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Up the stairs, down the elevator: valuation ratiosand shape predictability in the distribution of

stock returns ∗

Paolo Giordani † Michael Halling‡

ABSTRACTWhile a large literature on return predictability has shown a link between val-

uation levels and expected rates of returns, we document a robust link betweenvaluation levels and the shape of the distribution of cumulative (up to 24 months)total returns. Return distributions become more asymmetric and negatively skewedwhen valuation levels are high. In contrast, they are roughly symmetric when val-uation levels are low. These results shed some light on how equity prices regressback to their means conditional on valuation levels and have important practicalimplications for risk measurement and asset management.JEL Classifications: G12, G17, C22.Keywords: return predictability, valuation ratios, skewness.

∗Acknowledgments: We are grateful to John Cochrane, Paul Soderlind, Pietro Veronesi and sem-inar participants at the Swedish central bank and the University of St. Gallen. The views expressedare solely the responsibility of the authors and should not to be interpreted as reflecting the views ofthe Executive Board of Sveriges Riksbank. All errors are our responsibility.

†Central Bank of Sweden. Email: [email protected]‡Stockholm School of Economics. Email: [email protected]

1 Introduction

A large literature has looked at the time-variation in expected rates of returns andthe link between prices, dividends and discount rates (see, for example, Cochrane(2011) and Fama (2013) for recent discussions of that literature; Golez and Koudijs(2016) evaluate this link using four centuries of data). The goal of this paper is toprovide a fresh view on this important question in empirical asset pricing. Specif-ically, we look beyond the mean of the return distribution and focus on the shapeof the predictive return distribution. The key innovation of the paper is that wemodel the shape to depend on a valuation ratio such as the cyclically-adjustedprice-earnings ratio or the book-to-market ratio.

Our empirical analysis is motivated by a quick look into the data. Figure 2 re-ports histograms of observed cumulative 12-month total log-returns conditional onvaluation ratios being HIGH (top quartile) and LOW (bottom quartile) and high-lights a pronounced shift in the shape of the distribution: while it looks symmetricin the case of low valuation ratios, it becomes negatively skewed for higher valu-ation ratios. While many practitioners seem to be well aware of this conditionalasymmetry of the return distribution (for example, when they describe dynamicsin equity markets as “up the stairs, down the elevator”) we are not aware of anyacademic study that documents this strong and intuitive pattern and that models itin econometric terms.

Such an asymmetry has important economic implications. While the existingliterature on return predictability helps us understand the dynamics of time-varyingexpected rates of return, it does not explain how the reversion to that mean willactually occur. For example, when valuations are high (low), how will prices adjustreflecting the expected low (high) returns? Is this adjustment more likely to happensmoothly or rather abruptly? Another central question in this literature is aboutthe interpretation of long-lasting deviations of market values from fundamentals:do these patterns reflect (rational) bubbles? Put differently, why is the timing ofmarket reversals so difficult even at extremely high valuations? These are preciselythe questions we will address in this paper.

Specifically, we propose an econometric framework that is simple but flexibleenough to model the asymmetry of the return distribution as a function of a valu-ation ratio and, at the same time, nests the standard, linear predictive regressionsas a special case. In more detail, we compare the model with conditional skewnessto two benchmark models; one that implies a symmetric distribution, and one thatimplies a distributions with constant skewness. To reflect the well-known fact thatequity returns have fat tails and to avoid any confounding effects between excesskurtosis and skewness of the estimated return distributions, we use the (skew) T-distribution to model returns instead of the (skew) normal distribution. For some

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comparisons, however, we also refer back to the standard Gaussian model.Relative to the standard, linear predictive regression, our main empirical model

with conditional skewness has very similar implications for mean prediction. How-ever, the model is powerful enough to help us understand how regression to themean works. Using this framework and the standard US data, we find strong sta-tistical evidence that the shape of the return distribution varies conditional on thevaluation ratio and that the distribution becomes more negatively skewed whenvaluation ratios are high. Put differently, our empirical evidence documents that ifvaluations are high, regression to the mean is more likely to happen with extremelynegative returns (i.e., a crash); in contrast, if valuations are low it is more likely tohappen smoothly.

The model with conditional skewness is well-supported by the data. Its loglikelihood exceeds those of the competing benchmark models; and the parame-ter governing the link between valuation levels and the shape of the return dis-tribution is statistically significant. These results are very robust across differentsub-samples (pre-1945 and post-1945 samples), returns horizons (12-month and24-month returns) and proxies for valuation ratios (the cyclically-adjusted price-earnings ratio and the book-to-market ratio).

Interestingly, when valuation ratios are very high the most likely value of thefuture return (i.e., the mode of the predictive distribution) still remains positive (infact roughly unchanged) in our empirical analysis showing that timing the top ofa bull market is made inherently more difficult as a consequence of time-varyingskewness. Conversely, since at low valuations our predictive distributions becomeapproximately symmetric, very low valuations have higher power to forecast mar-ket direction.

Regarding the predictability of conditional mean returns, our results are in linewith the existing literature. One characteristic of a very asymmetric distribution,like the one we find when valuation ratios are high, is that it takes more observa-tions to learn about the conditional mean with sufficient accuracy. In contrast, forsymmetric distributions, like the ones we find for lowest valuation ratios, learningabout the conditional mean tends to happen faster. Thus, our results could pro-vide an explanation for the robust finding in the literature that evidence on meanpredictability is stronger during recessions than during expansions given that re-cessions (expansions) have some tendency to overlap with periods of low (high)valuations.

Our empirical results have important implications for investors, asset managersand risk managers. Obviously, ignoring that the return distribution becomes verynegatively skewed when valuation levels are high leads to severe underestimationof risk measures such as volatility, value-at-risk and expected-tail-loss. These is-sues of underestimation of risk hold for the standard Gaussian model as well as the

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two benchmark models that we evaluate empirically, a model assuming a symmet-ric T-distribution and a model assuming a T-distribution with constant skewness.For example, while a symmetric T-distribution, with parameters estimated fromthe full sample, implies a 1% value-at-risk of -45% (-36% simple returns) for 12-month cumulative total log returns when valuation levels are high, our model withconditional skewness implies a 1% value-at-risk of -71% (-51% simple returns) inthis case.

Interestingly, we observe the mirror-image of this pattern, albeit to a less ex-treme extent, when valuation levels are low. In this case, the Gaussian model andour benchmark models overestimate risk for any risk measure that we look at.For example, while a symmetric T-distribution estimates the 1% value-at-risk tobe -28% for 12-month cumulative total returns when valuation levels are low, ourmodel with conditional skewness implies a 1% value-at-risk of -23% in this case.Thus, from an investor’s point of view ignoring the conditional skewness is a lose-lose situation. For example, a mean-variance investor using a Gaussian model (or,more generally speaking, any estimate of realized volatility from a standard model)would invest too aggressively in the market when valuation ratios are already highbut too conservatively when valuation ratios are low.1

Several theories have been proposed to rationalize negative skewness in assetreturns. Among these the “leverage effect” (a drop in market valuations increasesleverage ratios and, as a consequence, increases volatility of subsequent returns)and the “volatility feedback effect” (bad news lowers future expected cash-flowsand increases the risk premium; good news, in contrast, increases future expectedcash-flows but, again increases the risk premium resulting in a dampened overalleffect) have been found to lack the quantitative importance to explain the data (see,for example, Bekaert and Wu (2000) and Poterba and Summers (1986)). Chen,Hong, and Stein (2001) propose and evaluate an alternative explanation basedon heterogeneous investors, differences in opinions and short-sale constraints forsome investors. Hueng and McDonald (2005), however, find no support for thisexplanation in the case of aggregate stock market returns.

