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2016 | OriginalPaper | Buchkapitel

5. The Full Financial Toolkit of Partial Second Moments

verfasst von : James Ming Chen

Erschienen in: Postmodern Portfolio Theory

Verlag: Palgrave Macmillan US

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Abstract

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

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Vorheriges Kapitel Seduced by Symmetry, Smarter by Half

Nächstes Kapitel Sortino, Omega, Kappa: The Algebra of Financial Asymmetry

See James S. Ang & Jess H. Chua, Composite Measures for the Evaluation of Investing Performance, 14 J. Fin. & Quant. Analysis 361–384 (1979); Robert C. Klemkosky, The Bias in Composite Performance Measures, 8 J. Fin. & Quant. Analysis 505–514 (1973); Hendrik Scholz, Refinements to the Sharpe Ratio: Comparing Alternatives for Bear Markets, 7 J. Asset Mgmt. 347–357 (2007).

See, e.g., Fred D. Arditti, Risk and the Required Return in Equity, 22 J. Fin. 19–36 (1967) (analyzing the relationship between expected return and skewness in the distribution of returns); Merton H. Miller & Myron S. Scholes, Rates of Return with Relation to Risk: A Reexamination of Some Recent Findings, in Studies in the Theory of Capital Markets 47–78 (Michael C. Jensen ed., 1972) (subjecting the CAPM to testing in response to asymmetry in the distribution of returns).

Justin Fox, The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street 86 (2009).

See Harry M. Markowitz, Portfolio Selection 188–194, 287–297 (1959); James C.T. Mao, Models of Capital Budgeting: E-V Vs. E-S, 4 J. Fin. & Quant. Analysis 657–675 (1970).

William F. Sharpe, Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, 19 J. Fin. 425–442, 428 n.8 (1964).

See James P. Quirk & Rubin Saposnik, Admissibility and Measurable Utility Functions, 29 Rev. Econ. Stud. 140–146 (1962).

James C.T. Mao, Survey of Capital Budgeting: Theory and Practice, 25 J. Fin. 349–360, 354 (1970).

See Philip L. Cooley, A Multidimensional Analysis of Institutional Investor Perception of Risk, 32 J. Fin. 67–78 (1977).

See Harry Markowitz, Peter Todd, Ganlin Xu & Yuji Yamane, Computation of Mean–Semivariance Efficient Sets by the Critical Line Algorithm, 45 Annals Oper. Research 307–317 (1993).

See Arthur D. Roy, Safety First and the Holding of Assets, 20 Econometrica 431–449 (1952).

Id. at 433; accord Levy, Chap. 4, supra note 13, at 45 n.1.

Harry M. Markowitz, Mean-Variance Analysis in Portfolio Choice and Capital Markets 37 (1987).

See Arthur D. Roy, Risk and Rank in Safety-First Generalized, 23 Economica 214–228 (1956). See generally Haim Levy & Marshall Sarnat, Safety First—An Expected Utility Principle, 7 J. Fin. & Quant. Analysis 1829–1834 (1972); David H. Pyle & Stephen J. Turnowsky, Safety-First and Expected Utility Maximization in Mean-Standard Deviation Portfolio Analysis, 52 Rev. Econ. & Stat. 75–81 (1970).

Id. The other paper was Harry M. Markowitz, Foundations of Portfolio Theory, 46 J. Fin. 469–477, 469–470 (1991). Other sources trace the origins of portfolio selection to Helen Makower & Jacob Marschak, Assets, Prices and Monetary Theory, 5 Economica 261–288 (1938) and Jacob Marschak, Money and the Theory of Assets, 6 Econometrica 311–325 (1938). See Fox, supra note 3, at 347.

See William J. Baumol, An Expected Gain-Confidence Limit Criterion for Portfolio Selection, 10 Mgmt. Sci. 174–182 (1963); Paul Halpern & Yehuda Kahane, A Pedagogical Note on Baumol’s Gain-Confidence Limit Criterion for Portfolio Selection and the Probability of Ruin, 4 J. Banking & Fin. 189–195 (1980).

Markowitz, Mean-Variance Analysis, supra note 12, at 38.

Id.

Mao, Survey of Capital Budgeting, supra note 7, at 354.

Leslie A. Balzer, Investment Risk: A Unified Approach to Upside and Downside Returns, in Managing Downside Risk in Financial Markets: Theory, Practice and Implementation 103–155, 115 (Frank A. Sortino & Stephen E. Satchell eds., 2001).

See Hersh Shefrin & Meir Statman, Behavioral Portfolio Theory, 35 J. Fin. & Quant. Analysis 127–151 (2000).

V.I. Norkin & S.V. Boyko, Safety-First Portfolio Selection, 48 Cybernetics & Sys. Analysis 180–191, 180 (2012).

Shefrin & Statman, Behavioral Portfolio Theory, supra note 20, at 130.

Id.

See id.

See Shinji Kataoka, A Stochastic Programming Model, 31 Econometrica 181–196 (1963).

