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2016 | OriginalPaper | Buchkapitel

10. A Four-Moment Capital Asset Pricing Model

verfasst von : James Ming Chen

Erschienen in: Postmodern Portfolio Theory

Verlag: Palgrave Macmillan US

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Abstract

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.

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Vorheriges Kapitel Asymmetric Volatility and Volatility Spillovers

Nächstes Kapitel The Practical Implications of a Spatially Bifurcated Four-Moment Capital Asset Pricing Model

Javier Estrada, Systematic Risk in Emerging Markets: The D-CAPM, 3 Emerging Mkts. Rev. 365–379, 366 (2002).

Levy, CAPM in the 21st Century, Chap. 8, supra note 13, at 4.

Id. (emphasis in original).

Id. at 184.

Don U.A. Galagedera & Robert D. Brooks, Is Co-Skewness a Better Measure of Risk in the Downside Than Downside Beta?, 17 J. Multinat’l Fin. Mgmt. 214–230, 215 (2007).

Timothy J. Nantell & Barbara Price, An Analytical Comparison of Variance and Semivariance Capital Market Theories, 14 J. Fin. & Quant. Analysis 221–242, 231 (1979).

Campbell R. Harvey & Akhtar Siddique, Conditional Skewness in Asset Pricing Tests, 55 J. Fin. 1263–1295, 1264 (2000).

Harry Markowitz, Portfolio Selection, 7 J. Fin. 77–91, 91 (1952); accord Campbell R. Harvey, John C. Liechty, Merrill W. Liechty & Peter Müller, Portfolio Selection with Higher Moments, 10 Quant. Fin. 469–485, 469 (2010).

See generally, e.g., Fred D. Arditti, Risk and the Required Return in Equity, 22 J. Fin. 19–36 (1967); Robert F. Dittmar, Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross-Section of Equity Returns, 57 J. Fin. 369–403 (2002); Irwin Friend & Randolf Westerfield, Co-Skewness and Capital Asset Pricing, 35 J. Fin. 897–913 (1980); Jonathan E. Ingersoll, Jr., Multidimensional Asset Pricing, 10 J. Fin. & Quant. Analysis 785–798 (1975) Alan Kraus & Robert L. Linzenberger, Skewness Preference and the Valuation of Risk Assets, 31 J. Fin. 1085–1100 (1976); Haim Levy, A Utility Function Depending on the First Three Moments, 24 J. Fin. 715–719 (1969); Kian-Guan Lim, A New Test of the Three-Moment Capital Asset Pricing Model, 24 J. Fin. & Quant. Analysis 205–216 (1989); Mark E. Rubinstein, The Fundamental Theorem of Parameter-Preference Security Valuation, 8 J. Fin. & Quant. Analysis 61–69 (1973); Paul Samuelson, The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances and Higher Moments, 37 Rev. Econ. Stud. 537–542 (1970).

Gustavo M. de Athayde & Renato G. Flôres, Jr., Finding a Maximum Skewness Portfolio—A General Solution to Three-Moments Portfolio Choice, 28 J. Econ. Dynamics & Control 1335–1352, 1336 (2004).

Nantell & Price, supra note 6, at 222.

See Kelly Price, Barbara Price & Timothy J. Nantell, Variance and Lower Partial Moment Measures of Systematic Risk: Some Analytical and Empirical Results, 37 J. Fin. 843–855, 854 (1982).

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

Id.

Harvey & Siddique, supra note 7, at 1264.

Estrada, Systematic Risk in Emerging Markets, supra note 1, at 366; Harvey, Liechty, Liechty & Müller, supra note 6, at 469 (describing the “us[e] [of] negative semi-variance in place of variance” as a “three moment optimization method[]”).

See, e.g., Malcolm P. Baker & Jeffrey Wurgler, Comovement and Predictable Relations Between Bonds and the Cross-Section of Stocks, 2 Rev. Asset Pricing Stud. 57–87 (2012); François Longin & Bruno Solnik, Extreme Correlation of International Equity Markets, 56 J. Fin. 649–676 (2001).