Importantly, however, the theories discussed in the previous paragraph fail torationalize that the shape of the return distribution varies with valuation ratios.A theoretical motivation that overcomes this shortcoming is linked to stochasticrational bubbles, as first developed by Blanchard and Watson (1982). In thesemodels, the stock price is the sum of a fundamental price and a bubble compo-nent. The bubble is stochastic, as it continues with a given probability p and bursts

1Recently asset management strategies based on volatility, such as risk parity and volatility tar-geting, have become increasingly popular. Obviously, such strategies are very sensitive to accuratepredictions of future volatilities.

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with probability (1-p). Importantly, the model explicitly links the shape of thepredictive distribution to the valuation ratio. If one, for example, assumes thatthe fundamental price follows a symmetric distribution then the Blanchard-Watsonmodel implies a symmetric predictive distribution at low valuations (in this case,the bubble component is zero); at high valuations, however, the predictive distri-bution becomes increasingly left skewed as a mixture of two distributions. Thiswould be consistent with our empirical results.

Note, however, that we do not view our empirical results as evidence in supportof the existence of price bubbles. Instead, we view them as being merely consistentwith some, but not all, of these models’ predictions. If rational bubbles existed,prices would change while expected returns would not (see, for example, Cochrane(2011)). In our case, however, we find both, time-varying asymmetry, consistentwith the Blanchard-Watson model, and predictability in mean returns, consistentwith Campbell and Shiller (1988) and Cochrane (2008).

Finally, David and Veronesi (2014) develop a dynamic equilibrium model oflearning that also provides a rationalization for the link between valuation ratiosand the shape of the return distribution. In their model investors learn about differ-ent regimes in the fundamental value. During a boom period, positive news aboutfundamentals has little impact on investors’ beliefs; negative news, however, maylead to a large downward revision in beliefs; thus, in that situation investors per-ceive greater downside risk than in bad times. As a consequence, stock returns willbe negatively skewed in good times.

Interestingly, David and Veronesi (2014) also provide evidence from optionmarkets that is consistent with our empirical results. They find that the ratio of theimplied volatilities of out-of-the-money puts over out-of-the-money calls, an indi-cator of the market’s assessment of downside risk versus upside risk, raises duringexpansions and drops during recessions (i.e., it is pro-cyclical). Their analysesfocuses on three-months options and the sample period of 1988 to 2011. Our anal-ysis, instead, documents consistent patterns for long-term returns up to two yearsand is based on a much longer sample period.

Our paper also relates, more broadly speaking, to the literature on the non-normality of asset returns. Looking at daily or even higher-frequency returns, thisliterature finds excess kurtosis and negative skewness. It usually models condi-tional skewness as following an autoregressive process and models it jointly withconditional volatility (see, for example, Harvey and Siddique (1999) and Jondeauand Rockinger (2003)). Our approach is very different as we look at longer hori-zon returns (in particular, 12-month and 24-month cumulative returns) and are in-terested in understanding how the shape of the predictive distribution depends onvaluation ratios. It is also important to emphasize that skewness in daily returnsmay have little or no connection to time-varying asymmetric shapes of cumulative

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returns.The remainder of this paper is organized as follows. Section 2 introduces our

empirical specification, Section 3 describes the data, and Section 4 summarizes ourempirical results. Section 5, finally, concludes.

2 Model Specification

A skew-T distribution with deterministically varying parameters. Thestandard predictive regression is a linear projection of cumulative log returns ona valuation (ratio), also in logs, so the implicit model is

yt,t+h = β0 +β1xt + εt ,

where yt,t+h = log((Pt+h +Dt+1:t+h)/Pt) are cumulative total log returns over hperiods, xt is a log valuation ratio, and OLS estimation is optimal under the as-sumption that εt is Gaussian.

The most parsimonious and interpretable way to extend this model to capturethe idea that valuation ratios may also affect the shape of the distribution is tomove from a symmetric to an asymmetric distribution, where the asymmetry is afunction of valuation levels. A skew-normal distribution would be the most imme-diate extension of the regression model, but we prefer to be slightly more generaland opt for a skew-T distribution. Allowing for fat tails is always good practice,particularly with financial data, and in our case it is particularly important to mit-igate the risk of interpreting one or a few outliers as asymmetry or time-varyingasymmetry. Forcing a Gaussian distribution on fat-tailed data results in extremelynoisy estimates of skewness in repeated samples, particularly if skeweness is mea-sured as the centered third moment. In our sample the key results are little changed(t-statistics are even higher) if we force a high value for the degrees of freedom.However, since this restriction is strongly rejected by the data, we show results forthe more general and robust model, which is

yt,t+h ∼ skewt(mt ,σ,v,γt)

where skewt is the skew-T distribution of Fernandez and Steel (1998). Here mt isthe mode (location parameters), σ is the dispersion parameter, v are the degrees offreedom, 0< γt <∞ is the asymmetry (shape) parameter, and the model parametersare deterministic functions of a constant and xt as follows:

mt = β0,m +β1,mxt

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logσ = β0,σ

logv = β0,v

logγt = β0,γ +β1,γxt .

Notice that we work with logs of the dispersion and degrees-of-freedom pa-rameters, σ and v, and also model logγt rather than γt as a linear function of log-valuation xt . This makes the distribution p(yt,t+h|xt) well-defined for any value ofβ1,m and β1,γ.

This model nests the standard predictive regression as a special case with β1,γ =β0,γ = 0 and v fixed at a large number. If v is freely estimated, we have a regressionwith T rather than Gaussian errors. We will refer to this model as the Symmetric-TModel. An interesting comparison is with a model where the skew is fixed, so β0,γis freely estimated but β1,γ = 0. We will refer to this model as the Constant-Skew-T model. The main model of interest, however, is one in which we also estimateβ1,γ to see whether valuation ratios affect the shape of the return distribution. Wewill refer to this model as the Conditional-Skew-T model. In this paper we presentresults for a simplified Conditional-Skew-T model by imposing β1,m = 0. In oursample this restriction is never rejected using any standard selection criteria likeBIC or AIC, and when β1,m and β1,γ are estimated jointly β1,m is always smallwith t-statistics much lower than one. What this implies is that the mode of thedistribution is fixed, and as the distribution becomes more left (right) skewed, itsmean is lower (higher). Of course this does not have to be the case for other assetsor samples, where mt could either move left or, as in the Blanchard-Watson rationalbubble model (Blanchard and Watson (1982)), shift right at higher valuations.

We could consider an even more general version of the model by estimatingβ1,σ — also allowing the variance to be a function of valuations. This improvesthe fit to the data but does not have important implications for the analysis of theshape of the return distribution which represents the focus of this paper. Thus,we decided to focus on the simpler model throughout the paper. If our goal wasto maximize the fit to the data we would indeed need to model the dispersion astime-varying, and include more variables; see Li and Villani (2010) for an exampleof such a model fitted to daily stock return data (without including any measure ofvaluation).

Skewness, asymmetry, and some features of the skew-T distribution. Thereare several skew-T distributions available in the literature. Jones (2014), with aunivariate emphasis, and Lee and McLachlan (2013), with a multivariate empha-sis, provide excellent reviews. Most proposals are fairly recent and there is stillvery little applied work to guide a choice (Jones (2014)). We have opted for the

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version of Fernandez and Steel (1998) because, in their model, the role of eachparameter is easy to interpret; in particular, our main hypothesis — that asymme-try varies with valuations — is captured by just one parameter. It also nests thestandard regression equation and its likelihood is available in closed form, whichaids in the estimation.

