Expected Correlation and Future Market Returns

Expected Correlation and Future Market Returns∗

Adrian Buss Lorenzo Schonleber Grigory Vilkov

This version: July 30, 2019

Abstract

Implied correlation, jointly extracted from index and stock options, is a robust pre-dictor of long-term market returns. We document that its predictive power stemsfrom its role as a leading procyclical state variable, predicting future investmentopportunities, that is, financial-market risks and macroeconomic conditions, for upto 18 months ahead. The predictability is inherited from the interplay between itsthree main components—implied market variance, implied idiosyncratic variance,and the implied dispersion of market betas—and not subsumed by measures of tailrisk. We also provide first empirical evidence of out-of-sample predictability, leadingto substantial economic gains for market-timing strategies.

Keywords: (option-)implied correlation, return predictability, idiosyncratic risk, option-implied information, contemporaneous betas

JEL: G11, G12, G13, G17

∗Adrian Buss is affiliated with INSEAD and CEPR, [email protected]; Lorenzo Schonleber andGrigory Vilkov are affiliated with the Frankfurt School of Finance & Management, [email protected] [email protected]. We received helpful comments and suggestions from Turan Bali, Bruno Biais, Nicole Branger(discussant), Agostino Capponi (discussant), Peter Carr, Mathijs Cosemans, Bernard Dumas, Thierry Foucault,Amit Goyal (discussant), Christian Heyerdahl-Larsen (discussant), Leonid Kogan, Petter Kolm, Hugues Lan-glois, Ding Luo (discussant), Philippe Muller, Viktor Todorov, Igor Pozdeev (discussant), Fabio Trojani, AndreaVedolin, and Hao Zhou. We also thank participants at the American Finance Association Meetings 2019, CityUniversity of Hong Kong International Finance Conference on Corporate Finance and Financial Markets 2019,China International Conference in Finance 2017, the 2nd Annual Eastern Conference on Mathematical Finance(ECMF) 2017, the BofAML Quant and Risk Premia Conference 2017, the Frontiers of Factor Investing Conference2018, the FMA Europe 2018, and the First Asset Pricing conference by LTI at the Collegio Carlo Alberto andseminar participants at Erasmus University (Rotterdam), ESSEC Business School, Frankfurt School of Finance& Management, HEC Paris, INSEAD, UNSW Sydney, the University of Sydney, the University of TechnologySydney, the University of Balearic Islands, and the University of Maastricht for useful comments. We gratefullyacknowledge financial support from the Canadian Derivatives Institute (CDI).

Expected Correlation and Future Market Returns

This version: July 30, 2019

Abstract

Implied correlation, jointly extracted from index and stock options, is a robust pre-

dictor of long-term market returns. We document that its predictive power stems

from its role as a leading procyclical state variable, predicting future investment

opportunities, that is, financial-market risks and macroeconomic conditions, for up

to 18 months ahead. The predictability is inherited from the interplay between its

three main components—implied market variance, implied idiosyncratic variance,

and the implied dispersion of market betas—and not subsumed by measures of tail

risk. We also provide first empirical evidence of out-of-sample predictability, leading

to substantial economic gains for market-timing strategies.

Keywords: (option-)implied correlation, return predictability, idiosyncratic risk, option-

implied information, contemporaneous betas

JEL: G11, G12, G13, G17

Option prices, by construction, reflect investors’ expectations about future price movements.

Hence, measures of risk extracted from option markets are natural candidates for predicting

future (aggregate) stock returns. In particular, one of the most robust option-implied predictors

of long-term market excess returns is implied correlation, that is, the average expected correlation

between individual stocks jointly extracted from the cross-section of individual stock options and

index options (see, e.g., Driessen, Maenhout, and Vilkov (2005, 2012), Cosemans (2011), and

Faria, Kosowski, and Wang (2016)).1 However, although this predictability is often linked to

downside (“tail”) risk or, more generally, economic uncertainty, its exact underlying economic

rationale is not yet well understood.

Consequently, the main objective of this paper is to study the economics behind the pre-

dictive power of implied correlation and the channels through which it is related to future

investment opportunities. Our main findings can be summarized as follows. The market return

predictability of option-implied correlation stems from its role as a leading procyclical state

variable, a role that differentiates it from variables like the VIX volatility index and the variance

risk premium, which are typically considered fear and tail risk proxies. In particular, option-

implied correlation forecasts future financial risks as well as future economic growth, economic

activity, and (economic) uncertainty at long horizons. It inherits its predictive power from its

three key determinants—aggregate risk, idiosyncratic risk, and the cross-sectional dispersion of

systematic risk—which, in combination, predict future financial and macroeconomic risks as

well as future market returns even better. Finally, we demonstrate that implied correlation

also predicts market excess returns out-of-sample, again for up to 12-month horizons. We now

discuss these findings in more detail.

In a first step, we use a stylized conceptual framework to illustrate how the average ex-

pected correlation depends on its three main determinants—aggregate risk, idiosyncratic risk,

and the cross-sectional dispersion of systematic risk—and demonstrate how these variables can

1Market return predictability using index -based option-implied variables is documented in Bollerslev, Tauchen,and Zhou (2009), Drechsler and Yaron (2011), and Bollerslev, Marrone, Xu, and Zhou (2014) for the variancerisk premium, as well as in Feunou, Jahan-Parvar, and Okou (2017) and Kilic and Shaliastovich (2017) forthe downside semivariance risk premium. Moreover, long-term market return predictability by past realizedcorrelation is discussed in Pollet and Wilson (2010).

1

be constructed from options data.2 Consistent with our framework, we find that, empirically,

variations in implied market variance and implied idiosyncratic variance can explain more than

60% (and up to 80%) of monthly variations in implied correlation. Intuitively, an increase in

implied market variance implies a stronger comovement among individual stocks, because the in-

crease strengthens the importance of aggregate shocks. Conversely, when implied idiosyncratic

variance increases, implied correlation declines, because stock-specific shocks are, to a larger

extent, responsible for fluctuations in individual stock returns.

Second, we study how implied correlation and its components are related to future finan-

cial risk and macro-economic conditions. We find that these variables act as state variables,

predicting different elements of the future investment opportunity set. In particular, implied

correlation significantly predicts future realized correlation for long horizons. It inherits its

predictive power from its three main components, which are strong predictors of their realized

counterparts, though some of the components’ predictive power is lost through aggregation (i.e.,

linear combinations of the option-implied components typically outperform implied correlation).

On the macro-economic side, implied correlation acts as a leading procyclical state variable. It

positively forecasts economic growth in the form of reduced unemployment and a stronger lead-

ing indicator (which captures the long-term growth rate in the coincident economic activity).

Also, it is negatively related to future economic uncertainty and a contrarian sentiment index,

and positively related to future market valuations. Again, most of this predictive power is in-

herited from its components, and linear combinations of the latter often demonstrate a superior

predictive performance (i.e., provide additional information).

Next, we present a variety of direct and indirect tests for market return predictability by

implied correlation. In-sample, we confirm the findings in the literature that implied correlation

significantly predicts future market excess returns for up to 1-year horizons, always with a

positive coefficient and the highest predictive power in univariate regressions. Importantly,

its predictive power is robust to controlling for other option-based variables built from the

2The existing literature has linked each of these variables—under the actual probability measure—to stockreturns. For example, Guo and Savickas (2006) show that idiosyncratic volatility predicts future market returns,and Herskovic, Kelly, Lustig, and Van Nieuwerburgh (2016) rely on the factor structure of idiosyncratic risk toexplain the cross-section of stock returns. Santos and Veronesi (2004) and Boloorforoosh, Christoffersen, Fournier,and Gourieroux (2017) relate the dispersion of market betas to the equity risk premium.

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marginal distribution of market returns, such as the risk premiums for market variance (e.g.,

Bollerslev, Tauchen, and Zhou (2009), Drechsler and Yaron (2011)) and downside semivariance

(e.g., Kilic and Shaliastovich (2017), Feunou, Jahan-Parvar, and Okou (2017)), both of which

are usually considered proxies for tail risk. Two of the option-implied components (idiosyncratic

variance and the dispersion of market betas) also predict future returns, though with lower R2s.

Consistent with our findings for risk predictability discussed above, linear combinations of its

components perform even better.

Using a vector autoregression (VAR) model similar to the one of Campbell (1991), we then

analyze the joint dynamics, in particular the shock propagation, of implied correlation, its com-

ponents, the market excess return, and dividend growth. Consistent with economic intuition, we

find that past dividend growth negatively affects implied correlation; that is, implied correlation

increases in adverse economic conditions. Decomposing implied correlation into components

again proves useful in improving the model’s fit. Finally, we document that, consistent with the

intuition from the present-value relation (see, e.g., Campbell and Shiller (1988) and Campbell

(1991)), revisions in expected market excess returns, as predicted by implied correlation, are

negatively correlated with contemporaneous market excess returns.

Lastly, we provide a novel analysis of out-of-sample return predictability by implied corre-

lation. Using traditional regression techniques, we document a significant predictive power of

implied correlation for horizons of up to 6 months. However, we also find that traditional out-of-

sample regression techniques cannot fully exploit the predictive power of many option-implied

predictors, because they require long historical estimation windows, whereas the sample period

of option-implied variables is rather limited. For example, for an estimation window shorter

than 5 years, only implied correlation and its risk premium show any predictive power, whereas

for a long estimation window of 10 years, also idiosyncratic variance and the risk premiums for

variance and downside semivariance deliver some predictability, though significance is difficult to

establish. In lieu of this evidence, we propose a novel out-of-sample predictability method that

extends the contemporaneous-beta approach (see, e.g., Cutler, Poterba, and Summers (1989),

Roll (1988)). Our method combines high-frequency increments in option-implied variables with

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the variables’ forward-looking risk premiums to predict future market excess returns.3 For

predictions based on correlation and downside semivariance risk, our approach leads to out-of-

sample R2s of around 8% for a 1-month horizon, 6% to 7.5% for 3- to 6-month horizons, and

of about 2.5% for 12-month horizons. Consistent with our initial motivation, we link the better

performance of our new approach to its stable (robust to outliers) and up-to-date regression co-

efficients. These results imply strong economic gains; for example, using the return predictions

by correlation risk in a market-timing strategy significantly improves investors’ risk-adjusted

returns for horizons up to 1 year.

Our work is related to several strands of the literature, for example, the literature on market

return predictability using information extracted from option markets.4 Driessen, Maenhout,

and Vilkov (2005, 2012), Cosemans (2011), and Faria, Kosowski, and Wang (2016) discuss the

in-sample market return predictability of correlation-related variables. Our key contribution to

this literature is that we carefully analyze the economic rationale behind the predictability of

implied correlation, tracing it back to the predictive power of its main determinants for future

financial and macroeconomic risks.5 Also, unlike these papers, we study out-of-sample return

predictability. For that purpose, we provide a novel extension to the contemporaneous-beta

approach that improves out-of-sample return forecasts for option-implied variables.

Bollerslev, Tauchen, and Zhou (2009), Drechsler and Yaron (2011), and Bollerslev, Marrone,

Xu, and Zhou (2014) show that the variance risk premium robustly predicts market returns for

horizons of up to one quarter. Longer-term predictability by various components of the variance

risk premium is presented in Fan, Xiao, and Zhou (2018), who rely on a decomposition into

a premium for variance risk and one for higher-order risks; Feunou, Jahan-Parvar, and Okou

(2017) and Kilic and Shaliastovich (2017), who use decompositions into upside (good) and down-

side (bad) semivariance risk premiums; and Bollerslev, Todorov, and Xu (2015), who focus on

jump components. These papers exclusively focus on information extracted from the marginal

distribution of the market, whereas we highlight the importance of information extracted from

3Pyun (2019) proposes a similar methodology, which relies on shocks to high-frequency realized variance toestimate the variance-risk-premium beta. Instead, we suggest using increments in option-implied variables.

4For an extensive literature survey, confer with Christoffersen, Jacobs, and Chang (2013). Goyal and Welch(2008) carefully analyze and discuss market return predictability more generally.

5We rely on techniques introduced in Buss and Vilkov (2012) to construct novel estimates of aggregate id-iosyncratic risk and the cross-sectional dispersion of market betas from options data.

4

the joint distribution of the market and individual stocks. In particular, we carefully document

that the predictability of implied correlation is not redundant relative to purely index-based

components. Finally, our work is related to Martin and Wagner (Forthcoming), who document

return predictability for individual stock returns using a combination of expected market vari-

ance and a stock’s expected (“excess”) variance, both of which are extracted from option prices.

We operate with a similar concept but on the aggregate level.

Our work also naturally connects to the literature studying correlation risk. Buraschi,

Porchia, and Trojani (2010) study optimal portfolios in the presence of correlation risk. Leippold

and Trojani (2010) propose a multivariate framework in which stochastic volatilities, stochas-

tic correlations, and jumps can be consistently modeled. In Buraschi, Trojani, and Vedolin

(2014), correlation risk is endogenously priced because of differences-in-beliefs. Relatedly, in

Piatti (2015), a correlation risk premium arises from agents’ disagreement about the likelihood

of systematic disasters. Empirically, Buraschi, Kosowski, and Trojani (2014) relate correlation

risk to a “no-place-to-hide” state variable; Chang, Christoffersen, Jacobs, and Vainberg (2012)

and Buss and Vilkov (2012) use option-implied correlations to measure systematic risk; and

Mueller, Stathopoulos, and Vedolin (2017) study correlation risk in foreign exchange markets.