James A. Xiong & Thomas M. Idzorek, The Impact of Skewness and Fat Tails on the Asset allocation Decision, 67:2 Fin. Analysts J. 23–35, 30 (March/April 2011).

Giovanni Barone Adesi, Patrick Gagliardini & Giovanni Urga, Testing Asset Pricing Models with Coskewness, 22 J. Bus. & Econ. Stat. 474–485, 474 (2004).

See Javier Estrada & Ana Paula Serra, Risk and Return in Emerging Markets: Family Matters, 15 J. Multinat’l Fin. Mgmt. 257–272, 259 (2004).

See generally Raymond Kan & Chu Zhang, GMM Test of Stochastic Discount Factor Models with Useless Factors, 54 J. Fin. Econ. 103–127 (1999); Raymond Kan & Chu Zhang, Two-Pass Tests of Asset Pricing Models with Useless Factors, 54 J. Fin. 203–235 (1999).

Barone Adesi, Gagliardini & Urga, supra note 19, at 474.

Don U.A. Galagedera, An Alternative Perspective on the Relationship Between Downside Beta and CAPM Beta, 8 Emerging Mkts. Rev. 4–19, 16 (2007).

Don U.A. Galagedera, Economic Significance of Downside Risk in Developed and Emerging Markets, 16 Applied Econ. Letters 1627–1632, 1627 (2009).

Galagedera & Brooks, supra note 5, at 215.

Compare Thierry Post & Pim Van Vliet, Downside Risk and Asset Pricing, 30 J. Banking & Fin. 823–849 (2006) (accepting downside beta as a risk measure) with Galagedera & Brooks, supra note 5, at 215 (dismissing Post and Van Vliet’s conclusions).

Don U.A. Galagedera, Elizabeth A. Maharaj & Robert Brooks, Relationship Between Downside Risk and Return: New Evidence Through a Multiscaling Approach, 18 Applied Fin. Econ. 1623–1633, 1631 (2008).

Estrada & Serra, supra note 20, at 268.

Don U.A. Galagedera, Relationship Between Systematic Risk Measured in the Second-Order and Third-Order Co-Moments in the Downside Framework, 3 Applied Fin. Econ. Letters 147–153, 152 (2007); see also Galagedera, Economic Significance, supra note 24, at 1632 (urging “further investigation of co-skewness as…a measure of risk in the downside especially in emerging markets”).

See Gordon Y.N. Tang & Wai Cheong Shum, The Risk-Return Relations in the Singapore Stock Market, 12 Pac.-Basin Fin. J. 179–195 (2004).

Andrew Ang & Joseph Chen, Asymmetric Correlations of Equity Portfolios, 63 J. Fin. Econ. 443–494, 445 (2002).

Barone Adesi, Gagliardini & Urga, supra note 19, at 482.

Id. at 483.

Id.

Ang & Chen, supra note 31, at 471; see also id. at 469 (“we are not capturing the same information in [correlation asymmetry] as skewness and co-skewness”).

Andrew Ang, Joseph Chen & Yuhang Xing, Downside Risk, 19 Rev. Fin. Stud. 1191–1239, 1193 (2006).

Id.

Id. Compare id. at 1194 (“stocks with very high volatility exhibit anomalously low returns”) with id. at 1232 (“it remains to be explored why the cross-sectional relation for downside risk does not hold for stocks with very high levels of volatility”).

Id. at 1193.

Id. at 1194.

Id.

Lakshman Alles & Louis Murray, Rewards for Downside Risk in Asian Markets, 37 J. Banking & Fin. 2501–2509, 2504 (2013).

Ang, Chen & Xing, supra note 36, at 1194.

Alles & Murray, supra note 42, at 2504; see also Ang, Chen & Xing, supra note 36, at 1199 (“Downside beta and coskewness may potentially capture different effects.”).

