Cross Sectional Asset Pricing Tests: Ex Ante versus Ex Post Approaches

Transcription

1 Cross Sectional Asset Pricing Tests: Ex Ante versus Ex Post Approaches Mahmoud Botshekan Smurfit School of Business, University College Dublin, Ireland John Cotter Smurfit School of Business, University College Dublin, Ireland and Financial Mathematics Computation Research Cluster (FMC 2 ) john.cotter@ucd.ie, Mathijs van Dijk Rotterdam School of Management, Erasmus University Rotterdam madijk@rsm.nl, We thank Michael Brennan, Gregory Connor, Rene Stulz and Matthew Spiegel for detailed and insightful comments. The authors acknowledge the support of Science Foundation Ireland under grant number 08/SRC/FMC1389.

2 Cross Sectional Asset Pricing Tests: Ex Ante versus Ex Post Approaches Abstract We compare commonly applied ex post realized returns with the implied cost of capital (ICC) as an ex ante proxy for expected returns in cross sectional asset pricing tests. We compare the Fama-Macbeth premia estimates using these two proxies for well-known systematic risk factors and firm characteristics, namely beta, size, book-to-market, momentum, idiosyncratic volatility, and illiquidity, in univariate and multivariate tests. The results show a robust and stable relationship between the ICC and the risk variables in both univariate and multivariate tests. However, the estimated premia associated with realized returns is much more volatile and also depends on the length of the period used to measure realized returns. Furthermore, positive estimated premia for size and idiosyncratic volatility using ICC stand in sharp contrast to the negative premia estimated by using realized returns and is consistent with a risk based interpretation for these variables. In contrast, the negative premia associated with momentum when we use ICC suggests overreaction as the main cause of the momentum effect. Keywords: Asset pricing tests, Implied cost of capital, Realized returns. JEL codes: G10, G12.

3 1 Introduction A core objective in asset pricing studies is to understand the determinants of expected returns. The capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965) provides a sound basis in modelling the tradeoff between risk and expected returns. However, the failure of the model to explain a number of anomalies has motivated many researchers to augment this model by additional variables that seem to capture part of the cross sectional variation in expected returns. A substantial body of research has been devoted to the empirical assessment of these models. The results of these studies are mixed and therefore cast doubt about the risk explanations of these additional factors 1. The bulk of these studies use realized returns as an ex post proxy for expected returns. However, As Elton (1999) also points out, realized returns can be a poor proxy of expected returns; Realized returns can deviate significantly from expected returns with prolonged periods of time (sometimes more than 10 years) where realized returns on average are less than risk free rate. Furthermore, realized returns can be negatively correlated with expected returns in the short run as positive or negative shocks to expected returns cause an opposite movement in contemporaneous realized returns. Therefore, the mixed evidence in estimating the price of different risk variables might be due to noise in realized returns 2. These arguments motivate some researchers to investigate the use of implied cost of capital (ICC) as an ex ante measure of expected returns in tests of asset pricing models 3. The ICC is defined as the internal rate of return that equates the present value of a stock s future expected 1 We will discuss this mixed evidence in Section 4. 2 See for example Elton (1999) and Pástor, Sinha, and Swaminathan (2008) for detailed discussions. This is also evident in the context of portfolio optimization, when the average realized returns is used as a proxy for expected returns and results in very poor out-of-sample performance (DeMiguel, Garlappi, and Uppal (2009)). 3 For example, Gebhardt, Lee, and Swaminathan (2001) use a residual income model to estimate the ICC for a large sample of US stocks and examine the relationship between the ICC and various firm risk characteristics. Lee, Ng, and Swaminathan (2009) use the ICC and test international asset pricing models at firm level. They show that the implied approach provides clear evidence of an economic relation between firm risk characteristics and equity premia, and support broader use of the ICC in the finance literature. More recently, Hanauer, Jäckel, and Kaserer (2013) use the same approach and test the Fama-French three factor model using the ICC as a proxy for expected returns. Pástor, Sinha, and Swaminathan (2008) use aggregate ICC (both equal- and valueweighted) to estimate the intertemporal relationship between risk and return and show that the ICC is useful in capturing time variation in expected returns, and outperforms realized returns in detecting a risk-return tradeoff. Li, Ng, and Swaminathan (2012) use the ICC to estimate the implied value premium as the difference between the implied cost of capital of value stocks and growth stocks. They argue that it is the best predictor of the ex post value premium during the time period. 3

