The Capital Asset Pricing Model: Comparison
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Please note this is a comparison between Version 2 by Nora Tang and Version 1 by James Chen.

The capital asset pricing model (CAPM) is an influential paradigm in financial risk management. It formalizes mean-variance optimization of a risky portfolio given the presence of a risk-free investment such as short-term government bonds. The CAPM defines the price of financial assets according to the premium demanded by investors for bearing excess risk.

  • beta
  • Sharpe ratio
  • three-factor model
  • momentum
  • intertemporal CAPM
  • Roll’s critique
  • investor heterogeneity
  • multifractality
  • efficient market hypothesis
  • fractal market hypothesis
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References

  1. Fama, E.F. Risk, return, and equilibrium: Some clarifying comments. J. Financ. 1968, 23, 29–40.
  2. Muth, J.A. Rational expectations and the theory of price movements. Econometrica 1961, 29, 315–335.
  3. Epstein, L.G.; Wang, T. Intertemporal asset pricing under Knightian uncertainty. Econometrica 1994, 62, 283–322.
  4. Ross, S.A. The arbitrage theory of capital asset pricing. J. Econ. Theory 1976, 13, 341–360.
  5. Anderson, E.W.; Ghysels, E.; Juergens, J.L. The impact of risk and uncertainty on expected returns. J. Financ. Econ. 2009, 94, 233–263.
  6. Supreme Court of the United States. Willcox v. Consolidated Gas Company; U.S. Reports; Supreme Court of the United States: Washington, DC, USA, 1909; Volume 212, pp. 19–55.
  7. Fama, E.F.; French, K.R. The capital asset pricing model: Theory and evidence. J. Econ. Perspect. 2004, 18, 25–46.
  8. Markowitz, H. Portfolio selection. J. Financ. 1952, 7, 77–99.
  9. Markowitz, H.M. Portfolio Selection: Efficient Diversification of Investments; John Wiley & Sons: New York, NY, USA, 1959.
  10. Sharpe, W.F. Capital asset prices: A theory of market equilibrium under conditions of risk. J. Financ. 1964, 19, 425–442.
  11. Lintner, J. The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat. 1965, 47, 13–37.
  12. Treynor, J.L. How to rate management of investment funds. Harv. Bus. Rev. 1965, 43, 63–75.
  13. French, C.W. The Treynor capital asset pricing model. J. Invest. Manag. 2003, 1, 60–72.
  14. Tobin, J. Liquidity preference as behavior towards risk. Rev. Econ. Stud. 1958, 25, 65–86.
  15. Black, F. Capital market equilibrium with restricted borrowing. J. Bus. 1972, 45, 444–454.
  16. Korajczyk, R.A. Introduction. In Asset Pricing and Portfolio Performance: Models, Strategy and Performance Metrics; Korajczyk, R.A., Ed.; Risk Books: London, UK, 1999; pp. xiii–xxxi.
  17. Dybvig, P.H.; Ross, S.A. Differential information and performance measurement using a security market line. J. Financ. 1985, 40, 383–399.
  18. Reilly, F.K.; Brown, K.C. Analysis of Investments and Management of Portfolios, 9th ed.; Cengage Learning: Farmington Hills, MI, USA, 2009.
  19. Sharpe, W.F. Mutual fund performance. J. Bus. 1966, 39, 119–138.
  20. Sharpe, W.F. Adjusting for risk in portfolio performance measurement. J. Portf. Manag. 1975, 1, 29–34.
  21. Sharpe, W.F. The Sharpe ratio. J. Portf. Manag. 1994, 21, 49–58.
  22. Modigliani, F.; Miller, M.H. The cost of capital, corporate finance, and the theory of investment. Am. Econ. Rev. 1958, 48, 261–297.
  23. Sharpe, W.F. A simplified model for portfolio analysis. Manag. Sci. 1963, 9, 277–293.
  24. Barberis, N.; Shleifer, A.; Wurgler, J. Comovement. J. Financ. Econ. 2005, 75, 283–317.
  25. Friend, I.; Blume, M. Measure of portfolio performance under uncertainty. Am. Econ. Rev. 1970, 60, 561–575.
  26. Leibowitz, M.L.; Bova, A.; Hammond, P.B. The Endowment Model of Investing: Return, Risk, and Diversification; John Wiley & Sons: Hoboken, NJ, USA, 2010.
