Accounting Valuation and Cost of Equity Capital Dynamics

Current theories of accounting valuation are based primarily on Ohlson (1995) or Vuolteenaho (2002). 1Accounting is incorporated in these models via the clean surplus relation. 2Whereas the Ohlson (1995) model is generated from standard non-arbitrage conditions, the Vuolteenaho (2002) model is derived from assumptions about the time series properties (such as stationarity) of the book-to-market (B/M) ratio. As has been shown recently by Lyle et al. (2013), these two models are conceptually related to each other despite their disparate technical underpinnings.

Both models have been extended in different ways. Feltham and Ohlson (1995, 1996) generalize the basic Ohlson (1995) model to include (unconditional) accounting conservatism by decomposing the clean surplus relation into operating and financial asset components. Feltham and Ohlson (1999) incorporate dynamic risk aversion into accounting valuation. Gode and Ohlson (2004) allow for dynamic risk-free rates of interest. Lyle et al. (2013) extend Feltham-Ohlson (1999) to incorporate aggregate risk and to generate time-varying dynamic costs of capital (discount rates). Callen and Segal (2004) extend the Vuolteenaho model utilizing the Feltham-Ohlson clean surplus decomposition.

The purpose of this paper is to selectively review accounting valuation and interrelated cost of capital models focusing on the dynamics that underlie these models and on the dynamics that are consequences of these models.

Theory Of Ohlson-Type Models

Risk Neutrality

The Residual Income Valuation Model (RIV), a precursor to Ohlson (1995), assumes (i) risk neutrality, (ii) non-arbitrage, (iii) clean surplus accounting, (iv) a transversality condition, (v) perfect capital markets, and (vi) an exogenous risk-free interest rate. The RIV implies that the value of the firm (S_t) is:

(1)

where E_t denotes the expectation operator conditioned on information available at time t, B_t book value of equity, Rf one plus the risk-free rate of interest, and abnormal earnings. 3

The Ohlson (1995) linear valuation model imposes on RIV the additional assumption that abnormal earnings follow an AR(1) dynamic. The Ohlson model also allows for an ‘other value-relevant’ variable that is also assumed to follow an AR(1) dynamic. Under these assumptions the value of the firm can be shown to be a linear function of book value plus a multiple of abnormal earnings plus a multiple of the other value-relevant variable. To simplify notation and with little loss of generality, I disregard other value-relevant variables in the theoretical discussion that follows. More formally, given the AR(1) dynamic for abnormal earnings:

(2)

Ohlson (1995) shows that the equity value of the firm equals:

(3)

where ω is the abnormal earnings persistence parameter assumed to lie between 0 and 1, and ε_t is a mean-zero error term. It is worth noting that the Ohlson (1995) linear valuation model incorporates uncertainty only through the ε_t+1 error term in the abnormal earnings dynamic.

Ohlson (1995) further shows that the cum-dividend equity return (r_t) that is a consequence of equation 3 necessarily takes on the simple dynamic:

(4)

The latter relation implies that equity returns are comprised of the risk-free rate plus a multiple of the shock to the (abnormal) earnings-to-price (E/P) ratio. Equation 4 also indicates, not surprisingly, that the more persistent are abnormal earnings, the greater the impact of any earnings shock on returns.

Taking expectations, equation 4 further implies that the firm's cost of equity (the expected return) is the constant risk-free rate of interest. The latter result is hardly surprising considering that investors are risk-neutral and the risk-free rate is an exogenous constant by assumption. It is well worth noting that many empirical applications of the Ohlson (1995) model replace the risk-free rate by some cost of capital measure usually based on the CAPM or the Fama-French three- or four-factor models. In addition to the fact that the CAPM (or Fama-French) and the Ohlson (1995) model are not generally consistent with each other (Morel, 2003), the risk neutrality assumption implies that it is conceptually incorrect to measure the cost of capital in an Ohlson (1995) framework other than by the risk-free rate, as follows directly from equation 4. 4

