Dynamic risk, accounting-based valuation and firm fundamentals

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1 Rev Account Stud (2013) 18: DOI /s x Dynamic risk, accounting-based valuation and irm undamentals Matthew R. Lyle Jerey L. Callen Robert J. Elliott Received: 20 May 2011 / Accepted: 8 March 2013 / Published online: 10 May 2013 Ó Springer Science+Business Media New York 2013 Abstract This study extends the accounting-based valuation ramework o Ohlson (Contemp Acc Res 11(2): , 1995) and Feltham and Ohlson (Acc Rev 74(2): , 1999) to incorporate dynamic expectations about the level o systematic risk in the economy. Our model explains recent empirical indings documenting a strong negative association between changes in economy-wide risk and uture stock returns. Importantly, the model also generates costs o capital that are solely a linear unction o accounting variables and other irm undamentals, including the book-to-market ratio, the earnings-to-price ratio, the orward earningsto-price ratio, size and the dividend yield. This result provides a theoretical rationale or the inclusion o these popular variables in cost o capital (expected return) computations by the accounting and inance literatures and obviates the need to estimate costs o capital rom unobservable (uture) covariances. The model also generates an accounting return decomposition in the spirit o Vuolteenaho (J Finance 57(1): , 2002). Empirically, we ind that costs o capital generated by our model are signiicantly associated with uture returns both in and out o sample in contrast to standard benchmark models. We urther obtain signiicantly M. R. Lyle (&) J. L. Callen Rotman School o Management, University o Toronto, Toronto, ON M5S 3E6, Canada matthew.lyle09@rotman.utoronto.ca J. L. Callen callen@rotman.utoronto.ca M. R. Lyle Kellogg School o Management, Northwestern University, Evanston, IL 60208, USA R. J. Elliott School o Mathematical Sciences, University o Adelaide, Adelaide 5005, Australia robert.elliott@adelaide.edu.au R. J. Elliott Centre or Applied Financial Studies, University o South Australia, Adelaide 5001, Australia

2 900 M. R. Lyle et al. lower valuation errors in out-o-sample tests than traditional models that ignore dynamic risk expectations. Keywords Dynamic risk Accounting valuation Cost o capital Firm undamentals JEL Classiication G12 G14 M41 1 Introduction This study extends the Ohlson (1995) and Feltham and Ohlson (1999) accountingbased valuation ramework to incorporate dynamic expectations about the level o risk in the economy. Our valuation model is comprehensive enough to include time varying risk, yet parsimonious enough to generate linear pricing equations. The linear pricing equation is comprised o accounting variables and a new actor that captures dynamic risk in the economy. The model yields the intuitive result that economy-wide risk and equity values are inversely related. The valuation model also generates an exact equation or equity returns and describes how returns evolve through time. These return dynamics provide a theoretical rationale or recent empirical evidence that changes in aggregate volatility are negatively associated with expected equity returns (Ang et al. 2006). The model also provides an explicit equation or the cost o capital (expected return) expressed solely as a linear combination o accounting variables and irm undamentals, including the book-to-market ratio, the earnings-to-price ratio, the orward earnings-to-price ratio, size, and the dividend yield. 1 Importantly, our indings provide theoretical justiication or the use o popular accounting ratios and irm characteristics as determinants o uture stock returns in the empirical accounting and inance literatures. In addition, we show that the equity return derived rom our accounting-based valuation model can be expressed as the expected return plus cash low news minus discount rate news, equivalent to the return decomposition o Vuolteenaho (2002). Thus we link the return decomposition literature based on book-to-mark dynamics with the accounting valuation literature based on non-arbitrage pricing. Without the inclusion o dynamic risk in the accounting-based valuation model, these results would not be possible. Empirically, we ind that costs o capital generated by our model are signiicantly associated with uture returns both in and out o sample, unlike standard benchmark models. We also ind that including an estimate o dynamic risk is important when pricing equity. More speciically, we irst ind that irm-level sensitivity to expected economy-wide risk is highly associated with average cross-sectional stock returns; more traditional sensitivities such as the CAPM beta or the Fama French 1 As we show urther below, the dividend yield can be eliminated using the clean surplus relation in which case the cost o capital can be expressed solely (in addition to size) as a linear unction o accounting numbers, namely, the past book-to-market ratio, the current book-to-market ratio, the earnings-price ratio, and the orward earnings-price ratio.

3 Firm undamentals 901 three-actor betas do not perorm as well. Second, we show that our theoretically derived cost o capital model based solely on accounting variables and irm undamentals provides an unbiased predictor o uture average stock returns. Using the model to ormulate a trading strategy generates average hedged monthly equity returns o 1.5 % and abnormal monthly returns o 1.18 % relative to the Fama French three-actor model. Third, in comparison to benchmark valuation models, a model that includes economy-wide risk expectations consistently produces signiicantly lower out-o-sample pricing errors than traditional valuation models. The remainder o the paper is organized as ollows: Sect. 2 provides motivation or this study. Section 3 develops the valuation model and derives the equity return dynamics as well as the cost o equity capital (expected return) dynamics. Section 4 estimates our model-driven costs o capital and equity prices empirically and compares these estimates to those derived rom benchmark models. Section 5 briely concludes. 2 Motivation Since the work o Ohlson (1995) and Feltham and Ohlson (1995), a large number o theoretical studies have explored how accounting data can be used to price equities. 2 This literature has largely ocused on increasing the sophistication o modeling irm undamentals, such as expected earnings, but not the dynamics o risk. There are exceptions. Feltham and Ohlson (1999) provide a theoretical oundation or more complete models that include dynamic risk and dynamic risk-ree rates. In the process, they extend the standard residual income model (RIM) to include dynamic (covariance) risk. 3 Gode and Ohlson (2004) integrate time varying interest rates in the Ohlson (1995) valuation ramework. However, their analysis assumes a risk neutral setting and does not incorporate dynamic risk adjustments. Motivated by Feltham and Ohlson (1999), Ang and Liu (2001) construct a generalized aine model that yields a nonlinear relation between market value and book value, which depends upon stochastic interest rates, irm level proitability, and growth in book value. Pástor and Veronesi (2003) develop a valuation model that incorporates the clean surplus relation and includes learning about accounting proits. However, these models do not provide closed-orm solutions or equity prices. Instead, the price o the equity is commonly expressed as an integral (or sum) o a unction that must be solved numerically. An exact closed-orm solution or equity prices that incorporates dynamic risk has proven elusive but, as we shall see, is clearly desirable. Our model provides such a solution. Empirically, the role o risk is oten modeled by a backward-looking capital asset pricing model (CAPM) and other actor models based on historical estimates that do not include inormation about expectations o risk or uture states o the economy. I historical estimates o risk are not representative o current expectations o uture risk, this could be problematic when valuing stocks. For example, an equity 2 See Ohlson (2009) and the citations therein. 3 See their Corollary 3.

