Essays on Empirical Asset Pricing

Gormsen, Niels Joachim Christfort Doctoral Thesis Essays on Empirical Asset Pricing PhD Series, No. 21.2018 Provided in Cooperation with: Copenhagen Business School (CBS) Suggested Citation: Gormsen, Niels Joachim Christfort (2018) : Essays on Empirical Asset Pricing, PhD Series, No. 21.2018, ISBN 9788793579897, Copenhagen Business School (CBS), Frederiksberg, https://hdl.handle.net/10398/9636 This Version is available at: https://hdl.handle.net/10419/209068 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/ ESSAYS ON EMPIRICAL ASSET PRICING Niels Joachim Christfort Gormsen PhD School in Economics and Management PhD Series 21.2018 PhD Series 21-2018 ESSAYS ON EMPIRICAL ASSET PRICING COPENHAGEN BUSINESS SCHOOL SOLBJERG PLADS 3 DK-2000 FREDERIKSBERG DANMARK WWW.CBS.DK ISSN 0906-6934 Print ISBN: 978-87-93579-88-0 Online ISBN: 978-87-93579-89-7 Essays on Empirical Asset Pricing Niels Joachim Christfort Gormsen Athesispresentedforthedegreeof Doctor of Philosophy Supervisor: Lasse Heje Pedersen Ph.D. School in Economics and Management Copenhagen Business School ii Niels Joachim Christfort Gormsen Essays on Empirical Asset Pricing 1st edition 2018 PhD Series 21.2018 © Niels Joachim Christfort Gormsen ISSN 0906-6934 Print ISBN: 978-87-93579-88-0 Online ISBN: 978-87-93579-89-7 The PhD School in Economics and Management is an active national and international research environment at CBS for research degree students who deal with economics and management at business, industry and country level in a theoretical and empirical manner. All rights reserved. No parts of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without permission in writing from the publisher. Abstract This thesis concerns the empirical relation between risk and return in equities. It studies why the expected return on stocks as a whole varies over time and why there are predictable cross-sectional di↵erences in the return on individual stocks. The thesis consists of three chapters which can be read independently. The first chapter addresses why the expected return on the market portfolio varies over time. The market portfolio is a claim to all future cash flows earned by the firms in the stock market. I study the expected return to these future cash flows individually. I find that the expected return to the distant-future cash flows increases by more in bad times than the expected return to near-future cash flows does. This new stylized fact is important for understanding why the expected return on the market portfolio as a whole varies over time. In addition, it has strong implications for which economic model that drives the return to stocks. Indeed, I find that none of the canonical asset pricing models can explain this new stylized fact while also explaining the previously documented facts about stock returns. The second chapter, called Conditional Risk, studies how the expected return on individual stocks is influenced by the fact that their riskiness varies over time. We introduce a new ”conditional-risk factor”, which is a simple method for determining how much of the expected return to individual stocks that can be explained by time variation in their market risk, i.e. market betas. Using this new factor, we find that around 20% of the cross-sectional variation in expected stock returns worldwide can be explained by such time variation in market betas. The third chapter studies why stocks with low market betas have high risk-adjusted returns. To shed light on this low-risk e↵ect, we decompose all stocks’ market betas into their volatility and their correlation with the market portfolio. We find that both stocks with lower volatility and stocks with lower correlation have higher risk-adjusted returns. The last fact, that stocks with low correlation have high risk-adjusted returns, is particularly important iii because it helps distinguish between competing theories of the low-risk e↵ect. Indeed, the high risk-adjusted returns to low-correlation stocks are consistent with leverage based theories of the low-risk e↵ect, but it is not immediately implied by competing behavioral theories we consider in the paper. iv Acknowledgements Writing this thesis has put me in debt to more people than I can mention here. The biggest debt is to Lasse Heje Pedersen who trained me as a financial economist and showed me how to do research, something I am profoundly grateful for. Working with Lasse has been the great privilege of my education. I am also indebted to much of the finance faculty at Harvard University. Most importantly, John Y. Campbell sponsored a one-year visit to Harvard and taught me asset pricing, and Robin Greenwood took me under his wings during the stay, which resulted in a great co-authorship and friendship. Finally, my fianc´ee Joanna deserves special thanks. Joanna has been far more involved in this finance thesis than any anthropologist would ever want to be. I am nonetheless glad she was, because it made writing this thesis a much greater pleasure than it would otherwise have been. With that said, the last four years of my life in Copenhagen can easily be summarized: David Lando built the FRIC center and made Copenhagen a great place to be a student of finance. Thomas Kjær Poulsen kept me in good standing with the PhD administration. My co-author Christian and I argued over everything we wrote. Friends and family made the time fly by. I could not have asked for four better years. v vi Introduction and Summaries The starting point for this thesis is the following two empirical observations: (1) the expected return on the market portfolio of stocks varies over time,1and (2) the expected return on individual stocks varies cross-sectionally.2Much of modern asset pricing is about understanding this time series and cross-sectional variation in expected returns, which are often referred to as discount rates (Cochrane,2011). All three chapters in this thesis document new empirical facts that help us understand this expected return variation in equities. The first paper improves our understanding of the economics behind time series variation in expected returns. The second and third paper improve our understanding of cross-sectional variation in expected returns. The next pages provide summaries of the individual papers in English and Danish. These summaries clarify the individual papers’ contribution. 