The sensitivity of beta to the time horizon when log prices follow an Ornstein–Uhlenbeck process

This paper provides a new theoretical approach to investigate the sensitivity of the familiar beta of the capital asset pricing model to the length of the return measurement interval; a phenomenon known as the intervalling effect. By setting the problem in a continuous time setting, and using exact results, we are able to generalize existing results in the literature. We derive an expression for beta as a function of the time horizon h, conditional on current time t. We show that beta is monotonic in h and derive conditions for it to be increasing or decreasing.     

The Role of Stock Size and Trading Intensity in the Magnitude of the "Interval Effect" in Beta Estimation: Empirical Evidence from the Polish Capital Market

In this paper, we present empirical evidence about the "interval effect" in estimation of beta parameters for stocks listed on the Warsaw Stock Exchange. We analyze models constructed for the returns calculated using intervals of different length—that is, 1, 5, 10, and 21 trading days (corresponding to, roughly, 1 day, 1 week, 2 weeks, and 1 month, respectively). In the cases in which heteroskedasticity was present, we estimated ARCH models. The results indicate that the estimates of betas for the same stock differ considerably when various return intervals are used. We further explore the source of differences in betas for every stock by investigating the relations between them and such factors as stock size and its trading intensity. The empirical results provide evidence that a statistically significant relationship exists between these two characteristics of stocks. This finding has important practical implications for beta estimation in practice.     

When Good News Is Not So Good: Economy‐wide Uncertainty and Stock Returns

This paper shows that variation in economy‐wide uncertainty causes asymmetric stock price responses to firm earnings surprises. The uncertainty that attends bad earnings news that arrives during expansions with greater economy‐wide uncertainty occasions larger price declines. This is because news inconsistent with investors’ prior beliefs about the state of the economy increases uncertainty, which amplifies the negative cash flow effects contained in bad earnings news. Asymmetrically, the positive cash flow effect of good earnings news that arrives during recessions is offset by increased investor uncertainty, which results in relatively smaller price reactions to the good news. This is consistent with Veronesi's rational expectations equilibrium model, which shows that investors demand higher expected returns in the face of greater uncertainty.     

THE DISTRIBUTION OF BETAS IN PRESENCE OF NONTRADED ASSETS

This study considers a capital assets pricing model (CAPM) in an incomplete financial market wherein not all risky assets are traded and the risk from non‐traded assets is not orthogonal to that of the existing or traded assets. The model shows the extent of the divergence of the CAPM betas (true betas) from the traditional CAPM betas (perceived betas) in market equilibrium conditions in an incomplete market. Specifically, it implies that the more incomplete a financial market is, the wider is the discrepancy between the true and perceived betas, and the distribution of the perceived betas tends to centre more around 1 in an incomplete market than that of true betas. Empirical evidence in various settings support these results.     

Stochastic portfolio theory and the low beta anomaly

Many studies have found that portfolios of low beta stocks have higher growth rates than portfolios of high beta stocks and have concluded that low beta stocks have higher growth rates than high beta stocks. Since rational investor behavior is thought to imply that additional risk is rewarded with additional return, the alleged higher growth rates of low beta versus high beta stocks has been termed a ‘Low Beta Anomaly’ (LBA). However, it is premature to conclude that these observed LBAs are due to stocks’ differential growth rates, because the tested portfolios are traded. Stochastic Portfolio Theory (SPT) shows that traded portfolios’ growth rates can exceed the growth rates of their stocks. This paper presents several SPT models of an LBA that do not require investment constraints, irrational investor behavior, or that low beta stocks have higher growth rates than high beta stocks. These LBAs are due to reconstitution relative volatility capture that favors portfolios of low vs. high beta stocks. They result from trading profit, not differential growth rates between low and high beta stocks. Monte Carlo simulations demonstrate a reconstitution relative volatility capture LBA that is consistent with the models and the literature.     

Market efficiency during the global financial crisis: Empirical evidence from European banks

This paper empirically investigates the asymmetric effect of news on the time-varying beta of selected banks from seven European countries during the current crisis period and also during the pre-crisis period. The paper applies daily data from thirteen large banks from France, Germany, Greece, Ireland, Italy, Portugal and Spain. The sample size ranges from 2002 to 2013 and includes the current global financial crisis (2007–2013). The BEKK GARCH model is first employed to estimate the time-varying beta and then linear regression is applied to investigate the asymmetric effect of news on the beta. The asymmetric effects are investigated based on both market and non-market shocks. Results show that some evidence of market efficiency can be witnessed via non-market shocks, however the market shocks indicate that the European banks foster a significant amount of uncertainty leading to asset mispricing. Results also show a clear rift in terms of quality of results between France and Germany taken as a group and the rest of the countries under study. These results shed light on the level of market efficiency and hedging strategies.     

NEGATIVE CORRELATION BETWEEN STOCK AND FUTURES RETURNS: AN UNEXPLOITED HEDGING OPPORTUNITY?

The negative correlation between equity and commodity futures returns is widely perceived by investors as an unexploited hedging opportunity. A Lucas (1982) asset‐pricing model is adapted to analyse the fundamentals driving equity and commodity futures returns. Using the model we argue that such a negative correlation could arise as an equilibrium relationship which reflects traders' perceptions about the shocks driving the fundamentals such as energy and consumables, and does not necessarily indicate any hedging opportunity.     

Regulation Effects on Company Beta Components*

This paper introduces a general model to analyse the effects of regulation on company risk. In particular, we consider two determinants of systematic risk: the company's overall risk and the correlation between the regulated company's value and the market. Theoretical findings indicate that, as regulation gets stricter, the company's abnormal returns will turn negative whereas the two systematic risk components will increase, and vice versa. We use event analysis elements and a time‐varying beta estimation to verify the regulation impact on risk and returns in the English electricity distribution industry. We find that systematic risk varies significantly during the period considered in our analysis. Furthermore, the analysis points to negative relationships between abnormal returns and both market correlation and overall risk variations.     

A saddlepoint approximation to the distribution of the sum of independent non‐identically beta random variables

The exact distribution of the sum of more than two independent beta random variables has not been known. Even in terms of approximations, only the normal approximation is known for the sum. Motivated by Murakami [Statistica Neerlandica, 2014, doi:10.1111/stan.12032], we derive here a saddlepoint approximation for the distribution of sum. An extensive simulation study shows that it always performs better than the normal approximation.     

Does the equity premium puzzle persist during financial crisis? The case of the French equity market

This paper examines the effects of the financial crisis that began in 2008 on the equity premium of 6 French sector indices. Since the systematic risk coefficient beta remains the most common explanatory element of risk premium in most asset pricing models, we investigate the impact of the crisis on the time-varying beta of the six sector indices cited. We selected daily data from January 2003 to December 2012 and we applied the bivariate MA-GARCH model (BEKK) to estimate time-varying betas for the sector indices. The crisis was marked by increased volatility of the sector indices and the market. This rise in volatility led to an increase in the systematic risk coefficient during the crisis and first post-crisis period for all the major indices. The results are intuitive and corroborate findings in the empirical literature. The increase of the time-varying beta is considered by investors as an additional risk. Therefore, as expected, investors tend to increase their equity premiums to b ear the impact of financial crisis.