Our contribution to this literature is to empirically link variations in implied correlation and

its components to variations in future financial market and macro-economic conditions by high-

lighting the role of implied correlation as a leading procyclical state variable.

Our analysis also relates to the literature on idiosyncratic risk. Ang, Hodrick, Xing, and

Zhang (2006, 2009) document a negative relation between idiosyncratic risk and stock returns

in the cross-section. Schneider, Wagner, and Zechner (2017) relate the pricing of idiosyncratic

risk to (implied) skewness. Goyal and Santa-Clara (2003) document that average variance (their

proxy for idiosyncratic risk) positively predicts future market returns, whereas Bali, Cakici, Yan,

and Zhang (2005) cannot find a significant relation between idiosyncratic volatility and future

market returns. Guo and Savickas (2006) show that, when combined with aggregate stock

market volatility, the value-weighted idiosyncratic volatility is significantly negatively related to

future market returns. Herskovic, Kelly, Lustig, and Van Nieuwerburgh (2016) rely on the factor

structure of idiosyncratic volatility to explore the cross-section of stock returns. We contribute

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to this literature by linking the predictive power of the implied correlation to idiosyncratic

risk. In particular, because implied correlation positively predicts future market returns but is

negatively related to idiosyncratic volatility, our evidence is consistent with a negative price for

idiosyncratic risk.

Consistent with our conceptual framework for expected correlation and our empirical find-

ings, Kogan and Papanikolaou (2012, 2013) demonstrate that idiosyncratic volatility and the

dispersion of market betas are positively related to growth opportunities, which, in turn, are

negatively related to the equity risk premium. Similarly, in the model of Santos and Veronesi

(2004), the equity risk premium is low when the dispersion of systematic risk is high, and, in

the model of Boloorforoosh, Christoffersen, Fournier, and Gourieroux (2017), market betas pos-

itively (negatively) covary with the pricing kernel for low (high) beta values, so that dispersion

of the market betas goes down (up) in bad (good) economic conditions.

The rest of this paper is organized as follows: Section 1 proposes a conceptual framework

to identify the main determinants of average expected correlation and shows how to construct

these components from options data. Section 2 explores the role of implied correlation as a state

variable, linking it and its components to future financial and economic conditions. Section 3

documents the in-sample predictive power of implied correlation and its components. Section 4

discusses out-of-sample predictability and introduces a novel forecasting approach based on

contemporaneous betas. Section 5 provides robustness tests. Section 6 concludes.

1 Expected Correlation and Its Determinants

In this section, we discuss the relation between correlation and its main components. For that

purpose, we introduce a stylized framework that connects expected correlation to aggregate risk,

idiosyncratic risk, and the dispersion of systematic betas. We then show how these variables

can be constructed from options data and empirically explore their relation.

6

1.1 Theoretical Framework

Our conceptual framework is based on that of Kelly, Lustig, and Van Nieuwerburgh (2016) and

relies on a discrete-time, composite one-factor structure for stock returns. In particular, we

consider a broad index consisting of a large number of stocks, denoted by i ∈ 1, . . . , N. The

annual log-return of each individual stock, ri,t+1, is described by the following dynamics:

ri,t+1 = µi,t + βi,t εA,t+1 + εi,Id,t+1, (1)

where εA,t+1 denotes an aggregate shock common to all stocks, and εi,Id,t+1 denotes an idiosyn-

cratic (stock-specific) shock, which is independent of the aggregate shock and of the idiosyncratic

shocks of the other stocks. Both aggregate and idiosyncratic shocks might comprise several com-

ponents, such as Gaussian shocks, jumps, or separate up and down jumps.6 Importantly, the

specific distribution of the aggregate and the idiosyncratic shocks is not important for our analy-

sis; what is relevant is the total expected variance of each factor, denoted by σ2A,t ≡ V art(εA,t+1)

and σ2i,Id,t ≡ V art(εi,Id,t+1). The exposure of the individual stocks to aggregate shocks (“market

betas”), βi,t, is assumed to be stochastic and distributed around the natural mean of 1.0 with

time-varying variance σ2β,t.

The total expected return variance of stock i, σ2i,t ≡ V art(ri,t+1), is equal to

σ2i,t = β2

i,t σ2A,t + σ2

i,Id,t, (2)

and its expected return is µi,t, which is the sum of the expected return on the index and risk

adjustments. The return on the index is given by rt+1 ≡∑N

i=1 ωi,t ri,t+1, where ωi,t denotes

the index weight of stock i.7 As the number of stocks becomes large (i.e., N → ∞), the

idiosyncratic shocks are diversified. Because the weighted average of the market betas is 1.0,

this implies that the conditional expected variance of the index is equal to σ2A,t. As it follows from

6For instance, a Merton jump setting similar to that used by Kelly, Lustig, and Van Nieuwerburgh (2016)can be captured by decomposing the aggregate shock into a systematic Gaussian shock εA,t+1 and a systematicjump JA,t+1 and by decomposing the idiosyncratic shock into an idiosyncratic Gaussian shock εi,Id,t+1 and anidiosyncratic jump Ji,Id,t+1. The stock return dynamics then would become

ri,t+1 = µi,t + βi,t εA,t+1 + βi,t JA,t+1 + εi,Id,t+1 + Ji,Id,t+1.

Intuitively, the exposure with respect to the aggregate jump, βi,t, captures the stronger reaction of high beta stocksto systematic jumps and is equivalent to “scaling” the mean and the volatility of their jump size distributions.

7For ease of exposition, we ignore non-linearities and volatility adjustments.

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expression (2), the cross-sectional average expected stock variance is given by (1+σ2β,t)σ

2A,t+σ

2i,Id,t

and, naturally, is an increasing function of the aggregate variance, idiosyncratic variance, and

the cross-sectional dispersion of systematic betas.

For the identification of the average correlation among stocks, note that index variance can be

computed in two ways: (1) directly from the index and (2) indirectly through the portfolio of its

constituents. Formally, this computation yields the restriction that the time-t expected variance

of the index, σ2A,t, must be equal to the expected variance of the portfolio of its constituents:

σ2A,t ,

N∑i=1

N∑i′=1

ωi,t ωi′,t σi,t σi′,t ρi,i′,t, (3)

where ρi,i′,t denotes the correlation between the returns of stocks i and i′. Under the assumption

of equal pairwise correlations (ρi,i′,t = ρj,j′,t,∀i, i′, j, j′), the “equicorrelation” is given by8

Corrt =

σ2A,t −

N∑i=1

ω2i,t σ

2i,t

N∑i=1

∑i′ 6=i ωi,t ωi′,t σi,t σi′,t

. (4)

In the case of a large number of stocks (i.e., N →∞), this correlation can be approximated as

Corrt ≈σ2A,t

σiσi′, (5)

where σiσi′ denotes the average product of individual variances.

Because of its non-linearity, deriving a more precise functional form of this correlation with-

out additional simplifying assumptions is not possible. However, simple simulations show that

the average correlation is an increasing function of the systematic variance (which enters both

numerator and denominator) and a decreasing function of the idiosyncratic variance and of the

dispersion of systematic betas.9

8Elton and Gruber (1973) are one of the first to use this average correlation under the physical measure.Driessen, Maenhout, and Vilkov (2005) and Skinzi and Refenes (2005) introduced it under the risk-neutral mea-sure, with the later literature referring to it as equicorrelation.

9We abstract from variations in index weights, which, over short periods of time, do not contribute muchto fluctuations in expected correlation. Under the risk-neutral probability measure, risk premiums could alsopotentially matter, for example, in the case of jumps, by altering the likelihood of jumps and the jump sizedistribution.

8

1.2 Variables Construction

Note that while computing implied correlation and its component under the physical probability

measure (P -measure) is relatively straightforward, our main interest is in their counterparts

under the risk-neutral probability measure (Q-measure). For this purpose, we rely on data from

the S&P 500 for a sample period from January 1996 to December 2017. We use daily data on

index and individual stock options from the Surface File of Ivy DB OptionMetrics and data on

the daily realized returns from CRSP.10 For the S&P 500, we also obtain intraday returns from

TickData. Data on the S&P 500 index composition and the stocks’ index weights (i.e., relative

market capitalizations) are obtained from Compustat and CRSP.11

In a first step, we compute option-implied variances (i.e., the expected variance under the

risk-neutral measure) for all stocks and the index. In particular, we use simple variance swaps,

as introduced by Martin (2013, 2017), to construct the day-t implied variance for options with

maturity T , IV (t, T ).12 Based on the implied variances of the index and the individual stocks,

we then compute implied correlation, IC(t, T ), using expression (4). Note that constructing

the option-implied idiosyncratic variance (IdIV) and the option-implied dispersion of market

(systematic) betas (IDMB) requires option-implied market betas, βQ(t, T ). For that purpose, we

rely on the technique introduced by Buss and Vilkov (2012), who show how one can construct a

heterogeneous option-implied correlation matrix for all components of an index. Their approach

relies on a parametric form that relates implied correlations to the physical correlation under

the P -measure and a heterogeneous correlation risk premium.13 As Buss and Vilkov show,

the market betas computed from this implied variance-covariance matrix predict future realized

betas significantly better than traditional rolling-window betas. The option-implied dispersion

10We select options with 1 to 12 months to maturity and an (absolute) delta of less than or equal to 0.5. Onaverage, options data are available for 491 of the 500 index constituents.

11We merge the two datasets through the CCM Linking Table using GVKEY and IID to link to PERMNO.Matching to options data is implemented through the historical CUSIP link, provided by OptionMetrics.

12Computing expected variances using log contracts, that is, model-free implied variances, like in Dumas (1995),Britten-Jones and Neuberger (2000), and Bakshi, Kapadia, and Madan (2003), does not affect our results. Theseresults are available on request.

13Appendix A discusses the details of this approach. Implied correlation, IC(t, T ), also can be constructedas the average correlation from the resultant heterogeneous correlation matrix. However, this average and theequicorrelation computed from (4) are almost perfectly correlated (ranging from 0.97 to 0.99), and the predictabil-ity results are the same. Hence, for ease of exposition, we only report the results for option-implied equicorrelationin the following.

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1 3 6 9 12Mean Std Skew Mean Std Skew Mean Std Skew Mean Std Skew Mean Std Skew

IC 0.378 0.129 0.546 0.417 0.114 0.164 0.444 0.105 -0.108 0.453 0.102 -0.170 0.459 0.097 -0.201IDMB 0.260 0.194 5.277 0.150 0.089 2.099 0.121 0.066 1.813 0.107 0.053 1.843 0.094 0.041 1.893IdIV 0.095 0.064 1.974 0.086 0.059 1.869 0.079 0.054 1.805 0.077 0.051 1.662 0.076 0.048 1.661IV 0.042 0.077 1.823 0.042 0.066 1.501 0.043 0.057 1.267 0.044 0.053 1.162 0.044 0.050 1.085

Table 1: Summary statistics. The table reports summary statistics (mean, standard deviation, and skewness)for implied correlation (IC), the implied dispersion of market betas (IDMB), implied idiosyncratic variance(IdIV ), and implied market variance (IV ). Statistics are reported for maturities of 1 to 12 months over a sampleperiod from January 1996 to December 2017 and, where applicable, in annual terms.

of market betas is then simply computed as the cross-sectional variance of these implied betas.

Finally, the option-implied idiosyncratic variance of each stock is computed as the difference

between a stock’s total variance and its systematic component, and its average value across all

stocks is used as the option-implied market-wide idiosyncratic variance:

IdIV (t) =1

N

N∑i=1

[(σQi,t)

2 − (βQi,tσQA,t)

2]. (6)

Table 1 reports summary statistics for the option-implied variables. At a 1-month maturity,

the average implied correlation is about 0.38. It is increasing with maturity and is highly

persistent (with a first-order autocorrelation AC(1) of 0.77) but still considerably varies over

time, with fluctuations being almost symmetric around the mean.14 The annualized S&P 500

implied index variance is 0.042% (which translates into an implied volatility of 20.5%). Also, it

is highly persistent (with a first-order autocorrelation of 0.81); it substantially fluctuates over

time; and it has a generally positive skewness. The implied idiosyncratic variance is extremely

persistent (with AC(1) of 0.93), decreasing with maturity, and is slightly positively skewed. The

dispersion of market betas is positively skewed, decreasing with maturity, and is slightly less

persistent than the correlation, with a monthly autocorrelation of 0.55.

1.3 Dynamics of Implied Correlation and Its Components

Figure 1 depicts the dynamics of implied correlation and its components over time for a 1-month

maturity. Implied correlation, being a bounded variable, displays (relatively) smaller fluctuations

than do the implied variances. Also, the most pronounced moves in implied correlation do not

14Tables A1 and A2 in Appendix D report the autocorrelations and the time-series correlations across variables.

