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.     

The Time-Series Behavior of IPO Betas

We examine individual IPO betas and provide further evidence that the documented decline in IPO betas results primarily from a seasoning or information effect and not from the delisting of high beta securities. We employ stochastic coefficient regression analysis which permits the estimation of individual IPO betas at all points in time, and therefore avoids disadvantages associated with grouped cross-sectional beta estimates and average individual time-series beta estimates. We find that IPO firms with the lowest betas are more likely to delist, and that individual IPO betas, on average, decline over time which provides support for the information hypothesis.     

Conditional beta pricing models: A nonparametric approach

We propose a two-stage procedure to estimate conditional beta pricing models that allows for flexibility in the dynamics of asset betas and market prices of risk (MPR). First, conditional betas are estimated nonparametrically for each asset and period using the time-series of previous data. Then, time-varying MPR are estimated from the cross-section of returns and betas. We prove the consistency and asymptotic normality of the estimators. We also perform Monte Carlo simulations for the conditional version of the three-factor model of Fama and French (1993) and show that nonparametrically estimated betas outperform rolling betas under different specifications of beta dynamics. Using return data on the 25 size and book-to-market sorted portfolios, we find that the nonparametric procedure produces a better fit of the three-factor model to the data, less biased estimates of MPR and lower pricing errors than the Fama–MacBeth procedure with betas estimated under several alternative parametric specifications.     

Regulator's Determination of Return on Equity in the Absence of Public Firms: The Case of Automobile Insurance in Ontario

In a regulated market, such as automobile insurance (AI), regulators set the return on equity that insurers are allowed to achieve. Most insurers are engaged in a variety of insurance lines of business, and thus the full information beta methodology (FIB) is commonly employed to estimate the AI beta. The FIB uses two steps: first, the beta of each insurer is estimated, and then the beta of each line of business is estimated, as the beta of an insurer is a weighted average of the betas of the lines of business. When there are a sufficient number of public companies, company and market returns are used. Otherwise, researchers have resorted to using accounting data in the FIB. Theoretically, the two steps are not separable and the estimation should be done with one step. We introduce the one‐step methodology in our article. The one‐step and two‐step methodologies are compared empirically for the Ontario market of AI. Insurers in Ontario are predominantly private companies; thus, accounting data are used to estimate the AI beta. We show that a significant bias is introduced by the traditional, two‐step FIB methodology in estimating the betas for different lines of business, while insurers’ betas are very similar under both methods. This has a significant application to the estimation of betas of “pure players” in classic corporate finance. It implies that their betas and hence the resulting, required rates of return used in the net present value calculations should be estimated based on the one‐step method that we develop in this article.     

Bias in estimating the systematic risk of extreme performers: Implications for financial analysis, the leverage effect, and long-run reversals

We show how bias can arise systematically in the beta estimates of extreme performers when long-run return reversals are present and partly, or wholly, due to sign changes in unanticipated factor realizations. Our evidence is consistent with this bias being responsible for the large shifts in the beta estimates of extreme performers, more so than the leverage effect, which has been the predominant explanation in prior literature. Bias in these contemporaneous realized betas, estimated with the same returns that are to be risk adjusted, arises due to the general problem of “overconditioning,” where betas are estimated conditional on information that is not yet known. Several methods for conditioning betas on out-of-sample returns are evaluated and found to be lacking, although some offer improvement under certain circumstances. We also show evidence of this bias in the Fama-French Three-factor loadings of extreme performers. Our findings indicate not only that previous studies of long-run reversals understate contrarian profits but that bias is prevalent in the OLS beta estimates of extreme performers, and this has implications for estimating the cost of capital and measuring long-run performance. We offer recommendations for identifying when this bias is likely present, as well as general methods to correct for it.     

Sampling interval and estimated betas: Implications for the presence of transitory components in stock prices

We provide a theoretical framework to explain the empirical finding that the estimated betas are sensitive to the sampling interval even when using continuously compounded returns. We suppose that stock prices have both permanent and transitory components. The discrete time representation of the beta depends on the sampling interval and two components labeled “permanent and transitory betas”. We show that if no transitory component is present in stock prices then no sampling interval effect occurs. However, the presence of a transitory component implies that the beta is an increasing (decreasing) function of the sampling interval for more (less) risky assets. In our framework, assets are labeled risky if their “permanent beta” is greater than their “transitory beta” and vice versa for less risky assets. Simulations show that our theoretical results provide good approximations for the estimated betas in small samples. We provide empirical evidence about the presence of negative serial correlation and mean reversion in the returns of the portfolios considered. We discuss why our model is better able to provide an explanation for this sampling interval effect than other models in the literature.     