The idea of Fernandez and Steel (1998) is to introduce an inverse scale factorin the positive and negative orthants, so that if the distribution f (εt) is unimodaland symmetric around zero, then we can create a skewed distribution p indexed byγt

p(εt |γt) =2

γt +1γt

{f (

εt

γtI[0,∞)(εt)+ f (γtεt)I(−∞,0)(εt)

}.

In our case εt = f (yt,t+h−mt) and f (εt) is a student T distribution with dispersionσ and degrees of freedom v. In Fernandez and Steel (1998) γ is fixed, but theextension is fairly straightforward. In our experience, this two-piece transformationfits moderate skewness well and is very convenient and robust in estimation, butmay not be the best choice for severe skewness.

Each parameter has a fairly straightforward interpretation: mt is the mode, σ

is the dispersion, v controls the fatness of the tails, and γt determines the amountof asymmetry. However, each statistical moment is in general a function of allfour parameters (see Fernandez and Steel (1998) for closed-form expressions). Inparticular, mt is the mode, which differs from the mean unless γt = 1, the varianceis a function of σ,v, and γt , and the most common measure of skewness as thecentered third moment divided by the cubed standard deviation is also a functionof σ,v, and γt .

For unimodal distributions, Arnold and Groeneveld (2010) propose a measureof skewness defined as one minus twice the probability mass left of the mode,which in our case is

γ2t −1

γ2t +1

,

since in the skew-T distribution of Fernandez and Steel (1998), γ alone controls theallocation of mass to each side of the mode as

P(yt ≥ mt |γt)

P(yt < mt |γt)= γ

2t .

Given our use of the Fernandez and Steel skew-T distribution, ”asymmetry”in this paper is a one-to-one function of the amount of probability mass on eachside of the mode. This definition is of course not free from shortcomings, but itis intuitive and far more stable than the centered third moment, particularly forfat-tailed distributions.

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Estimation. Since the likelihood and all derivatives are available in closedform, estimation by maximum likelihood is convenient and works well for thesmall models considered in this paper. When using overlapping data, the assump-tion of conditionally independent observations is incorrect and results can be in-terpreted as quasi-ML. A correction for autocorrelation should then be made tocompute standard errors and t-statistics.

A very effective Markov Chain Monte Carlo algorithm (Gamerman (1997))exists for generalized linear models, of which ours is a special case. Our versionis taken from Li and Villani (2010). The problem is broken into sequential stepsof estimating the coefficients associated with each parameter in separate blocks,with tailored proposal distributions obtained by maximizing the conditional likeli-hood at each step. The computational cost is compensated by increased reliability:in more complex problems and/or in less informative data, there can be multiplemodes that the MCMC is able to explore in our experience. The general version ofthe model in which explanatory variables can affect both the mode and the asym-metry is particularly prone to multimodality, requiring either MCMC or great carein optimization.

For all results presented in this paper, the posterior means from MCMC (whichwe report and which are obtained with very disperse priors) for the key parametersof interest, namely β0,γ and β1,γ, are nearly identical to ML estimates. Maximumlikelihood estimation gives consistently lower estimates of the degrees-of-freedomparameter than MCMC. This is not surprising: the data contain some very largeoutliers, and ML can only accommodate them with a fairly low v. In contrast,MCMC results in a posterior distribution for v. This distribution has a higher meanand mode than the ML estimate, but also a tail of very low draws of v which inducevery fat tails in the distribution of returns. We view this as a highly desirable featureof fully Bayesian MCMC inference, since it allows for very fat tails without forcingthe spikes in the center of the distribution that are associated with low degrees-of-freedom in a student-t distribution.

We are not aware of any fully Bayesian approach to inference with overlap-ping observations: the likelihood is technically misspecified. Common practiceis to either work with non-overlapping observations, which throws away a lot ofuseful data, or work as if observations were independent, which gives incorrectposteriors and over-confident results. We have employed an ad-hoc fix inspired byautocorrelation-consistent standard errors computed in a frequentist approach: thelog-likelihood within each MCMC step is divided by 1+0.5(h−1) in an attemptto account for overlapping observations. For the results reported in this paper, thisproduces standard deviations extremely close to autocorrelation consistent standarderrors when the posterior mean is close to the posterior mode (the ML estimate), asin the case for the key coefficients of interest β0,γ and β1,γ. We report results from

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these MCMC draws, but emphasize that the key findings are equally strong fromML estimates. For the full sample, t statistics for the main parameter of interestβ1,γ are close to three even if non-overlapping observations are used.

3 Data and Descriptive Statistics

The key variables of interest are the cumulative, overlapping (i.e., all possible12-month and 24-month periods are considered) 12-month and 24-month total re-turns. The only predictive variable is the cyclically adjusted price-to-earnings ratio(CAPE); as a robustness test, we replicate our main results using the market-to-book (MB) ratio as the only predictive variable. The sample period is from January1881 (January 1921) to December 2014 in the case of CAPE (MB). All variablesare in logs. We also standardize the valuation ratios in the model such that themean is zero and the standard deviation is one, which makes parameters easier tointerpret. Data is at the monthly frequency and is taken from Amit Goyal’s web-page.

Figure 1 shows the time-series graph of market-to-book and CAPE. As onewould expect, the two series are very closely related — noticeable differences canbe observed up to the 30ties and during the 60ties and 70ties. Negative valuescorrespond to periods of time during which book values exceed market prices whilelarge positive values correspond to market booms. One clearly observes the stockmarket crash before the great depression and the run-up and subsequent correctionassociated with the boom in technology stocks at the end of the last century.

Table 1 presents the summary statistics of 12-month (Panel A) and 24-month(Panel B) total returns including means, standard deviations and skewness. We re-port these statistics for the full sample period, and the pre-1945 and post-1945 sub-periods separately. Furthermore, we report them separately for the first, the pooledsecond and third, and the fourth valuation quartiles. There are several measures ofskeweness available in the literature, each attempting to quantify the asymmetry ina distribution. The most common measure is the centered and standardized thirdmoment. This statistic is known to suffer from large sampling errors in the case ofdistribution with fat tails and is therefore very susceptible to outliers.

The expected 12-month return is 16.2% in the case of low (lowest quartile ofCAPE) and 3.9% in the case of high (highest quartile of CAPE) valuation ratios(the unconditional mean is 8.7% with a standard deviation of 18.8%). The standarddeviation of 12-month returns is 16.9% for the case of lowest valuation ratios andonly slightly higher at 19.4% for highest valuation ratios. In the case of 24-monthreturns the full sample average is 17.3% with a standard deviation of 26.3%; ex-pected 24-month returns amount to 31.7% for periods with lowest and 7.3% for

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periods with highest valuation ratios; the corresponding standard deviations are18.7% and 32.9%, respectively. Note the mild increase in standard deviation (be-tween 12-month and 24-month returns) in the case of lowest valuation ratios andthe comparatively very stark increase in the case of highest valuation ratios. Thepatterns in average realized returns observed across valuation quartiles are consis-tent with the standard Campbell-Shiller argument that low (high) valuation ratiospredict high (low) expected rates of returns.

In terms of skewness, Table 1 shows that, as expected and consistent with ex-isting literature, cumulative 12-month and 24-month returns are, in general, nega-tively skewed. Most importantly, however, we find that they are more negativelyskewed when valuation ratios are high (top quartile) than when valuation ratiosare low (bottom quartile). In the case of bottom-quartile valuation ratios, we fre-quently find even positive or close-to-zero skewness. Thus, these simple descrip-tive statistics already imply a link between valuation ratios and the shape of thereturn distribution.

