On the systematic volatility of unpriced earnings

Author's Accepted Manuscript

On the systematic volatility of unpriced earnings Timothy C. Johnson, Jaehoon Lee

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S0304-405X(14)00127-5 http://dx.doi.org/10.1016/j.jfineco.2014.05.012 FINEC2431

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Journal of Financial Economics

Received date: 14 June 2013 Revised date: 5 September 2013 Accepted date: 2 December 2013 Cite this article as: Timothy C. Johnson, Jaehoon Lee, On the systematic volatility of unpriced earnings, Journal of Financial Economics, http://dx.doi.org/ 10.1016/j.jfineco.2014.05.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the systematic volatility of unpriced earnings$ Timothy C. Johnsona,∗, Jaehoon Leeb a University

of Illinois at Urbana-Champaign, United States of New South Wales, Australia

b University

Abstract Some important puzzles in macro finance can be resolved in a model featuring systematically varying volatility of unpriced shocks to firms’ earnings. In the data, the correlation between corporate debt and stock market valuations is low. The model accounts for this via the opposing effect of unpriced earnings risk on levered debt and equity prices. The model also explains the low (or nonexistent) risk-reward relation for the market portfolio of levered equity via the opposing effects of unpriced and priced uncertainty (both components of stock volatility) on the levered equity risk premium. Versions of the model calibrated to empirical measures of both types of fundamental risk can quantitatively substantiate these explanations. Variation in residual earning dispersion accounts for a significant fraction of observed disagreement between debt and equity valuations and of realized stock volatility. The implication that the two components of risk should forecast the levered equity risk premium with opposite signs is also supported in the data. The results are a notable advance for risk-based asset pricing. JEL classification: G12 Keywords: credit spreads, equity premium, volatility components

$ We are very grateful to Michael Brennan, Dana Kiku and an anonymous referee for their thoughtful comments and advice. We also thank seminar participants at Aalto University, Korea University, Stockholm School of Economics, University of British Columbia Summer Finance Conference, University of Missouri, and Yonsei University. ∗ Corresponding author at: University of Illinois at Urbana-Champaign, 343E Wohlers Hall, 1206 South Sixth Street, Champaign, IL 61820, United States. Tel: +1 217 333 4089. E-mail address: [email protected]

Preprint submitted to Elsevier

June 3, 2014

1. Introduction An unexplained fact in macro finance is that aggregate credit spreads often move in the same direction as the equity values of the underlying companies or, equivalently, debt and equity prices move in opposite directions. The correlation between the Standard & Poor’s (S&P) 500 price-earnings (P/E) ratio and Baa–Aaa credit spreads since 1927 is only about −0.50, for example. The two numbers move in the same direction in over 40% of the monthly observations. Equilibrium asset pricing models typically imply that this should almost never happen. Panel A of Fig. 1 shows rolling five-year correlations of the two series, which are positive for extended periods. A simple measure of disagreement between valuation levels is just to sum these series (after normalizing each). This statistic, shown in Panel B, has a range of over ±2. At times, equity valuations and credit spreads are simultaneously near historical highs (e.g., both normalized series ≥ 1) or lows (both ≤ −1). These observations are closely related to the finding of Collin-Dufresne, Goldstein, and Martin (2001) of a strong common component in changes in firms’ credit spreads that is not accounted for by changes in their stock prices or other controls. Explaining this finding has proved challenging and continues to be an active focus of research. We view it not as a puzzle about bond markets, but as a puzzle about the joint dynamics of bonds and stocks. We argue that resolving the low correlation of debt and equity valuations sheds important light on the understanding of fundamental risk. Recent advances in asset pricing theory have integrated modeling of debt and levered equity with equilibrium determination of discount rates. Yet it is not easy to pinpoint state variables in these models that could move debt and equity values in opposite directions. Changes in risk aversion or market prices of risk and shocks to profitability and growth rates should all affect aggregate stock and bond prices in the same fashion. Discrete corporate events that change leverage (e.g., leveraged buyouts) could move debt and equity in opposite directions

2

Panel A : Correlation between stock log P/E and corporate bond spreads

Panel B : Sum of normalized stock log P/E and corporate bond spreads

Fig. 1. Stock price-earnings (P/E) ratio and corporate bond spreads. Panel A shows the correlation between the Standard & Poor’s 500 index’s log P/E ratio and the Moody’s Baa–Aaa corporate bond spreads over the rolling window of past 60 months. Panel B shows the sum of the two normalized time series.

for individual firms. However, this channel is largely shut down for fixed credit rating portfolios. Thus, while the issue has yet to be examined in an equilibrium setting, we think it unlikely that the puzzle can be explained by dynamic capital restructuring. From a purely empirical standpoint, the literature following Collin-Dufresne, Goldstein, and Martin (2001) has ruled out leverage fluctuations as a source of the independent variation in credit spreads. The variation in the first principal component is also not explained by nominal interest rate fluctuations and a host of other candidate factors. In our empirical work, we likewise consider the degree to which measures of valuation divergence can be explained by controls,

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including inflation and debt quantities. Our conclusions are similar. One mechanism that can account for opposing movement in bond and stock valuations is fluctuations in the volatility of the unpriced component of firm cash flows. In partial equilibrium settings such as Merton (1974), it is well-known that increases in the (exogenous) volatility of unlevered asset values lower the risk premium of levered equity and raise its price, while having opposing effects on risky debt. Less well known is that the argument, in general, fails with equilibrium determination of asset values and their volatilities. Fig. 2 illustrates what happens to the valuation of a Merton (1974) firm’s claims under two otherwise highly successful models as fundamental uncertainty changes. Panel A shows the levered price-dividend ratio and credit spread as a function of volatility when the underlying asset is priced by the Campbell and Cochrane (1999) model.1 The horizontal axis in the figure is the volatility of the log-surplus consumption ratio, which is a monotonic function of the level of surplus consumption, the model’s main state variable. Panel B does the same calculation under the model of Bansal and Yaron (2004), varying the second moment of consumption (while holding the first moment fixed). In both panels, as uncertainty increases in the economy, credit spreads widen while an increase in discount rates lowers underlying asset values. This is more than enough to counteract the positive effect of convexity on the price of levered equity.2 What is needed in equilibrium, therefore, is a separate type of uncertainty shock: one that leaves discount rates unchanged. The present paper shows that this mechanism can work quantitatively at the aggregate level. The notion of systematic changes to firm operating risks has recently become the focus of much work in macroeconomics.

Starting with Bloom (2009),

1 We implement the correction to the model described in Chen, Collin-Dufresne, and Goldstein (2009). 2 The nonlinear valuation functions in these models mean that linear measures of association—correlation coefficients—do not pick up the perfectly monotonic relationship between debt and equity. We thank the referee for this observation. Other measures could be more revealing. Both these models imply, for example, that debt and equity price changes almost always have the same sign.

4

Fig. 2. Equilibrium debt and equity value: Merton model. Panel A shows the price-dividend ratio and credit spread (in percentage points) for the stock and bond in the Merton (1974) model when the firm value process is determined according to Campbell and Cochrane (1999) model, using the parameterization of Chen, Collin-Dufresne, and Goldstein (2009). The horizonal axis is the volatility of the log-surplus consumption ratio. Panel B does the same under the Bansal and Yaron (2004) model, using the parameters in that paper. The horizontal axis is the volatility of aggregate consumption. The valuation assumes the expected growth rate of the economy is at its steady-state value. In both cases, the bond has four years to maturity.

theoretical and empirical studies have delineated real channels through which stochastic volatility of idiosyncratic productivity affects aggregate output independent of changes in aggregate productivity. The literature sometimes refers to this distinct component of uncertainty as micro risk, which shows up in changes in cross-firm dispersion measures, in, e.g., profitability, earnings forecasts, or stock returns. This distinction has a familiar analog in the asset pricing literature stemming from Constantinides and Duffie (1996) that studies household-specific risk (e.g., labor income), whose dynamics could differ from that of aggregate consumption risk. We complement the recent macroeconomic advances by showing that the existence of the common uncertainty component in firm-specific risk has significant implications for aggregate financial variables, in particular the relative prices of debt and equity. In the context of our model, micro uncertainty corresponds to the volatility of the unpriced component of firms earnings, i.e., the component that does not affect the representative agent’s marginal utility. Intuitively, substantial

5

variability in profits simply represents a shifting distribution between firms or between capital and labor that do not have any net implications to consumers. This includes any purely idiosyncratic component of a given firm’s earnings and also encompasses the component of aggregate dividends that is unrelated to aggregate consumption. (Empirically, aggregate dividends are only weakly correlated with aggregate consumption.

The wedge between the two is

attributable to shifting labor share of income.) So, while it could help to think of our volatility channel as firm-level risk, we do not necessarily assume that this component of earnings is idiosyncratic, i.e., uncorrelated across firms. Instead, the crucial assumption is that the volatility of unpriced shocks is itself stochastic and common across firms. The idea of fluctuations in economic uncertainty, broadly defined, is now widely accepted. Distinguishing between uncertainty about priced and unpriced components of earnings, no obvious reason exists for why fluctuations should be limited to the former. Again in analogy with Constantinides and Duffie (1996), many reasons can be found for why the degree of uncertainty about the distribution of profits could vary. (Intuitively, this could come from a shock to competition or from labor market uncertainty.) Beyond the topic of debt and equity valuation, this hypothesis has important implications for the understanding of another fundamental financial quantity: stock market risk. A large body of academic and practitioner literature seeks to model the stochastic volatility of, for example, the S&P 500. In a capital asset pricing model (CAPM) world, the volatility of the market portfolio is the only volatility that investors should care about. Yet the connection between this quantity and fundamental risk has proven elusive. Researchers have long struggled to explain why increases in the risk of equity returns fail to predict increases in expected returns of the same equity. In the model that we propose, the explanation is simple: the two components of risk in dividends—priced and unpriced—have offsetting effects on the risk premium. Likewise, the two components affect the levered price-dividend ratio in the opposite direction, resolving a puzzle proposed by David and Veronesi (2013). 6

The asset pricing literature has previously recognized the important role of unpriced risk in affecting the risk premium on levered equity. Applications of this idea have focused on cross-sectional implications. [See, for example, Johnson (2004) and Barinov (2011).]

Systematic variation in idiosyncratic

volatility (of levered equity returns) has been successfully linked to common unexplained variation in credit spreads by Campbell and Taksler (2003). Consistent with our story, Jorgensen, Li, and Sadka (2012) find that increases in cross-firm earnings dispersion are associated with higher stock prices. Recently, Arellano, Bai, and Kehoe (2011) and Gilchrist, Sim, and Zakrajsek (2010) have proposed equilibrium models of debt and equity issuance by firms facing time-varying uncertainty that alters the relative costs of each via the same mechanism we model. The interaction of fluctuations in financing costs with financial constraints leads to amplified fluctuations in aggregate investment and output. Relative to this work, our contribution is to show that a story based on the fluctuations in the risk of unpriced earnings can work as an equilibrium explanation for aggregate financial variables, even without frictions. We show that realistically calibrated degrees of fluctuation in our state variable are large enough to explain the observed levels of debt and equity correlations and to resolve the puzzles relating to the ambiguity of stock volatility. We directly measure the evolution over time of our distinct component of uncertainty based on dispersion in operating (non–financial) variables, and we establish that it can account for a substantial component of the observed disagreement between debt and equity valuation and of stock market volatility. We also test and verify the model’s predictions for the role of the two types of uncertainty in forecasting returns for equity and corporate debt. The results are a noteworthy success for risk-based asset pricing. The outline of the paper is as follows. Section 2 describes the model of levered equity and credit spreads and deduces some key predictions. Section 3 studies the quantitative implications for valuation and return moments in some plausible calibrations and compares these with the data. Section 4 presents 7

empirical evidence on testable implications of the model.