4 cash flows to the stock s market price. There are several advantages for using implied cost of capital in asset pricing tests compared to using realized returns. First, ICC estimates are much more stable than realized returns, and even excess ICC remain positive for most stocks. Second, ICC trace changes in expected returns in a more timely manner. There is a continuous flow of information to the market where market participants incorporate this information in their analysis and continually update their perceptions towards the risk and value of risky assets. This updated information immediately affects market prices and so the ICC estimated at each point in time reflect the most recent information released to the market through changes in market prices. In contrast, average realized returns needs to be estimated over a period of time that is often of considerable length so that the average converges to expected value. The information released to the market over this period can be divided into two parts: cash flow news and discount rate news 4. The cash flow news released over the period makes average realized returns be different from expected values even when expected returns remain constant. The discount rate news released over the period causes time variation in expected returns so the expected returns at the end of period would be different from those at the beginning of the period. Therefore both kinds of news could be a source of noise in realized returns. In contrast, ICC are point estimates and reflect the discount rate that the market uses to discount future cash flows at a particular time 5. This is more important in an asset pricing setting, when we want to examine the relationship between expected returns and risk variables estimated at each point in time. Besides time variation in expected returns, the risk variables are also time-varying so realized returns that occur at the end of the period used to estimate realized returns are exposed to different risk variables compared to realized returns that occur at the beginning of the period. Therefore in each point in time, ICC has a higher validity to be used as the current market expected return to capture risk-return relationships for the current risk variables 6. 4 We know that stock return have two components; stock prices may move because investors revise future cash flows (so-called cashflow news) or because investors update their expectation towards risk of future cash flows and revise discount rates (so-called discount rate news). 5 Chen, Da, and Zhao (2013) use forecasts for future cash flows (from the IBES database) to back out ICC and then decompose stock returns over each period into cash flow news (defined as the price change over the period holding the implied cost of capital (ICC) constant, and discount rate news (defined as the price change over the period holding the cash flow forecasts constant). They conclude that stock returns have a significant cash flow news component (at both the firm and aggregate levels) whose importance increases with the investment horizon. 6 Note the use of ICC as a measure of expected returns is not without concern, especially as it is based on 4

5 In this paper, we contribute to the literature by providing a rigorous comparison between using ICC and realized returns in testing asset pricing models that aim at explaining cross sectional variations in firm-level expected returns. To perform our asset pricing tests for these two proxies of expected returns, we use the most common risk factors and firm characteristics (we refer to them as risk variables) that are proposed in the literature to capture part of the premia in the cross section of expected returns, namely, the market factor (Sharpe (1964) and Lintner (1965)), the size, book-to-market, and momentum factors (Fama and French (1993), Fama and French (1996), Jegadeesh and Titman (1993), and Carhart (1997)), the liquidity factor (Amihud (2002), Pastor and Stambaugh (2003)), and idiosyncratic volatility (Ang, Hodrick, Xing, and Zhang (2006)). We perform univariate and multivariate Fama-MacBeth regressions with time-varying risk variables estimated over 12-months rolling windows prior to the estimation of the expected return proxies for the period January 1981 to December In this way, we compute a time-series of estimated risk premia corresponding to the time-varying risk variables. We then test whether the time-series average of risk premia is positive and significantly different from zero. We perform our asset pricing tests using two different samples of stocks. The first sample is all individual common stocks in the IBES database over the period January 1981 to December 2010 for which we can compute both implied cost of capital and realized returns, allowing us to compare the results for the two proxies on an equal footing. We refer to this as the IBES sample. Note as larger firms are more likely to be followed by analysts, this sample might be biased toward big stocks. Therefore we also consider a second sample that consists of all individual common stocks traded on the NYSE, AMEX, and NASDAQ exchanges over the period January 1981 to December We refer to this as the CRSP sample which enables us to perform more comprehensive tests for realized returns. In our robustness checks, we also vary the sample to see whether our baseline results remain valid over different sub-periods and different sub-samples of the IBES database. We find sharp difference between results of asset pricing tests when we use ICC versus realized returns. First, all the risk variables are significant in explaining the variation in expected returns in our univariate tests when we use the ICC. However, this is not the case for realized returns for both IBES and CRSP samples. More specifically, the price of premia for different analyst earning s forecasts that could be biased (see for example Lin and McNichols (1998) and Easton and Zmijewski (1989)) or contain valuation errors (Abarbanell and Bushee (1997) and Francis, Olsson, and Oswald (2000)). For these and related discussions on the use of ICC see Hou, Van Dijk, and Zhang (2012). 5

