Expected Returns and Dividend Growth Rates Implied In Derivative Markets

7/28/2019 Expected Returns and Dividend Growth Rates Implied In Derivative Markets

1/73Electronic copy available at: http://ssrn.com/abstract=1271607

Expected Returns and Dividend Growth Rates Implied in

Derivative Markets

Benjamin Golez

University of Notre Dame

September, 2012y

Abstract

The dividend-price ratio is a noisy proxy for expected returns in the presence of time-varying

expected dividend growth rates. This paper uses a new and forward-looking measure of dividend

growth rates extracted from S&P 500 futures and options to correct the dividend-price ratio for

changes in expected dividend growth rates. For the period from January 1994 through June 2011,

I nd that the dividend growth rates implied in derivative markets reliably forecast future dividend

growth rates, and the corrected dividend-price ratio predicts S&P 500 returns substantially better

than the uncorrected dividend-price ratio. The forecasting relation between the corrected dividend-

price ratio and returns is statistically signicant and remarkably stable. Decomposing variance of

the dividend-price ratio, I show that a substantial portion of price movements is attributable to

the time-varying value of expected dividend growth rates. Expected dividend growth rates arealso highly correlated with expected returns. The empirical results depend importantly on the

simultaneous use of futures and options in estimating implied dividend growth rates.

Keywords: present value models, dividend-price ratio, return predictability, derivatives

JEL classication: G12, G13

Department of Finance, 256 Mendoza College of Business, Notre Dame, Indiana 46556-5646. Tel. (574) 631-1458.Email: [email protected].

yI would like to thank Mascia Bedendo, Geert Bekaert, Francesco Corielli, Martijn Cremers, Zhi Da, Jens CarstenJackwerth, Ralph Koijen, Gueorgui I. Kolev, Jos M. Marn, Francisco Pe naranda, Christopher Polk, Jesper Rangvid,Fabio Trojani, and Robert Zymek for invaluable comments. I also thank seminar participants at Universitat Pompeu

Fabra, Swiss Banking Institute, Stockholm School of Economics, Bocconi University, Carlos III University of Madrid,CNMV, Erasmus Research Institute of Management, FED Board, University of Notre Dame, Oslo BI, New EconomicSchool, Einaudi Institute for Economics and Finance as well as participants at Campus for Finance, 12th Conference ofthe Swiss Society for Financial Market Research, European Finance Association Doctoral Tutorial, Chicago QuantitativeAlliance Conference, and Eastern Finance Association. Earliest versions of the paper were circulated under the titleOptions Implied Dividend Yield and Market Returns.

1


2/73Electronic copy available at: http://ssrn.com/abstract=1271607

1 Introduction

The predictability of market returns is of great interest to market practitioners and has important

implications for asset pricing. At rst, any sign of market return predictability was understood as

evidence against market eciency. As evidence has accumulated that price multiples, such as the

dividend-price ratio, predict market returns (Fama and French, 1988), time-varying expected returns

have become an integral part of equilibrium models with ecient markets (Campbell and Cochrane,

1999; Bansal and Yaron, 2004). Yet return predictability remains debatable. The forecasting relation

between returns and the dividend-price ratio is subject to statistical biases (Stambaugh, 1999). It is

unstable and it also has poor out-of-sample forecasting power (Goyal and Welch, 2008). Koijen and

Van Nieuwerburgh (2011) provide a comprehensive survey of return predictability evidence.

This paper argues that the inconsistent performance of the dividend-price (DP) ratio in predicting

returns is attributable largely to the time-varying nature of expected dividend growth rates. I introduce

a novel measure of expected dividend growth rates that is extracted from index options and futures,

and derive a simple present value model to guide the empirical analysis. Using dividend growth rates

implied in derivative markets to correct the DP ratio for variation in expected dividend growth rates,

I nd that market returns are strongly predictable. Indeed, the corrected DP ratio predicts market

returns both in-sample and out-of-sample. It is robust as well to statistical biases that have been

shown to hinder the return predictive ability of the uncorrected DP ratio.

That time-varying expected dividend growth rates can reduce the ability of the DP ratio to predict

returns has long been recognized in the predictability literature (Campbell and Shiller, 1988; Fama

and French, 1988). According to the textbook treatment, the DP ratio may vary over time not only

because of changes in expected returns but also because of changes in expected dividend growth rates.

Therefore, Fama and French (1988) note that the DP ratio is a noisy proxy for expected returns

in the presence of time-varying expected dividend growth rates (see also Cochrane, 2008a; Rytchkov,

2008; Binsbergen and Koijen, 2010; Piatti and Trojani, 2012). Moreover, since DP ratio increases with

expected returns and decreases with expected dividend growth rates, problems caused by time-varying

expected dividend growth rates are pronounced when expected returns and expected dividend growth

rates are positively correlated (Lettau and Ludvigson, 2005). This positive correlation, which arises in

2


3/73

the general equilibrium model as a natural consequence of dividend growth rate predictability (Menzly

et al., 2004), osets changes in expected returns and in expected dividend growth rates, which further

hampers the DP ratios ability to predict returns.

Thus, if the task is to predict returns, and expectations as to dividend growth rates vary over

time, the DP ratio is insucient: We must also account for the changes in expected dividend growth

rates. Yet expected dividend growth rates are dicult to estimate because they aggregate investors

expectations about future growth opportunities.

Recent studies on return predictability typically assume that the future will be similar to the past,

and extract expected dividend growth rates from historical data. For example, Binsbergen and Koijen

(2010) use a latent variable approach within the present value model to lter out both expected returns

and expected dividend growth rates from the history of dividends and prices (see also Rytchkov, 2008).

Lacerda and Santa-Clara (2010) use a simple average of historical dividend growth rates as a proxy

for expected dividend growth rates. These authors conclude that improved prediction of dividend

growth rates enhances the predictability of longer-term (annual) returns. Nevertheless, their methods

use only the information that can be derived from past dividends and prices. Investors, however, base

expectations about future cash ows on a much richer and forward-looking information set.

This paper takes a dierent approach to estimating expected dividend growth rates. Rather than

rely on historical data, I extract a proxy for investors expected dividend growth rates from index

options and futures. Derivative markets provide a unique laboratory for estimating markets expec-

tations about future cash ows. As prices of derivatives depend on future dividend payments and are

related through no-arbitrage, one can invert pricing relations to extract a proxy for expected dividends

from observable market prices of options and futures. Because index derivatives are highly liquid, new

information about future cash ows should be rapidly incorporated into estimated implied dividends.

To provide an analytical framework for the empirical analysis, I rst derive a simple present valuemodel. Like Binsbergen and Koijen (2010), I combine the Campbell and Shiller (1988) present value

identity with a simple rst-order autoregressive process for expected returns and expected dividend

growth rates. In this model, future returns are a linear function of the DP ratio and the expected

dividend growth rates. We can therefore predict returns through a multivariate regression of returns

3


4/73

on the DP ratio and an estimate of expected dividend growth rates, or we can combine both values in

a single predictor, which I call the corrected DP ratio. The corrected DP ratio can be interpreted as

the dividendprice ratio adjusted for variation in expected dividend growth rates.

Following the present value model, I estimate the proposed proxy for expected dividend growth

rates. To extract dividend growth rates implied in index options and futures, I rst estimate an

implied dividend yield. Because any mispricing in options is easiest to arbitrage away by trading

futures, I estimate implied dividend yield simultaneously from options and futures. By combining

the no-arbitrage cost-of-carry formula for index futures and the putcall parity condition for index

options, I derive an expression that enables estimation of the implied dividend yield in terms of the

observable prices of derivatives and the underlying asset. Then I combine the implied dividend yield

with the realized DP ratio to calculate implied dividend growth rates and the corrected DP ratio.

I apply the empirical analysis to the S&P 500 index. As we require data on both options and

futures, the main analysis runs from January 1994 through June 2011.1 Like other studies, I nd

the standard DP ratio is a rather poor predictor of both future returns and dividend growth rates at

horizons ranging from one month to one year. Implied dividend growth rates, however, are reliable

predictors of future dividend growth rates for all the horizons. Accordingly, the ability to predict

market returns improves considerably when we include implied dividend growth rates as an additional

regressor in the standard DP ratio predictive regression. The results also conrm that the DP ratio

and the implied dividend growth rates can be replaced by a single predictor: the corrected DP ratio.