Alles & Murray, supra note 42, at 2508; see also id. at 2509 (assigning a 12 % price premium to downside beta and a 7 % price premium to negative coskewness).

Ang, Chen & Xing, supra note 36, at 1194; accord id. at 1227 (“the predictive pattern for cross-sectional returns from past coskewness is not picking up downside risk”).

Id. at 1193. On time-varying cokewness, see generally Campbell R. Harvey & Akhtar Siddique, Autoregressive Conditional Skewness, 34 J. Fin. & Quant. Analysis 465–487 (1999).

Alles & Murray, supra note 42, at 2507.

See Brian H. Boyer, Todd Mitton & Keith Vorkink, Expected Idiosyncratic Skewness, 23 Rev. Fin. Stud. 169–202, 172 (2010).

Id.

Alles & Murray, supra note 42, at 2507.

Id.

Massimo Guidolin & Allan Timmermann, Optimal Portfolio Choice Under Regime Switching, Skewness and Kurtosis Preferences, Federal Reserve Bank of St. Louis Working Paper 2005-006A, at 2 (Jan. 2005) (available at http://research.stlouisfed.org/wp/2005/2005-006.pdf).

See Tobias Adrian & Joshua Rosenberg, Stock Returns and Volatility: Pricing the Short-Run and Long-Run Components of Market Risk, 63 J. Fin. 2997–3030 (2008).

Campbell R. Harvey, Drivers of Expected Returns in International Markets, 1:1 Emerging Mkts. Q. 32–49, 46 (Fall 2000).

See, e.g., Rohan Christie David & Kukesh Chaudhry, Coskewness and Cokurtosis in Futures Markets, 8 J. Empirical Fin. 55–81 (2001); Kim Hiang Liow & Lanz Chan, Co-Skewness and Co-Kurtosis in Global Real Estate Securities, 22 J. Prop. Research 163–203 (2005).

See generally K.V. Mardia, Measures of Multivariate Skewness and Kurtosis with Applications, 57 Biometrika 519–530 (1970).

National Institute of Standards and Technology, eHandbook of Statistical Methods § 1.3.5.11 (April 2012) (“measures of skewness and kurtosis”) (available at http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm).

https://en.wikipedia.org/wiki/Skewness . See generally Barry C. Arnold & Richard A. Groeneveld, Measuring Skewness with Respect to the Mode, 49 Am. Statistician 34 (1995); Paul T. von Hippel, Mean, Median, and Skew: Correcting a Textbook Rule, 13:2 J. Stat. Educ. (July 2005) (available at http://www.amstat.org/publications/jse/v13n2/vonhippel.html); H.L. MacGillivray, Skewness and Asymmetry: Measures and Orderings, 14 Annals Stat. 994–1011 (1986); J.C.W. Rayner, D.J. Best & K.L. Matthews, Interpreting the Skewness Coefficient, 24 Communications in Stat.: Theory & Methods 593–600 (1995).

See Athayde & Flôres, supra note 10.3, at 1342; Harvey, Liechty, Liechty & Müller, supra note 8.3, at 476, 482; Harvey & Siddique, supra note 7, at 1271.

Kevin P. Balanda & H.L. MacGillivray, Kurtosis: A Critical Review, 42 Am. Statistician 111–119, 111 (1988).

https://en.wikipedia.org/wiki/Kurtosis.

Id.

Lawrence T. DeCarlo, On the Meaning and Use of Kurtosis, 2 Psych. Methods 292–307, 300 (1997).

Id.; cf. Hurst, Chap. 8, supra note 39; Koutsoyiannis, Chap. 8, supra note 39. See generally Thomas P. Hettmansperger & Michael A. Keenan, Tailweight, Statistical Inference and Families of Distributions—A Brief Survey, in 1 A Modern Course on Statistical Distributions in Scientific Work: Models and Structures 161–172 (G.P. Patel, S. Kotz & J.K. Ord eds., 1975).