6 risk variables also depends on the length of period used to measure realized returns. In our tests, we compare the results of using four different estimates for realized returns, namely onemonth, one-year, three-year, and five-year realized returns. There is a tradeoff for choosing the optimal period to estimate realized returns for use as a proxy in asset pricing tests. On the one hand, the longer periods contain more observations and therefore seem to be better proxies for expected returns as the average should converge to the expected value as more observations are used. However, as we estimate the risk variables prior to the estimation of realized returns, returns that occur say two years later may not fully be exposed to current risk variables. Therefore we should consider this tradeoff when we want to use realized returns as a proxy for expected returns. For example we find that whilst size and idiosyncratic volatility seems to be relevant factors for short-term realized returns, their effects decline when we use longer periods. In contrast, the effect of book-to-market on realized returns are more visible for longer estimates of realized returns. Second, the sign of obtained premia for size, momentum and idiosyncratic volatility when we use ICC stands in sharp contrast to the sign of premia when we use realized returns. Size and idiosyncratic volatility carry negative premia when we use realized returns but positive premia when we use ICC, consistent with a risk interpretation of these two anomalies. For momentum the opposite holds. Furthermore, the sign of premia remains stable (negative for momentum and positive for the other risk variables) when we use the ICC. This is not the case for realized returns, where the sign of estimated premia reverses several times during the last thirty years for all risk variables. This is evident when we perform sub-period analysis for six five-year periods between 1981 to Results of these tests strongly support a stable relationship between ICC as an ex ante measure of expected returns and the risk variables during the last 30 years which is lacking for realized returns. The volatile and unstable premia for the risk variables when we use realized returns may explain some of the mixed evidence about the price of these risk variables. For example, some studies suggest that the size premium has disappeared after the early 1980s or become negative across different periods (see Dimson and Marsh (1999), Chan, Karceski, and Lakonishok (2000), Horowitz, Loughran, and Savin (2000), Hirshleifer (2001), and also Van Dijk (2011) for a comprehensive review). There is also a debate about the price of idiosyncratic volatility in the cross section of individual stocks. Whilst Ang, Hodrick, Xing, and Zhang (2006) and Ang, Hodrick, Xing, and Zhang (2009) find a negative relationship between idiosyncratic volatility and subsequent realized returns, Bali and Cakici (2008) report that overall, no robust significant relation exists between idiosyncratic volatility and expected returns, and Fu (2009) finds a 6

7 positive significant relationship between the estimated conditional idiosyncratic volatilities and expected returns. Our results for realized returns are in line with Ang, Hodrick, Xing, and Zhang (2006) and Ang, Hodrick, Xing, and Zhang (2009). Further we suggest that the mixed evidence might be due to noise in realized returns (especially one-month realized returns) when it is being used as a proxy for expected returns. 7. The difference in the sign of premia for momentum (being negative when we use ICC and being positive when we use realized returns) highlights the different interpretations that exist regarding momentum as a risk factor. On one hand, the positive price of momentum for realized returns is consistent with Jegadeesh and Titman s (1993) findings on the effect of past three to twelve months returns on subsequent returns. In contrast, the negative price of momentum using ICC is in line with Cooper (1999) and Subrahmanyam (2005) who suggest overreaction and not risk as the main cause of the momentum effect. Our multivariate tests further support the results of the univariate tests. We test six different asset pricing models and for each we augment with an additional risk variable. The overall explanatory power of the models (R 2 ) using ICC are about two times larger compared to the models that use realized returns. For most of the risk variables included in these models, we observe a robust and significant premia when we use ICC that is not nearly as clear-cut using realized returns, and especially for the IBES sample. Again, the estimated premia is positive for momentum and negative for size and idiosyncratic risk for both IBES and CRSP realized returns whereas the opposite holds when using ICC. Comparing the results of using realized returns for the IBES and CRSP samples, we observe stronger relationships between realized returns and the risk variables for the CRSP sample. For example in our univariate tests over the whole sample, size, idiosyncratic volatility, and illiquidity are only significant when we use the CRSP sample. This might suggest that the risk attribute of these risk variables are more visible for small stocks which are less likely to be followed by analysts in the IBES sample. The results using ICC contradicts this conclusion as there are robust relationships between ICC and these risk variables even when using the IBES sample. We subject our results to a range of robustness checks. First, we test some additional firm characteristics (anomalies) that are detailed in the literature to have significant effects on the cross sectional variations in expected stock returns. The firm characteristics we consider 7 All of these studies use one-month realized excess returns as a proxy for expected returns in their main analysis. Ang, Hodrick, Xing, and Zhang (2006) also use 12-month realized excess returns as a robustness check and report the same results as for the one-month realized returns. 7

8 include investments (Titman, Wei, and Xie (2004); Fama and French (1996)), asset growth (Cooper (1999)), accruals (Sloan (1996); Hirshleifer, Hou, and Teoh (2012)), net operating assets (Hirshleifer, Hou, Teoh, and Zhang (2004), gross profitability (Novy-Marx (2013)), operating profitability (Fama and French (2015)), net stock issues (Pontiff and Woodgate (2008), return on equity (Hou, Xue, and Zhang (2014)) and return on assets (Balakrishnan, Bartov, and Faurel (2010). The majority of asset pricing studies that have examined these anomalies have employed realized returns as a proxy for expected returns. Our results show that investors demand lower ex ante expected returns to hold stocks with higher profitability and higher ex ante expected returns to hold companies with higher NOA and NSI. We observe a reverse pattern for ex post realized returns of these stocks in our sample. These could have considerable implications for testing asset pricing models that augment these firm characteristics and aim to explain cross sectional variations in expected returns. We also find that the our results are not sensitive to using different forecast horizons to estimate ICC and also improving the forecast quality by requiring earning forecasts to be estimated by at least 3 and 5 analysts. Finally we replace firm characteristics including size, book-to-market and momentum by covariance-based factor loadings β SMB, β HML, and β UMD that are factor loadings of our tests assets on factor mimicking portfolios SMB, HML and UMD, proposed by Fama and French (1993, 1996) and Carhart(1997). Although there is a stronger relationship between ICC and firm s characteristics than between ICC and covariancebased factor loadings but the sign and significance of risk variable remain stable also for most covariance-based factors. The results when we use realized returns is more blur as most of covariance-based factors are insignificant when we use covariance-based factors. Overall the results provide support for using implied cost of capital in tests of asset pricing models but is much more cautious about using realized returns. The estimated premia for different risk variables are robust and stable when we use ICC. This relationship is blurred when realized returns are used and is particularly dependent on the length of period used to estimate realized returns as well as the sample period. Furthermore the risk interpretation of different variables such as size, idiosyncratic volatility and momentum is quite different for the ex post and ex ante measure of expected returns. The remainder of the paper is organized as follows. In Section 2, we introduce the method used to estimate ICC and the empirical methodology for estimating the risk premia in our asset pricing tests. Section 3 describes the data. In Section 4 we present our empirical results from comparing the use of ICC and realized returns in asset pricing tests. Section 5 discusses robustness checks for using additional anomalies, alternative ICC estimates, and covariance- 8