A final section

summarizes and concludes. 2. A model of debt and equity To study movements in risky debt and equity prices, we follow Bhamra, Kuehn, and Strebulaev (2010), Chen (2010), and Gourio (2011) by solving for prices of particular debt contracts in an economy with time-varying cashflow moments and Epstein and Zin (1989) preferences. Our interest is particularly in the evolution of credit spreads on fixed-ratings portfolios (e.g., AAA or AA bonds), whose composition is largely determined by firm leverage ratios. Thus, we do not model the determination of debt quantities.3 Our design is most closely related to the study of Chen, Collin-Dufresne, and Goldstein (2009). Those authors are interested in the coevolution of credit spreads and equity return moments. They establish that a model including a habit-type pricing kernel and exogenously time-varying default boundaries and recovery rates can generate implied time series for credit spreads that match many features of the data, while also matching the main moments of equity returns. Our focus is on time-varying fundamental risk, not risk aversion.4 Our initial goal is to explain the divergence of debt and equity values for fixed-ratings portfolios even as both are driven by the same cashflow shocks and a common stochastic discount factor. The setup of the model mostly follows that of Bansal and Yaron (2004; hereafter BY). Time is discrete. There is a single perishable consumable good, the flow of which comes from an endowment asset. Our interest is in pricing claims to a second flow, the operating earnings of a particular firm. Letting C denote the consumption stream, D(i) denote the earnings stream of the ith 3 In

addition, we do not model any feedback from aggregate debt to the primitive cashflow moments driving prices. See Gourio (2011) for a general equilibrium treatment of aggregate debt. 4 The Appendix to Chen, Collin-Dufresne, and Goldstein (2009) also analyzes a model with time-varying cashflow moments and Epstein and Zin (1989) preferences. We compare our findings to theirs below.

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firm, and lowercase letters being logs, the stochastic specification is =

µc + xt + σc,t c,t+1 ,

∆dt+1

=

φ ∆ct+1 + σd,t d,t+1 ,

(2)

xt+1

=

ρ xt + σx,t x,t+1 .

(3)

∆ct+1 (i)

and

(1)

(i)

As in BY, changes in log consumption and earnings have a common expected growth term x, which evolves as an AR(1) process. Also as in that paper, the innovation terms, c , d , and x , are independent, standard normal random variables. Unlike BY, we do allow for contemporaneous correlation between ∆ct+1 and ∆dt+1 via the φ term. Thus, assuming d uncorrelated with c is without loss of generality. The extra volatility in earnings (or dividends) is a standard feature in (i)

consumption-based models. No assumption is made that the d shocks are uncorrelated across firms.

Aggregate corporate earnings are only weakly

correlated with consumption in the data, suggesting a strong common component (i)

across firms in d . Thus, the second term in earnings shocks should not be thought of as idiosyncratic. Instead, the crucial feature of these innovations is that they are unpriced in the sense that they do not covary with marginal utility. Intuitively, fluctuations in corporate profits that do not affect overall consumption (or its moments) can be thought of as just reallocations between labor and capital income shares or between profit shares within an industry that leave the representative investor or consumer indifferent. As the notation indicates, the volatilities of c, d, and x could all be stochastic. Another important departure from BY is that we do not assume the same process drives them all. Instead, we assume the variances evolve according to

and where σc =

√

vc,t+1

=

wc + αc vc,t + sc ηc,t+1

(4)

vd,t+1

=

wd + αd vd,t + sd ηd,t+1 ,

(5)

vc and σd =

√

vd . For parsimony, we take σx to be constant. We

take ηc and ηd to be independent and identically distributed gamma-distributed 9

sequence with mean zero and unit variance, and we restrict the distribution shape parameters to ensure the processes are non-negative.5

The variance

processes are assumed homoskedastic and stationary (the αs are between zero and one) and could be correlated with each other and with the growth rate process x. The stationary processes vc , vd , and x are the state variables that determine all aggregate quantities. (The assumption that the state variables are themselves homoskedastic is not necessary for our theory. While it limits the model’s ability to capture, e.g., a time-varying volatility risk premium, it makes the role of vd that we highlight simpler to understand.) For computational simplicity, we assume that these are not correlated with c and d , the cashflow shocks. It is worth clarifying that the state variable that we call unpriced risk is the risk of the unpriced cashflows, but this process itself could be priced. That is, vd could be correlated with vc and x, both of which directly enter marginal utility. While it is tempting to describe σd,t as firm-specific volatility, the crucial assumption of the model is that this process itself is not firm-specific. All firms’ unpriced earnings shocks are assumed to be scaled by this systematic factor (perhaps up to a constant, which we can normalize to unity).

We

provide evidence on the magnitude of this stochastic volatility component in Section 3. Most directly, we know σd,t varies by observing changes in measures of cross-firm dispersion in earnings or stock returns. Such fluctuations are well documented and strongly suggest the presence of a shared residual volatility term.6 Intuitively, sometimes greater uncertainty exists about the distribution of profits between capital and labor, or across sectors, regions, and competitors. In other words, σd,t can be seen as capturing a microeconomic element of uncertainty. √ √ ηc,t = uc,t − qc , where uc,t ∼ Γ(qc , 1/ qc ) has variance of one and mean √ qc . Positivity then requires wc ≥ sc qc . Analogous definitions apply for ηc,t . 6 Heterogeneity in common factor loadings can also produce changes in dispersion as the magnitude of realized factor shocks changes. In our empirical work, we partially control for this by working with residual earnings, i.e., after allowing for firm-specific loadings on aggregate shocks.

√

5 Technically,

10

While our analysis is concerned with firm-level asset prices and returns, to connect to aggregate quantities (such as index portfolios) it is helpful to define the common component of unpriced earnings via a simple specification such as (i)

d,t =

√

ρi,j ut +


(i) 1 − ρi,j ut ,

(6)

where ρi,j is the common cross-firm correlation (i.e., the same for any pair of firms i, j) and the u(i) component is purely idiosyncratic and effectively disappears when summing over a large number of firms. Under this assumption, for example, the variance of residual earnings across N identical firms is just (1 − ρi,j ) vd for large N . The representative agent in the economy has recursive utility with coefficient of risk aversion γ, elasticity of intertemporal substitution ψ, and time discount rate δ. Following BY, a log-linear approximation to the returns to a claim to consumption permits the log price-consumption ratio, zc , of the claim to be written as linear function of the state variables, zc,t = A0 + A1 xt + A2 vc,t , where the constant As can be found from the Euler equation for the claim via the method of undetermined coefficients. This makes the log pricing kernel also linear: mt+1 = log Mt,t+1 = θ log δ − γ ∆ct+1 − (θ − 1)[ A0 + A1 xt + A2 vc,t ] + (θ − 1) κ0 + (θ − 1) κ1 [ A0 + A1 xt+1 + A2 vc,t+1 ],

(7)

where θ ≡ (γ − 1) / ((1/ψ) − 1) and κ0 and κ1 are constants from the log-linearization. The expected growth rate of −m is the riskless one-period rate, denoted rtf . We next consider a division of the claims to the D stream between debt and equity. Specifically, we assume debt is of the Merton (1974) type: a single zero-coupon bond of face value K that must be paid off via liquidation at time T . We assume that the terminal value of assets is the value, VT , of a claim (unlevered) to the flow of dividends after T . Up to a second log-linear

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approximation, the quantity zd = log(V /D) is also linear in the state variables, zd,T = B0 + B1 xT + B2 vc,T + B3 vd,T .

(8)

We adopt this approximate valuation as an assumption. The Bs can again be directly obtained analytically in terms of the primitives. The expressions are omitted for brevity.7 Let F and S denote, respectively, the value of debt and levered equity. We assume that the terminal (and only) cashflow to the bond is FT = min[RT VT , K],

(9)

where RT is a recovery rate that could depend on the state of the economy at T . Equity holders are assumed to receive all earnings as a dividend stream8 until time T . Then, they receive a liquidation payment of ST = max [ VT − K, 0 ] FT = K.

(10)

Equity thus receives the residual asset value after debt has been fully repaid. The model does not require us to make any assumption about the optimality of the level of firm debt. This specification includes losses to debt and equity holders outside bankruptcy, i.e., in the region K < VT < K/RT . Several structural models have been proposed in the literature incorporating such losses in distress states with low cash flow. In our context, these help to raise credit spreads to realistic levels, but they do not otherwise drive our main results. Debt and levered equity prices and expected returns are not obtainable analytically but can be readily computed by Monte Carlo integration over any region of the state space.9 A solution is then a four-dimensional object, e.g., for 7 All model derivations are available in a supplemental Appendix available on Jaehoon Lee’s website (https://sites.google.com/site/jaehoon223/) or by request. 8 Because all earnings are paid as dividends, we use the terms “earnings”, “cashflow”, and “dividends” interchangeably. 9 This solution does not involve any linearization and is exact up to the numerical standard error of the integration. We impose a maximum percentage error tolerance of 0.001 in

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equity’s one period expected excess return, πS (Dt , xt , vc,t , vd,t ; K, T − t).

(11)

It is worth clarifying that, when we report unconditional moments of these quantities, they are computed over the distribution of the stationary state variables (xt , vc,t , vd,t ). For an individual firm or a fixed portfolio, the first state variable is nonstationary (and the time-to-maturity changes), so moments do not exist. The reported moments thus select fixed values for D (or D/K, the debt-coverage ratio) and for T − t. In effect then, these correspond to the pricing dynamics, not of an individual firm, but of a sequence of firms. (Likewise, when we report moments of a portfolio of firms, we need to assume a fixed distribution of D(i) /K (i) of the firms in the portfolio.) For example, following Chen, Collin-Dufresne, and Goldstein (2009), when we simulate a history of the AAA credit spread, we evaluate that spread each period for a (new) firm with constant debt maturity and with D/K chosen to be consistent with the historical AAA default probability. The model could be extended with some specification for a stationary law of motion for leverage, D/K, but we do not pursue that here. Before turning to numerical exploration of the model’s properties, let us outline the key features of the solutions. In fact, the relations we wish to highlight can readily be described in terms of the signs of partial derivatives. Let cr denote the credit spread of the risky bond in excess of the T -period f riskless zero-coupon rate, cr = − log(F/K)/T − rt,T . Fixing the current level

of firm earnings D, this spread satisfies ∂cr > 0, ∂vc

∂cr > 0. ∂vd

(12)

Increases in uncertainty unambiguously hurt debt holders, whose payoff is concave in VT . However, letting pd = log(S/D) be the levered log price-dividend valuations at the long-run values of the state variables.