  27. Miller, M.B. Mathematics and Statistics for Financial Risk Management, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2014.
  28. Tofallis, C. Investment volatility: A critique of standard beta estimation and a simple way forward. Eur. J. Oper. Res. 2008, 187, 1358–1367.
  29. Hui, C.-H.; Lo, C.-F.; Chau, P.-H.; Wong, A. Does Bitcoin behave as a currency? A standard monetary model approach. Int. Rev. Financ. Anal. 2020, 70, 101518.
  30. Farber, D.A. Uncertainty. Georget. Law J. 2011, 99, 901–960.
  31. Knight, F.H. Risk, Uncertainty, and Profit; Houghton Mifflin Co.: Boston, MA, USA; New York, NY, USA, 1921.
  32. Keynes, J.M. The general theory of employment. Q. J. Econ. 1937, 51, 209–223.
  33. Zhang, X.F. Information uncertainty and stock returns. J. Financ. 2006, 61, 105–137.
  34. Bloom, N. Fluctuations in uncertainty. J. Econ. Perspect. 2014, 28, 153–176.
  35. Epstein, L.G.; Schneider, M. Ambiguity, information quality, and asset pricing. J. Financ. 2008, 43, 197–228.
  36. Christiano, L.J.; Motto, R.; Rostagno, M. Risk shocks. Am. Econ. Rev. 2014, 104, 27–65.
  37. Fama, E.F. Mandelbrot and the stable Paretian hypothesis. J. Bus. 1963, 36, 420–429. Available online: https://www.jstor.org/stable/2350971 (accessed on 29 August 2021).
  38. Fama, E.F. Portfolio analysis in a stable Paretian market. Manag. Sci. 1965, 11, 404–416.
  39. de Athayde, G.M.; Flôres, R.G. Finding a maximum skewness portfolio—A general solution to three-moments portfolio choice. J. Econ. Dyn. Control. 2004, 28, 1335–1352.
  40. Estrada, J. Mean-semivariance behaviour: An alternative behavioural model. J. Emerg. Mark. Financ. 2004, 3, 231–248.
  41. Harvey, C.R.; Liechty, J.C.; Liechty, M.W.; Müller, P. Portfolio selection with higher moments. Quant. Financ. 2010, 10, 469–485.
  42. Jurczenko, E.; Maillet, B. The four-moment capital asset pricing model: Between asset pricing and asset allocation. In Multi-Moment Asset Allocation and Pricing Models; Jurczenko, E., Maillet, B., Eds.; John Wiley & Sons: Hoboken, NJ, USA, 2012; pp. 113–163.
  43. Jondeau, E.; Rockinger, M. Optimal portfolio allocation under higher moments. Eur. J. Financ. Manag. 2006, 12, 29–55.
  44. Campbell, J.Y.; Lo, A.W.; MacKinlay, C. The Econometrics of Financial Markets; Princeton University Press: Princeton, NJ, USA, 1997.
  45. Harvey, C.R.; Siddique, A. Conditional skewness in asset pricing tests. J. Financ. 2000, 55, 1263–1295.
  46. Bali, T.G.; Cakici, N.; Whitelaw, R. Maxing out: Stocks as lotteries and the cross-section of expected returns. J. Financ. Econ. 2011, 99, 427–446.
  47. Brunnermeier, M.K.; Gollier, C.; Parker, J.A. Optimal beliefs, asset prices, and the preference for skewed returns. Am. Econ. Rev. 2007, 97, 159–165.
  48. Scott, R.C.; Horvath, P.A. On the direction of preference for moments of higher order than the variance. J. Financ. 1980, 35, 915–919.
  49. Florida Public Service Commission. Returns on Common Equity for Water and Wastewater Utilities; Public Utilities Reports, 4th Series; Florida Public Service Commission: Tallahassee, FL, USA, 1999; Volume 194, pp. 81–91.
  50. Supreme Court of the United States. Bluefield Water Works v. Public Service Commission; U.S. Reports; Supreme Court of the United States: Washington, DC, USA, 1923; Volume 262, pp. 679–695.