9 based factors versus charactersitcs. Section 6 concludes. A number of additional results are detailed in the Appendix. 2 Empirical Methodology In this section we first explain the procedure followed for estimation of the implied cost of capital for each stock. Next, we describe the asset pricing models, and the methodology used for estimation of premia in our asset pricing tests. 2.1 Estimation of Implied Cost of Capital The implied cost of capital can be computed as the internal rate of return that equates the present value of future cash flows to the current stock price. For this purpose, we can use a general dividend discount model: P t = k=1 E[D t+k ] (1 + R) k (1) Where t is current time, E[D t+k ] is the expected cash flow to equity at t + k, P t is the current market price, and R is the implied cost of capital. To implement this method in practice, a three-stage model is used with three periods (explicit forecast period, transition period, and steady-state period) for estimation of future cash flows to equity. Based on the information available in the IBES database, Pástor, Sinha, and Swaminathan (2008) develop a version of this technique that with minor modifications is also used by Lee, Ng, and Swaminathan (2009), Li, Ng, and Swaminathan (2013), and Chen, Da, and Zhao (2013). We follow the approach outlined in Li, Ng, and Swaminathan (2013). In this method, a three-stage discount model can be written as P t = F E t+1(1 b t+1 ) (1 + R) + F E t+2(1 b t+2 ) (1 + R) 2 } {{ } explicit forecast + T k=3 F E t+k (1 b t+k ) (1 + R) k } {{ } transition period + F E t+t +1 R(1 + R) T }{{} terminal value, (2) where T is the forecast horizon, F E t+k is the earning forecast for year t+k, b t+k is the plowback rate forecast for year t + k and F E t+k (1 b t+k ) is the cashflow (dividend) for year t + k. In this formula, we forecast earnings and plowback rates up to a finite horizon of T years and the present value of future cashflows beyond year T is captured in the terminal value. Following 9

10 Pástor, Sinha, and Swaminathan (2008) and Li, Ng, and Swaminathan (2012) we use T = 15 years as the forecast horizon. We estimate F E t+1 and F E t+2 (in the explicit forecast period ) based on consensus earning forecasts that are reported in the IBES database for the current and next financial years (F Y 1 and F Y 2 in the IBES database). As we need 12-month ahead forecasts, we compute the earning forecast for the first year as F E t+1 = w F Y 1 + (1 w) F Y 2, where the weight w is the number of months remaining to the next fiscal year-end divided by 12. The growth rate implicit in F Y 1 and F Y 2, g 2 = F Y 2/F Y 1 1, is used to compute the earning forecast for year t + 2, so we have F E t+2 = F E t+1 (1 + g 2 ) 8. The implicit growth rate between F Y 1 and F Y 2 is considered as the starting value for the growth rate in the transition period. Over the transition period, that lasts for T 2 years, this rate mean-reverts exponentially to reach the steady-state growth rate, assumed to be equal to the nominal GDP growth of the economy (g) 9. After the transition period dividends grow constantly at the steady-state growth rate. Hence, for t = 3 to t = T + 1 we compute earnings growth rates and earning forecasts as g t+k = g t+k 1 exp[log(g/g 2 )/(T 2)], (3) F E t+k = F E t+k 1 (1 + g t+k ), (4) To estimate future cash flows, we also need to postulate plowback rates over the forecast horizon. For the first year in the explicit forecast period, we assume that the plowback rate (b t+1 ) is equal to last year s plowback rate, namely one minus the dividend payout ratio in the last financial year. Then, this rate reverts linearly to the steady state plowable rate (b) that is consistent with the growth rate in the steady state period, where the plowback rate times the estimated implied cost of capital should be equal to the steady state growth rate 10. Therefore, for t = 2 to t = T + 1 the plowback rate is computed as b t+k = b t+k 1 b t+1 b T 1 Having information on future earnings and plowback rates, we are able to compute the 8 Growth rates of more than 100% and less than 2% are given values of 100% and 2% respectively. 9 The intuition is that in the long run, competition will drive returns on all investments to the average growth rate of the economy. Here we use the US average GDP growth rate over the last 10 years as the steady state growth rate. 10 We assume that in the steady state, the return on investment is equal to the cost of equity or equivalently has zero economic profit. In this case we should have g = b R. (5) 10