1 Summaries in English Time Variation of the Equity Term Structure This paper studies the equity term structure, which is a novel way of studying the market portfolio. Usually we study the return to buying the market portfolio as a whole, which is really the return to buying the right to all future dividends. In contrast, when we study the equity term structure, we study the return to buying individual dividends on their own, which in turn allows us to get deeper insights into the economics of stock returns. More precisely, the equity term structure refers to how the expected return to dividends depends on how far into the future these dividends are paid out. The previous literature focuses on the average equity term premium, which is the average di↵erence in return on claims on longand short-maturity dividends. This literature finds that the equity term 1See e.g. Campbell and Shiller (1988); Fama and French (1988); Campbell and Thompson (2008). 2See Bondt and Thaler (1985); Fama and French (1992,2015); Pastor and Stambaugh (2003); Acharya and Pedersen (2005); Novy-Marx (2013) and more. vii Contents Abstract iii Acknowledgements v Introduction and Summaries vii 1 Summaries in English ............................... vii 2 Summaries in Danish ............................... xi 1 Time Variation of the Equity Term Structure 1 1 Motivating Theory ................................ 6 1.1 Model ................................... 7 1.2 Equity Term Premia and Their Cyclicality ............... 8 1.3 Equity Term Premia and Real Investments ............... 12 2 Data and Methodology .............................. 13 3 Counter-Cyclical Term Premia: A New Stylized Fact ............. 15 3.1 The Equity Term Premium Implied from Options Prices ....... 18 3.2 The Equity Term Premium Implied from the Cross-Section of Equities 18 3.3 Measurement Error Concerns ...................... 21 4 The Expectations Hypothesis .......................... 22 4.1 Defining Equity Yields and the Expectations Hypothesis ....... 22 4.2 Testing the Expectations Hypothesis .................. 24 5 Real E↵ects: Cyclicality in the Relative Investments by Longand ShortMaturity Firms .................................. 27 6 Testing Asset Pricing Models: Theory vs. Stylized Facts ........... 30 6.1 The Habit Model by Campbell and Cochrane (1999)......... 30 6.2 The Long-Run Risk Model by Bansal and Yaron (2004)........ 31 6.3 The Model by Lettau and Wachter (2007)............... 32 6.4 The Disaster Model by Gabaix (2012).................. 33 6.5 Reconciling the Facts: A Model with Negative and Counter-Cyclical Term Premia ............................... 33 7 Conclusion ..................................... 35 xiv 8 Proofs ....................................... 36 2 Conditional Risk 71 1 Conditional Risk Theory ............................. 75 1.1 A Simple Example: The CAPM ..................... 75 1.2 Conditional Risk in Factor Models ................... 79 1.3 Conditional Cash Flow and Discount Rate Risk ............ 81 2 Methodology ................................... 82 2.1 Identifying Conditional Moments .................... 82 2.2 Data .................................... 83 3 Conditional Risk in Stock Returns ........................ 84 3.1 Conditional Risk in the Cross-Section of Stock Returns ........ 86 3.2 Conditional Risk in Time-Series Strategies ............... 89 4 Arbitrage Trading as the Source of Conditional Risk .............. 91 5 Robustness .................................... 94 6 Relation to the Literature ............................ 96 7 Conclusion ..................................... 97 8 Appendix ..................................... 99 3 Bettin Against Correlation: Testing Theories of the Low-Risk E↵ect 131 1 Introduction ....................................132 2 Data and Methodology ..............................137 2.1 Constructing BAC and BAV factors ...................137 2.2 Constructing LMAX, SMAX, and IVOL factors ............139 2.3 Explanatory variables in factor regressions ...............140 2.4 Economic variables ............................141 3 Systematic Risk: Betting Against Correlation, Volatility, and Beta ......142 3.1 Double-sorting on correlation and volatility ...............142 3.2 Decomposing BAB into BAC and BAV .................142 3.3 The performance and factor loadings of BAC ..............143 4 Idiosyncratic Risk: LMAX, SMAX, and IVOL .................144 4.1 Double-sorting on MAX and volatility .................145 4.2 Decomposing LMAX into SMAX and BAV ...............145 4.3 The performance of idiosyncratic risk factors: LMAX, SMAX, and IVOL ...................................145 5 Testing the Underlying Economic Drivers ....................146 6 Horserace .....................................149 6.1 Horserace based on published factors ..................149 xv 6.2 Turnover and alpha decay ........................150 6.3 All factors constructed based on Fama-French methodology ......152 6.4 All factors constructed based on rank-weighting-BAB methodology .152 7 Conclusion ....................................153 Bibliography 183 xvi Chapter 1 Time Variation of the Equity Term Structure Abstract: Idocumentthatthetermstructureofholding-periodequityreturnsiscounter-cyclical:it is downward sloping in good times, but upward sloping in bad times. This new stylized fact implies that long-maturity risk plays a central role in asset price fluctuations, consistent with theories of long-run risk and habit, but these theories cannot explain the average downward slope. At the same time, the cyclical variation is inconsistent with recent models constructed to match the average downward slope. I present the theoretical source of the puzzle and suggest a new model as a resolution. My model also shows that the countercyclical term structure has implications for real activity, which I verify empirically: in bad times, long-duration firms decrease their investment and capital-to-labor ratio relative to short-duration firms. Keywords: asset pricing, equity term structure, time-varying discount rates. JEL classification: G10, G12. I am grateful for helpful comments from John Y. Campbell, Peter Feldh¨utter, Xavier Gabaix, Stefano Giglio, Robin Greenwood, Sam Hanson, Bryan Kelly, Ralph Koijen (discussant), Eben Lazerus, Martin Lettau, Dong Lou, Matteo Maggiori, Ian Martin, Tobias Moskowitz, Stefan Nagel, Lasse Heje Pedersen, Andrei Shleifer, Jeremy Stein, Adi Sunderam, and Paul Whelan, as well as seminar participants Berkeley Haas, Chicago Booth, Copenhagen Business School, London Business School, London School of Economics, Harvard University, Stockholm School of Economics, Oxford Sa¨ıd School of Business, and Yale School of Management, as well as participants at the 2017 NFN conference in Copenhagen. I gratefully acknowledge support from the European Research Council (ERC grant no. 312417) and the FRIC Center for Financial Frictions (grant no. DNRF102). 