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O R I G I N A L R E S E A R C H

The evolution of capital asset pricing models

Yi-Cheng Shih • Sheng-Syan Chen • Cheng-Few Lee • Po-Jung Chen

Published online: 23 February 2013 � Springer Science+Business Media New York 2013

Abstract The capital asset pricing models (CAPM) has been the benchmark of asset pricing models and has been used to calculate asset returns and the cost of capital for more

than four decades. Many researchers have tried to relax the original assumptions and

generalize the static CAPM. We survey the important alternative theoretical models of

capital asset pricing and provide a complete review of the evolution of asset pricing

models. We also discuss the interrelationships among these models and suggest several

possible directions for future research. Our results might be used as a guideline for future

theoretical and empirical research in capital asset pricing.

Keywords CAPM � Asset pricing models � Modern capital market theory

JEL Classification G11 � G12

This paper has benefited from comments of the seminar participants at the Eighteenth Annual Conference on Pacific Basin Finance, Economics, Accounting and Management Conference and the Eighth NTU Inter- national Conference on Economics, Finance and Accounting.

Y.-C. Shih (&) Department of Finance and Cooperative Management, College of Business, National Taipei University, 151, University Road, San-shia District, New Taipei City 237-41, Taiwan, ROC e-mail: [email protected]

S.-S. Chen � P.-J. Chen Department of Finance, College of Management, National Taiwan University, Taipei, Taiwan, ROC e-mail: [email protected]

P.-J. Chen e-mail: [email protected]

C.-F. Lee Department of Finance and Economics, School of Business, Rutgers University, Camden, NJ, USA e-mail: [email protected]

123

Rev Quant Finan Acc (2014) 42:415–448 DOI 10.1007/s11156-013-0348-x

1 Introduction

This paper surveys the evolution of Capital Asset Pricing Models (CAPM) during the last

four decades. The original CAPM of Sharpe (1964), Lintner (1965), and Mossin (1966) is

developed in a hypothetical world, where the following assumptions are made about

investors and the opportunity set: ‘‘(1) Investors are risk-averse individuals who maximize

the expected utility of their wealth; (2) Investors are price takers and have homogeneous

expectations about asset returns that have a joint normal distribution; (3) There exists a

risk-free asset such that investors may borrow or lend unlimited amounts at a risk-free rate;

(4) The quantities of assets are fixed, and all assets are marketable and perfectly divisible;

(5) Asset markets are frictionless, and information is costless and simultaneously available

to all investors; and (6) There are no market imperfections such as taxes, regulations, or

restrictions on short selling’’ (Copeland et al. 2005).

During the last four decades, the CAPM is the benchmark of asset pricing models and

most empirical studies apply it to calculate asset returns and cost of capital. Dybvig and

Ross (2003) think even though there are many more modern pricing models, the CAPM is

still the most important and it provides us most of our basic intuitions about the trade-off

between risk and return, about how market risk is priced, and about how idiosyncratic risk

is not priced. Because of the limitations of the six critical assumptions and possible model

misspecification, many researchers have tried to develop more general asset pricing models

by relaxing the assumptions of CAPM and testing the empirical implications. In this paper,

we survey the important literature of CAPM and organize it into the two research flows:

‘‘The Static CAPM’’ and ‘‘The Dynamic CAPM’’.

In the research flow of the static CAPM, Brennan (1970) first proposes an extended

form of the single period CAPM model that accounts for the differential taxation of

dividends over capital gains. Litzenberger and Ramaswamy (1979) extend the model of

Brennan (1970) to account for restrictions on investors’ borrowing. Besides the differential

taxation of dividends over capital gains, Black and Scholes (1974) also examine the effects

of dividend yield and policy on common stock prices and returns.

After Black and Scholes (1974), Sasson and Kolodny (1976) also examine the rela-

tionship between the CAPM and the question of dividend relevance. They argue that once

a security’s beta coefficient is given, the CAPM implies that knowledge of a firm’s divi-

dend policy is of no use in assessing the security’s return, or correspondingly, its market

value. However, they provide the evidence in their paper against this premise. Litzenberger

and Ramaswamy (1982) present some new empirical results to show a positive and non-

linear relationship between common stock returns and expected dividend yield. The pre-

diction rule for expected dividends is based solely on information that would have been

available to the investor ex-ante. Hagiwara and Herce (1997) consider dividend-based and

consumption-based capital asset pricing models. Their estimation results suggest that the

dividend asset pricing model provides a better explanation of the data than the con-

sumption asset pricing model.

Sharpe (1964), Lintner (1965), and Mossin (1966), following the work of Markowitz

(1959), develop the first formulations of the mean–variance capital asset pricing model.

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) contend that any system of upward sloping mean-standard devi-

ation 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 l-r indifference curves of a risk-averter are convex-downwards whenever the

416 Y.-C. Shih et al.

123

possible investment outcomes are assumed to follow a two-parameter probability distri-

bution. Although Tobin’s proof is correct for normal distributions, for a number of eco-

nomically interesting distributions, the indifference curves are not convex, showing that

when more than one asset has positive variance, an analysis in terms of only l and r is not strictly possible unless utility functions are quadratic or the possible subjective probability

distributions are severely restricted. Jean (1971) begins a general extension of the two-

parameter analysis to three or more parameters. Besides the effect of the higher moments

on utility function, traditional asset pricing theorists assume that investors seek to maxi-

mize expected utility. However, proponents of behavioral finance suggest that people

behave more in accordance with a psychologically based theory, such as prospect theory,

developed by Kahneman and Tversky (1979).

Levy et al. (2006) relax the homogeneous beliefs assumption of CAPM, which can be

derived under various sets of assumptions. However, one of the fundamental assumptions

that all of the above-mentioned models share, which is considered as the most critical

drawback of CAPM, is that all investors have homogeneous beliefs regarding the expected

returns and the variance–covariance matrix. Under the homogeneous beliefs assumption,

all investors invest in the same mix of risky assets, the CAPM holds, and a simple

relationship exists between risk and return, well known as the security market line (SML).

Levy et al. (2006) examine the robustness of the CAPM to the relaxation of one of its most

problematic assumptions: homogeneous beliefs. They prove that in a heterogeneous-belief

market with an infinite number of investors and an infinite number of risky assets, the

CAPM risk-return relationship precisely holds. Regarding the relaxation of homogeneous

investment horizon assumption, Lee et al. (1990) examine the effect of heterogeneous

investment horizons on the functional form of capital asset pricing and proposed a translog

model for estimating the risk-return relationship. In addition, their paper contends that

some empirical findings that are inconsistent with the traditional CAPM result from

misspecification of the CAPM by ignoring the discrepancy between the observed data

periods and the true investment horizons.

The other research flow of CAPM is about the dynamic CAPM, 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. Breeden (1979) utilizes the same

continuous-time economic framework as Merton, likewise permitting stochastic investment

opportunities. Merton’s multi-beta pricing equation can be collapsed into a single-beta

equation, where the instantaneous expected excess return on any security is proportional to

its beta with respect to aggregate consumption alone.

After Breeden (1979), it is important to emphasize, as Fama and French (1988), and

others note, that in the context of intertemporal models, predictability is not necessarily

inconsistent with the concept of market efficiency. Balvers et al. (1990) therefore present a

general equilibrium theory relating returns on financial assets to macroeconomic fluctua-

tions in a context that is consistent with efficient markets in that no excess-profit oppor-

tunities are available. The intuition underlying their theoretical model arises from

consumption smoothing by investors. Consumption opportunities are linked to output, and,

consistent with conventional macroeconomic models, output is serially correlated and

hence predictable. To maximize utility, investors attempt to smooth consumption by

adjusting their required rate of return for financial assets. For example, investors antic-

ipating lower output in the next period will attempt to transfer wealth to this anticipated

period of scarcity and therefore will accept a lower rate of return in order to smooth

consumption over time. Although Merton (1973) and Breeden (1979) relax the single-

The evolution of capital 417

123

period assumption of CAPM, they merely provide the demand-side models without supply-

side.

Black (1976) examines 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. After

Black (1976), Grinols (1984) extends Merton’s intertemporal capital asset pricing model

with multiple consumers to include a description of the supply of traded securities. Stulz

(1981a) also provides an intertemporal model of international asset pricing, which admits

differences in consumption opportunity sets across countries.

After we discuss these theoretical models to relax the assumptions of traditional CAPM,

we turn to the existence of equilibrium. Hart (1974) argues that in deriving these properties

of equilibrium prices, it is assumed that equilibrium does in fact exist.

The empirical evidence leads scholars to conclude that the pure theoretical form of the

traditional CAPM does not agree well with reality. For example, Roll (1977) argues ‘‘The

only legitimate test of the CAPM is whether or not the market portfolio is mean–variance

efficient…’’, ‘‘If performance is measured relative to an index that is ex post efficient, then from the mathematics of the efficient set, no security will have abnormal performance

when measured as a departure from the security market line.’’ Besides, most empirical

studies of the static CAPM assume that betas remain constant over time and that the return

on the value-weighted portfolio of all stocks is a proxy for the return on aggregate wealth.