11 ICC as the internal rate of return that equates the discounted future cash flows to the current market price of the stock using Equation (2). 2.2 Estimation Procedure To detect the relationship between risk variables and expected returns, we need to postulate a multi-factor asset pricing model. The benchmark asset pricing model that is proposed for the estimation of implied risk premiums is as follows 11 ˆR i = α i + β i λ β + S i λ S + BM i λ BM + M i λ M + IdioVol i λ IdioVol + ILLIQ i λ ILLIQ + e i, (6) Where ˆR i is the estimated implied cost of capital using Equation (2); β i is the sensitivity of asset i to the market factor and λ β is the market risk premium; S i is the size of asset i (market capitalization) and λ S is the size risk premium; BM i is the book-to-market of asset i and λ BM is the book-to-market risk Premium; M i is the momentum of asset i and λ M is the momentum risk premium; IdioV ol i is the idiosyncratic volatility of asset i and λ IdioV ol is the idiosyncratic volatility risk premium; ILLIQ i is the Amihud (2002) illiquidity measure of asset i and λ ILLIQ is the illiquidity risk premium; and e i is the error term. In our empirical analysis in Section 4 we test the out-of-sample relationship between the proxies we use for expected returns (ICC and realized average returns) and the above risk variables. We perform Fama-MacBeth regressions with time-varying risk variables estimated over 12-months rolling windows, prior to the estimation of the proxies for expected returns, from January 1981 to December In this way, we can compute a time-series of estimated risk premia corresponding to the time-varying risk variables. The test then considers whether the time-series average of risk premia associated with each risk variable is significantly different from zero. We compute Newey-West (1987) standard errors in our pricing tests, as we use overlapping windows to estimate the premia. We use individual stocks rather than portfolios as our test assets. Using individual stocks in asset pricing tests has been encouraged by Ang, Liu, and Schwarz (2008). They show 11 To investigate the additional effects for different risk variables in explaining the variation in expected returns we also consider other alternative factor models in Section 4. 11

12 analytically and empirically that the conclusions drawn from individual versus portfolio test assets can differ substantially due to a trade-off between bias and efficiency. They argue that use of portfolios leads to efficiency losses by shrinking the cross-sectional dispersion of factor loadings, and indicate that the use of individual stocks generally permits better asset pricing tests and estimates of risk premia especially when two pass cross-sectional regression coefficients are estimated. We therefore follow their recommendation to run the asset pricing tests in such cases based on individual stocks. 3 Data We perform our asset pricing tests using two different samples of stocks. The first sample is all individual common stocks in the IBES database over the period January 1981 to December 2010 for which we can compute both implied cost of capital and realized returns. We call this sample the IBES sample. As larger firms are more likely to be followed by analysts, this sample might be biased toward big stocks. Therefore we also consider a second sample that consists of all individual common stocks traded on the NYSE, AMEX, and NASDAQ exchanges over the period January 1981 to December We call the latter sample the CRSP sample which enables us to perform more comprehensive tests for the realized returns. In our robustness checks, we also vary the sample period for both these samples to see whether our baseline results remain valid over different periods. The risk variables for both samples are computed as follows. To compute β, momentum, and idiosyncratic volatility, we require that the firms have at least 180 valid daily returns during the 12-month period prior to the month that we estimate the expected return proxies (ICC and realized returns). To estimate the market beta, we use daily data on the value weighted market index from the CRSP database as the market index and the one-month treasury bill rate as the risk free rate 12. β for each firm is estimated as the slope coefficient resulting from regressing the firm s excess return on the market excess returns. The idiosyncratic volatility is the standard deviation of residuals from this regression. Momentum is defined as the geometric average return during the 12 months prior to the month we estimate the expected returns proxies. To estimate the size factor, we use the natural logarithm of the market value (share price times the number of share outstanding) at the end of the month prior to the estimation of ICC. 12 Treasury data is obtained from Kenneth R. French s database: 12