1 I study the term structure of equity returns and document a large cyclical variation. This cyclical variation is important for understanding which risks drive fluctuations in asset prices. Indeed, the cyclical variation documented in this paper suggests that price fluctuations are driven mainly by long-maturity risks such as persistent changes in dividend growth, and only less by short-maturity risks such as disaster risks. As such, the results are consistent with classical asset pricing models such as Campbell and Cochrane (1999)or Bansal and Yaron (2004), but they are inconsistent with the newer models that are designed to have downward sloping equity term structures. In addition, the cyclical variation of the equity term structure has important real consequences because it directly influences when capital flows to long-maturity firms such as biotech firms or short-maturity firms such as automobile firms and the extent to which these firms invest in production plants, R&D, or labor. By way of background, the previous research on the equity term structure has focused on its average slope, finding that it is downward sloping on average (Binsbergen, Brandt, and Koijen,2012), as indicated by the solid line in my Figure 1. This result is inconsistent with traditional models of long-run risk and habit which have upward sloping term structures. Addressing this challenge to traditional asset pricing models has become one of the most active areas in macro-finance (Cochrane,2017)andhasledtothedevelopmentofnew models with average downward sloping term structures.1 I contribute to the literature on the equity term structure by studying its time variation. My main result is that the equity term structure of holding-period returns is counter-cyclical: it is downward sloping in good times but upward sloping in bad times. As shown in Figure 1, this counter-cyclical variation is economically large. In good times, long-maturity equity has 4 percent lower expected annual return than short-maturity equity, but in bad times it has 5 percent higher expected return, meaning that the equity term premium varies by 9 percentage points between good and bad times. As shown in Figure 2, I document this new stylized fact using several di↵erent measures of term premia, sample periods, data sources, and by also using futures returns as opposed 1The reference model for a downward sloping term structure is Lettau and Wachter (2007), which precedes the empirical literature on the downward sloping equity term structure. More recent models include Eisenbach and Schmalz (2013); Andries, Eisenbach, and Schmalz (2015); Nakamura, Steinsson, Barro, and Urs´ua (2013); Belo, Collin-Dufresne, and Goldstein (2015); Croce, Lettau, and Ludvigson (2014); Hasler and Marfe (2016). Binsbergen and Koijen (2017) review the new theoretical models that have been motivated by the downward sloping terms structure. 2 Figre 1: The Term Structure of One-Year Equity Returns This figure shows the term structure of holding-period equity returns for the S&P 500. The figure shows the unconditional average return (solid line), the average return in bad times (dashed line), and the average return in good times (dash-dotted line). Good and bad times are defined by the ex ante dividend-price ratio. Short-maturity equity claims is the average return to dividend futures of 1 to 7 years maturity. The long-maturity claim is the average return to the market portfolio. Returns are annual spot returns, 2005 – 2016. to spot returns. Using dividend futures with maturities up to seven years, I find a positive relation between the ex ante dividend price ratio and the ex post one-year return di↵erence between longand short-maturity dividend futures (Panel A). The result also holds when using the market portfolio as the long-maturity claim, when considering Sharpe ratios instead of returns, when excluding the financial crisis, and when using other measures of bad times such as the CAPE ratio and the cay variable. The result holds in the U.S. for the S&P 500 and it holds internationally for Nikkei 225, Euro Stoxx 50, and the FTSE 100. Going beyond dividend futures, the result also holds when measuring the equity term structure using option implied dividend prices (Panel B) or the cross-section of stocks (Panel C).2 As shown in the first two columns of Table 1, the counter-cyclical equity term premia represent a puzzle for asset pricing theory: none of our canonical asset pricing models are able to produce both the counter-cyclical variation documented in this paper and the 2I estimate a term-premium mimicking portfolio in the cross-section of stocks by projecting the excess returns of characteristics-sorted portfolios onto the realized return di↵erence between longand short-maturity claims. 