There are many ways to generalize the Ohlson (1995) model. One obvious generalization (see Callen and Morel, 2001) is to assume that abnormal earnings follow an AR(q) process (q > 1) of the form:

(5)

The valuation equation can then be shown to take the form:

(6)

where the γ_k parameters are nonlinear functions of the risk-free rate of interest and the ωs. In this model too, the firm's cost of capital is the risk-free rate. This model shows that changes to the abnormal earnings dynamic have valuation consequences. 5Although hardly surprising, at least one influential empirical study appears to be unaware of this (Dechow et al., 1999).

Risk Aversion

An especially fruitful avenue of generalization is to allow for dynamic risk aversion in accounting valuation. Feltham and Ohlson (1999) extend the RIV to include dynamic risk aversion. In particular, they show that the value of the firm's equity can expressed as:

(7)

where cov_t is the time t conditional covariance and is the time t stochastic discount factor for cash flows expected at time (t+ τ). 6Equation 7 is the RIV extended to include dynamic stochastic discount factors as reflected in the dynamic conditional covariance terms.

Lyle et al. (2013) incorporate an extended system of dynamics in the Feltham-Ohlson (1999) RIV, conceptually similar to Ohlson's (1995) extension of the standard RIV. In addition to yielding a closed-form linear solution that is amenable to empirical estimation, their dynamic risk structure and empirical results are consistent with the extensive empirical evidence in the accounting and finance literatures that costs of capital (expected returns) are time varying. 7More specifically, Lyle et al. (2013) assume that abnormal earnings follow the linear autoregressive dynamic form:

(8)

so that the next period's abnormal earnings are a weighted average of this year's abnormal earnings and long-run abnormal earnings denoted by . This is essentially the same dynamic as in Ohlson (1995). 8However, the original Ohlson model is set in a risk-neutral world so that the firm's cost of capital is equal to the risk-free rate. If return on equity eventually converges to the firm's cost of capital because of competition, long-run abnormal earnings will converge to zero. By contrast, in a risk-averse world, the cost of capital is the risk-free rate plus a risk premium so that, if return on equity eventually converges to the firm's cost of capital, then abnormal earnings will converge to some long-run equilibrium value above zero. Hence, the long-run abnormal earnings variable is included in the dynamic.

In addition to their abnormal earnings dynamic, Lyle et al. (2013) also incorporate a linear dynamic for the stochastic discount factor of the form: 9

(9)

The error term e_t+1 is assumed to have a mean of zero and a unit variance and to be (positively) correlated with ∈ _t+1. 10The term σ_m,t is key in their analysis and represents the level of aggregate (systematic) risk in the economy. This formulation also assumes that risk-free rates are constant. Note that this dynamic is assumed to satisfy the general requirements of a stochastic discount factor, namely, that a stochastic discount factor must equal the inverse of the gross risk-free rate in expectation and take on non-negative values only (Cochrane, 2011). 11Although the form of this stochastic discount factor is fairly simple, the assumed volatility structure is a close discrete-time analog of the popular SABR stochastic volatility model commonly used in option pricing (Hagan et al., 2002). 12

Given this set of dynamics, Lyle et al. (2013) show that the price of equity is given by:

(10)

where α₁, α₂, λ₁ ≥ 0. The latter equation provides a parsimonious linear valuation equation similar to Ohlson's but with an additional economy-wide aggregate risk term. Consistent with intuition, this equation shows that equity values are positively related to firm fundamentals (e.g., abnormal earnings) but inversely related to economy-wide systematic risk. Intuitively, when uncertainty is high, such as during the 2007–08 economic crisis, equity values will be low relative to fundamentals. This effect is magnified for firms with highly persistent earnings, because persistence compounds systematic risk shocks. Ignoring the state of uncertainty in the economy by, say, discounting using an historical CAPM or some implied cost of capital model, could result in considerable model error relative to the market and foster the claim that a pricing anomaly has been discovered.

