considered himself an insurance theorist and then an economist, they all helped construct the foundations of finance Of the great minds that described financial equilibria, perhaps only Markowitz considered himself a financial theorist, at least when he was not a computer scientist concentrating on developing computing algorithms for the solving of financial problems or when he was not an operations research theorist

These great minds shared an important characteristic with all the great minds that came before them or would follow They each looked

at a familiar problem in a unique way and, through their unique perspective, produced stunning new insights Their insights also produced a new set of questions They provided a nascent finance literature with a new set of theoretical tools, but with little insight on how practitioners could use these tools The next task for the discipline

of finance would be to take these theoretical insights and determine how they could profitably price individual securities This is the topic

of the third book in the series

As we document the lives and times of these great minds, we answer the following questions: How was Harry Markowitz’s Modern Portfolio Theory extended to the pricing of a single security? And how could the insights of Markowitz’s PhD supervisor, Jacob Marschak, be used to quantify the unknown in a standard way so that risk could be priced? Finally, how can one better leverage and hedge his or her portfolio to reduce risk through the purchase of options, the instruments that derive their value from an underlying security? These are the questions that the pricing analysts sought to resolve

However, none of these great minds provided a satisfactory nation for how the price of individual securities evolve over time By the 1960s, the finance discipline was begging for a revolution that could turn the theoretical into the quantitative and practical To make this transition from theory to practice and to transform financial markets required a sequence of steps We shall discuss each of these steps in turn.

expla-First, the discipline had to quantify risk in a practical way Our first great mind in this volume, Jacob Marschak, proposed a measure of risk and a description of the return/risk trade- off based on what physicists then called the first and second moment of financial returns Physicists used such first and second moments to describe the center of gravity and the inertia of an object The discipline of finance used the same technique in what we now know as the mean and variance approach.With measures of the mean and variance at hand, scholars then described how these measures were used to price an individual security While we see that four scholars worked independently to develop the link between the mean return and the variance of a security and its mar-ket price, we will forever associate this new methodology of the Capital Asset Pricing Model (CAPM) with the great mind William Sharpe

A Roadmap to Resolve the Big Questions 5

However, while Sharpe’s insights helped us better understand how

an individual security is priced, the greatest need for the rapid pricing

of securities was in the derivatives market This new financial market, once the sleepy domain of farmers and food processors concerned about price stability for the future delivery of agricultural commodities, now represents an annual market value that rivals the combined size of the world’s economies There is now a much greater volume of trading

in these derivatives, in commodities futures and in options markets,

in credit default swaps and mortgage- backed securities, in foreign exchange futures and bond futures than in the traditional market for corporate securities Yet, before the publication of the theory from the great minds Fischer Black and Myron Scholes, we knew little about how

to price such financial derivatives

Meanwhile, Robert Merton, a disciple of the great mind Paul Samuelson, was rapidly extending the relatively static models of finance

to a dynamic context that more effectively included time In creating dynamic models of finance, he was able to more fully describe the evolu-tion of markets over time The techniques he developed, originally with the market for options in mind, even more clearly delineated finance from economics

By its very nature, finance must model the evolution of prices over time that is simply less relevant within the traditional study of econo-mics Consequently, finance now produces the most sophisticated models of dynamic decision- making and necessarily often requires the skills of those most adept at such dynamic modeling, produced by scholars trained as rocket scientists or applied mathematicians

Until we could create a science out of this financial art, financial derivatives markets could not develop fully, and myriad risks could not be hedged and traded efficiently However, with the advent of the Black- Scholes options pricing theory and its subsequent extensions, the options market burgeoned, primarily on the Chicago Board Options Exchange In addition, global financial markets traded amongst themselves, created mammoth global finance companies that became

too big to fail, and brought to the forefront concepts previously left

only to high financiers, until the failure of these markets affected us all and plunged the world into a global financial meltdown

Clearly, finance markets can be both blessings and curses However, there is no doubt that these financial markets benefited from the scien-tific tools of analyses and pricing that these great minds provided The genie is out of the bottle and financial worlds will never go back to

a more primitive and simplistic state The modern quants, and trillions

6 The Rise of the Quants

of dollars of financial investment each year, now rely on the pricing tools provided by William Sharpe, Fischer Black and Myron Scholes, and Robert Merton, based on the earlier foundational work of Jacob Marschak and a then obscure but brilliant French PhD student at the turn of the twentieth century named Louis Bachelier In our future, we shall inevitably rely even more on the products of these great minds

We will now turn to how the concepts came about and now affect us all so profoundly

Part I

Jacob Marschak

We can often discover the formative roots of one or two great insights that eventually culminated in a Nobel Prize for many of the great minds described in this series Others made brilliant observations or offered up techniques in finance with which they are forever associ-ated Jacob Marschak was different His legacy arose because he was

a bridge associated with so many that came before him who we now recognize as great contributors to finance, and so many that he inspired and subsequently became known as great minds themselves in finance

in the 1950s and 1960s Once one recognizes that Jacob Marschak was

a common denominator between the great minds of previous volumes that include Leonard Jimmie Savage and Milton Friedman, Kenneth Arrow and Harry Markowitz, and even Franco Modigliani, the root

of his influence on their work is compelling When we discover that Marschak made discoveries that were subtle and humble but were so timely and related to the essence of the work of William Sharpe, Fischer Black, and Myron Scholes, we must conclude that he was more than

a mentor of other great minds – he was a great mind himself We will begin with his story