In many cases, however, skewness does not monotonically decrease when val-uation levels increase; i.e., in some cases we find even lower skewness when valua-tion ratios are in the middle quartiles. While this seems to be at odds with our story,it is most likely related to the previously discussed shortcomings of the standardskewness measure that we report in Table 1. To get a better idea of the shapes ofthe empirical return distributions, Fig 2 (12-month returns) and Fig 3 (24-monthreturns) show histograms of realized returns for the full sample and conditional onvaluation ratios at the beginning of the return observation period. In both cases, weclearly see that the shape of the distribution of observed returns changes substan-tially conditional on the valuation ratio. While it looks slightly positively skewedin the case of the lowest valuation quartile, it becomes increasingly asymmetricand negatively skewed for higher valuation quartiles. These patterns also appear tobe somewhat more pronounced for realized 24-month than 12-month returns.

4 Empirical Results

In this section, we summarize our empirical results focusing on 12-month and 24-month cumulative total log returns and the cyclically-adjusted-price-earnings ratio(CAPE) as proxy for the valuation ratio.

4.1 Model Parameters

Table 2 summarizes parameter estimates for the three models of interest — theSymmetric-T model, the Constant-Skew-T model and the Conditional-Skew-T model

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— when 12-month returns are modeled. Panel A reports results based on the fullsample of data, Panel B focuses on the pre-1945 and Panel C on the post-1945sub-period.

The Symmetric-T model represents the standard, simple linear regression modelwith the only difference that we assume a T-distribution instead of a normal distri-bution for the residuals. Consistent with the literature we find that valuation ratiospredict expected returns: a one-standard deviation increase in log(CAPE) resultsin a, statistically and economically significant, drop in expected 12-month returnsof 4.226%. The corresponding full sample OLS estimates assuming a normal dis-tribution for the errors are 8.667 for β0,m and -4.817 for β1,m. Thus, the impact ofthe valuation ratio is slightly smaller once residuals are modeled to follow a fat-tailed distribution. Note also that parameters β0,m and β1,m predict the mode of thepredictive distribution rather than the mean in our framework. However, as long asthe predictive distribution is symmetric the mode is, obviously, equal to the mean.

The Constant-Skew-T model extends the basic model by allowing the predic-tive distribution to be skewed. This results in a substantial increase in fit as mea-sured by the log-likelihood. The constant asymmetry parameter of -0.316 impliesa negatively skewed predictive distribution, as one would expect. The coefficientof the valuation ratio in predicting the mode of the distribution stays essentiallyunchanged. Interestingly, however, the skewness parameter becomes insignificantand is cut in half, to -0.160, once we focus on the pre-1945 sample period imply-ing that returns were less negatively skewed on average according to that modelspecification.

Finally, Table 2 summarizes the parameter estimates for the Conditional-Skew-T model which models the predictive distribution’s asymmetry as a function of thevaluation ratio. We find that conditioning on the valuation ratio in the shape equa-tion improves the model fit (i.e., the log likelihood increases). We find a value of-0.175 for β1,γ indicating that, as expected, the distribution becomes more nega-tively skewed when valuation ratios increase. It also shows that the total estimateof the shape parameter, including the constant term, becomes essentially zero, im-plying a symmetric distribution, for valuation ratios that are close to two standarddeviation below zero (i.e., low valuation ratios).

As discussed before, we observe substantial variation in β0,γ across sub-periods.Interestingly, however, estimates of β1,γ do not share this behavior. Thus, while theoverall asymmetry in the return distribution has changed somewhat over time, thelink between valuation levels and skewness appears to have been stable and sta-tistically significant. Changes in the unconditional asymmetry (captured by β0,γ)across sub-periods do not necessarily imply a break in the relation we are inter-ested in, since β1,γ is stable and valuation proxies also have different sample meansacross sub-periods. The model then implies that periods of higher (lower) average

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valuations should have more (less) pronounced average skeweness. A formal testfor the null hypothesis that β0,γ is constant versus the alternative that it has changedpost-1945 has a borderline t-statistic of 1.9, with the AIC criterion picking the ex-tension and the more stringent BIC criterion chosing constant parameters.

Note that in the Conditional-Skew-T model we do not include the valuationratio in the mode equation (i.e., we set β1,m = 0). The main motivation to do sois for simplicity, as we focus on the impact of valuation ratios on the shape of thereturn distribution in this paper. We do, however, also estimate an extension of theConditional-Skew-T model, in which the valuation ratio affects both, the mode andthe shape, of the distribution. It turns out, however, that the more general modeldoes not fit better and that the estimated β1,m parameter is small and insignificant.For reasons of brevity, we decided not to report these results in detail.

Nevertheless, these results do have some further noteworthy implications: whilethe return distribution becomes much more negatively skewed when valuations arehigh, the most likely return of the distribution (i.e, the mode) is essentially un-affected by valuation levels. Given that β0,m > 0, this also means that the mostlikely return is positive even at very high valuation ratios reflecting the difficulty oftiming market reversals.

Table 3 contains parameter estimates of the three models when 24-month re-turns are used as dependent variable. Results look qualitatively very similar in thiscase. It is noteworthy to point out that the estimates of unconditional skewness inthe Constant-Skew-T model, β0,γ, are in all sample periods statistically insignifi-cant and smaller than in the case of 12-month returns. In contrast, however, thecoefficients capturing the conditional impact of valuation ratios on the shapes ofthe predictive distributions, β1,γ, are always statistically significant and increaserelative to Table 2.

Bottom line, we find — across all return definitions and sample periods — thatvaluation ratios have a statistically significant impact on the shape of the returndistribution. Specifically, distributions become more negatively skewed when val-uation ratios increase. In the following sections, we analyze the resulting shapes ofthe return distributions in more detail.

4.2 Predictive Distributions

The parameter estimates discussed in the previous section already document thatvaluation ratios help predict the shape of the distributions of 12-month and 24-month returns. Judging, however, how large this impact is in terms of the resultingasymmetry of the distributions directly from the parameter estimates is difficult.Thus, we take a detailed view at the predictive distributions implied by the variousmodels in this section.

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Figures 4 and 5 represent the main results of the paper. They show the con-ditional predictive distributions for 12-month (Figure 4) and 24-month (Figure 5)returns implied by the Conditional-Skew-T model2 using the full sample parameterestimates. The top graph in each figure represents the case of high and the bottomgraph the case of low valuation ratios. While the modes of the two distributionsare identical by design, the model implies very different shapes of the distributiondepending on the level of the valuation ratios: while predictive distributions lookpretty much symmetric for low valuation ratios, they become asymmetric and neg-atively skewed in the case of high valuation ratios. As discussed before, results areslightly stronger for 24-month returns than for 12-month returns.

Table 4 provides some further information on the conditional distributions im-plied by our models, namely the mean, standard deviation, the normalized thirdmoment (skewness), the probability mass left to the mode (asymmetry), the 1%Value-at-Risk and the 1% Expected Tail Loss. Most importantly, we are interestedin skewness and asymmetry. By construction, skewness is zero and asymmetryis equal to 50% in the case of the Symmetric-T Model. When we allow the dis-tribution to be skewed in the Constant-Skew-T Model, we find that the implieddistributions become asymmetric and that probability mass shifts to the left of themode; in the case of 12-month (24-month) returns 65% (61%) of the probabilitymass end up being below the mode.