  51. Lee, C.-F.; Lee, A.C.; Lee, J. (Eds.) Handbook of Quantitative Finance and Risk Management; Springer: New York, NY, USA, 2010; Volume 1.
  52. Jobson, J.D.; Korkie, B.M. Performance hypothesis testing with the Sharpe and Treynor measures. J. Financ. 1981, 36, 888–908.
  53. Chen, L.; He, S.; Zhang, S. When all risk-adjusted performance measures are the same: In praise of the Sharpe ratio. Quant. Financ. 2011, 11, 1439–1447.
  54. Sharpe, W.F. The arithmetic of active management. Financ. Anal. J. 1991, 47, 7–9.
  55. Jensen, M.C. The performance of mutual funds in the period 1945–1964. J. Financ. 1968, 23, 389–416.
  56. Canner, N.; Mankiw, N.G.; Weil, D.G. An asset allocation puzzle. Am. Econ. Rev. 1997, 87, 181–191.
  57. Kroll, Y.; Levy, H. Further tests of the separation theorem and the capital asset pricing model. Am. Econ. Rev. 1992, 82, 664–670. Available online: https://www.jstor.org/stable/2117330 (accessed on 29 August 2021).
  58. Zhang, L. The value premium. J. Financ. 2005, 60, 67–103.
  59. Bernstein, W.J. The Intelligent Asset Investor: How to Build Your Portfolio to Maximize Returns and Minimize Risk; McGraw-Hill Education: New York, NY, USA, 2000.
  60. Graham, B.; Dodd, D.L. Security Analysis, 6th ed.; McGraw-Hill Education: New York, NY, USA, 2008.
  61. Hiller, D.; Grinblatt, M.; Titman, S. Financial Markets and Corporate Strategy, 2nd ed.; McGraw-Hill Education: New York, NY, USA, 2011.
  62. Shalit, H.; Yitzhaki, S. An asset allocation puzzle: Comment. Am. Econ. Rev. 2003, 93, 1002–1008.
  63. Bajeux-Besnainou, I.; Jordan, J.V.; Portait, R. An asset allocation puzzle: Comment. Am. Econ. Rev. 2001, 91, 1170–1179.
  64. Gomez, J.-P.; Zapatero, F. Asset pricing implications of benchmarking: A two-factor CAPM. Eur. J. Financ. 2003, 9, 343–357.
  65. Mulvey, J.M.; Fabozzi, F.; Pauling, W.R.; Simsek, K.D.; Zhang, Z. Modernizing the defined-benefit pension system. J. Portf. Manag. 2005, 31, 73–82.
  66. Cowling, C.; Fisher, H.; Powe, K.; Sheth, J.; Wright, M. Funding Defined Benefit pension schemes: An integrated risk management approach. Br. Actuar. J. 2019, 24, E7.
  67. Lally, M. The valuation of GSF’s defined benefit pension entitlements. N. Z. Econ. Pap. 2000, 34, 183–199.
  68. Menoncin, F.; Vigna, E. Mean-variance dynamic optimality for DC pension schemes. Eur. Actuar. J. 2020, 10, 125–148.
  69. Pedersen, J.L.; Peskir, G. Optimal mean-variance portfolio selection. Math. Financ. Econ. 2017, 11, 137–160.
  70. Vigna, E.; Haberman, S. Optimal investment strategy for defined contribution pension schemes. Insur. Math. Econ. 2001, 28, 233–262.
  71. Basak, S.; Chabkauri, G. Dynamic mean-variance asset allocation. Rev. Financ. Stud. 2010, 23, 2970–3016.
  72. Zhou, X.Y.; Li, D. Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Appl. Math. Optim. 2000, 42, 19–33.
  73. Lovelock, J. Gaia: A New Look at Life on Earth; Oxford University Press: Oxford, UK, 1979.
  74. Peters, E.E. Chaos and Order in the Capital Markets—A New View of Cycles, Prices, and Market Volatility; John Wiley & Sons: New York, NY, USA, 1991.
  75. Peters, E.E. Fractal Markte Analysis—Applying Chaos Theory to Investment and Analysis; John Wiley & Sons: New York, NY, USA, 1994.