The predictive coecient on the corrected DP ratio is always statistically signicant. In the predictive

regressions with monthly returns, the corrected DP ratio has an in-sample adjusted R2 of 4:21% and

an out-of-sample R2OS of 5:33%, compared to 0:44% and 0:04% (respectively) for the uncorrected

DP ratio. Because the corrected DP ratio is less persistent than the uncorrected DP ratio, and as

innovations to the corrected DP are related only weakly to returns, the corrected DP ratio has the

additional advantage that it is robust to the small sample bias that has been shown to hinder the

predictive ability of the uncorrected DP ratio.

Contrary to the standard view that almost all the variation in the DP ratio corresponds to changes

1 Note that this period is not aected by structural breaks in the mean of the DP ratio, which have been shown toaect the forecasting relation of returns and the DP ratio over longer periods (Lettau and Van Nieuwerburgh, 2008;Favero et al., 2011).

4


5/73

in discount rates (Binsbergen and Koijen, 2010; Cochrane, 2011), a variance decomposition reveals

that a substantial portion of variation in the DP ratio is attributable to the time-varying value of

expected dividend growth rates. Expected dividend growth rates account for 34% of variation in

the DP ratio, and the discount rate eect accounts for 134% of its variation. I also nd that expected

returns and expected dividend growth rates are highly correlated at 0:74. This positive correlation

dampens the volatility of the DP ratio, making it less able to predict either returns or dividend growth

rates (see also Menzly et al. 2004; Lettau and Ludvigson, 2005). Once the DP ratio is corrected for

changes in the expected dividend growth rates, it becomes more volatile and better able to predict

returns.

The results are robust to a number of variations, such as exclusion of the recent nancial crisis or

inclusion of other return predictors, and are not driven by seasonality of dividend payments, dividend

risk premium or synchronicity issues in the end-of-day derivatives data. Further, the same results

cannot be obtained using historical dividend growth rates or information on announced dividends.

The results depend importantly, however, on the simultaneous use of options and futures in estimating

the implied dividend growth rates, which conrms that the arbitrage relations for S&P 500 derivatives

should best be analyzed simultaneously.

By emphasizing the role of dividend growth rates for price movements, the results have important

implications for the theoretical models (Campbell and Cochrane 1999; Bansal and Yaron, 2004). That

both time-varying expected returns as well as time-varying expected dividend growth rates should be

featured in the asset pricing models is also advocated by Koijen and Van Nieuwerburgh (2011) and

Chen et al. (2012).

The paper also draws on studies using implied dividends. Dividends implied in derivative markets

have been used as an input in the calculation of risk-neutral densities (Ait-Sahalia and Lo, 1998),

and to study empirical properties of dividend strips (Binsbergen et al. 2012a). I believe, however,

this paper is the rst to use implied dividends for predicting market returns. I also introduce a

new technique that enables extraction of implied dividends simultaneously from options and futures

without resort to the use of proxies for the implied interest rate.

Binsbergen et al. (2012a) document a substantial dividend risk premium in longer-maturity implied

5


6/73

dividends. Similar conclusion is provided by Binsbergen et al. (2012b), who analyze a growing

market for dividend futures.2 The dividend risk premium, however, is not a concern in our case

because I focus on derivatives with shorter maturities (six months), where a substantial portion of

dividends is preannounced and hence, riskless. Accordingly, I nd that the risk premium documented

by Binsbergen et al. (2012a) is absent in the 6-month implied dividends. The estimated implied

dividend ratios also exhibit very low correlations with other proxies for risk.

This work relates as well to the literature that uses other options-implied predictors to forecast

market returns. Bollerslev et al. (2009) use the variance risk premium and Bakshi et al. (2011)

employ forward variances. I focus instead on a cash ow-based predictor and show that information

about dividends implied in derivative markets conveys information about future returns beyond the

information in the variance risk premium. The proposed predictor is also unique in the sense that it

simultaneously exploits information embedded in two closely related derivatives markets, the options

market and the futures market.

Finally, several papers analyze equity options as opposed to index options and document that

measures of violations of no-arbitrage relations and implied skewness embody useful information for

predicting returns on individual stocks (Cremers and Weinbaum, 2010; Conrad et al. 2012; Xing et

al., 2010). In line with Pan and Poteshman (2006), however, I do not nd evidence that these results

carry over to the aggregate market. Also, while ndings on equity options suggest that informed

trading takes place in the derivative markets, results of this paper do not support that implication

and should be interpreted in the light of evidence on market integration. That is, information about

dividends implied in derivative markets can be useful for predicting returns on the underlying asset

only if markets are suciently integrated and investors use the same information to set prices in both

markets.

2 Dividend futures allow to trade dividends directly, are available across dierent regions, and have maturities up to 10

years. Thus, they are very suitable to study implied dividends. The disadvantage is the short sample. Dividend futureswere introduced around 2000 and are exchange traded since 2008.

6


7/73

2 Theoretical considerations

To provide an analytical framework for the empirical analysis, this section derives a simple log-linear

present value model and describes theoretical aspects of estimating dividend growth rates implied in

derivative markets.

2.1 Present value model

The model combines the Campbell and Shiller (1988) present value identity with time-series processes

for expected returns and expected dividend growth rates. Rytchkov (2008) and Binsbergen and Koijen

(2010) use a similar approach. The main innovation of this study lies in the empirical estimation of

this framework.

Dene log return rt+1, log dividend growth rate dt+1; and log dividend-price ratio dpt as

rt+1 = log

Pt+1 + Dt+1

Pt

; dt+1 = log

Dt+1

Dt

; dpt = log

DtPt

: (1)

Rewrite return as

rt+1 ' + dpt + dt+1 dpt+1; (2)

where = log

1 + exp(dp)

+ dp and = exp(dp)1+exp(dp)

are constants related to the long-run average

of the dividend-price ratio, dp. Iterate (2) forward to obtain the Campbell and Shiller (1988) present

value identity

dpt '

1 + Et

1Xj=0

j(rt+1+j)Et

1Xj=0

j(dt+1+j): (3)

Let t = Et(rt+1) be the conditional expected return, and let gt = Et(dt+1) be the conditional

expected dividend growth rate. Suppose that t and gt follow some time-series processes. For simplicity

assume that both t and gt follow AR(1) processes:3

3 The AR(1) structure for returns and dividend growth rates is considered in several recent contributions (Rytchkov,2008; Binsbergen and Koijen, 2010; Piatti and Trojani, 2012) and is motivated by growing evidence that both expected

7


8/73

t+1 = 0 + 1(t) + "t+1; (4)

gt+1 = 0 + 1(gt) + "gt+1; (5)

dt+1 = gt + "dt+1; (6)

where "t+1; "gt+1; and "

dt+1 are zero mean errors. Combine the present value identity in (3) with the

AR(1) assumptions to nd the dividend-price ratio:

dpt ' ' +

1

1 1

t

1

1 1

gt; (7)

where ' is a constant related to ;;0; 1; 0; and 1 (details are provided in the Appendix).

Equation (7) states that the log dividend-price ratio is related to expected returns and is therefore

a good candidate for predicting future returns. However, the dpt ratio also incorporates information

about expected dividend growth rates. Thus, if those rates vary over time, the dpt ratio is only a

noisy proxy for expected returns and an imperfect predictor of future returns (Fama and French,

1988; Rytchkov, 2008; Binsbergen and Koijen, 2010; Lacerda and Santa-Clara, 2010). Since the dpt

ratio increases with expected returns and declines with expected dividend growth rates, the problemis pronounced when expected returns and expected dividend growth are positively correlated (Menzly

et al., 2004; Lettau and Ludvigson, 2005). This positive correlation osets the changes in expected

returns and in expected dividend growth rates, further reducing its ability to predict returns.

This implies that the dpt ratio is insucient to capture variation in expected returns. We must also

account for the time-varying value of the expected dividend growth rates. To present this formally, I

combine (2), (6), and (7) to derive a return forecasting equation

rt+1 ' + (1 1)dpt +

1 11 1

gt + v

rt+1; (8)

returns and expected dividend growth rates are time-varying and persistent (Menzly et al., 2004; Bansal and Yaron, 2004;Lettau and Ludvigson, 2005). Assuming dierent time-series processes would preserve the main intuition (see Cochrane2008b for a discussion).

