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.     

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.     

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.     

Estimating Beta-Coefficients of German Stock Data: A Non-Parametric Approach

Abstract

Although the consumption based asset pricing theory appears to be theoretically superior and more elegant than the beta pricing model, in practice the beta pricing model is more widely applied. Indeed, beta pricing models are one of the most widely adopted tools in financial analysis. They readily allow handling systematic risk as priced in financial assets. However, accurately estimating beta-coefficients is not as straightforward as implicitly suggested by Sharpe's standard market model, i.e. simply using the ordinary least-squares (OLS) regression. This is primarily because beta-coefficients cannot generally be assumed to be stable over time. In order to overcome this deficiency, we present and apply a non-parametric estimation technique that allows capturing this time effect and promises both more reliable estimates than obtained with an OLS regression as well as better manageability compared with the existing time-series approaches dealing with time-varying beta-coefficients. Estimation results for constant and time-varying betas are presented for portfolios of German industries.     

SYSTEMATIC RISK IN A PURELY RANDOM MARKET MODEL: SOME EMPIRICAL EVIDENCE FOR INDIVIDUAL PUBLIC UTILITIES

A minimum norm quadratic (MINQU-) type of OLS estimator is derived. The estimator is used to test if the betas of the single factor market (SFM) model are random for a sample of utilities for two contiguous periods. The estimated betas for individual utilities vary considerably over time. The statistical significance of such nonstationarity depends on both the utilities and period studied. The relative reduction in the mean square error (MSE) from using a GLS (and not OLS) estimator of beta, when beta is purely random, can be substantial for some utilities but is modest on average.     

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.     

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.     

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.     

On the Cross-sectional Relation between Expected Returns and Betas

There is an exact linear relation between expected returns and true “betas” when the market portfolio is on the ex ante mean-variance efficient frontier, but empirical research has found little relation between sample mean returns and estimated betas. A possible explanation is that market portfolio proxies are mean-variance inefficient. We categorize proxies that produce particular relations between expected returns and true betas. For the special case of a zero relation, a market portfolio proxy must lie inside the efficient frontier, but it may be close to the frontier.     

Explaining the default risk anomaly by the two-beta model

This study attempts to explain the anomaly that firms with high-default risk earn low average realized returns. We measure default risk according to Ohlson's (1980) O-score and Campbell, Hilscher, and Szilagyi's (2008) failure probability and further implement Duffie, Saita, and Wang's (2007) doubly-stochastic intensity model to estimate default probabilities that incorporate the dynamics of the changes in covariates. We then employ Campbell and Vuolteenaho's (2004) two-beta model to estimate firms' cash-flow and discount-rate betas according to the default risk. The default risk anomaly persists when using Duffie el al.'s (2007) method. We show that cash-flow and discount-rate betas, respectively, earn a high and low premium and find that high-default firms tend to have relatively high discount-rate and low cash-flow betas. Hence, high-default firms deliver low expected returns. Importantly, 25.5% of the default risk anomaly can be explained by the two-beta model and that, on average, also accounts for 49.2% of the cross-sectional variation across the portfolios formed on default risk. This result implies that investors believe that high-default firms are unlikely to generate significantly extra cash flows when market-wide profitable opportunities improve.     

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.     

Estimating the Dynamics of Mutual Fund Alphas and Betas

This article develops a Kalman filter model to track dynamicmutual fund factor loadings. It then uses the estimates to analyzewhether managers with market-timing ability can be identifiedex ante. The primary findings are as follows: (i) Ordinary leastsquares (OLS) timing models produce false positives (nonzeroalphas) at too high a rate with either daily or monthly data.In contrast, the Kalman filter model produces them at approximatelythe correct rate with monthly data; (ii) In monthly data, thoughthe OLS models fail to detect any timing among fund managers,the Kalman filter does; (iii) The alpha and beta forecasts fromthe Kalman model are more accurate than those from the OLS timingmodels; (iv) The Kalman filter model tracks most fund alphasand betas better than OLS models that employ macroeconomic variablesin addition to fund returns.     