4 902 M. R. Lyle et al. valuation model may suggest that a stock is undervalued using cash low orecasts and a historical measure o risk expectations. But this could very well be the wrong conclusion and might (assuming that cash low orecasts are reasonable) be caused by poor discount rate (risk) orecasts, such as those computed rom historical betas and long term averages o excess market returns. Furthermore, there is oten a disconnect between the underlying valuation model and the empirical risk measurement model. Many empirical studies use the CAPM (or the market model) to measure the irm s cost o capital and then proceed to substitute their cost o capital estimate into another valuation model, disregarding the act that the valuation process and the return generating process are both determined simultaneously and logically should be estimated rom the same model. But there is the rub: how to value the irm and estimate its cost o capital rom the same model? 4 We solve this conundrum below. Another related issue revolves around the ubiquitous use o accounting numbers to predict costs o capital without much theoretical justiication. The well-known Fama and French (1993) asset pricing model is an obvious case, but there are many others. Fama and French (1993) oer no theoretical justiication or using the bookto-market actor to predict expected returns. They do so because it seems to works empirically. This is hardly satisactory rom a scientiic perspective. In contrast, this study oers an exact theoretical rationale or using accounting numbers to estimate costs o capital. The papers by Nekrasov and Shro (2009) and Penman and Zhu (2011) are directly related to ours but with substantive dierences. Nekrasov and Shro (2009) insightully simpliy the extended RIM model o Feltham and Ohlson (1999) so that it can be empirically estimated. Using historically based covariance risk measures, they compare value estimates based on their model with the CAPM and the Fama French three-actor model and ind that their model yields smaller out o sample pricing errors. The conceptual dierence between our pricing model and theirs is analogous to the dierence between the standard RIM and Ohlson (1995); namely, Ohlson (1995) imposes an abnormal earnings dynamic on the standard RIM, whereas we impose both an abnormal earnings dynamic and stochastic risk structure dynamic on the Feltham and Ohlson (1999) extended RIM. In contrast to Nekrasov and Shro (2009), our primary interest is in predicting expected returns rather than equity prices, although we predict equity prices too. Also, our risk measures are orward looking and obviate the need to estimate unobservable covariances. Penman and Zhu (2011) criticize the tendency by the literature to prematurely classiy high realized returns as anomalous. Based on an accounting identity, they cogently argue that accounting variables are likely predictive o risk and, hence, o returns so that the anomalous results ound by the literature may in act be driven by risk. Absent rom their analysis, as they acknowledge, is an equilibrium model o equity valuation so that they cannot provide an exact relation between equity returns and the accounting variables that determine returns. We provide such a relation. 4 To the best o our knowledge, Morel (2003) is the irst paper to decry this disconnect and attempt to deal with it.

5 Firm undamentals The valuation model Assuming that price is the discounted value o expected uture dividends and that the clean surplus relation holds, Feltham and Ohlson (1999) show that the price o equity at time t ( ) can expressed as: ¼ B t þ X1 R 1 ;t;tþi E t½x a tþi ŠþX1 cov t ðm t;tþi ; x a tþi Þ; ð1þ i¼1 i¼1 where B t is book value at time t; E t is the expectation operator conditional on inormation at time t; R,t,t?i is the return on a risk-ree bond rom time t to t? i; x t is earnings at time t; x a t (= x t - (R,t-1,t - 1)B t-1 ) is abnormal earnings at time t ; and m t,t?i is the time t stochastic discount actor or cash lows expected at time (t? i). Equation (1) is the RIM valuation model extended to include dynamic stochastic discount (risk) actors as maniested in the covariance terms. In order to derive a parsimonious linear accounting valuation model that incorporates dynamic risk, we ollow Ohlson (1995) and make explicit assumptions about the dynamics o abnormal earnings and the dynamics o the stochastic discount actor. Speciically, we assume that abnormal earnings and other value relevant inormation (v t ) ollow the linear autoregressive dynamic orm: x a tþ1 ¼ xxa t þð1 xþx a L þ v t þ tþ1 ð2þ v tþ1 ¼ cv t þ u tþ1: ð3þ With a minor exception, these are the same dynamics as in Ohlson (1995). The original Ohlson model is set in a risk-neutral world so that the irm s cost o capital is equal to the risk-ree rate. I return on equity eventually equals the irm s cost o capital, long-run abnormal earnings will converge to zero. In our risk-averse world, the cost o capital is the risk-ree rate plus a risk premium so that, i the return on equity eventually converges to the cost o capital, then abnormal earnings will converge to some long-run equilibrium value above zero, which we denote x a L.Asa consequence, we assume that next period abnormal earnings are a weighted average o this years abnormal earnings and long-run abnormal earnings. For mathematical convenience, we assume that the error term u t?1 is idiosyncratic and uncorrelated with the stochastic discount actor and that the error term tþ1 is homoscedastic with variance r 2 x. 5 Both error terms are assumed to be mean zero. In addition, we assume a linear dynamic or the stochastic discount actor o the orm: m t;tþ1 ¼ R 1 ð1 r m;t e tþ1 Þ: ð4þ The error term e t?1 is assumed to have a mean o zero and a unit variance and to be (positively) correlated with tþ1. The term r m,t is key and represents the level o aggregate (systematic) risk in the economy. This ormulation also assumes that riskree rates are constant. Note that this dynamic satisies the general requirements o a 5 The latter assumption is or simplicity. Relaxing this condition would increase the number o variables in the stock price equation by one.