Finally in the Conditional-Skew-T Model, we observe that valuation levelshave a strong impact on the shape of the distributions. In the case of 12-monthreturns, we find that, for low valuation levels, the distribution is nearly symmetricwith a skewness of zero and 47% of the probability mass being to the left of themode. In stark contrast, for high valuation levels, we find that nearly 80% of theprobability mass is below the mode and skewness is equal to -0.97. A similarlypronounced pattern prevails in the case of 24-month returns with the only differ-ence that, in the case of low valuation levels, the implied distribution seems to bepositively skewed with only 39% of the probability mass to the left of the mode.

These results illustrate, yet again, that the shape of the return distribution de-pends strongly on valuation levels. An equally important question, however, iswhether this shape dependence also has implications for other characteristics ofthe return distribution such as means or standard deviations. Table 4 reports sev-eral key characteristics of the return distributions implied by our models. In gen-eral, across all models, we find that expected returns are considerably lower whenvaluation levels are high, as one would expect. Not surprisingly, given the choiceof our models, estimates of expected rates of returns are also within rather narrow

2Note that the two benchmark models, by design, do not model conditional skewness and, thus,it makes no sense to draw these graphs for the two benchmark models.

13

bands.3

We also observe an interesting pattern for model-implied standard deviations.Both, the Symmetric-T and the Constant-Skew-T Model, show a tendency to over-estimate volatility when valuation levels are low and, at the same time, underesti-mate volatility when valuation levels are high, relative to the standard deviationsimplied by the Conditional-Skew-T Model. That means that the two benchmarkmodels are on the wrong side in both cases: when valuations are very low, a mean-variance investor using those standard deviation estimates would invest too cau-tiously while the same investor would invest too aggressively when valuation levelsare very high. Note that in our econometric setup these volatility patterns only ariseas a consequence of the change in the asymmetry of the distribution, as dispersionand degrees of freedom are modeled identically across all three models.

Similar patterns are observed when we move to risk measures beyond volatil-ity, such as value-at-risk and expected tail loss. In both cases, we find that thetwo benchmark models overestimate risk in the case of low valuation levels butunderestimate risk in the case of high valuation levels. For example, in the case of12-month returns the Constant-Skew-T Model implies a 1% value-at-risk of -37%(-53%) when valuation levels are low (high) while the Conditional-Skew-T Modelimplies values of -23% (-71%). While it is difficult to judge economic importanceof these differences without having a specific application or portfolio in mind, theycertainly look sizable and noteworthy to us.4

So far, we have focused on full sample evidence in the discussion of the predic-tive distributions. Figures 6 and 7 illustrate the model-implied distributions sepa-rately for the pre-1945 and post-1945 sample periods while Tables 5 and 6 providethe corresponding characteristics. Most importantly, the patterns that we discussedabove based on the full sample also hold for each sub-sample separately. Thus, ourresults are robust across different sample periods and do not seem to be driven byindividual years or particular events.

4.3 Robustness

As a robustness test, we replicate the main steps of our empirical analysis using themarket-to-book (MB) ratio instead of the CAPE ratio as our predictive variable.Note that in this case the sample period runs from January 1921 to December 2014.

3In the case of the Conditional-Skew-T Model, we find expected returns that are even slightlynegative when valuation levels are high. Obviously, negative expected returns of equity over 12 or24-month periods are not consistent with theory. In our simple models, however, there are no explicitmechanisms that ensure that expected returns are positive in all cases.

4In the case of the Conditional-Skew-T Model and 24-month returns, we find value-at-risk andexpected-tail-loss estimates of -103% and -126%, respectively. Note that throughout the paper weuse log-returns. Thus, these estimates correspond to -64% and -72% in terms of simple returns.

14

Table 7 shows the corresponding parameter estimates if we use 12-month (PanelA) and 24-month (Panel B) returns as dependent variables. We find results that areconsistent across panels and also confirm the results discussed in detail before: theConditional-Skew-T model gets most support in the data as reflected by the maxi-mum log-likelihoods and the coefficients of the MB-ratio in the shape equation arenegative and statistically significant. In fact, estimates of β1,γ reported in Table 7are quite comparable to the ones reported in Tables 2 and 3 in terms of value andstatistical significance. Thus, as valuation levels, proxied by the MB ratio, increasethe shapes of the return distributions, for both 12-month and 24-month returns,become more asymmetric and, in particular, more negatively skewed.

Figures 8 and 9 illustrate the corresponding model-implied return distributionsseparately for high (valuation levels two standard deviations above the average) andlow (valuation levels two standard deviation below the average) valuation levels.Again, one clearly observes how the shape of the distribution changes. While itlooks rather symmetric for low valuation levels, it becomes clearly asymmetric,featuring a long negative tail, in the case of high valuation levels.

Table 8, finally, reports detailed characteristics of these model-implied distri-butions. The patterns discussed for the main results also prevail when we usethe market-to-book ratio as our proxy for valuation levels. Most importantly, weobserve a stark change in the shape of the distributions conditional on valuationlevels. For example, in the case of 24-month returns the probability mass to theleft of the mode is 49% (83%) when valuation levels are low (high). Similarly, wefind that risk estimates (volatility, value-at-risk and expected-tail-loss) implied bythe benchmark models are underestimated (overestimated) when valuation levelsare low (high) relative to the two benchmark models. Again, magnitudes of thesedifferences are economically large.

5 Conclusion

In this paper, we document a robust link between valuation levels and the shapeof the distribution of cumulative (up to 24 months) total log returns in the SP500.Our key result is that return distributions become considerably more asymmetricand negatively skewed when valuation levels are high; in contrast they tend to besymmetric, sometimes even slightly positively skewed, when valuation levels arelow. These patterns are very robust across return horizons, proxies for valuationlevels and sample periods.

While the emphasis of the literature is usually on expected rates of return (pointprediction), we focus on the novel and important question of how asset prices ac-tually revert back to these time-varying means. Our empirical results indicate that

15

this reversion is rather smooth when valuation levels are low and rather abruptwhen valuation levels are high. Intuitively, this pattern is well summarized bypractitioners describing equity market as “up the stairs, down the elevator”.

The dependence of the shape of the return distribution on valuation levels hasseveral further interesting practical implications. Most importantly, it implies thatmeasures of risk (e.g., standard deviation, value-at-risk, expected-tail-loss), derivedfrom symmetric distributions or distributions with constant skewness, are underes-timated when valuation levels are high and overestimated when valuation levelsare low relative to a model with conditional skewness. This indicates a lose-losesituation for risk managers and asset managers relying on these risk measures. Im-portantly, magnitudes of these deviations are sizable.

Another noteworthy result of our empirical analysis is that we find the modeof the return distribution to be consistently positive and essentially unaffected byvaluation levels. This implies that even when valuation levels are extremely high,the most likely return over the next 12 to 24 months remains positive reflectingthe well-known difficulty of predicting turning points and market crashes. Overall,our empirical evidence on how valuations affect the asymmetry of the predictivedistribution of returns is qualitatively consistent with stochastic rational bubbles inthe spirit of Blanchard and Watson (1982). However, expected returns are constantin models of rational bubbles, whereas the introduction of time varying skewenessdoes nothing to change the relation between valuations and expected returns thathas been the focus of so much attention in the literature. The finding that themode of the predictive distribution (the most likely outcome) does not change andremains positive even at extremely high valuations may however provide usefulinsights into why such valuations can be reached at all.

16

References

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Bekaert, G., Wu, G., 2000. Asymmetric volatility and risk in equity markets. TheReview of Financial Studies 13, 1–42.

