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2016 | OriginalPaper | Buchkapitel

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

verfasst von : James Ming Chen

Erschienen in: Postmodern Portfolio Theory

Verlag: Palgrave Macmillan US

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Abstract

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.

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Vorheriges Kapitel The Full Financial Toolkit of Partial Second Moments

Nächstes Kapitel Sinking, Fast and Slow: Relative Volatility Versus Correlation Tightening

Javier Estrada, The Cost of Equity of Internet Stocks: A Downside Risk Approach, 10 Eur. J. Fin. 239–254, 241 (2004).

Hogan & Warren, Chap. 5, supra note 33, at 10; accord James Chong & G. Michael Phillips, Measuring Risk for Cost of Capital: The Downside Beta Approach, 4 J. Corp. Treas. Mgmt. 344–352, 347 (2012); see also Bawa & Lindenberg, Chap. 5, supra note 35, at 197 (noting that a “mean-lower partial moment framework … is identical in form to the traditional Capital Asset Pricing Model obtained in the mean-variance (MV) framework,” with the substitution of conditional beta for beta across the full spectrum of returns).

See William F. Sharpe, Mutual Fund Performance, 39 J. Bus. 119–138, 123 (1966); William F. Sharpe, Adjusting for Risk in Portfolio Performance Measurement. 1:2 J. Portfolio Mgmt. 29–34 (Winter 1975).

See Frank A. Sortino & Robert van der Meer, Downside Risk, 17:4 J. Portfolio Mgmt. 27–31 (Summer 1991). See generally Frank A. Sortino & Stephen Satchell, Measuring Downside Risk in Financial Markets: Theory, Practice and Implementation (2001).

See, e.g., James Clash, Focus on the Downside, Forbes, Feb. 12, 1999, at 162–163; Estrada, Cost of Equity in Internet Stocks, supra note 1, at 241.

See supra § 5.1, at 59–60.

“Moore’s law is the observation that over the history of computing hardware, the number of transistors on integrated circuits doubles approximately every two years.” http://en.wikipedia.org/wiki/Moore's_law. Computing power is often assumed to double every 18 months, based on “a doubling in chip performance,” which in turn combines the effects of more transistors and greater processing speed. Id. See generally Gordon E. Moore, Cramming More Components onto Integrated Circuits, Electronics Mag., Aug. 1965, at 4–7 (available at http://www.monolithic3d.com/uploads/6/0/5/5/6055488/gordon_moore_1965_article.pdf). For a journalistic assessment of the economic and sociological impact of Moore’s law, see Jonathan Rauch, The New Old Economy: Oil, Computers, and the Reinvention of the Earth, Atlantic Monthly, January 2001, at 35–49 (available at http://www.theatlantic.com/past/docs/issues/2001/01/rauch.htm).

See Markowitz, Todd, Xu & Yamane, Chap. 5, supra note 9.

See Fama & French, The Cross-Section of Stock Returns, Chap. 4, supra note 4; Fama & French, Size and Book-to-Market Factors, Chap. 4, supra note 49.

See generally, e.g., David Lamb & Susan M. Easton, Multiple Discovery: The Pattern of Scientific Progress (1984); Robert K. Merton, Resistance to the Systematic Study of Multiple Discoveries in Science, 4 Eur. J. Sociol. 237–282 (1963), reprinted in Robert K. Merton, The Sociology of Science: Theoretical and Empirical Investigations 371–382 (1973).

See Robert K. Merton, Singletons and Multiples in Scientific Discovery: a Chapter in the Sociology of Science, 105 Proc. Am. Phil. Soc’y 470–486 (1961), reprinted in Merton, The Sociology of Science, supra note 10, at 343–370.

Compare Roy, Safety First, Chap. 5, supra note 10, with Fishburn, Chap. 5, supra note 39.

See http://en.wikipedia.org/wiki/Moment_(mathematics)#Partial_moments.

See http://en.wikipedia.org/wiki/Moment_(mathematics).

See William F. Sharpe, The Sharpe Ratio, 21:1 J. Portfolio Mgmt. 49–58 (Fall 1994).

See id.

See Bawa, Optimal Rules, Chap. 5, supra note 39; Fishburn, Chap. 5, supra note 39; W.V. Harlow, Asset allocation in a Downside Risk Framework, 47:5 Fin. Analysts J. 28–40, 30 (Sept./Oct. 1991); Harlow & Rao, Chap. 5, supra note 37; http://en.wikipedia.org/wiki/Moment_(mathematics).

See Li Chen, Simai He & Shuzhong Zhang, When All Risk-Adjusted Performance Measures Are the Same: In Praise of the Sharpe Ratio, 11 Quant. Fin. 1439–1447 (2011).

See, e.g., Frank A. Sortino, From Alpha to Omega , Managing Downside Risk in Financial Markets, Chap. 5, supra note 19, at 3–25, 10; Sortino, van der Meer & Plantinga, Chap. 4, supra note 36. Others have also endorsed the use of downside semideviation as a risk measure. See, e.g., Clash, supra note 5; Estrada, Cost of Equity in Internet Stocks, supra note 1, at 241.