13 To estimate the book-to-market factor, we use the natural logarithm of the ratio of the book value to the last month s market value of equity. Following Fama and French (1992) we use the book value at the end of year t 1 to compute the book-to-market ratio for July of year t to June of year t + 1. For the illiquidity measure, we follow Amihud (2002) and compute illiquidity for each firm as the average of daily observations of a stock s absolute return divided by the stock s dollar volume trade over the year prior to the estimation of ICC. Like Amihud (2002), firms should have at least 200 daily observations of return and volume available during the last year and a price greater than 5 dollars at the start of the previous year so as to be included in our analysis. To estimate ICC for the IBES sample, we need estimates of earning per share for the next two years from the IBES database as well as last year s dividend payout ratio from the Compustat database. As we perform our asset pricing tests for realized returns on the same sample of stocks, we also require sufficient return data for these stocks from the CRSP database to be able to compute realized returns. We also need information such as book values from the Compustat database and returns, prices and volume information from the CRSP database to compute our risk variables, prior to estimation of both ICC and realized returns. Therefore, our test assets in the IBES sample include the intersection of common stocks in IBES, CRSP and Compustat databases that have sufficient information to compute ICC, realized returns, and the risk variables 13. Similarly our test assets in the CRSP sample include the intersection of common stocks in the CRSP and Compustat databases that have sufficient information to compute realized returns and the risk variables. Obviously this sample covers a larger number of stocks and would include more small stocks that are more likely to not be covered by the IBES database. Figure 1 presents the number of stocks for these two samples for the period January 1981 to December 2010, indicating the relative magnitude of the two samples and how this has varied over time. The average number of stocks in the IBES sample is equal to 2147 and in the CRSP sample it is equal to The dependent variable in our asset pricing test is either excess ICC (that is measured annually) or annualized excess realized returns. To compute excess returns, we use the oneyear Treasury bill rate. The excess returns are constructed at a monthly frequency from January 1981 to December 2010, comprising 360 monthly observations. As we need to estimate the risk variables over 12 months prior to the estimation of ICC and realized returns, and also we use different estimates for realized return (including three-year average realized returns) the actual 13 To match data between different databases, we use 8-digit CUSIP codes in IBES and CRSP databases and equivalent 9-digit CUSIPs in the Compustat database. 13

14 data we use covers the period of January 1980 to June Table 1 provides descriptive statistics for the expected returns proxies and risk variables for two samples. For the IBES sample the average ICC over the period 1981 to 2010 is equal to 15.85%, about 1% higher than average realized returns of 14.97%. The standard deviation of returns and also the range between minimum and maximum is much larger for realized returns. Comparing IBES and CRSP samples, the average realized return is higher for the IBES sample. As expected the average size of companies are higher for the IBES sample and is equal to $2913M compared to $2277M for the CRSP sample. The average book-to market is more than two times larger for the IBES sample, implying the existence of more growth companies in the CRSP sample. The average idiosyncratic risk and illiquidity is higher for the CRSP sample as it contains more small stocks. Lastly return seems to be more persistent for the IBES sample as we observe higher momentum for this sample. To get an overview of the degree of correlation in our risk variables, Table 1 also presents the average correlation of the risk variables for all windows applied in the Fama-Macbeth regressions performed in Section 4. We observe similar patterns for both samples. The highest average correlations between the risk variables for both samples are between size and idiosyncratic volatility (0.51 for the IBES sample and 0.57 for the CRSP sample) and between size and Amihud s illiquidity (0.48 for the IBES sample and 0.50 for the CRSP sample). The highest negative correlations are between book-to-market and momentum (-0.32 for the IBES sample and for the CRSP sample) and between beta and Amihud s illiquidity (-0.30 for the IBES and for the CRSP sample). There is a very low negative correlation between ICC and realized returns for the IBES sample. The correlation of risk variables with these two proxies indicate that there are higher linear associations between ICC and the risk variables compared to those for realized returns. We examine these relationships more rigorously in the next section. 14

15 Figure 1: Number of Companies in Two Samples The figure shows the average number of companies per year in the IBES and CRSP samples that have sufficient information to compute expected return proxies and risk variables over the period 1981 to

16 Table 1: Descriptive statistics This table reports descriptive statistics for our expected returns proxies (ICC and one-year realized returns) as well as the risk variables used in the Fama-Macbeth regressions for both IBES (Panel A) and CRSP (Panel B) samples. The panels also include average correlation between the variables. The values reported in the table are time series average of cross sectional statistics over the period January 1981 to December The risk variables include Beta (β), size (size), book-to-market (B/M), momentum (M), idiosyncratic volatility (IdioV ol), and Amihud s illiquidity measure (ILLIQ), respectively. Variables ICC 1Y RR β Size B/M M IdioV ol ILLIQ Panel A: IBES Sample Mean Median StdDev Skew Kurtosis Min Max Correlation with: 1Y RR β Size B/M M IdioV ol ILLIQ Panel B: CRSP Sample Mean Median StdDev Skew Kurtosis Min Max Correlation with: 1Y RR β Size B/M M IdioV ol ILLIQ