  76. Fama, E.F.; MacBeth, J.D. Risk, return, and equilibrium: Empirical tests. J. Political Econ. 1973, 81, 607–636.
  77. Basu, S. Investment performance of common stocks in relation to their price-earnings ratios: A test of the efficient market hypothesis. J. Financ. 1977, 12, 129–156.
  78. Bhandari, L.C. Debt/equity ratio and expected common stock returns: Empirical evidence. J. Financ. 1988, 43, 507–528.
  79. Rosenberg, B.; Reid, K.; Lanstein, R. Persuasive evidence of market inefficiency. J. Portf. Manag. 1985, 11, 9–17.
  80. Ball, R. Anomalies in relationships between securities’ yields and yield surrogates. J. Financ. Econ. 1978, 6, 103–126.
  81. Banz, R.W. The relationship between return and market value of common stocks. J. Financ. Econ. 1981, 9, 3–18.
  82. Reinganum, M.R. Misspecification of capital asset pricing: Empirical anomalies based on earnings’ yield and market values. J. Financ. Econ. 1981, 9, 19–46.
  83. Levy, H.; Levy, M. The small firm effect: A financial mirage? J. Portf. Manag. 2011, 37, 129–138.
  84. Fama, E.F.; French, K.R. The cross-section of expected stock returns. J. Financ. 1992, 47, 427–465.
  85. Fama, E.F.; French, K.R. Common risk factors in the returns on stocks and bonds. J. Financ. Econ. 1993, 33, 3–56.
  86. Fama, E.F.; French, K.R. Size and book-to-market factors in earnings and returns. J. Financ. 1995, 50, 131–155.
  87. Fama, E.F.; French, K.R. Multifactor explanations of asset pricing anomalies. J. Financ. 1996, 51, 55–84.
  88. Fama, E.F.; French, K.R. Value versus growth: The international evidence. J. Financ. 1998, 53, 1975–1999.
  89. Fama, E.F.; French, K.R. Size, value, and momentum in international stock returns. J. Financ. Econ. 2012, 105, 457–472.
  90. Jegadeesh, N.; Titman, S. Returns to buying winners and selling losers: Implications for stock market efficiency. J. Financ. 1993, 48, 65–91.
  91. Grinblatt, M.; Titman, S.; Wermers, R. Momentum investment strategies, portfolio performance, and herding: A study of mutual fund behavior. Am. Econ. Rev. 1995, 85, 1088–1105.
  92. Chan, L.K.C.; Jegadeesh, N.; Lakonishok, J. Momentum strategies. J. Financ. 1996, 51, 1681–1713.
  93. Carhart, M.M. On persistence in mutual fund performance. J. Financ. 1997, 52, 57–82.
  94. Rath, S.; Durand, R.B. Decomposing the size, value and momentum premia of the Fama–French–Carhart four-factor model. Econ. Lett. 2015, 132, 139–141.
  95. Fama, E.F.; French, K.R. Dissecting anomalies. J. Financ. 2008, 63, 1653–1678.
  96. Avramov, D.; Chordia, T. Predicting stock returns. J. Financ. Econ. 2006, 82, 387–415.
  97. Cochrane, J.H. Discount rates. J. Financ. 2011, 66, 1047–1108.
  98. Feng, G.; Giglio, S.; Xu, D. Taming the factor zoo: A test of new factors. J. Financ. 2020, 75, 1327–1370.
  99. Merton, R.C. On estimating the expected return on the market: An exploratory investigation. J. Financ. Econ. 1980, 8, 323–361.
  100. Merton, R.C. An intertemporal capital asset pricing model. Econometrica 1973, 41, 867–887.
  101. Chang, B.Y.; Christofferson, P.; Jacobs, K. Market skewness risk and the cross section of stock returns. J. Financ. Econ. 2013, 107, 46–68.
  102. Kim, H.K.; Kim, T. Capital asset pricing model: A time-varying volatility approach. J. Empir. Financ. 2016, 37, 268–281.
  103. Koutmos, D. An intertemporal capital asset pricing model with heterogeneous expectations. J. Int. Financ. Mark. Inst. Money 2012, 22, 1176–1187.
