Estimating the rank of a beta matrix: a GMM approach1

A full-rank beta matrix is a necessary condition for correctly estimating the risk premia in linear asset pricing models. However, the true values of betas are unobserved in practice and must be estimated. In this paper, we propose a straightforward testing method based on the generalised method of moments to assess whether the beta matrix is of full rank. We show that our method has desirable finite sample properties and performs better than available alternatives. We apply our method to several popular factor models and find that most models have rank deficiency in several datasets.     

A conditional regime switching CAPM

The standard Capital Asset Pricing Model (CAPM) is simple, intuitive, and grounded in sound economic theory. Yet, almost half a century's worth of empirical testing has so far failed to demonstrate its relevance. One major reason given for the CAPM's empirical failure is that beta is not the sole measure of systematic risk. In other words, the standard CAPM does not hold. Another important explanation is that the CAPM may hold conditionally rather than unconditionally. The standard CAPM fails to explain the cross-section of returns because it ignores the fact that both the risk and the price of risk are time-varying. The search for conditional models has led researchers to either disregard the theory behind the CAPM or to use statistical procedures that are too complex to be replicated by other researchers and practitioners. In this paper we propose a conditional model that is compatible with the standard CAPM while remaining simple and accessible to both researchers and practitioners. Beta and the risk premium are assumed to be time-varying, with the latter being associated with bull and bear states. We find strong support for the conditional CAPM with beta explaining both bull and bear markets. While the bear market ex-post risk premium is negative, the weighted average risk premium is positive and highly significant.     

Does tax convexity matter for risk? A dynamic study of tax asymmetry and equity beta

The purpose of this study is to explore the effect of tax convexity on firms’ market risk, where tax convexity measures the progressivity of firms’ tax function. We examine the relation between equity beta and tax convexity based on a standard contingent-claims model, in which firms face nonlinear tax schedules. We verify that in the presence of default and growth options, the effect of tax convexity on beta is significant and depends on several countervailing forces. Tax convexity has a direct, positive effect on beta, as well as two indirect countereffects through default and growth options. The overall effect is ambiguous and quantitatively significant. As asymmetric tax schedules are used in most countries, assuming a linear tax schedule in the valuation framework may misestimate beta and thus fail to assess risk accurately. Our theoretical model shows that tax convexity should be taken into consideration when estimating equity beta.     

Payment For Risk: Constant Beta Vs. Dual-Beta Models

Fama and French's (1992) assertion that investors receive premium payments for risk associated with the book value to market price (BE/ME) and size and not for holding beta risk has sparked a lively debate concerning risk factors that are priced in the market. Howton and Peterson (1998) use a dual-beta model to test the Fama and French conclusions. They conclude that the significant relationship between beta and returns depends on the use of the dual-beta model. This work, however, ignores the results reported by Pettengill, Sundaram, and Mathur (PSM, 1995). PSM find a significant relation between a constant risk beta and returns when data are segmented between up and down markets, but do not consider the impact of size and BE/ME. In this paper we show that the PSM (1995) market segmentation procedure alone provides a sufficient condition to identify a significant relation between beta and returns in the presence of size and BE/ME. Dual market betas may be relevant in explaining risk and return. However, the market segmentation procedure of PSM (1995) is the critical condition for finding a significant relationship between returns and betas.     

Unconditional Tests of Linear Asset Pricing Models with Time‐Varying Betas

In conditional affine factor models, estimated risk prices should satisfy certain unconditional constraints. Specifically, a cross‐sectional estimate of the unconditional slope associated with a risk factor should equal the average price of risk of the factor. The estimated slope associated with the product of a risk factor and an instrument should be equal to the covariance of the factor risk premium with the instrument. We show that the constraints only apply to the conditional models with time‐varying betas. We identify an unconditional constraint on unconditional betas for time‐varying beta models and incorporate it into model tests. We show that imposing this unconditional constraint changes estimates of unconditional betas and risk prices significantly.     

The Determinants of REIT Institutional Ownership: Tests of the CAPM

This study examines the determinants of institutional investment demand for REIT common stock. We estimate the demand function for financial institutions using the mean return and CAPM risk measures (beta and standard error) for REIT stocks. The objective is to determine whether institutional investment decisions are influenced by CAPM model attributes. In addition, we examine the predicatability of REIT institutional ownership based on the factors in our model. We employ conventional OLS forecasting techniques, as well as two neural network models in order to deal with possible nonlinearities in the relationships.     

