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The Evolution of Capital Asset Pricing Models

The Evolution of Capital Asset Pricing Models. Sheng-Syan Chen, National Taiwan University Cheng-Few Lee, Rutgers University Yi-Cheng Shih, National Taipei University Po-Jung Chen, National Taiwan University. Outline. 1. Introduction 2. Intertemporal Models

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The Evolution of Capital Asset Pricing Models

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  1. The Evolution of Capital Asset Pricing Models Sheng-Syan Chen, National Taiwan University Cheng-Few Lee, Rutgers University Yi-Cheng Shih, National Taipei University Po-Jung Chen, National Taiwan University

  2. Outline 1. Introduction 2. Intertemporal Models 3. Supply-Side Effect Models 4. International CAPM 5. Equilibrium Models with Heterogeneity 6. Dividend and Taxation Effect Models 7. Skewness Effect Models 8. Behavioral Finance 9. Liquidity-based Models 10. Existence of Equilibrium 11. Empirical Tests 12. Conclusion

  3. Abstract Based upon Markowitz (1952, 1959) Mean-Variance Portfolio Theory and six critical assumptions, Sharpe (1964), Lintner (1965), and Mossin (1966) have derived and developed the static Capital Asset Pricing Model. During the four past decades, the CAPM has been the benchmark of asset pricing models, and most empirical apply it to calculate asset returns and the cost of capital. To relax the original six assumptions, many researchers have tried to generalize the static CAPM by Sharpe, Lintner, and Mossin. In addition, many researchers have also tried to develop the dynamic Capital Asset Pricing Models.   In this paper, we survey the important alternative theoretical models of the Capital Asset Pricing for last four and half decades. We organize these theoretical models, as follows: (i) Merton’s Intertemporal CAPM, (ii) Consumption-based Intertemporal CAPM, (iii) Production-based Intertemporal CAPM, (iv) CAPM with Supply-side Effect, (v) International Equilibrium CAPM with Heterogeneity Beliefs and Investors, (vi) Equilibrium CAPM with Heterogeneity Investment Horizon, (vii) CAPM with Dividend and Taxation Effect, (viii) CAPM with Skewness Effect, and (ix) Behavioral Finance, and Liquidity-based CAPM. The interrelationship among these models is also discussed in some detail.   The results of this paper might be used as a guideline for future theoretical and empirical research in capital asset pricing. More specifically, we suggest three possible directions for future research. To the best of our knowledge, this is one of the most complete reviews of the evolution of theoretical capital asset models.

  4. 1. Introduction (1) (2)

  5. 2. Intertemporal Models 2.1 Merton Model 2.2 Consumption-based Models 2.3 Production-based Models

  6. 2. Intertemporal Models 2.1 Merton Model (i=1, 2,…,n), (3) where , is the covariance of the return on the ith asset with the return on the market portfolio and is the expected return on the market portfolio. (4)

  7. 2. Intertemporal Models 2.2 Consumption-based Models (5) (6) where is the intertemporal marginal rate of substitution of the investor, also known as the stochastic discount factor.

  8. 2. Intertemporal Models 2.2 Consumption-based Models (7) (8)

  9. 2. Intertemporal Models2.2 Consumption-based Models (9) (10)

  10. 2. Intertemporal Models2.2 Consumption-based Models (11) (12) (13)

  11. 2. Intertemporal Models2.2 Consumption-based Models (14) (15) (16)

  12. 2. Intertemporal Models2.2 Consumption-based Models (17) (18) (19)

  13. 2. Intertemporal Models2.2 Consumption-based Models (20) (21) (22)

  14. 2. Intertemporal Models2.2 Consumption-based Models (23) (24) (25)

  15. 2. Intertemporal Models2.2 Consumption-based Models (26) (27) (28)

  16. 2. Intertemporal Models2.2 Consumption-based Models (29) (30)

  17. 2. Intertemporal Models2.3 Production-based Models (31) (32) (33)

  18. 2. Intertemporal Models2.3 Production-based Models (34) (35) (36)

  19. 2. Intertemporal Models2.3 Production-based Models (37) (38)

  20. 2. Intertemporal Models2.3 Production-based Model (39) (40) (41)

  21. 2. Intertemporal Models2.3 Production-based Model (42) (43) (44) (45)

  22. 2. Intertemporal Models2.3 Production-based Models (46) (47) (48) (49)

