POSTMODERN PORTFOLIO

THEORY

JAMES MING CHEN

Navigating Abnormal Markets and

Investor Behavior

Quantitative Perspectives on Behavioral Economics and Finance

Series Editor James  Ming Chen

College of Law Michigan State University

East Lansing ,  Michigan , USA

The economic enterprise has fi rmly established itself as one of evaluating human responses to scarcity not as a rigidly rational game of optimiza-tion, but as a holistic behavioral phenomenon. The full spectrum of social sciences that inform economics, ranging from game theory to evolution-ary psychology, has revealed the extent to which economic decisions and their consequences hinge on psychological, social, cognitive, and emo-tional factors beyond the reach of classical and neoclassical approaches to economics. Bounded rational decisions generate prices, returns, and resource allocation decisions that no purely rational approach to optimiza-tion would predict, let alone prescribe.

Behavioral considerations hold the key to longstanding problems in economics and fi nance. Market imperfections such as bubbles and crashes, herd behavior, and the equity premium puzzle represent merely a few of the phenomena whose principal causes arise from the comprehensi-ble mysteries of human perception and behavior. Within the heterodox, broad-ranging fi elds of behavioral economics, a distinct branch of behav-ioral fi nance has arisen.

Finance has established itself as a distinct branch of economics by apply-ing the full arsenal of mathematical learning on questions of risk manage-ment. Mathematical fi nance has become so specialized that its practitioners often divide themselves into distinct subfi elds. Whereas the P branch of mathematical fi nance seeks to model the future by managing portfolios through multivariate statistics, the Q world attempts to extrapolate the present and guide risk-neutral management through the use of partial dif-ferential equations to compute the proper price of derivatives.

The emerging fi eld of behavioral fi nance, worthy of designation by the Greek letter psi (ψ), has identifi ed deep psychological limitations on the claims of the more traditional P and Q branches of mathematical fi nance. From Markowitz’s original exercises in mean-variance optimization to the Black-Scholes pricing model, the foundations of mathematical fi nance rest on a seductively beautiful Gaussian edifi ce of symmetrical models and crisp quantitative modeling. When these models fail, the results are often catastrophic.

The ψ branch of behavioral fi nance, along with other “postmodern” critiques of traditional fi nancial wisdom, can guide theorists and practi-tioners alike toward a more complete understanding of the behavior of capital markets. It will no longer suffi ce to extrapolate prices and forecast market trends without validating these techniques according to the full range of economic theories and empirical data. Superior modeling and

data-gathering have made it not only possible, but also imperative to har-monize mathematical fi nance with other branches of economics. Likewise, if behavioral fi nance wishes to fulfi ll its promise of transcending mere cri-tique and providing a more comprehensive account of fi nancial markets, behavioralists must engage the full mathematical apparatus known in all other branches of fi nance. In a world that simultaneously lauds Eugene Fama’s effi ciency hypotheses and heeds Robert Shiller’s warnings against irrational exuberance, progress lies in Lars Peter Hansen’s commitment to quantitative rigor. Theory and empiricism, one and indivisible, now and forever.

More information about this series at http://www.springer.com/series/14524

James  Ming Chen

Postmodern Portfolio Theory

Navigating Abnormal Markets and Investor Behavior

Quantitative Perspectives on Behavioral Economics and Finance ISBN 978-1-137-54463-6 ISBN 978-1-137-54464-3 (eBook) DOI 10.1057/978-1-137-54464-3

Library of Congress Control Number: 2016942679

© The Editor(s) (if applicable) and The Author(s) 2016 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfi lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specifi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the pub-lisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Cover illustration: © ZUMA Press, Inc. / Alamy Stock Photo Cover design by Oscar Spigolon

Printed on acid-free paper

This Palgrave Macmillan imprint is published by Springer Nature The registered company is Nature America Inc. New York

James  Ming Chen College of Law Michigan State University East Lansing , Michigan , USA