The general consensus is that the static CAPM is unable to explain satisfactorily the cross-

section of average returns on stocks. Therefore, Fama and French (1992, 1996) provide the

three-factor model to the cross-section of expected stock returns. So the great factor debate

rages on. Sharpe (1998) says, ‘‘I’d be the last to argue that only one factor drives market

correlation. There are not as many factors as some people think, but there’s certainly more

than one.’’

Thus, in this paper we survey the important literature of CAPM and organize it into the

following sections: ‘‘Dividend and Taxation Effect Models’’, ‘‘Equilibrium Models with

Heterogeneity Beliefs and Investors’’, ‘‘Equilibrium Models with Heterogeneity Invest-

ment Horizon ‘‘, ‘‘Skewness Effect Models’’, ‘‘Liquidity-based Models’’, ‘‘Intertemporal

CAPM-Merton Model’’, ‘‘Intertemporal CAPM-Consumption-based Model’’, ‘‘Intertem-

poral CAPM-Production-based Model’’, ‘‘Supply-side Effect Model’’, ‘‘Existence of

Equilibrium’’, ‘‘Behavioral Finance’’ and ‘‘Empirical Tests’’. Finally, we draw the evo-

lution of CAPM in Fig. 1.

2 The static CAPM

2.1 No riskless asset

First, how will CAPM change if investors cannot borrow and lending at the risk-free rate?

Black et al. (1972) provide the minimum-variance zero-beta portfolio to solve the problem

and the model is also called Black’s zero-beta CAPM.

2.2 Dividend and taxation effect models

Brennan (1970) first proposes an extended form of the single period CAPM model that

accounts for the differential taxation of dividends over capital gains. Litzenberger and

418 Y.-C. Shih et al.

123

Ramaswamy (1979) extend the model of Brennan (1970) to account for restrictions on

investors’ borrowing. Both these models assume that dividends and interest are taxed as

ordinary income and capital gains are taxed at more favorable rates. Brennan’s model is

under the assumption of proportional individual tax rate (not a function of income), certain

dividends, and unlimited borrowing at the riskless rate of interest as

EðRiÞ� rf ¼ b�bi þ sðdi � rfÞ; ð1Þ

where EðRiÞ is the before tax expected return to security i, bi is the covariance of asset i’s return with the market divided by the variance of the market, b� ¼ EðRmÞ� rf � sðdm � rfÞ � �

is the after-tax excess return of the market portfolio, rf is the

return on a riskless asset, di and dm is the dividend yield on security i and the market

portfolio, and s represents a tax differential between dividends and capital gains. Litzenberger and Ramaswamy (1979) extend this model by assuming both margin and

income constraints on borrowing. Their model is

EðRiÞ� rf ¼ a þ bbi þ cðdi � rfÞ; ð2Þ

where a ¼ EðRz�Þ� rf , b ¼ EðRmÞ� EðRz�Þ� cðdm � rfÞ, c is a tax differential reduced by a shadow price that reflects increases in investors’ ability to borrow from an additional

dollar of dividends, and EðRz�Þ is the expected return on a zero-beta portfolio with a dividend yield equal to the taxable riskless rate. These models are the standard two-

parameter pricing models adjusted for differential taxation of dividends and interest

income relative to capital gains. Besides the differential taxation of dividends over capital

gains, Black and Scholes (1974) examine the effects of dividend yield and policy on

common stock prices and returns. Their paper suggests that it is not possible to demon-

strate, using the best available empirical methods, that the expected returns on high-yield

The Original CAPM Sharpe (1964), Lintner (1965),and Mossin (1966)

The Static CAPM (single-period)

Existence of Equilibrium Hart (1974)

Nielsen (1989)

The Dynamic CAPM (multi-period)

Behavioral Finance Kahneman and Tversky (1979) Tversky and Kahneman (1992) Barberis, Huang and Santos (2001) Levy, De Giorigi and Hens (2003) Barberis and Huang (2008) Levy (2010)

Dividend and Taxation Effect Models Miller and Modigliani(1961) Brennan (1970) Black and Scholes (1974) Sasson and Kolodny(1976) Miller and Scholes (1978) Litzenberger and Ramaswamy (1979) Morgan (1982) Litzenberger and Ramaswamy (1982) Hagiwara and Herce (1997) Equilibrium Models with

Heterogeneity Investment Horizon Lee (1976) Levhari and Levy (1977) Lee, Wu, and Wei (1990)

Equilibrium Models with Heterogeneity Beliefs and Investors Constantinides (1982) Constantinides and Duffie (1996) Brav, Constantinides, and Geczy (2002) Basak (2005) Levy, Levy, and Benita (2006) Yoel(2009)

Liquidity-based Models Pastor and Stambaugh (2003) Acharya and Pedersen (2005)

Skewness Effect Models Borch (1969) Feldstein (1969) Jean (1971) Tsiang (1972) Ingersoll (1975) Schweser (1978) Sears and Wei(1988) Harvey and Siddique(2000)

Intertemporal CAPM

Merton Model Merton (1973)

Supply-Side Effect Models Black (1976) Grinols (1984) Lee, Tsai, and Lee(2009)

International CAPM Stulz (1981a) Stulz (1981b) Stulz (1982) Stulz (1984) Chang and Hung (2000)

Intertemporal CAPM Consumption-based Models Breeden (1979) Campbell (1993) Campbell and Cochrane (1999) Jagannathan and Wang (1996) Lettau and Ludvigson (2001a) Lettau and Ludvigson (2001b) Lewellen and Nagel (2006) BalversandHuang(2009)

Intertemporal CAPM Production-based Models Balvers, Cosimano, and McDonald (1990) Cochrane (1991,1996) Balvers andHuang (2007)

No Riskless Asset Black (1972)

Fig. 1 The evolution of capital assets pricing models

The evolution of capital 419

123

common stock differ from the expected returns on low-yield common stocks either before

or after taxes. After Black and Scholes (1974), Sasson and Kolodny (1976) also examine

the relationship between the CAPM and the question of dividend relevance. They argue

that once a security’s beta coefficient is given, the CAPM implies that knowledge of a

firm’s dividend policy is of no use in assessing the security’s return, or correspondingly, its

market value. However, they provide evidence in their paper against this premise. In the

tests they perform, knowledge of a firm’s dividend policy makes a significant contribution

to explaining the return received on the firm’s security. In particular, the results support the

position that investors have a net preference for receiving their return in the form of

dividends to receiving it in the form of capital gains. Therefore the results of their research

suggest that if, as shown, firms having greater payouts on average have less market risk and

offer the investor smaller risk-adjusted returns, the results of tests that classify firms

according to risk class would contain an underlying bias in favor of greater risk firms.

These firms would exhibit a greater risk-adjusted return than others because of the omitted

dividend policy variable. In such a situation, the CAPM should be modified or an alter-

native model used to eliminate the bias. Morgan (1982) summarizes three distinct views of

the importance of dividends to investors have received support at one time or another.

According to the two most important views, dividends have a neutral and a negative effect

on security prices respectively. Miller and Modigliani (1961) and Miller and Scholes

(1978) favor complete substitutability of dividends and capital gain. Brennan (1970) and

Litzenberger and Ramaswamy (1979) have developed models that incorporate differential

taxation of income and capital gain. In these models, the tax disadvantage of income

changes the investor’s portfolio problem from one of optimization of mean and variance of

portfolio before-tax return to optimization in after-tax terms. Stocks with relatively large

dividend yields should provide high expected returns (and so sell for low prices) to

compensate the investor for the tax. Litzenberger and Ramaswamy (1982) present some

new empirical results to show that a positive and non-linear relationship between common

stock returns and expected dividend yield. The prediction rule for expected dividends is

based solely on information that would have been available to the investor ex-ante.

Hagiwara and Herce (1997) consider dividend-based and consumption-based CAPMs.

Their estimation results suggest that the dividend asset pricing model provides a better

explanation of the data than the consumption asset pricing model.

2.3 Equilibrium models with heterogeneity

One of the fundamental assumptions that all of the above-mentioned models share, which

is considered as the most critical drawback of CAPM, is that all investors have homoge-

neous beliefs regarding the expected returns and the variance–covariance matrix. In the

consumption-based CAPM we need to assume that assets can be priced if there is a

representative agent who consumes aggregate consumption. Therefore, in this section we

will introduce the model of heterogeneous beliefs, agents, and investment horizon.

2.3.1 Heterogeneous beliefs and investors

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.

Equilibrium is determined in terms of a representative investor’s utility function with

stochastic weighting driven by the investor’s disagreement about the aggregate growth. In

addition, Levy et al. (2006) relax the homogeneous beliefs assumption of CAPM, which

420 Y.-C. Shih et al.

123

can be derived under various sets of assumptions. Under the homogeneous beliefs

assumption, all investors invest in the same mix of risky assets, the CAPM holds, and a

simple relationship exists between risk and return, well known as the security market line

(SML). They employ mathematical analysis and numerical simulations to study the effect

of the introduction of heterogeneity of beliefs on asset prices, they prove that in an infinite

market when number of investors and assets approach to infinite, with unbiased hetero-

geneous beliefs and bounded variance, the CAPM linear risk-return relationship precisely

holds.