  104. Roll, R. A critique of the asset pricing theory’s tests, part I: On past and potential testability of the theory. J. Financ. Econ. 1977, 4, 129–176.
  105. Gibbons, M.R.; Ross, S.A.; Shanken, J. A test of the efficiency of a given portfolio. Econometrica 1989, 57, 1121–1152.
  106. Stambaugh, R.F. On the exclusion of assets from tests of the two-parameter model: A sensitivity analysis. J. Financ. Econ. 1982, 10, 237–268.
  107. Browning, M.; Deaton, A.; Irish, M. A profitable approach to labor supply and commodity demands over the life-cycle. Econometrica 1985, 53, 503–543.
  108. Roy, A.D. Safety first and the holding of assets. Econometrica 1952, 20, 431–449.
  109. Breeden, D.T. An intertemporal asset pricing pricing model with stochastic consumption and investment opportunities. J. Financ. Econ. 1979, 7, 285–296.
  110. Grossman, S.J.; Shiller, R.J. The determinants of the variability of stock market prices. Am. Econ. Rev. 1981, 71, 222–227.
  111. Lucas, R.E., Jr. Asset prices in an exchange economy. Econometrica 1978, 46, 1429–1446.
  112. Rubinstein, M. The valuation of uncertain income streams and the pricing of options. Bell J. Econ. 1976, 7, 407–425.
  113. Cochrane, J.H. Asset Pricing; Princeton University Press: Princeton, NJ, USA, 1981.
  114. Paiella, M. Heterogeneity in financial market participation: Appraising its implications for the C-CAPM. Rev. Financ. 2004, 8, 445–480.
  115. Campbell, J.Y.; Cochrane, J.H. Explaining the poor performance of consumption-based asset pricing models. J. Financ. 2000, 55, 2863–2878.
  116. Breeden, D.T.; Gibbons, M.R.; Litzenberger, R.H. Empirical tests of the consumption-oriented CAPM. J. Financ. 1989, 44, 231–262.
  117. Hansen, L.P.; Singleton, K.J. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 1982, 50, 1269–1288.
  118. Hansen, L.P.; Singleton, K.J. Stochastic consumption, risk aversion, and the temporal behavior of asset returns. J. Political Econ. 1983, 91, 249–268.
  119. Wheatley, S. Some tests of international equity integration. J. Financ. Econ. 1988, 21, 177–212.
  120. Darrat, A.F.; Li, B.; Park, J.C. Consumption-based CAPM models: International evidence. J. Bank. Financ. 2011, 35, 2148–2157.
  121. Brock, W.A.; Hommes, C.H. Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J. Econ. Dyn. Control 1998, 22, 1235–1274.
  122. Glosten, L.R.; Milgrom, P.R. Bid, ask, and transaction prices in a specialist market with heterogeneously informed traders. J. Financ. Econ. 1985, 14, 71–100.
  123. Bhattacharya, U.; Daouk, H. The world price of insider trading. J. Financ. 2002, 57, 75–108.
  124. Cornell, B.; Sirri, E.R. The reaction of investors and stock prices to insider trading. J. Financ. 1992, 47, 1031–1059.
  125. Williams, J.T. Capital asset prices with heterogeneous beliefs. J. Financ. Econ. 1977, 5, 219–239.
  126. Detemple, J.; Murthy, S. Intertemporal asset pricing with heterogeneous beliefs. J. Econ. Theory 1994, 62, 294–320.
  127. Johnson, T. Forecast dispersion and the cross section of expected returns. J. Financ. 2004, 59, 1957–1978.
  128. Lintner, J. The aggregation of investor’s diverse judgments and preferences in purely competitive security markets. J. Financ. Quant. Anal. 1969, 4, 347–400.
  129. Varian, H. Divergence of opinion in complete markets: A note. J. Financ. 1985, 40, 309–317.
  130. Barberis, N.; Huang, M.; Thaler, R.H. Individual preferences, monetary gambles, and stock market participation: A case for narrow framing. Am. Econ. Rev. 2006, 96, 1069–1090.
  131. Polkovnichenko, V. Limited stock market participation and the equity premium. Financ. Res. Lett. 2004, 1, 24–34.
  132. Tversky, A.; Kahneman, D. Advances in prospect theory: Cumulative representation of uncertainty. J. Risk Uncertain. 1992, 5, 297–323.