8


9/73

where vrt+1 = "dt+1

"t+11

11

"gt+1

11

, and is a constant related to ;;0; 1; 0; and 1. According

to equation (8), to predict returns, we need both the dpt ratio and an estimate for the expected

dividend growth rates. Because dpt ratio and expected dividend growth rates are linearly related to

future returns, we can also replace them by a single predictor:

rt+1 ' + (1 1)dpCorrt + v

rt+1; (9)

where dpCorrt = dpt + gt

1

11

is the corrected dividend-price ratio, which can be interpreted as

the dpt ratio adjusted for variation in the expected dividend growth rates. The adjustment factor is

a constant that depends on the linearization parameter () and the persistence of expected dividend

growth rates (1). Because the linearization parameter is typically close to one and fairly stable overtime, the adjustment constant depends mainly on the persistence of the expected dividend growth

rates. The more persistent the expected dividend growth rates, the more important the role of the

expected dividend growth rates in the corrected dividend-price ratio.4

2.2 Estimating implied dividend growth rates

The present value model implies that the dividend-price ratio is not enough to capture variation in

expected returns. We also need an estimate for expected dividend growth rates.

I propose to extract an estimate for expected dividend growth rates from index derivatives. Prices

of derivatives, such as options and futures, depend on, among other factors, the dividends that the

index pays until expiration of the contracts. Therefore, we can invert pricing relations and estimate a

proxy for expected dividend growth rates from market prices of derivatives. All we need is absence of

arbitrage opportunities.

Following Ait-Sahalia and Lo (1998), I assume that the index pays a continuously compounded

dividend yield and describe how it can be extracted from derivatives using no-arbitrage relations. I

start with a simple case when future dividend yield is known, and then discuss the case of a stochastic

4 Lacerda and Santa-Clara (2010) derive a similar correction for their adjusted dividend-price ratio, dpAdjt = dpt +

gt

1

1t

: In their version, the adjusted dividend-price ratio (dpAdjt ) does not depend on the persistence of expected

dividend growth rates because they assume that expected dividend growth rates are equal to average historical dividendgrowth rates (gt).

9


10/73

dividend yield.

Under the assumption of a known dividend yield, the no-arbitrage futures price is given by the

cost-of-carry formula

Ft() = St exp[(rt() t())] ; (10)

where Ft is the futures price, St is the price of the underlying, rt() is the annualized continuously

compounded interest rate from t to t + , and t() is the annualized continuously compounded

dividend yield between t and t + . Similarly, by no-arbitrage, the dierence between a European call

and a European put written on the index is given by the put-call parity

Ct(K; ) Pt(K; ) = St exp[t()]Kexp[rt()] ; (11)

where Ct(K; ) and Pt(K; ) are the prices of a European call and a European put option with the

same maturity and the same strike price K.

Both no-arbitrage conditions relate prices of derivatives to the future dividend yield and the risk-

free rate. With a proxy for the implied risk-free rate, we can therefore estimate implied dividend

yield from either options or futures. Nevertheless, implied interest rate may be dierent from theobservable proxies for the interest rate (Naranjo, 2009), and choice of the proxy may adversely aect

results. Further, given that the basket of stocks in the index is dicult to trade, any mispricing of

options is easiest to arbitrage away by trading futures (Kamara and Miller, 1995). This suggests that

the no-arbitrage conditions in equations (10) and (11) should best be analyzed simultaneously. Thus,

I propose to combine both no-arbitrage relations, thereby also isolate the eect of the interest rate,

and estimate implied dividend yield simultaneously from index options and index futures.5

I rst solve for the implied interest rate by combining the no-arbitrage relations

rt() =1

log

Ft()K

Ct(K; ) Pt(K; )

: (12)

5 Section 8:2 considers alternative ways for estimating implied dividend yield and conrms that results are the strongestwhen implied dividend yield is estimated simultaneously from futures and options.

10


11/73

Then I plug (12) back in to (11) to obtain a formula for the implied dividend yield

t() = 1

log

Ct(K; ) Pt(K; )

St

+

K

St

Ct(K; ) Pt(K; )

Ft()K

: (13)

Following equation (13), all we need to estimate the implied dividend yield is a European call option

and a European put option with the same strike and the same maturity, the futures price with the

same expiration date as the options, and the price of the underlying asset.

Equation (13), however, is derived under the assumption of a known dividend yield. When the

future dividend yield is stochastic, as assumed in the present value model, the dividend yield in (13)

not only reects expectations about the future dividend yield, but may also include a dividend (yield)

risk premium.6

Recent studies show that the dividend risk premium plays an important role for

horizons longer than one year (Binsbergen et al. 2012a; 2012b). I focus, however, on derivatives with

six months to maturity, where a substantial portion of dividends is preannounced and hence riskless.

I later show that even at an annual horizon, dividend growth rates can be predicted with considerable

precision, given information on announced dividends. This largely mitigates the eect of the dividend

risk premium. Indeed, I nd that the risk premium documented by Binsbergen et al. (2012a) is absent

in the 6-month implied dividends. Therefore, in the main analysis, I use implied dividend yield as

an uncontaminated measure of expected dividend yield. Sensitivity analysis conrms that the risk

premium is not a concern for short-maturity implied dividend yields.

Once I have an estimate for the implied dividend yield, I combine it with the realized dividend-price

ratio to calculate implied dividend growth rates.

3 Data

I use the S&P 500 index as a proxy for the aggregate market. I obtain the daily S&P 500 price index

(without dividends) and the S&P 500 total return index (with dividends reinvested) from Datastream.

I purchase the S&P 500 futures data from the Chicago Mercantile Exchange and the S&P 500 options

6 Lioui (2006) derives the put-call parity relation in the case of stochastic dividend yield by assuming complete marketsand a mean-reverting stochastic price of market risk, Ct(K; ) Pt(K; ) = St exp

t() 1=2(

2S() ()

2)

Kexp[rt()] ; where 1=2(2S() ()

2) is a correction for the dividend risk premium.

11


12/73

data from Market Data Express. The futures data run from April 1982 through June 2011 and the

options data run from January 1990 through June 2011.

To analyze synchronicity issues in the derivative markets, I obtain for part of the sample intra-daily

data from Tick Data. The intra-daily cash index data cover the period from February 1983 through

June 2011. The intra-daily trade and quote data for S&P 500 options run from July 2004 through June

2011, and the newly released intra-daily trade and quote data for S&P 500 futures run from January

2010 to June 2011. To analyze demand imbalances in the options market, I also obtain for the sample

from January 1990 through June 2011 the SPX OpenClose data from Market Data Express. These

data provide separate daily observations for each option of opening and closing volume for dierent

investor groups (small customers, middle customers, large customers, and rm proprietary traders).

To compare implied dividends with dividend forecasts based on announced dividends, I obtain from

Standard & Poors the indicated dividend series for the S&P 500 index from February 1995 through

June 2011. To analyze dividend risk premium, I download implied dividend prices of Binsbergen et

al. (2012a) from Ralph Koijens website. To compare the period analyzed in this paper with longer

time periods, I download dividends data from Amit Goyals webpage. In some parts of the analysis I

also make use of other variables, I download daily constant-maturity 3-month and 6-month Treasury

bill rates, the default spread (dierence between Moodys BAA and AAA corporate bond spreads),

and the term spread (dierence between 10-year Treasury bond yield and 3-month Treasury bill rate)

from the Federal Reserve Bank of St. Louis. I obtain daily S&P 500 earnings-price ratio and 6-

month LIBOR data from Datastream. I download implied skewness from the Chicago Board Options

Exchange (CBOE), end-of-month variance risk premiums from Hao Zhous webpage, and the quarterly

consumption-to-wealth ratio from Sydney C. Ludvigsons website.

While European options on the S&P 500 index have been traded since April 1986, the Market

Data Express options data go back only to January 1990. Also, until 1994, the settlement procedures

for S&P 500 options and futures diered. While futures are settled in the opening value of the index

since June 1987, the most liquid S&P 500 options initially expired in the closing value of the index.

As liquid options and futures with matching expiration times are needed to estimate implied dividend

growth rates, I conne the main analysis to the period from January 1994 through June 2011. 7

7 Section 8:3 considers extending the time period back to 1990.

12


13/73

For all the derivatives data I regularly eliminate observations with missing values for any of the

main variables. With the help of a Market Data Express support team, I eliminate all but standard

S&P 500 options (that is, I eliminate quarterlies, weeklies, and mini options). I also eliminate options

that violate the basic arbitrage relations. All variables are sampled at the end-of-month.

3.1 Empirical estimation

Here I dene the main variables used in the empirical analysis, rst variables based on the realized

data and then variables related to dividends implied in the derivative markets.

Because dividend ratios are calculated by aggregating dividends over a given horizon (typically

one year), we need to take a stance on whether or not to reinvest the dividends in the market. Chen

(2009) and Binsbergen and Koijen (2010) show that when dividends are reinvested in the market, part

of the return properties are imputed in the dividend growth rates, and this adversely aects return

and dividend growth rate predictability results (see also Koijen and Van Nieuwerburgh, 2011). To

avoid such contamination, I aggregate realized dividends without reinvestment as in Ang and Bekaert

(2007). To ensure comparability with the implied dividend yield, as discussed below, I transform the

continuously compounded implied dividend yield in a raw dividend yield.