3 negative slope documented by Binsbergen, Brandt, and Koijen (2012). The counter-cyclical variation is consistent with the traditional macro-finance models such as Campbell and Cochrane (1999)andBansal and Yaron (2004), but inconsistent with the new models with average downward sloping term structures. Hence, traditional models explain the timevariation in the term premium, but not its average value, and vice versa for the newer models. The puzzle applies more generally than just the models in Table 1. To underline the generality of the puzzle and to identify its source, I study the cyclicality of term premia through a simple, essentially affine model that is sufficiently general to capture most of the dynamics of log-normal models. In the model, the term structure of returns may be either upward or downward sloping; but I show that if it is upward sloping it is counter-cyclical and if it is downward sloping it is pro-cyclical. To see the intuition behind this result, consider for instance a downward sloping model. The downwards sloping term structure suggests that short-maturity equity is riskier than long-maturity equity and commands a premium, meaning that the equity term premium is negative. In bad times, this premium on short-maturity equity increases because the price of risk increases and the term premium thus becomes even more negative, not positive as is observed empirically. To understand what is needed to resolve the puzzle and explain the stylized facts, I introduce a new model with a term premium that is both counter-cyclical and negative on average. In the model, investors trade o↵a demand for hedging investment opportunities with an aversion towards long-run risk: the required return on long-maturity equity is pushed down by investors’ demand for hedging investment opportunities, but it is pushed up by their aversion for long-run risk. The relative strength of the two e↵ects varies over time, and the model is specified such that demand for hedging dominates on average, meaning that the equity term premium is negative on average; but in bad times the aversion against long-run risk dominates so that the equity term premium becomes positive. The model is thus able to capture the two stylized facts of the equity term structure. The model is based on an exogenous stochastic discount factor and rooting it in a micro-foundation remains an interesting topic for future research. The counter-cyclical term premia documented in this paper may be surprising given the pro-cyclical ”equity yield curve” documented by Binsbergen, Hueskes, Koijen, and Vrugt (2013). An equity yield is the current dividends divided by the price of future dividends of a 4 given maturity, meaning that it is closely related to hold-to-maturity returns.3The authors document that the yield curve is steeply downward sloping in bad times, which might lead one to believe that during bad times, long-maturity claims are expected to have low returns relative to short maturity claims, i.e. that the one-period equity term premium is lower than usually. However, I directly study the one-period term premium and find that it is higher in bad times, even though the yield curve is downward sloping. To better understand this negative relation between equity term premia and the slope of the yield curve, I test an expectations hypothesis. The hypothesis is that equity term premia are constant, meaning that the expected development in yields can be inferred from the yield curve. I find that equity yields move in the direction suggested by the yield curve, but they move by more than suggested by the expectations hypothesis. I show that this excess movement in yields implies that the slope of the equity yield curve must be negatively correlated with equity term premia, thus reconciling my results with Binsbergen, Hueskes, Koijen, and Vrugt (2013). The result that yields move too much in the direction of what the yield curve suggests is surprising because it contrasts the results from the bond literature: for bonds, the expectations hypothesis is rejected because yields move in the opposite direction of what the yield curve suggests4. In addition, the test of the expectations hypothesis represents another tension between theory and the data. As shown in the third column of Table 1, none of the asset pricing models I consider are able to generate as strong a relation between the yield spread and future changes in yields as that observed in the data. The models fail in this regard because their term premium is pro-cyclical or because the models create too little predictability in equity yields relative to term premia. Finally, the counter-cyclical equity term structure is also important for understanding the cost of capital and how real resources are allocated in the economy. To better understand these real dynamics, I study firms‘ investment decisions in my model of the equity term structure. In the model, some firms have long-maturity cash flows and some have shortmaturity. These firms are di↵erently a↵ected by the equity term structure: in bad times, the counter-cyclical equity term structure incentivizes long-maturity firms to invest less and to apply less capital relative to labor compared to short-maturity firms because the long-maturity firms find capital relatively more expensive. 3Equity yields are equivalent to hold-to-maturity returns minus the hold-to-maturity growth rates. 4See e.g. Shiller (1979); Shiller, Campbell, and Schoenholtz (1983) and Campbell and Shiller (1991). 5 I verify the real implications of the model empirically, as summarized in Figure 3. I find that, in bad times, the long-maturity firms invest less in capital equipment and R&D than short-maturity firms do. On the other hand, they increase spending on wages relative to short-maturity firms. Taken together, the long-maturity firms thus decrease their capital to labor ratio relative to short-maturity firms. This pattern is consistent with long-maturity firms finding capital relatively more expensive than short-maturity firms do in bad times because the equity term structure is more upward sloping. In conclusion, this paper documents a new stylized fact that gives new insight into the drivers of the equity risk premium. The counter-cyclical term structure implies that the variation in the equity risk premium mainly comes from variation in long-term risk. Together with the observation that the equity term structure is downward sloping, the counter-cyclical term structure represents a puzzle for existing macro-finance models. I show theoretically that the canonical models are not able to reproduce both facts, and as a response I introduce a new model that can. Finally, I show empirically and theoretically that the cyclicality of the equity term structure is linked to the cylicality in real investments: in bad times where the equity term structure is upward sloping, long-maturity firms invest less than short-maturity firms. The paper proceeds as follows. Section I introduces a model of the equity term structure with implications for firm investment. Section II describes data sources. Section III documents the counter-cyclical equity term structure. Section IV tests the expectation hypothesis. Section V studies real consequences of the equity term structure. Section VI studies calibrations of several canonical asset pricing models individually as well as my model introduced in section II. Section VII concludes. 