Ronald L. Horswell & Stephen W. Looney, Diagnostic Limitations of Skewness Coefficients in Assessing Departures from Univariate and Multivariate Normality, 22 Communications in Stat.: Simulation & Computation 437–459, 437 (1993).

David P. Doane & Lori E. Seward, Measuring Skewness: A Forgotten Statistic?, 19:2 J. Stat. Educ. (July 2011) (available at http://www.amstat.org/publications/jse/v19n2/doane.pdf).

Id. See generally Richard A. Groeneveld & Glen Meeden, Measuring Skewness and Kurtosis, 33 J. Royal Stat. Soc’y, Series D 391–399 (1984); D.N. Joanes & C.A. Gill, Comparing Measures of Sample Skewness and Kurtosis, 47 J. Royal Stat. Soc’y: Series D 183–189 (1998).

Balanda & MacGillivray, supra note 62, at 111.

Id. at 119; see also id. at 111 (“Like location, scale, and skewness, kurtosis should be viewed as a ‘vague concept’ that can be formalized in many ways.”).

Frederick Mosteller & John W. Tukey, Data Analysis and Regression 17 (1977); see also id. at 19 (“Sometimes … specific concepts come first, and [a] vague concept is sought out to help …. More often, however, the vague concept comes first and guides us in identifying corresponding specific concepts.”).

See generally Emmanuel Jurczenko & Bertrand Maillet, The Four-Moment Capital Asset Pricing Model: Between Asset Pricing and Asset allocation, in Multi-Moment Asset allocation and Pricing Models 113–164 (Emmanuel Jurczenko & Bertrand Maillet eds., 2006); sources cited supra note 9.2.

Harvey, Liechty, Liechty & Müller, supra note 8.3, at 471; accord Athayde & Flôres, supra note 10, at 1342.

See Eric Jondeau & Michael Rockinger, Optimal Portfolio Allocation Under Higher Moments, 12 Eur. Fin. Mgmt. 29–55, 33 (2006); Jakša Cvitanić, Vassilis Polimenis & Fernando Zapatero, Optimal Portfolio Alocaiton with Higher Moments, 4 Annals Fin. 1–28 (2008).

See Yannick Malevergne & Didier Sornette, Higher-Moment Portfolio Theory, 31:4 J. Portfolio Mgmt. 49–55, 52 (Summer 2005) (evaluating the sixth- and eighth-order central moments of the distribution of returns on selected stocks).

Jondeau & Rockinger, supra note 75, at 30.

Harvey, Liechty, Liechty & Müller, supra note 8, at 470.

Id. at 469.

Jondeau & Rockinger, supra note 75, at 33.

See https://en.wikipedia.org/wiki/Taylor_series. Unless otherwise noted, background information on the mathematics of the Taylor series expansion comes from this source. The special case of a Taylor series where a = 0 is often designated a Maclaurin series.

Jondeau & Rockinger, supra note 75, at 33.

Javier Estrada, Mean-Semivariance Behaviour: An Alternative Behavioural Model, 3 J. Emerging Mkt. Fin. 231–248, 241 (2004).

See id.

John Y. Campbell, Andrew W. Lo & A. Craig MacKinlay, The Econometrics of Financial Markets 11 (1997).

Rajnish Mehra & Edward C. Prescott, The Equity Premium Puzzle in Retrospect, in Handbook of the Economics of Finance 888–936, 889 (George M. Constantinides, Milton Harris & René M. Stulz eds., 2003).

Id. at 888.

Campbell, Lo & MacKinlay, supra note 85, at 11.

See https://en.wikipedia.org/wiki/Taylor_series.

See http://www.wolframalpha.com/input/?i=taylor+series+for+ln%281%2Bx%29+at+x%3Dy.

Harvey & Siddique, supra note 7, at 1269.

Estrada, An Alternative Behavioural Model, supra note 83, at 241.

See https://en.wikipedia.org/wiki/Taylor%27s_theorem.

Estrada, An Alternative Behavioural Model, supra note 83, at 241.