Blanchard, O. J., Watson, M. W., 1982. Bubbles, rational expectations, and finan-cial markets. In: Crises in Economic and Financial Structure. Lexington Books12, 295–315.

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Cochrane, J. H., 2011. Presidential address: Discount rates. The Journal of FinanceLXVI (4), 1047–1108.

David, A., Veronesi, P., 2014. Investor’s and central bank’s uncertainty embeddedin index options. The Review of Financial Studies 27 (6), 1661–1716.

Fama, E. F., 2013. Two pillars of asset pricing, nobel Prize Lecture.

Fernandez, C., Steel, M. F. J., 1998. On bayesian modeling of fat tails and skew-ness. The Journal of the American Statistical Association 93 (441), 359–371.

Gamerman, D., 1997. Sampling from the posterior distribution in generalized lin-ear mixed models. Statistics and Computing 7, 57–68.

Golez, B., Koudijs, P., 2016. Four centuries of return predictability, working Paper.

Harvey, C. R., Siddique, A., 1999. Autoregressive conditional skewness. The Jour-nal of Financial and Quantitative Analysis 43 (4), 465–487.

Hueng, J. C., McDonald, J. B., 2005. Forecasting asymmetries in aggregate stockmarket returns: Evidence from condtional skewness. The Journal of EmpiricalFinance 12, 666–685.

17

Jondeau, E., Rockinger, M., 2003. Conditional volatility, skewness, and kurtosis:existence, persistence, and comovements. The Journal of Economic Dynamics& Control 27, 1699–1737.

Jones, M., 2014. On families of distributions with shape parameters. InternationalStatistical Review 83 (2), 175–192.

Lee, S. X., McLachlan, G. J., 2013. On mixtures of skew normal and skew t-distributions. Advances in Data Analysis and Classification 7 (3), 241–266.

Li, F., Villani, M., 2010. Flexible modelling of conditional distributions usingsmooth mixtures of asymmetric student t densities. The Journal of StatisticalPlanning and Inference 140, 3638–3654.

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18

Figure 1: Market-to-book and Cyclically-Adjusted-Price-Earnings (CAPE)RatiosThe figure shows the standardized — mean equal to zero, standard deviation equalto 1 — log market-to-book and log cyclically-adjusted-price-earnings ratio.

19

Tabl

e1:

Sum

mar

ySt

atis

tics

Thi

sta

ble

prov

ides

sum

mar

yst

atis

tics

—m

eans

,sta

ndar

dde

viat

ions

and

skew

ness

(the

norm

aliz

edth

ird

mom

ent)

—of

12-m

onth

and

24-m

onth

tota

lequ

ityre

turn

s.A

llva

riab

les

are

inlo

gs.I

nad

ditio

nto

unco

nditi

onal

estim

ates

,we

also

repo

rtth

ose

stat

istic

sco

nditi

onal

onva

luat

ion

quar

tiles

.We

furt

herr

epor

ttho

sest

atis

tics

sepa

rate

lyfo

rthe

pre-

1945

and

post

-194

5pe

riod

s.M

eans

and

stan

dard

devi

atio

nsar

ere

port

edin

perc

enta

gete

rms.

Pane

lA:1

2-m

onth

retu

rns

Full

Sam

ple

Pre-

1945

Post

-194

5M

ean

(%)

Std.

Dev

.(%

)Sk

ew.

Mea

n(%

)St

d.D

ev.(

%)

Skew

.M

ean

(%)

Std.

Dev

.(%

)Sk

ew.

All

Ret

urns

8.67

18.7

5-0

.83

6.34

21.6

5-0

.66

10.4

015

.65

-0.8

21s

tVal

uatio

nQ

uart

ile16

.21

16.8

50.

3016

.80

20.6

50.

3515

.47

13.0

3-0

.15

2nd

&3r

dV

alua

tion

Qua

rtile

7.30

18.2

1-1

.25

5.05

18.8

6-1

.56

10.4

614

.59

-0.5

64t

hV

alua

tion

Qua

rtile

3.88

19.4

3-0

.95

-1.4

723

.85

-0.5

05.

2418

.29

-1.0

2

Pane

lB:2

4-m

onth

retu

rns

Full

Sam

ple

Pre-

1945

Post

-194

5M

ean

(%)

Std.

Dev

.(%

)Sk

ew.

Mea

n(%

)St

d.D

ev.(

%)

Skew

.M

ean

(%)

Std.

Dev

.(%

)Sk

ew.

All

Ret

urns

17.3

326

.28

-1.1

312

.31

29.9

5-1

.09

20.9

221

.89

-0.7

41s

tVal

uatio

nQ

uart

ile31

.70

18.6

0-0

.13

30.9

321

.42

-0.0

330

.86

17.2

90.

132n

d&

3rd

Val

uatio

nQ

uart

ile15

.17

22.5

8-0

.37

12.2

722

.52

0.23

21.6

117

.76

-0.6

94t

hV

alua

tion

Qua

rtile

7.30

32.9

0-1

.41

-6.1

537

.88

-1.2

69.

6627

.64

-0.3

8

20

Table 2: Model Parameters when Predicting 12-month ReturnsThis table provides parameter estimates of three different models — theSymmetric-T Model, the Constant-Skew-T Model and the Conditional-Skew-TModel — when 12-month returns are used as dependent variable. All variablesare in logs. Details on the estimation of these parameters are summarized in theAppendix. The table reports mean parameter estimates and corresponding t-values.

Panel A: Full SampleSymmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat

β0,m 9.686 8.707 16.825 7.122 16.233 6.480β1,m -4.226 -3.721 -4.185 -4.007β0,σ 2.724 38.296 2.668 34.686 2.680 35.343β0,v 2.025 4.666 2.163 4.384 2.407 4.058β0,γ -0.316 -3.195 -0.297 -2.900β1,γ -0.175 -4.254Log-Likeli. -6780.1 -6747.2 -6742.3

Panel B: Pre-1945Symmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat


Panel C: Post-1945Symmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat


21

Table 3: Model Parameters when Predicting 24-month ReturnsThis table provides parameter estimates of three different models — theSymmetric-T Model, the Constant-Skew-T Model and the Conditional-Skew-TModel — when 24-month returns are used as dependent variable. All variablesare in logs. Details on the estimation of these parameters are summarized in theAppendix. The table reports mean parameter estimates and corresponding t-values.

Panel A: Full sampleSymmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat

β0,m 18.395 8.874 25.350 5.328 27.230 5.756β1,m -8.623 -4.030 -8.032 -3.765β0,σ 3.020 29.884 3.006 29.644 2.994 31.445β0,v 2.059 3.307 2.281 3.154 2.658 3.276β0,γ -0.219 -1.583 -0.263 -1.895β1,γ -0.240 -4.191Log-Likeli. -7190.4 -7178.0 -7162.7.01

Panel B: Pre-1945Symmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat


Panel C: Post-1945Symmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat


22

Tabl

e4:

Pred

ictiv

eD

istr

ibut

ions

(Ful

lSam

ple)

Thi

sta

ble

prov

ides

mea

ns,s

tand

ard

devi

atio

ns(S

D),

the

norm

aliz

edth

ird

mom

ent(

skew

ness

,SK

EW

),th

epr

obab

ility

mas

sbe

low

the

mod

e(a

sym

met

ry,

ASY

),th

e1%

Val

ue-a

t-R

isk

and

the

1%E

xpec

ted

Tail

Los

sof

pred

ictiv

edi

stri

butio

nsof

12-m

onth

and

24-m

onth

tota

leq

uity

retu

rns

impl

ied

byou

rm

odel

s.A

llva

riab

les

are

inlo

gs.

