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Über dieses Buch

This survey of portfolio theory, from its modern origins through more sophisticated, “postmodern” incarnations, evaluates portfolio risk according to the first four moments of any statistical distribution: mean, variance, skewness, and excess kurtosis. In pursuit of financial models that more accurately describe abnormal markets and investor psychology, this book bifurcates beta on either side of mean returns. It then evaluates this traditional risk measure according to its relative volatility and correlation components. After specifying a four-moment capital asset pricing model, this book devotes special attention to measures of market risk in global banking regulation. Despite the deficiencies of modern portfolio theory, contemporary finance continues to rest on mean-variance optimization and the two-moment capital asset pricing model. The term postmodern portfolio theory captures many of the advances in financial learning since the original articulation of modern portfolio theory. A comprehensive approach to financial 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 mathematical arsenal of the postmodern critique. Mastery of postmodern portfolio theory’s quantitative tools and behavioral insights holds the key to the efficient frontier of risk management.



Chapter 1. Finance as a Pattern of Timeless Moments

Quantitative finance traces its roots to modern portfolio theory. Despite the deficiencies of modern portfolio theory, mean-variance optimization nevertheless continues to form the basis for contemporary finance. The term postmodern portfolio theory captures many of the advances in financial learning since the original articulation of modern portfolio theory. A comprehensive approach to financial 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 mathematical arsenal of the postmodern critique.
James Ming Chen

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


Chapter 2. Modern Portfolio Theory

Portfolio theory may be the most fecund intellectual export from quantitative 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
James Ming Chen

Chapter 3. Postmodern Portfolio Theory

Modern portfolio theory, its name notwithstanding, needs a thorough renovation. The reaction of an informed contemporary critic to this venerable model of financial analysis would be comparable to that of a postmodern architect who encounters the naked geometry of a Brutalist monument for the first time: the edifice has nice “bones,” so to speak, but it needs to be rebuilt with human needs and emotions in mind before anyone will live in it.1
James Ming Chen

Bifurcating Beta in Financial and Behavioral Space


Chapter 4. Seduced by Symmetry, Smarter by Half

The capital asset pricing model (CAPM) remains the dominant paradigm in financial risk management—at least among practitioners, if not among scholars.1 Courts and regulators likewise depend on the CAPM, and in so doing confer legal significance on this model.2 Once upon a time, long long ago, “the hegemony of the CAPM” could be attributed “mostly to its apparent ease of applicability and, to a lesser extent, its empirical justifications.”3 The latter excuse, at least, has withered away. Despite evidence that beta is not positively related to returns on stock,4 to say nothing of beta’s failure to account for macroeconomic5 and idiosyncratic6 factors affecting security prices and returns, much of contemporary mathematical finance still hinges on the CAPM. Even Eugene Fama, beta’s leading nemesis, has conceded that “market professionals (and academics) still think about risk in terms of market β.”7
James Ming Chen

Chapter 5. The Full Financial Toolkit of Partial Second Moments

Traditional, two-tailed measurements of risk-adjusted performance, particularly the Sharpe ratio, give dangerous guidance during bear markets because they implicitly assume that returns are normally distributed and because they effectively treat upside and downside volatility as equal constituents of risk.1 The danger in assuming symmetry in the distribution of returns is neither new nor mysterious. Many of the architects of modern portfolio theory nevertheless adopted this statistical shortcut in grudging acceptance of that era’s computational limitations.2 Harry Markowitz’s theoretical call “for calculating the covariances of every security” initially posed a “monumental” barrier to practical implementation: under the constraints on computing power during the 1960s, “[c]alculating a single portfolio could eat up tens of thousands of dollars in computer time.”3
James Ming Chen