Asset Pricing with a Bank Risk Factor

This paper studies how the state of the banking sector influences stock returns of nonfinancial firms. We consider a two‐factor pricing model, where the first factor is the traditional market excess return and the second factor is the change in the average distance to default of commercial banks. We find that this bank factor is priced in the cross section of U.S. nonfinancial firms. Controlling for market beta, the expected excess return for a stock in the top quintile of bank risk exposure is on average 2.83% higher than for a stock in the bottom quintile.     

Determinants of market beta: the impacts of firm-specific accounting figures and market conditions

This article examines and extends research on the relation between the capital asset pricing model market beta, accounting risk measures and macroeconomic risk factors. We employ a beta decomposition approach that nests competing models with different business risk proxies and allows to frame cross-model comparison. Because model tests require estimated independent variables resulting in measurement error, we empirically estimate three comparable model specifications with instrumental variable estimators and for the first time provide thorough instrument diagnostics in this setting. Correcting for the heretofore neglected weak instruments problem we find that growth risk (i.e., the risk of firm sales variations that are inconsistent with the market wide trends), is the business risk that explains cross-sectional variations in market beta best.     

Can unpredictable risk exposure be priced?

We study the link between beta predictability and the price of risk. An investor who desires exposure to a certain risk factor needs to predict what next period’s beta will be. We use a simple model to show that an ambiguity averse agent’s demand is lower when betas are hard to predict, leading to a reduction in risk premiums. We test the implications for downside betas and VIX betas. We find that they have economically and statistically small prices of risk once we account for the fact that an investor cannot observe ex-post realized betas when determining asset demand.     

Does It Pay to Bet Against Beta? On the Conditional Performance of the Beta Anomaly

Prior studies find that a strategy that buys high‐beta stocks and sells low‐beta stocks has a significantly negative unconditional capital asset pricing model (CAPM) alpha, such that it appears to pay to “bet against beta.” We show, however, that the conditional beta for the high‐minus‐low beta portfolio covaries negatively with the equity premium and positively with market volatility. As a result, the unconditional alpha is a downward‐biased estimate of the true alpha. We model the conditional market risk for beta‐sorted portfolios using instrumental variables methods and find that the conditional CAPM resolves the beta anomaly.     

Systematic risk,total risk and size as determinants of stock market returns

This paper studies the historical relationship for the period 1962–1981 between stock market returns and the following variables: beta, residual standard deviation (or total variance), and size. We conclude that neither the traditional measure of risk (beta) nor the alternative risk measures (variance or residual standard deviation) can explain the cross-sectional variation in returns; only size seems to matter. When January returns are eliminated, even the size variable loses its statistical significance.     

Higher moments and beta asymmetry: evidence from Australia

We examine whether systematic higher moments capture beta asymmetry in an asset pricing model whereby the conditional beta of a risky asset increases (decreases) during a bear (bull) market state. We first provide a simple conceptual outline from the microeconomic literature to show that beta asymmetry is driven by time‐varying higher‐order risk preferences (prudence and temperance) across different market states. We then empirically relate these higher‐order risk preferences to systematic skewness and systematic kurtosis. We find that beta asymmetry in Australian stock returns cannot be explained by Carhart (1997) 4‐factor model but is subsumed by systematic higher moments.     

Equity duration and convexity when firms can fail or stagnate

Previous analysis of equity duration and convexity has either ignored the risk that firms can fail or stop growing, or else has incorporated the risk of failure in an ad hoc way. This paper derives equity duration and convexity rigorously corrected for the risk of failure or stagnation. We show that the correction is large enough to be important in standard applications. Further, equity duration is unaffected by any liquidation payout at the time of failure.     

The estimation and determinants of emerging market country risk and the dynamic conditional correlation GARCH model

Country risk assessment is central to the international investment, which recently has increasingly focused on emerging markets (EM). In this paper we proxy for country risk in EM by using time-varying beta. We extend existing literature by applying a dynamic conditional correlation GARCH model. After confirming beta is time varying in twenty EM over the period January 1995 to December 2008 we investigate the GARCH (1,1) model and find the t-distribution generates the lowest forecast errors compared to the normal error distribution and a generalised error distribution. In a comparison of previous modelling techniques the results of our modified Diebold-Mariano test statistics suggest that the Kalman Filter model outperforms the GARCH model and the Schwert and Seguin (1990) model. Using a DCC-GARCH model our evidence suggests that considering dynamic betas can improve beta out-of-sample predicting ability and therefore offers potential gains for investors. Finally, we find dynamic betas across EM are strongly associated with each nation's interest rates, US interest rates and the Consumer Price Index (CPI) and to a lesser extent the exchange rates. Our results have some similarities to those in previous studies of developed markets in the economic determinants of time-varying beta but differences exist in the results on best model to forecast time-varying beta. These findings have implications for estimating country risk for investment and risk management purposes in EM.     