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ratio, as long as αc is sufficiently large and the preference parameters ensure B2 < 0, results in ∂pd < 0, ∂vc

∂pd > 0. ∂vd

(13)

These comparative statics are explicitly provable for the case of one-period debt.10 They also obtain in the full model in all the numerical solutions that we have analyzed. Comparing the impact of vc , the credit spread response is the straightforward effect of vc on unlevered asset values. The effect on pd is not immediately obvious because of the convexity of equity’s claim on V . However, the first-order effect of asset values is larger than the convexity effect (unless vc shocks are short-lived and thus economically unimportant). This is not true for firm-specific earnings uncertainty, though. Now even the unlevered claim increases in value with uncertainty (B3 > 0) because this uncertainty is unpriced. With the option-like payoff of levered equity, this effect is magnified. The effect of vd is thus to move debt and equity in opposite directions. This observation, in partial equilibrium, is not new. Our contribution is to show that the argument holds with equilibrium determination of asset values and their volatilities and to show the magnitudes can be significant at the aggregate level. A crucial piece of supporting evidence for our story—and itself a salient and puzzling feature of the data—is the absence of a strong risk-reward ratio in the stock market. The risk premium πS and volatility σS of levered equity are defined as the first two conditional moments of the log excess returns,   Dt+1 + Pt+1 rS,t+1 − rf,t ≡ log − rf,t Pt = log(exp(pdt+1 ) + 1) − pdt + ∆dt+1 − rtf

(14)

In the numerical solutions, these have the following responses to the state 10 The

proofs appear in the Internet Appendix.

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variables (still holding D fixed): ∂πS >0 ∂vc ∂σS >0 ∂vc

∂πS < 0, ∂vd ∂σS > 0. ∂vd

and and

(15) (16)

Variation in σS is easy to understand. It comes mainly through current dividends, instead of the price-dividend ratio. In fact, because of our assumption of homoskedastic state variables, the volatility of the unlevered price-dividend ratio, zd , is constant. The levered price-dividend ratio, pd, does become more volatile as asset values decline due to the usual leverage effect. But the main variation in volatility is due to changing dividend risk. And dividend variance is just φ2 vc + vd . David and Veronesi (2013) point out that the rolling correlation between σS and pd frequently changes sign over time. Comparing the partials for σS and pd, we see why it is not puzzling that the two are not perfectly negatively correlated: The effect of vd on both quantities is positive. Turning to πS , increases in consumption risk raise the equity risk premium because this is systematic risk. But the d shocks are unpriced (they do not covary with c, vc , or x). As is well known (Johnson, 2004), increasing unpriced risk of an asset decreases the expected excess return of a call on that asset. Hence, it is clear that one should not, in general, expect the risk and the risk premium of levered equity to be perfectly positively correlated. In numerical solutions, the unconditional correlation can even be negative. While the assertions here pertain to the moments of an individual stock, they translate to portfolios as well. This can be shown immediately for the case of an equally weighted geometric index of identical firms. It has expected return equal to that of the individual stocks and variance equal to σS − (1 − ρi,j )vd . (i)

(This uses the assumption that c and d are not correlated with the state variables.) Thus, assuming a non-negative cross-sectional correlation, the signs of the volatility sensitivity are the same as for the individual stocks. In Section 4, we take these observations further and offer supporting evidence for the relations that the model implies for risk premia and volatility. First,

15

however, we turn to numerical results to demonstrate the plausibility of the mechanism embedded in the model. 3. Quantitative implications To illustrate the magnitude of the model’s effects, in this section we analyze two benchmark parameterizations of the economy and of firm-level cash flows that is in line with previous work with long-run risk type models. For comparison purposes we also present an alternative case without the model’s main mechanism. Our goal is to show that the model can successfully deliver the relations between debt and equity valuations, which we took as our primary motivation, and also account for the ambiguous nature of levered equity volatility as a measure of fundamental risk. To quantitatively assess the key mechanism in our model, we need to establish reasonable values for the parameters governing unpriced earnings uncertainty. Our hypothesis is that this component of cashflow volatility is itself volatile enough to play a significant role in the dynamics of aggregate financial quantities. As a first step in calibrating the model, therefore, we turn to the data to assess this issue. The model is about changes in fundamental, non-financial quantities, in particular, firm earnings volatility. Because firm earnings are observable only quarterly, it is difficult to try to fit a stochastic volatility process to time series data. However, as Section 2 described, under the model’s assumptions there is a one-to-one mapping between our vd process and the cross-sectional dispersion of residual firm earnings in any given quarter.11 Hence, we construct a time series of estimates of this dispersion. For each company in Compustat with a March, June, September, or December fiscal year, we regress the log changes in firm cashflows on log changes in quarterly consumption. Then, for each quarter, 11 This identification comes purely from the firm-specific component of unpriced earnings, not the aggregate part. Referring to Eq. (6), it would not work if ρij were close to one. To the extent that true fluctuations in unpriced risk are due to uncertainty about the aggregate term, our measure does not capture them. We thank the referee for this observation. In terms of the model, this implies we can underestimate sd .

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Fig. 3. Estimated volatility of residual earnings. The figure plots the time series of quarterly estimates of the standard deviation of firm operating earnings growth, after removing the component explained by aggregate growth. Details of the construction are provided in Appendix A.

we compute the cross-sectional standard deviation of these residuals.12 The resulting series, which we call σ ˆd (we call its square vˆd ), is plotted in Fig. 3. An alternative measurement of firm-specific uncertainty is undertaken by Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011) using the cross-sectional dispersion of total factor productivity shocks extracted from a structural estimation of firm-level production functions. The quantity measured does not correspond exactly to the earnings uncertainty of our model. However, it utilizes more information (firm-level capital and labor inputs) than our estimation and thus represents a complementary gauge of systematic changes to idiosyncratic uncertainty. (The series is only available at annual frequency.) As shown in Fig. 4, the Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011) measure mirrors the variation in our series closely, while also suggesting some greater low-frequency variation. Both series affirm that significant variability exists in earnings dispersion over time. This is direct evidence for the key feature of our model’s specification. We use our estimated series, vˆd , to calibrate the innovation parameter, sd , by matching the quarterly innovation volatility of this series to that of model 12 Full details of the procedure are given in Appendix A. The series can be downloaded from the journal’s website.

17

Fig. 4. Estimated dispersion of productivity shocks. The figure plots the time series of annual estimates of the dispersion of firm-level productivity shocks estimated by Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011).

simulations aggregated to quarterly frequency. In our first benchmark case, we choose the persistence parameter αd to give volatility shocks a half-life of one year. Our estimated quarterly series has somewhat less autocorrelation than this choice implies at low lags. The Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011) series has a half-life longer than one year, with statistically significant autocorrelation coefficients at five- and six-year lags. Our primary parameterization demonstrates that resolving the valuation puzzles does not require assuming long-horizon variability in vd . The full set of benchmark parameters for the main case, (II), is in Table 1. We also present results for a parameterization of the same economy but with no variability in vd . This case is referred to as case (I). To more easily distinguish the effects of each state variable, our primary case (II) assumes orthogonal innovations in x, vc , and vd . However, recent macroeconomic work emphasizes the countercyclicality of uncertainty shocks. We therefore include a case, (III), in which vd and vc are negatively correlated with expected growth and positively correlated with each other. The time unit in the economy is a month. The parameters for preferences and for debt are the same for all cases and are given in the table caption. The utility specification follows the long-run risk literature in taking ψ > 1 > 1/γ. The debt parameters are fairly standard. The calibrations do not employ a

18

Table 1 Parameter values. The table shows parameter values for three cases analyzed in Section 3. All cases use the preference specification γ = 7.5, ψ = 1.5, and δ = 0.996. All cases take firm debt to be defined by K = 1, T = 4 years, and R = 0.65.

(I)

(II)

(III)

Aggregate consumption µc σx ρ

0.0025 3.0 × 10−4 0.955

0.0025 3.0 × 10−4 0.955

0.00 3.0 × 10−4 0.955

wc αc sc qc

2.5 × 10−6 0.980 2.0 × 10−5 0.0156

2.5 × 10−6 0.980 2.0 × 10−5 0.0156

2.5 × 10−6 0.980 2.0 × 10−5 –

Firm earnings φ

2.5

2.5

2.5

wd αd sd qd

2.0 × 10−3 – 0 –

1.2 × 10−4 0.940 9.0 × 10−4 0.0178

6.0 × 10−5 0.970 6.4 × 10−4 –

0 0 0

-0.39 -0.48 0.32

Correlations ρvc ,x ρvd ,x ρvd ,vc

0 0 0

time-varying recovery rate. Parameters for the aggregate consumption process shown in the table also broadly follow the long-run risk literature. For each of the variance processes, the shape parameter governing the gamma innovations (qd and qc ) is chosen to be the largest value (hence the least skewed) consistent with the requirement that the process remain positive. For tractability, case (III) takes the state variables to be jointly Gaussian. This choice lowers the precautionary savings

19

motive in the economy and raises the riskless rate. To make the cases more comparable, we compensate by taking the mean growth rate of consumption to be lower for this case. For the volatility of unpriced earnings process, our choice of the dynamic parameters sd and αd for case (II) was described above. The intercept wd determines the unconditional level of vd via v¯d = wd /(1 − αd ). For each of the cases, wd is taken to yield v¯d = 0.002, corresponding to an annualized volatility of 15.5% for this component of firm cashflow. The correlated case employs a more persistent specification for vd than case (II), with the half-life of shocks rising from 11 to 23 months. It turns out this is needed for the independent effect of vd to overcome the masking by the other state variables. However, the innovation parameter sd is lowered correspondingly, so that cases (II) and (III) have the same unconditional variance of vd . The correlations between the state variables for case (III) were chosen using our estimated series vˆd together with proxies for x and vc , namely, the mean and dispersion of real gross domestic product (GDP) growth forecasts from the Survey of Professional Forecasters (SPF)13 . We fit AR(4) specifications to each quarterly series and then compute the sample correlations from the residuals. The numbers in the table are the sums of the contemporaneous correlations and those at lags +1 and -1 to compensate for possible time-aggregation biases.14 The same, fixed, firm characteristics are used throughout. We do not explore effects that arise from differences across firms (or ratings classes) in recovery rate, earnings volatility, volatility persistence, etc.

The only firm-level

heterogeneity that we examine comes through the ratio, D/K, of current earnings to debt face value. As described in Section 2, computing unconditional moments in our model 13 See http://www.phil.frb.org/research-and-data/real-time-center/ survey-of-professional-forecasters/ 14 Using the lead and lag correlation is conservative in the sense that the contemporaneous correlations are significantly lower in absolute value for each pair of variables. Also our procedure yields substantially higher correlations than fitting a vector autoregression (VAR) model to the joint time series.