6 904 M. R. Lyle et al. discount actor, namely, that a discount actor must equal the inverse o the gross risk-ree rate in expectation and take on nonnegative values only (Cochrane 2001). 6 Even though this stochastic discount actor is airly simple, the assumed volatility structure is a close discrete-time analog o the popular SABR stochastic volatility model (Hagan et al. 2002) commonly used in option pricing. Additionally, i one assumes that shocks to the discount actor are driven by shocks to the market portolio, this discount actor can be shown to produce standard CAPM-type models or expected returns. 7 Appendix 1 derives this dynamic rigorously rom irst principles and shows its relation to CAPM. To keep the analysis tractable, we also assume that the level o risk in the economy ollows a random walk: r m;tþ1 ¼ r m;t þ n tþ1 ; ð5þ where n t?1 is a mean zero random variable independent o e t?1. 8 Substituting the abnormal earnings and the stochastic discount actor dynamics into Eq. (1) yields a valuation equation that is a linear unction o accounting variables, other inormation, and a dynamic aggregate risk adjustment actor, as shown ormally in the ollowing proposition. Proos or all propositions are ound in Appendix 2. Proposition 1 Assume that the abnormal earnings dynamics are given by Eqs. (2) and (3) and the stochastic discount actor dynamics are given by Eq. (4) and (5). The price o equity is given by: ¼ B t þ a 1 x a L þ a 2x a t þ a 3 v t k 1 r m;t ; ð6þ or, equivalently, by: ¼ c 1 x a L þ c 2B t þ c 3 x t þ c 4 D t þ c 5 E t ½x tþ1 Š k 1 r m;t ; ð7þ where R ð1 xþ a 1 ¼ ðr xþðr 1Þ 0; a x 2 ¼ ðr xþ 0; a R 3 ¼ ðr xþðr cþ [0; R r x q k 1 ¼ ðr xþðr 1Þ [0;q ¼ E R ð1 xþð1 cþ tð tþ1 e tþ1 Þ[0; c 1 ¼ ðr xþðr cþðr 1Þ 0; c 2 ¼ R ð1 xþð1 cþ ðr xþðr cþ 0; c R cx 3 ¼ ðr xþðr cþ 0; c 4 ¼ xcðr 1Þ ðr xþðr cþ 0; and c 5 ¼ a 3 [0: 6 Because our analysis is in discrete time, this speciic orm o discount actor may yield negative values. We assume that does not occur 7 This result obtains by assuming that the market portolio has stochastic risk. This is a straightorward extension o the basic derivations in Cochrane (2001), Chapter 9. 8 We have also solved or a model where risk ollows a mean-reverting process. The implications o our results do not change even when risk is assumed to mean-revert.

7 Firm undamentals 905 Equation (6) o Proposition 1 provides a parsimonious linear valuation equation similar to Ohlson s but with an additional economy-wide aggregate risk term (and an intercept term). Consistent with intuition, Eq. (6) shows that equity prices are positively related to irm undamentals (e.g., abnormal earnings) but inversely related to economy-wide risk. 9 Intuitively, when uncertainty is high, such as during the economic crisis, market values will be low relative to undamentals. This eect is magniied or those irms with high a 2 coeicients, that is, irms with highly persistent earnings, because persistence compounds systematic risk shocks. Ignoring the state o uncertainty in the economy by, say, discounting using an historical CAPM or some other historical actor model, could result in considerable model error relative to the market and oster the claim that a pricing anomaly has been discovered. Two observations regarding this valuation equation are worth highlighting. First, in a risk neutral world k 1 = 0, resulting in the original Ohlson (1995) model (assuming x a L = 0). Second, k 1 is increasing in the level o abnormal earnings volatility (r x ). The latter result is consistent with the notion that increased uncertainty about irm undamentals should reduce stock values (or increase costs o capital). However, there is a standard caveat. Even i abnormal earnings volatility is high, stock values are unaected to the extent that shocks to abnormal earnings are uncorrelated with shocks to the stochastic discount actor (q = 0); that is, abnormal earnings volatility matters in the pricing o equities only i it is systematic. While Proposition 1 provides the equity pricing equation, it is not immediately obvious how equity returns change over time. This issue is important i only because returns based equations are commonly used in empirical research to relate the relevance o accounting variables both to equity returns and to costs o capital (expected equity returns). Our next proposition describes the equity return dynamics and cost o capital dynamics, which are a consequence o valuation Eq. (6). 3.1 Equity returns and costs o capital By using a stochastic discount actor approach to valuation, we need not speciy the returns or expected returns dynamics exogenously. Rather, they maniest as a byproduct o the underlying assumptions that determine the valuation equation. The ollowing proposition provides the dynamics o both returns and expected returns (costs o capital). Proposition 2 Let R tþ1 ¼ þ1þd tþ1 denote the cum dividend equity return and Dr m;t the change in expected economy-wide (systematic) risk. Given the equity valuation Eq. (6), the return generating process satisies the dynamic: 9 The result that k 1 is positive presumes a positive correlation (q) between shocks to abnormal earnings ( tþ1 ) and shocks to the stochastic discount actor (e t?1 ). On average it is unlikely to be otherwise as long as growth in the economy and growth in irm level abnormal earnings are positively correlated. To complete the argument note that the discount actor represents the marginal rate o consumption in the economy or growth in the economy see Appendix 1. Thereore shocks to the discount rate actor are driven by shocks to aggregate consumption or shocks to aggregate growth, represented by e t?1 which, in turn, should be positively related to shocks to (abnormal) earnings tþ1.