Nontraditional banking activities and bank failures during the financial crisis

We test whether income from nontraditional banking activities contributed to the failures of hundreds of U.S. commercial banks during the financial crisis. Estimates from a multi-period logit model indicate that the probability of distressed bank failure declined with pure fee-based nontraditional activities such as securities brokerage and insurance sales, but increased with asset-based nontraditional activities such as venture capital, investment banking and asset securitization. Banks that engaged in risky nontraditional activities also tended to take risk in their traditional lines of business, suggesting that deregulation was neither a necessary nor a sufficient condition for bank failure during the crisis.     

Pricing catastrophe risk in life (re)insurance

What is the catastrophe risk a life insurance company faces? What is the correct price of a catastrophe cover? During a review of the current standard model, due to Strickler, we found that this model has some serious shortcomings. We therefore present a new model for the pricing of catastrophe excess of loss cover (Cat XL). The new model for annual claim cost C is based on a compound Poisson process of catastrophe costs. To evaluate the distribution of the cost of each catastrophe, we use the Peaks Over Threshold model for the total number of lost lives in each catastrophe and the beta binomial model for the proportion of these corresponding to customers of the insurance company. To be able to estimate the parameters of the model, international and Swedish data were collected and compiled, listing accidents claiming at least twenty and four lives, respectively. Fitting the new model to data, we find the fit to be good. Finally we give the price of a Cat XL contract and perform a sensitivity analysis of how some of the parameters affect the expected value and standard deviation of the cost and thus the price.     

An Adjustment Procedure for Predicting Betas When Thin Trading is Present: Canadian Evidence

We have incorporated effects of the process that generates true betas for TSE stocks, as well as thin trading effects, into the beta adjustment model. We note the Blume and Dimson and Marsh beta adjustment techniques aim at eliminating beta forecast error through regression tendency bias. Effects of other sources of forecast error have been ignored. We show the process generating security betas affects both cross-sectional correlation coefficient and order bias, while thin trading affects only cross-sectional correlation coefficient. We demonstrate that when OLS beta estimates are used to forecast their future risk levels, order bias accounts for 86% of forecast error, while thin trading effects account for 14% of forecast error. A beta regression tendency model which properly accounts for effects of cross-sectional correlation (which is a function of thin trading) and order bias completely abates forecast error. Our results have implications for the use of correlation coefficient to measure stability of betas across time, for beta adjustment models proposed in the literature, and for event study methodologies that rely on prediction errors.     

In Search of Distress Risk

This paper explores the determinants of corporate failure and the pricing of financially distressed stocks whose failure probability, estimated from a dynamic logit model using accounting and market variables, is high. Since 1981, financially distressed stocks have delivered anomalously low returns. They have lower returns but much higher standard deviations, market betas, and loadings on value and small‐cap risk factors than stocks with low failure risk. These patterns are more pronounced for stocks with possible informational or arbitrage‐related frictions. They are inconsistent with the conjecture that the value and size effects are compensation for the risk of financial distress.     

Beta vs. characteristics: Comparison of risk model performances

We compare the beta model (a.k.a. covariance model) and the characteristics model in terms of their ability to reduce portfolio risk. When global-minimum-variance portfolios (GMVPs) are constructed out of the 500 largest US stocks for the 30-year period between 1981 and 2011, the beta-model-based GMVPs achieve lower volatility than the characteristics-model-based GMVPs. For robustness check, we consider alternative specifications of the characteristics model, and find that the advantage of the beta model persists. To make our analysis complete, we also consider some hybrid models. The performance of some hybrid models is comparable to that of the beta model.     

Beta and firm age

We document a robust pattern of beta declining over the age of a firm. We find that changes in systematic risk via firm characteristics and life-cycle stages are insufficient to explain this pattern. Moreover, standard proxies for the quantity and quality of information also explain this pattern only partially. To fully explain this pattern we rely on the increasingly important role of familiarity in financial decision making: familiarity is a determinant of beta and firm age is a proxy for the degree of familiarity that investors feel toward individual stocks. To illustrate the implication of our findings, we document that when we control for firm age there is support for the CAPM and its use as an input for the cost of equity capital calculation.