20

requires some stationary specification for the path of D/K of the observed firms or collection of firms. (Dt is nonstationary for any fixed firm.) For example, for purposes of comparing with data on A-rated firms, we need to decide how to represent within the model the D/K-path of such firms. The simplest approach, which we adopt, is to assume that D/K is fixed for the firms in the sample. This assumption is equivalent to a model of what a rating means. Rating agencies differ in their stated classification goals. However one common objective is to provide a stable mapping from ratings to true default probabilities, at least unconditionally. Tables of historical default frequencies by rating are widely viewed as meaningful predictors. We therefore interpret a firm with a given rating to be one whose earnings-to-debt ratio, D/K, yields an unconditional expected default probability equal to the historical frequency for that rating. This assumption means that the conditional default probability of a firm with a given rating will vary with the state of the economy. It also implicitly assumes an immediate response of rating to permanent changes in cash flow, i.e., shocks to D. While oversimplified, the ratings interpretation is not radically at odds with the data. Empirically, the actual quantity of debt for fixed-rating portfolios are reasonably stable over time. Fig. 5 shows time series measures of book debt to assets for A-rated firms. Plots of operating earnings to book debt are similarly smooth. To be clear, in presenting properties of, e.g., A-rated firms within the calibrations, our exercise is to build a time series in which a new firm (or collection of firms) is observed every month with fixed D/K (and constant maturity four-year debt). That is, we are not simulating histories of specific firms. So, no assumption is made that individual firms’ leverage ratios are constant (or stationary). Table 2 shows some basic unconditional moments for each of the calibrations together with corresponding moments taken from different data samples, as described in the caption. (Full details of the empirical moment estimation can be found in Appendix B.) 21

Fig. 5. Debt quantities. The figure shows three measures of debt quantities for A-rated issuers. The lowest line is the ratio of long-term debt to total assets. The middle line is long-term debt plus the debt component of current liabilities scaled by total assets. The top line is total liabilities scaled by total assets. For each series, the value reported is the median across A-rated companies in each month. Data are from Compustat quarterly files.

The first four lines show moderate real interest rates that are not too volatile. The next lines report the first and second moments of an equity index composed of identical A-rated firms having common pairwise correlation of unpriced earnings shock ρi,j = 0.25. This is not a bad representation of the market portfolio in the sense that A firms are by far the largest segment of the S&P 500 by market capitalization, making up between 30% and 40% of the index for the last 20 years. All cases in the table show reasonable equity premia. The index volatilities are somewhat too high because we have not included any heterogeneity in growth rates, for example. Thus all firms have identical price-dividend ratios. Hence there is no diversification in changes in valuation. The remaining lines in the table give moments for debt and equity of an individual A-rated company. Equity valuation levels are reasonable [though low in case (III) due to the higher riskless rate] and have reasonable autocorrelation. Leverage levels are moderate. Credit spreads are too low, but they are similar to the levels found in other long-run risk type models, e.g., Chen (2010) and Bhamra, Kuehn, and Strebulaev (2010). Gourio (2011) shows that similar models can be made consistent with the post-2007 level of credit spreads by incorporating time-varying disaster risk. Chen, Collin-Dufresne, and Goldstein (2009) evaluate credit spreads in some long-run risk models (in Appendix and in further detail in working paper versions) 22

Table 2 Calibration moments. (I), (II), and (III) show moments for the model calibrations defined in Table 1. Moments are computed in a sample of 120,000 monthly observations. Firm debt and equity moments assume a constant debt-to-earnings level that equates to steady-state default probability of an A-rated firm. Index statistics assume an index of identical firms all with this same debt level and pairwise earnings parameters ρi,j = 0.25. (i), (ii), and (iii) show similar moments computed in samples from December 1996, January 1954, and December 1927, respectively. Details are described in Appendix B.

Model

Data

(I)

(II)

(III)

(i)

(ii)

(iii)

E[rf,1mo ] std[rf,1mo ]

0.0132 0.0104

0.0132 0.0104

0.0388 0.0125

0.0012 0.0211

0.0106 0.0204

0.0042 0.0333

E[rf,4yr ] std[rf,4yr ]

0.0086 0.0055

0.0086 0.0055

0.0328 0.0045

0.0052 0.0171

0.0185 0.0189

0.0179 0.0191

E[πI ] E[σI ]

0.0421 0.2780

0.0477 0.2513

0.0379 0.2467

0.0475 0.1625

0.0500 0.1483

0.0465 0.1910

E[πS ] E[σS ]

0.0421 0.3099

0.0477 0.2787

0.0379 0.2866

0.0568 0.3284

0.0805 0.3070

0.0805 0.3070

E[pd] ac1[pd]

3.112 0.9681

3.134 0.9658

2.590 0.9649

2.483 0.9226

2.745 0.9868

2.745 0.9868

E[mkt lvg]

0.3411

0.3294

0.2657

0.4781

0.5053

0.5053

E[default prob]

0.0051

0.0066

0.0073

0.0052

0.0052

0.0052

E[cr] std[cr] std[∆cr] ac1[cr]

0.0069 0.0047 0.0011 0.9707

0.0058 0.0040 0.0011 0.9619

0.0061 0.0038 0.0010 0.9602

0.0145 0.0100 0.0024 0.9715

0.0099 0.0046 0.0014 0.9525

0.0114 0.0071 0.0017 0.9525

and report that default rates are too countercyclical. We assess cyclicality by regressing (ex ante) conditional default probabilities on the first two moments of consumption growth. Illustrating the critique, the R2 for case (I) is 91%. However, by incorporating risk fluctuations that are not spanned by these moments, cases (II) and (III) show a remarkable improvement on this dimension, yielding R2 s of 35% and 50%, respectively.

23

A primary objective of the calibration is to explain the degree to which bonds and stocks fail to co-move perfectly. As described in Section 2, credit spreads and equity prices are affected oppositely by the uncertainty of unpriced shocks to earnings. Fig. 6 shows these valuation measures as functions of the two types of uncertainty in the economy for an A-rated firm (with the expected growth rate, x, at its average level) in model (II). The plot verifies that levered equity can respond positively to an increase in uncertainty that does not affect discount rates, while the same increase raises credit spreads.

Fig. 6. Equity and debt valuation. The figure shows the price-dividend ratio and credit spread in calibration (II) for an A-rated issuer as a function of the two components of uncertainty when the economy’s expected growth rate is at its long-run level.

We confirm the importance of the vd effect in Table 3. Here we report measures of the time series association between credit spreads and log price-dividend ratios for different rating classes, in the three calibrations as well as in selected data samples. Our primary credit spread data are the Bank of America (BofA) Merrill Lynch US Corporate Option-Adjusted Spread series from http://research.stlouisfed.org/fred2/categories/32297.

These

series accurately account for call feature in the underlying bonds, but they begin only in 1996. For a longer sample, we also present results using the spread between Moody’s Baa and Aaa corporate bond yields and the price-dividend ratio of the S&P 500 index since January 1954. The top two panels report correlations in levels and differences. By these 24

Table 3 Correlation of debt and equity valuation metrics. (I)–(III) report measures of association between debt and equity valuation in the parameterizations given in Table 1. In the data column, credit spreads and price-dividend ratios for firms of rating AAA to BBB are computed from Bank of America Merrill Lynch Corporate Bond Spreads since December 1996. In the row labeled S&P 500, the credit spread is the difference between Moody’s Baa and Aaa corporate bond yields since January 1954 and the price-dividend ratio is that of the S&P 500 index.

Model (I)

(II)

(III)

Data

-0.79 -0.87 -0.92 -0.93

-0.49 -0.54 -0.63 -0.72

-0.59 -0.65 -0.71 -0.76

-0.40 -0.37 -0.48 -0.35 -0.45

-0.71 -0.82 -0.88 -0.90

-0.25 -0.28 -0.41 -0.53

-0.45 -0.51 -0.58 -0.65

-0.02 -0.16 -0.19 -0.21 -0.12

0.17 0.16 0.15 0.14

0.32 0.30 0.28 0.26

0.27 0.25 0.25 0.22 0.44

corr( pd, cr ) AAA AA A BBB S&P 500 corr(∆pd, ∆cr) AAA AA A BBB S&P 500

E[ sign∆pd = sign∆cr ] AAA AA A BBB S&P 500

0.04 0.03 0.03 0.03

corr( pd, cr ) over five-year rolling window of A-rated firms 90th percentile 95th percentile 99th percentile

-0.90 -0.85 -0.73

0.27 0.52 0.81

-0.27 -0.11 0.25

0.49 0.50 0.63

yardsticks, cases (II) and (III) do a good job of matching the level seen in the data, whereas the correlations in case (I), without vd variation, are much too high. It is true that the correlations in case (I) are also not perfectly negative.

25

This is due to the nonlinearity in the valuation functions. Linear measures of association, like sample correlation, can be misleading, however. The next panel of the table shows the percentage of monthly observations in which pd and cr changes are not of opposite sign. This can almost never happen without vd variation, while it is not uncommon in the data. Nonlinearity is also less of a factor in shorter samples. The bottom panel shows the percentile range of rolling five-year correlations for the models. Without vd variation, it is extremely unusual to see a five-year correlation more positive than −0.725 and a positive correlation is essentially impossible. In cases (II) and (III), by contrast, five-year periods of positive correlation are not unusual at all. Comparing the rolling correlations for case (II) shown in Fig. 7 to the analogous plot in the data Fig. 1, the model seems to accord well with observation. The model is able to reproduce the apparent time variation in the correlation despite the fact that the population correlation in the model is virtually constant (because of our simplifying assumption that the state variables vc , vd , and x are homoskedastic). The realized correlations vary simply because of variation in the realized magnitudes of the different types of shocks. When most realized variation in a given period is due to consumption uncertainty shocks, the standard intuition about co-movement holds and the correlation is close to −1. But when realized variation in consumption volatility is low compared with shocks to unpriced earnings volatility, the correlations approach zero and can even turn positive. Another key insight that emerges from the model is that the volatility of levered equity—an easily measured and widely studied quantity—is not an unambiguous measure of fundamental risk. This observation allows us to explain some puzzling non-relations. Several prominent studies (including Campbell, 1987; and Nelson, 1991) have reported negative empirical risk-reward relations for the market portfolio, while many others have reported insignificant findings. The first panel of Table 4 shows the population regression coefficients of realized index returns on the initial volatility of those returns. (As in Table 1, the index here is assumed to be 26

Fig. 7. Simulated conditional correlations. The figure shows sample correlations of credit spreads and log price-dividend ratios in rolling 60-month windows over a simulated 50-year history for A-rated issuers in calibration (II).

composed of identical A-rated firms with common pairwise cashflow correlation 0.25.) Without vd shocks, the coefficient in monthly regressions is 0.56, which is economically very large. An increase in volatility from 20% to 30% would increase equity expected returns by a sizable 560 basis points (all numbers are annualized). The corresponding numbers in the data, shown in cases (i) and (ii), are several times smaller at all horizons (and can be negative). Consistent with this, cases (II) and (III) imply no significant association at any horizon. The fact that the equity premium is not strongly related to equity volatility also explains why valuation ratios do not covary strongly with volatility. David and Veronesi (2013) point out that the lack of a negative relation is inconsistent with many equilibrium frameworks, including long-run risk type models. Moreover, Beeler and Campbell (2012) criticize long-run risk models on the grounds that they imply too strong an association between current price-dividend ratios and future stock market volatility. Neither critique applies to calibrations (II) and (III), however, because the vd component of σS is positively associated with pd. Table 4 shows the R2 from the regressions of future volatility on the price-dividend ratio and of the price-dividend ratio on lagged volatility. [The former is the regression run by Beeler and Campbell (2012). David and Veronesi (2013) report correlations of lagged volatility with the price-dividend ratio.] In 27

the calibration (I) the relations are much too strong. By contrast, cases (ii) and (III) yield numbers in line with the data. Table 4 Equity volatility and risk. In (I)–(III), the regressions are run in a ten thousand–year sample for an index of identical A-rated firms. The numbers in (i) are from Table 7B, Panel 2 of Beeler and Campbell (2012) and Table 3 of Bollerslev, Tauchen, and Zhou (2009), and those in (ii) are estimated from Standard & Poor’s 500 index excess returns and its log price-earnings ratio since January 1954.