However, it is also possible that utility-maximizing stock market investors are hetero-

geneous in important ways. If investors are subject to large idiosyncratic risks in their labor

income and can share these risks only indirectly by trading a few assets such as stocks and

Treasury bills, their individual consumption paths may be much more volatile than

aggregate consumption. Constantinides and Duffie (1996) assume an economy in which

heterogeneous investors k have different consumption levels ckt. The cross-sectional dis-

tribution of individual consumption is lognormal, and the change from time t to time t ? 1

in individual log consumption is cross-sectional uncorrelated with the level of individual

log consumption at time t. All investors have the same power utility function with time

discount factor d and coefficient of relative risk aversion c. In this economy each investor’s own intertemporal marginal rate of substitution is a valid stochastic discount factor. Hence

the cross-sectional average of investors’ intertemporal marginal rates of substitution, M�tþ1,

is a valid stochastic discount factor. The marginal utility of the cross-sectional average of

investors’ consumption, MRAtþ1, however, is not a valid stochastic discount factor when

marginal utility is nonlinear. This false stochastic discount factor would be used incorrectly

by an economist who ignores the aggregation problem in the economy. They state the

Euler equation of consumption of consumer k-th for security i:

Et d Ctþ1 Ct

� ��c exp

c cþ1ð Þ 2

Var�tþ1

h i li;tþ1

n o ¼ 1: ð3Þ

Here Var�tþ1 denotes a cross-sectional variance of individual log consumption growth

taken after aggregates at time t ? 1. Equation (3) adds the exponential term to the standard

consumption-based asset pricing equation. The difference between the logs of these two

stochastic discount factors is

M�tþ1 � M RA tþ1 ¼

cðc þ 1Þ 2

Var�tþ1Dck;tþ1; ð4Þ

therefore, they can construct a discount factor to represent any asset pricing anomalies.

Constantinides (1982) develops another heterogeneous investor’s model to claim that if

a complete set of markets exists and enables households to insure against idiosyncratic

income shocks, then heterogeneous households are able to equalize, state by state, their

marginal rates of substitution. Therefore, the equilibrium of a heterogeneous-household,

full-information economy is in its pricing implications to the equilibrium of a represen-

tative-household, full-information economy.

Under the full consumption insurance assumption implies that heterogeneous consumers

are able to equalize their marginal rate of substitution state by state. However, Brav et al.

(2002)’s finding is based on the set of Euler equation of household consumption rather than

the per capita consumption. Since the individual consumption data are reported with

substantial error and are difficult to test, they, therefore, test the hypothesis that the sto-

chastic discount factor given by the equally weighted sum of the household’s marginal

rates of substitution is a valid stochastic discount factor.

The evolution of capital 421

123

Yoel (2009) derives a general equilibrium asset pricing model, in which low-status

investors hold a single high volatility asset in order to move up the status ladder. Since

high-status investors are concerned about the risk of losing their status, they demand assets

that co-vary with high volatility assets as a hedge against low-status investors. The general

equilibrium asset pricing model derived in the paper is novel in at least two important

dimensions. First, it is not driven by the preferences of a single representative agent, but

instead it is a result of strategic interactions among heterogeneous investors. Second, the

general theme of asset pricing models is that factor risk premiums arise due to the fact that

risk-averse investors seek to limit their exposure to systematic risk factors. In the model,

the high-volatility factor premium arises due to the fact that status-conscious investors seek

exposure to this factor in order to hedge against their status risk.

2.3.2 Heterogeneous investment horizon

Lee (1976) derives two function forms for the CAPM, which will explicitly include the

investment horizon parameter to improve the explanatory power of the CAPM. To allow

the investment horizon parameter to be explicitly included in the CAPM, it will be

assumed that all investors have identical horizons. The main contributions of Lee (1976)

are to prove the observed function form of CAPM can become nonlinear and to show that

either the likelihood ratio method or the constant elasticity of substitution function methods

can be employed to improve the explanatory power of CAPM.

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

investment horizons. They have substantially contributed to the understanding of multi-

period investments, but they do not provide a generalized asset pricing model for the

equilibrium risk-return relationship under heterogeneous investment horizons. Lee et al.

(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. They argue that some empirical findings include inconsistence with traditional

CAPM from misspecification of the CAPM by ignoring the inconsistency between

observed data period and the true investment horizon. They assume that the holding period

returns of securities are serially independent and the return distributions are stationary.

Under these two assumptions, the expected returns and variance are identical over time,

and the covariance of returns between two periods equals zero. The translog model is a

suitable function for estimating the relationship between risk and expected return.

2.4 Skewness effect models

Sharpe (1964), Lintner (1965), and Mossin (1966), following the work of Markowitz

(1959), develop 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 l-r 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 l and r is not strictly

422 Y.-C. Shih et al.

123

possible unless utility functions are quadratic or the possible subjective probability dis-

tributions 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.

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.

2.5 Liquidity-based models

Financial economists have long suspected that less liquid securities might have to offer an

expected return higher than that justified by the covariance of the security’s return with the

market or other factors, and thus the security might have a lower price level.

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 (regression coefficients of stock returns on market

liquidity with other factors as controls), the portfolio of high-beta stocks earned 9 % more

than the portfolio of low-beta stocks, after accounting for market, size, and value-growth

effects with the Fama–French three factor model. Almost this entire premium is accounted

for by spread in liquidity betas and a factor risk premium estimated across 10 portfolios.

The standard asset pricing model is EðRiÞ¼ bik. Thus, a spread in average returns E(R i )

across portfolios i is explained if they are linearly related to the betas bi. A test whether

k ^ ¼ 0 is often performed to assess the significance of the model. Acharya and Pedersen (2005) perform a similar but more general investigation as

follows:

We are interested in how an asset’s expected return,

Rit ¼ Dit þ Pit

Pit�1 ; ð5Þ

depends on its relative illiquidity cost, T it represents transactions costs, such as broker fees

and bid-ask spread,

The evolution of capital 423

123

tit ¼ T it

Pit�1 ; ð6Þ

on the market return, S i

means total shares of security i,

RMt ¼ P

i S iðDit þ PitÞP i S

iPit�1 ; ð7Þ

and on the relative market illiquidity,

tMt ¼ P

i S iT itP

i S iPit�1

: ð8Þ

They rewrite the single-beta CAPM in net returns in terms of gross returns and get a

liquidity-adjusted CAPM for gross return.

EtðRitþ1 � t i tþ1Þ¼ rf þ kt

covtðRitþ1 � t i tþ1; R

M tþ1 � t

M tþ1Þ

vartðRMtþ1 � tMtþ1Þ ; ð9Þ

where kt ¼ EtðRMtþ1 � cMtþ1 � rfÞ is the risk premium. Thus, the conditional expected gross return is

EtðRitþ1Þ¼rf þ Etðt i tþ1Þþ kt

covtðRitþ1; RMtþ1Þ vartðRMtþ1 � tMtþ1Þ

þ kt covtðtitþ1; tMtþ1Þ

vartðRMtþ1 � tMtþ1Þ

� kt covtðRitþ1; tMtþ1Þ

vartðRMtþ1 � tMtþ1Þ � kt

covtðtitþ1; RMtþ1Þ vartðRMtþ1 � tMtþ1Þ

:

ð10Þ

They examine all four channels for a liquidity premium. First, a security might have to

pay a premium simply to compensate for its particular illiquidity or transactions cost.

Second, a security might have to pay a premium because it becomes more illiquid in bad

times, that is, when the market goes down. If you have to sell it (and sellers are the

marginal investor), this tendency amounts to a larger beta than would be measured by the

midpoint of a bid-asked spread. Third, the security’s price (the midpoint) might decline

when markets as a whole become less liquid. If ‘‘market liquidity’’ is a state variable, an

event that drives up the marginal utility of a marginal investor, then this tendency will also

result in a return premium. This is the mechanism that Pastor and Stambaugh (2003)

investigate. Fourth, the security could become more illiquid when the market becomes

more illiquid. Then, they examine whether the four sources of covariation described above

explain the variation in average returns. Interestingly, their largest premium k in EðRiÞ¼ bik is the covariance of liquidity with market return—the chance the stock may get more illiquid if the market goes down.

3 The dynamic CAPM

3.1 Intertemporal models

The static CAPM of Sharpe (1964), Lintner (1965), and Mossin (1966) states that the

expected premium on any risky asset is proportional to the premium on the market as a

whole. It is criticized for the requirement of additional assumption, especially

424 Y.-C. Shih et al.

123

homogeneous expectations and the single-period nature of the world. An intertemporal

model for the capital market is concluded from the multiperiod setting.

3.1.1 Merton model

The theoretical linear relationship on the static CAPM model, however, is subject to

criticism on the premise of a constant opportunity set. Merton (1973) subsequently relaxes

this assumption and shows that when the opportunity set fluctuates over time due to

changes in the state of the economy, individual securities are also priced with respect to

selected portfolios, providing a hedge against unanticipated fluctuations.

