Temporal Aggregation and the Continuous-Time Capital Asset Pricing Model

We examine how the empirical implications of the Capital Asset Pricing Model (CAPM) are affected by the length of the period over which returns are measured. We show that the continuous-time CAPM becomes a multifactor model when the asset pricing relation is aggregated temporally. We use Hansen's Generalized Method of Moments (GMM) approach to test the continuous-time CAPM at an unconditional level using size portfolio returns. The results indicate that the continuous-time CAPM cannot be rejected. In contrast, the discrete-time CAPM is easily rejected by the tests. These results have a number of important implications for the interpretation of tests of the CAPM which have appeared in the literature.     

ALTERNATIVE TESTS OF THE ZERO‐BETA CAPM

In this paper I develop an analytical Wald test of the zero‐beta capital asset pricing model (CAPM) in a simple iid (independent and identically distributed) setting and extend the Wald test to the generalized method of moments (GMM) framework that allows for a general form of serial correlation and conditional heteroskedasticity. The size and power of these tests, along with some existing tests, are investigated under normal errors and other alternative distributional specifications. The results show that, under alternative distributional assumptions for the error terms, the proposed Wald and GMM tests have reliable sizes for medium‐size samples, whereas the likelihood ratio test (LRT) rejects the efficiency too often, especially when the error terms significantly deviate from normality. However, the LRT is more powerful than both the Wald and GMM tests. JEL classification: C13, C53, G14.     

Robust estimation with flexible parametric distributions: estimation of utility stock betas

The distributions of stock returns and capital asset pricing model (CAPM) regression residuals are typically characterized by skewness and kurtosis. We apply four flexible probability density functions (pdfs) to model possible skewness and kurtosis in estimating the parameters of the CAPM and compare the corresponding estimates with ordinary least squares (OLS) and other symmetric distribution estimates. Estimation using the flexible pdfs provides more efficient results than OLS when the errors are non-normal and similar results when the errors are normal. Large estimation differences correspond to clear departures from normality. Our results show that OLS is not the best estimator of betas using this type of data. Our results suggest that the use of OLS CAPM betas may lead to erroneous estimates of the cost of capital for public utility stocks.     

UK stock returns and robust tests of mean variance efficiency

We test both the unconditional and conditional Mean Variance Efficiency of the UK stockmarket, paying particular attention to choosing a suitable set of instruments for the conditional version of the model. By considering more carefully than previous authors the pricing of economic risk within the mean-variance framework we show that certain instruments can enhance the basic model structure. Given the tendency for financial market data to display non-constancy in variance and non-normality we employ the GMM procedure described in Hansen (1982), which requires much weaker distributional assumptions than the more traditional OLS techniques. We discuss forming portfolios of stocks using both size and dividend yield as a criterion to achieve a suitable spread of risk and return, and find that our conclusions are sensitive both to the method of portfolio formation and to the choice of estimator. This is an important finding given the problem of thin trading associated with the size ordering of UK stocks. We find some support for both the unconditional and conditional version of the CAPM, though we are cautious about our conclusions given the instability of the parameter estimates.     

Tests of conditional asset pricing models in the Brazilian stock market

In this paper, we test a version of the conditional CAPM with respect to a local market portfolio, proxied by the Brazilian stock index during the period 1976–1992. We also test a conditional APT model by using the difference between the 30-day rate (Cdb) and the overnight rate as a second factor in addition to the market portfolio in order to capture the large inflation risk present during this period. The conditional CAPM and APT models are estimated by the Generalized Method of Moments (GMM) and tested on a set of size portfolios created from individual securities exchanged on the Brazilian markets. The inclusion of this second factor proves to be important for the appropriate pricing of the portfolios.     

Alternative beta risk estimators and asset pricing tests in emerging markets: The case of Pakistan

This paper tests and compares the applicability of two asset pricing models specifically, the CAPM and the Fama–French three factor models for an emerging stock market namely, Pakistan. The paper analyses a number of beta risk estimators, including OLS, the Dimson thin trading estimator, a trade-to-trade estimator and a sample selectivity estimator. To uncover any possible influence of the return interval and the type of the market index, the analysis is carried out on three data frequencies namely daily, weekly and monthly as well as for a value and an equally weighted market index. The alternative beta estimators appear to correct thin trading bias but their effects on asset pricing tests are not visible. Moreover contrary to the expectations the test results for monthly and weekly frequencies are not promising. Instead for daily data the cross-section of returns are explained by a number of risk factors and trading volume.     