Model

Data

(I)

(II)

(III)

(i)

(ii)

0.56

-0.05

-0.04

0.11

-0.357

proj (RI,t,t+3 | σI,t )

βˆ βˆ

0.54

-0.04

-0.03

0.055

proj (RI,t,t+12 | σI,t )

βˆ

0.51

-0.01

-0.01

0.192

proj (σI,t+1,t+12 | pdt )

R2

0.64

0.10

0.19

proj (pdt | σI,t−12,t−1 )

R2

0.68

0.11

0.21

proj (RI,t,t+1 | σI,t )

0.13

0.013 0.002

Summarizing, this section has shown that plausible calibrations of a model with systematic volatility in unpriced earnings shocks can quantitatively account for some fundamental puzzles in macro finance. Without such shocks, it is very hard for equilibrium models to account for the lack of a strong negative relations between aggregate credit spreads and price-dividend ratios or for lack of a strong positive relations between the risk of levered equity and its expected return. 4. Further tests We propose that variation in unpriced earnings risk is an important factor for understanding aggregate financial dynamics. In this section, we examine some testable implications of this hypothesis and provide direct supporting evidence. 4.1. Unpriced risk, disagreement, and stock volatility Section 3 shows that calibrations of our model can account for the important moments that motivated our study. The natural next step is to ask whether

28

observed fluctuations in the state variable that we introduce can be directly linked in the data to the financial quantities that we assert that it affects. Our primary motivation is to understand disagreement between debt and equity valuations, and we introduce the simple statistic pdcr, the sum of the normalized price-earnings ratio and the normalized credit spread for a fixed-rating portfolio, as a summary measure of that disagreement. Thus, an important test of the model is whether direct, exogenous measures of unpriced earnings volatility do, in fact, help to explain the evolution of pdcr. To assess this, we employ the estimator vˆd introduced in Section 3 whose time series is shown in Fig. 3. Details of the construction of this series are provided in Appendix A. Table 5 provides positive evidence that vˆd plays a significant role in explaining the evolution of disagreement observed in the data. In Panel A, Columns 1–4 show the projection of pdcr on the model’s state variables using the calibrations (II) and (III) of Section 3. (Throughout this section, model regressions are shown for the case of an A-rated firm. Calibration (I) had no variation in vd , and so the model regressions are not well defined.) These affirm that pdcr is strongly linked to vd in theory, with a univariate R2 as high as 62%. Columns 5–8 show empirical regressions using vˆd and other candidate explanatory variables when pdcr is constructed from the S&P 500 P/E ratio and the Moody’s Baa–Aaa credit spread. In all specifications, vˆd is statistically significant and explains an important amount of the variation in pdcr, though measurement error unsurprisingly lowers the R2 relative to the model regressions that use true values of the state variables. In terms of economic magnitude, the standard deviation of vˆd and pdcr are, respectively, 0.0347 and 0.994. Thus, based on the univariate regression in Column 5, a 1 standard deviation increase in vˆd implies a 0.436 standard deviation increase in pdcr. In Columns 6–8, the mean and dispersion of quarterly growth forecasts from the SPF are included as proxies for x and vc , because under the model these could be relevant for capturing nonlinearities. A number of other factors have been proposed in the empirical literature to explain the variation in credit spreads not due to changes in equity valuation. 29

Table 5 Explaining valuation disagreement. The table shows the results of regressions of valuation metrics on measures of fundamental uncertainty. Models (II) and (III) show the population values of the regressions in ten thousand–year simulations of the calibrations described in Section 3. In the data columns, pdt is the normalized log price-earnings ratio of Standard & Poor’s 500 index, crt , the normalized spread of Moody’s Baa and Aaa corporate bond yields, and pdcrt , their sum. Quarterly proxies x ˆ and vˆc are the mean and dispersion of real gross domestic product growth forecasts from the Survey of Professional Forecasters (SPF). Liabilities/Assets is the median for A-rated firms as described in the caption to Fig. 5. Expected inflation is next-quarter mean inflation forecast from the SPF. Numbers in parentheses are Newey-West t-statistics with four lags. ∗∗∗ , ∗∗ , and ∗ denote significance at the 1%, 5%, and 10% level, respectively. Panel A: Dependent variable: pdcrt Model (II)

vˆd,t

Model (III)

Data

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

23.7

23.1

13.3

19.1

12.8∗∗∗ (3.21)

16.6∗∗∗ (4.19)

14.7∗∗∗ (5.64)

14.6∗∗∗ (5.73)

vˆc,t

83.6

-65

-1,660∗∗∗ (-3.72)

64.7 (0.21)

2,857∗ (1.65)

x ˆt

28.9

29.9

6.9 (1.04)

1.6 (0.34)

-7.1 (-1.00)

-29.3∗∗∗ (-4.35)

-32.3∗∗ (-2.11)

E[ inflationt+1 ] Liabilities /Assets Number of observations R2

8.8 (1.41)

0.62

0.87

0.30

0.47

199

177

177

109

0.20

0.30

0.51

0.52

Panel B. Dependent variable: pdt Model (II)

vˆd,t

Model (III)

Data

(1)

(2)

(3)

(4)

(5)

(6)

9.41

9.16

-7.67

8.86

-5.56∗ (-1.84)

-1.04 (-0.31)

vˆc,t

-599

-494

-2,424∗∗∗ (-5.04)

x ˆt

53.3

55.0

7.23 (0.97)

Number of observations R2

0.06

0.99

0.06

30

0.88

199

177

0.03

0.26

Table 5 (continued) Panel C. Dependent variable: crt Model (II)

vˆd,t

Data

(1)

(2)

(3)

(4)

(5)

(6)

13.7

14.0

20.5

10.3

18.62∗∗∗ (5.46)

17.94∗∗∗ (4.55)

vˆc,t x ˆt Number of observations R2

Model (III)

0.12

683

429

764∗∗ (2.18)

-24.0

25.5

-0.31 (-0.07)

0.92

0.42

0.79

199

177

0.40

0.41

Here we verify that two such factors that are outside of the model—inflation and changes in debt quantities—are not driving our results. Column 7 includes the next-quarter mean inflation forecast from the SPF based on the idea that inflation is unambiguously bad for nominal debt, whereas equity could be less exposed to nominal price risk. We do find a significant explanatory role for expected inflation, although its sign is the opposite of that expected. A negative coefficient implies that higher inflation raises debt values (lowers credit spreads) relative to equity.15 Changes to debt levels may also, in theory, have opposite effects on firm bond and stock prices. We suggest above that these are unlikely to be a significant factor in aggregate indicies, especially when holding credit rating fixed. Column 8 verifies this by including an index of book debt that is plotted in Fig. 5. Its coefficient has the expected sign, but it is not statistically significant. Importantly, the additional controls increase the significance of vˆd . Decomposing the separate responses of pd and cr to vˆd sheds interesting light on the role of correlations among the state variables. Panel B of the table shows that, under the model, the univariate response of stock valuations to 15 This finding is due to the high correlation between inflation and aggregate uncertainty. Our inflation variable could be a better proxy for vc than the SPF real GDP forecast dispersion. We thank the referee for suggesting this test.

31

an increase in vd need not be positive when, as in case (III), these shocks are also associated with increases in vc and decreases in x. Here, the empirical regressions suggest that the correlations in case (III) could be important. We do not find evidence of a positive association between vˆd and pd.16 In Panel C, we verify that, regardless of the correlations, the model unambiguously predicts a positive relation between vˆd and cr. The empirical relations here accord well with this prediction. Finally, for robustness, Table 6 shows pdcr regressions using some alternative specifications and measures. The top panel continues to employ our primary earnings dispersion measure and shows annual and quarterly regressions with and without an extra lag. Panel B borrows two alternative proxies for vd proposed in the literature. First, vˆd in Columns 1 to 3 is taken from Gilchrist, Sim, and Zakrajsek (2010), who argue that the time fixed effect coefficients in a panel AR(1) model of individual stock volatility should proxy for firm-specific dispersion. Second, in Columns 4 to 6, vˆd is the one estimated by Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011) described in Section 3.

In all cases, vˆd is positively and significantly associated with observed

disagreement. The results in these two tables constitute strong evidence in favor of the paper’s primary hypothesis. Direct, exogenous measures of a particular type of uncertainty help account for observed divergences in stock and bond markets as predicted by the theory. This is a notable success for risk-based asset pricing in general, as well as for the model’s bivariate characterization of fundamental risk. A second key implication of the model is that vd plays a substantial role in the dynamics of stock return volatility. This relation is the basis for our assertion that the model can account for the lack of an association between stock volatility and valuation or expected returns. Table 7 reports results of 16 Jorgensen, Li, and Sadka (2012) report a positive association between stock returns and innovations to earnings dispersion, which is consistent with case (II).

32

Table 6 Disagreement regressions: alternative specifications. The dependent variable is pdcrt , the sum of normalized Standard & Poor’s 500 log price-earnings ratio and Moody’s Baa–Aaa credit spreads. The table uses its average over a quarter in Columns 1 to 3 and over a year in Columns 4 to 6. vˆd in Panel A is estimated as the variance of residuals from regressing each firm’s earnings growth on consumption growth. In Panel B, vˆd in Columns 1 to 3 is the micro-level uncertainty suggested by Gilchrist, Sim, and Zakrajsek (2010) and those in Columns 4 to 6 is the productivity dispersion provided by Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011). Numbers in parentheses are Newey and West t-statistics with four lags. ∗∗∗ , ∗∗ , and ∗ denote significance at the 1%, 5%, and 10% level, respectively.

Panel A: vˆd as the variance of earnings growth residuals (1) vˆd,t

(2)

∗∗∗

12.81 (3.21)

∗∗

8.60 (2.51)

R2

4.20 (2.20)

(4) ∗∗∗

17.86 (3.20)

(5)

(6) ∗

11.47 (1.85)

15.09∗∗∗ (4.54)

12.43∗∗∗ (2.76) 0.83∗∗∗ (15.35)

pdcrt−1

Number of observations

∗∗

7.15∗∗∗ (3.80)

vˆd,t−1

Frequency

(3)

0.58∗∗∗ (5.88)

Quarterly

Quarterly

Quarterly

Annual

Annual

Annual

199

198

198

50

49

49

0.196

0.234

0.797

0.250

0.337

0.576

Panel B: Alternative measures of vˆd Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011).