Chapter 6. Sortino, Omega, Kappa: The Algebra of Financial Asymmetry

This chapter traces the development of entire families of downside risk measures from partial statistical moments. The Sortino, omega, and kappa ratios provide credible, workable single-factor measures of financial dispersion below mean return. At a minimum, specifying these ratios provides a useful contrast with conventional, two-tailed measures such as the Sharpe and Treynor ratios. Because it is based on downside semideviation, the square root of the lower partial second moment, the Sortino ratio is particularly easy to reconcile with the more traditional and more familiar tools of modern portfolio theory. Indeed, closer examination of the Sortino ratio reveals Pythagorean relationships between single-sided risk measures and their counterparts within the conventional capital asset pricing model (CAPM). These relationships allow single-sided measures of volatility to be evaluated with trigonometric tools.
James Ming Chen

Chapter 7. Sinking, Fast and Slow: Relative Volatility Versus Correlation Tightening

Having exhausted the descriptive potential of semideviation and other elaborations of partial statistical moments, I now return to beta as a composite statistic combining relative volatility with correlation. The bifurcation of single-sided beta into distinct components measuring changes in volatility and in correlation reveals two very different aspects of market conduct, each with its own implications for investor behavior.
James Ming Chen

Τέσσερα, Τέσσερα: Four Dimensions, Four Moments


Chapter 8. Time-Varying Beta: Autocorrelation and Autoregressive Time Series

“[T]ime is the longest distance between two places.”1 This book has focused thus far on bifurcating beta in financial space—that is, on either side of mean rates of return or some other target. It has analyzed beta in recognition of two distinct but related departures from the conventions of modern portfolio theory, the capital asset pricing model (CAPM), and the efficient capital markets hypothesis. Financial markets are both abnormal and irrational. They are abnormal in the sense that they violate the central limit theorem and other properties of the normal, Gaussian distribution. Furthermore, real investors respond to such abnormalities in ways that deviate from the neoclassical assumption of perfect rationality and dispassionate maximization of individual or institutional welfare. Even if we cannot describe a mechanism by which human behavior causes market abnormalities or vice versa, or otherwise demonstrate a causal link, we can show that market abnormality and human irrationality travel together—that they are strongly correlated.
James Ming Chen

Chapter 9. Asymmetric Volatility and Volatility Spillovers

Why indeed is volatility asymmetrical? Wholly apart from their epochal methodological contributions, providing an answer to this question may be the greatest theoretical advance traceable to time series models. This chapter will explore three distinct accounts of asymmetrical volatility. Moreover, time series modeling can now detect volatility spillovers between markets. Even more remarkably, new methods can identify which markets transmit information and which ones, in effect, receive that information in the form of increased volatility. After noting those developments, this chapter will conclude with a quick survey of findings regarding asymmetrical volatility and volatility spillovers in developed and emerging markets.
James Ming Chen

Chapter 10. A Four-Moment Capital Asset Pricing Model

Having completed our interlude on time series methodology and the theoretical insights of time series modeling of asymmetric volatility, I now return to a strictly spatial topic left open by Part 2 of this book and not completely answered by the first chapters in Part 3. In this chapter and the next, I will use the behavior of single-sided beta to extrapolate a logical progression from the conventional two-moment specification of the CAPM to a four-moment CAPM.
James Ming Chen

Chapter 11. The Practical Implications of a Spatially Bifurcated Four-Moment Capital Asset Pricing Model

Javier Estrada has argued that differences attributable to the bifurcation of beta on either side of mean returns, at least in emerging markets, are too substantial to ignore.1 We should likewise expect to find considerable differences in financial performance arising from the deployment of a four-moment CAPM, the proper specification of this augmented CAPM according to cross moments, and the bifurcation of coskewness and cokurtosis along the upside and downside of mean returns.
James Ming Chen

Managing Kurtosis: Measures of Market Risk in Global Banking Regulation


Chapter 12. Going to Extremes: Leptokurtosis as an Epistemic Threat

“Time present and time past/are both present in time future/and time future contained in time past.”1 In projecting forward rather than backward over time, economic forecasting cannot escape “timeless” moments of a different sort: the mathematical moments of the distribution of financial returns.2 Of the four moments of greatest interest to financial institutions and their regulators3—mean, variance, skewness, and kurtosis—it is the fourth moment, kurtosis, that should pose the deepest epistemic concern. Kurtosis eludes detection where it counts most—in its fat tails. Our expectations and perceptions may underestimate the most extreme risks by a significant margin. The overarching goal in financial responses to leptokurtosis and “fat tails” is the accurate forecasting of extreme events. Simple accuracy in description, if attainable and attained, would be a fantastic accomplishment.
James Ming Chen