17 4 Empirical results We compare the performance of ICC versus realized returns in detecting the relationship between risk variables and expected returns. We present results using both univariate and multivariate tests and for both the IBES and CRSP samples. Our empirical tests are performed in an out-of-sample setting. Therefore, in each window of the Fama-Macbeth regressions, we regress the proxy used to estimate expected returns on the risk variables estimated over the 12 months prior to the estimation of expected returns. For the IBES sample in which we compare the results of using ICC and realized returns as two proxies for expected returns, we use completely equivalent data (the same stocks and the same risk variables) in each window of the Fama-Macbeth regressions so as to compare the results for the two proxies on an equal footing. We provide additional results for realized returns using the CRSP sample that includes a larger number of companies. 4.1 Univariate tests Table 2 presents results of our univariate tests using ICC for the IBES sample and one-year realized returns for the both IBES and CRSP samples 14. The first cross section estimates of ICC correspond to January 1981 that are regressed on the risk variables computed at the end of December The corresponding 12-months window for realized returns spans from January 1981 to December Then we roll over the estimates for ICC and realized returns by one month and regress on the risk variable estimated at the end of January By continuing this procedure, we compute risk premia estimates for 360 overlapping 12-months windows. Tests then consider if time series averages of premia estimates are significantly different from zero using Newey-West(1987) standard errors. In order to ensure that extreme outliers do not drive our findings, we winsorize the risk variables in each window at the 1% and 99% level. First we compare the results of using ICC and realized returns for the IBES sample using equivalent stocks for these two proxies. The first column in Table 2 shows that the standard beta has a significant and positive premium (at the 1% level significance) when we use ICC as a proxy for expected returns. However, when we use realized returns beta does not carry a significant premium. This is also the case for all other risk variables except momentum. This is the only variable that has a significant premium when we use realized returns for the IBES sample, although the level of significance and also the model s R 2 is higher when we use ICC. 14 Note that we cannot estimate ICC for the whole CRSP sample as we do not have analysts earnings forecasts for all stocks in this sample. 17

18 There is also one noticeable difference with the premia for momentum being negative when we use ICC and positive when we use realized returns. The positive price of momentum for realized returns is consistent with Jegadeesh and Titman s (1993) findings on the persistence of past three to twelve month stock return performance. In contrast, the negative price of momentum using ICC confirms Cooper (1999) and Subrahmanyam (2005) who suggest overreaction and not risk as the main source of this effect. We observe relatively large and strongly significant intercepts for most of the univariate regressions when use both ICC and realized returns. The intercept for each univariate regression indicates the part of average excess returns that can not be explained by the risk variable. This can be divided to two parts. The first part capture the cross sectional average of stocks abnormal returns which can not be explained by the risk factor (in the first stage of the Fama- Macbeth procedure). The second part captures possible errors in the estimation of premia (in the second stage of the Fama-Macbeth procedure) times the cross sectional average of the risk factor, which is caused by noise in the estimation of the risk factors or mispriced stocks. The latter part can be substantial especially when we use individual stocks in our asset pricing tests. If we compare the results of using realized returns for the CRSP sample (in panel C of Table 2) with the results of using realized returns for the IBES sample (in panel B of Table 2), we observe that size, idiosyncratic volatility, total volatility and illiquidity now becomes significant for the CRSP sample. As the CRSP sample covers a larger number of small stocks compared to the IBES sample, the evidence is supportive of the assumption that the pricing effect for these factors being more attributable to small stocks. For a more rigorous comparison between ICC and realized returns, we plot the time series of estimated premia for each risk variable in our univariate tests in Figure 2. To facilitate a comparison between ICC and realized returns, the left-hand side graphs correspond to the use of ICC and right-hand side graphs to the use of one-year realized returns for the IBES and CRSP samples. The common feature exhibited is that the estimated premia when we use ICC is more stable and much less volatile than those estimated by realized returns. For example, the estimated premia for β when we use one-year realized returns for the IBES sample changes from a minimum of % to a maximum of 100.4% whereas when we use ICC, it changes from a minimum of -4.7% to a maximum of 12.1%. A similar pattern is evident for other risk variables. The standard errors of premia estimates presented in Table 2 confirm this as well; volatility of the premia for β, size, momentum, and idiosyncratic volatility are more than six times larger when we use realized returns, and for book-to-market and Amihud s illiquidity this ratio is 3.3 and 5.6, respectively. The volatility of estimated premia for all risk variables using 18