Monthly returns. Monthly returns are dened as

rMt = log

Pt + Dt

Pt1

; (14)

where Pt and Dt denote price and monthly dividends at the end of month t. Realized dividends are

calculated from the S&P 500 price index and the S&P 500 total return index. Since Datastream

reinvests dividends daily, I rst calculate daily dividends and then sum them over the past month.

Dividend-price ratio. The dividend-price ratio is dened as a 12-month trailing sum of realized

dividends (D12t ) over the current price:

dpt = log [DPt] = log

D12tPt

: (15)

13


14/73

Dividend growth rates. Like Ang and Bekaert (2007), I calculate monthly dividend growth

rates based on a 12-month trailing sum of dividends:

dMt = log D12tD12t1 : (16)This approach to calculating dividend growth rates diminishes the eect of seasonality in dividend

payments, but introduces overlapping in dividend growth rates for any frequency higher than annual.

Implied dividend yield. I estimate implied dividend yield according to equation (13). I use

daily settlement prices for futures, the mid-point between the last bid and the last ask price for options,

and closing values for the S&P 500 price index.

To minimize eects of microstructure noise and market frictions, I calculate implied dividend yield

by aggregating information from a wide set of options and futures. At the end of each month, I use 10

days of (backward-looking) data and construct option pairs (put-call pairs with the same strike and

the same maturity) from all the reliable options (options with positive volume or open interest greater

than 200 contracts).8 Then I combine option pairs with futures of matching maturity and the current

value of the underlying index. To eliminate some extreme observations, I discard observations where

Ft()KCt(K;)Pt(K;)

is smaller than 0:5 or greater than 1:5, and where Ct(K; ) Pt(K; ) is zero.9 This

matching procedure results in several estimates of the implied dividend yield for a given maturity at

the end of each month (227 on average, with a minimum of 37 and a maximum of 935). The number of

estimates generally increases over time and declines with the maturity of the derivatives. I aggregate

these estimates at the end of each month by taking the median across all the implied dividend yields

with the same maturity.

As derivatives are most liquid at the short end, implied dividend yields estimated from near-

to-maturity options and futures should provide the most reliable estimates. Within year dividends

are highly seasonal, however, and a common approach to alleviate seasonality issues is to aggregate

dividends over one year. This suggests that the implied dividend yield should rather be calculated

from options and futures with one year to expiration. Considering both aspects, the choice of the

8 Untabulated results show that there is no strike price eect, i.e., the implied dividend yield does not depend on themoneyness level.

9 This lter eliminates a bit under 2% of the observations.

14


15/73

appropriate maturity of the implied dividend yield depends on the trade-o between the within-year

seasonality of dividend payments and the liquidity of derivatives.

To examine this trade-o, Table 1 reports summary statistics for annualized realized dividends

summed over either 1 month, 3 months, 6 months, or 12 months, and Table 2 reports summary

statistics on the liquidity of derivatives.

Table 1 conrms that within-year dividends are highly seasonal. The standard deviation of annual-

ized monthly dividends is 7.36, which compares to a standard deviation of 4.61 for dividends summed

over 12 months. Thus, annualized monthly dividends are 60% more volatile than annual dividends.

Nevertheless, the volatility of dividends decays rapidly with increase of the horizon. When dividends

are summed over 6 months, the standard deviation of annualized dividends is 4.70, which is only 2%

higher than the standard deviation of dividends summed over 12 months. This suggests that the

problem of seasonality in dividend payments has largely diminished at a semi-annual frequency.

[Insert Table 1 about here]

Table 2 conrms that short-maturity derivatives are the most liquid. The tilt toward short matu-

rities is especially pronounced for futures. While 60-80% of options trading is concentrated on options

with maturities of less than 3 months, as much as 95% of futures trading is in short-maturity futures.The market for S&P 500 futures, however, is among the most liquid derivative markets, and in notional

terms, there is a considerable amount of trading in the longer-maturity futures. The average daily

open interest of futures with times to maturity of between 3 and 6 months is $5.93 billion. Even open

interest of futures with maturities between 6 and 9 months amounts to $839 million per day. Hence,

although derivatives become less liquid with maturity, especially futures, there is a substantial amount

of trading in S&P 500 options and futures with maturities of 6 months.


Given the trade-o between the seasonality of dividends and the liquidity of derivatives, I conduct

the main analysis on options and futures with six months to expiration.10 Because options expire

10 Section 8.1 shows that the results are robust to using derivatives with either 5 or 7 months to expiration. Implieddividend ratios based on shorter-maturity derivatives are more volatile and yield somewhat weaker results.

15


16/73

monthly and futures expire quarterly, there are only four dates each year when options and futures

expire simultaneously (the third Friday morning in March, June, September, and December). To

obtain monthly implied dividend yields with a constant 6-month maturity, I proceed as follows. For

months on a quarterly cycle, I calculate implied dividend yield using options and futures with 6 months

to maturity. For the other two months in the quarter, I interpolate linearly between implied dividend

yields with maturities closest to 6 months.

By construction, the estimated implied dividend yield is annualized and continuously compounded.

To make it comparable to the realized dividend-price ratio, I transform it into a raw (eective) implied

dividend yield, IDYt = exp(bt) 1: The log implied dividend yield is simply idyt = log(IDYt).

Implied dividend growth rates. The implied dividend growth rate is dened as the dierence

between the log implied dividend yield and the log dividend-price ratio. Given that the implied

dividend yield is based on derivatives with maturities of 6 months, it should ideally be matched with

the 6 month dividend-price ratio construct. As the analysis of seasonality suggests that annualized

dividends calculated over 6 months behave much like dividends summed over 12 months, however, I

rely on the standard dividend-price ratio and calculate implied dividend growth rates by combining

annualized 6 month implied dividend yield with the annual dividend-price ratio:11

idgt = logIDYt

DPt

= idyt dpt: (17)

Corrected dividend-price ratio. The corrected dividend-price ratio is given by

dpCorrt = dpt +

1

1

b

c1

idgt; (18)

where b is the estimated linearization constant, and c1 is the AR(1) coecient of the implied dividendgrowth rates.

11 Section 8.1 shows that the return predictability results are robust to using a dividend-price ratio based on anannualized 6-month trailing sum of dividends.

16


17/73

3.2 Data description

Table 3 reports summary statistics for the main variables. All the variables are sampled monthly and

expressed in logs. Annualized returns and dividend growth rates are on average 7:84% and 3:78%,

respectively.

Figure 1 depicts construction of the implied dividend growth rates. Panel A plots implied dividend

yield along with the dividend-price ratio, and Panel B plots the dierence between both of them

the implied dividend growth rates. The implied dividend growth rates nicely reect general market

conditions. They are positive during the market booms of 1994-1997 and 2002-2007, when investors

were optimistic about future growth opportunities, and generally negative during stock market busts,

such as the Asian-Russian-LTCM crisis (1998), the dot.com bubble burst (2001), and the recent

nancial crisis (2008-2009), when investors were pessimistic about growth opportunities. The average

implied dividend growth rate is higher than the average realized dividend growth rate (7:31%), but

the dierence is not statistically signicant.

A somewhat disconcerting feature is rather high volatility, especially around the turn of the century

when implied dividend growth rate swings from 0:76 to +0:49. These uctuations are hard to justify

by changes in investors expectations and could be partially driven by data problems.12 Because

implied dividend yield is estimated from no-arbitrage relations that involve highly leveraged positions,

any errors would be magnied in the implied dividend growth rates (Boguth et al., 2012). The period

of these high uctuations also coincides with historically low dividend yields (approximately 1% per

year in raw terms) when even tiny deviations of implied dividend yield from the realized dividend-

price ratio caused big swings in the implied dividend growth rate. Thus, at least part of the observed

uctuations could be explained by a scaling eect, exacerbated by the log scale. Most important,

however, as documented later, implied dividend growth rates reliably predict future dividend growth

rates, and do not seem to be inuenced by either demand imbalances in derivative markets or other

measures of risk. Thus, despite volatility and possible noise in the data, implied dividend growth rates

appear to be a good proxy for expected dividend growth rates.

12 Golez and Jackwerth (2012) note an unusually high number of missing observations of S&P 500 options in 1998-2002.This pattern of missing observations is found in end-of-day options data provided by Market Data Express as well as inthe OptionMetrics data.