8 906 M. R. Lyle et al. r m;t R tþ1 ¼ R þðr 1Þk 1 þð1þa 2 Þ tþ1 u tþ1 Dr m;t þ a 3 k 1 : ð8þ Furthermore, the cost o capital (expected return), l t?1, is given by: r m;t l tþ1 ¼ R þðr 1Þk 1 ð9þ ¼ R 1 þ R x a tþ1 cov R x ; m t;tþ1 : ð10þ Equation (8) oers considerable insight into the behavior o stock returns and their relation to costs o capital. Equation (9), the expectation o (8), shows that higher values o k 1 increase expected equity returns (cost o capital). However, irms with the highest cost o capital or, equivalently, the largest k 1, will experience the most dramatic movements in their stock price when there is a change in expected economy-wide risk. In particular, the last term in (8) says that irms with the highest expected returns will experience the largest downward price movements when aggregate risk increases in the economy, and vice versa when aggregate risk decreases in the economy. Thus, irms with the highest (negative) covariance with changes in aggregate risk in the economy are expected to have the highest average stock returns. This is exactly the result documented by Ang et al. (2006) in their Table 1; irms with the most negative loadings on changes in aggregate risk (as proxied by the VIX) have the highest uture stock returns. Moreover, while other variables (such as beta in the CAPM) have not had much success at predicting stock returns, Ang et al. (2006) provide evidence that changes in aggregate risk robustly predict stock returns, adding credence to the theoretical results we have presented. Equation (9) shows that the cost o capital (expected return) is a unction o the risk-ree rate, volatility o abnormal earnings, earnings persistence, and the level o risk in the economy. 10 Higher values o each o the latter variables increase the return demanded by risk-averse investors. Furthermore, simultaneous inspection o Eqs. (6) and (9) suggests that our model has the potential or cost o capital estimation in a manner that is likely to be o keen interest to accounting scholars. Speciically, the act that the covariance term k 1 r m,t is present in both the expected return Eq. (9) and in the equity price Eq. (6) provides an opportunity to substitute observable price and accounting variables or the unobservable k 1 r m,t in solving or expected returns. This insight is investigated in the next section o the paper. An open question in the accounting literature is the relation, i any, between accounting valuation models based on non-arbitrage pricing and clean surplus, such as the Ohlson (1995) model, and the accounting return decomposition models o Vuolteenaho (2002) and Callen and Segal (2004), which are based on the 10 Note that k 1 is the irm level driver o expected returns being a unction o the persistence o abnormal earnings, the volatility o abnormal earnings, and the correlation between (shocks to) abnormal earnings and (shocks to) economy-wide systematic risk in the economy.

9 Firm undamentals 907 Table 1 Summary statistics R t?1 Price t B t x t E t [x t?1 ] Dt St Size t bm t Panel A: Firm speciic summary statistics Mean Std Max Min R t? B t / x t / E t [x t?1 ]/ D t / Panel B: Correlation matrix R t? B t / x t / E t [x t?1 ]/ D t / Mean Std. Max Min Panel C: VIX summary statistics VIX t DVIX t Panel A, reports descriptive statistics or 524, irm-months (6,778 irms) rom 1980 to Price t denotes price per share, R t?1 monthly cum-dividend (gross) returns, B t book value per share, x t earnings beore extraordinary items, D t dividends per share, Size t the log o market capitalization and bm t the log book to market ratio. E[x t?1 ] denotes the IBES consensus orecast or one-year-ahead earnings computed as the time-weighted mean consensus analyst orecast o year t? 1 and year t? 2 earnings multiplied by common shares outstanding Panel B, provides Pearson (the upper triangle) and Spearman (the lower triangle) correlations or the variables used in multivariate regressions. Bold numbers represent signiicance at the 1 % level. Nonbolded values are insigniicant Panel C, provides descriptive time-series statistics or the VIX (VXO) contracts provided by the CBOE or years at the end o each month. The sample consists o 300 monthly observations. The contracts represent implied volatilities and are presented as annualized percentages. The values are obtained rom the implied volatilities o contracts written on the S&P 100 (OEX) index time-series properties o the book-to-market ratio and clean surplus. The return decomposition literature proves that returns can be decomposed into expected returns, shocks to current and uture cash lows, called cash low news, and shocks to uture expected returns, called discount rate news. The ollowing corollary o Proposition 2 shows that returns derived rom our non-arbitrage based model ollow a similar return decomposition. Corollary Consistent with the return decomposition literature, equity returns in Eq. (8) decompose into expected returns plus cash low news minus discount rate news where