All

valu

esre

port

edar

ein

perc

enta

gete

rms

exce

ptfo

rsk

ewne

ss.

Spec

ifica

lly,

we

repo

rtth

ose

char

acte

rist

ics

sepa

rate

lyfo

rhi

gh(+

2st

anda

rdde

viat

ions

)and

low

(-2

stan

dard

devi

atio

ns)v

alua

tion

leve

ls.W

efu

rthe

rrep

ortt

hose

stat

istic

sse

para

tely

fort

heSy

mm

etri

c-T

Mod

el(P

anel

A),

the

Con

stan

t-Sk

ew-T

Mod

el(P

anel

B),

and

the

Con

ditio

nal-

Skew

-TM

odel

(Pan

elC

).

Pane

lA:S

ymm

etri

c-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LL

owva

luat

ion

18.1

318

.31

0.00

50.0

0-2

8.26

-40.

7635

.46

25.0

10.

0050

.00

-28.

30-4

6.68

Hig

hva

luat

ion

1.28

18.3

10.

0050

.00

-45.

09-5

7.22

1.23

25.0

10.

0050

.00

-62.

53-8

0.02

Pane

lB:C

onst

ant-

Skew

-TM

odel

12-M

onth

Ret

urns

24-M

onth

Ret

urns

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Low

valu

atio

n17

.06

18.4

3-0

.75

65.2

9-3

6.83

-51.

8333

.57

25.3

2-0

.56

60.7

8-3

7.59

-58.

27H

igh

valu

atio

n0.

2918

.43

-0.7

565

.29

-53.

35-6

7.74

1.40

25.3

2-0

.56

60.7

8-6

9.21

-88.

69

Pane

lC:C

ondi

tiona

l-Ske

w-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LL

owva

luat

ion

17.4

716

.60

0.00

47.3

5-2

3.04

-32.

7434

.14

22.7

40.

1639

.32

-17.

47-2

8.99

Hig

hva

luat

ion

-1.1

321

.68

-0.9

778

.48

-70.

76-8

6.44

-0.6

432

.09

-1.0

581

.55

-103

.37

-125

.70

23

Tabl

e5:

Pred

ictiv

eD

istr

ibut

ions

(Pre

-194

5)T

his

tabl

epr

ovid

esm

eans

,sta

ndar

dde

viat

ions

(SD

),th

eno

rmal

ized

thir

dm

omen

t(sk

ewne

ss,S

KE

W),

the

prob

abili

tym

ass

belo

wth

em

ode

(asy

mm

etry

,A

SY),

the

1%V

alue

-at-

Ris

kan

dth

e1%

Exp

ecte

dTa

ilL

oss

ofpr

edic

tive

dist

ribu

tions

of12

-mon

than

d24

-mon

thto

tal

equi

tyre

turn

sim

plie

dby

our

mod

els.

All

vari

able

sar

ein

logs

.A

llva

lues

repo

rted

are

inpe

rcen

tage

term

sex

cept

for

skew

ness

.Sp

ecifi

cally

,w

ere

port

thos

ech

arac

teri

stic

sse

para

tely

for

high

(+2

stan

dard

devi

atio

ns)a

ndlo

w(-

2st

anda

rdde

viat

ions

)val

uatio

nle

vels

.We

furt

herr

epor

ttho

sest

atis

tics

sepa

rate

lyfo

rthe

Sym

met

ric-

TM

odel

(Pan

elA

),th

eC

onst

ant-

Skew

-TM

odel

(Pan

elB

),an

dth

eC

ondi

tiona

l-Sk

ew-T

Mod

el(P

anel

C).

Pane

lA:S

ymm

etri

c-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LL

owva

luat

ion

19.1

221

.29

0.00

50.0

0-3

5.08

-50.

3235

.54

28.8

00.

0050

.00

-38.

96-6

2.25

Hig

hva

luat

ion

-4.9

821

.29

0.00

50.0

0-5

9.10

-73.

12-9

.63

28.8

00.

0050

.00

-83.

97-1

04.3

7

Pane

lB:C

onst

ant-

Skew

-TM

odel

12-M

onth

Ret

urns

24-M

onth

Ret

urns

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Low

valu

atio

n18

.46

21.4

8-0

.37

57.9

3-4

0.66

-57.

2234

.58

29.4

0-0

.31

53.9

4-4

5.32

-69.

26H

igh

valu

atio

n-5

.51

21.4

8-0

.37

57.9

3-6

4.24

-79.

22-9

.20

29.4

0-0

.31

53.9

4-8

8.53

-109

.58

Pane

lC:C

ondi

tiona

l-Ske

w-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LL

owva

luat

ion

18.3

421

.33

0.46

36.8

2-2

7.65

-38.

5036

.76

30.6

50.

6628

.29

-22.

38-3

5.88

Hig

hva

luat

ion

-5.9

324

.18

-0.8

373

.81

-81.

73-9

7.47

-11.

8236

.89

-1.0

681

.18

-130

.87

-150

.72

24

Tabl

e6:

Pred

ictiv

eD

istr

ibut

ions

(Pos

t-19

45)

Thi

sta

ble

prov

ides

mea

ns,s

tand

ard

devi

atio

ns(S

D),

the

norm

aliz

edth

ird

mom

ent(

skew

ness

,SK

EW

),th

epr

obab

ility

mas

sbe

low

the

mod

e(a

sym

met

ry,

ASY

),th

e1%

Val

ue-a

t-R

isk

and

the

1%E

xpec

ted

Tail

Los

sof

pred

ictiv

edi

stri

butio

nsof

12-m

onth

and

24-m

onth

tota

leq

uity

retu

rns

impl

ied

byou

rm

odel

s.A

llva

riab

les

are

inlo

gs.

All

valu

esre

port

edar

ein

perc

enta

gete

rms

exce

ptfo

rsk

ewne

ss.

Spec

ifica

lly,

we

repo

rtth

ose

char

acte

rist

ics

sepa

rate

lyfo

rhi

gh(+

2st

anda

rdde

viat

ions

)and

low

(-2

stan

dard

devi

atio

ns)v

alua

tion

leve

ls.W

efu

rthe

rrep

ortt

hose

stat

istic

sse

para

tely

fort

heSy

mm

etri

c-T

Mod

el(P

anel

A),

the

Con

stan

t-Sk

ew-T

Mod

el(P

anel

B),

and

the

Con

ditio

nal-

Skew

-TM

odel

(Pan

elC

).

Pane

lA:S

ymm

etri

c-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LL

owva

luat

ion

19.2

715

.52

0.00

50.0

0-1

9.05

-27.

6238

.13

21.3

90.

0050

.00

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11-2

7.54

Hig

hva

luat

ion

2.90

15.5

20.

0050

.00

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45-4

3.23

4.54

21.3

90.

0050

.00

-48.

35-5

7.92

Pane

lB:C

onst

ant-

Skew

-TM

odel

12-M

onth

Ret

urns

24-M

onth

Ret

urns

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Low

valu

atio

n18

.22

15.6

2-0

.71

70.3

9-2

6.74

-36.

6636

.61

21.7

6-0

.28

56.4

1-2

2.05

-35.

87H

igh

valu

atio

n2.

5815

.62

-0.7

170

.39

-41.

85-5

0.40

5.23

21.7

6-0

.28

56.4

1-5

1.43

-61.

31

Pane

lC:C

ondi

tiona

l-Ske

w-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LL

owva

luat

ion

18.2

213

.44

-0.1

949

.90

-15.