17


18/73

The AR(1) coecient of the implied dividend growth rates at monthly frequency is 0:53. Note

that because of the overlapping structure in the implied dividend growth rates, which invalidates the

AR(1) assumption of data, monthly AR(1) cannot be used to infer annual AR(1). Therefore, in the

the variance decomposition of the dividend-price ratio (Section 6), which is typically based on annual

variables, I use non-overlapping annual observations. In the predictability analysis, however, given

the relatively short time period, I use data sampled at monthly frequency, and make appropriate

adjustments for the overlap when necessary.


[Insert Figure 1 about here]

The corrected dividend-price ratio is given by a linear combination of the dividend-price ratio and

the implied dividend growth rates

dpCorrt = dpt +

1

1bc1

idgt

= dpt +

1

1 (0:9823 0:5335)

idgt = dpt + 2:1011 idgt; (19)

where the linearization constant b and the persistence of the implied dividend growth rates c1 arebased on the whole sample.13 Figure 2 plots dpCorrt ratio along with the dpt ratio. Both dividend

ratios exhibit strong comovement (a pairwise correlation coecient of 0:72), but the patterns dier in

three important aspects.


First, the dpCorrt ratio is on average higher than the dpt ratio in the boom periods and lower in the

bust periods. This means that the dpt ratio tends to predict returns that are too low to match the

markets optimism about growth opportunities during the boom periods. At the same time, the dpt

13 Because b and c1 are estimated using the whole sample, the corrected dividend-price ratio is subject to a look-aheadbias. Out-of-sample predictability results in Section 5:1 show, however, that look-ahead bias has a marginal inuence onthe results.

18


19/73

ratio tends to forecast returns that are too high during the crisis periods. This is particularly apparent

at the end of the sample, when the market experienced one of the greatest drops in the history of the

U.S. stock market, but the uncorrected dividend-price ratio rose and therefore predicted unrealistically

high returns.

Second, the corrected dividend-price ratio is notably more volatile than the uncorrected dividend-

price ratio. The standard deviation is 0:26 for the dpt ratio compared to 0:52 for the dpCorrt ratio. While

part of this dierence could be accounted for by measurement errors, in the context of the present

value model, such an increase in volatility implies that expected returns and expected dividend growth

rates are highly correlated. To see this formally, expand the variance of the corrected dividend-price

ratio as

var(dpCorrt ) = var(dpt) + 2

1

1 1

1

1 1

cov(t; gt)

1

1 1

2var(gt): (20)

Equation (20) says that the variance of the dpCorrt ratio can be higher than the variance of the dpt

ratio only if expected returns and expected dividend growth rates co-vary and the co-variation is high

enough

2

111

cov(t; gt) >

1

11

var(gt)

. Since dpt ratio increases with expected returns

and declines with expected dividend growth rates, this positive co-variation osets shocks to expected

returns and to expected dividend growth rates, and reduces the volatility of the dpt ratio (Lettau and

Ludvigson, 2005, Rytchkov, 2008; Binsbergen and Koijen, 2010). Once we correct the dividend-price

ratio for changes in the expected dividend growth rates, we restore part of the variation, which is

otherwise oset by comovement of the expected returns and the expected dividend growth rates (see

also Lacerda and Santa-Clara, 2010).

Finally, the dpCorrt ratio is also less persistent than the dpt ratio. The rst-order autocorrelation

coecient for the dividend-price ratio is 0:98, and the null hypothesis of a unit root cannot be rejected,

according to an augmented Dickey-Fuller (ADF) test. The AR(1) for the corrected dividend-price ratio

is 0:74, and the null of a unit root in the corrected dividend-price ratio is rejected at the 5% level. This

reduced persistence is important because highly autocorrelated predictors aggravate small-sample bias

(Stambaugh, 1999) and produce inaccurate inference results in the case of overlapping observations

19


20/73

(Boudoukh et al., 2008).14

By applying equation (12) and following the same estimation procedure as for the implied dividend

yield, I also estimate the implied interest rate (II Rt). Although IIRt is not of direct interest for this

study, it is worth noting that the IIRt exhibits the expected behavior. Figure 3 shows that IIRt

strongly co-varies with the T-bill rate and LIBOR, and is on average closer to LIBOR (see also

Naranjo, 2009). Still, IIRt is more volatile than the T-bill rate and LIBOR at the beginning of the

period analyzed, and it diverges from both proxies for the interest rate during the recent nancial

crisis, when it is notably lower than LIBOR. This points at the importance of isolating the eect of

interest rates when estimating the implied dividend yield.


4 Results

The present value model assumes an AR(1) structure for expected returns and expected dividend

growth rates. This implies that all the information is incorporated in the short-term (one horizon)

prices and dividends. But the model is silent about the appropriate length of the horizon. I consider

predicting dividend growth rates and market returns over horizons varying from one month to one

year. I use standard predictive regressions, in which returns or dividend growth rates are regressed on

lagged predictors. All regressions are based on monthly frequencies. The Hodrick (1992) t-statistics

reported explicitly take into account the overlapping structure of regressions.15 I also report adjusted

R2s. Note, however, that the R2s in the context of overlapping observations need to be interpreted

with caution because they tend to increase with the extent of the overlap even in the absence of true

predictability (Kirby, 1997; Valkanov, 2003; Boudoukh et al., 2008).

14

The reduced persistence also makes the corrected dividend-price ratio less prone to data snooping problems becausedata mining tends to produce highly persistent predictors (Ferson et al., 2003).15 Ang and Bekaert (2007) show that the Hodrick (1992) standard errors, which are based on summing predictors in

the past, perform better than other standards errors frequently employed in the literature, such as Newey-West (1987),or Hansen and Hodrick (1980) standard errors.

20


21/73

4.1 Predicting dividend growth rates

Figure 1 shows that implied dividend growth rates track general market conditions and therefore seem

to be a good proxy for expected dividend growth rates. Now I test whether implied dividend growth

rates also predict future dividend growth rates. For a comparison, I use annual realized dividend

growth rates and the dividend-price ratio. I use the latter as a competitive predictor for two reasons.

First, the dividend-price ratio is itself a function of expected dividend growth rates and could therefore

predict not only future returns but also future dividend growth rates. Second, implied dividend growth

rates are dened as the dierence between the implied dividend yield and the dividend-price ratio.

Therefore, it is necessary to show that implied dividend growth rates do not predict future dividend

growth rates simply because they duplicate information contained in the dividend-price ratio.

The main dividend growth rates regression is

dt+h = a0 + a1(Xt) + "t+1; (21)

where dt+h =hX

i=1

dMt+h is the dividend growth rate with h = 1; 3; 6; or 12 months, and Xt is a set

of predictor variables. Because of the overlapping structure in the realized dividend growth rates, I

use lagged annual dividend growth rates as a competitive predictor only in predictive regressions for

annual dividend growth rates.

Table 4 reports the results. The dividend-price ratio is always negatively related to future dividend

growth ratesjust as the theory suggestsbut the associated t-statistics are relatively low and vary

between 1:88 and 1:54. Accordingly, the associated adj: R2s are low, ranging from 1:59% for

predicting monthly dividend growth rates to 4:39% for predicting annual dividend growth rates. The

implied dividend growth rates by contrast are positively related to future dividend growth rates,

explaining 4:99% of the variation in monthly dividend growth rates and 19:05% of the variation in

annual dividend growth rates. Also, all the estimated parameters on the implied dividend growth

rates are highly statistically signicant, with t-statistics varying between 3:70 and 4:98.

The implied dividend growth rates remain signicant in bivariate predictive regressions with the

dividend-price ratio, which conrms that the implied dividend growth rates provide an independent

21


22/73

source of information about the future dividend growth rates. Moreover, in bivariate regressions, the

statistical signicance of both predictors (the implied dividend growth rates and the dividend-price

ratio) is higher than in the univariate regressions, and the adj: R2 rises to 9:73% in a regression for

monthly dividend growth rates and to 32:86% in a regression for annual dividend growth rates.16

Further, the implied dividend growth rates predict future dividend growth rates also better than

do the lagged realized dividend growth rates. The implied dividend growth rates exhibit higher adj:

R2 and t-statistics than the realized dividend growth rates in the univariate regressions, and remain

highly signicant in the bivariate regression with realized dividend growth rates. While the t-statistic

on the estimated parameter on realized dividend growth rates in the bivariate regression is 2:50, it is

4:48 for implied dividend growth rates.