Not surprisingly, the dynamics of the valuation equation drive the dynamics of firm returns. Given their equity valuation equation above, Lyle et al. (2013) show that (cum-dividend) returns necessarily satisfy the dynamic:

(11)

where Δσ_m,t is the shock to economy-wide (systematic) risk. As a result, the firm's cost of capital (expected return), E_t(r_t+1), differs from the risk-free rate and is given by: 13

(12)

Importantly, Lyle et al. (2013) show that the firm's cost of capital can be re-expressed as a linear function of firm fundamentals: 14

(13)

where η₁, η₂, η₄ ≥ 0 and η₃ > 0; that is, the firm's cost of capital can be expressed as a non-negative linear function of its long-run E/P ratio, the B/M ratio, contemporaneous E/P ratio, and dividend yield. Whereas it has long been hypothesized in the finance literature that the firm's cost of capital is a function of firm fundamentals (e.g., Daniel and Titman, 1997), this is the first paper to show this on theoretical grounds. In addition, equation 13 provides a sorely lacking theoretical justification for the well-known Fama-French size and the B/M ratio risk factors.

Ohlson-type Model with Imperfect Parameter Information

Typically, the accounting valuation literature presupposes that parameters are perfectly observable to investors. Suppose, however, that investors have only imperfect information about the firm's long-run earnings ( ) and they learn about long-run earnings over time. Accounting valuation models that involve learning were analyzed by Pastor and Veronesi (2003) in a continuous time framework and more recently by Lyle (2013) in a discrete time framework. The discussion that follows is based on an Ohlsonian version of the Lyle (2013) model. 15

Consider an Ohlson-type model in the spirit of Lyle et al. (2013) where initially investors have perfect information about the model parameters. Specifically, assume the dynamics:

(14)

(15)

Here the abnormal earnings dynamic is a function of long-run abnormal earnings as well as an explicit function of both systematic risk ∈ _S,t+1 and idiosyncratic risk ∈ _I,t+1. The latter random variables are assumed to be normally and independently distributed with mean zero and variance of 1. In contrast to Lyle et al. (2013), the volatility parameter (σ_m) in the stochastic discount factor is assumed to be time-independent for simplicity.

Now assume that investors cannot perfectly observe long-run abnormal earnings. Let denote investors’ conditional belief at time t about long-run abnormal earnings. Investors are assumed to learn about long-run earnings over time by observing the time series of past abnormal earnings (which imperfectly incorporate long-term earnings) and the stochastic discount factor. 16In particular, investors optimally update past information about long-run abnormal earnings using a Bayesian updating mechanism as encapsulated in a Kalman filter (see Liptser and Shiryaev, 1977).

It is shown in the Appendix that the evolution of investors’ beliefs about expected long-run abnormal earnings, y_t, takes the form: 17

(16)

where ν_t denotes the conditional variance of y_t. The evolution of the conditional variance is shown in the Appendix to take the form:

(17)

(18)

where . Solving equation 18 forward yields:

(19)

where ν₀ is conditional variance of y₀.

The Appendix also shows the updated abnormal earnings dynamic takes the form:

(20)

Equation 20 indicates that investors’ beliefs about future abnormal earnings with learning is more volatile than without learning as . Solving for firm value with imperfect information—see the Appendix—yields:

(21)

The valuation equation with imperfect information about long-run abnormal earnings is almost identical to the valuation with perfect information except that now firm value is a function of investors’ conditional expectations of long-term earnings (y_t).

Although imperfect information about long-run earnings in a learning environment has little effect on the form of the valuation equation, it has a marked impact on returns. It is shown in the Appendix that cum-dividend returns (r_t) under imperfect information about long-run earnings take the form: 18

(22)

In contrast to the perfect information case, returns are now an explicit function of idiosyncratic risk where the coefficient on idiosyncratic risk is a nonlinear but decreasing function of the firm's age t and increasing function of abnormal earnings persistence.