To Heather Elaine Worland Chen, with all my love

ix

This book incorporates ideas from papers I have presented at the University of Cincinnati, Florida State University, Georgetown University, Michigan State University, the University of Pennsylvania, the University of Virginia, and the Faculty of Economics of the University of Zagreb (Ekonomski Fakultet Sveučilišta u Zagrebu). The International Atlantic Economic Society and the ACRN Oxford Academic Research Network have pro-vided multiple platforms for the work underlying this book. Along the way, I have benefi ted from scholarly and professional interactions with Anna Agrapetidou, Abdel Razzaq Al Rababa’a, Moisa Altar, Christopher J. Brummer, Irene Maria Buso, Adam Candeub, Seth J. Chandler, Felix B.  Chang, Tendai Charasika, César Crousillat, David Dixon, Robert Dubois, John F.  Duffy, Daniel A.  Farber, Christopher C.  French, Santanu K.  Ganguli, Tomislav Gelo, Periklis Gogas, Gil Grantmore, Andy Greenberg, Losbichler Heimo, Hemantha Herath, Jesper Lyng Jensen, Jagoda Kaszowska, Daniel Martin Katz, Yuri Katz, Imre Kondor, Carolina Laureti, Cordell Lawrence Jr., Cordell Lawrence Sr., Matthew Lee, Othmar Lehner, Heimo Losbichler, Gerry Mahar, Milivoj Marković, L.  Thorne McCarty, Steven C.  Michael, Ludmila Mitkova, José María Montero Lorenzo, Kevin Lynch, Laura Muro, Vivian Okere, Merav Ozair, Elizabeth Porter, Mobeen Ur Rehman, Carol Royal, Bob Schmidt, Jeffrey A. Sexton, Galen Sher, Ted Sichelman, Jurica Šimurina, Nika Sokol Šimurina, Robert Sonora, Lisa Grow Sun, Elvira Takli, Peter Urbani, Robert R. M. Verchick, Benjamin Walther, Karen Wendt, Gal Zahavi, and Johanna F. Ziegel. Christian Diego Alcocer Argüello, of Michigan State University’s Department of Economics, provided very capable research

ACKNOWLEDGMENTS

x ACKNOWLEDGMENTS

assistance. I am also grateful for contributions by several students at Michigan State’s College of Law: Angela Caulley, Yuan Jiang, Morgan Pitz, Emily Strickler, Paul M.  Vogel, and Michael Joseph Yassay. The research services of the Michigan State University Law Library and admin-istrative support by Marie Gordon were indispensable. Special thanks to Heather Elaine Worland Chen.

xi

CONTENTS

1 Finance as a Pattern of Timeless Moments 1 1.1 Introduction 1

Part 1 Perpetual Possibility in a World of Speculation: Portfolio Theory in Its Modern and Postmodern Incarnations 3

2 Modern Portfolio Theory 5 2.1 Mathematically Informed Risk Management 5 2.2 Measures of Risk; the Sharpe Ratio 6 2.3 Beta 6 2.4 The Capital Asset Pricing Model 9 2.5 The Treynor Ratio 10 2.6 Alpha 12 2.7 The Effi cient Markets Hypothesis 13 2.8 The Effi cient Frontier 15

3 Postmodern Portfolio Theory 27 3.1 A Renovation Project 27 3.2 An Orderly Walk 28 3.3 Roll’s Critique 29 3.4 The Echo of Future Footsteps 31

xii CONTENTS

Part 2 Bifurcating Beta in Financial and Behavioral Space 39

4 Seduced by Symmetry, Smarter by Half 41 4.1 Splitting the Atom of Systematic Risk 41 4.2 The Catastrophe of Success 44 4.3 Reviving Beta’s Dead Hand 45 4.4 Sinking, Fast and Slow 47

5 The Full Financial Toolkit of Partial Second Moments 59 5.1 A History of Downside Risk Measures 59 5.2 Safety First 60 5.3 Semivariance, Semideviation, and Single-Sided Beta 62 5.4 Traditional CAPM Specifi cations of Volatility, Variance,

Covariance, Correlation, and Beta 64 5.5 Deriving Semideviation and Semivariance from Upper

and Lower Partial Moments 67

6 Sortino, Omega, Kappa: The Algebra of Financial Asymmetry 79 6.1 Extracting Downside Risk Measures from Lower Partial

Moments 79 6.2 The Sortino Ratio 80 6.3 Comparing the Treynor, Sharpe, and Sortino Ratios 81 6.4 Pythagorean Extensions of Second-Moment Measures:

Triangulating Deviation About a Target Not Equal to the Mean 85

6.5 Further Pythagorean extensions: Triangulating Semivariance and Semideviation 87

6.6 Single-Sided Risk Measures in Popular Financial Reporting 89 6.7 The Trigonometry of Semideviation 91 6.8 Omega 94 6.9 Kappa 95 6.10 An Overview of Single-Sided Measures of Risk Based on 

Lower Partial Moments 97 6.11 Noninteger Exponents Versus Ordinary Polynomial

Representations 99

CONTENTS xiii

7 Sinking, Fast and Slow: Relative Volatility Versus Correlation Tightening 107 7.1 The Two Behavioral Faces of Single-Sided Beta 107 7.2 Parameters Indicating Relative Volatility and 

Correlation Tightening 111 7.3 Relative Volatility and the Beta Quotient 115 7.4 The Low-Volatility Anomaly and Bowman’s Paradox 116 7.5 Correlation Tightening 120 7.6 Correlation Tightening in Emerging Markets 122 7.7 Isolating and Pricing Correlation Risk 126 7.8 Low Volatility Revisited 129 7.9 Low Volatility and Banking’s “Curse of Quality” 131 7.10 Downside Risk, Upside Reward 132

Part 3 Τέσσερα, Τέσσερα: Four Dimensions, Four Moments 153

8 Time-Varying Beta: Autocorrelation and Autoregressive Time Series 155 8.1 Finding in Motion What Was Lost in Time 155 8.2 The Conditional Capital Asset Pricing Model 157 8.3 Conditional Beta 158 8.4 Conventional Time Series Models 160 8.5 Asymmetrical Time Series Models 162

9 Asymmetric Volatility and Volatility Spillovers 173 9.1 The Origins of Asymmetrical Volatility;

the Leverage Effect 173 9.2 Volatility Feedback 174 9.3 Options Pricing and Implied Volatility 176 9.4 Asymmetrical Volatility and Volatility Spillover

Around the World 177

10 A Four-Moment Capital Asset Pricing Model 189 10.1 Harbingers of a Four-Moment Capital Asset

Pricing Model 189 10.2 Four-Moment CAPM as a Response to the 

Fama–French–Carhart Four-Factor Model 190

xiv CONTENTS

10.3 From Asymmetric Beta to Coskewness and Cokurtosis 192 10.4 Skewness and Kurtosis 196 10.5 Higher-Moment CAPM as a Taylor Series Expansion 198 10.6 Interpreting Odd Versus Even Moments 202 10.7 Approximating and Truncating the Taylor Series

Expansion 204 10.8 Profusion and Confusion Over Measures of 

Coskewness and Cokurtosis 205 10.9 A Possible Cure for Portfolio Theory’s Curse of 

Dimensionality: Relative Lower Partial Moments 210

11 The Practical Implications of a Spatially Bifurcated Four-Moment Capital Asset Pricing Model 225 11.1 Four-Moment CAPM Versus the Four- Factor Model 225 11.2 Correlation Asymmetry 226 11.3 Emerging Markets 227 11.4 Size, Value, and Momentum 228

Part 4 Managing Kurtosis: Measures of Market Risk in Global Banking Regulation 235

12 Going to Extremes: Leptokurtosis as an Epistemic Threat 237 12.1 VaR and Expected Shortfall in Global

Banking Regulation 237 12.2 Leptokurtosis, Fat Tails, and Non-Gaussian

Distributions 240

13 Parametric VaR Analysis 247 13.1 The Basel Committee on Bank Supervision and the 

Basel Accords 247 13.2 The Vulnerability of VaR Analysis to Model Risk 249 13.3 Gaussian VaR 251 13.4 A Simple Worked Example 252

CONTENTS xv

14 Parametric VaR According to Student’s t -Distribution 261 14.1 Choosing Among Non-Gaussian Distributions 261 14.2 Stable Paretian Distributions 262 14.3 Student’s t -Distribution 264 14.4 The Probability Density and Cumulative Distribution