69-2

3.19

37.0

821

.41

0.44

35.3

9-8

.14

-18.

11H

igh

valu

atio

n1.

9318

.11

-0.7

281

.99

-55.

14-6

2.05

5.91

25.8

6-0

.75

80.8

1-6

8.11

-74.

42

25

Table 7: Model Parameters when Using the Market-to-Book Ratio (Robust-ness Test)This table provides parameter estimates of three different models — theSymmetric-T Model, the Constant-Skew-T Model and the Conditional-Skew-TModel — when 12-month returns and 24-month returns are used as dependentvariable and the Market-to-Book ratio is used as proxy for the valuation level. Allvariables are in logs. All results summarized in the table are based on the full sam-ple of data. Details on the estimation of these parameters are summarized in theAppendix. The table reports mean parameter estimates and corresponding t-values.

Panel A: 12-Month ReturnsSymmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat


Panel B: 24-Month ReturnsSymmetric-T Constant-Skew-T Conditional-Skew-TMean t-stat Mean t-stat Mean t-stat


26

Tabl

e8:

Rob

ustn

essT

est:

Pred

ictiv

eD

istr

ibut

ions

usin

gM

arke

t-to

-Boo

kR

atio

asPr

edic

tive

Vari

able

(Ful

lSam

ple)

Thi

sta

ble

prov

ides

mea

ns,s

tand

ard

devi

atio

ns(S

D),

the

norm

aliz

edth

ird

mom

ent(

skew

ness

,SK

EW

),th

epr

obab

ility

mas

sbe

low

the

mod

e(a

sym

met

ry,

ASY

),th

e1%

Val

ue-a

t-R

isk

and

the

1%E

xpec

ted

Tail

Los

sof

pred

ictiv

edi

stri

butio

nsof

12-m

onth

and

24-m

onth

tota

lequ

ityre

turn

sim

plie

dby

our

mod

els

usin

gth

eM

arke

t-to

-Boo

kra

tioas

pred

ictiv

eva

riab

le.

All

vari

able

sar

ein

logs

.A

llva

lues

repo

rted

are

inpe

rcen

tage

term

sex

cept

for

skew

ness

.Sp

ecifi

cally

,w

ere

port

thos

ech

arac

teri

stic

sse

para

tely

for

high

(+2

stan

dard

devi

atio

ns)

and

low

(-2

stan

dard

devi

atio

ns)

valu

atio

nle

vels

.W

efu

rthe

rre

port

thos

est

atis

tics

sepa

rate

lyfo

rth

eSy

mm

etri

c-T

Mod

el(P

anel

A),

the

Con

stan

t-Sk

ew-T

Mod

el(P

anel

B),

and

the

Con

ditio

nal-

Skew

-TM

odel

(Pan

elC

).

Pane

lA:S

ymm

etri

c-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LL

owva

luat

ion

18.7

319

.88

0.00

50.0

0-3

3.31

-49.

8833

.87

26.0

60.

0050

.00

-39.

24-6

3.23

Hig

hva

luat

ion

4.40

19.8

80.

0050

.00

-47.

63-6

3.16

7.22

26.0

60.

0050

.00

-66.

52-8

7.49

Pane

lB:C

onst

ant-

Skew

-TM

odel

12-M

onth

Ret

urns

24-M

onth

Ret

urns

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Mea

nSD

SKE

WA

SY1%

-VaR

1%-E

TL

Low

valu

atio

n16

.96

20.0

1-0

.99

68.4

8-4

5.41

-64.

2930

.45

27.7

1-0

.92

64.8

9-5

4.57

-79.

18H

igh

valu

atio

n2.

7020

.01

-0.9

968

.48

-59.

18-7

6.86

8.30

27.7

1-0

.92

64.8

9-7

5.62

-97.

31

Pane

lC:C

ondi

tiona

l-Ske

w-T

Mod

el12

-Mon

thR

etur

ns24

-Mon

thR

etur

nsM

ean

SDSK

EW

ASY

1%-V

aR1%

-ET

LM

ean

SDSK

EW

ASY

1%-V

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LL

owva

luat

ion

18.5

117

.77

-0.1

752

.20

-28.

75-4

3.18

34.3

223

.13

-0.4

848

.85

-27.

95-4

6.29

Hig

hva

luat

ion

0.12

24.0

7-1

.25

80.4

7-7

9.47

-96.

503.

0435

.61

-1.1

183

.45

-110

.67

-127

.72

27

Figure 2: Histograms of Realized 12-month ReturnsThe figure shows four histograms of realized 12-month returns. All returns are inlogs. The left graph in the top row shows the full sample unconditional distributionof realized 12-month returns. The remaining graphs show, in clock-wise direc-tion, the full sample distributions conditional on being in the (i) lowest valuationquartile, (ii) the top valuation quartile, and (iii) the two middle valuation quartiles.

28

Figure 3: Histograms of Realized 24-month ReturnsThe figure shows four histograms of realized 24-month returns. All returns are inlogs. The left graph in the top row shows the full sample unconditional distributionof realized 24-month returns. The remaining graphs show, in clock-wise direc-tion, the full sample distributions conditional on being in the (i) lowest valuationquartile, (ii) the top valuation quartile, and (iii) the two middle valuation quartiles.

29

Figure 4: Model-implied Conditional 12-month Return DistributionThe figure shows the model-implied return distributions of 12-month returns forlow (two standard deviations below the mean) and high (two standard deviationsabove the mean) valuation levels. All returns are in logs. The parameters governingthe distributions are summarized in Table 2.

30

Figure 5: Model-implied Conditional 24-month Return DistributionThe figure shows the model-implied return distributions of 24-month returns forlow (two standard deviations below the mean) and high (two standard deviationsabove the mean) valuation levels. All returns are in logs. The parameters governingthe distributions are summarized in Table 3.

31

Figure 6: Sub-sample Results: Model-implied Conditional 12-month ReturnDistributionThe figure shows the model-implied return distributions of 12-month returns forlow (two standard deviations below the mean) and high (two standard deviationsabove the mean) valuation levels separately for the pre-1945 and the post-1945sample periods. All returns are in logs. The parameters governing the distributionsare summarized in Table 2.

Panel A. Pre-1945

Panel B. Post-1945

32

Figure 7: Sub-sample results: model-implied, conditional 24-month return dis-tributionThe figure shows the model-implied return distributions of 24-month returns forlow (two standard deviations below the mean) and high (two standard deviationsabove the mean) valuation levels separately for the pre-1945 and the post-1945sample periods. All returns are in logs. The parameters governing the distributionsare summarized in Table 3.

Panel A. Pre-1945

Panel B. Post-1945

33

Figure 8: Robustness Test: Model-implied Conditional 12-month Return Dis-tribution using the Market-to-Book Ratio as Predictive VariableThe figure shows the model-implied return distributions of 12-month returns forlow (two standard deviations below the mean) and high (two standard deviationsabove the mean) valuation levels when the Market-to-Book ratio is used as predic-tive variable. All returns are in logs. The parameters governing the distributionsare summarized in Table 5.

34

Figure 9: Robustness Test: Model-implied Conditional 24-month Return Dis-tribution using the Market-to-Book Ratio as Predictive VariableThe figure shows the model-implied return distributions of 24-month returns forlow (two standard deviations below the mean) and high (two standard deviationsabove the mean) valuation levels when the Market-to-Book ratio is used as predic-tive variable. All returns are in logs. The parameters governing the distributionsare summarized in Table 5.

35

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