10 908 M. R. Lyle et al. expected returns ¼ R þðr 1Þk 1 r m;t ; cash low news ¼ð1þa 2 Þ tþ1 Dr m;t discount rate news ¼ k 1 : þ a 3 u tþ1 ; As in the return decomposition literature, a positive shock to cash lows, measured by shocks to abnormal earnings and other inormation, increases equity returns, whereas a positive shock to expected returns drives down equity returns. This result cannot be demonstrated in standard accounting valuation models or which risk is constant over time. 3.2 Determinants o the cost o capital-accounting and irm undamentals In this section, we present one o the main indings o our paper, namely, that expected returns (costs o capital) can be expressed as a linear unction o accounting variables and other irm undamentals delated by price. Much o the work in inance has ocused on using covariances (such as beta) to measure expected returns. However, estimating these values has proven to be extremely diicult, and their success in predicting out o sample stock returns has been elusive despite extensive eorts by the literature (Daniel and Titman 1997; Subrahmanyam 2010). As opposed to ocusing on unobservable covariances as the driving orce o expected returns, our model allows us to substitute observable irm characteristics or unobservable covariances as in the next proposition. Proposition 3 The irm s costs o capital (expected returns) can be expressed as: x a L B t x t l tþ1 ¼ 1 þ g 1 þ g S 2 þ g t S 3 t or, equivalently, as: where l tþ1 ¼ 1 þ g 1 x a L þ g 0 2 g 1 ¼ R ð1 xþð1 cþ ðr xþðr cþ 0; g 3 ¼ ðr 1ÞR cx g 5 ¼ ðr xþðr cþ 0; g 4 ¼ þ g 4 E t ½x tþ1 Š B t x þ g 0 t E t ½x tþ1 Š 3 þ g S 4 t þ g 5 D t ; g 2 ¼ R ðr 1Þð1 xþð1 cþ ðr xþðr cþ ðr 1ÞR ðr xþðr cþ [ 0; ð11þ þ g 5 B t 1 ; ð12þ 0; ðr 1Þ 2 xc ðr xþðr cþ 0; g0 2 ¼ g 2 g 5 ; and g 0 3 ¼ g 3 g 5 0:

11 Firm undamentals 909 This proposition represents one o the key theoretical indings in our paper and has a number o important implications or measuring costs o capital. First, Eq. (11) shows that the cost o capital (expected return) can be expressed solely as a linear unction o irm undamentals: the book-to-market ratio, the earnings-price ratio, the orward earnings-price ratio, (the inverse o) size, and the dividend yield. Alternatively, urther eliminating dividends via clean surplus, Eq. (12) shows that the cost o capital can be expressed solely as a linear unction o accounting variables and size where the accounting variables include the current bookto-market ratio, the past book-to-market ratio, the earnings-price ratio, and the orward earnings-price ratio. These variables have been used in empirical research as predictors o expected returns see Subrahmanyam (2010), or example particularly size and the book-to-market ratio, which have generated signiicant attention since the empirical work o Fama and French (1992). Second, there are no betas or other covariance terms on the right-hand side o these equations to estimate. Firm undamentals alone determine costs o capital. Third, the result shows that accounting variables play a vital role in asset pricing and cost o capital measurement. Fourth, our theory not only give strong theoretical guidance or which speciic irm undamentals should be used to determine costs o capital but also how these irm undamentals are to be combined as joint determinants o expected returns. The next section tests the empirical validity o our model. 4 Empirical estimation Our main empirical ocus is on using irm undamentals and current equity prices to predict uture equity returns. We initially test whether equity returns derived rom our accounting-based model are signiicantly associated with uture realized returns in the cross-section. We then compare the equity returns orecasted by our model with the return orecasts rom standard benchmarking models. Finally, we calibrate the model to real world data and test to see how well the calibrated model perorms in predicting out o sample returns. In addition to returns, we also test whether the model can predict security prices by predicting out o sample stock prices relative to standard benchmarking models. 4.1 The sample Our sample consists o a large cross-section o publicly traded irms with December iscal year-ends rom 1980 to Firm undamentals are obtained rom the annual Xpresseed Compustat database. Analyst orecasts are obtained rom the IBES summary statistics database. Stock return data are collected rom CRSP. We restrict our sample to irms with positive book values, price per share greater than $5, and at least 2 years o consecutive data. As in Nekrasov and Shro (2009), we require that irms have 1- and 2-year-ahead analyst orecasts with a positive second year orecast, book-to-market ratios between 0.01 and 100, we also require expected

12 910 M. R. Lyle et al. earnings growth to be between 0 and 100 %. 11 Ater these restrictions, we are let with a total o 524, irm-months (6,778 irms) in the sample. In some o our analysis, we use VIX option contract data. These data are obtained rom the Chicago Board o Options Exchange (CBOE) website. The CBOE has two VIX contracts, one based on the S&P 100 (the OEX), which is the original VIX contract, and the new VIX, which is constructed rom options written on the S&P 500. Although the new S&P 500 VIX has replaced the OEX based version, data or the new VIX are available only since Data or the old VIX contracts, renamed the VXO, are available rom January We ollow previous research (e.g., Ang et al. 2006) and use the old VIX contracts in order to maximize the number o observations in our sample. When we use VIX option data our sample size reduces to 441,290 irm-months (6227 irms) observations. Table 1 provides general summary statistics. 4.2 Cross-sectional stock return tests Our initial set o tests is meant to determine how well our accounting-based cost o capital measure perorms in the cross-section. The empirical regression model ollows Eq. (11) and takes the regression orm: 12 R tþ1 1 ¼ a þ g 1 þ g 2 B t þ g 3 x t þ g 4 E t ½x tþ1 Š þ g 5 D t þ e tþ1 : ð13þ x t is measured by income beore extraordinary items and B t is book value. D t is measured by dividends paid to common shareholders. Expected uture earnings, E t [x t?1 ], is measured by the time-weighted mean consensus analyst orecast o year t? 1 and year t? 2 earnings multiplied by the number o common shares outstanding as per Compustat. 13 The delator,, is last periods stock price adjusted or stock spits multiplied by the number o shares outstanding in Compustat. From Proposition 3, we expect the coeicients or the inverse o size, book-to-market ratio, expected earnings-price ratio, and dividend yield (g 1, g 2, g 4 and g 5 )tobe positive and the coeicient on the earnings-price ratio, g 3, to be negative. 14 To test the model, we estimate monthly Fama MacBeth regressions o realized next-period equity returns on irm undamentals delated by stock price. We initially regress returns on each explanatory variable separately and then on all o the explanatory variables simultaneously as per Eq. (13). The mean cross-sectional 11 The restriction on earnings growth does not aect any o our results, i anything our results are stronger when we remove this restriction. 12 We preer to use equation (11), which includes the dividend yield, rather than Eq. (12) because o the high correlation between current and past book-to-market ratios. We reer to Eq. (11) as the accountingbased model although size and the dividend yield are not accounting variables. 13 Particularly, we measure next period expected earning by multiplying 1 year ahead concensus IBES earnings by w d and 2 year ahead concensus IBES earnings by 1 - w d where w d is the dierence between the irms icscal year end date and the current orecast date divided by We assume that long-run abnormal earnings are cross-sectionally constant, consistent with the notion that, in the long-run, a irm will grow to the point where it resembles a cross-section o irms.