3

The Early Years

Jacob Marschak was not at all unusual among the cadre of great minds that formed the discipline of finance in the first half of the twentieth century Like the families of Milton Friedman, Franco Modigliani, Leonard Jimmie Savage, Kenneth Arrow, John von Neumann, and Harry Markowitz, Marschak’s family tree was originally rooted in the Jewish culture and derived from the intellectually stimulating region of Eastern, Central and Southern Europe at the beginning of the twentieth century This region, comprising what is now Ukraine, Hungary, Poland, Romania, and parts of Italy, was under the influence of the Austro- Hungarian Empire in the late nineteenth and early twentieth centuries.The Austro- Hungarian Empire thrived from 1867 until the end of the First World War The region was multi- national, culturally diverse, and politically sophisticated, as its leaders tried to navigate the obvious problems of plurality created by such a geographically and culturally varied region It was perhaps this diversity, and the dual pride its citizens took in their own region but also, rather uncomfortably,

at times in the accomplishments of the Empire as a whole, that fueled its intellectual ambition The liberal, innovative, and progressive nation that such a diverse population forged exhibited rapid growth and industrialization, and spectacular intellectualization The Empire was in the shadow of the German Republic and Great Britain, and hence, like

a younger sibling, it tried harder

Budapest and Vienna were the intellectual capitals of the Hungarian Empire The intellectual tradition, combined with the strong belief in education among Jewish families, produced perhaps the great-est number of scientific geniuses for at least two generations in the early 1900s Jacob Marschak was a prime example of this tradition

Austro-10 The Rise of the Quants

Marschak was born in Kiev, Ukraine on July 23, 1898 His father was a jeweler in an upper- middle- class family that valued education and intel-lectual activism However, unlike Friedman, Arrow, Savage, Modigliani, von Neumann, and Markowitz, each of whom are documented in the first two volumes of this series and who otherwise shared some of his heritage, Marschak was not the eldest son Indeed, he was the last of five children and, like many youngest siblings, he had a rebellious streak in his youth

Marschak’s family did not practice their religion strictly Rather, they devoted their energies more into social issues, even if they were quite comfortable economically Marschak learned French and German from governesses, but had to be educated both at home and at the First Kiev School of Commerce when his Jewish heritage prevented him from being admitted into the local Gymnasium

Marschak was still a teenager when the Russian Revolution cally charged the region He had been active in the Marxist movement

politi-at the time and had even served briefly as the Minister of Labor in the short- lived revolutionary social democratic party, the Menshevik International caucus, in the Soviet republic of Terek He had been a hero

of the Revolution because of his youthful anti- Tsarist radicalism that had caused him to be imprisoned at the age of 18 until he was liberated after the overthrow of the Tsarist regime

Figure 3.1 The Marschak family tree

The Early Years 11

However, while Marschak could attribute his liberty to the new Bolshevik regime that replaced the Tsarists, he did not share the revo-lutionary zeal of the new Moscow- centric politburo His family was forced to flee growing unrest in Kiev, but he instead found himself embroiled in politics in their adopted home in the Northern Caucasus His activism in defense of an independent and democratic Caucasus again threatened his freedom as the new Soviet regime gained strength and consolidated its power over the regions He made the difficult deci-sion to leave Ukraine and continue his studies in another intellectual hotbed, first the University of Berlin, where he was exposed to statistical methods in economics, and then the University of Heidelberg in the German Republic

Marschak had discovered economics as an outlet for his superior analytic and statistical mind, just as economics was metamorphosing from a political economy to a decision science He received his PhD from Heidelberg in 1922 and became an academic vagabond as he held temporary teaching positions at a number of German universities At the same time, and to earn a steadier living, he wrote on economic

policy for a leading German newspaper, the Frankfurter Zeitung.

It was while writing for the newspaper that Marschak (still known

at this point as Jascha rather than by his anglicized name Jacob) met Marianne Kamnitzer They married in 1927 and had a daughter, Ann,

in 1928 and a son, Thomas, in 1930

Just a few years after the birth of his son, Marschak was again forced

to flee an omnipotent and unforgivingly ideological regime He had suffered the initially subtle oppression of Jewish intellectuals under the Hitler regime of his adopted land Once the Nazis took power

in 1933, he succumbed to academic consequences because of his heritage – he could not as a Jew be granted a permanent university position at a German university Consequently, he moved his family

to the University of Oxford in England and served as the director of the Oxford Institute of Statistics for four years, beginning in 1935, under the auspices of the funding of the Rockefeller Foundation based

in the USA

The Laura Spelman Rockefeller Memorial Foundation was endowed by

US industrialist John D Rockefeller, Jr The purpose of the Foundation was to promote research in the social and decision sciences, but it was also seen as a way to expose European scholars to American schools of thought The Foundation was run by Beardsley Ruml, an experimental psychology with a PhD from the University of Chicago Ruml espoused