  23. 2. Intertemporal Models2.3 Production-based Models (50) for all i, (51)

  24. 3. Supply-Side Effect Models 3.1 Demand function of capital assets 3.2 Supply function of securities 3.3 Multiperiod Equilibrium Models

  25. 3. Supply-Side Effect Models3.1 Demand function of capital assets (52) j = 1,…,N, (53) j = 1,…,N, (54)

  26. 3. Supply-Side Effect Models3.1 Demand function of capital assets (55) (56) (57)

  27. 3. Supply-Side Effect Models3.1 Demand function of capital assets (58) (59) (60)

  28. 3. Supply-Side Effect Models3.2 Supply function of securities (61) (62)

  29. 3. Supply-Side Effect Models3.2 Supply function of securities (63) where

  30. 3. Supply-Side Effect Models 3.3 Multiperiod Equilibrium Models (64) (65)

  31. 3. Supply-Side Effect Models 3.3 Multiperiod Equilibrium Models (66) (66´) (67)

  32. 4. International CAPM Without a model showing how assets are priced in a world in which asset markets are fully integrated, it is impossible to determine whether asset markets are segmented internationally or not. Stulz (1981a) provide an intertemporal model of international asset pricing, which admits differences in consumption opportunity sets across countries. The model shows that the real expected excess return on a risky asset is proportional to the covariance of the return of that asset with changes in the world real consumption rate. It has no barriers to international investment, but it is compatible with empirical facts, which contradict the predictions of earlier models and which seem to imply that asset markets are internationally segmented. Besides, Stulz (1981b) also presents a simple model in which it is costly for domestic investors to hold foreign assets. The implications of the model for the composition of optimal portfolios at home and abroad are derived. It is shown that all foreign assets with a beta larger than some beta plot on either one of two security market lines. Some foreign assets with a beta smaller than are not held by domestic investors even if their expected return is increased slightly.

  33. 4. International CAPM After the above two papers, Stulz (1982) examines the conditions under which a risk premium is incorporated in the forward exchange rate. A new condition for the existence of a risk premium is proposed. He shows that earlier models of the risk premium, which emphasize either the role of net foreign investment or of the relative supplies of “outside” assets, are not suited for assessing the effects of changes in macroeconomic policy. Finally, Stulz (1984) summarizes that how differences across countries of 1) inflation rate, 2) consumption baskets of investors, and 3) investment opportunity sets of investors matter when one applies capital asset pricing models in an international setting. In particular, the fact that countries differ is shown to affect the portfolio held by investors, the equilibrium expected returns of risky assets, and the financial policies of firms. In empirical studies, Chang and Hung (2000) employ a two-factor international equilibrium asset pricing model to examine pricing relationships among the world's five largest equity markets. Their paper suggests that the intertemporal asset pricing model proposed by Campbell (1993) can be used to explain the returns on the five largest stock market indices.

  34. 5. Equilibrium Models with Heterogeneity 5.1 Heterogeneous Beliefs and Investors 5.2 Heterogeneous investment horizon

  35. 5. Equilibrium Models with Heterogeneity5.1 Heterogeneous Beliefs and Investors (68) (69)

  36. 6. Dividend and Taxation Effect Models (70) (71)

  37. 7. Skewness Effect Models Sharpe (1964), Lintner (1965), and Mossin (1966), following the work of Markowitz (1959), develope the first formulations of the mean-variance CAPM. However, many researchers criticize the widely used mean-variance analysis of portfolio selection and argue that assets pricing models should subsume the effects of the higher moments. Borch (1969) contends that any system of upward sloping mean-standard deviation indifference curves can be shown to be inconsistent with the basic axiom of choice under uncertainty. Feldstein (1969) shows that Tobin (1958, 1965) is incorrect in asserting that the μ-σ indifference curves of a risk-averter are convex-downwards whenever the possible investment outcomes are assumed to follow a two-parameter probability distribution. Although Tobin‘s proof is correct for normal distributions, for a number of economically interesting distributions, the indifference curves are not convex, showing that when more than one asset has positive variance, an analysis in terms of only μ and σ is not strictly possible unless utility functions are quadratic or the possible subjective probability distributions are severely restricted. Tsiang (1972) argues that although the mean-standard deviation analysis was at first introduced by Tobin to explain liquidity preference in the sense of an investment demand for cash, in his defense of it against its critics, he actually finds that it is quite incapable of doing what Tobin has expected of it. Furthermore, he claims that the importance of skewness preference for major risk-takers should obviously be taken into consideration in problems of investment incentives.