Functions of Student’s t -Distribution 266 14.5 Adjusting Student’s t -Distribution According to 

Observed Levels of Kurtosis 268 14.6 Performing Parametric VaR Analysis with Student’s

t -Distribution 270

15 Comparing Student’s t -Distribution with the Logistic Distribution 281 15.1 The Logistic Distribution 281 15.2 Equal Kurtosis, Unequal Variance 284

16 Expected Shortfall as a Response to Model Risk 291 16.1 VaR Versus Expected Shortfall 291 16.2 The Incoherence of VaR 292 16.3 Extrapolating Expected Shortfall from VaR 296 16.4 A Worked Example 298 16.5 Formally Calculating Expected Shortfall from 

VaR under Student’s t -Distribution 299 16.6 Expected Shortfall Under a Logistic Model 302

17 Latent Perils: Stressed VaR, Elicitability, and Systemic Effects 307 17.1 Additional Concerns 307 17.2 Stressed VaR 308 17.3 Expected Shortfall and the Elusive Ideal of Elicitability 310 17.4 Systemic Risk 312 17.5 A Dismal Forecast 315

18 Finance as a Romance of Many Moments and Plural Views 327

Index 331

xvii

LIST OF FIGURES

Fig. 2.1 The security market line 10 Fig. 2.2 The security characteristic line 13 Fig. 2.3 The effi cient frontier 16 Fig. 6.1 The Pythagorean relationship between upside and downside

semideviation 92 Fig. 6.2 Fitting an ordinary polynomial function to the equation

y = x a + b, where the exponent a + b is a noninteger. The coeffi cients for each term in the polynomial equation are analytically rather than computationally determined 101

Fig. 14.1 A plot of the probability density function for several members of the Student’s t family of distributions 264

Fig. 15.1 The Gaussian and logistic distributions 282 Fig. 15.2 Logistic (blue), Student’s t (red), and Gaussian (gold)

distributions 285 Fig. 15.3 Logistic (blue), Student’s t (red), and Gaussian (gold)

distributions: A deeper look at the left tail 286

xix

Table 6.1 The Treynor, Sharpe, and Sortino ratios 82 Table 6.2 Comparing the Sharpe ratio with a standard score 82 Table 6.3 A closer comparison of the Sharpe and Sortino ratios 85 Table 6.4 Trigonometric measurements of asymmetry in single-sided

volatility 93 Table 6.5 Risk-adjusted measures of fi nancial performance drawn

from lower partial moments of the distribution of returns 98 Table 7.1 Relative volatility and correlation tightening in developed and

emerging markets, 1988–2001 124 Table 10.1 The fi rst four orders of a family of risk measures drawn from

statistical cross moments 212 Table 10.2 The fi rst four orders of a family of single-sided risk

measures drawn from statistical cross moments, specifi ed on the downside of mean returns 213

Table 11.1 Coeffi cients of determination indicating the statistical power of Fama and French’s three-factor model, CAPM with coskewness, and conventional (single-factor) CAPM 229

Table 13.1 Gaussian VaR at commonly used intervals 254 Table 14.1 VaR according to Student’s t-distribution for different

numbers of degrees of freedom (ν) 271 Table 14.2 Kurtosis in returns on 13 noncash asset classes,

February 1990 through May 2010 273 Table 15.1 VaR at γ2 = 1.2 under the logistic distribution and Student’s

t-distri bution, ν = 9 284 Table 16.1 VaR and expected shortfall under Student’s t-distribution,

ν ∈ {4, 6, 10, 12} 301

LIST OF TABLES

xx LIST OF TABLES

Table 16.2 VaR and expected shortfall under the logistic distribution 303 Table 17.1 VaR and expected shortfall under logistic, Student’s t,

and Gaussian models at a confi dence level of p = 2 –15 316

1© The Editor(s) (if applicable) and The Author(s) 2016J.M. Chen, Postmodern Portfolio Theory, DOI 10.1057/978-1-137-54464-3_1

CHAPTER 1

1.1 INTRODUCTION Quantitative fi nance traces its roots to modern portfolio theory. Despite the defi ciencies of modern portfolio theory, mean-variance optimization nevertheless continues to form the basis for contemporary fi nance. The term postmodern portfolio theory captures many of the advances in fi nancial learning since the original articulation of modern portfolio theory. A com-prehensive approach to fi nancial risk management must address all aspects of portfolio theory, from the beautiful symmetries of modern portfolio theory to the disturbing behavioral insights and the vastly expanded math-ematical arsenal of the postmodern critique.