a more clinical and empirical approach to the social sciences he

12 The Rise of the Quants

believed were becoming increasingly theoretical and esoteric A greater integration of pragmatic statistics into economics was his objective in funding the Oxford Institute of Statistics which Marschak directed.After four years at the Oxford Institute of Statistics, Marschak was attracted to the New School for Social Research in New York City just as

the USA joined the Second World War He had joined The University in

Exile, a group of more than 180 mostly Jewish anti- fascist scholars who

were offered a home at the New School for the turbulent years between

1933 and 1943 Marschak again found himself in an eclectic and disciplinary framework for his expanding economic research agenda.Marschak joined not just a university but also a philosophy and school of thought The New School for Social Research was co- founded

cross-by Alvin Saunders Johnson (1874–1971), an American economist from Homer, Nebraska who had completed his PhD education at Columbia in

1902 and had taught in a number of universities across the USA before

he returned to New York to edit the New Republic in 1917 The next

year he helped start the New School with the objective of providing

a rich and rigorous multi- disciplinary approach to the development of economic theory and social sciences As the New School’s first director, Johnson created the division that offered refuge to Jewish scholars in the social sciences and humanities Marschak joined this eclectic divi-sion of the New School and, while there, was influential as the mentor and supervisor of Franco Modigliani and many others

A New School and beyond

While at the New School, Marschak transformed his youthful ism into passionate advocacy on behalf of other Russian and Eastern European refugee scholars He was particularly instrumental in attrac-ting intellectual refugees seeking refuge from Nazi Germany during the Nazi era and later on the Soviet Union during the Cold War

activ-While at the New School, Marschak was reunited with other mists in exile, including Emil Lederer (1882–1939), his former mentor

econo-at Heidelberg, and Hans Neisser (1895–1975) Both these kindred spirits were political and intellectual activists in Germany Neisser had been instrumental in the formation of the influential Vienna Colloquium, which helped motivate, hone, and publicize some of John von Neumann’s most significant work in the early to mid-1930s Neisser, who the renowned economist Joseph Schumpeter once described

as “one of the most brilliant economic minds (of his generation),” remained at the New School until he died in 1975.1

The Early Years 13

Lederer and Neisser were most influential in Marschak’s early work These former participants of what was known as the “Kiel School” had exposed Marshak to an approach to economics that was developed

by some reform- oriented economists in the Kiel Institute of World Economics from 1914 until the rise of fascism forced them to seek refuge elsewhere These scholars, and others, were reunited at the New School, where they continued their research into economic growth and the business cycle

This approach to economic growth was timely, for many reasons The world was experiencing the first global depression in the 1930s and only economic growth could offer any salvation Moreover, economics was at this point moving away from the classical simplistic and static models of individual markets and was recognizing the need to model economics and finance within a richer general equilibrium, multi- sector approach that changed dynamically over time Scholars of the Kiel School were leading the quest for a better understanding of how economies evolve and grow

The Cowles Commission

One of Marschak’s uncanny abilities was to associate himself with world- class economic institutions, often in the earliest stages of their development After a few years at the New School, he joined the Cowles Commission for Research in Economics as its director in 1943 At that time, the Cowles Commission resided at the University of Chicago, where he remained until he moved with the Commission to Yale University in 1955

The Cowles Commission was a grand academic experiment Its founder, Alfred Cowles III, was a prominent Colorado businessman and financial advisor whose financial instincts convinced him of the need to improve the level of science and quantitative rigor in econo-mics and finance His mission was especially relevant following the economic discipline’s colossal inability to predict the Great Crash in

1929 or to solve the Great Depression during the 1930s He actually

produced original work on the random walk and lamented whether

stock prices could be forecast.2 He was pondering the efficient market hypothesis as early as 1933, well before Eugene Fama helped coin the expression and a new finance paradigm in the 1960s

The grandson of Alfred Cowles, the founder of the Chicago Tribune

newspaper, and the son of newspaperman and corporate board director Alfred Cowles Jr., Cowles III’s insights and his wealth motivated him to

14 The Rise of the Quants

form the Econometric Society and fund its journal, Econometrica Most notably, he also set up the Cowles Commission for Economic Research in

1932, first in Colorado Springs, Colorado and then in Chicago in 1939;

it now resides at Yale University, his 1913 alma mater, in New Haven, Connecticut First used as a resource to analyze and model stock market indices, the Cowles Commission pursued the integration of mathema-tics and statistics into economic and financial theory, especially through general equilibrium theory and econometrics

Four years after the Cowles Commission moved to Chicago, Marschak was appointed its director He headed the Commission through the incredibly dynamic and progressive period to 1948, at which point the directorship passed to Tjalling Koopmans, who was a subsequent Nobel laureate As its Director, Marschak was responsible for assembling perhaps the most accomplished and visionary group of economists that ever worked under one academic umbrella