  38. 7. Skewness Effect Models Therefore, Jean (1971) begins a general extension of the two-parameter analysis to three or more parameters; however, Ingersoll (1975) corrects several errors in Jean’s model (1971) and derives a normative, individual pricing model for risky securities analogous to the capital market line within the framework of a perfect market. Finally, Schweser (1978) clarifies and corrects certain parts of Ingersoll’s correction of Jean’s work. Although many researchers pay more attention to the skewness effect on capital asset pricing models, Lee (1977) first employs the transformation technique developed by Box and Cox (1964) to determine the true functional form for testing the risk-return relation and to examine the possible impact of the skewness effect on capital asset pricing. According to Sears and Wei (1988) although the estimated coefficient of co-sknewness gives important information on the marginal rate of substitution between skewness preferences, that is independent of the effects of the market risk premium. Moreover, Harvey and Siddique (2000) suggest that if asset returns have systematic skewness, expected returns should include rewards for accepting this risk. They formalized this intuition with an asset pricing model that incorporates conditional skewness. Their results show that conditional skewness helps to explain the cross-sectional variation of expected returns across assets and is significant even when factors based on size and book-to-market are included.

  39. 8. Behavioral Finance (72) (73)

  40. 9. Liquidity-based Models (74) (75) (76)

  41. 9. Liquidity-based Models (77) (78) (79)

  42. 10. Existence of Equilibrium Hart (1974) argues that in deriving the properties of equilibrium prices, it has been assumed that equilibrium does in fact exist. Surprisingly, no attempt appears to have been made to establish the existence of equilibrium in the basic Lintner-Sharpe model or in more general versions of the model. Yet, the existence of equilibrium is not implied by any of the standard existence theorems because these theorems assume that consumption sets are bounded below. By contrast the assumption that investors can hold securities in unlimited negative amounts implies that consumption sets are unbounded below. In his paper, he finds the conditions for the existence of equilibrium in a very general version of the Lintner-Sharpe model; moreover, Nielsen (1989)presents simple conditions and a simple proof of the existence of equilibrium in asset markets where short-selling is allowed and satiation is possible. Unlike standard non-satiation assumptions, the one used here is weak enough to be reasonable in the mean-variance CAPM and in asset market models where investors maximize expected utility and where total returns to individual assets may be negative.

  43. 11. Empirical Tests Black et al. (1972) and Fama and MacBeth (1973) test the implication of CAPM and find empirical evidence to support the linear relationship between risk and return and efficient market; therefore, their empirical studies support the CAPM. Roll (1977), however, criticizes their empirical results by declaring that (a) no correct and unambiguous test of the theory has appeared in the literature, and (b) there is practically no possibility that such a test can be accomplished in the future. Besides, Cheng and Grauer (1980) also criticize the tests of Black et al. (1972) and Fama and MacBeth (1973) based only on the assumption of constant β and stationarity of the distribution of return; therefore, their paper argues that it makes no sense to attempt a test of the CAPM based on stationarity because the validity of the CAPM over time implies stationarity cannot hold in any but a very degenerate sense. Thus, they find the CAPM generally does poorly in their tests. Finally, Fama and French (1992) conclude that market capitalization (a measure of size) and the ratio of the book to the market value equity should replace beta altogether.

  44. 12. Conclusion We have surveyed the evolution of CAPM from 1964 to 2009. We use both figures and a table to summarize this paper. Figure 1 shows the research flow chart, and Table 1 provides the literature summary. Sharpe (1964), Lintner (1965), and Mossin (1966) derive their original static CAPM according to the six critical assumptions. Many scholars have tried to get more generalized asset pricing models by relaxing the assumption to meet the real world situation. Because of the limitation of six critical assumptions and possible model misspecification, we should carefully use the original static CAPM to acquire the required return of an asset and calculate its abnormal return. Fama and French (2004) argue that the CAPM’s empirical problems may reflect theoretical failings, the result of many simplified assumptions; however, they may also be caused by difficulties in implementing valid tests of the model. Fama and French’s empirical research is based only upon the original static CAPM, but we believe that empirical research should not only be based upon the original static CAPM.