See Leslie G. Telser, Safety-First and Hedging, 23 Rev. Fin. Stud. 1–16 (1955).

See Enrique R. Arzac & Vijay S. Bawa, Portfolio Choice and Equilibrium in Capital Markets with Safety-First Investors, 4 J. Fin. Econ. 277–288 (1977).

Shefrin & Statman, Behavioral Portfolio Theory, supra note 20, at 131.

See, e.g., Yuanyao Ding & Bo Zhang, Optimal Portfolio of Safety-First Models, 139 J. Stat. Planning & Inference 2952–2962 (2009); Haim Levy & Moshe Levy, The Safety First Expected Utility Model: Experimental Evidence and Economic Implications, 33 J. Banking & Fin. 1494–1506 (2009) (proposing an optimization approach that uses the weighted average of expected utility and a safety-first maximization of utility).

See D. Li, T.F. Chan & W.L. Ng, Safety-First Dynamic Portfolio Selection, 4 Dynamics Continuous, Discrete & Impulsive Sys. 585–600 (1998); Wei Yan, Continuous-Time Safety-First Portfolio Selection with Jump-Diffusion Processes, 43 Int’l J. Sys. Sci. 622–628 (2012).

See Dennis W. Jansen, Kees G. Koedijk & Casper G. de Vries, Portfolio Selection with Limited Downside Risk, 7 J. Empirical Fin. 247–269 (2000).

See Mahfuzul Haque, M. Kabir Hassan & Oscar Varela, Safety-First Portfolio Optimization for U.S. Investors in Emerging Global, Asian and Latin American Markets, 12 Pac. Basin Fin. J. 91–116 (2004); Mahfuzul Haque, Oscar Varela & M. Kabir Hassan, Safety-First and Extreme Value Bilateral U.S.-Mexican Portfolio Optimization Around the Peso Crisis and NAFTA in 1994, 47 Q. Rev. Econ. & Fin. 449–469 (2007).

William W. Hogan & James M. Warren, Toward the Development of an Equilibrium Capital-Market Model Based on Semivariance, 9 J. Fin. & Quant. Analysis 1–11 (1974).

See id. at 5 & n.2; R. Burr Porter, Roger P. Bey & David C. Lewis, The Development of a Mean-Semivariance Approach to Capital Budgeting, 10 J. Fin. & Quant. Analysis 639–649 (1975).

See Vijay S. Bawa & Eric B. Lindenberg, Capital Market Equilibrium in a Mean-Lower Partial Moment Framework, 5 J. Fin. Econ. 189–200, 191, 198 (1977).

See id. at 197; Hogan & Warren, supra note 33, at 10; cf. Javier Estrada, Systematic Risk in Emerging Markets: The D-CAPM, 3 Emerging Mkts. Rev. 365–377, 370 (2002).

See W.V. Harlow & Ramesh K.S. Rao, Asset Pricing in a Generalized Mean-Lower Partial Moment Framework: Theory and Evidence, 24 J. Fin. & Quant. Analysis 285–311 (1989).

See id. at 291 (demonstrating how a second-order mean lower partial moment model assuming normally distributed returns and adopting the risk-free rate as the target return yields the conventional CAPM).

See Harlow & Rao, supra note 37, at 286–292. On lower partial moments, compare Vijay S. Bawa, Optimal Rules for Ordering Uncertain Prospects, 2 J. Fin. Econ. 95–121 (1975) with Peter C. Fishburn, Mean-Risk Analysis with Risk Associated with Below-Target Returns, 67 Am. Econ. Rev. 116–126 (1977). See generally Bruce J. Feibel, Investment Performance Measurement 155–164 (2003); Stephen E. Satchell, Lower Partial-Moment Capital Asset Pricing Models: A Re-Examination, in Managing Downside Risk in Financial Markets, supra note 19, at 156–168.

Compare Harlow & Rao, supra note 37, at 286 with Estrada, Systematic Risk in Emerging Markets, supra note 36, at 369.

See Estrada, Systematic Risk in Emerging Markets, supra note 36, at 368.

See id. at 369–370 & n.2; Javier Estrada, Mean-Semivariance Behavior: Downside Risk and Capital Asset Pricing, 16 Int’l Rev. Econ. & Fin. 169–185, 174 (2007); Hsin-Jung Tsai, Ming-Chi Chen & Chih-Yuan Yang, A Time-Varying Perspective on the CAPM and Downside Betas, 29 Int’l Rev. Econ. & Fin. 440–454, 441 (2014).

Don U.A. Galagedera, An Alternative Perspective on the Relationship Between Downside Beta and CAPM Beta, 8 Emerging Mkts. Rev. 4–19, 7 (2007). For fuller mathematical elaboration of Estrada’s measure alongside those of Bawa & Lindenberg, supra note 35, and Harlow & Rao, supra note 37, see Galagedera, supra, at 6–7, 17–19.