However, by the mid-1950s, the Cowles Commission’s progressive and activist policy prescriptions began to rile the traditional neoclas-sical approach of the University of Chicago’s economics department Koopmans petitioned the Cowles family to allow the Commission to move to Yale in 1955, where it was renamed the Cowles Foundation.Scholars associated with the Cowles Commission developed an incredible number of techniques that were groundbreaking at the time Cowles researchers developed new methods such as the indirect least squares and instrumental variable methods, the full information max-imum likelihood estimation method, and the limited information maximum likelihood estimation method All of these methods are now used extensively in finance

The Cowles Commission colleagues also pioneered sophisticated eral equilibrium modeling, as represented by the work of Cowles scholars Kenneth Arrow and Gerard Debreu Beyond Koopmans, Arrow, and Debreu, each of whom were honored with Nobel Prizes, Cowles Commission scholars Trygve Haavelmo, Lawrence Klein, Harry Markowitz, Franco Modigliani, Herbert Simon, and James Tobin were all likewise recognized with awards by the Nobel Committee for work initiated at Cowles.Despite his passion and dedication on behalf of those escaping perse-cution, Marschak remained a sought- after colleague and mentor He was gracious, modest, and fair- minded, and created a nurturing environment that allowed his PhD students in economics to excel As he contributed first to the University in Exile at the New School, and then helped to found, direct, and define the research agenda of those assembled at the Cowles Commission, Marschak was the common denominator of

gen-The Early Years 15

a new analytic movement and, indeed, in the formation of decision sciences

Marschak was much more than a lone cog in a growing academic machine In the various areas of asset choice and portfolio theory, the axiomatic approach to decision sciences, the modeling of uncer-tainty, and optimal investment theory, he made either the first or the second academic volley Ever humble, though, he typically used his insights and innovations to instead motivate and nurture the work of other younger scholars Many of these scholars, influenced and inspired

by him, went on to make a lifetime of contributions to finance and economics At least two of his students, Modigliani and Markowitz, eventually won Nobel Prizes for their work while at the Cowles Commission, as would his Cowles collaborators Friedman, Arrow, Koopmans, and Debreu

Marschak’s wildly innovative and successful approach arose as nomics and finance was in a state of flux and reinvention His timing was impeccable and his insights were profound He took these insights

eco-to unexpected heights that redefined economics and finance in ways that still remain relevant today

The Kiel School

Before the First World War, our understanding of economics took one

of two forms For some, the analysis was rhetorical and straddled the boundary between politics and economics The political economy of Karl Marx (1818–1883), John Stuart Mill (1806–1873), David Ricardo (1772–1823), or even Adam Smith (1723–1790) treated such topics

as trade, economic systems, and the ownership of resources and the means of production with unsophisticated graphical tools and with the strength of philosophical argument and logic Alternatively, others, most notably Léon Walras (1834–1910), Antoine Augustin Cournot (1801–1877), Francis Ysidro Edgeworth (1845–1926), and Irving Fischer (1867–1947), enhanced our understanding of individual markets by introducing to the discipline increasingly sophisticated mathematical tools

The Times 17

While the insights of these early great minds in economics remain valid today, their theories were not sufficiently rigorous and analytic to answer questions in modern finance Indeed, the persistent recession in Europe in the aftermath of the First World War, the unexpected Great Crash of 1929, and the stubborn Global Depression in 1930 demon-strated that the prevailing faith in a simplistic market equilibrium at the micro level did not translate into a better understanding of more complex and aggregated financial markets

This breakdown of the prevailing classical school over the Great Depression arose because of a number of oversimplifications inherent

in the simplistic classical model First, markets may not behave in

a predictable manner if our tools of prediction are based on rationality, but the actors in markets do not at times behave rationally Second,

an element may evolve in isolation in a way that departs from its evolution within a broader system Specifically, a complex system may reach an equilibrium that diverges wildly from the natural equilibria of each of its parts in isolation For instance, one financial security may normally converge toward a predictable price in isolation, but this price might oscillate over time when its market is coupled with another By the 1930s, economic luminaries were arriving at the conclusion that the Classical School was inadequate to explain a complex modern economy

To remedy this inadequacy, members of the Kiel School and the Vienna Colloquia acknowledged the need to use much more sophisticated tools

to model the interactions between markets and characterize markets in the aggregate

Scholars like Marschak and John von Neumann used the Vienna Colloquia to expand their understanding of economics and introduce

to economics tools from physics and applied mathematics At the same time, these scholars realized that there must emerge a much better understanding of the motivations and decisions of agents as diverse as individuals and households, firms and organizations, and even govern-ment and society

The Kiel School approach shifted the economic debate in two tant ways Up to that point, economics at the macro level was considered

impor-a triviimpor-al extension of economics of the smimpor-all, or microeconomics In the microeconomics of a market, a price that is too high results in supply that exceeds demand and a reduction in the price until the surpluses

in the market clear The trivial extension to the macroeconomy mends that such symptoms of excess supply as unemployment should result in lower wages and market clearing In other words, individual

com-18 The Rise of the Quants

markets are either at or converging toward equilibrium at all times, and

so must the aggregation of all markets at the macroeconomic level.The Kiel School also recognized that aggregates of markets do not behave as simplistically as classically trained microeconomists might wish For instance, an aggregate market, such as a stock exchange, must acknowledge all the subtle interactions between its component mar-kets for individual securities These complex intersectoral interactions obviate the simplicity of the classical microeconomic approach

This more sophisticated and nuanced approach to markets created room for more elaborate understandings and approaches to economic gyrations It also called for greater sophistication in the management

of balanced economic growth Marschak emerged as the intellectual leader of his Kiel School colleagues and brought new ideas to a new world, first to the New School for Social Research in New York City and then to a cadre of young scholars at the Cowles Commission at Chicago and Yale

a movement that began in the late 1920s.