13 Firm undamentals 911 Table 2 Cross-sectional return regressions Regression model: R tþ1 1 ¼ Intercept þ g 1 Bt xt Et½xtþ1Š Dt þ g 2 þ g 3 þ g 4 þ g 5 St Intercept 0.012*** 0.007** 0.014*** 0.009** 0.014*** (3.42) (1.97) (3.95) (2.53) (3.57) (1.53) -1 St 0.933*** 0.670*** (6.90) (5.76) B t / 0.012*** 0.007*** (5.68) (3.98) x t / *** (-0.19) (-3.11) E t [x t?1 ]/ 0.061*** 0.034** (3.33) (2.03) D t / (0.85) (0.09) adj-r This table reports mean coeicients and t-statistics rom Fama and MacBeth (1973) cross-sectional regressions o one-month-ahead excess equity returns on the variables shown. The sample consists o 524,091 irm-months rom years 1980 to denotes split adjusted price per share multiplied by common shares outstanding, B t book value, x t earnings beore extraordinary items, and D t is dividends. E[x t?1 ] denotes the IBES consensus orecast or one-year-ahead earnings computed as the time-weighted mean consensus analyst orecast o year t? 1 and year t? 2 earnings multiplied by common shares outstanding. The t-statistics are calculated rom Fama MacBeth standard errors. ** and *** denote twotailed statistical signiicance at the 5 and 1 % signiicance levels, respectively St St St coeicients and Fama MacBeth t-statistics are presented in Table 2. Moving rom let to right across the table, the three most important irm undamental variables or determining expected returns are the inverse o size, the book-to-market ratio, and the uture earnings-price ratio. Importantly, ocusing on the last column o the table in which all model-driven variables are included in the regression, the signs o all o the coeicients match the theoretical predictions o the model although the dividend yield is insigniicant Comparison with standard covariance risk proxies In this section, we explore how our returns model compares with benchmark CAPM and Fama French (FF) three actor models. We ollow standard estimation procedures. The irm-speciic CAPM betas and FF betas are estimated using 5-year rolling windows. 15 Betas are updated every April. We estimate monthly Fama MacBeth regressions o realized next-period equity returns on irm betas. We also include CAPM betas and FF betas in our accounting-based model to determine i these covariance variables subsume the explanatory power o irm undamentals in explaining stock returns. Table 3 presents the results. The only beta to prove 15 The results in this and the ollowing tables are not sensitive to the size o the estimation window.

14 912 M. R. Lyle et al. Table 3 Cross-sectional return regressions with covariance risk actors Regression model: R tþ1 1 ¼ Intercept þ g 1 Bt xt Et½xtþ1Š Dt þ g 2 þ g 3 þ g 4 þ g 5 þ P St i b i St Intercept *** 0.011*** (1.53) (4.38) (4.55) (0.92) (0.96) -1 St 0.670*** 0.683*** 0.659*** (5.76) (5.90) (6.03) B t / 0.007*** 0.007*** 0.007*** (3.98) (4.15) (3.95) x t / *** *** *** (-3.11) (-3.11) (-3.14) E t [x t?1 ]/ 0.034** 0.033** 0.032** (2.03) (2.10) (2.15) D t / (0.09) (0.26) (0.60) b (1.21) (1.30) b m (1.05) (1.44) b h (0.86) (-0.18) b s 0.003*** (2.88) (1.41) adj-r This table reports mean coeicients and t-statistics rom Fama and MacBeth (1973) cross-sectional regressions o one-month-ahead excess equity returns on the variables shown. The sample consists o 524,091 irm-months rom years 1980 to denotes split adjusted price per share multiplied by common shares outstanding, B t book value, x t earnings beore extraordinary items, and D t is dividends. E[x t?1 ] denotes the IBES consensus orecast or one-year-ahead earnings computed as the time-weighted mean consensus analyst orecast o year t? 1 and year t? 2 earnings multiplied by common shares outstanding. b denotes the irm-speciic beta rom the CAPM. b m, b h, and b s denote irm-speciic betas obtained rom the Fama and French (1993) three-actor model where the betas represent the irm speciic slopes on the market portolio, the high book-to-market over low book-to-market portolio (SMB), and the portolio consisting o small irms over large irms (HML), respectively. The t-statistics are calculated rom Fama MacBeth standard errors. ** and *** denote two-tailed statistical signiicance at the 5 and 1 % signiicance levels, respectively St St signiicant is the size beta in the third regression. All other betas are insigniicant. When we include the betas together with our accounting model-driven variables, the results are basically unchanged. Firm undamentals carry signiicant associations with realized stock returns, whereas the betas do not Generating cost o capital estimates In the previous section, we evaluated whether the variables in our theoretical model are associated with uture stock returns. However inding an association does not necessarily imply that the equation actually predicts stock returns in the