Cross-sectional analysis of Swedish stock returns with time-varying beta: the Swedish stock market 1983–96

This paper analyses the ability of beta and other factors, like firm size and book-to-market, to explain cross‐sectional variation in average stock returns on the Swedish stock market for the period 1983–96. We use a bivariate GARCH(1,1) process to estimate time-varying betas for asset returns. The estimated variances of these betas, derived from a Taylor series approximation, are used for correcting errors in variables. An extreme bound analysis is utilized for testing the sensitivity of the estimated coefficients to changes in the set of included explanatory variables.
Our results show that the estimated conditional beta is a more accurate measure of the true market beta than the beta estimated by OLS. The coefficient for beta is not significantly different from zero, while the variables book-to-market and leverage have significant coefficients, and the latter coefficients are also robust to model specification. Excluding the down turn 1990–92 from the sample shows that the significance of the risk premium for leverage might be considered as an industry effect during this extreme period. Finally, we find a close dependence between the risk premium for beta and that for size and book-to-market. The omission of each of these variables may cause statistical bias in the estimated coefficient for beta.     

The extent of nonstationarity of beta

This paper investigates the extent of nonstationarity of beta across the firm size and the beta magnitude by suggesting the sequential parameter stationarity model and estimating change-points of betas. The high-beta firm has shorter stationary interval, which means that its beta changes more frequently than do the low-beta firm's. The firm size, however, does not have a monotonic relation with the length of stationary interval. The small and large firms have relatively shorter stationary interval than do the mid-sized firms. The average length of stationary interval is estimated about five years (exactly 54.19 months). This fact could support the currently widely-used arbitrary 5-year assumption of beta stationarity. The fluctuation of the large firm's beta is more severe than the small firm's, and the high- and low-beta firms have the relatively greater fluctuating betas than do the mid-beta firms. The frequency of detected change-points is found to be positively related to market returns. When the market return is high, the systematic risk changes more frequently, and vice versa.     

The size effect in the Mexican stock market

This paper tests for a firm size effect in the Mexican stock market using data from January 1987 to December 1992. Our initial tests indicate that average stock returns are positively related to market betas. We also find, however, that average returns are negatively related to firm size. To measure the effects on average return of betas that are unrelated to firm size, we examine portfolios formed on the basis of size and beta We find that beta is priced in addition to firm size for the Mexican stock market, even after carefully separating the effects of beta and size.     

Forecasting Beta: How Well Does the 'Five-Year Rule of Thumb' Do?

CAPM betas are generally estimated from historical data and applied to a future period. There is widespread evidence that the CAPM betas vary considerably over time and this raises two questions: can this variation be explained and can it be forecast better than the 'five-year rule of thumb' (i.e using the most recently estimated beta)? We estimate time-varying betas and explain the time-variation in the betas using regression models which we subsequently use for forecasting. We find that forecasting equations have good explanatory power but that their forecasts are dominated, on average, by the five-year rule of thumb.     

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.     

EARNINGS NEWS AND MARKET RISK: IS THE MAGNITUDE OF THE POSTEARNINGS ANNOUNCEMENT DRIFT UNDERESTIMATED?

Postearnings announcement drift is the tendency for cumulative abnormal returns to drift in the direction of earnings surprise after the earnings news is released. I show that a standard approach to measuring abnormal returns by using preannouncement estimates of market risk (betas) causes the magnitude of this phenomenon to be significantly underestimated. I find that stock beta tends to rise (fall) following the release of bad (good) earnings news. In addition, I find that by not taking into account postannouncement shifts in betas, prior studies are likely to have underestimated the magnitude of the drift. My results are robust to different model specifications, as well as to different earnings surprise measures.     

THE RELATIONSHIP BETWEEN MARKET AND ACCOUNTING BETAS FOR COMMERCIAL BANKS

This study examines the correlation between market equity betas and accounting asset and equity betas in the commercial banking sector. A sample of banks was taken from the COMPUSTAT tapes and annual equity and asset accounting betas (calculated with various indices) were estimated over varying time periods. Cross-sectional correlations were then determined for individual banks and five-bank portfolios. The results indicate that the correlations are comparable to those found in other non-banking studies of accounting and market betas. A noticeable difference in our study was the sensitivity of the correlations to the length of the estimation interval for the betas. The longest estimation intervals (16–18 years) produce few significant correlations in our sample. These correlations were significantly lower than those obtained using fifteen or fewer years of data. Construction of portfolios generally increases all the correlations but the longer time period correlations are still significantly lower. The correlations are largely invariant, however, to the choice of index used in the estimation of the betas.     