4.2 Predicting market returns

I employ three specications for the return predictive regressions. I start with the standard predictive

regression, in which returns are regressed on the lagged dividend-price ratio. Then I augment the

standard predictive regression with the implied dividend growth rates. Finally, I replace the dividend-

price ratio and the implied dividend growth rates with the corrected dividend-price ratio. The main

return regression is

rt+h = b0 + b1(Xt) + "t+1; (22)

where rt+h =

hXi=1

rMt+i is the market return with h = 1; 3; 6; or 12 months, and Xt is a set of predictor

variables.

Note that the adjustment factor in the corrected dividend-price ratio depends on the persistence

16 This improvement in predicting dividend growth rates can be accounted for by the partial resolution of the omittedvariable bias. According to the present value model, a regression of dividend growth rates on the dividend-price ratiois subject to the omitted variable bias because it lacks a control for expected returns. When expected dividend growthrates are highly correlated with expected returns, a proxy for expected dividend growth rates (implied dividend growthrates) can help to diminish the omitted variable bias. That omitted variable bias plays an important role in the dividendgrowth rate predictive regressions is also documented in Binsbergen and Koijen (2010) and Koijen and Van Nieuwerburgh(2011).

22


23/73

of expected dividend growth rates, which, according to the present value model, is measured over

the same horizon as returns. As all return regressions are based on monthly frequency, I let the

adjustment factor in the corrected dividend-price ratio vary with the return horizon as dpCorrt =

dpt + 1= h1b\1(h)i idgt, where \1(h) = AR(h) is estimated at a monthly frequency. Thus, in thepredictive regression for monthly returns, the adjustment factor is based on the AR(1) of the implied

dividend growth rates. The corrected dividend-price ratio in this case is the same as in Table 3 and

Figure 2. For longer return horizons, the impact of the implied dividend growth rates on the corrected

dividend-price ratio decreases with the length of the horizon in line with the decline in the persistence

of the implied dividend growth rates.

Table 5 reports the results. In accordance with the theory, the estimated parameter on the

dividend-price ratio is always positive, but the associated t-statistics are relatively low, especially

at the short end. The estimated parameter on the dividend-price ratio is signicant at the conven-

tional 5% signicance level only in the predictive regression for annual returns. The associated adj:

R2s are relatively low and vary from 0:44% for predicting monthly returns to 17:62% for predicting

annual returns.

When the implied dividend growth rates are included as an additional regressor to the dividend-

price ratio, return predictability improves for all the horizons considered. The adj: R2 rises from

0:44% to 4:49% in the regression for monthly returns and from 17:62% to 19:87% in the regressions

for annual returns. Results also conrm that we can replace the dividend-price ratio and the implied

dividend growth rate by a single predictor, the corrected dividend-price ratio. The corrected dividend-

price ratio predicts returns approximately as well as the dividend-price ratio and the implied dividend

growth rates together. The adj: R2 amounts to 4:21% at the monthly horizon and to 19:04% at the

annual horizon. Estimated parameters on the dpCorrt ratio are always statistically signicant, with

t-statistics varying between 2:42 and 3:10.17

Because the corrected dividend-price ratio incorporates expectations about future dividend growth

rates, it should also be less prone to statistical biases found in regressions with the dividend-price ratio.

17 Untabulated results show that the F-test fails to reject restrictions regarding predictive coecients in the bivariate

return regressions with the dividend-price ratio and the expected dividend growth rates \bdpt1 (h) = (1b\1(h)) \bidgt1 (h),at all the considered horizons.

23


24/73

Specically, positive news about future dividends produces high returns and low dividend-price ratios.

This induces negative correlation between shocks to returns and shocks to the dividend-price ratio,

contributing to an upward bias in the predictive coecient (Fama and French, 1988; Stambaugh,

1999; Amihud and Hurvich, 2004, Polk et al. 2006; Boudoukh et al., 2008). The more persistent the

predictor, the greater the bias. The dividend-price ratio is highly persistent and shocks to returns and

shocks to the dividend-price ratio are highly negatively correlated at 0:97. Simulations indicate that

the upward bias in this case can be just as high, or even higher, then the predictive coecient on the

dividend-price ratio found in the data.18 Unlike the dividend-price ratio, the corrected dividend-price

ratio incorporates expectations about future dividends, which reduces negative correlation between

shocks to returns and shocks to the predictor variable. I nd that the correlation between shocks

in the case of the corrected dividend-price ratio is only 0:23. Since the corrected dividend-price

ratio is also less persistent, this evidence suggests (and simulations conrm) a very low bias in the

predictive coecient (2% 3%, depending on the return horizon). Simulations also conrm that the

predictive coecient on the realized dividend-price ratio is insignicant, with bootstrapped p-values

between 29% and 48%, depending on the horizon, compared to p-values of between 0% and 1% for

the corrected dividend-price ratio.19

Further, the same results cannot be obtained using lagged dividend growth rates as opposed to the

implied dividend growth rates. Inclusion of the realized dividend growth rates in the return predictive

regressions with the dividend-price ratio and the implied dividend growth rates leaves the adj: R2

largely unchanged. The realized dividend growth rate is statistically insignicant at all the horizons.


18 The simulated model is

rt+h = + 0xt + ut+1;

xt+1 = + xt + vt+1;

where h is either 1, 3, 6, or 12, and xt is the predictor variable. Error terms are bivariate-normal. I simulate monthlyreturns under the null of no predictability ( = 0). Longer-horizon returns are constructed from simulated monthlyreturns. In 100,000 simulations, bias is measured as the mean of the predictive coecients. Bootstrapped p-values arecalculated as the percentage of simulated coecients that are farther away from zero than the coecients found in thedata.

19 For the case of non-overlapping monthly return regressions, approach proposed by Amihud and Hurvich (2004)conrms that the bias-adjusted estimated parameter on the corrected dividend-price ratio is essentially identical to theOLS estimated parameter (0:0183 versus 0:0188), and the adjusted t-statistic is 3:10, which is identical to the Hodrick(1992) t-statistic. This compares to a slightly negative estimated parameter on the realized dividend-price ratio, afteradjustment (0:0010) and an adjusted t-statistic of0:07.

24


25/73

5 Additional tests

To further validate the ability of the corrected dividend-price ratio to predict returns, I next examine

stability of in-sample forecasting relations and consider predicting returns out-of-sample. I also analyze

how the corrected dividend-price ratio compares to other popular return predictors, including measures

of risk and demand imbalances in the derivative markets. In addition, I report results for the subsample

analysis. To avoid statistical problems inherent in the use of overlapping observations (Kirby, 1997;

Boudoukh et al., 2008), but to maximize the number of observations, I restrict the analysis henceforth

to prediction of non-overlapping monthly returns.

5.1 Parameter stability and out-of-sample predictability

The forecasting relation of returns and the realized dividend-price ratio is found to be unstable, and

the dividend-price ratio typically fails to predict returns out-of-sample (Lettau and Van Nieuwerburgh,

2008; Goyal and Welch, 2008).20

To examine whether the corrected dividend-price ratio produces a more stable forecasting relation,

Figure 4 shows predictive coecients and t-statistics for predictive regressions of monthly returns on

either the dividend-price ratio or the corrected dividend-price ratio using 120-month rolling windows.

As expected, the forecasting relation between returns and the dividend-price ratio is very unstable.

During the recent nancial crisis, the dividend-price ratio even became negatively related to future

returns. The predictive coecient on the dividend-price ratio can be anywhere between 0:03 and

0:07. The estimated parameter on the corrected dividend-price ratio, however, is much more stable,

as well as always positive, between 0:01 and 0:05. Its t-statistics are always well above the t-statistics

associated with the dividend-price ratio. Thus, the forecasting relation between returns and the

corrected dividend-price ratio is both strong and stable.


Next, I examine the ability of the corrected dividend-price ratio to predict returns in real time. I

calculate the out-of-sample R2 as in Campbell and Thompson (2008) and Goyal and Welch (2008).

20 An in-depth discussion on the possibility of predicting returns out-of-sample is provided in a special issue of theReview of Financial Studies (Spiegel, 2008).

25


26/73

R2OS = 1MSEFVMSEHA

; (23)

where M SEFV =1

T1

T

Xt=1

(rt+1bt)2 is the mean-squared error based on the tted values (bt) from

a predictive regression estimated through period t, and M SEHA =1

T1

TXt=1

(rt+1 rt)2 is the mean-

squared error based on the historical average return (rt) estimated through period t. A positive

out-of-sample R2 indicates that the predictive regression has a lower mean-squared prediction error

than the historical average return. I measure the statistical signicance of results using the M SEF

statistic proposed by McCraken (2007).

To make out-of-sample forecasts, I split the sample to two subperiods. I use the period fromJanuary 1994 through December 1999 for estimation of the initial parameters and the period from

January 2000 through June 2011 for calculation of the R2OS. All out-of-sample forecasts are based on

a recursive scheme using all the available information up to time t.