1 Motivating Theory In this section, I introduce a simple extension of the model of the equity term structure by Lettau and Wachter (2007). In the special case of the original Lettau and Wachter model, Ishowthatthereisalinkbetweenthesignandcyclicalityofthetermpremiuminthe sense that term premia are either positive on average and counter-cyclical or negative on average and pro-cyclical (Proposition 1.a). In the more general version of the model, one can capture the empirical regularities that I uncover, that is, one can have term premia that are negative on average and counter-cyclical (Proposition 1.b). Finally, I study the link 6 between the equity term structure and the investment decisions of individual firms, finding that long-maturity firms use less capital to labor when the equity term structure is more upward sloping (Proposition 2). 1.1 Model The economy has an aggregate equity claim with dividends at time tdenoted by Dt,where dt=ln(Dt)evolvesas dt+1 =µg+zt+d✏d,t+1 (1.1) Here µg2Ris the unconditional mean dividend growth and ztdrives the conditional mean: zt+1 ='zzt+z✏z,t+1 (1.2) where 0 <' z<1. Further, ✏d,t+1 and ✏z,t+1 are normally distributed mean-zero shocks with unit variance and d,zare their volatilities. The risk-free rate rfis constant and the stochastic discount factor is given by Mt+1 =exp✓rf1 2x2 txt✏d,t+1 a✓1 2a+xt⇢dx +✏x,t+1◆◆ (1.3) where a2Rand the state variable xtdrives the price of risk: xt+1 =(1'x)¯x+'xxt+x✏x,t+1 (1.4) The parameter ¯x2R+is the long-run average, 0 <' x<1, and ✏x,t+1 is a normally distributed mean-zero shock with unit variance and xis the volatility. The three shocks have correlations denoted ⇢dx,⇢dz,and⇢zx,where⇢zx =0,⇢dxx'x,and⇢dzz< d(1 'z). The first assumption is also made by Lettau and Wachter (2007) and the latter two hold in their empirical calibration. To understand the intution behind the stochastic discount factor, consider first the case where a=0asinLettau and Wachter (2007). In this case, investors are averse towards shocks to dividends, ✏d,t+1. A negative shock to dividends increases the marginal utility and thus increases the value of the stochastic discount factor. The e↵ect of a given shock on the stochastic discount factor depends on the price-of-risk variable xt,whichinthissensecan 7 tied to the calendar year. The payo↵on the contract is the declared dividends that go ex-dividend during the given calendar year. The contracts are forward contracts, meaning everything is settled at the expiration date. For example, on February 11th 2011, the 2013 forward contract for S&P 500 trades at $31. In this contract, the buyer agrees to pay the seller $31 by the end of December 2013, and the seller agrees to pay the buyer the sum of the dividends that have gone ex-dividend between January 1st 2013 and the end of December 2013. Because the expiration dates of the contracts are fixed in calendar time, the maturity of the available contracts varies over the calendar year. To get constant maturity prices I thus interpolate across the prices of di↵erent contracts each month, following the norm in the literature on dividend futures prices (see e.g. Binsbergen, Hueskes, Koijen, and Vrugt (2013); Binsbergen and Koijen (2017); Cejnek and Randl (2016b,a)). Option implied equity term premium: Binsbergen, Brandt, and Koijen (2012) make their estimated time series of dividend prices and returns available online. The dividend prices are for the S&P 500 and the sample runs from 1996-2009. Binsbergen, Brandt, and Koijen (2012) estimate both the return to buying next year’s dividends and the return to buying the dividend two years ahead, which they call the dividend steepener. The first strategy’s returns are based on the collected dividends whereas the second strategy’s returns are pure capital gains. Because dividend returns and capital gains are taxed di↵erently, I use the dividend steepener because these returns are more easily compared to the returns to the market portfolio and to the returns in the remainder of the paper (see Schulz (2016) for an analysis of the impact of taxes on the returns to dividends). Cross-section of equity: Stock returns are from the union of CRSP and the XpressFeed Global Database. For companies traded in multiple markets, I use the primary trading vehicle identified by XpressFeed. Fundamentals are from the XpressFeed Global Database. I consider standard characteristics that may be related to the duration of cash-flow. I measure book-to-market, profitability, and investment following Fama and French (2015). Portfolio breakpoints are calculated each June using the most recent characteristics starting from the end of the previous year. Portfolios are rebalanced at the end of each calendar month. Portfolio breakpoints are based on NYSE firms and returns are equal-weighted. Dividends: The dividends for the S&P 500 index are from Shiller’s webpage. For the international indexes, I get dividends from Bloomberg. I measure dividends as the running 14 annual dividends instead of end of year dividends. I do so to avoid omitting easily available information about the final annual dividends. Returns: I measure equity term premia in log-returns to mitigate measurement error issues, as advocated by Boguth, Carlson, Fisher, and Simutin (2012). In addition, the expectations hypothesis makes assumptions about log-returns, and using log-returns in the entire analysis thereby ensures consistency. The results are not sensitive to this choice. 