  133. Del Vigna, M. Financial market equilibria with heterogeneous agents: CAM and market segmentation. Math. Financ. Econ. 2013, 7, 405–429.
  134. Chiarella, C.; Dieci, R.; He, X.-Z.; Li, K. An evolutionary CAPM under heterogeneous beliefs. Ann. Financ. 2013, 9, 185–215.
  135. Black, F. Noise. J. Financ. 1986, 41, 529–543.
  136. Shefrin, H.; Statman, M. Behavioral capital asset pricing theory. J. Financ. Quant. Anal. 1994, 29, 323–349.
  137. He, X.-Z.; Li, Y. Heterogeneity, convergence, and autocorrelations. Quant. Financ. 2008, 8, 58–79.
  138. Lux, T.; Alfarano, S. Financial power laws: Empirical evidence, models, and mechanisms. Chaos Solitons Fractals 2016, 88, 3–18.
  139. Westerhoff, F.H. Multiasset market dynamics. Macroecon. Dyn. 2004, 8, 591–616.
  140. Westerhoff, F.H.; Dieci, R. The effectiveness of Keynes-Tobin transaction taxes when heterogeneous agents can trade in different markets: A behavioral finance approach. J. Econ. Dyn. Control 2006, 30, 293–322.
  141. Brown, S.J. The efficient market hypothesis, the Financial Analysts Journal and the professional status of investment management. Financ. Anal. J. 2020, 76, 5–14.
  142. Fama, E.F. Efficient capital markets: A review of theory and empirical work. J. Financ. 1970, 33, 3–56.
  143. Fama, E.F. Efficient capital markets: II. J. Financ. 1991, 46, 1575–1617.
  144. Vasicek, O.A.; McQuown, J.A. The efficient market model. Financ. Anal. J. 1972, 28, 71–84.
  145. Faber, M.; Manstetten, R.; Petersen, T. Homo Oeconomicus and Homo Politicus. Political economy, constitutional interest and ecological interest. Kyklos 1997, 50, 457–483.
  146. McMahon, J. Behavioral economics as neoliberalism: Producing and governing homo economicus. Contemp. Political Theory 2015, 14, 137–158.
  147. Di Matteo, T.; Aste, T.; Dacorogna, M.M. Scaling behavior in differently developed markets. Phys. A Stat. Mech. Its Appl. 2003, 324, 183–188.
  148. Bianconi, M.; MacLachlan, S.; Sammon, M. Implied volatility and the risk-free rate of return in options markets. N. Am. J. Econ. Financ. 2015, 31, 1–26.
  149. Deng, Z.-C.; Yu, J.-N.; Yang, L. An inverse problem of determining the implied volatility in option pricing. J. Math. Anal. Appl. 2008, 340, 16–31.
  150. Pasquini, M.; Serva, M. Multiscaling and clustering of volatility. Phys. A Stat. Mech. Its Appl. 1999, 269, 140–147.
  151. Alexander, C.; Lazar, E.; Stanescu, S. Analytic moments for GJR-GARCH (1, 1) processes. Int. J. Forecast. 2021, 37, 105–124.
  152. Bollerslev, T.; Wooldridge, J.M. Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariance. Econ. Rev. 1992, 11, 143–172.
  153. Nugroho, D.B.; Kurniawati, D.; Panjaitan, L.P.; Kholil, Z.; Susanto, B.; Sasongko, L.R. Empirical performance of GARCH, GARCH-M, GJR-GARCH and log-GARCH models for returns volatility. J. Phys. Conf. Ser. 2019, 1307, 012003.
  154. Bacry, E.; Delour, J.; Muzy, J.F. Multifractal random walk. Phys. Rev. E 2001, 64, 026103.
  155. Morales, R.; Di Matteo, T.; Aste, T. Non-stationary multifractality in stock returns. Phys. A Stat. Mech. Its Appl. 2013, 392, 6470–6483.
  156. Castiglioni, P.; Lazzeroni, D.; Coruzzi, P.; Faini, A. A multifractal-multiscale analysis of cardiovascular signals: A DFA-based characterization of blood pressure and heart-rate complexity by gender. Complexity 2018, 2018, 4801924.