See, e.g., Fischer Black, Capital Market Equilibrium with Restricted Borrowing, 45 J. Bus. 444–455 (1972); Fischer Black, Michael C. Jensen & Myron S. Scholes, The Capital Asset Pricing Model: Some Empirical Tests, in Studies in the Theory of Capital Markets, supra note 2, at 79–121; John Lintner, Security Prices, Risk and Maximal Gains from Diversification, 20 J. Fin. 587–615 (1965); John Lintner, The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets, 73 Rev. Econ. & Stats. 13–37 (1965); Jan Mossin, Equilibrium in a Capital Asset Market, 34 Econometrica 768–783 (1966); William F. Sharpe, Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, 19 J. Fin. 425–442 (1964); Jack L. Treynor, Toward a Theory of Market Value of Risky Assets, in Asset Pricing and Portfolio Performance: Models, Strategy and Performance Metrics 15–22 (Robert A. Korajczyk ed., 1999); Jack L. Treynor & Fischer Black, Corporate Investment Decisions, in Modern Developments in Financial Management 310–327 (Stewart C. Myers ed., 1976). See generally Bernell K. Stone, Risk, Return, and Equilibrium: A General Single-Period Theory of Asset Selection and Capital Market Equilibrium (1970); Eugene F. Fama & Kenneth R. French, The Capital Asset Pricing Model: Theory and Evidence, 18:3 J. Econ. Persp. 25–46 (Summer 2004).

See generally William H. Press, Brian P. Flannery, Saul A. Teukolsky & William T. Vetterling, Numerical Recipes in Fortran 77: The Art of Scientific Computing “§ 14.1, at 604–609” (2d ed. 1992) (Moments of a Distribution: Mean, Variance, Skewness, and So Forth.).

See Liebowitz, Bova & Hammond, Chap. 4, supra note 29, at 14 (defining beta as “the correlation between the asset (or portfolio) return and the market return, multiplied by the ratio of their volatilities”); Michael B. Miller, Mathematics and Statistics for Financial Risk Management 198, 213, 292 (2d ed. 2014) (defining beta as the product of correlation between the returns on two assets and the ratio of their volatilities); Shannon P. Pratt & Roger J. Grabowski, Cost of Capital: Applications and Examples 305–306 (4th ed. 2010).

See Estrada, Downside Risk and Capital Asset Pricing, supra note 42, at 172; Estrada, Systematic Risk in Emerging Markets, supra note 36, at 368–369.

See Estrada, Downside Risk and Capital Asset Pricing, supra note 42, at 171; Estrada, Systematic Risk in Emerging Markets, supra note 36, at 367.

See Galagedera, An Alternative Perspective, supra note 43, at 16 (concluding that beta as defined by Bawa & Lindenberg, supra note 35, “appears to be a better measure of systematic risk” when “securities have abnormal returns” and that beta as defined by Harlow & Rao, supra note 37, “is more suitable as a measure of systematic risk” in “markets whose returns distributions have high kurtosis”); Tsai, Chen & Yang, supra note 42, at 446 (suggesting that definitions of beta based on Hogan & Warren, supra note 33, and Harlow & Rao, supra note 37, “had more explanatory power for the expected stock market compared with … other” definitions of beta).

See Estrada, Systematic Risk in Emerging Markets, supra note 36, at 366; Bawa & Lindenberg, supra note 35, at 191, 198.

See Javier Estrada, Mean-Semivariance Behaviour: An Alternative Behavioural Model, 3 J. Emerging Mkt. Fin. 231–248, 242 (2004) (validating this analytical observation through empirical data).

See Turan G. Bali, Nusret Cakici & Robert F. Whitelaw, Hybrid Tail Risk and Expected Stock Returns: When Does the Tail Wag the Dog?, 4 Rev. Asset Pricing Stud. 206–246 (2014) (concluding that a measure of stock return tail covariance risk reports a positive, significant relationship with expected returns only when covariance is measured across the left tail of the distribution of returns, and not over entirety of the distribution).

One salient exception is Andrew Ang, Joseph Chen & Yuhang Xing, Downside Risk, 19 Rev. Fin. Stud. 1191–1239, 1199–1200 (2006) (introducing “two additional measures” beyond regular, unconditional beta: relative upside beta and relative downside beta).

See Fishburn, supra note 39, at 116. Partial moments are Lebesgue–Stieltjes integrals. See generally Paul R. Halmos, Measure Theory §§ 15.9, 18.11, 25.4, at 67, 80, 106 (2d ed. 2013).

See Estrada, Downside Risk and Capital Asset Pricing, supra note 42, at 173; Estrada, Systematic Risk in Emerging Markets, supra note 36, at 369–370.

See Estrada, Systematic Risk in Emerging Markets, supra note 36, at 369–370 & n.2.

Titel: The Full Financial Toolkit of Partial Second Moments
verfasst von: James Ming Chen
Verlag: Palgrave Macmillan US
Buch: Postmodern Portfolio Theory
Print ISBN: 978-1-137-54463-6

Electronic ISBN: 978-1-137-54464-3

Copyright-Jahr: 2016
DOI: https://doi.org/10.1057/978-1-137-54464-3_5