Merton (1973) first develops 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. Consider K investors, who are concerned

with maximizing the expected utility of wealth at the end of a period, where the utility

functions are twice-differentiable concave functions. The k-th investor is assumed to be

able to invest in a riskless asset with instantaneous rate of return, rf, and in i-th risky asset

with instantaneous rate of return, given by the stochastic differential equation

dPi

Pi ¼ lidt þ ridzi; ð11Þ

where dzi is the increment to a standard Brownian motion. Processes such as Eq. (11) are

call Itô processes, and they are continuous and are not differentiable. Thus the stochastic

process for the investor’s wealth is shown as

dW ¼ Xn

i¼1 wi li � rf � �

þ rf

" #

Wdt þ Xn

i¼1 wiWridzi þ y � cð Þdt; ð12Þ

where wi is the fraction of the wealth invested in the i-th asset, y is the wage income, and

c is the consumption. Merton (1973) develops the simplest form of the model, which

occurs when the investment opportunity set is constant through time and the distributions

for price per share will be log-normal for all assets. Using the condition that the market

portfolio is efficient in equilibrium, it can be shown that, for this version of the model, the

equilibrium returns will satisfy

ai � r ¼ bi aM � rð Þ i ¼ 1; 2; . . .; nð Þ; ð13Þ

where bi ¼ riM=r2M , riM is the covariance of the return on the ith asset with the return on the market portfolio and aM is the expected return on the market portfolio. Equation (13) is the continuous-time analog to the security market line of the classical CAPM. Hence, the

additional assumption of a constant investment opportunity set is a sufficient condition for

investors to behave as if they were single-period maximizers and for the equilibrium return

relationship specified by the CAPM to obtain. Except for some singular cases, this

assumption is also necessary.

Unfortunately, the assumption of a constant investment opportunity set is not consistent

with the facts because at least one element of the opportunity set is directly observable:

namely, the interest rate, and it is definitely changing stochastically over time. The simplest

form of the model consistent with this observation occurs if it is assumed that a single state

variable is sufficient to describe changes in the opportunity set. Merton (1973) further

assumes that this variable is the interest rate (i:e:; ai ¼ ai rð Þ and ri ¼ ri rð Þ ). The interest

The evolution of capital 425

123

rate has always been an important variable in portfolio theory, general capital theory, and

to practitioners. It is observable, satisfies the condition of being stochastic over time, and

although it is surely not the sole determinant of yields on other assets, it is an important

factor.

Merton (1973) derives the equilibrium market clearing conditions for the model of the

stochastic opportunity set and the equilibrium relationship between the expected return on

an individual asset and the expected return on the market.

ai � r ¼ ri qiM � qinqnM½ �

rM 1 � q2nMð Þ aM � rð Þþ

ri qin � qiM qnM½ � rn 1 � q2Mnð Þ

an � rð Þ: ð14Þ

Let the first fund hold the same proportions as the risky fund. Let the second fund hold only

the nth asset and the third fund only the riskless asset. Given (a, an, r, r, rn, q) where a and a2 is the expected return and variance on the first fund’s portfolio and q is its covariance with the return on the second fund and q = r(an - r)/rn(a - r). M is the (equilibrium) value of all assets, the market.

The (instantaneous) expected return = aM � Pn

1 wjðaj � rÞþ r; covariance ¼ riM �Pn 1 Wjrij; variance of market portfolio = r

2 M �

Pn 1 wjrjrjM. Equation (14) states that, in

equilibrium, investors are compensated in terms of expected return for bearing market risk

and for bearing the risk of unfavorable shifts in the investment opportunity set; and it is a

natural generalization of the security market line of the classical capital asset pricing

model. Note that if a security has no market risk i:e:; bi ¼ 0 ¼ qiMð Þ, its expected return will not be equal to the riskless rate as forecast by the usual model.

Merton (1973) shows that if investment opportunities vary over time, then long-term

investors generally care about shocks to investment opportunities—the productivity of

wealth—and not just about wealth itself. They may seek to hedge their exposures to wealth

productivity shocks, and this gives rise to intertemporal hedging demands for financial

assets.

3.1.2 Consumption-based models

The intertemporal CAPM model of Merton (1973) with stochastic investment opportunities

states that the expected return on assets is derived from a multi-beta version of the CAPM

with numbers of betas being equal to one plus the number of state variables needed to

describe the relevant characteristics of the investment opportunity set. Since all of those

state variables are not easily identified, Breeden (1979) utilizes the same continuous-time

economic framework as that used by Merton, likewise permitting stochastic investment

opportunities, showing that the expected return on any asset is proportional to its beta with

respect to aggregate consumption alone. He argues that a single beta relative to a specific

variable, given certain stationary assumptions on the joint distributions of rates of return

and aggregate consumption, make the model easier to test and implement. Therefore,

Breeden (1979) is the main theory of consumption-based CAPM, explaining return vari-

ation across assets from an optimizing intertemporal perspective under the assumption that

if agents have time-additive utility, that is, locally quadratic, then the expected return of

assets are is linear with their aggregate consumption.

Breeden (1979) derives a single-beta asset pricing model in a multi-good, continuous-

time model with uncertain consumption-goods prices and uncertain investment opportu-

nities; furthermore, he considers the intertemporal choice problem of an investor k who can

trade in some asset i and can obtain a gross simple rate of return (1 ? Ri,t?1) on the asset

426 Y.-C. Shih et al.

123

held from time t to time t ? 1. If the investor consumes Ck,t at time t and has time-

separable utility with discount factor d and period utility U(Ck,t), then the first-order condition of utility function is

U0ðCk;tÞ¼ d Et 1 þ Ri;tþ1 � �

U0ðCk;tþ1Þ � �

: ð15Þ

The left-hand side of Eq. (15) is the marginal utility cost of consuming one real dollar less

at time t; the right-hand side is the expected marginal utility benefit from investing the

dollar in asset i at time t, sell it at time t ? 1, and consuming the proceeds. The investor

equates marginal cost and marginal benefit, so Eq. (15) must describe the optimum.

Dividing Eq. (15) by U0ðCk;tÞ yields

1 ¼ Et ð1 þ Ri;tþ1Þd U0ðCk;tþ1Þ U0ðCk;tÞ

h i ¼ Et 1 þ Ri;tþ1

� � Mk;tþ1

� � ; ð16Þ

where Mk;tþ1 ¼ d U0ðCk;tþ1Þ=U0ðCk;tÞ � �

is the intertemporal marginal rate of substitution

of the investor, also known as the stochastic discount factor. Equation (16) assumes the

existence of an investor maximizing a time-separate utility function. The existence of a

positive stochastic discount factor is guaranteed by the absence of arbitrage in markets in

which nonsatiated investors can trade freely without transaction costs. In general, different

investors k whose marginal utilities follow different stochastic process will have different

Mk,t?1, but each stochastic discount factor must satisfy Eq. (16). It is common practice to

drop the subscript k from the equation and rewrite

1 ¼ Et 1 þ Ri;tþ1 � �

Mtþ1 � �

: ð17Þ

In complete markets the stochastic discount factor Mt?1 is unique because investors can

trade with one another to eliminate any idiosyncratic variation in their marginal utilities.

Equation (17) is written as the expectation of the product equals the product of expecta-

tions plus the covariance

Et 1 þ Ri;tþ1 � �

Mtþ1 � �

¼ Et 1 þ Ri;tþ1 � �� �

Et Mtþ1½ �þ Covt Ri;tþ1; Mtþ1 � �

: ð18Þ

Substituting Eq. (18) into Eq. (19) and rearranging produces

1 þ Et Ri;tþ1 � �

¼ 1 � Covt Ri;tþ1; Mtþ1

� �

Et Mtþ1½ � : ð19Þ

Equation (19) must hold for any asset, including a riskless asset whose gross simple return

is 1 ? Rf,t?1. Since the simple riskless return has zero covariance with the stochastic

discount factor, that is,

1 þ Rf ;tþ1 ¼ 1

Et Mtþ1½ � : ð20Þ

Equation (20) can be used to rewrite Eq. (19) as

1 þ Et Ri;tþ1 � �

¼ 1 þ Rf ;tþ1 � �

1 � Covt Ri;tþ1; Mtþ1 � �� �

ð21Þ

because the intertemporal marginal rate of substitution of the investor, also known as the

stochastic discount factor is Mk;tþ1 ¼ d U0ðCk;tþ1Þ=U0ðCk;tÞ � �

: : Under the assumption of

the form of utility function, we can get the stochastic discount factor from the first order

conditions of maximizing investors’ utilities. Then, according to Eq. (21), we can describe

any expected excess return on risky assets over the riskless rate.