Id. at 246.

See, e.g., Balzer, Chap. 8, supra note 68, at 121.

See, e.g., NIST eHandbook of Statistical Methods, supra note 59, § 1.3.5.11 (available at http://itl.nist.gov/div898/handbook/eda/section3/eda35b.htm); https://en.wikipedia.org/wiki/Kurtosis.

Cf. Jondeau & Rockinger, supra note 75, at 34 (adopting a functionally equivalent definition of the Taylor series expansion of expected returns).

Id. at 34 n.5.

100

Compare sources cited supra note 9 (especially Friend and Westerfield; Ingersoll; and Kraus and Linzenberger, who rely on the traditional definition of skewness) with Harvey & Siddique, supra note 7, at 1268 & n.4 (relaxing the definition of skewness according to the third standardized central moment)

101

Jondeau & Rockinger, supra note 75, at 33.

102

Id. at 34.

103

Robert C. Scott & Philip A. Howath, On the Direction of Preference for Moments of Higher Order Than the Variance, 35 J. Fin. 915–919, 917 (1980); see also Markus K. Brunnermeier, Christian Gollier & Jonathan A. Parker, Optimal Beliefs, Asset Prices, and the Preference for Skewed Returns, 97 Am. Econ. Rev. 159–165 (2007).

104

Levy, CAPM in the 21st Century, Chap. 8, supra note 13, at 61 n.4.

105

Id.

106

See id. at 70–71. “Skewness preferences and positive third derivatives are related to third degree Stochastic Dominance.” Id. at 61 n.4. See generally G.A. Whitmore, Third-Degree Stochastic Dominance, 60 Am. Econ. Rev. 457–459 (1970).

107

See Robert Dittmar, Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross-Section of Equity Returns, 57 J. Fin. 369–403 (2002).

108

Scott & Howath, supra note 103, at 917–918.

109

Estrada, An Alternative Behavioural Model, supra note 83, at 241.

110

Id.

111

Balzer, Chap. 8, supra note 68, at 130.

112

Athayde & Flôres, supra note 10, at 1336.

113

Id.

114

See Turan G. Bali, Nusret Cakici & Robert Whitelaw, Maxing Out: Stocks as Lotteries and the Cross-Section of Expected Returns, 99 J. Fin. Econ. 427–446 (2011).

115

Athayde & Flôres, supra note 10, at 1336.

116

Jondeau & Rockinger, supra note 75, at 35.

117

Id.

118

Balzer, Chap. 8, supra note 68, at 141.

119

See generally id. at 140–142, 150.

120

Malevergne & Sornette, supra note 76, at 53.

121

Id.

122

See Jørgen Vitting Andersen & Didier Sornette, Have Your Cake and Eat It Too: Increasing Returns While Lowering Large Risks!, 2 J. Risk Fin. 70–82 (2001).

123

See Valeri Zakamouline & Steen Koekebakker, Portfolio Evaluation with Generalized Sharpe Ratios: Beyond the Mean and Variance, 33 J. Banking & Fin. 1242–1254 (2009).

124

Ang, Chen & Xing, supra note 36, at 1199.

125

Jondeau & Rockinger, supra note 75, at 33.

Id.

Id. at 30.

Id.

Ang, Chen & Xing, supra note 36, at 1199.

130

Jondeau & Rockinger, supra note 75, at 30. See generally Patrick L. Brockett & James R. Garven, A Reexamination of the Relationship Between Preferences and Moment Orderings by Rational Risk-Averse Investors, 23 Geneva Risk & Ins. Rev. 127–137 (1998).

131

See Ravi Bansal, David A. Hsieh & S. Viswanathan, A New Approach to International Arbitrage Pricing, 48 J. Fin. 1719–1747, 1733 (1993); A. Ronald Gallant, Peter E. Rossi & George Tauschen, Stock Prices and Volume, 5 Rev. Fin. Stud. 199–242, 214 (1992). See generally Ravi Bansal & S. Viswanathan, No Arbitrage and Arbitrage Pricing: A New Approach, 48 J. Fin. 1231–1262 (1993).

