Specification tests of asset pricing models using excess returns

In this paper, we discuss the impact of different formulations of asset pricing models on the outcome of specification tests that are performed using excess returns. We point out that the popular way of specifying the stochastic discount factor (SDF) as a linear function of the factors is problematic because (1) the specification test statistic is not invariant to an affine transformation of the factors, and (2) the SDFs of competing models can have very different means. In contrast, an alternative specification that defines the SDF as a linear function of the de-meaned factors is free from these two problems and is more appropriate for model comparison. In addition, we suggest that a modification of the traditional Hansen–Jagannathan distance (HJ-distance) is needed when we use the de-meaned factors. The modified HJ-distance uses the inverse of the covariance matrix (instead of the second moment matrix) of excess returns as the weighting matrix to aggregate pricing errors. Asymptotic distributions of the modified HJ-distance and of the traditional HJ-distance based on the de-meaned SDF under correctly specified and misspecified models are provided. Finally, we propose a simple methodology for computing the standard errors of the estimated SDF parameters that are robust to model misspecification. We show that failure to take model misspecification into account is likely to understate the standard errors of the estimates of the SDF parameters and lead us to erroneously conclude that certain factors are priced.     

Historical simulation approach to the estimation of stochastic discount factor models

We propose an approach to the estimation of the parameters of stochastic discount factor (SDF) models which is based on the idea that the next period joint distribution of the variables in a SDF and asset returns can be well approximated by their joint historical distribution. The estimates of the SDF parameters may therefore be found as the values of the parameters at which the mean of the historical distribution of the product of the SDF with an asset return equals one. Each time period, the estimates are updated using the most recent periods of data and hence can change over time. This method can be viewed as an alternative to the approaches that specify a particular functional form relating the SDF parameters to proxies for the state of the world.     

Idiosyncratic Consumption Risk and the Cross Section of Asset Returns

This paper investigates the importance of idiosyncratic consumption risk for the cross‐sectional variation in asset returns. We find that besides the rate of aggregate consumption growth, the cross‐sectional variance of consumption growth is also a priced factor. This suggests that consumers are not fully insured against idiosyncratic consumption risk, and that asset returns reflect their attempts to reduce their exposure to this risk. The resulting two‐factor consumption‐based asset pricing model significantly outperforms the CAPM, and its performance compares favorably with that of the Fama–French three‐factor model.     

Diagnosing Asset Pricing Models Using the Distribution of Asset Returns

This paper develops a set of diagnostic tests which can shed light on why a particular model is failing and indicate what steps might be taken to make the model consistent with asset returns. Theoretical bounds on the moments of a stochastic discount factor are derived as a function of the moments of observed asset returns. Particular attention is paid to restrictions on moments other than the variance. These bounds can also be used to measure the information about the distribution of the discount factor contained in the moments of various asset returns. As an application of this methodology, bounds on the discount factor are estimated using size-based portfolios, and the results are used to analyze the small firm effect. Empirical results indicate, for the period 1926–1975, that moments of the returns of small firms contain information about the discount factor that is not contained in the moments of the returns of large firms and/or a proxy of the aggregate wealth portfolio. However, this difference disappears when more recent data is included.     

Common asset pricing factors in volatilities and returns in futures markets

Factor-based asset pricing models have been used to explain the common predictable variation in excess asset returns. This paper combines means with volatilities of returns in several futures markets to explain their common predictable variation. Using a latent variables methodology, tests do not reject a single factor model with a common time-varying factor loading. The single common factor accounts for up to 53% of the predictable variation in the volatilities and up to 14% of the predictable variation in the means. S&P500 futures volatility predicted by the factor model is highly correlated with volatility implied in S&P500 futures options. But both the factor and implied volatilities are significant in predicting future volatility. In derivatives pricing, both implied volatility from options and factors extracted from asset pricing models should be employed.     

INTANGIBLE ASSETS,BOOK‐TO‐MARKET,AND COMMON STOCK RETURNS

I examine two anomalies where the Fama and French three‐factor model fails to adequately explain monthly industry and index returns. Both anomalies are consistent with a bad model problem where the book‐to‐market factor introduces a negative bias in the intercepts. I propose the intangibles model as an alternative where the three‐factor model is known to have difficulty. This alternative model, which replaces the book‐to‐market factor with zero investment portfolio returns based on prior investments in intangible assets, is well specified in random samples, has comparable power, and fully explains both anomalies.     