The Return Performance of Cubic Market Model: An Application to Emerging Markets

This article studies the performance of the high-order moment capital asset pricing model (CAPM) market models in emerging markets. We apply the cubic market model (4-moment CAPM) to 16 emerging market stock indices ranging from January 2010 to September 2015. Performance of the model is evaluated through the Fama and MacBeth’s two-step regression and through different corrections proposed in the literature, as well as generalized method of moments (GMM) estimation. According to Fama–MacBeth’s procedure, CAPM, the quadratic and cubic market models seem to be insignificant for the analyzed sample; however, the GMM estimation shows that quadratic model is valid for Indian, Polish, and Thai country indices, whereas cubic market model is accurate for Indian country index.     

Pension benefits, wealth accumulation, and capital market equilibrium: a life cycle hypothesis-based dynamic model

This paper examines the impacts of pension benefits on capital asset pricing in conjunction with wealth accumulation and retirement, and derives and tests a dynamic capital asset pricing model (CAPM) within the framework of a life cycle hypothesis-based dynamic model. The life cycle hypothesis-based dynamic model maximizes the expected utility of the individual's lifetime wealth in a continuous time process. An optimal solution of the individual's wealth path, incorporating the ages of retirement and death, is obtained and, based on the optimal wealth path, an analysis of comparative dynamics is pursued. The dynamic CAPM is then derived from the optimal wealth path; simulation and nonparametric tests are undertaken to evaluate the performance of the dynamic CAPM as compared to the traditional model which does not consider the impacts of pension benefits and the static model that incorporates the effects of pension benefits. The test results suggest that the proposed dynamic CAPM closely states the expected rate of return for a capital asset; that the new dynamic CAPM is preferable over the static model that is preferable over the traditional model; and that the three models considered are statistically distinguishable from one another.     

w-MPS risk aversion and the shadow CAPM: theory and empirical evidence

This paper presents the shadow capital asset pricing model (CAPM) of Ma [2011a. Advanced Asset Pricing Theory. London: Imperial College Press] as an intertemporal equilibrium asset pricing model, and tests it empirically. In contrast to the classical CAPM – a single-factor model based on a strong behavioral or distributional assumption – the shadow CAPM can be represented as a two-factor model, and only requires a modest behavioral assumption of weak form mean-preserving spread risk aversion. The empirical tests provide support in favor of the shadow CAPM over the classical CAPM, the consumption CAPM, or the Epstein and Zin [1991. “Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis”. Journal of Political Economy 99, 263–286] model. Moreover, the shadow CAPM provides a consistent explanation for the cross-sectional variations of expected returns on the stocks and for the time-varying equity premium.     

Omitted debt risk, financial distress and the cross-section of expected equity returns

The study of Ferguson and Shockley (2003) shows that, if the Merton (1974) model can reflect reality, the omission of debt claims from the market portfolio proxy may explain the poor pricing ability of the CAPM in empirical tests. We critically re-assess this argument by first reviewing existing, but also new avenues through which the Merton (1974) model can point to a systematic bias in market beta estimates. However, we also show that some avenues are diversifiable, and that they all rely on excessive economy-wide default risk to create a non-negligible bias. We then use the Merton (1974) model to proxy for the total debt portfolio, but find that its application in empirical tests cannot improve pricing performance. We conclude that there are (so far) no valid theoretical reasons to believe that omitted debt claims undermine CAPM tests.     

On the power of cross-sectional and multivariate tests of the CAPM

This paper examines the power of the cross-sectional and multivariate tests of the CAPM under ideal conditions. When the CAPM is true the positively weighted market portfolio is MV-efficient and securities plot on the security market line. When the CAPM is false an alternative asset pricing model determines prices. An examination of the population intercepts, slopes and R² from cross-sectional regressions of expected returns on betas indicates that all three are unreliable indicators of whether the CAPM holds. Simulation analysis of the power of the cross-sectional tests expands on and reinforces the analysis based on the population values. The Gibbons et al. (1989) multivariate test fares much better.     

The performance of cross-sectional regression tests of the CAPM with non-zero pricing errors

We show that in the presence of non-zero pricing errors, the Fama–MacBeth (FM) cross-sectional regression test is very likely to either reject the Capital Asset Pricing Model (CAPM) when it (almost) holds or accept the model when it grossly fails. We investigate the case when pricing errors are correlated with betas and demonstrate that the test performance depends crucially on the correlation, cross-sectional distribution of betas, and several other parameter values. Even when the CAPM holds exactly (pricing errors are zero) the FM test is equally likely to either reject or accept the model when typical sample sizes are used.     