Although learning about the firm's long-run abnormal earnings affects returns, it has no impact on the firm's cost of capital (expected returns). Indeed, taking expectations of equation 22 yields the same cost of capital as in the case of perfect information, namely:

(23)

This result appears counter-intuitive in light of the theoretical literature on parameter uncertainty and estimation risk, which maintains that a reduction in parameter uncertainty reduces the firm's cost of capital (Bawa et al., 1979; Barry and Brown, 1984). However, the result follows from the fact that the cost of capital in this model is solely a function of systematic risk and investors learn all that they can about systematic risk by observing the stochastic discount factor. Thus, the assumption that costs of capital necessarily fall with a reduction in investor uncertainty about valuation parameters is not correct in general.

The Term Structure of Costs of Equity Capital

Should cash flows that the firm expects to receive one year from now be discounted at the same cost of capital as cash flows expected to be received in five years or in 10 years? What is the relation between the firm's costs of equity for different time periods? In other words, what does the term structure of the firm's costs of capital look like? These issues are addressed by Callen and Lyle (2013). To derive a term structure of implied costs of equity, they borrow from the multi-period valuation model of Ang and Liu (2004) in which costs of capital are allowed to vary by period. Ang and Liu (2004) show that firm value can be expressed as:

(24)

Here each period's expected dividend is discounted at a potentially different discount rate (μ_t,t+T), yielding a term structure of discount rates.

Given the latter equation, Callen and Lyle (2013) prove that costs of equity capital (discount rates) for period t to t+T can be expressed recursively as:

(25)

To solve for any μ_t,t+T, T > 1, one must first solve for μ_t,t+1, which can then be used to solve for μ_t,t+2 and so on up to μ_t,t+T. This method of successive substitution is conceptually similar to deriving a term structure of bond yields recursively using the price of a one-period pure discount bond to solve for a one-period yield and then using that yield together with the price of a two-period pure discount bond to solve for the second-period yield and so on.

Unfortunately, this recursive method cannot readily be applied to market data because it depends upon expectations of future equity values. To finesse this issue, Callen and Lyle (2013) express unobservable expected future equity prices E_t[S_t+T] by the prices of observable traded instruments. Specifically, they denote the value of a forward contract at time t defined on the market value of the firm's equity to be paid out at time t + T by F_t,t+T. In the absence of arbitrage opportunities, the value of the forward contract is given by:

(26)

so that

(27)

where Rf_t,t+T is the time t discount rate for riskless cash flows to be received at time t+T (or the return from holding a risk-free bond from time t to t+T), assumed for simplicity to be known at time t, and m_t,t+T is the stochastic discount factor at time t for cash flows to be received at time t+T.

Equation 27 cannot be used to estimate the term structure of implied costs of equity empirically without specifying the discount factor's functional form. Callen and Lyle elect to specify the same parsimonious stochastic discount factor form as used to derive the Conditional Capital Asset Pricing Model (CCAPM) (Cochrane, 2005). In this case, the stochastic factor discount factor can be replaced by a linear function of the return on the market portfolio so that the theoretical recursive relation 25 can be replaced by the estimable recursive relation :

where denotes the return on the market portfolio from period t to t+T, is the conditional correlation between the return on the market portfolio and the return on firm equity over the period t to t+T, is the conditional volatility of the return on the market portfolio from time t to t+T, and is the conditional volatility of the return on the firm's equity from time t to t+T. Hence, estimating a term structure of implied costs of equity from equation (28) requires estimates of equity futures prices, future risk-free rates, expected future dividends, future volatilities of the return of the market portfolio and of firm-level equity returns, and future correlations between the return on the market portfolio and equity returns.