Exchange Rates and the Conversion of Currency‐Specific Risk Premia

How do the risk factors that drive asset prices influence exchange rates? Are the parameters of asset price processes relevant for specifying exchange rate processes? Most international asset pricing models focus on the analysis of asset returns given exchange rate processes. Little work has been done on the analysis of exchange rates dependent on asset returns. This paper uses an international stochastic discount factor (SDF) framework to analyse the interplay between asset prices and exchange rates. So far, this approach has only been implemented in international term structure models. We find that exchange rates serve to convert currency‐specific discount factors and currency‐specific prices of risk – a result linked to the international arbitrage pricing theory (IAPT). Our empirical investigation of exchange rates and stock markets of four countries presents evidence for the conversion of currency‐specific risk premia by exchange rates.     

Asset Pricing at the Millennium

This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the trade-off between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor (SDF) that prices all assets in the economy. The behavior of the term structure of real interest rates restricts the conditional mean of the SDF, whereas patterns of risk premia restrict its conditional volatility and factor structure. Stylized facts about interest rates, aggregate stock prices, and cross-sectional patterns in stock returns have stimulated new research on optimal portfolio choice, intertemporal equilibrium models, and behavioral finance.     

Communication,Excess Comovement and Factor Structures

This paper develops a model in which investors communicate before trading in a general equilibrium. Investors repeatedly communicate in a social network but have limited knowledge of the network structure and thus do not fully realize the consequences of their communication and belief updating. As a result, asset returns contain excess comovement and more concentrated factor structures than fundamental values do. The model generates testable empirical predictions that are consistent with the empirical literature on excess comovement in asset returns.     

Market-wide illiquidity and the distribution of non-parametric stochastic discount factors

Employing out-of-sample non-parametric estimation techniques, we show that market-wide liquidity risk matters for asset pricing independently of the specific functional form of the stochastic discount factor (SDF) and, therefore, of the asset pricing model specification. Market-wide illiquidity significantly affects the distribution of the SDF. Specifically, it boosts up the volatility of the SDF, causing minor effects on higher moments of its distribution. This finding is robust to the use of different sets of test assets in the estimation of the SDF, including equity and corporate bond portfolios, and the use of a high-dimensional data estimation procedure.     

Are the Latent Variables in Time-Varying Expected Returns Compensation for Consumption Risk?

Multibeta asset pricing models are examined using proxies for economic state variables in a framework which exploits time-varying expected returns to estimate conditional betas. Examples include multiple consumption-beta models and models where asset returns proxy for the state variables. When the state variables are not specified, the tests indicate two or three time-varying expected risk premiums in the sample of quarterly asset returns. Conditional betas relative to consumption generate less striking evidence against the model than betas relative to asset returns, but both the consumption and the market variables fail to proxy for the state variables.     

Asset pricing with Second-Order Esscher Transforms

The purpose of the paper is to introduce, in a discrete-time no-arbitrage pricing context, a bridge between the historical and the risk-neutral state vector dynamics which is wider than the one implied by a classical exponential-affine stochastic discount factor (SDF) and to preserve, at the same time, the tractability and flexibility of the associated asset pricing model. This goal is achieved by introducing the notion of exponential-quadratic SDF or, equivalently, the notion of Second-Order Esscher Transform. The log-pricing kernel is specified as a quadratic function of the factor and the associated sources of risk are priced by means of possibly non-linear stochastic first-order and second-order risk-correction coefficients. Focusing on security market models, this approach is developed in the multivariate conditionally Gaussian framework and its usefulness is testified by the specification and calibration of what we name the Second-Order GARCH Option Pricing Model. The associated European Call option pricing formula generates a rich family of implied volatility smiles and skews able to match the typically observed ones.     

Conditional performance evaluation for German equity mutual funds

We investigate the conditional performance of a sample of German equity mutual funds over the period from 1994 to 2003 using both the beta-pricing approach and the stochastic discount factor (SDF) framework. On average, mutual funds cannot generate excess returns relative to their benchmark that are large enough to cover their total expenses. Compared to unconditional alphas, fund performance sharply deteriorates when we measure conditional alphas. Given that stock returns are to some extent predictable based on publicly available information, conditional performance evaluation raises the benchmark for active fund managers because it gives them no credit for exploiting readily available information. Underperformance is more pronounced in the SDF framework than in beta-pricing models. The fund performance measures derived from alternative model specifications differ depending on the number of primitive assets taken to calibrate the SDF as well as the number of instrument variables used to scale assets and/or factors.     

Explaining abnormal returns in stock markets: An alpha-neutral version of the CAPM

This paper develops a behavioural asset pricing model in which traders are not fully rational as is commonly assumed in the literature. The model derived is underpinned by the notion that agents’ preferences are affected by their degree of optimism or pessimism regarding future market states. It is characterized by a representation consistent with the Capital Asset Pricing Model, augmented by a behavioural bias that yields a simple and intuitive economic explanation of the abnormal returns typically left unexplained by benchmark models. The results we provide show how the factor introduced is able to absorb the “abnormal” returns that are not captured by the traditional CAPM, thereby reducing the pricing errors in the asset pricing model to statistical insignificance.     