A Test of the Conditional CAPM with Simultaneous Estimation of the First and Second Conditional Moments

Recent studies of the conditional Capital Asset Pricing Model (CAPM) are extended by modeling the first and second conditional moments separately, but estimating them jointly, allowing the isolation of time-varying influences in the different moments. Results support evidence that both conditional moments and conditional risk parameters are time varying, and help shed light on previous rejections of the conditional CAPM. The GMM test of the over-identifying restrictions fails to reject the null that the CAPM holds through time when a non-linear specification is used, and only weakly rejects the null with a linear specification.     

NONNESTED PROCEDURES IN ECONOMETRIC TESTS OF ASSET PRICING THEORIES

In this paper nonnested tests are used to contrast the performance of the capital asset pricing (CAPM) and consumption capital asset pricing (CCAPM) theories in describing the U.S. stock market. The procedures employed include the N‐test, the NT‐test, the W‐test, the J‐test, and the Encompassing test. The tests are carried out using data on firms as well as portfolios based on beta, capitalization, and Standard Industrial Classification codes. The findings indicate that although during 1973–82 the CAPM dominates the CCAPM, during 1978–87 the results are mixed, and during 1983–92 the CCAPM dominates. The finding in favor of the CCAPM in 1983–92 conflicts with much of the existing literature, which favors the CAPM.     

Evaluating asset pricing models in the Korean stock market

This paper evaluates and compares asset pricing models in the Korean stock market. The asset pricing models considered are the CAPM, APT-motivated models, the Consumption-based CAPM, Intertemporal CAPM-motivated models, and the Jagannathan and Wang conditional CAPM model. By using various test portfolios as well as individual stocks, we conduct time-series tests and cross-sectional regression tests based on individual t-tests, the joint F-tests, the Hansen and Jagannathan (1997) distance, and R-squares. Overall, the Fama and French (1993) five-factor model performs most satisfactorily among the asset pricing models considered in explaining the intertemporal and cross-sectional behavior of stock returns in Korea. The Fama and French (1993) three-factor model, the Chen et al. (2010) three-factor model, and the Campbell (1996) model are the next. The results indicate that the two bond portfolios, term spread and default spread, play an important role in explaining stock returns in Korea.     

On multivariate tests of the CAPM

This paper evaluates the power of multivariate tests of the Capital Asset Pricing Model. The results indicate that when employing an unspecified alternative hypothesis, the ability of the tests to distinguish between the CAPM and other pricing models is poor. An upper bound is derived for the distance the alternative distribution of the test statistic can be from the null distribution when the deviations from the CAPM are due to missing factors. This upper bound explains the low power of the tests.     

Market proxies as factors in linear asset pricing models: Still living with the roll critique

A new model misspecification measure for linear asset pricing models is proposed for the case where misspecification maps to latency of one of the pricing factors; in this case, the market return. This measure is suited both for testing models that include the market return as a pricing factor in a traditional sense (i.e., whether the chosen model does or does not price a collection of risky assets) and ranking those models (i.e., determining which model performs best). The proposed measure is used in pricing portfolios reflecting the size, value, and momentum premia. The conditional CAPM of Jagannathan and Wang (1996) is found to best the performance of both the simple CAPM and the ICAPM of Petkova (2006). Moreover, it is discovered that winner stocks in a momentum portfolio may have higher market betas than loser stocks.     

Testing multi-beta asset pricing models

This paper presents a complete solution to the estimation and testing of multi-beta models by providing a small sample likelihood ratio test when the usual normality assumption is imposed and an almost analytical GMM test when the normality assumption is relaxed. Using 10 size portfolios from January 1926 to December 1994, we reject the joint efficiency of the CRSP value-weighted and equal-weighted indices. We also apply the tests to analyze a new version of Fama and French's [Fama, E.F., French, K.R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics. 33, 3–56.] three-factor model in addition to two standard ones, and find that the new version performs the best.     

Monotonicity in asset returns: New tests with applications to the term structure,the CAPM,and portfolio sorts

Many theories in finance imply monotonic patterns in expected returns and other financial variables. The liquidity preference hypothesis predicts higher expected returns for bonds with longer times to maturity; the Capital Asset Pricing Model (CAPM) implies higher expected returns for stocks with higher betas; and standard asset pricing models imply that the pricing kernel is declining in market returns. The full set of implications of monotonicity is generally not exploited in empirical work, however. This paper proposes new and simple ways to test for monotonicity in financial variables and compares the proposed tests with extant alternatives such as t-tests, Bonferroni bounds, and multivariate inequality tests through empirical applications and simulations.