This survey of portfolio theory, from its modern origins through more sophisticated, “postmodern” incarnations, evaluates portfolio risk accord-ing to the fi rst four moments of any statistical distribution: mean, variance, skewness, and excess kurtosis. Postmodern Portfolio Theory also evaluates the challenge that prospect theory and behavioral fi nance pose to portfo-lio theory and, more broadly, to quantitative fi nance. The effi cient capital markets hypothesis and the conventional two-moment capital asset pricing

Finance as a Pattern of Timeless Moments

T.S. ELIOT , Little Gidding , in FOUR QUARTETS 49–59, 58 (Harcourt, Brace & Co. 1971; 1st ed. 1943) (“for history is a pattern/Of timeless moments”).

model now compete with the postmodern alternative of an expanded four-moment capital asset pricing model and its behavioral extensions. Mastery of postmodern portfolio theory’s quantitative tools and behav-ioral insights holds the key to the effi cient frontier of risk management.

This book proceeds in four parts. Part 1 introduces portfolio theory. Chapter 2 expounds modern portfolio theory as a framework for assess-ing risk-adjusted fi nancial returns. Conventional mean-variance analysis, the foundation of modern portfolio theory, emphasizes expected return, standard deviation, and beta. These quantitative measures are drawn from the lower moments of statistical distributions.

Chapter 3 outlines new approaches to portfolio theory that account for market abnormalities and investor behavior. The foundational theory of contemporary fi nance is riddled not only with mistakes in measurement, but also with mistakes in perception. At its most ambitious, the postmod-ern critique seeks ways to account for the destructive potential of systemic coordination and cascades. At its most modest, postmodern portfolio the-ory respects fundamental limits on human knowledge.

Parts 2 and 3 pursue the postmodern agenda for risk management by emphasizing asymmetry in fi nance and the higher statistical moments of fi nancial returns. This book’s approach to postmodern portfolio theory emphasizes single-sided statistical moments, the statistical notions of skewness and kurtosis, and behavioral responses to these decidedly abnor-mal fi nancial phenomena. Beginning with a bifurcation of beta on either side of mean returns, Part 2 of this book tours fi nancial and behavioral space in search of a mathematically cogent account of risk on either side of mean returns. After a brief exploration of time series models that measure asymmetry in volatility alongside intertemporal changes in volatility, Part 3 specifi es a four-moment capital asset pricing model based upon a Taylor series expansion of log returns.

Part 4 devotes additional attention to the problem of fat tails and kur-tosis risk in fi nance. It does so by examining the treatment of value-at-risk and expected shortfall as measures of market risk in the trading book of fi nancial institutions observing the Basel Accords on international bank-ing regulation. Risk management, even when undertaken by some of the world’s largest fi nancial institutions under central bank supervision, cannot fully escape mathematically dictated limitations on economic forecasting.

2 J.M. CHEN

PART 1

Perpetual Possibility in a World of Speculation: Portfolio Theory in Its

Modern and Postmodern Incarnations

See ELIOT , Burnt Norton , in FOUR QUARTETS 13–22, 13 (“What might have been is an abstraction/Remaining perpetual possibility/Only in a world of speculation.”).

5© The Editor(s) (if applicable) and The Author(s) 2016J.M. Chen, Postmodern Portfolio Theory, DOI 10.1057/978-1-137-54464-3_2

Chapter 2

2.1 MatheMatically inforMed risk ManageMent

portfolio theory may be the most fecund intellectual export from quanti-tative finance to other sciences. Social sciences outside the strictly financial domain have applied portfolio theory to subjects as diverse as regional development,1 social psychology,2 and information retrieval.3 proper understanding of portfolio theory and its place in finance and cognate sciences begins with a return to the origins of modern portfolio theory. For “the end of all our exploring/Will be to arrive where we started/and know the place for the first time.”4

Modern portfolio theory offers a mathematically informed approach to financial risk management.5 Modern portfolio theory assumes that inves-tors are rationally risk averse.6 Given two portfolios with the same expected return, investors prefer the less risky one.7 although idiosyncratic risks are hard to identify, let alone manage, diversification reduces the systemic risk that market forces will swamp an entire portfolio of highly correlated assets.8 reward follows risk:9 though a riskier investment is not necessarily more rewarding, modern portfolio theory does predict that an investor will demand a higher expected return in exchange for accepting greater risk.10 a direct, positive relationship between risk and return is canonical to conventional theories of finance.