Gilchrist, Sim, and Zakrajsek (2010)

vˆd,t

(1)

(2)

(3)

(1)

(2)

(3)

2.87∗∗∗ (3.60)

1.75∗∗∗ (2.79)

1.12∗ (1.91)

14.44∗∗∗ (4.48)

7.78∗ (1.94)

10.54∗∗∗ (5.20)

2.34∗∗∗ (5.83)

vˆd,t−1

0.83∗∗∗ (15.11)

pdcrt−1

Frequency Number of observations R2

10.45∗∗∗ (3.74) 0.49∗∗∗ (5.85)

Quarterly

Quarterly

Quarterly

Annual

Annual

Annual

192

191

192

37

36

37

0.189

0.286

0.809

0.450

0.560

0.631

33

Table 7 ˆc . Regression of stock return volatility on σ ˆd and σ In Panel A, the dependent variables in Columns 5 and 6 are the average standard deviation of A-rated firms’ daily stock returns; those in Column 7 the standard deviation of Standard & Poor’s (S&P) 500 index daily returns; and those in Column 8 the average Chicago Board Options Exchange Market Volatility Index (VIX). In Panel B, the regressions in Columns 6–8 of Panel A are repeated with different sources of σ ˆc as described in the text. The regressions are quarterly. All variables are annualized. The samples start in 1985Q4 in Columns 5 and 6, 1968Q4 in Column 7 and 1990Q1 in Column 8. Numbers in parentheses are Newey and West t-statistics with four lags. ***, **, and ∗ denote significance at the 1%, 5%, and 10% level, respectively.

Panel A: σ ˆc from the forecast dispersion of real gross domestic product growth Model (II)

Model (III)

Data

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

σ ˆd,t

1.17

1.15

1.18

1.08

2.03∗∗∗ (3.47)

1.86∗∗∗ (3.51)

0.79∗∗ (2.20)

1.06∗∗∗ (4.17)

σ ˆc,t

–

3.12

–

2.18

8.20∗∗ (2.56)

-0.82 (-0.94)

8.35∗∗∗ (4.27)

A-firms

A-firms

S&P 500

VIX

108

108

176

91

0.38

0.42

0.15

0.44

Dependent variable Number of observations R2

0.78

0.97

0.86

0.98

Panel B: Alternative measures of σ ˆc Baker, Bloom, and Davis (2012)

standard deviation of real consumption growth

(6)

(7)

(8)

(6)

(7)

(8)

1.823∗∗∗ (3.440) 0.082∗∗ (1.974)

1.159∗∗∗ (3.206) 0.098∗∗∗ (3.334)

1.059∗∗∗ (4.142) 0.084∗∗∗ (3.427)

2.045∗∗∗ (3.521) -0.555 (-0.518)

0.731∗∗ (2.569) -0.393 (-0.392)

1.274∗∗∗ (4.213) 0.163 (0.102)

A-firms

S&P 500

VIX

A-firms

S&P 500

VIX

Number of observations

108

111

91

108

198

91

R2

0.42

0.37

0.50

0.38

0.17

0.38

σ ˆd,t σ ˆc,t

Dependent variable

34

regressions using levered equity volatility as the dependent variable. Panel A shows the role of the state variables in the models and in the data. The first four columns affirm that vd is the dominant factor in stock volatility in both calibrations. The four regressions in Columns 5–8 use differing measures of stock volatility in quarterly regressions. The results verify that 1/2

our primary proxy, σ ˆd = vˆd , does have a significant association with stock volatility, accounting for as much as 38% of the observed variation. Moreover, the magnitude of the coefficient is similar to that found in the model simulations and economically large. Using the estimates in Column 5, a 1 standard deviation (4.42%) increase in σ ˆd implies an increase of 4.68% in annualized stock return volatility. (The sample mean and standard deviation of the annualized stock return volatility is 28.7% and 9.72%, respectively.) The finding here is noteworthy given the conventional wisdom that much of stock market volatility is not associated with fundamentals. Our uncertainty proxy is constructed directly from profitability data, without reference to any financial market information. Like all consumption-based models, our model also implies that consumption volatility should be positively related to stock volatility. In this regard, the result are mixed. In Panel A, which uses the SPF forecast dispersion for σ ˆc , a positive relation is found in two of three specifications. Panel B tries σ ˆc measures from two alternative sources. In the first three columns we use a series taken from Baker, Bloom, and Davis (2012), who create an economic policy uncertainty index as a weighted average of news coverage about economic uncertainty, tax code expiration data, and economic forecaster dispersion.17

The last three

columns use the standard deviation of real consumption expenditure growth rate over the past 12 months. The predicted positive relation with σ ˆc shows up strongly with the former measure, but not at all with the latter. However, unpriced earnings risk, σ ˆd , continues to enter significantly in all specifications. 17 See

http://www.policyuncertainty.com/us_monthly.html.

35

4.2. Unpriced earnings risk and expected returns Having verified the predicted associations between unpriced earnings risk and the main financial quantities that we originally sought to understand, we now turn to the model’s distinct predictions regarding risk premia. As described in Section 2, the model implies that vc and vd are, respectively, positively and negatively related to equity expected returns for fixed-rating portfolios. (For corporate bond returns, weaker forecasting relations are implied, with little or no effect from vˆd .) The equity prediction for vˆd is noteworthy. A rational, frictionless asset pricing model suggests a negative association between a type of risk and a measure of expected return. To begin, Table 8 builds on the preceding results and assesses the forecasting ability of our direct measures of uncertainty, vˆd and vˆc . (II) and (III) in Panel A verify the comparative statics of Section 2 and show the predicted strength of the relations in the two calibrations.18 In Column 5–7 of Panel A, the dependent variable is the S&P 500 excess returns in the next quarter. Column 5 uses the SPF dispersion proxy for vˆc , and Columns 6 and 7 use the Baker, Bloom, and Davis (2012) measure that was successfully tied to stock volatility above. Column 7 replaces our quarterly dispersion measure for vˆd with the annual measure of Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011). All specifications yield the predicted signs of the coefficients, and two of the three produce significant estimated coefficients on vˆd . Moreover, even in the statistically weak result in Column 5, the economic magnitude of the equity premium response is not negligible. Using the coefficient estimate there, a 1 standard deviation increase in vˆd (0.0347) implies a decrease in the equity risk premium of 167 basis points. Panel B shows the relation between the uncertainty measures and corporate bond expected excess returns in the data and the model. The model columns establish that, even though vd has a significant positive association with credit 18 In the model columns, the dependent variable is the true risk premium on levered equity, not realized returns. So the R2 numbers are not comparable to the data regressions for which the true risk premium is unobserved.

36

Table 8 Stock market return forecast by vˆd and vˆc . The table shows regressions of stock and bond market excess returns on lagged measures of fundamental risk. In the model columns, the dependent variables are the true risk premia, πS,t in Panel A and πF,t in Panel B. vˆc,t is the dispersion of real gross domestic product forecast from the Survey of Professional Forecasters in Column 5, and it is taken from Baker, Bloom, and Davis (2012) in Columns 6 and 7. vˆd,t is the dispersion of residual earnings in Columns 5 and 6, and it is taken from Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011) in Column 7. Regressions in Columns 5 and 6 are quarterly, while those in Column 7 are annual. Stock market returns in Panel A are for the Standard & Poor’s 500 indexes, and the corporate bond returns in Panel B the quarterly returns of the Vanguard Short-Term Investment-Grade Fund (VFSTX). Excess returns are their returns over three-month Treasury bill rates one quarter ago. Numbers in parentheses are Newey and West t-statistics with four lags. ∗∗∗ , ∗∗ , and ∗ denote significance at the 1%, 5%, and 10% level, respectively.

Panel A: Forecasting stock market excess returns Model (II)

Model (III)

Data

(1)

(2)

(3)

(4)

(5)

(6)

(7)

vˆd,t

-1.12

-1.1

-0.68

-1.0

-0.481 (-0.548)

-2.352∗ (-1.881)

-5.473∗∗ (-2.065)

vˆc,t

–

25.4

–

27.7

111.113 (0.952)

2.371∗∗∗ (3.341)

3.359 (0.365)

177

112

24

0.01

0.06

0.15

Number of observations R2

0.44

0.99

0.22

0.90

Panel B: Forecasting corporate bond excess returns vˆd,t

-0.02

-0.02

0.04

0.01

0.343 (1.249)

0.379 (1.558)

0.576 (1.498)

vˆc,t

–

2.1

–

2.13

63.947 (0.800)

0.310∗∗ (2.545)

3.734∗ (1.741)

121

112

24

0.04

0.09

0.20

Number of observations R2

0.02

0.52

0.02

0.13

spreads, this does not translate into a significant relation with bond risk premia. Instead, in the model, bond risk premia are more responsive to aggregate consumption risk. The data columns, which employ the excess returns of an investment-grade bond mutual fund as dependent variable, appear to bear this out. There is a positive significant relation with vˆc in two of the three estimations and no significant relation with the vˆd proxies. 37

To this point, we rely exclusively on our non-financial, exogenous measures of uncertainty. However, the model also delivers falsifiable predictions about the relations among financial quantities that have not previously been examined in the literature. In particular, the converse of our assertion that the volatility of unpriced earnings drives the level of disagreement between debt and equity is the implication that the divergence itself should be a good proxy for vd . Therefore, the model predicts that it should be negatively related to the risk premium of levered equity. We assess this prediction utilizing versions of pdcr for different portfolios in monthly forecasting regressions.