The evolution of capital 427

123

Campbell and Cochrane (1999) present a habit persistence model to explain the dynamic

pricing phenomena, that is, using lagged consumption as the state variable. They replace the

utility function U(Ct) with U(Ct - Xt). The identical agents maximize the utility function

E X1

t¼1 dt ðCt � XtÞ1�c � 1

1 � c ; ð22Þ

where Xt denotes the level of habit and d belongs to the time discount factor. They use surplus consumption ratio St �ðCt � XtÞ=Ct to capture the relation between consumption and habit. The surplus consumption ratio increases with consumption. St = 0 means a bad

state in which consumption is equal to habit, and St = 1 as consumption rises relative to

habit under the assumption that the habit is the external specification and the habit is

determined by the history of aggregate consumption instead of the history of individual

consumption. Therefore, the marginal utility is

UcðCt; XtÞ¼ ðCt � XtÞ�c ¼ C�ct S �c t : ð23Þ

In this model, the intertemporal marginal rate of substitution depends on change in the ratio

of consumption to habit as well as on consumption growth,

Mtþ1 ¼ d UcðCtþ1; Xtþ1Þ

UcðCt; XtÞ ¼ d Ctþ1

Ct

� ��c Stþ1 St

� �c : ð24Þ

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; the stochastic discount factor is

Mtþ1 ¼ at þðb0 þ b1ztÞ� Dctþ1: ð25Þ

The innovation allows the slope coefficient b, which acts as the risk-aversion coefficient in

the model, to vary over time; Dct?1 is consumption growth, the single factor in the asset pricing model. However, the risk parameters at and bt will depend on risk premium, thus

they seek a scaling variable, log consumption-wealth ratio, to measure zt (as in Lettau and

Ludvigson 2001a). In the condition version of CAPM models, Jagannathan and Wang

(1996) argue that the CAPM holds in a conditional sense; that is, betas and the market

premium vary over time. They add the labor income to explain the cross-section asset

returns. Furthermore, Kumar et al. (2008) examine the information-dependent conditional

CAPM by adding the innovations in market volatility, oil prices, exchanges rates, and

dispersion of analysts’ forecasts to explain the cross section of stock returns.

Balvers and Huang (2009) add money to the standard consumption-based CAPM of

Breeden (1979). Balvers and Huang (2009) consider asset pricing in a monetary economy

that the consumption CAPM driving from real money growth as an additional factor to

determine the asset return. They argue that the availability of money as a source of

liquidity improves transactions and affects the marginal value of wealth in generating

consumption. To isolate the contribution of the money factor, Balvers and Huang (2009)

exclude Merton’s (1973) factors by assuming no changes occur over time in the exogenous

dividend processes, ruling out shifts in the investment opportunities set and concluding that

real money growth as an additional factor determines asset returns.

However, Campbell (1993) substitutes consumption out of the model to get a discrete-

time version of Merton’s (1973) intertemporal CAPM. The representative agent’s dynamic

budget constraint can be written as

428 Y.-C. Shih et al.

123

Wtþ1 ¼ Rm;tþ1ðWt � CtÞ; ð26Þ

where Wt is total wealth, Ct is consumption at period t, and Rm,t?1 represents gross return

on the portfolio of all invested wealth from period t to period t ? 1. The subscript m

denotes the fact that total invested wealth is the market portfolio of assets. The loglinear

approximation begins by dividing Eq. (26) by Wt to obtain

Wtþ1 Wt ¼ Rm;tþ1 1 �

Ct

Wt

� ; ð27Þ

or in logs

Dwtþ1 � rm;tþ1 þ a þ 1 � 1

q

� ct � wtð Þ; ð28Þ

indicating that if consumption to aggregate wealth ratio is stationary, the budget constraint

maybe be approximated by taking a first-order Taylor expansion of the equation. In Eq. (28),

when the log consumption-wealth ratio is constant, then q can be interpreted as (W - C)/W, and a is a constant. Solving Eq. (28) by assuming that limi!1 q

i ctþi � wtþið Þ ¼ 0, the log consumption-wealth ratio may be written as follows:

ct � wt ¼ X1

i¼1 qiðrm;tþi � DctþiÞþ

qk 1 � q

: ð29Þ

Taking the conditional expectations of both sides of Eq. (29) to obtain

ct � wt ¼ Et X1

i¼1 qiðrm;tþi � DctþiÞþ

qk 1 � q

; ð30Þ

Equation (30) shows that if the aggregate consumption-wealth ratio is not constant, it must

forecast changing returns to the market portfolio or changing consumption growth. In other

words the consumption-wealth ratio can vary only if consumption growth or returns or both

are predictable.

Building on the work of Krep and Porteus (1978), Epstein and Zin (1989, 1991) and

Weil (1989) develop a more flexible version of the basic power utility model. The

objective function is

Ut ¼ 1 � dð ÞC 1�c h

t þ d Et U 1�c tþ1

� �1 h

� h 1�c

: ð31Þ

Here c is the coefficient of relative risk aversion, h ¼ 1 � cð Þ= 1 � 1=rð Þ½ �, and r is the elasticity of intertemporal substitution. The Euler equation for asset i’s return, Ri,t?1 can be

written as

1 ¼ Et d Ctþ1 Ct

� �1=r( )h 1

1 þ Rm;tþ1 � �

( )1�h 1 þ Ri;tþ1 � �

2

4

3

5: ð32Þ

For the market portfolio itself, this takes the simpler form of

1 ¼ Et d Ctþ1 Ct

� �1=r Rm;tþ1

( )h2

4

3

5: ð33Þ

The evolution of capital 429

123

The log version of general Euler Eq. (32) can be used for cross-sectional asset pricing. It

takes the more complicated form of

0 ¼ h log d � h r

EtDctþ1 þ h � 1ð ÞEt rm;tþ1 þ Et ri;tþ1

þ 1

2

h r

� 2 Vcc þðh � 1Þ2Vmm þ Vii �

2h r ðh � 1ÞðVcmÞ�

2h r

Vci þ 2ðh � 1ÞVim

" #

:

ð34Þ

When the asset under consideration is a risk-free real return rf,t?1, this expression simplifies

because some variance and covariance drop out; then

Et ri;tþ1 � rf ;tþ1 ¼� Vii

2 þ h

Vic

r þð1 � hÞVim: ð35Þ

After substituting, Eq. (30) comes to

ct � wt ¼ 1 � rð ÞEt P1

i¼1 qiðrm;tþiÞ þ

q k � lmð Þ 1 � q

: ð36Þ

Equation (36) means the log consumption-wealth ratio is a constant, plus (1 - r) times the discounted value of expected future returns on invested wealth. Instead of substituting the

wealth return out of the Epstein–Zin–Weil model, Campbell (1993) substitutes con-

sumption out of the model to get a discrete-time version of Merton’s (1973) intertemporal

CAPM. The innovation in consumption is

ctþ1 � Et ctþ1 ¼ rm;tþ1 � Et rm;tþ1 þ 1 � rð Þ Etþ1 � Etð Þ X1

i¼1 qirm;tþ1þi: ð37Þ

Thus, the covariance of any asset return with consumption growth must satisfy

Cov ri;tþ1; Dctþ1 � �

� Vic ¼ Vim þ 1 � rð ÞVih; ð38Þ

where Vih denotes the covariance of asset return i with revisions in expected future returns

on wealth:

Vih � Cov ri;tþ1 � Et ri;tþ1;ðEtþ1 � EtÞ X1

i¼1 qirm;tþ1þi

!

: ð39Þ

The letter h here is used as a mnemonic for hedging demand (Merton 1973), a term

commonly used in the finance literature to describe the component of asset demand that is

determined by investors’ responses to changing investment opportunities. Substituting Eq.

(38) into Eq. (35) and using the definition h ¼ 1 � cð Þ= 1 � 1=rð Þ½ �,

Et ri;tþ1 � rf ;tþ1 ¼� Vii

2 þ cVim þðc � 1ÞVih: ð40Þ

The only parameter of the utility function that enters Eq. (40) is the coefficient of relative

risk aversion c. The elasticity of intertemporal substitution r does not appear once con- sumption has been substituted out of the model. Intuitively, this result comes from the fact

that r plays two roles in the theory. A low value of r reduces anticipated fluctuations in consumption, but it also increases the risk premium required to compensate for any

430 Y.-C. Shih et al.

123

contribution to the fluctuations (Eq. 35). These offsetting effects lead r to cancel out of the pricing Eq. (40).

A large number of models amount to situations in which the discount factor generalizes

the power-utility case by adding another state variable (Cochrane 2005). It is very danger

in such models that they often work well for short-run returns, but not in the long run. The

trouble is that the marginal utility of consumption depends not only on consumption but

also on an additional variable. Even if they assume the function form of the utility function,

there is no general agreement on how to measure the marginal utility of the consumption:

Any combination of state variables may have a quantitatively significant impact on mar-

ginal utility (Balvers and Huang 2007). Lewellen and Nagel (2006) also criticize the

consumption model because of the low covariance between the risk premium and the betas.

The covariance between consumption (market) betas and the consumption (market) risk

premium obtained from a series of estimates over small time windows is too small to

support the importance of any conditional variable.