Robert Libby & Peter C. Fishburn, Behavioral Models of Risk Taking in Business Decisions, 15 J. Accounting Research 272–292, 277 (1977); accord Harlow & Rao, Chap. 5, supra note 37, at 292.

See, e.g., Bawa & Lindenberg, Chap. 5, supra note 35, at 192 n.3 (acknowledging that portfolio optimization according to semivariance “can be solved for any fixed point”); Harlow & Rao, Chap. 5, supra note 37, at 286 (devising a “generalized Mean-Lower Partial Moment” model “consistent with any prespecified target rate of return” [emphasis in original]); id. at 287 (obtaining portfolio equilibrium “for arbitrary τ” as part of “a generalized…asset pricing framework” making use of mean lower partial lower moments to any order n).

Feibel, Chap. 5, supra note 39, at 160.

E.g., Chong & Phillips, supra note 2, at 347. Other sources take pains to specify that it is semivariance rather than semideviation that is straightforwardly additive. See, e.g., Estrada, An Alternative Behavioural Model, Chap. 5, supra note 51, at 231, 237; Estrada, Downside Risk and Capital Asset Pricing, Chap. 5, supra note 42, at 177 n.4.

http://www.morningstar.com/InvGlossary/upside-downside-capture-ratio.aspx.

See generally sources cited, Chap. 4, supra note 14.

Paul D. Kaplan & James A. Knowles, Kappa: A Generalized Downside Risk-Adjusted Performance Measure 2 (2004) (available at http://corporate.morningstar.com/NO/documents/MethodologyDocuments/ResearchPapers/KappaADownsideRisk_AdjustedPerformanceMeasure_PK.pdf).

See William F. Shadwick & Con Keating, A Universal Performance Measure, 6:3 J. Performance Measurement 59–84 (Spring 2002).

See Kaplan & Knowles, supra note 26, at 15.

See http://en.wikipedia.org/wiki/Moment_(mathematics).

See Kaplan & Knowles, supra note 26, at 15.

See id. at 3.

See Hossein Kazemi, Thomas Schneeweis & Raj Gupta, Omega as a Performance Measure (June 15, 2003) (available at http://faculty.fuqua.duke.edu/~charvey/Teaching/BA453_2006/Schneeweis_Omega_as_a.pdf) (developing a closely related measure called Sharpe-Omega).

See Kaplan & Knowles, supra note 26, at 3 n.1.

See, e.g., Philippe Bertrand & Jean-Luc Prigent, Omega Performance Measures and Portfolio Insurance, 35 J. Banking & Fin. 1811–1823 (2011); Theofanis Darsinos & Stephen Satchell, Generalising Universal Performance Measures, Risk, June 2004, at 80–84 (available at http://www.risk.net/data/Pay_per_view/risk/technical/2004/0604_tech_investment.pdf) (proposing an entire family of measures, like the omega ratio, that directly compare upper and lower partial moments of the same degree); S.J. Kane, M.C. Bartholomew-Biggs, M. Cross & M. Dewar, Optimizing Omega, 5 J. Global Optimization 153–167 (2009); Helmut Mausser, David Saunders & Luis Seco, Optimising Omega, Risk, Nov. 2006, at 88–92 (available at http://www.risk.net/data/risk/pdf/technical/risk_1106_Mausser.pdf).

Kaplan & Knowles, supra note 26, at 3.

Cf. Denisa Cumova & David Nawrocki, Portfolio Optimization in an Upside Potential and Downside Risk Framework 9 (Oct. 2003) (available at http://www90.homepage.villanova.edu/michael.pagano/DN%20upm%20lpm%20measures.pdf) (proposing the possibility of applying separate exponents to upper and lower partial moments to reflect different levels of risk-seeking or risk-averse behavior in individual investors).

See Harlow & Rao, Chap. 5, supra note 37.

David Nawrocki, A Brief History of Downside Risk Measures, 8:3 J. Investing 9–25 (Fall 1999).

Campbell R. Harvey, John C. Liechty, Merrill W. Liechty & Peter Müller, Portfolio Selection with Higher Moments, 10 Quant. Fin. 469–485, 471 (2010).

See http://www.wolframalpha.com/input/?i=plot+%283%5E.6-1%29%2F%283-1%29*x%5E3+%2B+%283-3%5E.6%29%2F%283-1%29*x%5E2+and+x%5E2.6+for+x%3D0+to+3.

Harvey, Liechty, Liechty & Müller, supra note 39, at 471.

Subrahmanyan Chandrasekhar, Truth and Beauty: Aesthetics and Motivations in Science 148 (1987).

See Richard P. Feynman, The Feynman Lectures on Physics 22 (1977) (describing Euler’s identity as “our jewel”—indeed, as “the most remarkable formula in mathematics”).

Serge Lange, The Beauty of Mathematics: Three Public Dialogues 3 (1985).

Titel: Sortino, Omega, Kappa: The Algebra of Financial Asymmetry
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_6