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A: Implied Correlation

19982002

20062010

20142018

Year

0.2

0.3

0.4

0.5

0.6

0.7

IC

B: Implied Market Variance

19982002

20062010

20142018

Year

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

IV

C: Implied Dispersion of Market Betas

19982002

20062010

20142018

Year

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

IDMB

D: Implied Idiosyncratic Variance

19982002

20062010

20142018

Year

0.05

0.10

0.15

0.20

0.25

0.30

0.35

IdIV

Figure 1: Time-series of option-implied variables. The figure shows the time series of implied correla-tion (Panel A) and its components: implied market variance (Panel B), the implied dispersion of market betas(Panel C), and implied idiosyncratic variance (Panel D). The sample period spans from January 1996 to December2017, with variables being computed from 1-month options and sampled daily.

necessarily coincide with those in the implied variances. Consistent with this, the time-series

correlations between the implied correlation and the implied variances (which behave in a similar

way most of the time) are not that high; for example, the correlation with implied market

variance is 0.58 in levels (or 0.64 in differences); the correlation with idiosyncratic variance is

−0.16 in levels (−0.18 in differences). Finally, the dispersion of market betas is quite volatile.

To evaluate empirically how the different components affect implied correlation, we rely on

simple regressions of monthly changes in implied correlation on changes in its components. To

highlight the incremental power of each of the components, we first identify the most important

variable; then the most important pair of variables; and, finally, include all three explanatory

variables (“hierarchical regression”). Table 2 provides results for all available maturities.15

15We have carried out the same exercise for realized correlation and its components. Table A2 in Appendix Dprovide results, which are comparable. In addition to using monthly increments in regressions, we also used logdifferences. Doing so leads to comparable results, with slightly worse explanatory power.

11

1 month 3 months 6 months 9 months 12 months

IV 2.52∗∗∗ 3.14∗∗∗ 2.98∗∗∗ 2.84∗∗∗ 3.44∗∗∗ 3.32∗∗∗ 2.73∗∗∗ 3.47∗∗∗ 3.45∗∗∗ 2.69∗∗∗ 3.28∗∗∗ 3.24∗∗∗ 3.11∗∗∗ 3.44∗∗∗ 3.44∗∗∗

IdIV - -1.79∗∗∗ -1.73∗∗∗ - -1.89∗∗∗ -1.89∗∗∗ - -1.94∗∗∗ -1.94∗∗∗ - -1.99∗∗∗ -1.99∗∗∗ - -2.06∗∗∗ -2.06∗∗∗

IDMB - - -0.12∗∗∗ - - -0.13∗∗ - - -0.04 - - -0.07 - - -0.002R2 40.27 58.13 61.57 47.94 69.90 70.59 45.17 72.82 72.75 43.55 72.77 72.74 54.65 81.34 81.27

Table 2: Determinants of implied correlation. The table reports the results for regressions of monthlychanges in implied correlation on respective monthly changes in its components, namely, the dispersion of marketbetas (IDMB), idiosyncratic variance (IdIV ), and market variance (IV ). For each maturity, we report the speci-fications with one, two, and three independent variables with the highest R2. The data are sampled monthly fromJanuary 1996 to December 2017. ∗∗∗, ∗∗, and ∗ indicate significance at the 1%, 5%, and 10% level, respectively;based on Newey and West (1987) standard errors.

As shown in Table 2, implied market variance captures the largest fraction of variations

in implied correlation, with R2s between 40% and 55% across the different horizons. Implied

idiosyncratic variance is the second most important component, adding around 20%–25% of R2.

The implied dispersion of market betas is less important and typically explains only additional

1%–2% of the variations in implied correlation and is insignificant for horizons of 6 months and

longer. Consistent with our stylized framework, implied market and idiosyncratic variance affect

implied correlation with opposite signs.

2 Implied Correlation and Future Investment Opportunities

In this section, we analyze the potential role of implied correlation and its components as

state variables describing future investment opportunities. First, we concentrate on variables

capturing future financial risks, and, second, we study variables capturing future economic

conditions.

2.1 Implied Correlation and Future Financial Risks

Future aggregate and idiosyncratic variances as well as the future correlations among individual

stocks are key elements of the future investment opportunity set of investors. Hence, in a first

step, we now analyze the predictive power of implied correlation and its components for these

financial variables. For that purpose, we rely on a series of predictive regressions of future

12

A: Realized Correlation

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C: Realized Idiosyncratic Variance

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D: Realized Dispersion of Market Betas

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Figure 2: Financial risk predictions: Predictive power. The figure reports the R2 of predictive regressionsof future realized financial risk measures on current option-implied variables for horizons of 1 to 12 months.Panels A to D report the R2 for realized correlation, realized market variance, realized idiosyncratic variance, andthe realized dispersion of market betas, respectively. Results are based on monthly-sampled data from January1996 to December 2017.

realized risk measures (correlation, market variance, idiosyncratic variance, and dispersion of

market betas) on the respective date-t option-implied variables (with matching maturity).16

The predictive power of these regressions (in terms of R2) is illustrated in Figure 2 across

various horizons. For all horizons, implied correlation is the best predictor of future realized

correlation, outperforming even the linear combination of its components (which affect future

realized correlation in a way that one would expect). In univariate regressions, realized market

variance is best predicted by implied market variance (Panel B), whereas both idiosyncratic

variance and the dispersion of market betas are best predicted by implied idiosyncratic variance

16Realized variance is computed as the sum of squared returns; using demeaned daily returns for individualstocks and 5-minute intraday returns for the S&P 500 (as suggested by Liu, Patton, and Sheppard 2015). Con-sistent with our construction of implied correlation, we compute realized correlation using expression (4), nowbased on realized variances instead of implied ones. The corresponding realized idiosyncratic variance and thedispersion of market betas are computed from market model regressions for each stock.

13


Panel A: Realized CorrelationsIC 0.796∗∗∗ - 0.808∗∗∗ - 0.800∗∗∗ - 0.790∗∗∗ - 0.854∗∗∗ -IDMB - 0.182∗∗∗ - 0.157∗ - 0.185 - 0.267∗ - 0.437∗

IdIV - -1.470∗∗∗ - -1.395∗∗∗ - -1.431∗∗∗ - -1.533∗∗∗ - -1.753∗∗∗

IV - 3.634∗∗∗ - 3.810∗∗∗ - 4.122∗∗∗ - 4.070∗∗∗ - 4.074∗∗∗

R2 50.74 44.68 56.16 51.88 52.53 50.48 51.92 47.99 55.99 48.12

Panel B: Realized VarianceIC 0.170∗∗∗ - 0.139∗∗∗ - 0.086 - 0.067 - 0.078 -IDMB - 0.031∗∗∗ - -0.045 - 0.005 - 0.074∗∗ - 0.146∗∗∗

IdIV - 0.054 - 0.198∗ - 0.099 - 0.023 - -0.041IV - 1.372∗∗∗ - 1.289∗∗∗ - 1.378∗∗∗ - 1.313∗∗∗ - 1.246∗∗∗

R2 12.74 70.94 8.21 72.60 3.52 71.81 2.42 62.45 3.87 55.91

Panel C: Realized Idiosyncratic VarianceIC 0.030 - -0.102 - -0.276∗∗∗ - -0.324∗∗∗ - -0.309∗∗∗ -IDMB - -0.018 - 0.062 - 0.118 - 0.102 - 0.057IdIV - 1.093∗∗∗ - 1.090∗∗∗ - 1.066∗∗∗ - 1.080∗∗∗ - 1.082∗∗∗

IV - 0.706∗∗∗ - 0.665∗∗∗ - 0.597∗ - 0.555 - 0.590R2 -0.20 80.00 1.10 66.66 9.50 58.66 13.72 53.62 12.14 49.31

Panel D: Realized Dispersion of Market BetasIC -0.848∗∗∗ - -0.502∗∗∗ - -0.625∗∗∗ - -0.617∗∗∗ - -0.595∗∗ -IDMB - 0.646∗∗∗ - 0.591∗∗∗ - 0.333 - -0.022 - -0.365∗∗

IdIV - 1.227∗∗ - 1.323∗∗∗ - 1.827∗∗∗ - 2.121∗∗∗ - 2.345∗∗∗

IV - -1.701∗∗∗ - -0.375 - -0.446 - -0.408 - -0.342R2 7.62 11.26 12.69 44.63 23.71 60.65 26.56 67.83 25.54 76.52

Table 3: Financial risk predictions. The table reports the results for selected specifications of predictiveregressions of future realized financial risk measures (correlation, market variance, idiosyncratic variance, andthe dispersion of market betas) on current option-implied variables for horizons of 1 to 12 months. The estima-tion results are based on monthly-sampled data from January 1996 to December 2017. ∗∗∗, ∗∗, and ∗ indicatesignificance at the 1%, 5%, and 10% level, respectively; based on Newey and West (1987) standard errors.

(Panels C and D). Also, for all these variables, the linear combination of the components is

usually superior to any single predictor.

Table 3 contains the corresponding regression coefficients. Notably, some components have

signs opposite to those one would expect from their relation to implied correlation and the rela-

tion between implied correlation and future financial risks. For example, whereas the regression

coefficient of implied correlation for future idiosyncratic variance is negative and the correlation

between implied correlation and implied market variance is positive, the regression coefficient

of implied market variance for future idiosyncratic variance is actually positive. This indicates

that, on occasions, implied correlation and its main components convey different information

about future investment opportunities.

14

2.2 Implied Correlation and Future Economic Conditions

Naturally, future investment opportunities are also driven by the underlying economic conditions.

Hence, in a second step, we now study the predictive power of implied correlation for future

economic fundamentals by regressing future (log)changes in macro-economic variables on the

date-t-implied variables.

For the macro-economic variables, we focus on variables that have been shown to be linked

to the state of the economy and are observed at a monthly frequency. These include (1) the

Industrial Production Index (INDPRO), a measure of the real output of all facilities in the United

States; (2) the Civilian Unemployment Rate (UNRATE); (3) the Coincident Economic Activity

Index (CEA), an aggregate measure of economic activity; (4) the Leading Index (LI), which aims

to predict the 6-month growth rate in CEA; (4) the TED spread (TED), capturing the difference

between interbank rates and short-term government yields;17 (5) the price-dividend ratio on the

broad CRSP index (PD), a measure of market valuations; (6) an indicator of investor sentiment

(SENT) following Baker and Wurgler (2007); and (7) the Baseline Overall Index (BOI), an

indicator of economic policy uncertainty by Baker, Bloom, and Davis (2016).18

Figure 3 illustrates the predictive power (in terms of R2) of the option-implied variables

across various horizons. Table 4 contains the corresponding regression coefficients. At horizons

of 6 months or longer, implied correlation significantly predicts many of the macro-economic

variables, with the exception of industrial production (INDPRO), the TED spread, and coin-

cident economic activity (CEA). In particular, being positively related to the leading indicator

(LI) and negatively related to unemployment (UNRATE), economic uncertainty (BOI), and

sentiment (SENT), high implied correlation predicts long-term improvements in economic fun-

damentals.19 Moreover, it predicts higher market valuation (i.e., a higher price-dividend ratio).

Hence, overall, the empirical evidence indicates that implied correlation is a leading procyclical

state variable that predicts economic growth for 6 to 18 months ahead.

17These variables are all obtained from the Federal Reserve Bank of St. Louis.18We appreciate having an opportunity to download updated series of the Investor Sentiment index from the

website of Jeffrey Wurgler (http://people.stern.nyu.edu/jwurgler/) and that of the Economic Policy Uncertaintyindex from the website of the authors (www.policyuncertainty.com/).

19Note that sentiment (SENT) is a contrarian predictor; that is, when sentiment is high, future returns are lowand vice versa (see, e.g., Baker, Wurgler, and Yuan 2012).

15

http://people.stern.nyu.edu/jwurgler/

http://www.policyuncertainty.com/

A: Industrial Production

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Figure 3: Economic condition predictions: Predictive power. The figure reports the R2 of predictiveregressions of future (log)changes in macro-economic variables on current option-implied variables for horizons of1 to 12 months. Results are based on monthly-sampled data from January 1996 to December 2018.