In the absence of equity futures markets, Callen and Lyle (2013) estimate futures prices synthetically using traded options and the put-call price parity relation:

(29)

where C denotes the call price, P the put price, and both instruments have the same strike price K and expiry date, t+T. R_t+ T is the gross risk-free rate at time t+ T. Assuming that the price of the put, the price of the call, and the t+ T risk-free interest rate are observable and known at time t, then the price of the synthetic futures contract is also known and non-stochastic at time t. One potential problem with this approach is that put-call price parity holds strictly only for European options, whereas available firm-level data are of the American option variety. To address this concern, following Bakshi et al. (2003), Callen and Lyle (2013) first compute synthetic European option prices from American options data. Specifically, implied volatilities derived from American option prices for each maturity t+ T (controlling for dividends and early exercise premia) are substituted into a standard Black-Scholes equation to generate synthetic European call and put option prices. These synthetic European option prices are substituted in turn into equation 29 to obtain the synthetic futures price.

Implied volatilities, strike prices, (future) risk-free rates, and times to expiry are obtained from the OptionMetrics implied volatility surface data set (Yan, 2011) and their coupon bond file. Implied volatilities estimated from option contracts traded on the S&P 500 Index proxy for expected market volatilities. Correlations between the firm-equity returns and the return on the market portfolio (the S&P 500 Index) are estimated using rolling 36 months of observations. 19

Dynamics and the Empirical Ohlson Literature

Many empirical papers make reference to and/or apply Ohlson-type valuation models. Generally speaking, these papers assume the same information dynamics as in Ohlson (1995) or Feltham and Ohlson (1995, 1996). However, empirical evidence suggests that the simple Ohslon (1995) and Feltham-Ohlson (1995) dynamics are not supported by the data. Bar-Yosef et al. (1987) find that their data are better described by a multiple lagged information dynamic than a single lag information dynamic. Morel (1999) finds that the (optimal) lag structures of the Ohlson (1995) information dynamic and the pricing relation do not yield consistent results.

Often, empirical papers that reference Ohlson-type models incorporate ‘other value-relevant’ variables in their valuation equation. The choice of these latter variables is necessarily ad hoc given that the model itself fails to specify which other variables are in fact value-relevant. However, Liu and Ohlson (2000) show how to finesse the ‘other value-relevant’ variables but at the cost of adding more expectational variables to the valuation equation. Callen and Segal (2005) use their technique to evaluate the Ohlson (1995) and Feltham-Ohlson (1995) valuation models empirically. They find that the latter models neither load properly nor predict particularly well out of sample. In fact, they seem to predict no better than a simple (forward) earnings per share model.

Three papers to date have addressed empirically the issue of accounting valuation when investors are risk-averse. Two of the latter have further addressed the issue of time-varying discount rates. Nekrasov and Shroff (2009) estimate the Feltham and Ohlson (1999) RIV valuation equation (7 above) using an historically-based covariance risk measure. Comparing value estimates based on the Feltham-Ohlson RIV with the CAPM and the Fama-French three-factor model, they find that their RIV yields smaller out-of-sample pricing errors.

Lyle et al. (2013) test both their valuation model and their related cost of capital dynamics. Comparing their valuation model with Ohslon (1995), the CAPM, and the Fama-French three-factor models, they find that their model yields significantly smaller prediction errors compared to the latter three benchmark models. They also find that the prediction errors of their model are significantly smaller than the prediction errors reported by Nekrasov and Shroff (2009) when comparing their Feltham-Ohlson RIV with the CAPM and Fama-French factor models.

Lyle et al. (2013) also evaluate the dynamic cost of equity capital estimates derived from their valuation model. They find that their costs of equity estimates load on firm fundamentals in the direction predicted by their theory. They further test their dynamic cost of equity estimates by comparison to future realized returns in accordance with standard asset pricing tests. In a cross-sectional analysis, they find that their one-month ahead costs of capital are highly significantly related to future realized monthly returns in contrast to monthly expected return estimates derived from standard factor models. They also employ time series portfolio tests and show that both portfolio average returns and risk-adjusted returns increase monotonically with their cost of equity estimates. Further, in contrast to standard factor models, portfolio return alphas are highly significant for their cost of equity estimates.