3.1.3 Production-based models

After Breeden (1979), it is important to emphasize, as Fama and French (1988), and others

note, that in the context of intertemporal models, predictability is not necessarily incon-

sistent with the concept of market efficiency. Therefore, Balvers et al. (1990) present a

general equilibrium theory relating returns on financial assets to macroeconomic fluctua-

tions in a context that is consistent with efficient markets in that no excess-profit oppor-

tunities are available. Balvers et al. (1990) argue that 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. The advantage to this approach is that output growth is likely

measured more accurately than consumption growth. Balvers et al. (1990) indicate that

changes in aggregate output lead to attempts by agents to smooth consumption, which

affect the required rate of return on financial assets. Output yt occurs at time t, and at that

time the firm divides the output into dividends dt and investment it. The investment

becomes productive as capital one period after, kt?1, and leads to production yt?1 after the

random productivity shock ht is revealed. The dividends dt?1 are paid to the investors and together with changes in share prices pt and pt?1 determine the gross realized return Rt?1 on stock held in the period from t to t ? 1. The representative firm determines its level of

investment each period to maximize shareholder wealth

E0 X1

t¼0

Yt

i¼0 Rið Þ�1

" #

dt ð41Þ

subject to

dt ¼ yt � it ¼ yt � ktþ1 ð42Þ

yt ¼ ABtht kat ; ð43Þ

where A and B are positive constants, a is less than one, and Rt is one plus the appropriate discount rate. Substituting Eqs. (41) and (42) into (43) and differentiating with respect to

kt?1 yields the stochastic Euler condition:

Et Rtþ1ð Þ�1 a ytþ1=ktþ1ð Þ h i

¼ 1: ð44Þ

The evolution of capital 431

123

That is, the expectation of the marginal product of investment must equal the one unit of

the consumption good sacrificed in favor of investment. The consumer maximizes the

present value of time-additive utility function from the consumption of all goods. The

utility function indicates concave one-period consumption ct and q as the consumer’s discount factor for utility. The consumer maximizes

E0 X1

t¼0 qt u ctð Þ; ð45Þ

subject to

ct þ pt ytð Þstþ1 ¼ pt ytð Þþ dt ytð Þ½ �st: ð46Þ

In the budget constraint, dt(yt) represents the dividends per share, paid at the beginning of

the period; pt(yt) is the price per share in state yt immediately after dividends dt are paid;

and st is the number of shares held at the beginning of period t. Maximization by the

representative consumer yields the following Euler equation with respect to the choice

variable stþ1:

pt ytð Þu0 ctð Þ¼ qEt ptþ1 ytþ1ð Þþ dtþ1 ytþ1ð Þ½ �u0 ctþ1ð Þf g: ð47Þ

The Euler equation relates the price and return of a share to the cost of delaying con-

sumption, where they define

Rtþ1 ytþ1; ytð Þ� ptþ1 ytþ1ð Þþ dtþ1 ytþ1ð Þ½ �=pt ytð Þ ð48Þ

as the realized holding period return on shares. Equation (47) represents the consumer will

choose current consumption such that the utility of current consumption equals the

expected discount return of buying a stock times the marginal utility of consumption when

the stock is sold in the next period. However, returns will vary because consumption

carries due to the randomness in aggregate output. Solving Eq. (47) forward yields a

general expression for ex-dividend share prices

pt ¼ Et X1

i¼1 qi u0 ctþið Þ=u0 ctð Þ½ �dtþi: ð49Þ

Under the assumption of logarithmic utility function to solve the general equilibrium

model,

u ctð Þ¼ a ln ctð Þ: ð50Þ

Using the fact that u0 ctð Þ¼ a=ct and the market-clearing condition ct = dt, the demand for consumption goods equals the net supply to the consumers of the multi-purpose good, the

share pricing Eq. (39) yields

pt ¼ Et X1

i¼1 qidt ¼ q= 1 � qð Þ½ �dt: ð51Þ

Under the logarithmic form of the utility function, pt does not depend on future dividends.

From Eq. (48) for share prices and (51) to define returns,

Rtþ1 ¼ 1=qð Þ dtþ1=dtð Þ: ð52Þ

432 Y.-C. Shih et al.

123

Equations (46), (47), (48), and (52) characterize the dynamic path for dt, kt, yt, and Rt.

Where investment is proportional to output, then yields ktþ1 ¼ abyt , which implies from Eq. (46) that dt = (1 - ab)yt. Therefore, Eq. (39) becomes

Rtþ1 ¼ 1=qð Þ ytþ1=ytð Þ; ð53Þ

then substituting Eq. (53) into (44) verifies that the posited solution is correct. Since future

output depends on current investment, it can be predicted from current observations. The

solution kt?1 = abyt implies that

ytþ1 ¼ htþ1cbt yat ; ð54Þ

where c � AB abð Þa. Equations (53) and (54) together explain the return predictability in our model; therefore, Balvers et al. (1990) develop an intertemporal equilibrium model that

relates financial asset returns to movements in aggregate output.

Cochrane (1991, 1996) is an attempt to extend the production-based ideas to describe

the asset returns. Somewhat similar to the consumption-based model, the production-based

model ties asset returns to marginal rates of transformation, which are inferred from the

investment through a production function. It is derived from the producer’s first order

condition for optimal intertemporal investment demand. The discount factor of the form is

Mtþ1 ¼ b0 þ zt b1ð Þ� ftþ1; ð55Þ

which shows that the discount factor might be a linear combination of factors f with

weights that vary as a vector of instrument z varies across different information set. Scaling

the factors f by the instrument z achieves the same result. So Eq. (55) is equivalent to

Mtþ1 ¼ b0ftþ1 þ b1 ftþ1 � ztþ1ð Þ: ð56Þ

Here ft?1 denotes the investment return, functions of investment and capital only, that is,

ftþ1 ¼ fðIitþ1=Kitþ1; Iit=KitÞ; therefore, given the choice of instruments, performing the GMM estimation and testing with scaled factors is a completely general test of a dynamic,

conditional factor pricing model based on the instruments.

Balvers and Huang (2007) derive the stochastic discount factor in which the produc-

tivity shock is the single factor in asset pricing model. The social planner maximizes

V kt; htð Þ¼ Max nt;ktþ1

u ct; �n � ntð Þþ d Et V ktþ1; htþ1ð Þ½ �f g ð57Þ

subject to

htþ1 ¼ H ht; etþ1ð Þ ð58Þ

ct ¼ f ht; nt; ktð Þ� ktþ1: ð59Þ

The lifetime utility of the representative consumer is maximized subject to a standard

production function f(ht, nt, kt), where the inputs are labor nt and capital kt and an exog- enous technology level ht. The productivity level is assumed to follow a Markov process as given by Eq. (58) with et representing the zero-mean white noise productivity shock. Per- capita consumption is given in Eq. (59) as the part of capita production that is not invested.

Through Eq. (57) to Eq. (59) and the stochastic discount factor, Mtþ1 ¼ d uc ctþ1; �n � ntþ1ð Þ=uc ct; �n � ntð Þ½ � to obtain

Mtþ1 ¼ gtþ1

fk htþ1; ntþ1; ktþ1ð Þ ; ð60Þ

The evolution of capital 433

123

with gtþ1 ¼ Vk ktþ1; htþ1ð Þ=Et Vk ktþ1; htþ1ð Þ½ �. The gtþ1 represents a fundamental, nondiv- ersifiable risk inherent in capital accumulation that cannot be ignored and should be needed

for asset pricing. Balvers and Huang (2007), therefore, relate gt?1 to shocks in the marginal value of capital to obtain an explicit production-based expression for the asset pricing

model. They examine whether the pricing kernel, derived from a model that has been

successful in characterizing key macroeconomic moments, is also useful in pricing the

cross-section of financial assets.

Et r iE tþ1Vk ktþ1; htþ1ð Þ=Fk htþ1; ntþ1; ktþ1ð Þ

� � ¼ 0 for all i; ð61Þ

where in this excess returns formulation we can eliminate the Et[Vk(kt?1, ht?1)] term in the pricing kernel.

In principle, the production-based pricing kernel should price assets exactly as well as

the consumption-based pricing kernel; however, as Campbell (1993) points out, the nec-

essary consumption data may not be measured well, and the consumption-based kernel

may include additional mysterious ‘‘x’’ variables that are not easily identified. Thus,

application of the consumption-based pricing kernel is not straightforward.

Accordingly, Balvers and Huang (2007) propose evaluating the usefulness of the pro-

duction-based kernel empirically. Balvers et al. (1990) and Cecchetti et al. (1990) merely

replace consumption by output and do not consider cross-sectional asset pricing

implications.

3.2 Supply-side effect models

Although Merton (1973) and Breeden (1979) relax the single-period assumption of CAPM,

they merely provide demand-side models without the supply-side effect. Black (1976)

examines 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 CAPM is constructed by assuming rational expectations and adding a supply

side to the static model of capital asset pricing. After Black (1976), Grinols (1984) extends

Merton’s intertemporal CAPM with multiple consumers to include a description of the

supply of traded securities. The production decisions of firms are described in a model with

stochastic investment opportunities and incomplete markets. Moreover, Lee et al. (2009)

argue that Black’s theoretically elegant model has never been empirically tested for its

implications in dynamic asset pricing. They first theoretically extend Black’s CAPM, then

use price, dividend per share, and earnings per share to test the existence of supply effect

with U.S. equity data. They find the supply effect important in U.S. domestic stock mar-

kets. Lee et al. (2009) theoretically extend the dynamic, simultaneous CAPM model of

Black (1976) to the existence of the supply effect in the asset pricing process.

3.2.1 Demand function of capital assets

The demand equation for the assets is derived under the standard assumptions of the

CAPM. An investor’s objective is to maximize the expected utility in terms of the negative

exponential function of wealth:

U ¼ a � h � ef�bWtþ1g; ð62Þ

where the terminal wealth Wt?1 = Wt(1 ? Rt); Wt is initial wealth; and Rt is the rate of

return on the portfolio. The parameters, a, b and h, are assumed to be constants.