16


Panel A: Industrial ProductionIC -0.002 - 0.001 - 0.029 - 0.053 - 0.066 -IDMB - -0.010∗∗∗ - -0.027 - -0.019 - 0.024 - 0.128IdIV - -0.003 - -0.038 - -0.145∗∗ - -0.256∗∗∗ - -0.358∗∗∗

IV - -0.066∗∗∗ - -0.103∗ - -0.018 - 0.147 - 0.289R2 -0.22 17.56 -0.38 17.30 1.15 11.11 2.07 8.14 1.87 7.07

Panel B: Unemployment RateIC 0.017 - -0.005 - -0.181∗∗ - -0.379∗∗∗ - -0.549∗∗∗ -IDMB - 0.023∗∗ - 0.165∗∗∗ - 0.415∗∗ - 0.483∗ - 0.352IdIV - 0.053 - 0.154 - 0.445∗ - 0.992∗∗∗ - 1.642∗∗∗

IV - 0.211∗∗∗ - 0.555∗∗∗ - 0.801∗∗ - 0.681 - 0.387R2 0.31 14.42 -0.37 35.59 4.46 40.24 9.35 35.56 10.88 30.83

Panel C: Coincident Economic ActivityIC -0.003∗∗∗ - -0.003 - 0.008 - 0.024 - 0.036∗ -IDMB - -0.003∗∗∗ - -0.024∗∗∗ - -0.054∗∗∗ - -0.084∗∗∗ - -0.108∗∗∗

IdIV - 0.000 - 0.003 - -0.014 - -0.049 - -0.089∗

IV - -0.028∗∗∗ - -0.087∗∗∗ - -0.154∗∗ - -0.185∗∗ - -0.197R2 4.42 42.70 0.29 53.14 0.60 47.26 3.00 40.11 3.74 32.60

Panel D: Leading IndicatorIC 0.059 - 0.426 - 1.292∗∗∗ - 1.593∗∗ - 1.567∗ -IDMB - -0.019 - 0.671 - 2.579∗∗ - 3.364∗∗ - 4.498∗

IdIV - -0.335 - -2.374∗∗∗ - -5.560∗∗∗ - -5.846∗∗∗ - -5.861∗

IV - -0.371 - 1.690 - 7.759∗∗ - 10.084∗∗∗ - 11.425∗∗

R2 -0.33 -0.38 1.40 4.08 7.69 15.52 8.98 15.96 6.21 13.17

Panel E: Price-Dividend RatioIC -0.054 - 0.062 - 0.285∗∗ - 0.437∗∗ - 0.508∗∗ -IDMB - -0.278 - -0.217 - -0.768∗∗∗ - -1.116∗∗∗ - -1.272∗∗∗

IdIV - 0.419 - -0.047 - 0.100 - -0.195 - -0.564IV - -0.943 - 0.250 - 1.370 - 2.289∗∗∗ - 2.650∗∗

R2 -0.36 -0.52 -0.23 0.19 1.58 6.48 3.99 12.50 5.05 13.65

Panel F: SentimentIC -0.684∗ - -2.965∗∗∗ - -3.829∗∗∗ - -4.518∗∗ - -6.805∗∗ -IDMB - 0.218 - -1.578∗∗ - -4.538∗∗∗ - -8.170∗∗∗ - -18.280∗∗∗

IdIV - 0.537 - 6.184∗∗∗ - 13.158∗∗∗ - 17.725∗∗∗ - 25.574∗∗∗

IV - -3.146 - -27.244∗∗∗ - -49.158∗∗∗ - -58.443∗∗∗ - -78.001∗∗∗

R2 0.84 -1.15 17.28 28.23 17.15 36.25 15.95 39.01 25.72 48.87

Panel G: Baseline Overall IndexIC 0.044 - -0.230∗ - -0.378∗∗ - -0.518∗∗ - -0.651∗∗ -IDMB - -0.169∗∗ - 0.175 - -0.025 - -0.379 - -0.914IdIV - 0.311 - 0.250 - 0.819∗ - 1.402∗∗ - 1.794∗

IV - -0.591 - -1.075∗ - -2.479∗∗∗ - -3.054∗∗∗ - -3.238∗∗

R2 -0.27 0.69 0.91 0.83 1.95 2.71 3.40 3.67 4.32 3.94

Panel H: TED SpreadIC 0.056 - -0.273 - -0.330 - -0.074 - 0.151 -IDMB - -0.187∗∗ - -0.648 - -1.694∗∗∗ - -1.465∗ - -1.671IdIV - -0.075 - 0.179 - 0.278 - -1.571 - -2.923∗

IV - -0.398 - -3.728∗∗ - -8.991∗∗∗ - -10.732∗∗∗ - -11.980∗∗

R2 -0.26 1.27 0.42 11.27 0.21 31.22 -0.36 33.30 -0.33 38.39

Table 4: Economic condition predictions. The table reports the results for predictive regressions of future(log)changes in macro-economic variables on current option-implied variables for 1- to 12-month horizons. Esti-mation results are based on monthly-sampled data from January 1996 to December 2017. ∗∗∗, ∗∗, and ∗ indicatesignificance at 1%, 5%, and 10% level, respectively; based on Newey and West (1987) standard errors.

17

Individually, implied market variance and the implied dispersion of market betas (for hori-

zons of less than 3 months) are significant predictors of industrial production (INDPRO) and the

coincident economic activity (CEA). Also, the implied idiosyncratic variance predicts financial

market valuations (especially over the long term). Notably, however, linear combinations of the

three correlation components almost always outperform implied correlation (in fact, any single

predictor) and, in most cases, do so substantially. This result indicates that implied correlation

actually inherits its predictability from its three components and that (a large fraction of the

available) information is lost during aggregation. For example, in the case of the coincident eco-

nomic activity indicator (CEA), the joint regressions deliver high R2s, and regression coefficients

for implied market variance and the implied dispersion of market betas are usually significant.

Yet implied correlation typically has a quite low predictive power and an insignificant coefficient

for the coincident economic activity (CEA). Interestingly, the components of implied correla-

tion can often predict future investment opportunity at different time horizons; for example,

in predictions of industrial production implied idiosyncratic variance gains significance for long

horizons, while implied market variance loses its significance.

3 Forecasting Market Returns

We now focus on the ability of implied correlation (and its components) to predict future mar-

ket excess returns in-sample. We first discuss the empirical evidence from standard in-sample

regressions, then present evidence from a VAR model, and, finally, study whether revisions in

return forecasts are consistent with contemporaneous market excess returns.

3.1 Market Return Predictability

To study the in-sample predictability of market excess returns for various horizons, we rely on

standard in-sample predictive regressions of the following form:

rt→t+τr = γ +∑k≤K

βk PREDk(t, t+ τr) + εt, (7)

18

where rt→t+τr denotes the market excess return from t to t+ τr and PREDk(t, t+ τr) denotes

a set of predictors (known at time t).20 We use market excess returns from the end of a month,

and, when predicting returns for horizons longer than 1 month, we compute Newey and West

(1987) standard errors to correct for autocorrelation due to overlapping observations.

To differentiate the predictability of implied correlation and its components from well-known

proxies of downside risk, we expand the set of predictors to include risk premiums for correlation

(CRP ), variance (V RP ), and upside and downside semivariances (V RP u and V RP d). In par-

ticular, the ex-ante variance risk premium, V RP (t, T ), is computed as the difference between

the day-t implied variance from options with maturity T and the realized variance for the period

from t− (T − t) + 1 to t. Similarly, the correlation risk premium CRP (t, T ) is computed ex-ante

using forward-looking implied and realized correlation from matching periods. The upside and

downside semivariance risk premiums, V RP u(t, T ) and V RP d(t, T ), are computed as the dif-

ference between the risk-neutral expectations of upside and downside semivariances, IV u(t, T )

and IV d(t, T ) (following the corridor variance methodology of Andersen and Bondarenko 2007

and Andersen, Bondarenko, and Gonzalez-Perez 2015), and realized semivariances (constructed

following Andersen, Bollerslev, Diebold, and Ebens 2001 and Feunou, Jahan-Parvar, and Okou

2017). Table A3 in Appendix D contains summary statistics for these variables, all of which are

consistent with the findings in the literature.

Figure 4 reports the results for regression (7). Panel A shows that, in univariate regressions,

implied correlation has the strongest predictive power (R2) for future market excess returns,

except for the 1-month horizon. Its predictive power increases from about 3.5% at the 1-month

horizon to around 25% for the 9- and 12-month horizons. Moreover, its regression coefficient is

always highly significantly positive, with t-statistics consistently above 2 (Panel B). The compo-

nents of implied correlation also perform quite well, especially the implied dispersion of market

betas, which is highly significant and delivers R2s of up to 17% for 6 and 9 months. Implied

idiosyncratic variance is also significant, whereas implied market variance shows almost no pre-

dictability at any horizon. As shown in Panel A of Figure 4, the three components of implied

correlation outperform implied correlation by a very large margin, boosting the R2 by 15% (for

20We always rely on variables extracted from options with a maturity matching the forecasting horizon.

19

A: Regression, Implied Variables: R2

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Figure 4: In-sample return predictability. The figure depicts the results of the in-sample market excessreturn predictability regressions (7) for 1- to 12-month horizons. Panels A and B report the adjusted R2 andt-statistics for regressions with implied correlation and/or its components as explanatory variables, respectively.Panels C and D report the adjusted R2 and t-statistics for regressions with risk premiums for correlation (CRP ),market variance (V RP ), and upside and downside semivariances (V RPu and V RP d) as explanatory variables,respectively. The sample period spans from January 1996 to December 2017, and the data are sampled monthly.Standard errors are corrected for autocorrelation following Newey and West (1987), and the black dotted linesrepresent 5% significance bounds.

a 1-year horizon) to almost 90% (for a 1-month horizon) in relative terms. The components are

significant for most horizons, with consistent coefficient signs (Panel A of Table 5).

Panels C and D of Figure 4 illustrate that the predictive power of the risk premiums is

typically weaker than that of implied correlation, though the downside semivariance risk pre-

mium delivers the highest R2 at the shortest horizon of 1 month. Also, whereas the variance

risk premium significantly predicts returns at short horizons, its predictive power and statis-

tical significance quickly vanish as the horizon lengthens. Notably, the downside semivariance

risk premium delivers a better predictability than the total variance risk premium, with R2s of

7%–11% at short horizons and around 4% at longer horizons (consistent with the findings in

20


Panel A: Implied VariablesIC 0.059∗∗ - 0.243∗∗∗ - 0.501∗∗∗ - 0.708∗∗∗ - 0.873∗∗∗ -IDMB - -0.017 - -0.121 - -0.402∗∗ - -0.569∗ - -0.519IdIV - -0.201∗ - -0.600∗∗∗ - -0.938∗∗∗ - -1.430∗∗∗ - -1.993∗∗∗

IV - 0.204 - 0.909∗∗ - 2.085∗∗∗ - 2.848∗∗∗ - 3.253∗∗∗

R2 2.62 4.95 12.52 18.72 20.69 29.02 24.66 32.09 24.20 28.44

Panel B: Implied Variables with Risk Premium ControlsIC 0.031 - 0.174∗∗∗ - 0.537∗∗∗ - 0.738∗∗∗ - 0.927∗∗∗ -IDMB - -0.015 - -0.147 - -0.369∗∗ - -0.522∗ - -0.566IdIV - -0.152∗ - -0.438∗∗∗ - -0.996∗∗∗ - -1.519∗∗∗ - -2.170∗∗∗

IV - 0.121 - 0.565∗ - 2.717∗∗∗ - 4.026∗∗∗ - 3.942∗∗∗

CRP 0.017 -0.001 0.072 -0.032 0.253 0.249 0.355 0.372 0.239 0.022V RP d 0.677∗∗∗ 0.680∗∗∗ 1.030∗∗ 1.311∗∗∗ -1.373 -2.004∗ -1.355 -2.678∗∗∗ -1.040 -1.458R2 7.02 8.80 15.89 21.34 23.23 31.47 28.15 35.99 25.00 28.71

Table 5: In-sample return predictability. The table reports the results of the in-sample market excess returnpredictability regressions (7) for 1- to 12-month horizons. Panel A shows the results for univariate regressions withimplied correlation (IC) or its three components as explanatory variables. Panel B shows the results for the samespecifications with two additional controls: the correlation risk premium (CRP ) and the downside semivariancerisk premium (V RP d). The sample period spans from January 1996 to December 2017, and the data are sampledmonthly. ∗∗∗, ∗∗, and ∗ indicate significance at the 1%, 5%, and 10% level, respectively; based on Newey andWest (1987) standard errors.

Feunou, Jahan-Parvar, and Okou (2017) and Kilic and Shaliastovich (2017)), with a regression

coefficient that is always significantly positive.

Importantly, when jointly used with the risk premiums, implied correlation and its com-

ponents always remain significant, with sign and significance mostly unchanged (Panel B of

Table 5). In particular, although these regressors always improve the R2 compared to the origi-

nal specifications in Panel A, the correlation risk premium is actually never significant, and the

downside semivariance risk premium counterintuitively changes sign and becomes insignificant

for long horizons.

3.2 VAR Models

Studying the joint dynamics of the options-based variables and the market excess return can

provide additional information on in-sample return predictability. In that regard, we follow

Campbell (1991) and estimate a variety of VAR models of the following form:

zt+1 = Azt + εt+1, (8)

21

Model 1 2 3 4 5 6

IC 0.157∗∗ - - - - -IDMB - -0.129∗∗ - - -0.052 -IdIV - - -0.120∗ - -0.184∗∗ -0.215∗∗∗

IV - - - 0.056 0.155∗ 0.182∗∗

DIV -0.017 -0.017 -0.007 -0.005 -0.019 -0.015MKTRF 0.130∗∗ 0.093 0.063 0.118∗ 0.109 0.112∗

R2 3.952 3.023 3.106 0.569 6.209 5.889

Table 6: VAR models. The table reports the coefficients, together with p-values, and the R2 of the marketreturn equation for various specifications of the VAR(1) in (8). The results are based on monthly data fromJanuary 1996 to December 2017. ∗∗∗, ∗∗, and ∗ indicate significance at 1%, 5%, and 10% level, respectively.

where zt denotes a k-dimensional vector consisting of the market excess return (MKTRF ) and

the “predictors.” Consistent with the literature, we always include the standard predictor—log

dividend growth (DIV )—in addition to various combinations of options-based variables.