Callen and Lyle (2013) estimate firm-level term structures of implied costs of equity for the years 1996–2012 from equation (28) above using synthetic forward prices estimated from traded options data. They validate their method for calculating synthetic forward prices by estimating synthetic prices of traded S&P 500 futures index on the CME. They compare their synthetic index prices with actual prices and find that the synthetic prices approximate the actual prices to a very close degree.

Based on firm-level data, Callen and Lyle (2013) are able to reject the ubiquitous assumption that term structures of implied costs of equity are flat. Rather, they find that for most years and almost all Fama-French industries, term structures of equity capital and term structures of risk premia are significantly upward-sloping and concave. Their empirical results indicate that the upward slope in the implied cost of equity is an economically meaningful 70 basis points per year on average. They also find that during the crisis period of 2008–09 term structures flattened, consistent with sharp increases in interest rates and limited lending during this period. In addition to estimating firm-level term structures for maturities up to two years using options data, Callen and Lyle (2013) also estimate a long-run firm-level implied cost of equity based on extrapolating the shape of the options-based term structure using spline functions.

Callen and Lyle (2013) further compare the ability of their term structure model to predict realized returns with reference to standard factor models using monthly data and with reference to standard static implied cost of equity models, including Claus and Thomas (2001), Easton's (2004) PEG and the forward E/P ratio, using annual data. 20Both cross-sectional and time-series asset pricing tests indicate that their time-varying implied costs of equity capital, including the long-run estimate, are positively and significantly associated with future realized returns. In contrast, they find that traditional static implied costs of equity and factor model costs of equity are largely unassociated with future realized returns.

Finally, recent literature has shown that impending earnings announcements are perceived to be risky by the market in that equity returns tend to be predictably higher during earnings announcement months (e.g., Barber et al., 2015). Callen and Lyle (2013) find that their forward-looking implied term structure estimates capture this phenomenon ex ante. Specifically, they find that their term structure estimates predict higher future returns (conditional on information in the month before the announcement) in the month of the earnings announcement. In contrast, standard (backward-looking) finance factor models do not forecast future equity returns or anticipate predictable earnings announcement-month return premia.

The Accounting Variance Decomposition Model

The accounting variance decomposition literature is based on Vuolteenaho (2002). 21Assuming the clean surplus relation and stationarity of the market-to-book (M/B) ratio, Vuolteenaho (2002) derives a return decomposition such that unexpected (cum-dividend) returns can be expressed as the shock to the present value of future normalized earnings over the lifetime of the firm (earnings news) less the shock to the present value of expected future returns over the lifetime of the firm (discount rate news). More formally,

(30)

where r_t denotes the (log) cum-dividend stock return at time t, E_t−1(r_t) denotes the market's expectation at time t–1 of the stock return at time t, ΔE_t =E_t − E_t−1 denotes the revision (shock) from period t–1 to period t, ρ is a discount rate and roe_t is the (log) return on book value equity at time t. 22Intuitively, if price is the present value of expected future dividends then any shock (revision) to returns must come either from the shock to future dividends—or earnings, given clean surplus—or the shock to discount rates (expected returns) over the lifetime of the firm.

Taking variances of the latter return decomposition yields the accounting variance decomposition:

(31)

where Ne_t is earnings news:

(32)

and Nr_t is discount rate news:

(33)

The accounting variance decomposition analysis indicates the extent to which the volatility in returns is driven either by the volatility of earnings news or by the volatility of discount rate news or both. Thus, the accounting variance decomposition indicates the extent to which earnings and/or discount rates are value-relevant. However, the variance decomposition concept of value relevance differs from standard value relevance obtained by regressing the revision in returns on shocks to earnings in the earnings response coefficient (ERC) literature. ERC measures the value relevance of earnings from an ex post conditional perspective; that is, conditional on a shock to earnings of, say, 1%, what is the percentage expected revision to returns? In this ERC perspective, the larger the revision to returns the more value-relevant are earnings. But, because the ERC is an ex post metric, it fails to inform about the ex ante expected frequency of return shocks generated by earnings shocks. In contrast, the variance metric of information content measures the ex ante likelihood that earnings shocks will occur and generate return shocks. The greater the volatility of earnings shocks and the more persistent are earnings, the greater the ex ante likelihood that shocks to returns will occur.

