CAPM for estimating the cost of equity capital: Interpreting the empirical evidence

We argue that the empirical evidence against the capital asset pricing model (CAPM) based on stock returns does not invalidate its use for estimating the cost of capital for projects in making capital budgeting decisions. Because stocks are backed not only by projects in place, but also by the options to modify current projects and undertake new ones, the expected returns on stocks need not satisfy the CAPM even when expected returns of projects do. We provide empirical support for our arguments by developing a method for estimating firms' project CAPM betas and project returns. Our findings justify the continued use of the CAPM by firms in spite of the mounting evidence against it based on the cross section of stock returns.     

Does the Conditional CAPM Work? Evidence from the Istanbul Stock Exchange

Using a sample of common stocks traded on the Istanbul Stock Exchange from February 1997 to April 2008, we test whether the conditional capital asset pricing model (CAPM) accurately prices assets. In our empirical analysis, we closely follow the methodology introduced in Lewellen and Nagel (2006). Our results show that the conditional CAPM fares no better than the static counterpart in pricing assets. Although market betas do vary significantly over time, the intertemporal variation is not large enough to drive average conditional alphas to zero.     

Risk Interpretation of the CAPM's Beta: Evidence from a New Research Method

This study tests the validity of using the CAPM beta as a risk control in cross‐sectional accounting and finance research. We recognize that high‐risk stocks should experience either very good or very bad returns more frequently compared to low‐risk stocks, that is, high‐risk stocks should cluster in the tails of the cross‐sectional return distribution. Building on this intuition, we test the risk interpretation of the CAPM's beta by examining if high‐beta stocks are more likely than low‐beta stocks to experience either very high or very low returns. Our empirical results indicate that beta is a strong predictor of large positive and large negative returns, which confirms that beta is a valid empirical risk measure and that researchers should use beta as a risk control in empirical tests. Further, we show that because the relation between beta and returns is U‐shaped, that is, high betas predict both very high and very low returns, linear cross‐sectional regression models, for example, Fama–MacBeth regressions, will fail on average to reject the null hypothesis that beta does not capture risk. This result explains why previous studies find no significant cross‐sectional relation between beta and returns.     

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.     

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.     

The Conditional Beta and the Cross-Section of Expected Returns

We examine the cross-sectional relation between conditional betas and expected stock returns for a sample period of July 1963 to December 2004. Our portfolio-level analyses and the firm-level cross-sectional regressions indicate a positive, significant relation between conditional betas and the cross-section of expected returns. The average return difference between high- and low-beta portfolios ranges between 0.89% and 1.01% per month, depending on the time-varying specification of conditional beta. After controlling for size, book-to-market, liquidity, and momentum, the positive relation between market beta and expected returns remains economically and statistically significant.     

Pension benefits, wealth accumulation, and capital market equilibrium: a life cycle hypothesis-based dynamic model

This paper examines the impacts of pension benefits on capital asset pricing in conjunction with wealth accumulation and retirement, and derives and tests a dynamic capital asset pricing model (CAPM) within the framework of a life cycle hypothesis-based dynamic model. The life cycle hypothesis-based dynamic model maximizes the expected utility of the individual's lifetime wealth in a continuous time process. An optimal solution of the individual's wealth path, incorporating the ages of retirement and death, is obtained and, based on the optimal wealth path, an analysis of comparative dynamics is pursued. The dynamic CAPM is then derived from the optimal wealth path; simulation and nonparametric tests are undertaken to evaluate the performance of the dynamic CAPM as compared to the traditional model which does not consider the impacts of pension benefits and the static model that incorporates the effects of pension benefits. The test results suggest that the proposed dynamic CAPM closely states the expected rate of return for a capital asset; that the new dynamic CAPM is preferable over the static model that is preferable over the traditional model; and that the three models considered are statistically distinguishable from one another.     

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.     