Table 6 summarizes the estimation results at a monthly frequency. As expected from the

ability of the implied variables to predict future investment opportunities, most of these vari-

ables have significant predictive power in the market return equation of the VAR model. As a

single add-on to dividend growth and the market excess return, implied correlation, the implied

dispersion of market betas and implied idiosyncratic variance significantly predict expected re-

turns, with R2s of about 3%–4%. In contrast, implied market variance is not significant when

used without other implied variables. When used jointly, the three correlation components out-

perform implied correlation, having an R2 of 6.2%. Taking only the two variance components of

implied correlation slightly decreases the explanatory power of the model (to 5.89%), but, again,

the results are stronger than with implied correlation alone. The corresponding R2s for longer

horizons, depicted in Figure 5, confirm these conclusions.21 Quantitatively, the model featuring

implied correlation delivers R2s of up to 17.5% for 5 to 9 months and 14% for longer periods,

but models including various correlation components deliver a substantially higher predictive

power for most horizons with R2s of up to 24%.

21The long-term implied R2s are computed by regressing realized returns on VAR-implied forecasts for marketexcess returns. R2’s are potentially biased at longer horizons because of autocorrelation in the predictive variables.

22

1 2 3 4 5 6 7 8 9 10 11 12horizon

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ICIDMBIdIVIVIV+IDMB+IdIVIV+IdIV

Figure 5: VAR-implied R2 for market return forecasts. The figure shows the implied R2 (expressed asa percentage) for predicting market excess returns based on various VAR(1) specifications of (8) for forecastinghorizons of 1- to 12-month. All specifications include the market excess return and the log dividend growth ratefor the CRSP aggregate index. The implied R2 is based on regressing realized monthly market excess returns onVAR-implied return forecasts for a sample period from January 1996 to December 2017.

In contrast with the individual predictive regressions, the VAR model also allows us to un-

derstand how shocks to the various variables propagate through the system.22 Shocks to implied

correlation positively affect market returns for up to 10 months (consistent with the results for

the implied variance and implied idiosyncratic variance components). Shocks to market returns

significantly (and negatively) affect implied idiosyncratic variance and the implied dispersion of

market betas, while not significantly propagating to either implied market variance or implied

correlation. Interestingly, implied correlation captures shocks to dividend growth with a nega-

tive sign, so that high current dividend growth leads to a lower implied correlation in the future.

This negative link between dividend growth and implied correlation is generated by a positive

link between dividend growth and implied idiosyncratic variance and the implied dispersion of

market betas (to a lesser degree). Thus, a positive shock to today’s expected return, due to a

positive dividend growth shock, lowers implied correlation by boosting the latter two, all leading

to a lower expected return in the future.

22For the sake of brevity, we omit the impulse response functions (IRFs) here; they are reported in Figures A1and A2 in Appendix C for Models 1 and 5 of Table 6, respectively.

23

3.3 Return Forecasts and Contemporaneous Returns

Economic theory provides a tight link between revisions in expected market excess returns and

contemporaneous market excess returns. In particular, using the Campbell and Shiller (1988)

log-linearization, one can relate contemporaneous, unexpected excess returns, et+1 − Et[et+1],

to revisions in expected dividend growth rates and revisions in expected future excess returns:

et+1 − Et[et+1] = (Et+1 − Et)∑n

ρn−1∆dt+n − (Et+1 − Et)∑n

ρnrf,t+1+n

− (Et+1 − Et)∑n

ρnet+1+n, (9)

where et+1 denotes the time-t+ 1 excess return, ∆dt denotes the log dividend-growth rate, rf,t

denotes the time-t risk-free rate, and ρ denotes a parameter with a value slightly less than one.

All else equal, (9) implies that, because of the negative sign in front of the revisions to

future expected excess returns, an increase in the forecasts of future market excess returns,

(Et+1 − Et) et+1+n, causes a lower contemporaneous excess return, et+1−Et[et+1]. Consequently,

to be consistent with economic theory, revisions in the forecasts of future market excess returns

from (7) must negatively correlate with contemporaneous excess returns.

Empirically, we find that the correlation between increments in the predicted market excess

returns and contemporaneous daily market excess returns is negative for most options-based

variables. For example, the correlation between daily excess returns and changes in forecasts

based on implied correlation is −0.61. For implied market variance and the implied disper-

sion of market betas, it is also negative, whereas it is positive (0.17) for implied idiosyncratic

variance.23 For joint predictions based on all three components, the correlation with contempo-

raneous returns is also negative (−0.41). Hence, there is strong and robust empirical evidence

that predictions based on the implied correlation are consistent with the relation stipulated by

economic theory in the form of the present value relation.

23For longer horizons, this correlation turns (modestly) negative.

24

4 Out-of-Sample Predictability

Although many variables have strong predictive power in-sample, hardly any evidence exists for

out-of-sample predictability, as convincingly shown by Goyal and Welch (2008). Consequently,

we now concentrate on the out-of-sample performance of implied correlation, its components,

and various risk premiums.

4.1 Measuring Predictability

To evaluate the performance of a specific forecasting model s, we compare its predictive power

to that of a model based on the historical mean of the market excess return (s = 0). This

forecast serves as a natural benchmark, because, as documented by Goyal and Welch (2008) and

Campbell and Thompson (2008), almost all predictive variables fail to beat it out-of-sample.

To measure the out-of-sample predictive power, we rely on two criterion.24 First, we consider

the out-of-sample R2 relative to the forecasts from the (benchmark) historical average return

model:

R2s,τr = 1− MSEs,τr

MSE0,τr

, (10)

where MSEs,τr = 1N

(e>s,τr × es,τr

)denotes the mean-squared error of model s computed from the

N×1 vector of prediction errors es,τr for horizon τ .25 A particular model, s ≥ 1, outperforms the

benchmark model based on the average historical return if the out-of-sample R2s,τr is significantly

positive.

Second, we assess a model’s predictive power by studying the certainty-equivalent gain of

a mean-variance investor who follows a market-timing strategy based on the expected return

signal (similar to the setting of Campbell and Thompson 2008). That is, we compute the gain

24Usually, out-of-sample performance is evaluated using a loss function that is either an economically meaningfulcriterion, such as utility or profits (e.g., Leitch and Tanner (1991), West, Edison, and Cho (1993), Della Corte,Sarno, and Tsiakas (2009)), or a statistical criterion (e.g., Diebold and Mariano (1995), McCracken (2007)). Theseapproaches have been recently unified and extended by Giacomini and White (2006), who develop out-of-sampletests to compare the predictive ability of competing forecasts, given a general loss function under conditions ofpossibly misspecified models.

25We also consider the Diebold and Mariano (1995) loss function, that is, the average squared-error loss relativeto the predictions from the benchmark model: δs,τr = MSEs,τr −MSE0,τr . The inferences are unchangedrelative to the out-of-sample R2, with the results being available on request.

25

in certainty-equivalent return for a myopic mean-variance investor with a horizon of τr and

risk aversion γ = 1, who uses the forecasts of model s instead of the historical average return

(s = 0). Formally, to compute the portfolio weight wt,τ,s =rj,s,τrσ2 of the market portfolio, we use

a model’s predicted excess return (rs,t,τr) and the 1-year historical variance (σ2).26 From the

resultant time-series of realized returns rMVs,τ , one can then compute the mean-variance certainty

equivalent CEj,τr and the respective gain in the certainty equivalent return (CE gain)

∆CEs,τr = CEs,τr − CE0,τr , where CEs,τr = E[rMVs,τr ]− γ

2σ2(rMV

s,τr ). (11)

Intuitively, this measures the economic benefit of a better performing portfolio resulting from a

better market return forecast.

Because of the limited availability of options data, our sample period spans about 20 years.

Consequently, asymptotic standard errors may not be accurate, so we resort to bootstrapping.

Specifically, we use the moving-block bootstrap procedure of Kunsch (1989),27 to randomly

resample with replacement from the time series of a model’s forecasts to construct bootstrapped

distributions for our performance measures. Using the bootstrapped distribution, we compute

the p-value for the null hypothesis: R2 = 0.

4.2 Traditional Out-of-Sample Approach

Traditionally, out-of-sample predictions are based on rolling-window estimations of the predictive

regression (7) (with the addition of a time-specific intercept):

rt−τr→t = γt +∑k∈K

βk,t PREDk(t− τr, t) + εt, (12)

The estimated coefficients βk,t, together with the time-t value of the “predictors” PREDk(t, t+

τr) then form the out-of-sample return forecast, rt→t+τ . Note that, at date t, one uses only

observations from the past to avoid any look-ahead bias.

26Following Campbell and Thompson (2008), we restrict the optimal weights to be in the [0, 1.5] range, whichis technically similar to restricting the forecast to a (nonextreme) positive value.

27The moving-block bootstrap is shown (see, e.g., Lahiri (1999)) to be comparable in performance to otherwidely used methods, for example, the stationary bootstrap by Politis and Romano (1994) and the circular blockbootstrap of Politis and Romano (1992). However, constant block sizes lead to smaller mean-squared errorscompared with random block sizes used in a stationary bootstrap. Technically, we draw 10,000 random samplesof 200 blocks, with blocks of 12 observations (i.e., 1-year blocks), to preserve the autocorrelation in the data.

26

A: Implied Variables, R2

1 3 6 9 12horizon

40

30

20

10

0

10R

2

IC IDMB IdIV IV

B: Implied Variables, Significance of R2

1 3 6 9 12horizon

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

pva

lue

IC IDMB IdIV IV

C: Risk Premiums, R2

1 3 6 9 12horizon

40

30

20

10

0

10

R2

CRP VRP VRPu VRPd

D: Risk Premiums, Significance of R2

1 3 6 9 12horizon

0.0

0.1

0.2

0.3

0.4

0.5

pva

lue

CRP VRP VRPu VRPd

Figure 6: Traditional approach: Out-of-sample R2. The figure shows the out-of-sample R2 (definedin (10)) and its p-value for predictions based on the traditional approach with a short (3-year) estimation window.Panels A and B depict the results for implied correlation (IC) and its components: the dispersion of market betas(IDMB), idiosyncratic variance (IdIV ), and market variance (IV ). Panels C and D depict the results for riskpremiums for correlation (CRP ), market variance (V RP ), and upside and downside semivariances (V RPu andV RP d). p-values are computed from bootstrapped distributions, and the dotted lines represent 5% significancebounds. The R2 axis is cut at the −45% level for better readability. Predictions are made at a monthly frequencyfrom January 1996 to December 2017.

Figure 6 reports the out-of-sample R2 for univariate predictions using the traditional pre-

dictability approach and a 3-year estimation window.28 Implied correlation delivers positive R2s

of 1.6%, 6.0%, and 10.1% for 1-, 3-, and 6-month horizons (which are significant at the 10% level

for the first and at the 5% level for the two longer horizons). In contrast, the components of

implied correlation show no significant predictive power (Panels A and B). Also, there is almost

no predictability in the case of the risk premiums for correlation and (semi)variances (Panels C

28Note, to avoid overfitting, we always rely on a single predictor; optimizations (for example, model selection,overfitting corrections, and so on) of the out-of-sample predictions are left to future research. Results from a simplemodel averaging the three correlation components (as a way to reduce overfitting following Rapach, Strauss, andZhou 2009) show stronger predictability relative to a model with joint predictors; however, most out-of-sampleR2s are still negative.

27

A: Implied Variables, CE

1 3 6 9 12horizon

0.00

0.01

0.02

0.03

0.04CE

IC IDMB IdIV IV

B: Implied Variables, Significance of CE

1 3 6 9 12horizon

0.0

0.1

0.2

0.3

0.4

pva

lue

IC IDMB IdIV IV

C: Risk Premiums, CE

1 3 6 9 12horizon

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

CE

CRP VRP VRPu VRPd

D: Risk Premiums, Significance of CE

1 3 6 9 12horizon

0.0

0.1

0.2

0.3

0.4

0.5

pva

lue

CRP VRP VRPu VRPd

Figure 7: Traditional approach: Certainty-equivalent gain. The figure shows the out-of-sample certainty-equivalent (CE) gain (defined in (11)) and its p-value for predictions based on the traditional approach with a short(3-year) estimation window. Panels A and B depict the results for implied correlation (IC) and its components:the dispersion of market betas (IDMB), idiosyncratic variance (IdIV ), and market variance (IV ). Panels C andD depict the results for risk premiums for correlation (CRP ), market variance (V RP ) and upside and downsidesemivariances (V RPu and V RP d). p-values are computed from bootstrapped distributions and the dotted linesindicate 5% significance bounds. Predictions are made at a monthly frequency from January 1996 to December2017.

and D), with the exception of the correlation risk premium at the 3-month horizon and the

downside variance risk premium at the 6-month horizon.