Recall that the corrected dividend-price ratio is dened as dpCorrt = dpt +

11bc1

idgt; where

(a linearization constant) and 1 (persistence of the implied dividend growth rates) are estimated

using the whole sample period, therefore introducing a slight look-ahead bias. To alleviate the concern

that the look-ahead bias could inuence results, I additionally estimate a so-called no-look-ahead-bias

corrected dividend-price ratio, dpCorrNLABt = dpt +

11bc1

idgt; where bt and bt are time-varying

and estimated using the same recursive scheme as in calculation of the out-of-sample R2OS.

Table 6 reports the results. The dpt ratio with its poor ability to predict returns in-sample also fails

to predict returns out-of-sample. The R2OS for the dpt ratio is 0:04%, compared to an out-of-sample

R2OS for the dpCorrt of 5:33% (statistically signicant at the one percent signicance level). Thus, the

dpCorrt ratio both predicts returns in-sample and delivers better out-of-sample forecasts of monthly

returns than forecasts based on the historical average.21 Moreover, approximately the same R2OS, if

not even slightly higher, is obtained with a corrected dividend-price ratio adjusted for the look-ahead

bias (R2OS 5:35%): Hence, look-ahead bias is not a concern, and the dpCorrt ratio can be eectively be

21 Untabulated results show that the dpCorrt ratio predicts returns out-of-sample better than a constant of 6% per year,which has been found to be the best forecasting variable in Simin (2008).

26


27/73

used in real time for portfolio allocation decisions.22



To illustrate the relative success of the dpCorrt ratio in predicting returns out-of-sample, Figure 5

plots out-of-sample forecasts and realized returns. Although realized returns are signicantly more

volatile than any of the forecasted returns, there are considerable dierences across the forecasts.

The forecasts based on the realized dividend-price ratio and the forecasts based on the historical

average return are both relatively smooth. Forecasts based on the corrected dividend-price ratio vary

signicantly more, and the changes of the forecasts are often of the same sign as the changes of the

realized returns.

5.2 Alternative predictors

Next I analyze how corrected dividend-price ratio compares to other return predictors. For the rst

alternative predictor, I construct a dividend-price ratio corrected for average historical dividend growth

rates. Lacerda and Santa-Clara (2010) show that correcting the dividend-price ratio for the 10-year

moving average of dividend growth rates improves the predictability of longer-horizon (annual) returns.

As my focus is on short-horizon predictability, and the 10-year moving average of dividend growth rates

evolves rather slowly, I calculate average historical dividend growth rates as a moving average of 12

months of annualized monthly dividend growth rates. For comparability with the dividend-price ratio

corrected for implied dividend growth rates, the persistence of the historical dividend growth rates is

presumed the same as the persistence of the implied dividend growth rates. Thus, the dividend-price

ratio adjusted for the historical dividend growth rates is dened as dpHistt = dpt + 2:10 dgMt , where

dgMt is the moving average of annualized monthly dividend growth rates over the past 12 months. In

addition, I use two other standard return predictors, the earnings-price ratio ept, and the consumption-

22 The rather small dierence in the R2OS between the two corrected dividend-price ratios occurs because the p ersistenceof the implied dividend growth rates bt and the linearization constant bt are very stable (bt ranges from 0:4713 to 0:6185;and bt is between 0:9814 and 0:9833). This makes dpCorrt and dpNLABCorrt highly correlated (0:9988) and almostindistinguishable.

27


28/73

to-wealth ratio cayt proposed by Lettau and Ludvigson (2001).23

Because the implied dividend growth rates and consequently the corrected dividend-price ratio

could be inuenced by the risk premium, I analyze how the corrected dividend-price ratio relates to

other risk related predictors, such as the default spread deft, the term spread termt, and the variance

risk premium as implied in the S&P 500 index. The variance risk premium is especially relevant

because it is also based on information implied in the S&P 500 options. I employ two versions of

the variance risk premium. The rst version, dened as in Bollerslev et al. (2009), measures the

dierence between the variance implied in the S&P 500 options and the realized variance based on the

intra-daily data for the S&P 500 Index vrpRV Ht . The second version is based on the dierence between

the implied variance and the forecasted variance (forecasts are based on the VIX and the intra-daily

realized variance) vrpFRVHt .

Implied dividend yield is estimated from no-arbitrage relations. Consequently, any violations of

no-arbitrage relations would impact implied dividend growth rates. Since violations of put-call parity

and related measures, such as implied skewness, are shown to predict returns in the cross-section (Xing

et al. 2010; Cremers and Weinbaum, 2010; Conrad et al., 2012), I consider whether these results carry

over to the aggregate market. For this purpose, I use the skewness implied in 30 days-to-maturity

S&P 500 options as provided by the CBOE skewt. Because violations of no-arbitrage relations are

related to the transactions volume in puts and calls to open new positions (Cremers and Weinbaum,

2010), I also calculate open-buy volume based on the open-close data for S&P 500 options. I dene

an open-buy variable obt as the dierence between the opening volume for calls and puts.

If demand imbalances aect prices of derivatives, then these imbalances could aect not only

implied dividend growth rates, but also the estimates of implied interest rate, especially because they

are estimated simultaneously and are based on the identical data. Therefore, I additionally use the

spread between 6-month LIBOR and the 6-month implied interest rate sprLIRt . Because LIBOR is an

indicated average rate for all the large banks, and the implied interest rate reects the interest rate for

arbitrageurs (presumably those with the cheapest access to money), the spread may not reect only

23 Lettau and Van Nieuwerburgh (2008) adjust the dividend-price ratio for structural breaks in its mean. Using datafrom 1927 through 2004, they nd strong evidence in favor of a structural break in the early 1990s (and possibly anotherone in the 1950s). When I consider the period from January 1994 until June 2011 in isolation, however, I do not ndevidence for structural breaks, according to tests developed by Bai and Perron (1998). SupF tests of zero versus onebreak, or zero versus two breaks, are both insignicant, and sequential test chooses the number of breaks as zero.

28


29/73

market frictions, but also credit risk.

Summary statistics. Table 7 reports the basic summary statistics and the unconditional cor-

relation structure for all the variables sampled monthly.24 The standard return predictors are highly

persistent, and the hypothesis of a unit root for most of them cannot be rejected. The persistence

of the dividend-price ratio corrected for average historical dividend growth rates is only marginally

lower than the persistence of the realized dividend-price ratio. Dividend-price ratio corrected for im-

plied dividend growth rates, variance risk premiums, and measures of market frictions, however, are

substantially less persistent, and the null of a unit root is strongly rejected in most cases.

The realized dividend-price ratio, the dividend-price ratio corrected for historical dividend growth

rates, and the dividend-price ratio corrected for implied dividend growth rates are all highly correlated.

Also, they are all positively correlated with the earnings-price ratio and the consumption-to-wealth

ratio. Importantly, the dividend-price ratio corrected for implied dividend growth rates exhibits overall

much lower correlations with the risk-related predictors than the realized dividend-price ratio.

Both dividend ratios also exhibit low correlations with the variables measuring demand imbalances

and market frictions. The only exception is the relation with the spread between LIBOR and the

implied interest rate. While sprLIRt is positively correlated with the realized dividend-price ratio,

it is negatively correlated with the corrected dividend-price ratio. This dierence, however, can be

explained by the sprLIRt capturing information about the credit risk. Indeed, sprLIRt is strongly

correlated with the default spread, which is related to the dividend-price ratio, but unrelated to the

corrected dividend-price ratio.


Predicting market returns. Table 8 reports results for predicting monthly returns. Since in-

sample and out-of-sample results are largely consistent, I evaluate predictors mainly on the in-sample

evidence. The traditional predictors based on the realized data explain only a small portion of the

variation in future monthly returns. The univariate return regressions with either the earnings-price

24 Since cayt is available only quarterly, monthly observations ofcayt are dened by the most recently available quarterlyobservation.

29


30/73

ratio, the consumption-to-wealth ratio, the term spread, or the default spread exhibit adj: R2s between

minus 0:5% and 1%. The dividend-price ratio, as already documented, explains 0:44% of the variation

in future monthly returns.