Modern portfolio theory

2.2 Measures of risk; the sharpe ratio

Measures of risk abound within modern portfolio theory. harry Markowitz’s original formulation used the volatility of returns, as mea-sured by their standard deviation, as a proxy for risk.11 William Sharpe proposed a measure of “reward to variability” that relied squarely on stan-dard deviation:12

Sharpe ratio =

-R Rf

s

where R represents expected return, Rf represents the return from a risk- free baseline such as treasury bonds, and σ represents standard deviation. the Sharpe ratio bears an obvious resemblance to the definition of a stan-dard score in ordinary statistics:13

z

x=

- ms

2.3 Beta

an alternative measure for risk, beta, compares returns on an individual asset or a portfolio of assets with returns realized from a broader bench-mark, based on the entirety or at least some significant portion of the financial market.14 the beta of an asset within a portfolio measures the (1) the covariance between the rate of return on the asset and the rate of return on the portfolio as a whole (2) divided by the variance of returns on the portfolio.15 More formally:

baa b

p

r r

r=

( )( )

cov ,

var

Beta may be most intuitively understood by relation to standard statis-tical measures of correlation. pearson’s r is the standard measure of cor-relation between two sets of data. When specified for an entire population

6 J.M. Chen

rather than a sample, pearson’s correlation coefficient is designated as ρ(x, y):

rs s

x yx y

x y

,cov ,

( ) = ( )

By contrast, beta is a measure of covariance. For any two sets of data, rep-resented by independent variable x and dependent variable y, beta for y is the ratio of the covariance between the two data sets to the variance of x:

bsy

x

x y

x

x y=

( )( )

=( )cov ,

var

cov ,2

the mathematical relationship between beta and pearson’s correlation coefficient can be reduced to the verbal description of beta as “correlated relative volatility”:16

b r

bss s

ss s s

r

y

yx

y x

x

y x y

x y

x y x yx y

µ ( )

× =( )

× =( )

= ( )

,

cov , cov ,,

2

Whether designated as correlated relative volatility or (even more col-orfully) as financial elasticity, beta reports the sensitivity of an individual asset or an entire asset class to market returns.17 Because beta measures nondiversifiable, systematic risk, it supplies information on volatility and liquidity in the broader marketplace.18 By measuring covariance between a single security and the market as a whole, beta presents the simplest model of market behavior that does not “assum[e] away the existence of interrelationships among securities,” but nevertheless “captures a large part of such interrelationships.”19 In this sense, beta is less comprehensive than standard deviation, which, as used in the Sharpe ratio, captures both systematic risk and the idiosyncratic risk inherent in a single asset.20 this limitation on beta can prove useful in fund management, since a measure-ment of beta may help separate an active portfolio manager’s skill from her or his willingness to take risk.21

Modern portFolIo theory 7

Zero beta indicates a lack of correlation between an asset and its benchmark. negative beta indicates inverse correlation; positive mar-ket movement means a loss in value for the asset, and vice versa.22 For certain assets, negative beta may represent successful performance. For instance, over an appropriately limited time frame, an inverse exchange-traded fund (etF) that uses derivatives to profit from a decline in the Standard & poor’s 500 (S&p 500) would report complete success in that endeavor if it is able to report a beta of −1 relative to the S&p 500. By holding that etF, an investor can hedge against a decline in the S&p 500 without carrying the margin account needed to engage in the short-selling of securities.23

there is a notable exception to this interpretation of zero beta. Under unusual conditions, zero beta may indicate zero correlation with the market despite rampant volatility in the price of an asset. even if the underlying asset is quite volatile in absolute terms, a complete lack of correlation with the market yields a beta of zero.24 a purely speculative asset, such as Bitcoin, may exhibit wild swings in price with zero or nearly zero correlation with changes in the price of other trad-able assets. estimates of the absolute value of Bitcoin’s correlation with other assets range from a high of 16–17 % for gold and inflation-linked bonds25 to a low of practically zero for gold and leading world curren-cies.26 In these circumstances, beta would report little or no risk, while standard deviation would report a palpable degree of risk. Formally: β ≈ 0; σ ≫ 0.

