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.     

Cross-sectional tests of the CAPM and Fama–French three-factor model

Grouping does not produce a wide range of betas. Consequently, cross-sectional tests of the CAPM are bound to lack power. This paper provides a simple way to alleviate the problem by repackaging the data with zero-weight portfolios. When the CAPM is true and the data are repackaged, simulation shows that the average values of the intercept and slope converge to their true values more rapidly and there are striking increases in R² and the power of the tests. Empirical results are dramatically different in datasets with and without the zero-weight portfolios.     

Return Distributions and Improved Tests of Asset Pricing Models

We compare and contrast some existing ordinary least squares(OLS)- and generalized method of moments (GMM)-based tests ofasset pricing models with a new more general test. This newtest is valid under the assumption that returns are ellipticallydistributed, a necessary and sufficient assumption of the linearcapital asset pricing model (CAPM). This new test fails to rejectthe CAPM on a dataset of stocks sorted by market valuations,whereas similar tests constructed from OLS and GMM estimationmethods reject the linear CAPM. We also find that outliers reducethe OLS-estimated mispricing of the linear CAPM on monthly returnssorted by previous performance, that is, momentum. Monte Carloevidence supports superior size and power properties of thenew test relative to OLS- and GMM-based tests.     

A conditional higher-moment CAPM

This paper investigates whether dynamic and moment extensions to the traditional CAPM can improve its empirical performance and offer some alternative explanation to the cross-section of average returns on portfolios of stocks double sorted on book-to-market ratios and size. We consider three extensions. First, we introduce time-varying factor loadings obtained from a multivariate GARCH and dynamic conditional correlations. Second, we extend the model to a four-moment CAPM, which incorporates coskewness and cokurtosis. Finally, we allow for time-varying risk premia, based on a Markov-switching process. Our results confirm that the higher-moment CAPM does not perform well in its unconditional version, but its performance is significantly improved when we introduce a conditional version that accounts for both time-varying factor loadings and time-varying risk premia. The four-moment CAPM tests lead to a positive total risk premium estimate of 0.67% per month over the period 1926–2021, with all risk premia (beta, coskewness, and cokurtosis) exhibiting the expected theoretical signs.     

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.     

A LIKELIHOOD RATIO TEST OF THE ZERO-BETA CAPM IN AUSTRALIAN EQUITY RETURNS

This paper tests the zero-beta CAPM with Australian equity returns, using the multivariate approach developed by Gibbons (1982). For the period 1958 to 1987, based on its asymptotic distribution, the likelihood ratio test (LRT) statistic indicates a strong rejection of the model when an equally weighted market index is used. However, small sample adjustments to the test suggested by Jobson and Korkie (1982) and by Shanken (1985) place the validity of this conclusion in some doubt. When a value weighted market index is used for the period 1974 to 1987, the tests reveal at least moderate support for the zero-beta CAPM.     

Portfolio choice and equilibrium in capital markets with safety-first investors

This paper develops optimal portfolio choice and market equilibrium when investors behave according to a generalized lexicographic safety-first rule. We show that the mutual fund separation property holds for the optimal portfolio choice of a risk-averse safety-first investor. We also derive an explicit valuation formula for the equilibrium value of assets. The valuation formula reduces to the well-known two-parameter capital asset pricing model (CAPM) when investors approximate the tail of the portfolio distribution using Tchebychev's inequality or when the assets have normal or stable Paretian distributions. This shows the robustness of the CAPM to safety-first investors under traditional distributional assumptions. In addition, we indicate how additional information about the portfolio distribution can be incorporated to the safety-first valuation formula to obtain alternative empirically testable models.     

CAPM and Option Pricing With Elliptically Contoured Distributions

This article offers an alternative proof of the capital asset pricing model (CAPM) when asset returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption for modeling asset returns. The class of elliptically contoured distributions, which includes the more familiar Normal distribution, provides flexibility in modeling the thickness of tails associated with the possibility that asset returns take extreme values with nonnegligible probabilities. As summarized in this article, this class preserves several properties of the Normal distribution. Within this framework, we prove a new version of Stein's lemma for this class of distributions and use this result to derive the CAPM when returns are elliptical. Furthermore, using the probability distortion function approach based on the dual utility theory of choice under uncertainty, we also derive an explicit form solution to call option prices when the underlying is log‐elliptically distributed. The Black–Scholes call option price is a special case of this general result when the underlying is log‐normally distributed.     

Multifactor Asset Pricing Models

Since the early 1960s, the mean-variance Capital Asset Pricing Model (CAPM) has been a dominant paradigm in modern finance. Recently, the accumulation of anomalous evidence, and a realisation that empirical tests of the model are tautologically related to the efficiency of the market index, have pushed that paradigm to a point of crisis. This paper reviews alternative asset pricing models which coexisted with the CAPM and may provide plausible substitutes. The major distinguishing feature of these models is that they predict multiple risk factors and, with the exception of the Arbitrage Pricing Theory (APT), are extensions of the CAPM.     