Importantly from an economic perspective, implied correlation improves the portfolio perfor-

mance relative to a naive prediction by 3%–4% p.a. in certainty equivalent terms (Panels A and

B of Figure 7), outperforming all its components for long horizons. Within the set of risk pre-

miums, the downside semivariance risk premium performs best with certainty-equivalent gains

between 2.5% and 3.5% p.a., followed by the correlation risk premium for intermediate horizons

(Panels C and D).

28

Intuitively, when choosing the estimation horizon for the rolling-window regressions (12),

one typically faces a trade-off between the precision of the coefficients βk,t and their ability

to reflect the current economic conditions. Variations in the estimation window show that

traditional out-of-sample regression techniques cannot fully exploit the predictive power of many

variables, because they require a relatively long historical estimation window for the regression

coefficients.29 In addition, the traditional approach has another drawback; that is, to avoid any

look-ahead bias, when forecasting returns from t to t+ τr, the last observations of the predictors

used in the estimation of the regression coefficients in (12) are from time t − τr. Hence, for an

annual forecasting horizon, this implies that the predictors and the estimated predictive relation

will be 1 year old and, in the case of changing economic conditions, severely outdated.

4.3 Contemporaneous-Beta Approach

In light of these findings, we now propose a new approach for out-of-sample forecasts that re-

lies on contemporaneous betas computed from higher-frequency data and significantly improves

return predictability. The approach can be applied to state variables with observable risk pre-

miums and, hence, is especially suited to option-implied variables.

4.3.1 Intuition

The approach effectively extends the contemporaneous-regression idea, introduced by Cutler,

Poterba, and Summers (1989) and Roll (1988). Their approach traditionally has been used to

study contemporaneous innovations in market excess returns and the respective variables (similar

to our VAR analysis in Section 3.2). In contrast, our objective is to combine the coefficients

obtained from the contemporaneous regressions using forward-looking option-implied variables

with time-t forward-looking risk premiums to efficiently forecast future market excess returns.30

Intuitively, the idea can be motivated by a simple linear framework within the arbitrage

pricing theory. For example, consider a setting in which the market excess return is described

29For a variety of rolling-window horizons ranging from 1 to 5 years, the results are qualitatively the same.30Pyun (2019) employs a similar idea, using shocks to expected daily variance under the physical probability

measure to estimate contemporaneous-beta coefficients.

29

by a linear factor model with (time-varying) factor exposures:31

rt+1 =K∑k=1

βk,t fk,t+1 + εt+1, (13)

where fk,t+1 denotes the return on factor k ∈ 1, . . . ,K, and εt+1 denotes the “noise.” In this

case, the equity risk premium is given by

ERP (t, t+ 1) =K∑k=1

βk,tRPk,t+1 + pt(εt+1), (14)

where RPk,t+1 denotes the risk-premium on the kth factor, and pt(εt+1) denotes the pricing

error, for example, because of omitted factors.32 Essentially, to arrive at a good estimate for the

future market excess return, one needs precise estimates of the time-t factor exposures βk,t and

the corresponding risk premiums RPk,t+1.

Our approach relies on first estimating a regression of the type (13) using daily (or even

intraday) market excess returns and, at the same frequency, shocks to the (option-implied) vari-

ables, such as implied correlation or implied variance. Doing so delivers accurate and up-to-date

estimates of βk,t. Intuitively, the higher frequency observations have two advantages: (1) they

improve the properties of the regression coefficients (betas), which are essentially functions of

second moments, and (2) they allow us to shorten the estimation window so that betas quickly

adjust to new information. In the second step, one can then use the “pricing equation” (14)

to predict future market excess returns by combining the estimated coefficients with the time-t

forward-looking risk premiums on the respective variable (e.g., the correlation risk premium or

the variance risk premium).

4.3.2 Implementation

For estimating regression (13), the contemporaneous-beta approach requires estimating shocks

in the variables, that is, estimates of the “factors.” In the case of option-implied variables, these

31A similar structure endogenously arises in Bollerslev, Tauchen, and Zhou (2009), Bollerslev, Todorov, andXu (2015), Kilic and Shaliastovich (2017), and Feunou, Jahan-Parvar, and Okou (2017).

32Although early applications of the arbitrage pricing theory (e.g., Ross 1976) derived bounds on the pricingerrors by assuming a correct specification of a linear factor model, recent studies (e.g., Raponi, Uppal, andZaffaroni 2018) show that the errors are bounded even when the model is misspecified or factors are omitted.

30

shocks can be proxied for using shocks to integrated expected variables. For example, note

that the time-t expected integrated correlation obtained from options with maturity T can be

decomposed as follows:

IC(t, T ) = EQt

[EQt+∆t

[∫ t+∆t

tρ(s)ds+

∫ T

t+∆tρ(s)ds

]]

= EQt

[∫ t+∆t

tρ(s)ds

]+ EQ

t [IC(t+ ∆t, T )] ,

where ρ denotes the stochastic process of correlation, and Q denotes the risk-neutral probability

measure. Hence, increments of implied correlation are given by

∆IC(t+ ∆t, T ) = IC(t+ ∆t, T )− IC(t, T )

= IC(t+ ∆t, T )− EQt [IC(t+ ∆t, T )]︸︷︷︸

“true” shock to the state variable

−EQt

[∫ t+∆t

tρ(s)ds

]. (15)

If the last term in equation (15)—the expected integrated correlation over a short period of time

∆t—is small, that is, if risk-neutral expected integrated correlation can be well approximated

by a martingale,33 short-interval increments are indeed proxies for random shocks to correlation.

The same argument holds for shocks to (semi)variances.

Importantly, our approach allows us to use increments from any option maturity; in partic-

ular, one can use variables extracted from the most liquid options at short maturities. Doing so

reduces the potential bias in betas arising from nonlinearities in response to random shocks in

a variable. One only needs to correct the resultant betas for the difference in variability of the

regressors used for beta estimation (i.e., increments in variables) in (13) and the variability of

the risk premiums for the pricing equation (14). Appendix B derives a simple procedure that

does exactly that.

33Empirical evidence supports this approximation. For example, Filipovic, Gourier, and Mancini (2016, p. 58)find that a “martingale model provides relatively accurate forecasts for the one-day horizon variance.” Moreover,integrated expected variance and the integrated expected correlation are highly persistent, with first-order au-tocorrelations between 0.97 and 0.994 for variance and between 0.97 and 0.993 for correlation at various optionmaturities in our sample period.

31

4.3.3 Results

To illustrate the advantages of the contemporaneous-beta approach relative to the traditional

approach, we now compare its performance for the case of a 3-year historical rolling-window es-

timation period. Recall that in this case, only implied correlation significantly predicted returns

out-of-sample for the traditional approach, whereas there is no out-of-sample predictability for

the risk premiums (cf. Figure 6).

To estimate the contemporaneous betas, we use daily market excess returns and daily incre-

ments in implied correlation and implied (semi)variances from options with 1-month maturity.34

For the out-of-sample predictions, we then combine these betas with the time-t expected risk

premium on the predictors, that is, the correlation and the (semi)variance risk premiums, for

an option maturity matching the forecasting horizon.

As shown in Figure 8, despite the short estimation window, the new approach leads to a stable

out-of-sample return predictability for most horizons. The correlation risk premium delivers the

highest out-of-sample R2 for 3- to 9-month horizons, whereas the downside semivariance risk

premium performs best at the very short (1-month) horizon. Interestingly, even the variance

risk premium displays a decent performance with an R2 of 7% at a 1-month horizon, declining

to 2.2% for 6 months. The R2s for the correlation risk premium are significant for horizons up

to 9 months, for the downside semivariance risk premium up to 6 months, and for the variance

risk premium for 1 and 3 months.

Results shown in Figure 9 confirm that the market return predictions by the correlation

risk premium based on the contemporaneous-beta approach indeed prove useful for a myopic

mean-variance investor. The certainty-equivalent gain for the correlation risk premium is above

4% for horizons of 1 to 6 months and significant for up to 9 months. The downside semivariance

risk premium is now slightly inferior to the correlation risk premium for short horizons and loses

significance for longer ones. Overall, the gains in the risk-adjusted return are stronger with the

contemporaneous-beta approach compared to the traditional one.35

34These are then adjusted by the ratio of the 3-year historical volatilities of the implied variables and therespective risk premiums, as described in Appendix B.

35Notably, even for an extremely short estimation window of 1 month, the predictive performance is impressive(results are available on request).

32

A: Risk Premiums, R2

1 3 6 9 12horizon

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

R2

CRP VRP VRPd VRPu

B: Risk Premiums, Significance of R2

1 3 6 9 12horizon

0.0

0.1

0.2

0.3

0.4

0.5

pva

lue

CRP VRP VRPd VRPu

Figure 8: Contemporaneous-beta approach: Out-of-sample R2. The figure shows the out-of-sample R2

(defined in (10)) and its p-value for predictions based on the contemporaneous-beta approach with a short (3-year)estimation window at various forecast horizons. The results are shown for risk premiums for correlation (CRP ),market variance (V RP ), and upside and downside semivariances (V RPu and V RP d). p-values are computedfrom bootstrapped distributions and the dotted lines indicate 5% significance bounds. Predictions are made at amonthly frequency from January 1996 to December 2017.

A: Risk Premiums, CE

1 3 6 9 12horizon

0.04

0.02

0.00

0.02

0.04

CE

CRP VRP VRPd VRPu

B: Risk Premiums, Significance of R2

1 3 6 9 12horizon

0.00

0.05

0.10

0.15

0.20

0.25

0.30

pva

lue

CRP VRP VRPd VRPu

Figure 9: Contemporaneous-beta approach: Certainty-equivalent gain. The figure shows theout-of-sample certainty equivalent (CE) gain (defined in (11)) and its p-value for predictions based on thecontemporaneous-beta approach with a short (3-year) estimation window at various forecast horizons. The resultsare shown for risk premiums for correlation (CRP ), market variance (V RP ), and upside and downside semivari-ances (V RPu and V RP d). p-values are computed from bootstrapped distributions, and the dotted lines represent5% significance bounds. Predictions are made at a monthly frequency from January 1996 to December 2017.

To understand what drives the differences in the approaches’ predictive power, recall that

the pricing equation is the same for the traditional and the contemporaneous-beta approaches.

However, the two approaches considerably differ in the estimation of the regression coefficients,

βk,t. Figure 10 depicts the estimated betas for the various variables and forecasting horizons.

Notably, the resultant betas show very different dynamics. That is, the traditional betas are

far more volatile (pay attention to the y-axis scales) and exhibit more abrupt jumps in re-

33

action to “extreme” observations. Also, whereas the traditional betas often change sign, the

contemporaneous betas, though substantially fluctuating over time, remain positive (consistent

with theory). Moreover, the traditional betas fluctuate much more with the forecasting hori-

zon, with standard deviations across horizons that are 5 to 25 times larger than those of the

contemporaneous approach.

5 Robustness

In the following, we carry out a series of additional tests that confirm the robustness of our main

findings for various alternative specifications. Appendix E houses figures and tables containing

the detailed results of these exercises.

First, to remove potential biases in the R2s of the predictive regressions for the macro-

economic variables induced by overlapping observations, we repeat these predictive regressions

with non-overlapping predictions. In particular, for a given forecasting horizon and predictor, we

run regressions for each starting month and then compute the average R2. Overall, the results

are similar to those presented in Section 2, with an expected downward correction in R2s for

longer horizons (see Figure RF1). The economic policy uncertainty indicator (BOI), for which

only implied correlation delivers a positive R2 for long horizons, is most affected.

Second, for similar reasons (overlapping observations),36 we also redo the in-sample market

return predictive regressions using non-overlapping regressions (in the same way as we did for

the macro-economic variables above). Also, we compute the univariate R2s adjusted for the

autocorrelations of the predictors using the procedure outlined in Boudoukh, Richardson, and

Whitelaw (2008). The R2s substantially decline in both cases; however, notably, implied correla-

tion still delivers the strongest predictive power at longer horizons (Figures RF2 and Figure RF3,

respectively). These findings are supported by additional tests based on the instrumental vari-

able approach by Kostakis, Magdalinos, and Stamatogiannis (2015).37 In most cases, our initial

corrections for autocorrelations based on Newey-West standard errors are stricter than those

from the instrumental variable approach; hence, the results remain unchanged.

36In particular, as shown by Boudoukh, Richardson, and Whitelaw (2008), for highly persistent regressors,in-sample R2s mechanically increase with the predictability horizon in the case of overlapping observations.

37We are grateful to the authors for providing the code on their website.