Multifactor Explanations of Asset Pricing Anomalies

Previous work shows that average returns on common stocks are related to firm characteristics like size, earnings/price, cash flow/price, book-to-market equity, past sales growth, long-term past return, and short-term past return. Because these patterns in average returns apparently are not explained by the CAPM, they are called anomalies. We find that, except for the continuation of short-term returns, the anomalies largely disappear in a three-factor model. Our results are consistent with rational ICAPM or APT asset pricing, but we also consider irrational pricing and data problems as possible explanations.     

A conditional higher-moment CAPM

This paper investigates whether dynamic and moment extensions to the traditional CAPM can improve its empirical performance and offer some alternative explanation to the cross-section of average returns on portfolios of stocks double sorted on book-to-market ratios and size. We consider three extensions. First, we introduce time-varying factor loadings obtained from a multivariate GARCH and dynamic conditional correlations. Second, we extend the model to a four-moment CAPM, which incorporates coskewness and cokurtosis. Finally, we allow for time-varying risk premia, based on a Markov-switching process. Our results confirm that the higher-moment CAPM does not perform well in its unconditional version, but its performance is significantly improved when we introduce a conditional version that accounts for both time-varying factor loadings and time-varying risk premia. The four-moment CAPM tests lead to a positive total risk premium estimate of 0.67% per month over the period 1926–2021, with all risk premia (beta, coskewness, and cokurtosis) exhibiting the expected theoretical signs.     

Heteroskedasticity in Stock Returns

We use predictions of aggregate stock return variances from daily data to estimate time-varying monthly variances for size-ranked portfolios. We propose and estimate a single factor model of heteroskedasticity for portfolio returns. This model implies time-varying betas. Implications of heteroskedasticity and time-varying betas for tests of the capital asset pricing model (CAPM) are then documented. Accounting for heteroskedasticity increases the evidence that risk-adjusted returns are related to firm size. We also estimate a constant correlation model. Portfolio volatilities predicted by this model are similar to those predicted by more complex multivariate generalized-autoregressive-conditional-heteroskedasticity (GARCH) procedures.     

Semiparametric estimation of a characteristic-based factor model of common stock returns

We introduce an alternative version of the Fama–French three-factor model of stock returns together with a new estimation methodology. We assume that the factor betas in the model are smooth nonlinear functions of observed security characteristics. We develop an estimation procedure that combines nonparametric kernel methods for constructing mimicking portfolios with parametric nonlinear regression to estimate factor returns and factor betas simultaneously. The methodology is applied to US common stocks and the empirical findings compared to those of Fama and French.     

An International Dynamic Asset Pricing Model

We examine the ability of a dynamic asset-pricing model to explain the returns on G7-country stock market indices. We extend Campbell's (1996) asset-pricing model to investigate international equity returns. We also utilize and evaluate recent evidence on the predictability of stock returns. We find some evidence for the role of hedging demands in explaining stock returns and compare the predictions of the dynamic model to those from the static CAPM. Both models fail in their predictions of average returns on portfolios of high book-to-market stocks across countries.     

Some extensions of the CAPM for individual assets

There is ample evidence that stock returns exhibit non-normal distributions with high skewness and excess kurtosis. Experimental evidence has shown that investors like positive skewness, dislike extreme losses and show high levels of prudence. This has motivated the introduction of the four-moment capital asset pricing model (CAPM). This extension, however, has not been able to successfully explain average returns. Our paper argues that a number of pitfalls may have contributed to the weak and conflicting empirical results found in the literature. We investigate whether conditional models, whether models that use individual stocks rather than portfolios and whether models that extend both the moment and factor dimension can improve on more traditional static, portfolio-based, mean–variance models. More importantly, we find that the use of a scaled coskewness measure in cross-section regression is likely to be spurious because of the possibility for the market skewness to be close to zero, at least for some periods. We provide a simple solution to this problem.     

Residual momentum

Conventional momentum strategies exhibit substantial time-varying exposures to the Fama and French factors. We show that these exposures can be reduced by ranking stocks on residual stock returns instead of total returns. As a consequence, residual momentum earns risk-adjusted profits that are about twice as large as those associated with total return momentum; is more consistent over time; and less concentrated in the extremes of the cross-section of stocks. Our results are inconsistent with the notion that the momentum phenomenon can be attributed to a priced risk factor or market microstructure effects.     