3 Counter-Cyclical Term Premia: A New Stylized Fact In this section, I document that equity term premia are counter-cyclical. I first show this using the full sample of dividend futures. I afterwards document the robustness using other sample periods, other measures of cyclicality, and other measures of equity term premia. Istudythecyclicalityofequitytermpremiabyregressingtherealizedreturndi↵erence between longand short-maturity equity on the ex ante dividend price ratio. That is, for each index, I run the following regression for di↵erent maturity pairs nand m,wheren>m: rn t,t+12 rm t,t+12 =n,m 0+n,m 1(dtpt)+✏t,t+12 (1.18) where rn t,t+12 is the log-return on the nmaturity claim between period tand t+12, and dtpt is the log of the dividend price ratio of the index at time t.Theregressionisimplemented on the monthly level using rolling one-year log returns.7Accordingly, I use Newey-West standard errors corrected for 18 lags. Panel A in Table 2 shows the estimates of n,m 1for the S&P 500. The parameter estimates are positive for all maturity pairs. The positive parameter estimates suggest that term premia are larger when the dividend price ratio is high, which is to say that the term premia are counter-cyclical. The estimates are highly significant for low nand mbut the significance becomes weaker as nand mincreases. The estimates of n,m 1are large in magnitude. Consider for instance the premium of the five-year claim in excess of the two-year claim. The loading on the dividend price ratio is around 0.2, suggesting that the term premium increases by 20 percentage points annually 7Throughout the analysis I work with rolling annual returns. Working with an annual horizon allows me to calculate realized Sharpe ratios and easily compare with the results on the expectations hypothesis. The results are similar when using quarterly horizon (Table A2), but the statistical significance is lower partly because of noise in the dividend futures data. 15 when the log dividend price ratio increases with 1. In the sample, the log dividend price ratio varies by 0.6, implying that this one-year term premium varies by more than 12 percentage points over the sample. The results in the international sample are similar to those in the U.S.. Across almost all indexes and maturity pairs, the parameter estimates are positive. The exception is the long premium in excess of the three-year claim for FTSE 100 and Euro Stoxx 50; the estimate for these term premia are negative. In the rightmost column, I include the market portfolio as the long-maturity claim. Because the return to the market portfolio is not a futures contract, I must correct for the e↵ect of interest rates. Following Binsbergen and Koijen (2017), I subtract from the market portfolio the 30 year bond return over the same period. Across the four indexes, the term premia that have the market as the long-maturity claim are all counter-cyclical, except for the term premium in excess of the three year claim for Euro Stoxx 50. The statistical significance is highest in the U.S. and highest at low m. Together, the results provide both statistically and economically significant evidence that equity term premia are counter-cyclical. Given that equity term premia are negative on average (Binsbergen, Brandt, and Koijen,2012;Binsbergen and Koijen,2017), the results thus reject a large class of model (see Proposition 1.a and Section VI). I consider several robustness checks. First, one possible concern is that the results are driven by the financial crisis during which prices on dividends may have deviated from fundamentals. To address this concern, I run the regression again, excluding observations starting in 2008 and 2009. Table 3 reports these results. The parameter estimates are still positive, and they are generally larger and more statistically significant, underlining that the results are not driven by the financial crisis. Another way to see that the results are not driven by the financial crisis is by considering the time series of the term premium and the dividend price ratio in Figure 4. The figure shows on each date the dividend price ratio and the future realized return di↵erence between longand short-maturity claims. Consider for instance Euro Stoxx 50 in Panel C. As can be seen on its dividend price ratio, the Euro Stoxx 50 goes through two crises: the financial crisis in 2008 and the sovereign debt crisis in 2011. In both instances, the term premium increases substantially. The results are similar for Nikkei 225 and FTSE 100, both of which also see an increase in the dividend price ratio around 2011. Finally, Panel A shows the 16 S&P 500, for which the time series goes all the way back to 1996. The figure shows that the term premium also tracked the dividend price ratio through the tech bubble and the subsequent recession, again underlining the generality of the counter-cyclical term premium. The pre-2005 S&P 500 results are based on implied dividend prices from options, which I analyze in depth in Section 3.1. I next test the cyclicality of the equity term premia using the cay measure (Lettau and Ludvigson,2001a) instead to ensure that the cyclicality is not driven by the choice of conditioning variable. The results, reported in Table 4, are similar: the term premia are highly counter cyclical. The cylicality is slightly weaker in the sample excluding the financial crisis, but term premia remain counter-cyclical. Binsbergen and Koijen (2017) document that both expected returns and Sharpe ratios on equity claims are downward sloping in maturity. In a similar spirit, I study how the Sharpe ratios of the term premia vary over time. To this end, I calculate the time-varying realized variance using 12 months of monthly returns and use it to estimate realized Sharpe ratios as:8 SRn,m t,t+12 =rn t,t+12 rm t,t+12 vart(rnm t,t+12)=rn t,t+12 rm t,t+12 q1 121P12 i=1 (rn t+irm t+i)(¯rn t,t+12 ¯rm t,t+12)2(1.19) InextregresstheSharperatioontheexantedividendpriceratiotoestimatethe cyclicality. The results of this regression are reported in Table 5. For the S&P 500 in Panel A, the term premia are all counter-cyclical. The cyclicality is statistically significant for almost all maturity pairs, but the statistical significance decreases as mincreases. Panels B through D of Table 5 report similar results for the international indexes: the Sharpe ratios are generally counter-cyclical, and the e↵ect is strongest for the term premia with low m. The exception is the Sharpe ratios of term premia measured in excess of the three-year claim for Euro Stoxx 50 and FTSE 100; these parameter coefficients are negative but statistically insignificant. The counter-cyclical Sharpe ratios are consistent with the model covered earlier. In the model, changes in the term premium come from changes in the price of risk and not from changes in volatility. Accordingly, we would expect higher term premia to be associated with higher Sharpe ratios, which is indeed what Table 5 suggests. 