34

A: Traditional approach: CRP

19992001

20032005

20072009

20112013

20152017

Year

3

2

1

0

1

2

CRP, 1 month CRP, 6 months CRP, 12 months

B: Contemporaneous approach: CRP

19992001

20032005

20072009

20112013

20152017

Year

0.05

0.10

0.15

0.20

0.25

0.30

CRP, 1 month CRP, 6 months CRP, 12 months

C: Traditional approach: V RP

19992001

20032005

20072009

20112013

20152017

Year

20

15

10

5

0

5

10

VRP, 1 month VRP, 6 months VRP, 12 months

D: Contemporaneous approach: V RP

19992001

20032005

20072009

20112013

20152017

Year

0.25

0.50

0.75

1.00

1.25

1.50

1.75

VRP, 1 month VRP, 6 months VRP, 12 months

E: Traditional approach: V RP d

19992001

20032005

20072009

20112013

20152017

Year

20

10

0

10

20

VRPd, 1 month VRPd, 6 months VRPd, 12 months

F: Contemporaneous approach: V RP d

19992001

20032005

20072009

20112013

20152017

Year

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

VRPd, 1 month VRPd, 6 months VRPd, 12 months

G: Traditional approach: V RPu

19992001

20032005

20072009

20112013

20152017

Year

40

30

20

10

0

10

20

VRPu, 1 month VRPu, 6 months VRPu, 12 months

H: Contemporaneous approach: V RPu

19992001

20032005

20072009

20112013

20152017

Year

1

2

3

4

VRPu, 1 month VRPu, 6 months VRPu, 12 months

Figure 10: Dynamics of out-of-sample regression coefficients. This figure shows the time series of theregression coefficients for univariate models of out-of-sample market return forecasts at forecasting horizons of 1,6, and 12 months. The panels on the left show the regression coefficients from the traditional approach, and theones on the right show the regression coefficients from the contemporaneous approach. For both approaches, weuse a 3-year historical estimation window.

35

Third, to ensure that our findings for in-sample market return predictability are robust to the

addition of “traditional predictors,” we run regressions controlling for the Earnings Price Ratio

(EP ), the Term Spread (TMS), the Default Yield Spread (DFY ), the Book-to-Market Ratio

(BTM), and the Net Equity Expansion (NTIS).38 To avoid overfitting, we select, for each

predictive horizon, the relevant variables by applying a LASSO procedure to regression (7),

selecting the optimal regularization parameter using the Akaike information criterion (AIC).

Overall, the predictive power of implied correlation and its components is not much affected

by the additional controls. In particular, implied correlation is always selected by the LASSO

procedure and remains highly statistically significant for longer horizons (Table RT1). Also, the

components of implied correlation are practically always selected by the procedure, and their

significance is unchanged.

Fourth, to analyze the sensitivity of the in-sample predictability results with respect to the

sample period, we carry out a number of subsample analyses. This includes (1) conditioning

on implied correlation being below and above its median (Panels A and B in Table RT2), (2)

splitting the sample in two halves (Panels C and D), and (3) conditioning on NBER recession

(Panels E and F). At short horizons, the predictive power of implied correlation is better when

its value is low and vice versa for long horizons. Also, the predictive power is considerably better

in the first half of the sample, and long-term predictability improves during recessions.

Fifth, we study out-of-sample return predictability using a long (10-year) estimation window

for the traditional approach (Figure RF4).39 Although the out-of-sample R2 for predictions

based on implied correlation is initially negative, it demonstrates a steady increase from about

1% for 3 months to about 14% at the 9- and 12-month horizons, delivering the highest predictive

power among the predictors at 6-month and longer horizons. For 1- and 3-month horizons, id-

iosyncratic variance performs the best with R2s around 3%–4%. In the case of the risk premiums,

hardly any variable delivers an R2 that is statistically different from zero.

38We construct the variables following Goyal and Welch (2008). That is, EP is defined as the log ratio ofearnings to prices; TMS is the difference between the long term yield on government bonds and the Treasurybill; DFY is the difference between BAA and AAA-rated corporate bond yields; BTM is the ratio of book valueto market value for the Dow Jones Industrial Average; and NTIS is the ratio of 12-month moving sums of netissues by NYSE listed stocks divided by the total end-of-year market capitalization of NYSE stocks.

39Here, we follow Kilic and Shaliastovich (2017), who use a 10-year window to show that decomposing thevariance risk premium into good and bad components improves the out-of-sample predictability of market returns.

36

6 Conclusion

The correlation among individual stocks implied by option prices (“implied correlation”) is one

of the most robust predictors of long-term market excess returns. In this paper, we further

explore its predictive power and, most importantly, study its underlying economic sources.

In particular, we show that implied correlation is a leading procyclical state variable, gen-

erally predicting future investment opportunities, that is, financial-market risks and macro-

economic conditions for 6 to 18 months ahead. We provide empirical evidence that implied cor-

relation inherits this predictability from its three main components—implied market variance,

implied idiosyncratic variance, and the implied dispersion of market betas—which, individually,

also predict future investment opportunities. Indeed, linear combinations of its three main com-

ponents usually outperform implied correlation in predicting future financial risk measures and

macro-economic variables, indicating that some information is lost when aggregated into the

specific functional form of correlation.

We also provide additional (new) evidence of long-term market excess return predictability

by implied correlation. First, we document that its three main components can also predict

future market returns in-sample and, when used jointly, outperform implied correlation in terms

of predictive power. Second, we document that the predictability by implied correlation is not

subsumed by measures of tail risk, such as the variance, correlation or the downside semivariance

risk premium. Third, the predictability is also robust to traditional return predictors. Fourth,

we provide first empirical evidence of robust out-of-sample return predictability by implied

correlation, leading to high out-of-sample R2s even at long horizons and substantial economic

gains for market-timing strategies.

We also make a methodological contribution in proposing a novel extension of the contempo-

raneous-beta approach that overcomes the problem of the high sensitivity of the traditional

approaches to the length of the estimation window and to anomalous observations (outliers) in

the data. Our approach combines high-frequency increments in option-implied variables with

their respective risk premium. We demonstrate that the approach can improve out-of-sample

return forecasts in many applications because of its stable and up-to-date regression coefficients.

37

Moreover, because of its straightforward implementation, the contemporaneous-beta approach

naturally lends itself to other settings.

By highlighting the importance of information not spanned by the marginal distribution of

the market, our results have important implications for theoretical work building pricing models

from individual stock dynamics (see, e.g., Leippold and Trojani (2010)). Moreover, by providing

evidence of the pricing of idiosyncratic variance at the aggregate level, our work connects to the

ongoing debate about the importance and pricing of idiosyncratic risk.

38

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42

Appendix

A Option-Implied Beta Construction

To construct option-implied market betas, we follow the approach proposed by Buss and Vilkov

(2012). In a first step, we compute an option-implied correlation matrix for all index components;

based on the following parametric form for implied correlations ρQi,i′(t):

ρQi,i′(t) = ρPi,i′(t)− α(t)(1− ρPi,i′(t)). (A1)

Intuitively, this expression models the implied correlation, ρQi,i′(t), as the expected physical corre-

lation, ρPi,i′(t), plus a heterogeneous correlation risk premium which is driven by the correlation-

risk-premium parameter α(t) and inversely proportional to the physical correlation.40 Impor-

tantly, this specification guarantees the positive-definiteness of the resultant correlation matrix.

Substituting the implied correlations (A1) into restriction (3), one arrives at the following

expression for α(t):

αt = −(σQI (t))2 −

N∑i=1

N∑i′=1

ωiωi′σQi (t)σQi′ (t)ρ

Pi,i′(t)

N∑i=1

N∑i′=1

ωiωi′σQi (t)σQi′ (t)(1− ρPi,i′(t))

,

which, together with (A1), is used to construct the “full” implied correlation matrix ΩQ(t) = ρQi,i′(t).

Pre- and postmultiplying the correlation matrix ΩQ(t) by a diagonal matrix of option-implied

stock volatilities, one gets the option-implied variance-covariance matrix ΣQ(t). Finally, the

vector of option-implied market betas can be obtained as

βQ(t) =ΣQ(t)×W(σQM (t))2

,

where W denotes the N × 1 vector of index weights ωi and σQM (t) denotes the implied volatility

of the market (index).

40For the empirical implementation, we rely on a historical 1-year rolling-window correlation matrix for indi-vidual stocks, though the results are robust to variations in the window length.

43

B “Normalizing” Variance and Correlation Betas

The variance and correlation betas, β∆IV,t and β∆IC,t estimated by regressing market excess

returns on increments in option-implied variables in (13), differ from the exposures βV,t and βρ,t

in the pricing equation (14). In particular, they need to be adjusted for the difference in the

variability of the regressors used for beta estimation (i.e., increments in risk-neutral expected

variance and correlation) and the variability of the predictors in the pricing equation (i.e., the

variance and correlation risk premium).

Intuitively, the variance beta in the pricing equation, βV,t, can be decomposed into (i) the

correlation between the market excess return and the variance risk premium, and (ii) the ratio

of their volatilities. Similarly, the variance beta in the estimation equation, β∆IV,t, can be

decomposed into (i) the correlation between the market excess return and the increments in

implied variance, and (ii) the ratio of their volatilities.

Consequently, if we assume that the correlation between returns and increments in im-

plied variance equals the correlation between returns and the variance risk premium, that is,

Corr (rt+τ , V RP (t, t+ τ)) = Corr (rt+τ ,∆IV (t, t+ τ)), one gets:

βV,t = Corr (rt+τ , V RP (t, t+ τ))× Vol (rt+τ )

Vol (V RP (t, t+ τ))

= Corr (rt+τ ,∆IV (t, t+ τ))× Vol (rt+τ )

Vol (∆IV (t, t+ τ))× Vol (∆IV (t, t+ τ))

Vol (V RP (t, t+ τ))

= βDeltaIV,t ×Vol (∆IV (t, t+ τ))

Vol (V RP (t, t+ τ)).

Accordingly, one can simply adjust the variance beta, β∆IV,t, by the ratio of the volatility

of the increments in implied variance and the volatility of the variance risk premium. Both

variables are observable, so that the ratio can easily be estimated from the data. The same

principle works for up and down semivariance betas, and for correlation betas. For example,

computations for the correlation risk premium lead to a comparable adjustment:

βρ,t = β∆IC,t ×Vol (∆IC(t, t+ τ))

Vol (CRP (t, t+ τ)).

44

C Additional Figures

A: IC to Other Variables B: Variables to MKTRF

0.0

0.2

0.4

0.6

0.8

1.0IC IC

0.05

0.00

0.05

0.10

0.15

IC DIV

0 2 4 6 8 10 120.00

0.05

0.10

0.15

0.20

0.25

IC MKTRF

Impulse responses

0.00

0.05

0.10

0.15

0.20

0.25

IC MKTRF

0.15

0.10

0.05

0.00

0.05

0.10

DIV MKTRF

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0MKTRF MKTRF

Impulse responses

C: DIV to Other Variables D: MKTRF to Other Variables

0.20

0.15

0.10

0.05

0.00DIV IC

0.60.40.20.00.20.40.60.81.0

DIV DIV

0 2 4 6 8 10 120.15

0.10

0.05

0.00

0.05

0.10

DIV MKTRF

Impulse responses

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

MKTRF IC

0.10

0.05

0.00

0.05

MKTRF DIV

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0MKTRF MKTRF

Impulse responses

Figure A1: VAR specification 1: Impulse response functions. The figure shows a set of selected impulseresponse functions for a VAR(1) specification of (8) featuring the market excess return (MKTRF ), the logdividend-growth (DIV ), and the implied correlation (IC); for horizons of 1 to 12 months. The sample periodspans from January 1996 to December 2017. The dotted lines indicate 5% confidence bounds.

45

A: IMDB to Other Variables B: Variables to MKTRF

0.10

0.05

0.00

0.05

IDMB IV

0.00

0.25

0.50

0.75

1.00IDMB IDMB

0.05

0.00

0.05

IDMB IdIV

0.1

0.0

0.1

IDMB DIV

0 2 4 6 8 10 120.2

0.1

0.0

0.1IDMB MKTRF

Impulse responses

0.0

0.1

0.2

0.3

IV MKTRF

0.2

0.1

0.0

0.1IDMB MKTRF

0.3

0.2

0.1

0.0IdIV MKTRF

0.1

0.0

0.1DIV MKTRF

0 2 4 6 8 10 120.00

0.25

0.50

0.75

1.00MKTRF MKTRF

Impulse responses

C: IV to Other Variables D: IdIV to Other Variables

0.00

0.25

0.50

0.75

1.00IV IV

0.4

0.2

0.0IV IDMB

0.2

0.1

0.0

IV IdIV

0.1

0.0

0.1

0.2IV DIV

0 2 4 6 8 10 120.0

0.1

0.2

0.3

IV MKTRF

Impulse responses

0.0

0.2

0.4

IdIV IV

0.0

0.2

0.4

0.6IdIV IDMB

0.00

0.25

0.50

0.75

1.00IdIV IdIV

0.2

0.1

0.0

0.1IdIV DIV

0 2 4 6 8 10 12

0.3

0.2

0.1

0.0IdIV MKTRF

Impulse responses

Figure A2: VAR specification 5: Impulse response functions. The figure shows a set of selected impulseresponse functions for a VAR(1) specification of (8) featuring the market excess return (MKTRF ), the logdividend-growth (DIV ), the implied dispersion of market betas (IDMB), idiosyncratic variance (IdIV ), andmarket variance (IV ); for horizons of 1 to 12 months. The sample period spans from January 1996 to December2017. The dotted lines indicate 5% confidence bounds.