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.     

Accounting‐based Risk Management and the Capital Asset Pricing Model: An Empirical Comparison

Accounting‐based risk management (ABRM) is a theoretically consistent and practical tool for calculating the cost of capital from underlying financial ratios. In this paper, a sample of ABRM‐generated discount factors is used to generate risk‐adjusted returns, which are compared to CAPM equivalent discount factors. In view of the debates about CAPM's validity, alternative models, the nature and scale of the equity risk premium, and the importance of discount rates in capital budgeting and asset valuation, ABRM's characteristics and resulting discount rates offer a potentially useful alternative. Results suggest that although average discount rates are comparable, their cross‐sectional distributions are dissimilar, so that investors in an average risky firm are overcompensated for systematic risk when using CAPM discount rates, because CAPM discount factors overestimate risk arising from fixed costs in most firms.     

On the exclusion of assets from tests of the two-parameter model: A sensitivity analysis

This study investigates the sensitivity of tests of the CAPM to different sets of asset returns. Tests are conducted with market portfolios that include returns for bonds, real estate, and consumer durables in addition to common stocks. Even when stocks represent only 10% of the portfolio's value, inferences about the CAPM are virtually identical to those obtained with a stocks-only portfolio. In contrast, inferences are sensitive to the set of assets used in the tests.     

The role of co-skewness in the pricing of real estate

The current study investigates whether systematic skewness offers an alternative perspective as to why the risk-adjusted returns on real estate should be similar to that for stocks. This is not a trivial issue since an affirmative finding implies that we might be incorrectly measuring real estate risk from both a pricing and a portfolio allocation perspective. A multivariate test of the Kraus-Litzenberger model is used to investigate this skewness proposition with the K-L CAPM tested against several alternative versions of the CAPM. The study finds that the Kraus-Litzenberger model offers additional insights into the measurement of real estate risk. Evidence is also found that both the zero beta and the consumption-oriented CAPM hold, which is consistent with the recent literature in real estate.     

A MULTIVARIATE TEST OF THE GENERALISED MEAN-LOWER PARTIAL MOMENT ASSET PRICING MODEL: AUSTRALIAN EVIDENCE

This paper tests the Mean-Lower Partial Moment (MLPM) model of asset pricing, using the Gibbons (1982) multivariate methodology, as developed by Hariow and Rao (1989). The MLPM model specifies risk to be a measure of the downside deviations of return relative to a prespecified and exogenous target rate of return. The MLPM model of Hariow and Rao is a new model which has only been tested once, using US data. Therefore, further tests using independent data will help to assess whether the model deserves more serious consideration as a possible alternative to the Capital Asset Pricing Model (CAPM). In general, tests in this paper using Australian data confirm the results of Hariow and Rao (1989). The MLPM model cannot be rejected against an unspecified alternative, nor can it be rejected against the zero-beta CAPM. Conditional on the MLPM model's validity, the optimal target rate appears to be more closely related to mean market returns than either to a risk-free return or to a zero return. In addition, there is some evidence to support an intertemporally constant target rate of around 3 percent per month, over the 30 year period examined.     

Sensitivity of Multivariate Tests of the Capital Asset-Pricing Model to the Return Measurement Interval

The capital asset-pricing model's (CAPM) primary empirical implication is a positively sloped linear relation between a security's expected rate of return and its relative risk (beta). Recent research indicates that inferences about the risk-return relation are sensitive to the choice of the return measurement interval. We perform multivariate tests of the Sharpe-Lintner CAPM using monthly and annual returns on market-value-ranked portfolios. The CAPM is rejected using monthly returns, a result consistent with previous research. In contrast, we fail to reject the CAPM when annual holding period returns are used.     

The cost of capital in internationally integrated markets: The case of Nestlé

This paper argues that the cost of capital for firms in small countries should be estimated using the global CAPM rather than a local CAPM. Two related formulas showing the mistake made when using a local CAPM rather than a global CAPM are presented. the global CAPM is implemented for the case of Nestlé and the results are compared to the cost of capital estimate one obtains for Nestlé using a local CAPM when the global CAPM is appropriate.     

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.     

Yes,The APT Is Testable

The Arbitrage Pricing Theory (APT) has been proposed as an alternative to the mean-variance Capital Asset Pricing Model (CAPM). This paper considers the testability of the APT and points out the irrelevance for testing of the approximation error. We refute Shanken's objections, including his assertion that Roll's critique of the CAPM is applicable to the APT. We also explain the testability of the APT on subsets, and we explore the relationship between the APT and the CAPM.     

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.     

The Conditional Relationship Between Beta and Returns: A Reassessment

Abstract:  Several recent empirical tests of the Capital Asset Pricing Model have been based on the conditional relationship between betas and market returns. This paper shows that this method needs reconsideration. An adjusted version of this test is presented. It is then demonstrated that the adjusted technique has similar, or lower, power to the more easily implemented CAPM test of Fama and MacBeth (1973) if returns are normally distributed.