434 Y.-C. Shih et al.

123

The dollar returns on N marketable risky securities can be represented as follows:

Xj;tþ1 ¼ Pj;tþ1 � Pj;t þ Dj;tþ1; j ¼ 1; . . .; N; ð63Þ

where Pj,t?1 = (random) price of security j at time t ? 1, Pj,t = price of security j at time

t, and Dj,t?1 = (random) dividend or coupon on security at time t ? 1. These three vari-

ables are assumed to be jointly normal distributed. After taking the expected value of Eq.

(63) at time t, the expected returns for each security, xj,t?1, can be rewritten as follows:

xj;tþ1 ¼ Et Xj;tþ1 ¼ Et Pj;tþ1 � Pj;t þ Et Dj;tþ1; j ¼ 1; . . .; N; ð64Þ

where Et Pj;tþ1 ¼ E Pj;tþ1 Xtj � �

; Et Dj;tþ1 ¼ E Dj;tþ1 Xtj � �

; Et Xj;tþ1 ¼ E Xj;tþ1 Xtj � �

; Xt is the given information available at time t. Then, a typical investor’s expected value of end-of-

period wealth is

wtþ1 ¼ Et Wtþ1 ¼ Wt þ r� Wt � q0tþ1Pt � �

þ q0tþ1xtþ1; ð65Þ

where Pt ¼ P1;t; P2;t; P3;t; . . .; PN;t � �0

; xtþ1 ¼ x1;tþ1; x2;tþ1; x3;tþ1; . . .; xN;t � �0¼ Et Ptþ1�

Pt þ Et Dtþ1; qtþ1 ¼ q1;tþ1; q2;tþ1; q3;tþ1; . . .; qN;t � �0

; qj;tþ1 = number of units of security j

after reconstruction of his portfolio; r *

= risk-free rate. In Eq. (65), the first term on the

right hand side is the initial wealth, the second term is the return on the risk-free invest-

ment, and the last term is the return on the portfolio of risky securities. The variance of

Wt?1 can be written as

V Wtþ1ð Þ¼ E Wtþ1 � wtþ1ð Þ Wtþ1 � wtþ1ð Þ0¼ q0tþ1Sq;tþ1; ð66Þ

where S ¼ E Xtþ1 � xtþ1ð Þ Xtþ1 � xtþ1ð Þ0 = the covariance matrix of returns of risky secu- rities. Maximization of the expected utility of Wtþ1 is equivalent to

Max wtþ1 � b

2 VðWtþ1Þ: ð67Þ

By substituting Eqs. (65) and (66) into Eq. (67), Eq. (67) can be rewritten as Eq. (68)

Max 1 þ r�ð ÞWt þ q0tþ1 xtþ1 � r � Ptð Þ� b

2

� q0tþ1S qtþ1 : ð68Þ

Differentiating Eq. (68), one can solve the optimal portfolio as

qtþ1 ¼ b�1S�1 xtþ1 � r � Ptð Þ: ð69Þ Under the assumption of homogeneous expectation, or by assuming that all the investors

have the same probability belief about future return, the aggregate demand for risky

securities can be summed as

Qtþ1 ¼ Xm

k¼1 qktþ1 ¼ cS

�1 Et Ptþ1 �ð1 þ r�ÞPt þ Et Dtþ1½ �; ð70Þ

where c = R(bk)-1. In the standard CAPM, the supply of securities is fixed, denoted as Q

* . Then, Eq. (70)

can be rearranged as Pt ¼ 1=r�ð Þ xtþ1 � c�1SQ�ð Þ , where c-1 is the market price of risk. In fact, this equation is similar to the Lintner’s (1965) well-known equation in capital asset

pricing.

The evolution of capital 435

123

3.2.2 Supply function of securities

It is assumed that there exists a solution to the optimal capital structure and that the firm

has to determine the optimal level of additional investment. The one-period objective of

the firm is to achieve the minimum cost of capital vector with adjustment costs involved in

changing the quantity vector, Qi,t?1:

Min Et Di;tþ1Qi;tþ1 þ 12 � �

DQ0i;tþ1AiDQi;tþ1 � �

subject to Pi;tDQi;tþ1 ¼ 0; ð71Þ

where Ai is a ni 9 ni positive definite matrix of coefficients measuring the assumed qua-

dratic costs of adjustment. If the costs are high enough, firms tend to stop seeking to raise

new funds or retire old securities. The solution to Eq. (71) is

DQi;tþ1 ¼ A�1i kiPi;t � Et Di;tþ1 � �

; ð72Þ

where ki is the scalar Lagrangian multiplier. Aggregating Eq. (74) over N firms, the supply function is given by

DQtþ1 ¼ A�1i BPt � Et Dtþ1ð Þ; ð73Þ

where A�1 ¼

A�11 A�12

. . .

A�1N

2

6 6 6 4

3

7 7 7 5

, B ¼

k1I k2I

. . .

kN I

2

6 6 6 4

3

7 7 7 5

, and Q ¼

Q1 Q2

..

.

QN

2

6 6 6 4

3

7 7 7 5 :

Equation (73) implies that a lower price for a security will increase the amount retired of

that security. In other words, the amount of each security newly issued is positively related

to its own price and negatively related to its required return and the prices of other

securities.

3.2.3 Multiperiod equilibrium models

The aggregate demand for risky securities presented by Eq. (74) can be seen as a difference

equation. The prices of risky securities are determined in a multiperiod framework.

Clearly, the aggregate supply schedule has similar structure. As a result, the model can be

summarized by the following equations for demand and supply, respectively:

Qtþ1 ¼ cS�1 Et Ptþ1 � 1 þ r�ð ÞPt þ Et Dtþ1ð Þ and ð74Þ

DQtþ1 ¼ A�1i BPt � Et Dtþ1ð Þ: ð75Þ

Differencing Eq. (74) for period t and t ? 1 and equating the result with Eq. (75), a new

equation relating demand and supply for securities is

cS�1 Et Ptþ1 � Et�1Pt � 1 þ r�ð Þ Pt � Pt�1ð Þþ Et Dtþ1 � Et�1Dt½ � ¼ A�1 BPt � Et Dtþ1ð Þþ Vt; ð76Þ

where Vt is included to take into account the possible discrepancies in the system. Here, Vt is assumed to be random disturbance with zero expected value and no autocorrelation.

Obviously, Eq. (76) is a second-order system of stochastic differential equation in Pt,

and conditional expectations Et-1Pt and Et-1Dt. By taking the conditional expectation at

436 Y.-C. Shih et al.

123

time t - 1 in Eq. (76), and because of the properties of Et-1[EtPt?1] = Et-1Pt?1 and

Et�1E Vtð Þ¼ 0, Eq. (76) becomes

cS�1 Et�1Ptþ1 � Et�1Pt � 1 þ r�ð Þ Et�1Pt � Pt�1ð Þþ Et�1Dtþ1 � Et�1Dt½ � ¼ A�1 BEt�1Pt � Et�1Dtþ1ð Þ:

ð760Þ

Subtracting Eq. (760) from Eq. (76),

1 þ r�ð ÞcS�1 þ A�1B � �

Pt � Et�1Ptð Þ ¼ cS�1 Et Ptþ1 � Et�1Ptþ1ð Þ þ cS�1 þ A�1 � �

Et Dtþ1 � Et�1Dtþ1ð Þ� Vt: ð77Þ

Equation (77) shows that prediction errors in prices (the left hand side) due to unexpected

disturbance are a function of expectation adjustments in price (first term on the right-hand

side) and dividends (the second term on the right-hand side) two periods ahead. This

equation can be seen as a generalized capital asset pricing model.

Lee et al. (2009) first theoretically extend Black’s CAPM. Then they use price, dividend

per share, and earnings per share to test the existence of supply effect with U.S. equity data.

They find the supply effect is important in U.S. domestic stock markets and the existence

of the supply effect in the asset pricing.

3.3 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 b* plot on either one of two security market lines. Some foreign assets with a beta smaller than b* are not held by domestic investors even if their expected return is increased slightly. 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 equi-

librium 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.

The evolution of capital 437

123

4 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.

5 Behavioral finance

The feature that distinguishes the behavioral approach and traditional approach to asset

pricing is the assumption of expected utility. Traditional asset pricing theorists assume that

investors seek to maximize expected utility; however, proponents of behavioral finance

suggest that people behave more in accordance with a psychologically based theory, such

as prospect theory, developed by Kahneman and Tversky (1979).

Preference-based behavioral models often work with the prospect theory of Kahneman

and Tversky (1979), according to which, people do not judge outcomes on an absolute

scale but compare outcomes with an initial reference point. Their objective function has a

kink at the reference point, so risk aversion is locally infinite at that point. The objective

function is concave for gains (outcomes above the reference point) but is convex for losses

(outcomes below the reference point).