Soon after he arrived at the New School, the movement to formalize and mathematize economics was progressing at full stride Economics began to incorporate the new quantitative tools to resolve the new ques-tions that were beginning to be raised At the New School, Marschak championed new tools of mathematics and helped develop the field

of econometrics that soon became the standard data analytic tool The seminars he organized, first at the New School and then at the Cowles Commission, developed a broad academic community of unparalleled potency and potential

Especially at Cowles, Marschak became the central figure for the redefinition of economics and, in turn, the development of finance

in the post- War era A better understanding of risk and uncertainty was at its core Augmenting Frank Knight’s treatise on risk and uncer-

tainty, Risk, Uncertainty and Profit,1 Marschak and Helen Makower in

1938 used the subject of monetary theory to introduce uncertainty

to economic modeling.2 Then, in Money and the Theory of Assets,

he set the stage for what would later become Modern Portfolio Theory following the subsequent work of his graduate student, Harry Markowitz

20 The Rise of the Quants

The modeling of uncertainty

Frank Hyneman Knight (1885–1972) was the first to differentiate between the known probabilities that might affect one’s fortune and the unknowable uncertainties that frustrate our decision- making Knowable risk, embodied in such probabilities as the odds of a coin turning up heads or coming up red on a Roulette wheel, has been the subject of mathematical treatments as early as Bernoulli’s resolution of the St Petersburg Paradox in 1738 However, until Knight differentiated between risk and uncertainty in his classic 1921 book, risk had only been described superficially in economic models, and uncertainty had not even been defined, much less incorporated

Knightian uncertainty is differentiated from risk in that risk involves

outcomes unknown but with known probability distributions With knowledge of these known risks, John von Neumann and Oskar

Morgenstern, in their 1944 magnum opus Theory of Games and

Economic Behavior, constructed expected utilities for which

makers could maximize If these probabilities are known, it is not difficult to construct simple decision rules to optimize returns in financial markets

However, as every financial advisor warns, past patterns may not be predictive of future returns The price of financial securities has less well- defined unknowns and outcomes than may be the case for the flip

of a coin The random, and unknown or unknowable, uncertainties of financial markets are a much higher degree than the predictable vagar-ies of the toss of a coin Knight placed into play a much more subtle and less tractable definition of the unknown

It took another generation of scholars, though, to begin to incorporate Knightian uncertainty into decision theories While some great minds, such as Irving Fischer and John Maynard Keynes, had the prescience

to note the need to more formally incorporate uncertainty, the matical intractability of incorporating the unknown into formal models vexed scholars then, and still does to this day, to a lesser degree

mathe-For instance, Keynes noted in 1937:

By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable The game of roulette is not subject, in this sense, to uncertainty The sense in which I am using the term is that in which the prospect

of a European war is uncertain, or the price of copper and the rate

of interest twenty years hence About these matters there is no

The Theory 21

scientific basis on which to form any calculable probability whatever

We simply do not know.3

Knight and Keynes are well remembered for their addition to our standing of risk and uncertainty However, neither scholar provided the way by which finance could successfully incorporate risk into our models.There was one relatively obscure and successful exception to this combined inability to successfully model uncertainty A notable, but little known and too short- lived, scholar named Frank Plumpton Ramsey (1903–1930) contributed to our understanding of uncertainty in an unpublished paper entitled “Truth and Probability.”4 Ramsey described how our subjective beliefs about unknown probabilities influence our decisions He postulated that the rational decision- maker will align his

under-or her beliefs of unknown probabilities to the consensus bets of impartial

bookmakers, a technique often called the Dutch Book Thirty later, the great

mind Leonard “Jimmie” Savage (1917–1971) elaborated his concept into

an axiomatic approach to decision- making under uncertainty using ments remarkably similar to Ramsey’s logic The concepts of Ramsey and Savage also formed the basis for the theory of Bayesian statistics and are important in many aspects of financial decision- making

argu-Marschak’s great insight

While Ramsey created and Savage broadened the logical landscape for the inclusion of uncertainty into decision- making, it was not possible

to incorporate their logic until the finance discipline could develop actual measures of uncertainty Of course, modern financial analysis depends crucially even today on such a methodology to measure uncertainty