U.S. Banking Sector Risk in an Era of Regulatory Change: A Bivariate GARCH Approach

This paper assesses the impact of regulatory change on the risk and returns of the U.S. banking industry. The impact of five major regulatory changes on banking sector risk was assessed using daily data for eighteen major U.S. regional banks, money center banks and savings and loan type depository institutions. Risk in this case was proxied via the use of an M-GARCH model which generates time dependent conditional beta estimates. The evidence obtained suggests that the impact of deregulation and reregulation on banking sector risk is case specific. Further, the results obtained show that the market model incorporating dummy variables, which has proven so popular amongst existing studies, discards important information about the variability of beta which the time varying conditional betas capture.     

CAPM over the long run: 1926–2001

A conditional one-factor model can account for the spread in the average returns of portfolios sorted by book-to-market ratios over the long run from 1926 to 2001. In contrast, earlier studies document strong evidence of a book-to-market effect using OLS regressions over post-1963 data. However, the betas of portfolios sorted by book-to-market ratios vary over time and in the presence of time-varying factor loadings, OLS inference produces inconsistent estimates of conditional alphas and betas. We show that under a conditional CAPM with time-varying betas, predictable market risk premia, and stochastic systematic volatility, there is little evidence that the conditional alpha for a book-to-market trading strategy is different from zero.     

Markov Chains and Regression toward the Mean

Using the theory of stationary Markov chains, we uncover a previously unknown property of the behavior of betas. Specifically, if the cross-sectional distribution of betas is stationary over time, then the set of firms that remain in an arbitrarily chosen beta interval between one period and the next will not regress toward the mean. This surprising result occurs in spite of the well-known fact that the set of all the firms in the interval will exhibit the regression tendency. Our empirical tests indicate that betas behave in remarkable accordance with this prediction.     

Resilience to the financial crisis in customer-supplier networks

Inspired by the Capital Asset Pricing Model (CAPM) beta, we construct customer and supplier betas to separately investigate potentially different properties of downstream and upstream linkages. With the adjacency matrix acting as a ‘filter’ to extract each company's return covariances with its trading partners, the cross-sectional dependence contained in the customer-supplier network is summarized by our betas. We explore how these two betas are related to a company's resilience to the financial crisis of 2008–2009. We observe that a higher customer beta is generally associated with more resilience during the crisis. Therefore, investors could construct the customer beta to gain insights into the relative negative impact of a potential crisis on a stock's performance.     

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.     

OLIVE: A SIMPLE METHOD FOR ESTIMATING BETAS WHEN FACTORS ARE MEASURED WITH ERROR

We propose a simple and intuitive method for estimating betas when factors are measured with error: ordinary least squares instrumental variable estimator (OLIVE). OLIVE performs well when the number of instruments becomes large, whereas the performance of conventional instrumental variable methods becomes poor or even infeasible. In an empirical application, OLIVE beta estimates improve R² significantly. More important, our results help resolve two puzzling findings in the prior literature: first, the sign of average risk premium on the beta for market return changes from negative to positive; second, the estimated value of average zero‐beta rate is no longer too high.     

Return attribution analysis of the UK insurance portfolios

We examine the attribution of premium growth rates for the five main insurance sectors of the United Kingdom for the period 1969–2005; in particular, Property, Motor, Pecuniary, Health and Accident, and Liability. In each sector, the growth rates of aggregate insurance premiums are viewed as portfolio returns which we attribute to a number of factors such as realized and expected losses and expenses, their uncertainty and market power, using the Sharpe (Determining the Fund’s Effective Asset Mix. Investment Management Review, November–December, pp. 59–69, 1988; J. Portfolio Manag. 18:7–19, 1992) Style Analysis. Our estimation method differs from the standard least squares practice which does not provide confidence intervals for style betas and adopts a Bayesian approach, resulting in a robust estimate of the entire empirical distribution of each beta coefficients for the full sample. We also perform a rolling analysis of robust estimation for a window of seven overlapping samples. Our empirical findings show that there are some main differences across industries as far as the weights attributed to the underlying factors. Rolling regressions assist us to identify the variability of these weights over time, but also across industries.