Assessing Project Risk

Finding the appropriate discount rate, or cost of capital, for evaluating investment projects requires an accurate estimate of project risk. This can be challenging because project risk cannot be estimated directly using the CAPM, but must instead be inferred from a set of traded securities, typically the equity betas of comparable firms in the same industry. These equity betas are then unlevered to undo the effect of comparable companies' financial leverage and obtain estimates of “asset” betas, which are then used to estimate project risk. The authors show that asset betas estimated in this way are likely to overestimate project risk. The equity returns of companies are risky not only because of their existing projects but also because of their growth opportunities. Such growth opportunities often include embedded “real options,” such as the option to delay, expand, or abandon a project. Because such real options are similar to leveraged positions in the underlying project, a company's growth opportunities are typically riskier than its existing projects. Therefore, to properly assess project risk, analysts must also unlever the asset betas derived from comparable company stock returns for the leverage contributed by their growth options. The authors derive a simple method for unlevering asset betas for growth options leverage in order to properly assess project risk. They then show that standard methods for assessing project risk significantly overestimate project costs of capital—by as much as 2–3% in industries such as healthcare, pharmaceuticals, communications, medical equipment, and entertainment. Their method should also be applied to stock return volatility to derive project volatility, an important input for determining the value of a firm's growth opportunities and the appropriate time for investing in these opportunities.     

Measuring performance in a dynamic world: Conditional mean–variance fundamentals

We develop conditional alpha performance measures that are consistent with conditional mean–variance analysis and the magnitude and sign of the implied true conditional time-varying alphas. The sequence of conditional alphas and betas is estimable from surprisingly simple unconditional regressions. Other common performance measures are derivable from the conditional investment opportunity set based on its conditional asset return moments. Our bootstrap analysis of Morningstar mutual fund returns data demonstrates that the differences between existing conditional alpha measures and our proposed alpha are substantive for typical parameterizations.     

Further evidence on the stationarity of betas and errors in their estimates

The objective of this paper is to study the stationarity of beta-coefficients of common stocks through time. Special attention is paid to the influence of errors in the estimated betas on various measures of stationarity. The results indicate that even for small porfolios the true betas (as opposed to estimated betas) are reasonably stable.     

The performance of cross-sectional regression tests of the CAPM with non-zero pricing errors

We show that in the presence of non-zero pricing errors, the Fama–MacBeth (FM) cross-sectional regression test is very likely to either reject the Capital Asset Pricing Model (CAPM) when it (almost) holds or accept the model when it grossly fails. We investigate the case when pricing errors are correlated with betas and demonstrate that the test performance depends crucially on the correlation, cross-sectional distribution of betas, and several other parameter values. Even when the CAPM holds exactly (pricing errors are zero) the FM test is equally likely to either reject or accept the model when typical sample sizes are used.     

NON-REPRESENTATIVE TRADING FREQUENCIES AND THE DETECTION OF ABNORMAL PERFORMANCE

Numerous studies employ betas computed with the ordinary least squares technique and daily returns. However, betas computed with OLS and daily returns are biased and inconsistent due to nonsynchronous trading periods or differences in trading frequency. The purpose of this study is to evaluate the effect of trading frequency on event studies. Brown and Warner (1985) investigated this and several other problems associated with daily returns and found no effect. However, they did not analyze the trading frequencies of the securities in their sample. This study uses a computer simulation for which trading frequency is an input, and, thus, tests stocks with known trading frequency.     

Corporate Mergers and Shareholder Wealth Effects: 1977-1986

The paper investigates the distribution of returns to shareholders of UK companies involved in acquisitions during the period 1977-1986. Three control models were used in the analysis: the market model with parameters identified through OLS regression, a model based on adjusted betas, and finally an index-relative model. Abnormal returns were identified around both bid announcement and outcome dates for bidders and targets in completed and abandoned bids. Examination was also made of the distribution of wealth changes for bidders and targets separately and for both in combination. The results demonstrate that, although there is no net wealth decrease to shareholders in total as a result of takeover activity, shareholders of bidder firms do suffer wealth decreases. By contrast, shareholders in target firms obtained significant, positive wealth increases in both completed and abandoned bids.     

Linear beta pricing with efficient/inefficient benchmarks and short-selling restrictions

We provide two security pricing models that can be used when short sales of risky securities are not permitted. The first model uses a benchmark located on the expected return-standard deviation efficient frontier without short sales and presents security expected returns as a weighted linear function of two betas, one induced by the benchmark and the other adjusting for the short-sale constraints. The second model uses a benchmark that is inefficient relative to the efficient frontier, does not allow short sales, and expresses security returns as a weighted linear function of three betas:one induced by the inefficient benchmark, and the other two adjusting simultaneously for the short-sales restrictions and the benchmark's inefficiency.These results complement the linear pricing models that link expected returns and betas by allowing, for efficient or inefficient expected return-standard deviation, benchmarks with restricted short sales.     

On the Cross-Sectional Relation between Expected Returns, Betas, and Size

In this paper, I set up scenarios where the mean-variance capital asset pricing model is true and where it is false. Then I investigate whether the coefficients from regressions of population expected excess returns on population betas, and expected excess returns on betas and size, allow us to distinguish between the scenarios. I show that the coefficients from either ordinary least squares or generalized least squares regressions do not allow us to tell whether the model is true or false.