Tversky and Kahneman (1992) modify their prospect theory by using a cumulative

distribution function for the domain of gains and a cumulative distribution function for the

domain of losses rather than separate decisions called Cumulative Prospect Theory. The

value function is a utility function defined over gains and losses. The investor maximizes a

value function of the form,

VðxÞ¼ w a if x 0 �kð�wbÞ if x\0 ;

ð78Þ

where 0 \ a \ 1, 0 \ b \ 1, k [ 1, and w is the change of wealth rather than total wealth. The investor employs decision weights estimated by the formula,

xþðPÞ¼ Pc

Pc þð1 � PÞc½ �1=c

x�ðPÞ¼ Pd

Pd þð1 � PÞd h i1=d ;

ð79Þ

where P stands for cumulate probability, and x� and x? denote the transformed cumulative probability in the negative and positive domain, respectively.Although we know that the

438 Y.-C. Shih et al.

123

mean–variance analysis and the CAPM are based on the expected utility theory framework,

Tversky and Kahneman (1979, 1992) show that the investor is not always as rational as

assumed by expected utility theory economists. Barberis et al. (2001) study asset prices in

an economy where investors derive direct utility not only from consumption but also from

fluctuations in the value of their financial wealth. They argue that investors are loss averse

over these fluctuations, and the degree of loss aversion depends on their prior investment

performance. The design of their model is influenced by prospect theory and by experi-

mental evidence on how prior outcomes affect risky choice.

Levy et al. (2003) show that under the assumption of normally distributed returns, the

cumulative prospect theory is consistent with the CAPM in every financial market equi-

librium; however, they also show that under the specific functional forms suggested by

Tversky and Kahneman (1992) financial market equilibria do not exist.

Levy (2005) suggests experimental research is very important because it allows us to

control variables and sometimes to study issues that cannot be studied empirically, for

example, testing the CAPM with ex-ante parameters. Experimental findings, in particular

prospect theory and cumulative prospect theory, contradict expected utility theory.

Barberis and Huang (2008) study the asset pricing implications of Tversky and

Kahneman’s (1992) cumulative prospect theory with a particular focus on its probability

weighting component. Their main result, derived from a novel equilibrium with nonunique

global optima, is that, in contrast to the prediction of a standard expected utility model, a

security’s own skewness can be priced: A positively skewed security can be ‘‘overpriced’’

and can earn a negative average excess return. Levy (2010) and Barberis et al. (2001) have

shown that the prospect theory can be converted to CAPM when the rate of return is

normally distributed. Therefore, the relationship between prospect theory and the skewed

type of CAPM is still an open question.

Levy (2010) establishes a very interesting result; the prospect theory investor will

choose a portfolio that is mean–variance efficient. He also suggests 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.

The behavioral models cannot be tested using data on aggregate consumption or the

market portfolio because rational utility-maximizing investors neither consume aggregate

consumption nor hold the market portfolio. This makes it hard to test behavioral models

without having detailed information on the investment strategies of different market

participants.

6 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, crit-

icizes 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 b 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.

The evolution of capital 439

123

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.

7 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 pro-

vides 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.

In this paper, we have carefully reviewed papers which have extended the original static

CAPM. These papers have been classified into (1) Merton’s Intertemporal CAPM, (2)

Consumption-based Intertemporal CAPM, (3) Production-based Intertemporal CAPM, (4)

CAPM with Supply-side Effect, (5) International Equilibrium CAPM with Heterogeneity

Beliefs and Investors, (6) Equilibrium CAPM with Heterogeneity Investment Horizon, (7)

CAPM with Dividend and Taxation Effect, (8) CAPM with Skewness Effect, and (9)

Behavioral Finance, and (10) 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.

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. Holmström and

Tirole (2001) suggest, for example, new determinants of asset prices, such as the distri-

bution 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 vari-

ables 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.

440 Y.-C. Shih et al.

123

Table 1 Literature summary

Models Literature Results and contributions

The static CAPM

No riskless asset Black et al. (1972) Black et al. (1972) provide the minimum-variance zero-beta portfolio to solve the problem if there is no risk-free asset that has constant returns in every state of nature

Dividend and taxation effect models

Miller and Modigliani (1961)

Miller and Modigliani (1961) present a cogent argument for the fact that the value of the firm is unaffected by dividend policy in a world without tax or transaction costs

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

Black and Scholes (1974)

Black and Scholes (1974) suggest that it is not possible to demonstrate, using the best available empirical methods, that the expected returns on high yield common stock differ from the expected returns on low yield common stocks either before or after taxes

Sasson and Kolodny (1976)

Sasson and Kolodny (1976) argue that once a security’s beta coefficient is given, the CAPM implies that knowledge of a firm’s dividend policy is of no use in assessing the security’s return, or correspondingly, its market value. However, they provide the evidence in their paper against this premise

Miller and Scholes (1978)

Miller and Scholes (1978) favor complete substitutability of dividends and capital gain

Litzenberger and Ramaswamy (1979)

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

Morgan (1982) Morgan (1982) summarize three distinct views of the importance of dividends to investors have received support at one time or another. According to the two most important views, dividends have a neutral and a negative effect on security prices respectively. Miller and Modigliani (1961) and Miller and Scholes (1978) favor complete substitutability of dividends and capital gain. Brennan (1970) and Litzenberger and Ramaswamy (1979) have developed models which incorporate differential taxation of income and capital gain

Litzenberger and Ramaswamy (1982)

Litzenberger and Ramaswamy (1982) present some new empirical results to show that a positive and non-linear relationship between common stock returns and expected dividend yield

Dividend and taxation effect models

Hagiwara and Herce (1997)

Hagiwara and Herce (1997) consider dividend-based and consumption-based capital asset pricing models. Their estimation results suggest that the dividend asset pricing model provides a better explanation of the data than the consumption asset pricing model

The evolution of capital 441

123

Table 1 continued

Models Literature Results and contributions

Equilibrium models with heterogeneity beliefs and investors

Constantinides (1982)

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)

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 et al. (2002)

Brav et al. (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) 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 et al. (2006)

Levy et al. (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

Yoel (2009) Yoel (2009) derives a general equilibrium asset pricing model, low-status investors hold a single high volatility asset in order to move up the status ladder. Since high-status investors are concerned about the risk of losing their status, they demand assets that co-vary with high volatility assets, as a hedge against low-status investors

Equilibrium models with heterogeneity investment horizon

Lee (1976) 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)

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

Equilibrium models with heterogeneity investment horizon

Lee et al. (1990) Lee et al. (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

Skewness effect models Borch (1969) 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

442 Y.-C. Shih et al.

123

Table 1 continued

Models Literature Results and contributions

Feldstein (1969) Feldstein (1969) showed that Tobin (1958, 1965) was incorrect in asserting that the l - r 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 l and r is not strictly possible unless utility functions are quadratic or the possible subjective probability distributions are severely restricted

Jean (1971) Jean (1971) began a general extension of the two-parameter analysis to three or more parameters

Tsiang (1972) 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

Ingersoll (1975) 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) Schweser (1978) clarified and corrected certain parts of Ingersoll’s correction of Jean’s work

Sears and Wei (1988)

Sears and Wei (1988) indicated that although the estimated coefficient of co-skewness 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)

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

Liquidity-based models

Pastor and Stambaugh (2003)

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)

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

The dynamic CAPM

Intertemporal CAPM-Merton model

Merton (1973) 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

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Table 1 continued

Models Literature Results and contributions

Intertemporal CAPM- consumption-based models

Breeden (1979) 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) 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)

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

Intertemporal CAPM- consumption-based models

Jagannathan and Wang (1996)

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)

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

Lettau and Ludvigson (2001b)

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)

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)

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

Intertemporal CAPM— production-based models

Balvers et al. (1990)

Balvers et al. (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

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Table 1 continued

Models Literature Results and contributions

Cochrane (1991, 1996)

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)

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) 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

Supply-side effect models

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

Lee et al. (2009) Lee et al. (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

International CAPM

Stulz (1981a) Stulz (1981a) provided an intertemporal model of international asset pricing which admits differences in consumption opportunity sets across countries

Stulz (1981b) Stulz (1981b) also presented 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

Stulz (1982) Stulz (1982) examined the conditions under which a risk premium is incorporated in the forward exchange rate

Stulz (1984) Stulz (1984) summarized 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

Chang and Hung (2000)

Chang and Hung (2000) suggest that the intertemporal asset pricing model proposed by Campbell (1993) can be used to explain the returns on the five largest stock market indices

Existence of equilibrium

Existence of equilibrium

Hart (1974) Hart (1974) argues that in deriving the properties of equilibrium prices, it has been assumed that equilibrium does in fact exist

Nielsen (1989) 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

Behavioral finance

Behavioral finance

Kahneman and Tversky (1979)

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)

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

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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.

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Barberis et al. (2001)

Barberis et al. (2001) argue that investors are loss averse over these fluctuations, and the degree of loss aversion depends on their prior investment performance

Levy et al. (2003) Levy et al. (2003) show that under the assumption of normally distributed returns, the cumulative prospect theory is consistent with the CAPM in every financial market equilibrium

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Levy (2010) 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

Empirical Tests

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Black et al. (1972) Black et al. (1972) find the empirical evidence to support the linear relationship between risk and return and efficient market

Fama and MacBeth (1973)

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Cheng and Grauer (1980) argue that it makes no sense to attempt a test of the CAPM based on stationarity, since 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

Fama and French (1992)

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

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