In addition, motivated by the model’s

prediction that the component of levered equity’s volatility not due to vd is related to aggregate risk, we build a monthly proxy for vc , denoted σ⊥ , by orthogonalizing realized stock return volatility with respect to pdcr. From an empirical standpoint, these measures are observable more often and with greater precision than is the case for the accounting-based fundamental measures utilized above. (I)–(III) of Table 9 show the predicted relations in the three calibrations of the model. Cases (II) and (III) verify that the proxies work as conjectured. While not reported in the table, it is the case that pdcr is responsible for almost all of the predictability for equity returns, whereas the predictability for bonds comes entirely through σ⊥ . In model (I), where there is no variability in vd , virtually all of the forecastability for both debt and equity returns comes through σ⊥ . Moreover, this case predicts that both forecasting relations should be extremely strong. (It also implies a positive coefficient on pdcr in both regressions.) The data strongly support the implications of calibrations (II) and (III). Columns 1–6 of Panel A show results for six different sets of firms as described in the table caption. For each column, pdcr and σ⊥ are constructed from the credit spreads, valuation ratios, and stock volatility of the respective firms. As in cases (II) and (II) but not case (I), for all the samples, pdcr is negatively associated with future returns, and the association with σ⊥ is positive. Both 38

39 0.17

0.22

0.014

-0.027

(III)

R2 0.90

0.004

σ⊥

Number of observations

0.008

pdcr

0.08

0.001

0.001

0.10

0.003

0

Panel B: Forecasting corporate bond excess returns

R2 0.99

0.009

0.03

σ⊥

Number of observations

-0.023

0.031

(II)

pdcr

(I)

Model

Panel A: Forecasting stock market excess returns

0.09

0.06

181

0.03∗∗ (2.33)

0.05∗∗∗ (2.65) 181

0.01 (1.11)

0.06

0.01 (0.89)

0.07

181

0.20∗∗∗ (3.62)

0.21∗∗ (2.29) 181

-0.08 (-2.15)

∗∗

(2)

-0.10 (-2.13)

∗∗

(1)

0.16

181

0.06∗∗∗ (3.54)

-0.00 (-0.24)

0.08

181

0.20∗∗∗ (2.60)

-0.11 (-2.34)

∗∗

(3)

Data

0.07

181

0.03∗∗∗ (3.47)

0.01 (1.23)

0.13

181

0.16∗∗∗ (3.38)

-0.16 (-3.10)

∗∗∗

(4)

0.08

181

0.02∗∗ (2.12)

0.02∗∗ (1.96)

0.03

181

0.11 (1.36)

-0.06 (-1.22)

(5)

0.13

179

0.04∗∗∗ (3.02)

-0.01 (-0.57)

0.09

179

0.10∗∗ (2.09)

-0.10∗∗ (-2.54)

(6)

Table 9 Risk premium forecasts. The table shows regressions of stock and bond market excess returns on lagged financial variables. In the model columns, the dependent variables are the true risk premia, πS,t in Panel A and πF,t Panel B. For each regression, pdcr is the sum of normalized log price-earnings ratios and credit spreads for a specified portfolio of firms, and σ⊥ is the realized portfolio equity return volatility orthogonalized with respect to pdcr. The variables are estimated from Standard & Poor’s (S&P) 500 stocks in Column 1, from AAA/AA/A/BBB-rated firms in Columns 2 to 5, and from financial firms in Column 6. Details of each are given in Appendix B. The dependent variables are, in Panel A, the excess returns of the S&P 500 index and, in Panel B, those of the Vanguard Short-Term Investment-Grade Fund (VFSTX). Returns are the excess over one-month Treasury bill rates from Ibbotson Associates and are annualized. Numbers in parentheses are Newey and West t-statistics with 12 lags. ∗∗∗ , ∗∗ , and ∗ denote significance at the 1%, 5%, and 10% level, respectively.

of the relations are statistically significant in most samples, with pdcr being stronger than σ⊥ . It is notable that purging equity volatility of the component associated with pdcr is sufficient to restore a positive risk-reward relation, as predicted. The predictability of stock returns by pdcr is a new result that verifies a central implication of the model.19 Moreover, the results are not merely rediscovering the well-documented forecasting power of the price-dividend ratio itself. In similar monthly regressions using pd alone or in combination with untransformed stock volatility we find statistically insignificant coefficients for both predictors. In the S&P 500 sample in Column 1, for example, the R2 from such regressions is 0.01. Moreover, the R2 s from unrestricted regressions using all three variables (pd, cr, and realized volatility) are insignificantly higher (0.08 versus 0.07 in Column 1) than using the two predictors constructed from them as per the model, supporting the restrictions implied by the model.20 Turning to bond returns, in Panel B the same independent variables are used in Columns 1–6 but the dependent variable is the monthly excess return from holding the Vanguard Short-Term Corporate Bond Fund. We again find the pattern predicted by calibrations (II) and (III). Consistent with the model, a strong positive association exists with σ⊥ , while there is no consistent relation with pdcr. For both bond and stock returns the magnitudes of the predictability effects in the data are economically large, exceeding the model values by several times. The estimation in Table 9 covers the period from December 1996, which is the period spanned by our cleanest credit spread data, the BofA series. To check whether the findings are specific to this time span and other measurement details, Table 10 repeats the regressions with alternative data. The table focuses only on the S&P 500 portfolio firms because the index P/E 19 The finding here is robust to correction for small sample predictability bias (Stambaugh, 1999). Bias-adjusted p-values for the ordinary least squares t-statistics in the A-regression, for example, are 0.0029 and 0.0133 for the coefficients on pdcr and σ⊥ , respectively. Full results are available upon request. 20 These results are omitted for brevity but are available upon request.

40

41

1990m1 ∼2013m6

Sample

295

1988m11 ∼2013m6

Realized volatility

five-year interest rate swap spreads

(2)

252

1991m1 ∼2011m12

Realized volatility

Moody’s intermediate-term AA spreads

(3)

0.019 (0.742)

0.073∗∗ (1.971) 0.032

σ⊥

R2 0.051

0.106∗∗∗ (2.635)

-0.086∗∗ (-2.537)

0.045

0.037 (1.425)

-0.087∗∗∗ (-3.683)

331

1985m11 ∼2013m6

Realized volatility

Moody’s Aaa yields less 30-year T-bond yields

(4)

R2

σ⊥

pdcr

0.020

0.001 (0.106)

0.012∗∗ (2.382)

0.054

0.021∗∗∗ (3.351)

0.017∗∗∗ (4.161) 0.065

0.009∗ (1.703)

-0.006∗ (-1.724)

0.055

0.016∗∗∗ (4.206)

-0.004 (-1.214)

Panel C: Dependent variable: Vanguard short-term corporate bond fund returns

0.047

-0.085∗∗∗ (-3.721)

pdcr

-0.076∗∗ (-2.295)

Panel B: Dependent variable: S&P 500 excess returns

281

VIX

Volatility

Number of observations

Moody’s Baa–Aaa spreads

Spreads

Panel A: Sample characteristics

(1)

0.047

0.019∗∗∗ (4.018)

0.002 (0.384)

0.059

0.084∗∗∗ (2.741)

-0.094∗∗∗ (-3.586)

331

1985m11 ∼2013m6

Realized volatility

Moody’s Baa yields less 30-year T-bond yields

(5)

0.025

0.013∗∗∗ (2.743)

0.002 (0.467)

0.003

0.030 (1.362)

-0.003 (-0.192)

713 (368)

1954m1 ∼2013m6

Realized volatility

Moody’s Baa–Aaa

(6)

0.032

0.012∗∗∗ (2.643)

0.005 (1.014)

0.026

0.050 (1.642)

-0.098∗∗∗ (-2.867)

331

1985m11 ∼2013m6

Realized volatility

Moody’s Baa–Aaa

(7)

Table 10 Alternative data sources. The table repeats the regressions of Table 9 with different data sources. Sample characteristics are specified in Panel A. Sources are given in Appendix B. The dependent variable in Panel B is Standard & Poor’s (S&P) 500 index excess returns in the next month, and that in Panel C is Vanguard intermediate-term corporate bond fund excess returns. The excess returns are annualized. Realized volatility is estimated as the standard deviation of the S&P 500 index daily returns over the past six months. VIX denotes the Chicago Board Options Exchange Market Volatility Index. Numbers in parentheses are ordinary least squares t-statistics. ∗∗∗ , ∗∗ , and ∗ denote significance at the 1%, 5%, and 10% level, respectively.

ratio is available starting in January 1954. In Column 1, the Chicago Board Options Exchange Market Volatility Index (VIX) is used in place of realized volatility to construct σ⊥ . In Columns 2–7, different credit spread series are employed over different time periods, as specified in Panel A. (Data sources are given in Appendix B.) The dependent variables in Panel A are the S&P 500 index’s excess returns in the next month, and those in Panel B are again the Vanguard fund returns, which start in January 1983. The results in Panels B and C confirm those of the previous tests, regardless of the data source and time period.

Equity returns are always negatively

associated with pdcr. Both equity and debt returns are positively associated with σ⊥ . The strength of the pdcr effect continues to be large and significant, with one exception: Stock return forecastability is weaker when the sample is extended back to 1954.21 For the other samples, the point estimates for both predictors are not as extreme as in the previous table and come closer to the levels implied by the model. Summarizing the results of this section, we show that the model introduced here can go beyond its success with unconditional moments. We present positive evidence of time series associations between observed fluctuations in financial quantities and direct measures of exogenous uncertainty or proxies for them implied by the model. The results are consistent with our assertion that variation in the risk of unpriced earnings shocks is an important driver of debt and equity valutions and expected returns. 5. Conclusion This paper introduces a simple addition to a standard model of asset prices that has powerful consequences. Motivated by the often persistent divergence in debt and equity valuations (holding credit rating fixed), we propose distinguishing 21 A possible explanation is that the Moody’s credit spread data do not adjust for bond call options, which induces significant measurement error during periods such as the late 1970s having high volatility of nominal rates. In (unreported) results, we find that the predictive power of the raw, untransformed variables (credit spreads, volatility, and the price-dividend ratio) disappears in this sample as well.

42

between two components of systematic stochastic volatility corresponding to the priced and unpriced components of firms’ cash flows. The finance literature has previously appreciated that variation in unpriced risk could account for a variety of effects in the cross section of stock returns. Recent macroeconomic literature has advanced the idea that firm-specific uncertainty could have a systematic component that evolves separately from aggregate uncertainty. Putting these ideas together not only can account quantitatively for the degree of bond-stock disagreement, but it also can explain the absence of a positive risk-reward relation for the market portfolio.

Given the power and parsimony of this

one idea, we believe it should become a standard feature of our description of aggregate financial quantities. The new dimension of risk that we highlight is not merely a modeling device. We directly measure it in the data and establish that its fluctuations are economically large and distinct from fluctuations in measures of aggregate consumption risk. Its movements are linked to movements in bond and stock valuations and return moments in the way that the model says they should. We describe unpriced earnings shocks as ones that come from rearrangements of output shares between firms or between capital and labor. These can occur at the firm, sector, or economywide levels. Empirically, our identification of uncertainty about these shocks comes from the idiosyncratic part (via cross-firm dispersion in earnings). An intriguing topic for future investigation is to try to measure the component we are missing: fluctuations in uncertainty about the capital share of aggregate profits. This would be an important first step in understanding the economic forces driving distinct components of uncertainty.

43

Appendix A. Estimation method of σ ˆ d and σ ˆc This Appendix explains how we estimate the two components of fundamental uncertainty for our empirical analysis. Start with σ ˆd , which is supposed to measure the volatility of earnings growth shocks that are not explained by consumption growth. First, for each firm, earnings growth is regressed on consumption growth. ∆dt+1 = α + φ1 ∆ct+1 + β0 ∆dt + β1 ∆dt−1 + β2 ∆dt−2 + β3 ∆dt−3 + φ2 σd,t edt+1 (17) where ∆ct+1 ≡

Earnings t ∆dt+1 ≡ log CPIt+1t+1 − log Earnings (log real earnings growth) and CPIt Consumptiont+1 Consumptiont log − log (log real consumption growth). CPIt+1 CPIt

Earnings data are collected from Compustat item oibdpq, which represents quarterly operating income before depreciation, and nominal consumption expenditure data are provided by the Federal Reserve Economic Data (FRED) database.22 Lags of ∆dt are added on the right hand side to control for seasonality and other omitted factors in the conditional mean. Observations in which any of the oibdpq is negative are excluded from the regression. (Approximately 87.3% of firm quarters have positive values of oibdpq.) In the specification above, φ2 and σd are not separately identified. We estimate φ2 for each firm as its residual standard deviation. Dividing each firm’s residuals by this quantity gives a time series of normalized residuals in which the unconditional level of σd is normalized to unity. We then measure time series variation in the cross-sectional dispersion of these normalized residuals and then un-normalize this series by multiplying by the average φ2 across firms. For this second step, we restrict the set of stocks by imposing the minimum inflation-adjusted market capitalization of 1963Q4 ($27,143,000 in 1963 dollars), which is the quarter having the smallest set of firm observations on Compustat. The goal here is to control for changing composition of the Compustat sample over time, which would induce spurious changes in cross-firm dispersion. (In 22 See

http://research.stlouisfed.org/fred2/series/PCE.