to compound the difficulty in interpreting results of this sort, sources point in diametrically opposite directions on the investment potential of Bitcoin. one source, persuaded that Bitcoin’s “high risk is compensated by low correlations with other assets,” recommends “[i]ncluding even a small proportion of Bitcoins in a well diversified portfolio” as a step toward “dramatically improv[ing] risk-return characteristics.”27 another source concludes that Bitcoin behaves more like “a speculative investment than a currency”28 and “is completely ineffective as a tool of risk man-agement.”29 Consistent with the notion of confirmation bias in behav-ioral economics,30 one’s perspective on Bitcoin may hinge on whether the interpreter is (or at least is predisposed to be) a breathless enthusiast,31 or else dismisses cryptocurrency as a technological fantasy for gamblers, libertarians, polemicists, and criminals.32

although there is no upper or lower bound on the value of beta, a useful analytical baseline is represented by a beta of 1. Beta of 1 indicates

8 J.M. Chen

an asset whose systemic volatility, or sensitivity to risk, is exactly the same as that of the broader market.33 positive values for beta below 1—that is, 0 < β < 1—indicate an asset that moves in the same direction as the broader market but is not as sensitive to the market’s movements. Values for beta greater than 1 indicate greater sensitivity.34

2.4 the capital asset pricing Model

Beta plays a pivotal role in one of the most important expressions of modern portfolio theory, the capital asset pricing model (CapM).35 the CapM expresses return on an asset as a function of risk, which in turn can be expressed as volatility, beta, or some other measure. the independent development, particularly by William Sharpe and John lintner,36 of “gen-eral models represent[ing] equivalent approaches to the problem of capital asset pricing under uncertainty” gave rise to what we recognize today as the CapM.37

the CapM quantifies the premium demanded by the market for shoul-dering that asset’s volatility over a benchmark represented by the return on a risk-free investment:38

R R R Ra f a m f= + -( )b

where Ra, Rm, and Rf respectively represent returns on the asset, on the broader market of comparable investments, and on a risk-free investment, and where βa represents the individual asset’s beta vis-à-vis a portfolio based on the broader market.39 this formula takes the form of a linear equation where the return on an asset (Ra) is expressed as a function of the premium over a risk-free baseline (Rm − Rf).40

Within the CapM, beta (βa) represents the slope of the linear func-tion, and the risk-free return (Rf) is a constant that defines the function’s y-intercept.41 this graphical representation of the CapM is known as the security market line (Fig. 2.1).42

Modest algebraic rearrangement of the CapM yields the following relationship:

R R

R Rm f

a f

a

- =-

b

Modern portFolIo theory 9

the left side of the foregoing equation represents the risk premium demanded for the entire asset class represented by a particular segment of the market.43 Modern portfolio theory expresses the risk premium as the difference between returns on a specific investment or class of investments and some sort of risk-free benchmark.44 this premium dictates a firm’s cost of capital; indeed, capital asset pricing, in its original incarnation, offered a solution to the problem of determining the price that investors would demand for bearing risk in excess of a risk-free alternative.45

another common application of the CapM compares an index of equities designed to track the S&p 500 against the putatively risk-free baseline of short-term treasury bills.46 this market-wide risk premium is equivalent to the risk-adjusted premium expressed on the right side of the equation—namely, the risk premium for the asset vis-à-vis a risk-free investment, divided by the individual asset’s beta.47

2.5 the treynor ratio

this ratio between risk-adjusted return and volatility bears closer examina-tion. recall that the foregoing equation is merely an algebraically refor-mulated version of the basic CapM:

Fig. 2.1 the security market line

10 J.M. Chen