The value relevance of accounting information cannot be measured solely by reference to an ERC-type analysis. A variance decomposition is also de rigeur. To illustrate the complementary nature of ERC and variance decomposition, Callen (2009) provides an example for which earnings are highly persistent, β = 0.90, but the variance of earnings is small, only 0.1% and there is no shock to ‘other information’. In this case, it can be shown that the ERC is 10 but the volatility of unexpected returns generated by the earnings shock is only 1.9%. This means that although shocks to earnings, when they occur, have a dramatic impact on returns, nevertheless, shocks to earnings, and hence shocks to returns, are expected to be relatively infrequent ex ante. Callen (2009) subsequently considers the diametrically opposite case where earnings are not very persistent, β = 0.1, and the variance of earnings is relatively large, say, 5%. Here, the ERC is only 1.11 but the volatility of unexpected returns generated by the earnings shock is 6.1%. In contrast to the first case, shocks to earnings are expected to occur relatively frequently ex ante but each shock that occurs ex post will have a relatively small impact on equity returns. Based on a typical ERC analysis, the researcher would have concluded that earnings in the first case (ERC = 10) are far more ‘value-relevant’ than in the second case (ERC = 1.11). However, as this example illustrates, the information contained in an ERC analysis together with the variance decomposition paint a very different picture about the true value relevance of earnings than that obtained from the ERC analysis alone.

Before the variance decomposition can be estimated, one must specify the time series dynamics of returns, normalized earnings, and any other variables assumed to be value-relevant. Typically, the literature assumes that all (value-) relevant variables follow a log-linear vector autoregressive system (VAR) of the form:

(34)

where z_t is a vector of variables including returns, earnings (i.e., return on book equity), B/M ratio, and other value-relevant variables. Γ is a matrix of parameters and η_t is a vector of zero-mean shocks. The choice of other value-relevant variables is not driven by theory, similar conceptually to the other value-relevant variables in the standard Ohlson (1995) model. The multivariate VAR assumption implies that all variables in z_t interact with each other so that, say, the persistencies of returns, earnings, and the B/M ratio are inter-related. This is in contrast to standard ERC analysis (or the standard Ohlson model) in which the persistencies of the different variables are assumed to be independent of each other.

One of the weaknesses of the VAR approach is that realistically one cannot estimate VAR parameters at the firm level given the paucity of sufficient time series data. Moreover, the more variables included in z_t the more unstable are the coefficient estimates (Chen and Zhao, 2009). Instead, the VAR parameters are typically estimated at the industry level (or even at the economy-wide level) and the parameter estimates are then assumed to be constant across firms in the industry. Methods for estimating the VAR and the subsequent variance decomposition, including computer programs, are provided by Callen and Segal (2010).

In an influential attack on the variance decomposition methodology, Chen and Zhao (2009) claim that the relative variance contributions of different variables to unexpected returns in a variance decomposition analysis depend critically upon the method of estimation. They find empirically that whether discount rate news or earnings news drives the volatility of unexpected (bond) returns depends crucially upon which of these (discount rate news or earnings news) is estimated residually and which is estimated directly. In turns out, however, that their critique is flawed on a number of grounds as shown by Campbell et al. (2010) and especially Engsted et al. (2012). Importantly, Engsted et al. (2012) show inter alia that the VAR model is not meaningful unless it incorporates the fundamental valuation variable that gave rise to the variance decomposition in the first place (in their case the dividend–price ratio and in Vuolteenaho's case the M/B ratio), something which Chen and Zhao (2009) fail to do. Incorporating this variable in the VAR, Engsted et al. (2012) obtain results that are consistent with the extant literature and contrary to Chen and Zhao (2009).