8These are not technically Sharpe ratios because they are based on log-returns to ensure consistency with the rest of the paper. The results are, however, similar when using simple returns. 17 For additional robustness, I next confirm that my results are similar when using other measures of equity term premia over other sample periods. In particular, I estimate the equity term premium by using implied dividend prices from Binsbergen, Brandt, and Koijen (2012) and by using the cross-section of stock returns. Neither of these measures are as direct as the dividend futures, but using them allows me to consider a sample that goes as far back as 1964. 3.1 The Equity Term Premium Implied from Options Prices Binsbergen, Brandt, and Koijen (2012) use options prices to estimate the present value of future dividends. The intuition behind their method is simple. When you buy the index you get next year’s dividends plus next year’s resale price. By going short a call option and buying a put option you can hedge the resale price such that you are certain only to get next year’s dividends. The price of buying the stock and hedging the resale price thus reflects the price of the dividends. To measure the equity term premium, I compare the return to these implied dividends with the return to the market portfolio. To measure the cyclicality, I again regress the rolling one-year realized return di↵erence between longand short-maturity claims onto the ex ante dividend price ratio. The results are shown in the first two columns of Table 6. The term premium estimated from options prices is highly counter-cyclical. The realize return di↵erence has a loading of 1 on the dividend price ratio, which is approximately twice as large as the loadings in Table 2 that are based on the dividend futures. The results thus support the notion that term premia are highly counter-cyclical. The second column shows that the results are robust to controlling for the five Fama and French (2015) factors as well as the yield spread and the short yield. Because the returns used in this regression are spot returns and not future returns, I include the treasury yield spread and the treasury short yield to control for potential interest rate e↵ects. 3.2 The Equity Term Premium Implied from the Cross-Section of Equities I next use the cross-section of equities to study the cyclicality of the term premium. I first identify a portfolio that mimicks the equity term premium that I observe in the 19962015 sample. I then study the cyclicality of this portfolio in the full sample running from 18 1964 to 2015. Consistent with the previous results, I find that the mimicking portfolio has counter-cyclical abnormal returns. Iuse30characteristics-sortedportfoliosasthefoundationofthemimickingportfolio. Iusecharacteristics-sortedportfoliosratherthanindividualequitiesbecausetheduration of characteristics-sorted portfolios is more stable than the duration of individual stocks.9I use ten portfolios sorted on book-to-market, ten portfolios sorted on profitability, and ten portfolios sorted on investment. The portfolios are based on NYSE breakpoints and returns are equal-weighted. To construct the mimicking portfolio, I first project the equity term premium onto the 30 characteristics-sorted portfolios. I do so by regressing the monthly excess return to these portfolios onto the equity term premium between 1996 and 2015. Before 2005 I use option implied dividend returns, from 2005 to 2009 I use the average of the option implied dividend returns and the dividend futures returns, and after 2009 I use the dividend futures returns.10 Ithenusethesebetastoconstructthemimickingportfolio. Foreachstyle(e.g. bookto-market), I rank the ten portfolios based on the term premium betas. I assign the two portfolios with highest betas to the long-duration portfolio and I assign the two portfolios with the lowest betas to the short-duration portfolio. I then equal weight the six low-beta portfolios into a short-duration portfolio and I equal weight the six high-beta portfolios into a long-duration portfolio. The mimicking portfolio is then long the long-duration portfolio and short the short-duration portfolio. The term premium betas generally line up with expectations. For instance, the literature argues that value stocks have short cash-flow maturity, and, consistent with this, I find that value stocks have low term premium betas and growth stocks have high term premium betas.11 I also find that term premium betas are decreasing in profitability and increasing in investment. The term premium betas are, however, not linearly correlated with characteristics. For instance, the portfolio with highest book-to-market does not have a particularly low term premium beta, which suggests that the characteristics pick up other signals than only duration. 9Indeed, over the life-cycle, stocks may start as growth stocks with long cash flow duration and evolve into value stocks with short cash flows duration. 10Iusethe(mkt,2) premia as the monthly term premium because this term premium is available both for dividend futures and option implied dividend prices. 11It is worth noting, however, that a long cash-flow maturity does not mean that the term premium beta must be high (for instance, Hansen, Heaton, and Li (2008) find that short-maturity value stocks behave like long-maturity claims in the sense that they load highly on long-run consumption shocks). 