44

Table A1 Summary statistics of σ ˆd and σ ˆc .

Mean

Standard deviation

5th percentile

50th percentile

95th percentile

σ ˆd

0.3796

0.0449

0.3079

0.3762

0.4528

vˆd

0.1461

0.0353

0.0948

0.1415

0.2050

σ ˆc

0.0108

0.0065

0.0038

0.0089

0.0253

vˆc

0.0002

0.0002

0.0000

0.0001

0.0006

particular, Compustat coverage of small stocks increases drastically in the 1970s.) Then in the final step, for each quarter, σ ˆd is estimated as the standard deviation of the normalized residuals of the included stocks. It is then re-scaled √ by the median φ2 , annualized by 2 (= 4), and then seasonality-adjusted by subtracting quarterly means. We define vˆd as the square of σ ˆd . Table A1 shows summary statistics for σ ˆd and vˆd . σ ˆc is estimated from the forecast dispersion of real GDP growth rates by the Survey of Professional Forecasters.23 Because the SPF provides the difference of 25 and 75 percentiles of forecasts, we scale the dispersion by 1 / (2 × 0.67) because P (X ≤ 0.67 σ) = 0.75 and P (X ≤ −0.67 σ) = 0.25 for X ∼ N ( 0, σ 2 ). Appendix B. Data samples This Appendix describes the samples for which empirical results are reported in Tables 2–10. We have two principal sources of credit spreads. One is BofA Merrill Lynch Corporate Bond Spreads and the other is Moody’s Corporate Bond Baa–Aaa Spreads.24 The pros of BofA spreads are that they are adjusted for option values of corporate bonds, are well diversified, and are provided for each credit rating. However, one pitfall is its short history dating back only up to December 23 See http://www.phil.frb.org/research-and-data/real-time-center/ spf-forecast-dispersion.cfm. 24 See http://research.stlouisfed.org/fred2/categories/32297 and http://research. stlouisfed.org/fred2/categories/119.

45

1996. In contrast, Moody’s spreads have been available for nearly one hundred years, since 1919, but they do not take into account option-value adjustment. Moreover, Moody’s spreads set its maturity as close as possible to 30 years despite the fact that even the US government had not issued 30-year Treasury bonds until 1985. In addition, five-year interest rate swap spreads (downloaded from Bloomberg) are used as the credit spreads of financial-firm portfolio in Table 9. Also, Moody’s US Intermediate-Term Corporates Spreads are used for a robustness check in Table 10.25 Table B1 shows the summary statistics of credit spreads from these four data sources along with their available sample horizons. Table B1 Summary statistics of credit spreads. The table shows summary statistics of credit spreads from different sources along with their available sample horizons. The summary statistics are in percentages.

Rating

Sample horizon

Mean

Standard deviation

Bank of America Merrill Lynch Corporate Bond Spreads AAA AA A BBB

1996m12 1996m12 1996m12 1996m12

∼ ∼ ∼ ∼

2013m8 2013m8 2013m8 2013m8

0.854 1.083 1.446 2.120

0.626 0.791 1.005 1.217

5.847 7.044 1.197

2.709 2.863 0.715

Moody’s Corporate Bond Yields Aaa Baa Baa–Aaa

1919m1 ∼ 2013m8 1919m1 ∼ 2013m8 1919m1 ∼ 2013m8

Moody’s US Intermediate-Term Corporate Spreads AAA AA A

1991m1 ∼ 2011m12 1991m1 ∼ 2011m12 1991m1 ∼ 2011m12

0.825 1.169 1.350

0.391 0.804 0.861

0.436

0.208

Interest rate swap spreads five-year

1988m11 ∼ 2013m8

Table 2 reports sample moments of real risk-free interest rates (rf ). We 25 See http://credittrends.moodys.com/pro/chartroom_chart.asp?status=1&script_ name=/pro/chartroom_chart.asp&cid=77.

46

compute real rates as nominal interest rates less expected inflation, the latter of which had been unobservable in early periods due to the lack of forecast surveys. Thus, we approximate the expected inflation as the predicted value of the following regression: ∆cpit,t+s = β0 +β1 ∆cpit−12,t +β2 ∆cpit−24,t−12 +β3 ∆cpit−24,t−36 +β4 ∆cpit−48,t−36 (18) where ∆cpit,t+s ≡ log

CP It+s CP It .

CPI data are provided by the St. Louis Fed.26

Nominal one-month risk-free interest rates are by the Ibbotson Associates and the nominal four-year risk-free rates are from Fama-Bliss Discount Bond Yields of the Center for Research in Security Prices (CRSP).27 The moments of rf,1mo and rf,4yr in (i) to (iii) of Table 2 are estimated identically from these sources but differ in values due to the difference of sample periods. The moments of stock market excess returns, πI and σI , are estimated from S&P 500 index monthly returns less one-month nominal risk-free rates and then annualized. The moments of individual stock excess returns, πS and σS , are estimated for each A-rated firm and then annualized and averaged. The stock market valuation ratio, pd, is the log of S&P 500 index’s P/E ratio, which is provided by the Bloomberg field indx adj pe and dates back to January 1954. Market leverage is estimated as the ratio of the book value of total liabilities divided by the sum of the total liabilities and market capitalization for each firm using Compustat and CRSP data, and then averaged among A-rated firms. πS , σS , and market leverage in (ii) and (iii) are equal because Compustat provides the credit rating data only since December 1985. E[default prob] is taken from Moody’s average cumulative default rates for four-year maturity debts between 1970 and 2009.

Credit spreads (cr) are

provided by the BofA Merrill Lynch Corporate Bond Spreads in (i) and by the Moody’s Corporate Bond Baa–Aaa Spreads in (ii) and (iii). 26 See

http://research.stlouisfed.org/fred2/series/CPIAUCNS. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_ factors.html. 27 See

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Table 3 reports the correlation of debt and equity valuations. For each credit rating of AAA to BBB, pd is estimated from Compustat and CRSP and cr is from the BofA Merrill Lynch Corporate Bond Spreads. For S&P 500 index, pd is from Bloomberg and cr from Moody’s Baa–Aaa spreads. The benefit of using S&P 500 is that the sample horizon can be substantially expanded. Table 4 demonstrates the elusiveness of risk-reward relation. Its dependent variable, stock market excess return, is estimated using the S&P 500 total return index including dividends less nominal risk-free rate for a given period s. The nominal risk-free rate is from Ibbotson Associates for s = 1, three-month Treasury bill yields from the St. Louis Fed for s = 3, and one-year Fama-Bliss Discount Bond Yields for s = 12.28 σI,t and σI,t,t+12 are estimated as the standard deviation of daily S&P 500 index returns in one month and one year, respectively. pd is the log valuation ratio of S&P 500 index. All numbers are then annualized. Table 5 explains the valuation-disagreement variable, pdcr, which is the sum of normalized pd and cr. pd is from the log S&P 500 P/E ratio and cr from the Moody’s Baa–Aaa spreads. The estimates of vˆd and vˆc are explained in Appendix A. x ˆt and E[ inflationt+1 ] are the mean forecasts of real GDP (RGDP) and its price deflator (PDGP) from the Survey of Professional Forecasters. The book leverage ratio, Liabilities / Assets, is the ratio of total liabilities over total assets from Computstat. Table 6 reassures the results of Table 5. For example, the numbers in (1) of Panel A, Table 6, are equal to those in Column 5 of Panel A, Table 5. Panel A deals with alternative model specifications and Panel B with alternative estimation sources of vˆd . The first source is provided by Gilchrist, Sim, and Zakrajsek (2010), who estimate micro-uncertainty by the following four steps: (1) regress stock returns on Fama and French three factors, (2) estimate the quarterly standard deviation of its residuals, (3) run a panel regression of the standard deviations with firm and time fixed effects, and (4) take the estimated 28 See

http://research.stlouisfed.org/fred2/series/TB3MS.

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time fixed effects as the proxy of micro-uncertainty. The second source of vˆd is from Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry (2011), who estimate total factor productivity (TFP) from the residuals of regressing log outputs on log capital and labor inputs with time fixed effects, run a panel regression of the TFP, and use the cross-sectional standard deviation of the panel regression residuals as the proxy of micro-uncertainty. Table 7 looks to the covariation of stock return volatility with σ ˆc and σ ˆd . Stock return volatility is estimated in three different ways: the average standard deviation of A-rated firms’ daily stock returns, the quarterly standard deviation of S&P 500 index daily returns, and the average VIX. The last two are also downloaded from Bloomberg. σ ˆd is also employed in three different versions: our benchmark estimate in Panel A; Baker, Bloom, and Davis (2012)’s economic policy uncertainty index as a weighted average of news coverage about economic uncertainty, tax code expiration data, and economic forecaster dispersion in Panel B; and the standard deviation of real consumption growth rate over the past 12 months in Panel C.29 Table 9 uses six sets of firms and credit spreads to compute the return predictors denoted pdcr and σ⊥ . Column 1 uses the S&P 500 index. Its pdcr is computed from the 12-month trailing P/E ratio and the BofA option-adjusted A-credit spread. Columns 2–5 also use the BofA spreads for ratings AAA through BBB. For each rating tier, P/E ratios are computed for the portfolio of firms having that credit rating on Compustat, dividing each portfolio’s market capitalization by the sum of net income and depreciation over the past year. Column 6 is based on financial firms whose standard industrial classification codes begin with 6xxx and having credit ratings either AAA or AA except the Federal National Mortgage Association (Fannie Mae) and the Federal Home Loan Mortgage Corporation (Freddie Mac). Its P/E ratios are estimated from Compustat, and credit spreads are proxied by five-year interest rate swap spreads. For each column, stock return volatility is estimated from each stock’s daily 29 See

http://www.policyuncertainty.com/us_monthly.html.

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stock returns over the past six months and then their equal-weighted average is taken for each portfolio. Table 10 repeats the regressions with alternative data. In Column 1, the VIX is used in place of realized volatility to construct σ⊥ . In Column 2, five-year interest rate swap spreads substitute for A credit spreads. In Column 3, Moody’s US intermediate-term corporate spreads are used. Columns 4 and 5 use the spread of Moody’s Aaa/Baa corporate bond yields over 30-year Treasury bond yields. In Columns 6 to 7, Moody’s Baa–Aaa corporate bond spreads are used. Their corresponding sample periods are specified in Panel A.

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