  45. 12. Conclusion In this paper, we have carefully reviewed papers which have extended the original static CAPM. These papers have been classified into (i) Merton’s Intertemporal CAPM, (ii) Consumption-based Intertemporal CAPM, (iii) Production-based Intertemporal CAPM, (iv) CAPM with Supply-side Effect, (v) International Equilibrium CAPM with Heterogeneity Beliefs and Investors, (vi) Equilibrium CAPM with Heterogeneity Investment Horizon, (vii) CAPM with Dividend and Taxation Effect, (viii) CAPM with Skewness Effect, and (ix) Behavioral Finance, and Liquidity-based CAPM. As a result of our review, we believe that some important issues remain for future researchers. Now we discuss these potential important research issues as follows: First, we can try to subsume behavioral finance into asset pricing models, for example, investor sentiment. Obviously, many noise traders affect stock returns, but we still have no theoretical asset pricing model that includes their behaviors into a pricing factor.

  46. 12. Conclusion Second, we can further explore the supply side of asset pricing models. In the past, there was relatively few literature on the supply side; however, it is important. Holmstrom and Tirole (2001) suggest, for example, new determinants of asset prices, such as the distribution of wealth within the corporate sector and between the corporate sector and the consumers. Also, leverage ratios, capital adequacy requirements, and the composition of saving affect the corporate demand for liquid assets and, thereby, interest rates. Third, although Fama and French’s (1996) three-factor model has good empirical performance, they acknowledge that there are important limitations in their model. Their empirical results still do not cleanly identify the two consumption-investment state variables of special hedging concern to investors that would provide a neat interpretation of their results in terms of Merton’s (1973) ICAPM or Ross’ (1976) APT. Merton’s (1973) ICAPM not only has a complete and solid theoretical framework but also provides better empirical performance than the static CAPM, such as Fama and French’s (1996) three-factor model if we can find those solid and robust state variables. We suggest that future researchers should pay more attention to how to identify those solid and robust state variables. Moreover, it will make bring Merton’s (1973) ICAPM closer to real world, and its implication will be useful for empirical studies. Fourth, the relationship between perspective theory and CAPM needs further research in both theoretically and empirically, and especially the relationship between skewness type of CAPM and perspective theory needs to be carefully investigated.

  47. The Original CAPM Sharpe (1964), Lintner (1965),and Mossin (1966) Behavioral Finance Kahneman and Tversky (1979, Econometrica) Tversky and Kahneman (1992,Journal of Risk and Uncertainty) Levy (2010, European Financial Management) The Static CAPM (single-period) Existence of Equilibrium Hart (1974, Journal of Economic Theory) Nielsen (1989, Review of Economic Studies) The Dynamic CAPM (multi-period) Dividend and Taxation Effect Models Miller and Modigliani(1961,Journal of Business) Brennan (1970, National Tax Journal) Black and Scholes (1974,Journal of Financial Economics) Sasson and Kolodny(1976, The Review of Economics and Statistics) Miller and Scholes (1978,Journal of Financial Economics) Litzenberger and Ramaswamy (1979, Journal of Financial Economics) Morgan (1982,The Journal of Finance) Litzenberger and Ramaswamy (1982,The Journal of Finance) Hagiwara and Herce (1997, The American Economic Review) Equilibrium Models with Heterogeneity Investment Horizon Lee (1976, The Review of Economics and Statistics) Levhari and Levy (1977, The Review of Economics and Statistics) Lee, Wu, and Wei (1990, Journal of Financial and Quantitative Analysis) Supply-Side Effect Models Black (1976, American Economic Review) Grinols (1984, Journal of Finance) Lee, Tsai, and Lee (2009, Quarterly Review of Economics and Finance) Intertemporal CAPM－Merton Model Merton (1973,Econometrica) International CAPM Stulz (1981a, Journal of Finance) Stulz (1981b, Journal of Financial Economics) Stulz (1982, Journal of International Economics) Stulz (1984, Journal of International Business Studies) Equilibrium Models with Heterogeneity Beliefs and Investors Constantinides (1982, Journal of Business) Constantinides and Duffie (1996, Journal of Political Economy) Brav, Constantinides, and Geczy (2002, Journal of Political Economy) Basak (2005, Journal of Banking and Finance) Levy, Levy, and Benita (2006, Journal of Business) Intertemporal CAPM－Consumption-based Models Breeden (1979,Journal of Financial Economics) Campbell (1993, American Economic Review) Campbell and Cochrane (1999, Journal of Political Economy) Jagannathan and Wang (1996, Journal of Finance) Lettau and Ludvigson (2001a, Journal of Finance) Lettau and Ludvigson (2001b, Journal of Political Economy) Lewellen and Nagel (2006, Journal of Financial Economics) Balvers and Huang (2009, Journal of Financial and Quantitative Analysis) Skewness Effect Models Borch (1969, Review of Economics Studies) Feldstein (1969, Review of Economics Studies) Jean (1971,Journal of Financial and Quantitative Analysis) Tsiang (1972, American Economic Review) Ingersoll (1975,Journal of Financial and Quantitative Analysis) Schweser (1978,Journal of Financial and Quantitative Analysis) Liquidity-based Models Pastor and Stambaugh (2003, Journal of Political Economy) Acharya and Pedersen (2005, Journal of Financial Economics) Yoel(2009,working paper) Intertemporal CAPM－Production-based Models Balvers, Cosimano, and McDonald (1990, Journal of Finance) Cochrane (1991, Journal of Finance) (1996, Journal of Political Economy) Balvers and Huang (2007, Journal of Financial Economics) Figure1. Flow Chart