15 Firm undamentals 913 Table 4 Cross-sectional cost o capital tests l acct l l capm r t?1 Panel A: Summary statistics Mean Std Max Min l acct l l capm Coe. t-stat Coe. t-stat Coe. t-stat Panel B: Cross-sectional regressions o realized returns on costs o capital l 1.181*** (6.88) (1.14) (1.35) Intercept (1.06) 1.177*** (3.07) (1.53) adj-r Panel A, reports summary statistics or model-determined expected return (cost o capital) orecasts and realized net equity returns. l acct denotes the accounting-based expected returns [Eq. (13)], l the Fama French three-actor expected returns, and l capm the CAPM expected returns. r t?1 = R t?1-1 denotes net realized equity returns Panel B, reports mean coeicients and t-statistics rom Fama and MacBeth (1973) cross-sectional regressions o realized returns on each o three dierent costs o equity capital estimates. The sample consists o 282,810 irm-month observations rom years 1990 to The t-statistics are calculated rom Fama MacBeth standard errors. *** denotes two-tailed statistical signiicance at the 1 % signiicance level cross-section. Rather, i the model is a good proxy or expected returns, we should ind a strong association between ex ante (expected) and ex post (average) stock returns. Following Lewellen (2011), we calibrate our model (equation 13) using 10-year (one hundred and twenty month) rolling Fama MacBeth coeicients (with a minimum o 5 years data) along with current irm undamentals and stock prices to predict next period s out o sample equity returns. The methodology insures that only historical data are used to orm expectations about uture stock returns. Returns are predicted rom 1985 to Table 4, Panel A, provides summary statistics or realized stock returns and or expected returns derived rom our accounting-based model, the CAPM, and FF three-actor model. Results are again shown or 5-year rolling betas. To test the association o each o the expected return orecasts with realized equity returns, we regress monthly realized returns on each o the cost o capital orecasts separately as ollows: R tþ1 1 ¼ A 1 þ A 2 l t;tþ1 þ w tþ1 : ð14þ l t,t?1 denotes period (t? 1) expected return orecasts conditioned on inormation at time t. IA 1 = 0 and A 2 = 1, then the null that l t,t?1 represents the true expected equity return (cost o capital) cannot be rejected. However, veriying that A 2 = 0is oten regarded by the literature as suicient evidence that l t,t?1 is a good proxy or the true expected return

16 914 M. R. Lyle et al. Table 4, Panel B, shows that expected returns based on irm undamentals (l acct ) as per Eq. (13) are strongly associated with realized returns. In particular, Fama MacBeth regressions o Eq. (14) yield an insigniicant intercept and a highly signiicant slope coeicient that is statistically indistinguishable rom one. By contrast, cost o capital estimates using the CAPM (l capm ) and the FF three actor model (l ) have virtually no ability to predict realized equity returns. These results imply that our model, which is driven by irm undamentals, carries considerable inormation about uture realized returns in contrast to historical covariance estimates Portolio returns The cross-sectional irm-level results in the prior section indicate a statistically signiicant association between realized stock returns and expected returns derived rom irm undamentals. To minimize potential concerns associated with measurement error at the irm level, we next turn to a portolio analysis. We sort irms into Table 5 Portolio-based time-series tests E t [r t?1 l acct ] r tþ1 E t [r t?1 l ] r tþ1 E t [r t?1 l capm ] r tþ1 Panel A: Portolio-based returns p p p p p p 5 - p 1 1.5*** t-stat l acct l l capm Panel B: Portolio-based alphas p p p p p p 5 - p *** t-stat Panel A, reports monthly average excess returns in percentages or ive equally weighted quintile portolios, p 1 to p 5, sorted by (expected return) cost o capital measures. r t?1 = R t?1-1 denotes net realized returns.e t [r t?1 l ] denotes net expected stock returns conditional on the cost o capital model, and r tþ1 the average realized net return. l acct denotes the accounting-based expected return [Eq. (13)], l the Fama French three actor expected return, and l capm the CAPM expected return Panel B, reports portolio-adjusted excess returns (alphas) or each quintile based on the Fama and French (1993) three-actor model. t-statistics are based on heteroscedasticity consistent standard errors. *** denotes two-tailed statistical signiicance at the 1 % signiicance level

17 Firm undamentals 915 expected return equally weighted quintile portolios and then determine whether an investor who allocated resources based on our expected return accounting model would have realized gains. Table 5, Panel A, shows raw portolio returns rom investing in the three proxies or expected returns. The irst column lists expected next-period portolio returns derived rom our accounting-based model. The second column lists the realized next-period portolio returns. Columns 3 through 6 show similar results based on expected return quintile portolios derived rom the FF three-actor model and the CAPM. Average realized returns are monotonically increasing or all expected return metrics. The last row in the table shows the results o a long-short strategy; that is, buying the highest expected return quintile and shorting the lowest expected return quintile. Only our irm undamentals expected return model yields signiicant positive returns. To determine whether our theoretically derived measure o expected returns is subsumed by standard risk actors, we regress each o the portolio returns on the FF three actors. Table 5, Panel B. shows the resulting alphas; that is, the abnormal hedged returns earned by investing based on each o the expected return (cost o capital) metrics. Both o the covariance-based expected return measures generate insigniicant abnormal returns. In contrast, our expected return measure generates an economically signiicant 1.18 % abnormal return per month. This test is important because it illustrates that our metric o (rational) expected returns is not anomalous. Instead, the excess returns are generated because o risk rather than, say, lack o attention by investors or some behavioral bias. Yet traditional asset pricing models would likely classiy this excess return as anomalous. Our results are consistent with Penman and Zhu (2011), who criticize the tendency to prematurely classiy high realized returns as anomalous. 4.3 Stock price estimation The prior analysis ocused on exploring whether our undamentals-based returns equation yields a reasonable proxy or the cost o equity capital. In this section we explore whether our model does a good job o predicting equity values. First, we explore the cross-sectional explanatory power o the model. To overcome the problem that the primary variable o interest in our model, namely, economy-wide risk, is cross-sectionally constant, we estimate the sensitivity to economy-wide risk instead using a two-pass regression approach. In the irst-pass, we regress next period s excess return on the VIX (and the change in the VIX) using irm-level timeseries data. The regression coeicient measures the sensitivity o the irm s returns to economy-wide risk where the VIX is our empirical proxy or expected economywide (systematic) risk. In the second pass cross-sectional regression, we regress excess returns on the estimated sensitivity to aggregate risk coeicients rom the irst-pass regressions and on the FF actor betas over the same window as or the time-series estimation. This two-pass type estimation procedure is common to the asset pricing literature (see Core et al. 2008, or example). The details o this estimation procedure ollow.