132

Jondeau & Rockinger, supra note 75, at 33.

133

Id. at 34.

134

Id.

135

Harvey & Siddique, supra note 7, at 1264.

136

Id. at 1291.

137

See Gordon Y.N. Tang & Wai Cheong Shum, The Relationship Between Unsystematic Risk, Skewness and Stock Returns During Up and Down Markets, 12 Int’l Bus. Rev. 523–541 (2003).

138

See Bo Young Chang, Peter Christofferson & Kris Jacobs, Market Skewness, Risk, and the Cross-Section of Stock Returns, 107 J. Fin. Econ. 46–68 (2013).

139

See Brian H. Boyer & Keith Vorlink, Stock Options as Lotteries, 69 J. Fin. 1485–1527 (2014) (finding differences in average returns on option portfolios ranging from 10 % to 50 % per week, suggesting that intermediaries collect large premiums for bearing risk that they cannot hedge, even as they accommodate investor demand for options promising positively skewed, lottery-like payouts).

140

See Diego Amaya & Aurelio Vasquez, Skewness from High-Frequency Data Predicts the Cross-Section of Cross-Returns (March 15, 2009) (available at http://www.ssrn.com/abstract=1351885).

141

See, e.g., Alexandros Kostakis, Kashif Muhammad & Antonios Siganos, Higher Co-Moments and Asset Pricing on London Stock Exchange, 36 J. Banking & Fin. 913–922 (2012).

142

See, e.g., Md. Zobaer Hasan & Anton Abdulbasah Kamil, Contribution of Co-Skewness and Co-Kurtosis of the Higher Moment CAPM for Finding the Technical Efficiency, 2014 Econ. Research Int’l 253527 (discussing the Bangladeshi stock market) (available at http://dx.doi.org/10.1155/2014/253527).

143

See Roman Kozhan, Anthony Neuberger & Paul Schneider, The Skew Risk Premium in the Equity Index Market, 26 Rev. Fin. Stud. 2174–2203, 2184 (2013).

144

Id. at 2196.

145

Id. at 2192.

146

See Benoît Carmichael & Alain Coën, Asset Pricing with Skewed-Normal Return, 10 Fin. Research Letters 50–57 (2013); Kostakis, Muhammad & Siganos, supra note 141; Alan Kraus & Robert H. Litzenberger, Skewness Preference and the Valuation of Risk Assets, 31 J. Fin. 1085–1100 (1976).

147

Harvey, supra note 56, at 38.

148

Chi-Fu Huang & Robert H. Litzenberger, Foundations for Financial Economics 156 (1988).

149

Harvey, supra note 56, at 38.

150

Ang, Chen & Xing, supra note 36, at 1193.

151

Markus K. Brunnermeier, Stefan Nagel & Lasse H. Pedersen, Carry Trades and Currency Crashes, in NBER Macroeconomics Annual 2008, at 313–347, 315 (Daron Acemoglu, Kenneth Rogoff & Michael Woodford eds., 2009).

152

Harvey, supra note 56, at 38.

153

Id.

154

Ang & Chen, supra note 31, at 473.

155

Harvey, supra note 56, at 38.

156

Id. at 46; see also id. (conceding the absence of an “attempt to measure … cokurtosis” despite the presence of “a positive relation between kurtosis and returns” in emerging markets, albeit “not in developed markets”).

157

Rohan Christie-David & Mukesh Chaudhry, Coskewness and Cokurtosis in Futures Markets, 8 J. Empirical Fin. 55–81, 57 (2001).

158

See Dittmar, Non-Linear Pricing Kernels, supra note 107, at 380. See generally Tobias Moskowitz & Mark Grinblatt, Do Industries Explain Momentum?, 54 J. Fin. 1249–1290, 1252–1254 (1999) (using Standard Industrial Classification codes to generate 20 value-weighted portfolios sorted by industry).