  48. The Dynamic CAPM (multi-period) Supply-Side Effect Models Black (1976, American Economic Review) Grinols (1984, Journal of Finance) Lee, Tsai, and Lee (2009, Quarterly Review of Economics and Finance) Intertemporal CAPM－Merton Model Merton (1973,Econometrica) International CAPM Stulz (1981a,Journal of Finance) Stulz (1981b,Journal of Financial Economics) Stulz (1982,Journal of International Economics) Stulz (1984,Journal of International Business Studies) Chang and Hung (2000, Review of Quantitative Finance and Accounting) Intertemporal CAPM－Consumption-based Models Breeden (1979,Journal of Financial Economics) Campbell (1993, American Economic Review) Campbell and Cochrane (1999, Journal of Political Economy) Jagannathan and Wang (1996, Journal of Finance) Lettau and Ludvigson (2001a, Journal of Finance) Lettau and Ludvigson (2001b, Journal of Political Economy) Lewellen and Nagel (2006, Journal of Financial Economics) Balvers and Huang (2009, Journal of Financial and Quantitative Analysis) Intertemporal CAPM－Production-based Models Balvers, Cosimano, and McDonald (1990, Journal of Finance) Cochrane (1991, Journal of Finance) (1996, Journal of Political Economy) Balvers and Huang (2007, Journal of Financial Economics) Figure2. The Dynamic CAPM

  49. The Static CAPM (single-period) Dividend and Taxation Effect Models Miller and Modigliani(1961,Journal of Business) Brennan (1970, National Tax Journal) Black and Scholes (1974,Journal of Financial Economics) Sasson and Kolodny(1976, The Review of Economics and Statistics) Miller and Scholes (1978,Journal of Financial Economics) Litzenberger and Ramaswamy (1979, Journal of Financial Economics) Morgan (1982,The Journal of Finance) Litzenberger and Ramaswamy (1982,The Journal of Finance) Hagiwara and Herce (1997, The American Economic Review) Equilibrium Models with Heterogeneity Investment Horizon Lee (1976, The Review of Economics and Statistics) Levhari and Levy (1977, The Review of Economics and Statistics) Lee, Wu, and Wei (1990, Journal of Financial and Quantitative Analysis) Equilibrium Models with Heterogeneity Beliefs and Investors Constantinides (1982, Journal of Business) Constantinides and Duffie (1996, Journal of Political Economy) Brav, Constantinides, and Geczy (2002, Journal of Political Economy) Basak (2005, Journal of Banking and Finance) Levy, Levy, and Benita (2006, Journal of Business) Skewness Effect Models Borch (1969, Review of Economics Studies) Feldstein (1969, Review of Economics Studies) Jean (1971,Journal of Financial and Quantitative Analysis) Tsiang (1972, American Economic Review) Ingersoll (1975,Journal of Financial and Quantitative Analysis) Schweser (1978,Journal of Financial and Quantitative Analysis) Liquidity-based Models Pastor and Stambaugh (2003, Journal of Political Economy) Acharya and Pedersen (2005, Journal of Financial Economics) Yoel(2009,working paper) Figure 3. The Static CAPM

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