Estimation of expected return: CAPM vs. Fama and French

Most practitioners favour a one-factor model (CAPM) when estimating expected return for an individual stock. For estimation of portfolio returns, academics recommend the Fama and French three-factor model. The main objective of this paper is to compare the performance of these two models for individual stocks. First, estimates for individual stock returns based on CAPM are obtained using different time frames, data frequencies, and indexes. It is found that 5 years of monthly data and an equal-weighted index, as opposed to the commonly recommended value-weighted index, provide the best estimate. However, performance of the model is very poor; it explains on average 3% of differences in returns. Then, estimates for individual stock returns are obtained based on the Fama and French model using 5 years of monthly data. This model, however, does not do much better; independent of the index used, it explains on average 5% of differences in returns. These results therefore bring into question the use of either model for estimation of individual expected stock returns.     

Pricing assets with stochastic cash-flow growth

We model the time series behavior of dividend growth rates, as well as the profitability rate, with a variety of autoregressive moving-average processes, and use the capital asset pricing model (CAPM) to derive the appropriate discount rate. One of the most important implications of this research is that the rate of return beta changes with the time to maturity of the expected cash flow, and the degree of mean reversion displayed by the growth rate. We explore the consequences of this observation for three different strands of the literature. The first is for the value premium anomaly, the second for stock valuation and learning about long-run profitability, and the third is for the St. Petersburg paradox. One of the most surprising results is that the CAPM implies a higher rate of return beta for value stocks than growth stocks. Therefore, value stocks must have higher expected returns, and this is what is required theoretically in order to explain the well-known value premium anomaly.     

RETURN INTERVAL,FIRM SIZE AND SYSTEMATIC RISK ON THE DUTCH STOCK MARKET

In this paper we investigate the behavior of betas of 50 Dutch firms as a function of the return measurement interval. We find beta estimates measured from different intervals differ significantly from each other. As the sample mainly contains stocks that are relatively thin compared to the index, beta estimates from short intervals are on average lower than those obtained from longer intervals. The results further indicate that there exists some variability in the beta coefficients for each interval length. Betas depend on the manner daily prices are juxtaposed to calculate the returns. A way to account for this variability is to average the different betas for each interval length. Asymptotic betas are also computed to show the appropriateness of this method. Finally we show that the size effect is reduced when the interval length is increased, although it remains statistically significant.     

New ESG rating drivers in the cross-section of European stock returns

We assess the performance of two quantitative signals based on ESG scores across a large, multi-national cross-section of European stock returns. We test whether the cost of equity capital is more influenced by the upward momentum (measured over time) of the ESG scores of the firms issuing stocks or by their stability (identified as the volatility of the scores over time), measured around a changing mean level. We find that short-term ESG momentum over 1 month has a significant impact on the cross-section of stock returns, lowering the anticipated cost of capital and leading to positive average abnormal returns. This suggests that short-term ESG momentum may represent a novel, priced systematic risk factor. Furthermore, we find strong evidence that an ESG volatility spread strategy which buys low ESG score volatility stocks and sells high volatility ones, generates a substantial alpha and affects the ex-ante cost of capital. Both quantitative ESG signals result in portfolio sorting and long-short strategies that enhance the overall sustainability profile of the issuing firms without compromising the raw average of their ESG scores.     

The return impact of realized and expected idiosyncratic volatility

We show that the negative relation between realized idiosyncratic volatility, measured over the prior month, and returns is robust in non-January months. Controlling for realized idiosyncratic volatility, we show that the relation between returns and expected idiosyncratic volatility is positive and robust. Realized and expected idiosyncratic volatility are separate and important effects describing the cross-section of returns. We find the negative return on a zero-investment portfolio that is long high realized idiosyncratic volatility stocks and short low realized idiosyncratic volatility stocks is dependent on aggregate investor sentiment. In cross-sectional tests, we find the negative relation is weaker for stocks with a large analyst following and stronger for stocks with high dispersion of analyst forecasts. The positive relation between expected idiosyncratic volatility and returns is not due to mispricing.