Much of what is now standard in the measurement of financial tainty originated with a paper by Marschak in 1938 He was the first to describe and advocate how to combine asset valuation with uncertainty The approach he developed allowed us to augment our ordering of preferred outcomes in a certain world with a measure of uncertainty in the real world

uncer-By the 1930s, scholars were beginning to describe how to incorporate uncertainty into decision- making and asset management For instance,

in 1935, Nobel laureate Sir John Hicks noted:

By investing only a proportion of total assets in risky enterprises, and investing the remainder in ways which are considered more safe, it

22 The Rise of the Quants

will be possible for the individual to adjust his whole risk situation to that which he most prefers, more closely than he could do by invest-ing in any single enterprise.5

Hicks’ approach commended a number of features for an emerging model of asset pricing He anticipated models that include a risk- free return rf, a risky financial asset, and a set of preferences that incorporate both return and risk

By 1938, and six years before von Neumann and Morgenstern had

established the expected utility hypothesis in their 1944 book Theory of

Games and Economic Behavior, Marschak proposed and explored a

ordi-nal theory of decision- making under uncertainty.6 He was also the first

to propose that these decisions be made over the mean and the variance (or standard deviation) of the asset value

Marschak’s work to formulate preferences in the now- familiar variance space formed the basis for much of financial asset pricing theory He did so in his 1950 paper entitled “Money and the Theory of Assets.” He formalized Keynes’ concept of liquidity preferences in his observation:

mean-[I]n the actual uncertain world, the future production situation [technique, weather, etc.] and future prices are not known …

in the mind of the producer, to each combination of assets there corresponds one and only one n- dimensional set of yield combinations … [T]o each combination of assets there cor-responds in [the decision-maker’s] mind and n- dimensional joint- frequency distribution of the yields [of financial assets and commodities] Thus, instead of assuming an individual who thinks

he knows the future events we assume an individual who thinks he

knows the probabilities [emphasis added] of future events We may

call this situation the situation of a game of chance, and consider

it as a better although still incomplete approximation to reality … than the usual assumption that people believe themselves to be prophets…7

Marschak’s recognition of the interplay between what people know and believe and the decisions they make came well before von Neumann and Morgenstern framed their expected utility hypothesis under risk, Savage outlined an axiomatic approach to decision- making under sub-jective uncertainty, or Kenneth Arrow described decision- making under various states of nature in financial markets Marschak had framed the

as the mathematical expectation of first year’s meat consumption,

y may be its standard deviation, z may be the correlation cient between meat and salt consumption … etc … It is sufficiently realistic, however, to confine ourselves, for each [return] to two parameters only: the mathematical expectation … and the coefficient

coeffi-of variation [“risk”].8

Marschak proposed a simple approach to the consideration of the interplay between return and risk by confining its description to first moments, known as means, and second moments of returns, labeled variances and covariances He also proposed how the variation of one asset may affect another through their covariances and their coefficient

of variation

Marschak also established a new set of terms to describe the general equilibrium of interrelated markets under uncertainty well before Arrow and Debreu subsequently adopted his vocabulary Finally, his mean- variance approach emerged as the basis of Modern Portfolio Theory at the hands of his PhD supervisee and subsequent Nobel Prize winner, Harry Markowitz Markowitz later professed that he was unfamiliar with this groundbreaking work of his supervisor

Marschak’s 1950 paper also preceded the work of Savage on an omatic approach to utility under subjective uncertainty In a paper entitled “Rational Behavior, Uncertain Prospects, and Measurable Utility,” Marschak exhibited characteristic clarity and prescience in the emerging theory of asset pricing.9 In this work, he modified the von Neumann and Morgenstern expected utility hypothesis to include subjective probabilities In doing so, he also graciously tipped his hat

axi-in a footnote to his Cowles Commission colleagues Arrow and Savage for their comments on his manuscript In turn, these two great minds would take still further Marschak’s desire to incorporate uncertainty into models of finance

In his paper, Marschak returned to the mean- variance approach by noting that “the average amounts of goods are not alone relevant to the man’s decision,” and he anticipated Modern Portfolio Theory through comments on “the advantages of diversification.”10

24 The Rise of the Quants

In the same year, Marschak produced a series of lectures on utility and subjective probability.11 These lectures described various states of nature in an uncertain world, in much the same way as Arrow would later incorporate in his Nobel Prize- winning work on the existence of market general equilibrium, and described an approach to the expected utility hypothesis that would later be employed in Markowitz’s Nobel Prize- winning formulation of Modern Portfolio Theory He was also paving the way for his Cowles Commission colleague, Leonard Jimmie

Savage, to produce his seminal work, The Foundations of Statistics, in

1954 This work established a set of axioms by which financial markets can incorporate our human subjective sense of probabilities

Ever graceful, in his lectures Marschak even noted the important work

of Frank Plumpton Ramsey a quarter of a century earlier and the bility that personal probabilities could be deduced based on the actions

possi-of decision- makers He wrote:

The probabilities on which the subject bases his action need not be identical with some objective properties of chance devices (cards, dice) which the experimenter uses This was observed by the English mathematician and logician, F P Ramsey He shows that manifest decisions can be thought of as revealing both the subject’s probabili-ties and utilities.12