The Empirical Accounting Variance Decomposition Literature

Vuolteenaho (2002) was the first to develop and estimate an accounting variance decomposition. He finds that earnings news rather than discount rate news is the main driver of unexpected stock return volatility at the firm level. In contrast, earnings news is largely diversified away in aggregate portfolios so that discount rate news is the main driver of unexpected returns at the aggregate level.

Callen and Segal (2004) incorporate the Feltham-Ohlson (1995, 1996) breakdown of the clean surplus relation into the Vuolteenaho model. They find that cash flow news, accrual news, and discount rate are all value-relevant and significant drivers of return volatility at the firm level. They further find that accrual and cash flow news have fairly similar impacts in driving returns and both dominate discount rate news.

Callen et al. (2005) analyze the relative value relevance of foreign and domestic earnings in driving returns using a variance decomposition analysis. Consistent with the literature, they find that both foreign and domestic earnings are value-relevant from an ERC perspective. They also find that foreign earnings are more persistent than domestic earnings, suggesting that foreign earnings are more value-relevant than domestic earnings from an ERC point of view. 23However, on the basis of a variance decomposition analysis, Callen et al. (2005) find that domestic earnings, rather than foreign earnings, are the primary driver of unexpected equity returns. Intuitively, the variance metric involves a trade-off between the persistencies of the earnings components and the variances of the earnings components. Although foreign earnings are more persistent than domestic earnings, domestic earnings are far more volatile than foreign earnings so that the variance of unexpected equity returns—the value relevance metric in a variance decomposition framework—is higher for domestic earnings than foreign earnings.

As noted above, Callen (2009) argues that value relevance requires both ERC and variance decomposition analyses. In the case of foreign versus domestic earnings, Callen et al. (2005) find that while a 1% shock to foreign earnings has a larger impact on equity returns than a 1% shock to domestic earnings, the likelihood of shocks occurring to domestic earnings is far larger than the likelihood of shocks occurring to foreign earnings. As this case illustrates, both ERC and variance decomposition metrics are required (at a minimum) in order to make adequate value-relevance statements.

Callen et al. (2006) evaluate the value relevance of discount rate news, cash flow news, and accruals around Securities and Exchange Commission (SEC) filing dates. The variance decomposition framework is a natural framework for determining the value relevance of these informational news components around SEC filings because, unlike preliminary earnings announcements which can reasonably be estimated and classified as good or bad news, SEC filings provide information that (potentially) revise the information conveyed by preliminary earnings announcements. Hence, the researcher is usually unable to determine ex ante whether the information conveyed by SEC filings is good or bad news. As a consequence, the value relevance metric for SEC filings has to be non-directional (unsigned). The variance decomposition methodology provides such an unsigned metric, namely, the variance of unexpected returns. Although other non-directional metrics have been employed by the literature, these fail to control for changes in costs of capital (discount rate news) that may be generated by the news event. By contrast, the variance decomposition methodology provides an empirically estimable relation between the non-directional value relevance metric and the variances of the informational news components and discount rate news. A variance decomposition explicitly estimates the impact of changes in costs of capital on security returns during the news event. Callen et al. (2006) show in their study that discount rate news increases significantly on SEC filing dates.

Conclusion

This paper provides a selective review of accounting valuation models and inter-related cost of capital models focusing on the dynamics that underlie these models and on the dynamics that are consequences of these models. Surprisingly, despite the relative simplicity of Ohlsonian and variance decomposition models, both can be generalized in various ways to address issues of concern both to theorists and empiricists. Of these, the most important in the author's view are issues involving the dynamics of valuation and the dynamics of returns such as estimating time-varying discount rates, estimating term structures of costs of equity capital, and addressing the effect of parameter uncertainty on firm valuation and costs of capital.