the evolution of capital asset pricing models sheng-syan chen, national taiwan university cheng-few...

The Evolution of Capital Asset Pricing Models

Sheng-Syan Chen, National Taiwan UniversityCheng-Few Lee, Rutgers UniversityYi-Cheng Shih, National Taipei UniversityPo-Jung Chen, National Taiwan University

Outline

1. Introduction2. Intertemporal Models3. Supply-Side Effect Models4. International CAPM5. Equilibrium Models with Heterogeneity6. Dividend and Taxation Effect Models7. Skewness Effect Models8. Behavioral Finance 9. Liquidity-based Models10. Existence of Equilibrium11. Empirical Tests12. Conclusion

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.

1. Introduction

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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.

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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.

5. Equilibrium Models with Heterogeneity

5.1 Heterogeneous Beliefs and Investors

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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.

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.

8. Behavioral Finance

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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.

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.

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.

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.

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.

Table 1. Literature Summary

Models Literature Results and Contributions

Intertemporal CAPM－ Merton Model

Merton (1973, Econometrica) Merton (1973) relaxes the single-period assumption to develop the intertemporal CAPM model with stochastic investment opportunities, stating that the expected return on any asset is deduced from a multi-beta version of CAPM in a continuous-time model.

Intertemporal CAPM－ Consumption-based Models

Breeden (1979, Journal of Financial Economics) Breeden (1979) utilizes the same continuous-time economic framework as used by Merton (1973), shows Merton’s multi-beta pricing equation can be collapsed into a single-beta equation. The expected return on any asset is proportional to its beta with respect to aggregate consumption alone.

Campbell (1993, American Economic Review) Campbell (1993) substitutes consumption out of the model to get a discrete-time version of the intertrmporal CAPM of Merton (1973).

Campbell and Cochrane (1999, Journal of Political Economy) Campbell and Cochrane (1999) present a habit persistence model to explain the dynamic pricing phenomena, that is, using lagged consumption as the state variable to explain the procyclical variation of stock prices, the long-horizon predictable of excess stock returns, and the countercyclical variation of stock market volatility.

Jagannathan and Wang (1996, Journal of Finance) Jagannathan and Wang (1996) argue that the CAPM holds in a conditional sense that betas and the market premium vary over time. They add the labor income to explain the cross-section asset returns.

Lettau and Ludvigson (2001a, Journal of Finance) Lettau and Ludvigson (2001a) investigate the power of fluctuations in the log consumption-wealth ratio for forecasting asset returns.

Lettau and Ludvigson (2001b, Journal of Political Economy) Lettau and Ludvigson (2001b) is the first reexamination of a consumption-based factor model, the first recent paper that finds some success in pricing the value premium from a macro-based model. They examine a conditional version of the linear consumption-based CAPM model with time-varying coefficients.

Lewellen and Nagel (2006, Journal of Financial Economics) Lewellen and Nagel (2006) criticize consumption model on the argument that the low covariance between the risk premium and the betas. The covariance between consumption betas and the consumption risk premium obtained from a series of estimates over small time windows is too small to support the importance of any conditional variable.

Balvers and Huang (2009, Journal of Financial and Quantitative Analysis)

Balvers and Huang (2009) exclude Merton (1973) factors by assuming that there are no changes over time in the exogenous dividend processes, ruling out shifts in the investment opportunities set and conclude that real money growth as an additional factor determine asset returns.

Table 1. (Continued)


Intertemporal CAPM－ Production-based Models

Balvers, Cosimano, and McDonald (1990, Journal of Finance) Balvers, Cosimano and McDonald (1990) present a general equilibrium theory relating returns on financial assets to macroeconomic fluctuations in a context that is consistent with efficient markets in that no excess-profit opportunities are available. Aggregate output is equal or proportionate to aggregate consumption and that one can evaluate the marginal utility of consumption at the observed level of output so that aggregate output growth becomes the key asset pricing factor.

Cochrane (1991, Journal of Finance) (1996, Journal of Political Economy)

Cochrane (1991, 1996) extend the production-based CAPM by deriving from producer ’s first order condition for optimal intertemporal investment demand to describe the asset returns.

Balvers and Huang (2007, Journal of Financial Economics) Balvers and Huang (2007) derive the productivity shocks in the marginal value of capital to obtain an explicit production-based CAPM expression for the asset pricing model.

Supply-Side Effect Models

Black (1976, American Economic Review) Black (1976) examined the effects of disequilibrating shocks on individual behavior in financial markets and the effects of such modified behavior on market outcomes. A short-run dynamic, multi-period capital asset pricing model is constructed by assuming rational expectations and adding the supply side to the static model of capital asset pricing.

Grinols (1984, Journal of Finance) Grinols (1984) extended Merton's intertemporal capital asset pricing model with multiple consumers to include a description of the supply of traded securities.

Lee, Tsai, and Lee (2009, Quarterly Review of Economics and Finance) Lee, Tsai, and Lee (2009) first theoretically extend the dynamic, simultaneous CAPM model of Black (1976) to the existence of the supply effect in the asset pricing process. They use price, dividend per share and earnings per share to test the existence of supply effect with U.S. domestic stock markets.

Equilibrium Models with Heterogeneity Beliefs and Investors

Constantinides (1982, Journal of Business) Constantinides (1982) argue the equilibrium model of a heterogeneous-household, full-information economy under the assumption that the households insure against idiosyncratic income shocks.