Marschak’s work was groundbreaking in a number of ways and a number

of fields of particular importance to finance theory His simple observation and recommendation that we extend the traditional individual choice model to include the variance of assets and consumption goods as well as their means was unprecedented It was a profound extension of Keynes’ prescription that money, assets, and uncertainty must all be included in our models, and hence incorporated into our fundamental approach to asset pricing Marschak’s contemporary Kenneth Arrow noted:13

If we take the Keynesian construction seriously, that is, as of a world with a past as well as a future and in which contracts are made in terms of money, no equilibrium may exist From all this, as well

as from our existence discussions, we conclude that the Keynesian revolution cannot be understood if proper account is not taken of the powerful influence exerted by the future and the past on the present and by the large modifications that must be introduced into both value theory and stability analysis, if the requisite future markets are missing

The mean- variance approach to utility theory

We can demonstrate how this mean- variance approach translates wealth into utility While we often attribute the most common exposi-tion to the derivation of utility from uncertain wealth to the work of Harry Markowitz, Marschak’s student, and James Tobin, his colleague at the Cowles Commission, Marschak motivated this foundation

Consider the decision of an investor who must pay a fee F out of a

wealth W in anticipation of an uncertain return R ˜ over the risk- free return

We know that these uncertain levels of wealth give rise to the utility that motivates the investor’s decisions Then, utility can be written as:

U(W  F  R˜).

Were the investor risk- neutral, utility would be linearly proportional to the uncertain return- augmented wealth The risk- neutral investor would remain unconcerned about symmetric variations of the random return about its mean We can demonstrate the result on expected utility for the pattern of returns by representing utility as an infinite series A Taylor’s series expansion takes into account the deviation of a variable and its effect on the dependent variable through the various derivatives of the relationship between the two variables In the case of utility, a Taylor’s series expansion measures how deviations in wealth affects utility through its slope and the curvature of utility It is calculated as follows:

U(W ˜ )  U(W  F  E(R˜))/0!

 U'(W  F  E(R˜))(R  E(R)) /1!

 U"(W  F  E(R˜))(R  E(R))2 /2!

U"'(W  F  E (R˜)) (R E(R))3 /3!

If this infinite series were calculated for an infinite number of terms, the series of means, first, second, third, and so on derivatives would

26 The Rise of the Quants

precisely determine the value of utility for any deviation arising from

the random return R However, subsequent terms generally become

less significant for five reasons First, notice that each subsequent term incorporates a higher level derivative of utility If utility rose linearly with income, i.e., U’( )  constant > 0, there is no curvature of the utility

curve and hence the second derivative is zero Alternatively, if utility is increasing but at a steadily declining rate of increase, then the second derivative U’’( ) is constant and negative, while the third derivative U’’’( ) is zero These third- order and higher effects are arguably of less significance in defining how the translation between income and utility depart from linearity In addition, these higher order terms are zero if

we accept the commonly assumed quadratic utility function.

Second, notice that each subsequent term is divided by a factorial that grows rapidly The first term is divided by 0!, or 1, and the second term by 1!, or 1, while the third term is divided by 2!, or 2 However, the fourth term is divided by 3!, or 6, and the fifth by 4!, or 24, etc These factorials rapidly diminish the importance of higher order terms.Third, each term contains an expression of the departure of the ran-dom return from its mean, first to the 0 power, then to the 1st power, then to the 2nd power, etc If these deviations of the random variables from their means are comparatively small, then these deviations to

a power are relatively even smaller yet

Fourth, when the power terms are odd, and the random variable is symmetric about its mean, these higher terms cancel out as the devia-tion in one direction counteracts the deviation in the other

Finally, one of the most common distributions in nature is the mal distribution It can be easily demonstrated that such a distribution can be described completely by only its first and second moments, or its mean and variance If one is willing to assume that investment returns are distributed according to this common normal distribution, the ignoring of higher order distributions is not inappropriate

nor-Mathematicians and physicists call weighted deviations from a tral mean value as the first moment, such deviations squared from the central value as the second moment, etc Using this mathematical vocabulary, the most important measures of the increase in utility aris-ing from the random component of wealth is the rate of increase of utility multiplied by the first moment, or the mean value of the random return, and one half of the second derivative multiplied by the second moment, also known as the variance of the random return

cen-The cen-Theory 27

The second moment above can be simplified as follows:

E(R  E(R))2 E(R2)  E(R)2This calculation is typically called the variance of returns, or designated

sR It is also labeled the sigma- squared or the mean squared

When Jacob Marschak proposed that we characterize the relationship between investment returns and utility as most significantly represented

by the investment return’s first and second moments, he was offering

a compact way for finance theorists and practitioners to measure the impact of various distributions of returns Two investments with equal returns but with different distributions of these returns are not equal

in their effect on an investor’s utility Because the variance is always a positive number, but enters the Taylor’s series calculation as a product with the (negative) second derivative U’’, higher variance ultimately subtracts from the resulting utility High variance is a measure of greater return deviations from the mean, or greater uncertainty This uncer-tainty, as characterized by the variance, detracts from utility Marschak proposed that we measure investments both by their expected returns

as well as by their historical variance for that reason His theory of ity under uncertainty then established the link between the mean and variance of returns and the utility of a decision- maker

util-In fact, it was Marschak’s colleagues Arrow, Tobin, Markowitz, and others who more fully described and extended his simple characteriza-tion of investment Even as he took great pains to acknowledge and celebrate the work of those who came before him, Marschak quietly motivated and nurtured others in a most humble manner, without any expectation of credit or notoriety Such intellectual modesty and gener-osity may have been his most gracious and endearing quality