18 916 M. R. Lyle et al Estimating sensitivity to aggregate risk Equation (8) o Proposition 2 links equity returns to both the level and changes in expected economy-wide risk. We re-arrange this equation into the orm: R tþ1 R ¼ðR 1Þk 1 r m;t k 1 Dr m;t þ e r;tþ1 ; ð15þ where e r,t?1 is a zero mean error term that contains the cash low shocks. We urther re-arrange Eq. (15) into the empirical regression orm: R tþ1 R ¼ k 0 þ k 1 ðr 1Þ r m;t Dr m;t þ e r;tþ1 : ð16þ Because economy-wide risk is not observable, we use the CBOE VIX contract as a proxy. Thus, to estimate k 1, we use the ollowing empirical model: R tþ1 R ¼ k 0 þ k 1 ðr 1Þ VIX t DVIX t þ e r;tþ1 : ð17þ We limit our sample to irms with at least 120 trading days o time-series data using daily CRSP stock returns, CBOE daily VIX contract data, and risk-ree rates obtained rom Ken French s website. To control or the well-known microstructure issues that arise rom using daily data, we ollow prior research (see Dimson 1979 and Bali et al. 2009, or example) and include the lagged independent variable in the regression. 16 The estimate o k 1 is the sum o the coeicients on the contemporaneous and lagged independent variables, denoted by ^k 1. We then incorporate ^k 1 as an additional variable in the second-pass cross-sectional regression representing a irm-level metric o sensitivity to risk in the economy. A similar two-pass regression approach is ollowed to estimate each o the market beta and the FF three-actor betas over the same time period. Summary statistics or annual estimates o ^k 1, irm betas, and FF three-actor betas are provided in Table 6, Panel A. The average sensitivity to changes in the VIX is 0.16, consistent with the sign predicted by our theoretical derivation. Multiplying the latter by the VIX contract value delated by price per share yields an average value o 1.55, implying an average risk premium o around 1.55 o the riskree rate. Given that risk-ree rates rom Ken French s website averaged around 4.8 % over the sample period, our model implies an average risk premium over the sample period o approximately 7.4 % and an average gross return o 12.2 %. Since the actual gross return averaged around 17 % over the sample period, our empirical model appears to underestimate average returns Does sensitivity to aggregate risk predict stock returns? Ang et al. (2006) show that sensitivity to economy-wide risk is associated with equity returns. Nevertheless, in order to determine i sensitivity to economy-wide 16 Letting Z t ¼ðR 1Þ VIXt St DVIXt ; we also include Z St t-1 in the regression.

19 Firm undamentals 917 Table 6 Time-series risk estimates and association with uture returns ^k 1 b k VIXt St b b m b h b s Panel A: Summary statistics Mean Std Max Min Panel B: Cross-sectional return regressions c k1 = *** (8.01) b 0.39* (1.94) b m 0.42** (1.98) b h (-0.07) b s 0.07 (0.33) adj-r Panel A, reports summary statistics based on time-series estimates o risk actors. The sample consists o 441,290 irm-months rom April 1986 to May ^k 1 denotes the irm speciic sensitivity to aggregate risk, and b the irm speciic beta rom the CAPM. b m, b h, and b s denote irm speciic betas rom the Fama and French (1993) three-actor model, where the betas represent the irm speciic slopes on the market portolio, the high book-to-market over low book-to-market portolio (SMB), and the portolio consisting o small irms over large irms (HML), respectively Table 6, Panel B, reports mean coeicients and t-statistics rom Fama and MacBeth (1973) crosssectional regressions o one-month-ahead excess equity returns on estimated risk actors. The sample consists o 441,290 irm-months rom April 1986 to May The t-statistics are calculated rom Fama MacBeth standard errors. *, ** and *** denote two-tailed statistical signiicance at the 10, 5, and 1 % signiicance levels, respectively risk is a reasonable risk actor or our sample, we conduct an asset pricing test and regress stock returns on this estimated risk actor (delated by price). We urther compare our sensitivity to economy-wide risk actor with the CAPM and with Fama French risk actors. Speciically, we regress separately monthly excess equity returns on the sensitivity to economy-wide risk actor (^k 1 ) and on the FF actor betas to determine which variables predict stock returns in the cross-section. 17 Table 6, Panel B, presents the results. The sensitivity to economy-wide risk delated by price is highly signiicant suggesting that ^k 1 proxies or priced risk. Our risk actor has a much higher level o statistical signiicance by comparison to the market beta. The beta coeicients or the HML and SMB actors are insigniicant. 17 The irm-level ^k 1 ; estimated market beta, and estimated FF actor betas are updated annually each April.