Evaluating stochastic discount factors from term structure models

This paper examines the feasibility of applying the stochastic discount factor methodology to fixed-income data using modern term structure models. Using this approach the researcher can examine returns on bond portfolios whose exact composition is unknown, as is often the case. This paper proposes an observable proxy for the SDF from continuous-time models and documents via Monte Carlo methods the properties of the GMM estimator based on using this proxy.     

Option Coskewness and Capital Asset Pricing

This article shows how the market coskewness model of Rubinstein(1973) and Kraus and Litzenberger (1976) is altered when a nonredundantcall option is optimally traded. Owing to the option’snonredundancy, the economy’s stochastic discount factor(SDF) depends not only on the market return and the square ofthe market return but also on the option return, the squareof the option return, and the product of the market and optionreturns. This leads to an asset pricing model in which the expectedreturn on any risky asset depends explicitly on the asset’scoskewness with option returns. The empirical results show thatthe option coskewness model outperforms several competing benchmarkmodels. Furthermore, option coskewness captures some of thesame risks as the Fama–French factors small minus big(SMB) and high minus low (HML). These results suggest that thefactors that drive the pricing of nonredundant options are alsoimportant for pricing risky equities.(JEL G11, G12, D61)     

Constructing Asset Pricing Models With Specific Factor Loadings

We demonstrate how one can build pricing formulae in which factors other than beta may be viewed as determinants of asset returns. This is important conceptually as it demonstrates how the additional factors can compensate for a market portfolio proxy that is mis‐specified, and also shows how such a pricing model can be specified ex ante. The procedure is implemented by first selecting an ‘orthogonal’ portfolio which falls on the mean‐variance efficient frontier computed from the empirical average returns, variances and covariances on the equity securities of a large sample of firms. One then determines the inefficient index portfolio which leads to a vector of betas that when multiplied by the average return on the orthogonal portfolio, and which when subtracted from the vector of average returns for the firms comprising the sample, yields an error vector that is equal to the vector of numerical values for the variables that are to form the basis of the asset pricing formula. There will then be a perfect linear relationship between the vector of average returns for the firms comprising the sample, the vector of betas based on the inefficient index portfolio and such other factors that are deemed to be important in the asset pricing process. We illustrate computational procedures using a numerical example based on the quality of information contained in published corporate financial statements.     

Tax-Induced Intertemporal Restrictions on Security Returns

This article derives testable restrictions on equilibrium asset prices when investors have the option to time the realization of their capital gains and losses for tax purposes. The tax-timing option alters both the magnitude and timing of equity returns relative to those in a tax-free model. The tax-induced restrictions are empirically examined, and the tax rates and preference parameters are estimated. While the tax-free model can be rejected in favor of the tax-based model as the specified alternative, the tax-based model is still unable to adequately explain cross-sectional differences in asset returns.     

Findings from a Cross-Sectional Housing Risk-Factor Model

Housing data from the last 25 years show that returns to residential real estate in the U.S. can be volatile and vary significantly among locations. The variations in returns are driven by economically as well as geographically and psychologically motivated factors, but so far, no asset pricing model that adequately explains systematic risks in cross-sectional housing returns is widely accepted. This paper proposes an asset pricing model for housing returns that includes a market-wide return factor, an economically motivated factor derived from income growth, a geographically based factor derived from land supply elasticity and a momentum factor, which is psychological in nature. The model explains well the systematic risks in housing returns and is robust to different portfolio segmentations. Moreover, the model illustrates that local risk factors indirectly capture the risk previously attributed to market-wide price changes. While housing is not actively traded when compared to other financial assets, understanding the risk-factors that explain housing return in cross-section provides important insight for real estate investors, builders, real estate future traders, homeowners, banks and other mortgage lenders.     

The Cross Section of Expected Real Estate Returns: Insights from Investment-Based Asset Pricing

Motivated by investment-based asset pricing, we show that two firm fundamentals, investment and profitability, have substantial predictive power for REIT returns. The return predictability of investment and profitability is not subsumed by conventional models and can be useful for understanding the cross section of expected REIT returns. To illustrate, we construct an investment-based factor model for REITs that consists of a market factor, an investment factor, and a profitability factor. The investment-based model outperforms conventional models in capturing well-known cross-sectional patterns in REIT returns. Our findings suggest that incorporating investment-based asset pricing can be a promising direction for future real estate finance research.