To see this, let us determine a measure of the cost of risk Let us

denote this risk penalty as p Then, the utility a decision- maker can expect from wealth and an uncertain return is equivalent to the utility from the wealth and the expectation of the mean return, net of the risk penalty p Mathematically:

EU(W R)  U(W  E(R)  p)

Taking a Taylor’s series expansion of the left- and right- hand side, and neglecting higher order terms, gives:

U(W  E(R))  U'(R  E (R))  U"(R  E(R))2/2!

= U(W  E(R))  U'p

Applications 29

Embedded in this expression is the measure of variance of returns

(R˜ E(R˜))2 We can simplify and solve for the risk premium p to find:

p  (1/2)(U"/U')Var(R)

Various measures of risk aversion

Marschak’s contemporaries Kenneth Arrow1 and John Pratt2 proposed such a measure for the calculation of a risk premium In his 1965 paper, Pratt denoted the ratio of the curvature to the slope of the utility func-tion U’’/U’ as a measure of absolute risk aversion However, he noted that his results were also contained within seminars given a year before

by Arrow

In fact, for a dozen years from 1952 to 1964, Arrow had published seminal work on the functioning of securities markets as the ways in which investors could use financial instruments to reduce uncertainty

In his 1964 paper, he used the concept of risk aversion to explore the optimal investment strategy for investors who could hold cash or an actuarially fair security He showed that an investor’s optimal strategy depends critically on his or her level of risk aversion He also demon-strated that the investor will hold less in cash and purchase more of the risky asset if the level of risk aversion is lower While he approached the problem from the investor’s perspective, his conclusions dovetailed nicely with those of Pratt Hence, we now classify variations of measures

of risk aversion as flowing from the Arrow- Pratt measure

The more risk- averse, by this measure, the more the decision- maker will be willing to pay the premium p to avoid the risk of an investment

If the investor has diminishing marginal utility or an absolute aversion toward risk, the second derivative U’’ is negative and the investor would

be willing to pay a positive premium to avoid the risk Relating his or her degree of risk aversion to the variance of the risky activity gives the amount the decision- maker is willing to pay to avoid the risk On the other hand, one who is risk- loving has a second derivative of utility U’’ that is positive and hence is willing to pay a premium to take a risk This result helps explain why gamblers will pay $1.00 to earn a mean of

$0.97 or less, on average, in gambling halls that take at least 3 per cent

of each game for “the house.”

This Arrow- Pratt measure of risk aversion is a simple and elegant way to bridge the observables of finance with the decisions of humans motivated in ways more complex than the simple return on financial investments The intuitive and widely accepted principle of diminishing

30 The Rise of the Quants

marginal utility, combined with the easy- to- calculate historical measures

of a security’s variability, yields a measure of a decision-maker’s ingness to pay to avoid the volatility This risk premium behaves qualitatively in ways that reinforce our intuition

will-However, this intuitively appealing construct still presents some problems First, measures of past variability may not be representative

of future volatility Second, utility remains immeasurable, even if its properties – if these could be measured – have been well described.Finally, utility theory predicts that one’s measure of risk aversion

is not a constant Because the slope and possibly the curvature of an investor’s utility function should depend on income, so would both the measure of risk aversion and of one’s risk premium Indeed, large swings in wealth due to equally large variations in the return on a siz-able investment can cause an investor to swing to dramatically different positions on risk aversion

A cursory inspection of the risk premium p also shows that the risk premium to avoid financial volatility depends on the curvature of the utility function U’’ relative to its slope It is widely accepted, but is not beyond dispute, that investors have a decreasing marginal utility U’

as wealth increases If the marginal utility increases with wealth more than does the second derivative U’’, then the premium an investor would be willing to pay to avoid risk should rise However, the wealth partitioned to risky assets will also typically increase with income As

a consequence, a better relative measure of risk aversion may be the risk premium p one would be willing to pay as a share of wealth The resulting calculation of wealth- adjusted relative preferences toward risk avoidance is then:

p relative  w(U"/U')Var(R)/2

There could then be six possible regimes of risk aversion Absolute risk aversion could increase, remain constant, or decrease with income and assets, as could relative risk aversion However, some of these pos-sibilities defy intuition Increasing absolute risk aversion requires that,

as wealth increases, an individual chooses to hold fewer funds in risky assets Decreasing absolute risk aversion better fits observed behavior As wealth increases, more wealth is held in risky assets

Measures of relative risk aversion are also more or less appealing

to intuition Increasing relative risk aversion predicts that one holds

a smaller proportion of wealth in risky assets as wealth increases More appealing is the notion of decreasing relative risk aversion, in which,