  157. Gieraltowski, J.; Żebrowski, J.J.; Baranowski, R. Multiscale multifractal analysis of heart risk variability recordings with a large number of occurrences of arrhythmia. Phys. Rev. E 2012, 85, 021915.
  158. Barunik, J.; Aste, T.; Di Matteo, T.; Liu, R. Understanding the source of multifracticality in financial markets. Phys. A Stat. Mech. Its Appl. 2012, 391, 4234–4251.
  159. Carbone, A.; Castelli, G.; Stanley, H.E. Time-dependent Hurst exponent in financial time series. Phys. A Stat. Mech. Its Appl. 2004, 344, 267–271.
  160. Domino, K. The use of the Hurst exponent to investigate the global maximum of the Warsaw stock exchange WIG20 index. Phys. A Stat. Mech. Its Appl. 2012, 391, 156–159.
  161. Grech, D.; Pamuła, G. The local Hurst exponent of the financial time series in the vicinity of crashes on the Polish stock exchange. Phys. A Stat. Mech. Its Appl. 2008, 387, 4299–4308.
  162. Morales, R.; Di Matteo, T.; Gramatica, R.; Aste, T. Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series. Phys. A Stat. Mech. Its Appl. 2012, 391, 3180–3189.
  163. Gneiting, T.; Schlather, M. Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 2004, 46, 269–282.
  164. Salat, H.; Murcio, R.; Arcaute, E. Multifractal methodology. Phys. A Stat. Mech. Its Appl. 2017, 473, 467–487.
  165. Kristoufek, L. Measuring correlations between non-stationary series with DCCA coefficient. Phys. A Stat. Mech. Its Appl. 2014, 402, 291–298.
  166. Kristoufek, L. Detrending moving-average cross-correlation coefficient: Measuring cross-correlations between non-stationary series. Phys. A Stat. Mech. Its Appl. 2014, 406, 169–175.
  167. Kristoufek, L. Detrended fluctuation analysis as a regression framework: Estimating dependence at different scales. Phys. Rev. E 2015, 91, 022802.
  168. Kristoufek, L. Scaling of dependence between foreign exchange rates and stock markets in central Europe. Acta Phys. Pol. 2016, 129, 908–912.
  169. Kristoufek, L. Fractal market hypothesis and the global financial crisis: Scaling, investment horizons and liquidity. Adv. Complex Syst. 2012, 15, 1250065.
  170. Kristoufek, L. Fractal market hypothesis and the global financial crisis: Wavelet power evidence. Sci. Rep. 2013, 3, 2857.
  171. Weron, A.; Weron, R. Fractal market hypothesis and two power-laws. Chaos Solitons Fractals 2000, 11, 289–296.
  172. Jiang, Z.-Q.; Xie, W.-J.; Zhou, W.-X.; Sornette, D. Multifractal analysis of financial markets: A review. Rep. Prog. Phys. 2019, 82, 82–125901.
  173. Ihlen, E.A.F.; Vereijken, B. Multifractal formalisms of human behavior. Hum. Mov. Sci. 2013, 32, 633–651.
  174. Kahneman, D.; Tversky, A. Choices, values, and frames. Am. Psychol. 1984, 39, 344–350.
  175. Kristoufek, L.; Ferreira, P. Capital asset pricing model in Portugal: Evidence from fractal regressions. Port. Econ. J. 2018, 17, 173–183.
  176. Tilfani, O.; Ferreira, P.; El Boukfaoui, M.Y. Multiscale optimal portfolios using CAPM fractal regression: Estimation for emerging stock markets. Post-Communist Econ. 2020, 32, 77–112.
  177. Ma, F.; Wei, Y.; Huang, D. Multifractal detrended cross-correlation analysis between the Chinese stock market and surrounding stock markets. Phys. A Stat. Mech. Its Appl. 2013, 392, 1659–1670.
  178. Cao, G.; Xu, L.; Cao, J. Multifractal detrended cross-correlations between the Chinese exchange market and stock market. Phys. A Stat. Mech. Its Appl. 2012, 391, 4855–4866.