46

D Additional Tables

1 3 6 9 12

IC 0.766 0.832 0.892 0.913 0.914IDMB 0.552 0.854 0.934 0.940 0.946IdIV 0.928 0.952 0.965 0.972 0.977IV 0.808 0.857 0.891 0.906 0.915CRP 0.152 0.648 0.835 0.867 0.881V RP 0.378 0.705 0.875 0.901 0.922V RPu 0.480 0.792 0.924 0.948 0.962V RP d 0.266 0.444 0.678 0.753 0.802

Table A1: Autocorrelations. The table reports autocorrelations for option-based variables, computed frommonthly observations; for maturities from 1 to 12 months. The variables include implied correlation (IC),implied variance (IV ), and the risk premiums for correlation (CRP ), variance (V RP ) and upside and downsidesemivariances (V RPu and V RP d). The sample period spans from January 1996 to December 2017.

A: Levels

IC IDMB IdIV IV CRP V RP V RPu V RP d

IC 1.000 -0.521 -0.162 0.579 0.253 -0.011 -0.157 0.333IDMB -0.521 1.000 0.343 -0.099 -0.494 -0.086 -0.042 -0.178IdIV -0.162 0.343 1.000 0.602 -0.216 -0.337 -0.396 -0.125IV 0.579 -0.099 0.602 1.000 -0.013 -0.377 -0.551 0.130CRP 0.253 -0.494 -0.216 -0.013 1.000 0.362 0.308 0.408V RP -0.011 -0.086 -0.337 -0.377 0.362 1.000 0.971 0.844V RPu -0.157 -0.042 -0.396 -0.551 0.308 0.971 1.000 0.695V RP d 0.333 -0.178 -0.125 0.130 0.408 0.844 0.695 1.000

B: Differences

IC IDMB IdIV IV CRP V RP V RPu V RP d

IC 1.000 -0.334 -0.176 0.636 0.233 0.064 0.001 0.220IDMB -0.334 1.000 0.002 -0.188 -0.488 -0.014 -0.011 -0.044IdIV -0.176 0.002 1.000 0.347 -0.263 -0.371 -0.386 -0.304IV 0.636 -0.188 0.347 1.000 0.050 -0.172 -0.274 0.109CRP 0.233 -0.488 -0.263 0.050 1.000 0.202 0.182 0.235V RP 0.064 -0.014 -0.371 -0.172 0.202 1.000 0.981 0.938V RPu 0.001 -0.011 -0.386 -0.274 0.182 0.981 1.000 0.858V RP d 0.220 -0.044 -0.304 0.109 0.235 0.938 0.858 1.000

Table A2: Time-series correlations. This table reports time-series correlations between option-based vari-ables. Panel A shows correlations in levels and Panel B in the first differences. The variables include impliedcorrelation (IC), implied variance (IV ), and the risk premiums for correlation (CRP ), variance (V RP ) and up-side and downside semivariances (V RPu and V RP d). All variables are computed from one-month options andare sampled daily.

1 3 6 9 12Mean Std Skew Mean Std Skew Mean Std Skew Mean Std Skew Mean Std Skew

CRP 0.059 0.106 -0.170 0.099 0.083 0.209 0.126 0.079 0.056 0.134 0.079 -0.162 0.136 0.075 -0.101V RP 0.006 0.028 -7.706 0.006 0.027 -5.737 0.008 0.020 -3.236 0.009 0.023 -3.509 0.009 0.025 -2.782V RP u -0.001 0.017 -8.265 -0.001 0.016 -6.162 0.000 0.012 -4.159 0.000 0.013 -3.818 -0.000 0.014 -2.974V RP d 0.009 0.013 -0.653 0.011 0.013 0.204 0.012 0.012 2.347 0.012 0.014 1.493 0.013 0.015 1.486

Table A3: Risk premiums: Summary statistics. The table reports summary statistics (mean, standarddeviation, and skewness) for the risk premiums on correlation (CRP ), variance (V RP ), and upside and downsidesemivariances (V RPu, and V RP d). Statistics are annualized and reported for maturities of 1 to 12 months overa sample period from January 1996 to December 2017.

47

E Robustness

A: Industrial Production

1 3 6 9 12horizon

0

5

10

15

R2


B: Leading Index

1 3 6 9 12horizon

2.5

0.0

2.5

5.0

7.5

10.0

12.5

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Figure RF1: Economic condition predictions: Non-overlapping predictions. This figure reports theaverage R2 of predictive regressions of future (log)changes in macro-economic variables on current option-impliedvariables for horizons of 1 to 12 months; sampled at a frequency equal to the predictive horizon (i.e., non-overlapping). The sampe period spans from January 1996 to December 2017.

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A: Regression, implied variables: R2

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Figure RF2: In-sample return predictability: Non-overlapping predictions. The figure depicts theresults of the in-sample market excess return predictability regressions (7) for horizons of 1 to 12 months; sampledat the frequency equal to the predictive horizon (i.e., non-overlapping). Panels A and B report the adjusted R2 andt-statistics for regressions with implied correlation and/or its components as explanatory variables, respectively.Panels C and D report the adjusted R2 and t-statistics for regressions with risk premiums for correlation (CRP ),market variance (V RP ) and upside and downside semivariances (V RPu and V RP d) as explanatory variables,respectively. Panels E and F report the adjusted R2 and t-statistics for multivariate regressions, respectively.The sample period spans from January 1996 to December 2017, and the data are sampled monthly. Standarderrors are corrected for autocorrelation following Newey and West (1987) and the black dotted lines indicate 5%significance bounds.

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Figure RF3: In-sample return predictability: R2 Adjustment. The figure depicts the R2 of the in-samplemarket excess return predictability regressions (7) adjusted for autocorrelation following Boudoukh, Richardson,and Whitelaw (2008); for horizons of 1 to 12 months. Panel A shows the R2 for univariate regressions withimplied correlation (IC), the dispersion of market betas (IDMB), idiosyncratic variance (IdIV ), and marketvariance (IV ) as explanatory variables. Panel B reports the R2 for univariate regressions with risk premiums forcorrelation (CRP ), market variance (V RP ) and upside and downside semivariances (V RPu and V RP d). Thesample period spans from January 1996 to December 2017.


Panel A: IC with ControlsIC 0.033 0.228∗∗∗ 0.455∗∗∗ 0.463∗∗∗ 0.581∗∗∗

CRP 0.013 0.033 0.216∗ 0.552∗∗∗ 0.526V RP d 0.716∗∗∗ 0.628 -0.884 -1.470 -2.468TMS - -1.560∗∗∗ -3.250∗∗∗ -2.876∗∗ -1.946DFY -1.394 -4.462 - 8.523 14.137EP - -0.071∗∗ -0.125∗∗∗ -0.104∗ -0.048BTM 0.078∗ 0.445∗∗∗ 0.851∗∗∗ 0.996∗∗∗ 1.041∗∗

NTIS - 0.716 2.441∗∗∗ 3.390∗∗∗ 4.132∗∗

R2 8.52 23.03 41.43 47.66 43.68

Panel B: IC Components with ControlsIDMB -0.024 -0.133 -0.200 -0.271 -0.964∗

IdIV -0.098 -0.641∗∗∗ -1.102∗∗∗ -1.038∗∗∗ -1.424∗∗∗

IV - 0.598∗∗ 3.025∗∗∗ 3.624∗∗∗ 1.015CRP - -0.085 0.087 0.504∗ -V RP d 0.726∗∗∗ 1.088∗∗∗ -1.974∗∗ -3.412∗∗∗ -TMS - -1.362∗∗ -2.091∗ -1.586 -DFY - - -7.795 1.694 11.513EP - -0.062∗ -0.120∗∗∗ -0.060 -0.080BTM - 0.184 0.842∗∗∗ 1.040∗∗∗ 0.561NTIS - 0.990∗ 2.021∗∗ 2.963∗∗ 4.269∗∗

R2 9.00 27.12 45.92 51.67 45.72

Table RT1: In-sample return predictability: Traditional predictors. The table reports the results of thein-sample market excess return predictability regressions (7); for horizons from 1 to 12 months. Panels A and Bshow the results for regressions using implied correlation (IC) and its three components as (potential) explanatoryvariables, respectively. The regressions include a variety of “traditional” predictors; selected, for each predictivehorizon, by applying a LASSO procedure to regression (7) (with an optimal regularization parameter based onthe Akaike information criterion). ∗∗∗, ∗∗, and ∗ indicate significance at 1%, 5%, and 10% level, respectively,based on Newey and West (1987) standard errors.

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Panel A: IC below medianIC 0.022 - 0.197 - 0.543∗∗∗ - 0.984∗∗∗ - 1.265∗∗∗ -IDMB - -0.013 - -0.308∗∗∗ - -0.668∗∗∗ - -0.806∗∗∗ - -0.872∗

IdIV - -0.122 - -0.120 - -0.186 - -1.045 - -1.421IV - 0.020 - -0.079 - -0.912 - 0.657 - 0.176R2 -0.63 2.84 2.35 23.71 9.78 35.08 18.04 31.65 18.60 27.60

Panel B: IC above medianIC 0.049 - 0.320∗∗∗ - 0.463∗∗∗ - 0.573∗∗ - 0.438∗ -IDMB - -0.009 - 0.412∗ - 0.766∗∗∗ - 0.644 - 1.243∗

IdIV - -0.261 - -1.268∗∗∗ - -1.060∗ - -0.786 - -0.296IV - 0.218 - 1.421∗∗∗ - 2.361∗∗∗ - 2.457∗∗∗ - 1.761∗

R2 0.03 1.18 9.24 16.48 7.56 21.36 7.61 18.40 2.73 13.77

Panel C: First Half of the SampleIC 0.105∗∗∗ - 0.331∗∗∗ - 0.651∗∗∗ - 0.959∗∗∗ - 1.178∗∗∗ -IDMB - -0.059∗ - -0.208∗∗ - -0.542∗∗∗ - -0.845∗∗∗ - -1.202∗∗∗

IdIV - -0.115 - -0.446∗∗∗ - -0.870∗∗∗ - -1.445∗∗∗ - -1.946∗∗∗

IV - 0.344∗ - 1.267∗∗∗ - 2.001∗∗∗ - 2.403∗∗∗ - 2.127∗

R2 6.59 9.03 24.25 32.05 40.74 53.89 46.93 60.52 42.57 56.07

Panel D: Second Half of the SampleIC 0.026 - 0.183∗∗ - 0.525∗∗ - 0.811∗∗ - 1.013∗∗ -IDMB - 0.042∗∗ - 0.099 - -0.368 - -0.433 - 1.039IdIV - -0.624∗∗ - -2.008∗∗∗ - -2.695∗∗∗ - -3.519∗∗ - -4.792∗

IV - 0.529∗ - 2.165∗∗∗ - 4.133∗∗∗ - 5.293∗∗∗ - 6.095∗

R2 -0.15 8.79 4.70 25.46 11.66 24.30 15.68 23.66 15.95 20.45

Panel E: Recession Indicator = 0IC 0.071∗∗∗ - 0.228∗∗∗ - 0.431∗∗∗ - 0.616∗∗∗ - 0.744∗∗∗ -IDMB - -0.009 - -0.087 - -0.384∗ - -0.610 - -0.373IdIV - -0.219∗∗∗ - -0.539∗∗∗ - -0.867∗∗∗ - -1.331∗∗∗ - -1.873∗∗∗

IV - 0.412∗∗∗ - 1.209∗∗∗ - 2.120∗∗∗ - 2.570∗∗∗ - 2.616∗∗∗

R2 4.90 8.04 15.52 22.20 23.19 34.19 26.70 37.12 23.39 28.75

Panel F: Recession Indicator = 1IC 0.018 - 0.505 - 1.053∗ - 1.233 - 1.592∗ -IDMB - 0.025 - 0.105 - 1.400∗∗∗ - 1.665∗∗∗ - 1.554∗∗

IdIV - 0.076 - -1.362 - -2.913∗∗∗ - -0.801 - 0.186IV - -0.016 - 1.931 - 7.259∗∗∗ - 7.429∗∗∗ - 7.842∗∗∗

R2 -4.07 -12.89 12.32 3.10 22.46 41.68 23.67 70.12 33.78 75.99

Table RT2: In-sample return predictability: Subsamples. The table reports the results of the in-samplemarket excess return predictability regressions (7); for horizons from 1 to 12 months. Panels A and B conditionon implied correlation being below and above its median value (for the whole sample), Panels C and D split thesample in two halves, and Panels E and F condition on the NBER recession indicator. ∗∗∗, ∗∗, and ∗ indicatesignificance at 1%, 5%, and 10% level, respectively, based on Newey and West (1987) standard errors. The sampleperiod spans from January 1996 to December 2017.

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Figure RF4: Traditional approach: Out-of-sample R2. The figure shows the out-of-sample R2 (as definedin (10)) and its p-value for predictions based on the traditional approach with a long (10-year) estimation window.Panels A and B depict the results for implied correlation (IC) and its components: the dispersion of market betas(IDMB), idiosyncratic variance (IdIV ), and market variance (IV ). Panels C and D depict the results for riskpremiums for correlation (CRP ), market variance (V RP ) and upside and downside semivariances (V RPu andV RP d). p-values are computed from bootstrapped distributions and the dotted lines indicate 5% significancebounds. Predictions are made at a monthly frequency from January 1996 to December 2017.

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Expected Correlation and Future Market Returns

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