Correcting the dividend-price ratio for average historical dividend growth rates does not seem to

improve the predictability of monthly market returns in the period analyzed. The adj: R2 in the return

regression with the dpHistt is approximately the same, if not slightly lower, than the adj: R2 in the

regression with the realized dividend-price ratio.25

Furthermore, predictive regressions with the skewness and open-buy exhibit slightly negative adj:

R2s. Thus, the results as to the predictive ability of those variables from the cross-section do not seem

to carry over to the aggregate market. This is in line with Pan and Poteshman (2006), who nd that

put-call ratios from option volume initiated by buyers to open new positions predict stock returns,

but they fail to predict returns on the aggregate market. Like the default spread, the spread between

LIBOR and the implied interest rate is negatively related to future returns. It is marginally signicant

at 10% level and explains 1:89% of the variation in monthly returns.

The dividend-price ratio corrected for implied dividend growth rates dpCorrt and the variance risk

premium based on the realized variance vrpRV Ht , however, explain substantially more of the variation

in future monthly returns. The vrpRVHt is positively related to future returns, with an adj: R2 of

4:62%. The dpCorrt , as already documented, has an adj: R2 of 4:21%. The vrpRV Ht and the dp

Corrt

are also the only predictors that are signicant at conventional levels of statistical signicance. The

t-statistics are 2:13 on the variance risk premium and 3:10 on the corrected dividend-price ratio. Note,

however, that the predictability results using the variance risk premium depend on how it is dened.

Unlike the variance risk premium based on the realized variance vrpRVHt , the variance risk premium

based on the forecasted variance vrpFRVHt fails to predict returns in the period analyzed.

All in all, the corrected dividend-price ratio and the variance risk premium based on the realizedvariance are the strongest return predictors of the variables considered. Since they are both based

on information extracted from the derivative markets, the relative success of the dpCorrt in predicting

25 Untabulated results show that this result is robust to using alternative proxies for historical dividend growth rates,such as lagged monthly dividend growth rates or the moving average of 10 years of monthly dividend growth rates.Furthermore, results are as well robust to using the adjusted dividend-price ratio proposed by Lacerda and Santa-Clara(2010), in which the adjustment factor does not depend on the persistence of expected dividend growth rates.

30


31/73

future returns could be driven by the fact that the dpCorrt is duplicating information in the variance

risk premium. To address this concern, I show that the corrected dividend-price ratio remains highly

signicant in a bivariate regression with the variance risk premium. Moreover, in a bivariate regression,

the statistical signicance of both predictors increases and the adj: R2 rises to as much as 10:03%.

Both predictors also predict returns out-of-sample with an R2OS of 11:37%. The corrected dividend-

price ratio as well remains highly signicant in bivariate regressions with other measures of risk and

market frictions.


5.3 Subsample Analysis

The period of analysis runs from January 1994 through June 2011 and covers several of turmoil periods,

including the recent nancial crisis. Such periods have an important impact on forecasting relations.

Figure 4 has shown that the forecasting relation between returns and the dividend-price ratio at the

peak of the nancial crisis momentarily even changed sign.

Table 9 repeats all the predictive regressions by truncating the period at December 2007. The

variance risk premium based on the realized variance vrpRVHt , which performs very well in the full

sample, now becomes an insignicant predictor for future returns and loses its advantage to the variance

risk premium based on the forecasted variance.

Cash-ow related predictors, however, such as the realized dividend-price ratio, the earnings-price

ratio, and the consumption-to-wealth ratio, now become signicantly related to future monthly returns.

The realized dividend-price ratio alone exhibits an adj: R2 of 2:83%. Correcting the dividend-price

ratio for the historical dividend growth rates improves the predictability of returns, with an adj: R2

of 4:02%, although still lower than the adj: R2 of 6:49% for the dividend-price ratio corrected for

implied dividend growth rates. Interestingly, just as in the full period, the corrected dividend-price

ratio predicts returns especially well jointly with the variance risk premium vrpRVHt . The in-sample

adj: R2 and the out-of-sample R2OS in the bivariate regression are 9:50% and 12:64%, respectively.

All in all, the corrected dividend-price ratio emerges as the strongest predictor of monthly returns

when the nancial crisis is excluded. But its relative advantage over the realized dividend-price

31


32/73

ratio is greater when the nancial crisis is included. This is intuitive, as the outbreak of the crisis

completely altered expectations about future cash ows and made lagged realized dividend growth rates

an overoptimistic proxy for expected dividend growth rates.26 As a result, the standard dividend-price

ratio as well as the dividend-price ratio corrected for the historical dividend growth rates are both too

high to be justied by the general pessimism in the market. The dividend-price ratio corrected for the

implied dividend growth rates, however, incorporates forward-looking expectations, and is thus better

able to predict future returns.


6 What moves the prices?

The general conclusion in the literature is that price movements are due primarily to changes in dis-

count rates and not so much to changes in expected dividend growth rates (Cochrane, 1992; Cochrane,

2005; Ang and Bekaert, 2007; Binsbergen and Koijen, 2010; Koijen and Van Nieuwerburgh, 2011).

Yet the predictability results of this paper suggest that variation in expected dividend growth rates is

important for price movements. Therefore, I now explore further the implications of the present value

model to provide additional insights into the interpretation of results.

According to equation (9), the estimated parameter on the corrected dividend-price ratio equals

1 bt b1. The persistence of expected returns b1 can therefore be calculated from the estimatedparameter on the corrected dividend-price ratio and the linearization constant bt.27 Further, we canuse equation (7) to decompose the variance of the dividend-price ratio to two covariance terms as

var(dpt) '

1

1 1

cov(dpt; t)

1

1 1

cov(dpt; gt); (24)

where the rst term on the right-hand side represents the contribution of expected returns (discount

26 Untabulated results conrm that the crisis also had an important impact on the dividend growth rate predictability.While implied dividend growth rates reliably predict future dividend growth rates (including or excluding the crisis), theforecasting relation between the dividend-price ratio and the dividend growth rates changes sign and the dividend-priceratio becomes positively (wrongly) related to future dividend growth rates when the crisis is excluded. Further, asexpected, lagged dividend growth rates predict future dividend growth rates better when the crisis is excluded.

27 Alternatively, the persistence of expected returns can be identied from the estimated parameter on the dividend-price ratio or the implied dividend growth rates, according to equation (8). Both appraoches yield very similar results.

32


33/73

rates), and the second term the contribution of expected dividend growth rates (cash ow). I stan-

dardize the right-hand side by the left-hand side, so that both terms sum up 100%. If we alternatively

decompose the variance of the dividend-price ratio to two variance terms and a covariance term, we

can also calculate the correlation between expected returns and expected dividend growth rates.28

The variance of expected returns can be estimated by inverting equation (7).

So far all results have been based on data sampled at monthly frequencies. Dividend ratios, how-

ever, are based on realized dividends summed over 12 months, which invalidates the AR(1) assumption

of data sampled monthly. Therefore, the analysis should best be conducted on non-overlapping an-

nual observations (see also Binsbergen and Koijen, 2010). As the time period in the main analysis is

rather short, but the data are available monthly, I rely on non-overlapping annual observations and

estimate all parameters 12 times using successive annual samples starting at the end of each month

(end of Januarynext January, end of Februarynext February, and so on). I report the mean and

the interquartile range for the estimated parameters. This approach does not avoid the problem of

overlapping observations, but lets us use more data and examine how stable the estimated parameters

are.

The relatively short time period also raises a concern that the results may be sample specic.

Therefore, for a comparison with the implied dividend growth rates, I rst estimate the parameters

using lagged annual dividend growth rates as a proxy for expected dividend growth rates. Using

realized dividend growth rates I also examine how the sample period analyzed in this paper compares

to longer time periods. In Model (1), I report results based on the period from January 1926 through

June 2011. Model (2) collects results for the post-war period from January 1946 through June 2011

and Model (3) reports results for the period from January 1994 through June 2011.

Next I use implied dividend growth rates as a measure of expected dividend growth rates. As dis-

cussed earlier, implied dividend growth rates could be contaminated by market frictions, risk premium,

and other measurement errors. Because measurement errors aect the volatility and the persistence of

expected dividend growth rates, they could importantly inuence the variance decomposition results.

28 The variance decomposition of the dividend-price ratio in this case is

var(dpt) '

1

11

2var(t) +

1

11

2var(gt) 2

1

11

1

11

cov(t; gt):

Note that assigning half of the covariance term to expected returns and half to expected dividend growth rates wouldyield the same results as the variance decomposition in equation (24).

33


34/73

To address this concern, I use four dierent versions of implied dividend growth rates. In Model (4),

I use the implied dividend growth rate as is. In Model (5), I clean implied dividend growth rates

of variation due to demand imbalances

Expected Returns and Dividend Growth Rates Implied In Derivative Markets

Documents

Transcript of Expected Returns and Dividend Growth Rates Implied In Derivative Markets