Chapter 13. Parametric VaR Analysis

Leptokurtosis poses an especially keen threat to the economically informed evaluation of market risk in the trading books of major financial institutions (including, but not limited to, those deemed systemically important to global financial stability). Why this should be so warrants a quick look at global banking regulation. I will then describe VaR analysis and its parametric implementation under Gaussian assumptions.
James Ming Chen

Chapter 14. Parametric VaR According to Student’s t-Distribution

Parametric VaR “generalizes to other distributions as long as all the uncertainty is contained in σ.”1 If we are concerned that reliance on the Gaussian distribution systematically and inappropriately underestimates tail risk, we could substitute any “distribution [with] fatter tails than the normal.”2 Phillipe Jorion recommends the use of Student’s t-distribution with six degrees of freedom.3 I shall do my best to provide a theoretical justification for conducting parametric VaR according to the family of Student’s t-distributions. Even more importantly, I shall suggest an empirical basis for a more precise estimate of the number of degrees of freedom needed to calibrate Student’s t-distribution in response to observed levels of kurtosis.4
James Ming Chen

Chapter 15. Comparing Student’s t-Distribution with the Logistic Distribution

Bell curves come in different configurations. Like the family of multivariate Student’s t-distributions, the entire family of multivariate logistic distributions belongs to the same class of jointly elliptical distributions that includes the standard normal distribution.1 The simplest version of the logistic distribution2 provides an instructive contrast with the t-distributions we have used thus far to enhance the robustness of our parametric VaR analysis, relative to the Gaussian baseline.
James Ming Chen

Chapter 16. Expected Shortfall as a Response to Model Risk

Reducing the vulnerability of parametric VaR to model risk by improving its robustness addresses merely one threat to the reliability of VaR analysis. The most serious menace to VaR—and to the technique’s viability as the Basel Accords’ preferred approach to financial risk management—lies in VaR’s failure to satisfy the theoretical rigors demanded of “coherent” measures of risk.1 The expected shortfall for any confidence interval, which is derived directly from VaR for that interval, is subadditive and coherent. VaR itself is not.2 The allure of subadditivity and coherence supports Basel III’s embrace of expected shortfall as the international banking system’s preferred measure of market risk.
James Ming Chen

Chapter 17. Latent Perils: Stressed VaR, Elicitability, and Systemic Effects

Model risk, as demonstrated by the gap between VaR and its corresponding values for expected shortfall, is hardly the only threat to proper financial risk assessment. Even if we have properly modeled risk, whether by engaging in thorough nonparametric VaR, by specifying the proper parameters in a more accurate parametric model of value at risk, or by substituting more conservative (and coherent) values for expected shortfall in place of VaR, we cannot eliminate the problem of straightforward mistakes in estimation.1 Despite considerable advances in computation, the “fat finger” persists, in typography and in finance.2
James Ming Chen

Chapter 18. Finance as a Romance of Many Moments and Plural Views

The postmodern revival of portfolio theory is the story of an intellectual discipline coming to embrace the richness of economic accounts transcending its origins in austerely beautiful but excessively rigid mathematical models. Quantitative risk management harmonizes the making of policy in a wide range of domains, from the regulation of systemically important financial institutions to natural disaster prevention, mitigation, and recovery. Postmodern portfolio theory exploits the full range of sophisticated quantitative methods known in contemporary finance. Perhaps as importantly, postmodern portfolio theory acknowledges its own methodological limits. Comprehensive understanding of these tools, from their origins in early modern finance through contemporary postmodern critiques, places quantitative finance on the efficient frontier of risk management.
James Ming Chen


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