Constantinides and Duffie (1996, Journal of Political Economy) Constantinides and Duffie (1996) construct a discount factor to represent any asset pricing anomalies under the assumption that investors have the same power utility function.

Brav, Constantinides, and Geczy (2002, Journal of Political Economy) Brav, Constantinides, and Geczy (2002) test the stochastic discount factor given by the equally weighted sum of the household’s marginal rates of substitution to be a valid stochastic discount factor based on the set of Euler equation of household consumption.

Basak (2005, Journal of Banking and Finance) Basak (2005) provides a continuous-time pure-exchange framework to study asset pricing implication of the present of heterogeneous beliefs, within a rational Bayesian setting.

Levy, Levy, and Benita (2006, Journal of Business) Levy, Levy, and Benita (2006) relax the homogeneous beliefs assumption of CAPM. They employ the mathematical analysis and numerical simulations to study the effect of the introduction of heterogeneity of beliefs on asset prices.



Equilibrium Models with Heterogeneity Investment Horizon

Lee (1976, The Review of Economics and Statistics) Lee (1976) first prove the observed function form of CAPM can become nonlinear and show that either the likelihood ratio method or constant elasticity of substitution function methods can employed to improve the explanatory power of CAPM.

Levhari and Levy (1977, The Review of Economics and Statistics)

Levhari and Levy (1977) investigate the empirical implications of heterogeneous investment horizons.

Lee, Wu, and Wei (1990, Journal of Financial and Quantitative Analysis)

Lee, Wu, and Wei (1990) examine the effect of heterogeneous investment horizons on the functional form of capital asset pricing and suggest a translog model for estimating the relation between risk and return.

Taxation Effect Models

Brennan (1970, National Tax Journal) Brennan (1970) first propose an extended form of the single period CAPM model that accounted for the differential taxation of dividends over capital gains.

Litzenberger and Ramaswamy (1979, Journal of Financial Economics)

Litzenberger and Ramaswamy (1979) extend the model of Brennan (1970) to account for restrictions on investors’ borrowing. The model is the standard two-parameter pricing models adjusted for differential taxation of dividends and interest income relative to capital gains.

Skewness Effect Models

Borch (1969, Review of Economics Studies) Borch (1969) contended 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, Review of Economics Studies) Feldstein (1969) showed that Tobin (1958, 1965) was 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 shows 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.

Jean (1971, Journal of Financial and Quantitative Analysis) Jean (1971) began a general extension of the two-parameter analysis to three or more parameters.

Tsiang (1972, American Economic Review) Tsiang (1972) argued 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 claimed that the importance of skewness preference for major risk-takers should obviously be taken into consideration in problems of investment incentives.

Ingersoll (1975, Journal of Financial and Quantitative Analysis) 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.



Schweser (1978, Journal of Financial and Quantitative Analysis) Ingersoll (1975) developed a normative multidimensional security pricing model for individual investor in which he corrected errors in an earlier attempt by Jean (1971) at developing such a model. Schweser (1978) clarified and corrected certain parts of Ingersoll’s correction of Jean’s work.

Skewness Effect Models Sears and Wei (1988, The Financial Review) Sears and Wei (1988) indicated that 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.

Harvey and Siddique (2000, Journal of Finance) Harvey and Siddique (2000) suggested that if asset returns have systematic skewness, expected returns should include rewards for accepting this risk. They formalized an asset pricing model that incorporates conditional skewness. Their results showed 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.

Behavioral Finance

Kahneman and Tversky (1979, Econometrica) Kahneman and Tversky (1979) developed the prospect theory to describe that people behave more in accordance with a psychologically based theory rather than seek to maximize the expected utility.

Tversky and Kahneman (1992, Journal of Risk and Uncertainty) Tversky and Kahneman (1992) modified the prospect theory by using a cumulative distribution function for the domain of gains and losses rather than separate decisions called Cumulative Prospect Theory.

Levy (Forthcoming, European Financial Management) Levy suggested that a modified version of mean-variance analysis and the traditional CAPM can be justified in the Cumulative Prospect Theory framework, despite the fact that under the Cumulative Prospect Theory, the expected utility theory is invalid.

Liquidity-based Models

Pastor and Stambaugh (2003, Journal of Political Economy) Pastor and Stambaugh (2003) find that stocks whose prices decline when the market gets more illiquid receive compensation in expected returns. Dividing stocks into 10 portfolios based on liquidity betas, the portfolio of high-beta stocks earned more than the portfolio of low beta stocks, after accounting for market, size, and value-growth effects.

Acharya and Pedersen (2005, Journal of Financial Economics) Acharya and Pedersen (2005) performed a similar but more general investigation on four channels for a liquidity premium. Their largest premium is the covariance of liquidity with market return — the chance the stock may get more illiquid if the market goes down.

Existence of Equilibrium

Hart (1974, Journal of Economic Theory) Hart (1974) argued that in deriving these 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.

Nielsen (1989, Review of Economic Studies) 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 Capital Asset Pricing Model and in asset market models where investors maximize expected utility and where total returns to individual assets may be negative.

the evolution of capital asset pricing models sheng-syan chen, national taiwan university cheng-few...

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intertemporal models

consumptionbased models

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productionbased models

liquiditybased models

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skewness effect models

taxation effect models