159

Chang, Christofferson & Jacobs, supra note 138, at 54.

160

Jennifer Conrad, Robert F. Dittmar & Eric Ghysels, Ex Ante Skewness and Expected Stock Returns, 68 J. Fin. 85–124, 106 (2013); cf. Dittmar, Non-Linear Pricing Kernels, supra note 107, at 380.

161

See Christie-David & Chaudhry, supra note 157, at 58.

162

Ang, Chen & Xing, supra note 36, at 1199.

163

Id. (emphasis added).

164

See generally Miller, Chap. 8, supra note 43, at 53–56.

165

Ang, Chen & Xing, supra note 36, at 1197.

166

See Harvey, supra note 56, at 46.

167

See https://en.wikipedia.org/wiki/Coskewness.

168

See Ang & Chen, supra note 31, at 469.

169

See Ang, Chen & Xing, supra note 36, at 1197.

170

See Harvey & Siddique, supra note 7, at 1276 (equation 11) (defining β _SKD as a “direct measure of coskewness”); see also id. (“As defined, standardized coskewness is unit-free and analogous to a factor loading.”).

171

See https://en.wikipedia.org/wiki/Coskewness.

172

Harvey, supra note 56, at 33 (adopting this formulation as a “secondary measure of coskewness” to supplement the more traditional definition).

173

See https://en.wikipedia.org/wiki/Coskewness.

174

Galagedera & Brooks, supra note 5, at 217 (equation 6) (citing Estrada, Systematic Risk in Emerging Markets, supra note 1).

175

Galagedera & Brooks, supra note 5, at 217 (equation 2) (citing William W. Hogan & James M. Warren, Toward the Development of an Equilibrium Capital-Market Model Based on Semivariance, 9 J. Fin. & Quant. Analysis 1 (1974)); accord Galagedera, Downside Framework, supra note 29, at 147; Galagedera, Economic Significance, supra note 24, at 1627; Galagedera, Maharaj & Brooks, supra note 27, at 1625.

176

Galagedera & Brooks, supra note 5, at 229.

177

Id. at 218.

178

See generally https://en.wikipedia.org/wiki/Cokurtosis.

179

Miller, Chap. 8, supra note 43, at 56.

180

Id.

181

Id.

182

See Engle & Kroner, Chap. 8, supra note 73.

183

See Bollerslev, Engle & Wooldridge, Chap. 8, supra note 11 (specifying a GARCH model expressed in terms of the diagonal vector of conditional variance matrixes, which came to be known as VECH). See generally Wolfgang Scherrer & Eva Ribarits, On the Parameterization of Multivariate GARCH Models, 23 Econ. Theory 464–484 (2007) (reviewing the structure and parameterization of VECH and BEKK models).

184

See Zhuaxin Ding & Robert F. Engle, Large Scale Conditional Covariance Modelling, Estimation and Testing, 29 Acad. Econ. Papers 157–184 (2001).

185

See generally Richard Ernest Bellman, Dynamic Programming (1957); Richard Ernest Bellmann, Adaptive Control Processes (1961); R.B. Marimont & M.B. Shapiro, Nearest Neighbour Searches and the Curse of Dimensionality, 24 IMA J. Applied Math. 59–70 (1979).

186

Miller, Chap. 8, supra note 43, at 56.

187

Id.

188

Id.

189

Javier Estrada, Mean-Semivariance Optimization: A Heuristic Approach, 18 J. Applied Fin. 57–72, 60 (2008).

190

Leslie A. Balzer, Measuring Investment Risk: A Review, 3:3 J. Investing 47–58, 57 (Fall 1994) (emphasis in original); accord Balzer, Chap. 8, supra note 68, at 129. The balance of the information in this paragraph is derived from these sources.

191

Balzer, Chap. 8, supra note 68, at 129.

Titel: A Four-Moment Capital Asset Pricing Model
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_10

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