19 realized returns for the CRSP sample are very similar to those of the IBES sample, although the averages of the time series premia are different. Figure 2 also shows that the time series of premia when using realized returns follow a close pattern for the IBES and CRSP samples for all risk variables. However the average of the time series premia are different resulting in insignificant premia estimates for most of the risk variables for the IBES sample. Another important feature is that the sign of the premia remains stable (positive or negative) for most of the risk variables when we use ICC during the last thirty years (except the premia for β that became negative for two years in the late 1990s). In contrast, for all the risk variables, the sign of the premia reverses several times using realized returns (for both IBES and CRSP samples). These results shed some light on the mixed empirical results in the literature for the price of different risk variables, and especially the price of the size effect and idiosyncratic volatility. The results for the size premium when using realized returns are in line with studies that suggest the size effect has disappeared after the early 1980s (see for example Dichev (1998), Chan, Karceski, and Lakonishok (2000), Horowitz, Loughran, and Savin (2000)) or show that the size premium becomes negative across different time periods (Dimson and Marsh (1999)). All of these studies use realized returns as a proxy for the risk premium. By way of contrast, when we use ICC the size premium is positive and stable during the last 30 years. We also observe a robust and stable positive premium for idiosyncratic volatility using ICC and a volatile and negative premia when we use realized returns. Our results for realized returns are compatible with Ang, Hodrick, Xing, and Zhang (2006) and Ang, Hodrick, Xing, and Zhang (2009) who find a negative relationship between idiosyncratic volatility and subsequent realized returns. On the other hand Bali and Cakici (2008) report no robust significant relation between idiosyncratic volatility and expected returns in the cross section of expected returns. This mixed evidence might be due to the use of the noisy proxy of realized returns in these studies. In contrast, we find a robust positive relationship between idiosyncratic volatility and ICC as a proxy for expected returns that is intuitive where we expect stocks with more idiosyncratic risk requiring some sort of compensation in terms of higher expected returns. Up to now we estimate realized returns over a one year period. We can measure realized returns using different periods, for example, one-month or three years. The question that can be raised is whether the estimated premia for different risk variables also dependent on the length of time used for measuring realized returns and if an optimal length exists. This is a relevant empirical question as there is a trade-off in choosing the optimal period for realized returns when we want to use them as a proxy for expected returns in out-of-sample asset pricing tests. 19

20 Intuitively, we should use longer periods for measuring realized returns, as more observations result in average returns being better representatives of expected returns. More importantly, shocks to expected returns are negatively correlated with realized returns in the short run. On the other hand, if we use longer periods, returns within the time frame are less exposed to the risk variables that are estimated before the period, so the detection of a risk-return relationship would be harder using longer periods. To investigate this issue, we compare the price of premia, using four different estimates of realized returns, namely one-month, one-year, three-year, and five-year. Table 3 presents results of univariate asset pricing tests using our four estimates of realized returns for the CRSP sample. There are some visible patterns in the price of different risk variables as we increase the length of period for measuring realized returns from one month to five years. Size, Momentum, idiosyncratic risk and illiquidity are strongly priced (at the 1% level) for one-month returns. The level of significance decreases when we move from one-month to one-year and then to three-years returns, so that size and illiquidity become insignificant for three years returns. Using five year realized returns, not one of these four variables are priced. On the other hand, book-to-market is only significantly priced for three-year and five-year realized returns. Thus it appears that book-to-market has a long-term effect on subsequent returns whereas the relationship between realized returns and risk variables like size and illiquidity is more visible for short to medium term realized returns. More importantly, unlike ICC, this relationship is negative for size and illiquidity that might be due to the negative correlation between realized returns and shocks to expected returns in the short run. To save space, the results for the four estimates of realized returns using the IBES sample are presented in Table A7 of the appendix. In general we observe weaker results for the IBES sample as only momentum (for all four estimates) and idiosyncratic volatility (for one-month realized returns) are significantly priced. Again we observe a positive premia for momentum and a negative premia for idiosyncratic volatility. Comparing the average of the R 2 for the univariate tests across the four estimates, most of the risk variables have the highest explanatory power when we use one-year realized returns for both the IBES and CRSP samples. Therefore, we consider one-year returns as our benchmark period in the subsequent analysis.. 20

21 Table 2: Univariate Tests for Different Samples This table shows the time-series averages and their Newey-West (1987) standard errors (in parentheses) of premia estimates λ jt, using univariate Fama-MacBeth regressions, where t denotes the time that expected return proxies are estimated, and j denotes the risk variable, namely Beta (β), size (size), book-to-market (B/M), Momentum (M), idiosyncratic volatility (IdioV ol), total volatility (σ), and Amihud s illiquidity measure (ILLIQ), respectively. The sample consists of all companies in the IBES, CRSP and Compustat databases that have the required data for computation of expected return proxies and risk variables from January 1981 to December Panel A correspond to using ICC estimates for stocks in the IBES sample, and Panel B and C correspond to one-year realized returns for stocks in the IBES and CRSP samples, respectively. There are months overlapping estimation windows in the sample. Stocks with missing information in a specific estimation window are deleted from the cross-sectional regression for that window. Returns and risk variables in each window have been winsorized at the 1% level and 99% level.,, and denote significance at the 1, 5, and 10 percent levels, respectively. Time series averages of R 2 are also reported for each univariate regression. β Size B/M M IdioV ol σ ILLIQ Panel A: ICC, IBES Sample α (0.303) (0.685) (0.406) (0.342) (0.387) (0.401) (0.359) λ (0.232) (0.050) (0.290) (0.005) (0.158) (0.162) (0.054) Average R Panel B: 1-year Realized Return, IBES Sample α (3.184) (9.241) (2.829) (2.682) (3.410) (3.728) (2.691) λ (2.869) (0.587) (0.900) (0.032) (1.529) (1.506) (0.305) Average R Panel C: 1-year Realized Returns, CRSP Sample α (3.177) (8.572) (3.250) (2.861) (3.311) (3.612) (2.919) λ (2.855) (0.528) (1.283) (0.033) (1.353) (1.366) (0.217) Average R