  179. Ferreira, P.; Silva, M.F.D.; Santana, I.S.D. Detrended correlation coefficients between exchange rate (in dollars) and stock markets in the world’s largest economies. Economies 2019, 7, 9.
  180. Sun, L.; Xiang, M.; Marquez, L. Forecasting the volatility of onshore and offshore USD/RMB exchange rates using a multifractal approach. Phys. A Stat. Mech. Its Appl. 2019, 532, 121787.
  181. Fan, Q.; Liu, S.; Wang, K. Multiscale multifractal detrended fluctuation analysis of multivariate time series. Phys. A Stat. Mech. Its Appl. 2019, 532, 121864.
  182. Pagnottoni, P.; Spelta, A.; Pecora, N.; Flori, A.; Pammolli, F. Financial earthquakes: SARS-CoV-2 news shock propagation in stock and sovereign bond markets. Phys. A Stat. Mech. Its Appl. 2021, 582, 126240.
  183. Kristoufek, L. Fractality in market risk structure: Dow Jones Industrial components case. Chaos Solitons Fractals 2018, 110, 69–75.
  184. Tilfani, O.; Ferreira, P.; El Boukfaoui, M.Y. Building multi-scale portfolios and efficient market frontiers using fractal regressions. Phys. A Stat. Mech. Its Appl. 2019, 532, 121758.
  185. Fernández-Martínez, M.; Sánchez-Granero, M.A.; Muñoz Torrecillas, M.J.; McKelvey, B. A comparison of three Hurst exponent approaches to predict nascent bubbles in S&P500 stocks. Fractals 2017, 25, 1750006.
  186. Preis, T.; Schneider, J.J.; Stanley, H.E. Switching processes in financial markets. Proc. Natl. Acad. Sci. USA 2011, 108, 7674–7678.
  187. Bekaert, G.; Erb, C.B.; Harvey, C.R.; Viskanta, T.E. Distributional characteristics of emerging market returns and asset allocation. J. Portf. Manag. 1998, 24, 102–116.
  188. Peiró, A. Skewness in financial returns. J. Bank. Financ. 1999, 23, 847–862.
  189. Aparicio, F.M.; Estrada, J. Empirical distributions of stock returns: European securities markets, 1990–95. Eur. J. Financ. 2001, 7, 1–21.
  190. Kon, S.J. Models of stock returns—A comparison. J. Financ. 1984, 19, 147–165.
  191. Gray, J.B.; French, D.W. Empirical comparisons of distributional models for stock index returns. J. Bus. Financ. Account. 1990, 17, 451–459.
  192. Mandelbrot, B.B.; Hudson, R.L. The (Mis) Behavior of Markets: A Fractal View of Risk, Ruin, and Reward; Basic Books: New York, NY, USA, 2004.
  193. Fama, E.F.; French, K.R. The CAPM is wanted, dead or alive. J. Financ. 1996, 51, 1947–1958.
  194. Kaplanski, G. Traditional beta, downside risk beta, and market risk premiums. Q. Rev. Econ. Financ. 2004, 44, 636–653.
  195. Koller, T.; Goedhart, M.; Wessels, D. Valuation: Measuring and Managing the Value of Companies, 7th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2020.
  196. Levy, H. The Capital Asset Pricing Model in the 21st Century: Analytical, Empirical, and Behavioral Perspectives; Cambridge University Press: Cambridge, UK, 2012.
  197. Jagannathan, R.; Wang, Z. The conditional CAPM and the cross-section of expected returns. J. Financ. 1996, 51, 3–53.
  198. Chan, L.K.C.; Lakonishok, J. Are reports of beta’s death premature? J. Portf. Manag. 1993, 19, 51–62.
  199. Lai, T.-Y.; Stohs, M.H. Yes, CAPM is dead. Int. J. Bus. 2015, 20, 144–158.
  200. Levy, H. The CAPM is alive and well: A review and synthesis. Eur. Financ. Manag. 2009, 16, 43–71.
  201. Lopes, L.L. Between hope and fear: The psychology of risk. Adv. Exp. Soc. Psychol. 1987, 20, 255–295.
  202. Preis, T.; Stanley, H.E. Switching phenomena in a system with no switches. J. Stat. Phys. 2010, 138, 431–446.
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