19 Table 6 reports results on cyclicality of the mimicking portfolio. The third column reports results from a regression of the mimicking portfolio on the ex ante dividend price ratio. The parameter estimate is positive, suggesting that the returns to the mimicking portfolio are counter-cyclical. The e↵ect is, however, statistically insignificant. In the fourth column, I augment the regression with a series of controls. I control for the five Fama and French (2015) factors, the one-year treasury yield, and the treasury yield spread. I control for the five Fama and French factors to ensure that I do not pick up welldocumented cyclicality to one of the other factors. For instance, the mimicking portfolio has a positive beta, and since the market returns are counter-cyclical, one might worry that the counter-cyclical returns simply come from this positive beta. Controlling for the market, and the other factors, mitigates such concerns.12 Because the returns are spot and not forward returns, I also include the treasury yield spread and the short treasury yield to control for potential interest rate e↵ects. As can be seen in the fourth column, the returns to the mimicking portfolio remain counter-cyclical even after including the controls. Including the controls mainly decrease the standard error of the parameter estimate on the dividend price ratio. Accordingly, the parameter estimate is now statistically significant with a t-statistic of 3.67. The parameter estimate is, however, an order of magnitude smaller than when using the equity term premium from options (also Table 6) or when using the dividend futures (Table 2). One reason for this could be that the actual maturity of the short-maturity firms are not as short as the short-maturity claims in Table 2. Indeed, the average firm has a maturity above 20 years, which is substantially higher than the maturities of the dividends futures. In the fifth and sixth columns, I separate the sample into two parts: before and after 1996. Recall that the mimicking portfolio is identified in the 1996-2015 sample, so the returns should be counter-cyclical in this sample almost by construction because the term premium it mimicks is counter-cyclical. As can be seen in the fifth column, the term premium is indeed counter-cyclical in this sample. More interestingly, the mimicking portfolio is also counter-cyclical in the pre 1996 sample. As can be seen in the sixth column, the parameter estimate on the dividend price ratio is the same in the early sample as it is in the full sample, 12Yogo (2006) for instance argues that the value premium can be explained by cyclical properties that are unrelated to duration and the equity term premium. In addition, Asness, Liew, Pedersen, and Thapar (2017) argue that the time-variation in the value premium mostly comes from potentially behavioral drivers, which are also unrelated to duration. More generally, Gormsen and Greenwood (2017) document that most risk factors related to fundamentals have counter-cyclical returns. 20 although the statistical significance is only around half. 3.3 Measurement Error Concerns The research on the equity term structure is based on prices of either option implied dividends or dividend futures, which one might worry are measured with error. One concern in this regard is that potential measurement error will bias returns upwards, as argued by Blume and Stambaugh (1983): when computing returns, one divides end-of-period price with beginning-of-period price, and if there is white noise measurement error in the beginning-of-period price, then the average returns will be biased upwards because the inverse of the price is convex over positive prices. This potential upward bias is a serious concern when working with option implied dividend prices (se e.g. Boguth, Carlson, Fisher, and Simutin (2012)).13 One-month returns on option implied dividends are indeed highly volatile and have negative autocorrelation, which suggest that there might be measurement error in prices. Such measurement error does, however, not influence the main results in this paper. Indeed, while measurement error influence average returns, they do not influence the covariance with the dividend price ratio. The parameter estimate on the dividend price ratio is thus unbiased, even when working with noisy data.14 A second advantage of the method in this paper is related to inference in the relatively short time-series of available data. As pointed out by Merton (1980), estimating average returns requires longer horizons than estimating covariances. The reason is that dividing the sample into shorter parts increases the precision of the estimate of covariances while it generally does not improve the estimate of the average returns. However, this advantage of estimating covariances only partly applies, because one of the variables, the dividend price ratio, is quite persistent, thereby making estimating the covariance more like estimating a mean. 13Schulz (2016) and Song (2016) also underline potential tax and microstructure issues related to the option implied dividend prices. 14To see this, consider a normally distributed measurement error "⇠N(µ, ") in returns such that the observed returns ˆrtis equal to the true return rtplus the measurement error. Assuming the dividend price ratio for the market portfolio is observed correctly, the observed parameter estimate is thus ˆ =cov(rt+1 +✏t+1;dtpt) var(dtpt)(1.20) =+cov(✏t+1;dtpt) var(dtpt)=(1.21) where is the true parameter coefficient in regression (1.18). 21 Finally, the methodology in this paper potentially produces a Stambaugh bias. Stambaugh (1999)showsthatregressioncoefficients are upwards biased when one predicts returns with a persistent predictor that has innovations that are negatively correlated with realized returns. The Stambaugh bias is, however, not as serious in the regressions in this paper as in usual predictive regressions for two reasons. First, realized return di↵erences between longand short-maturity claims are not as strongly linked to innovations in the dividend yield as the realizations of the market portfolio are, because the equity term premia are both long and short an equity claim. Second, the dividend price ratio is much less persistent in this sample compared to the full 1930-2017 U.S. sample.15 Accordingly, I find that the biases are insufficiently small to significantly alter the inference. For the results reported in Table 2, the bias is around 20 percent for m= 1 and it quickly decays to a few percent for m=3 (see Table A4). 4 The Expectations Hypothesis Inextaddresshowthecounter-cyclicaltermpremiainfluencetherelationbetweentheequity yield curve and the future development of equity yields. The benchmark for this relation is the expectations hypothesis. The expectations hypothesis is that equity term premia are constant, and that the future development of yields therefore can be inferred from the equity yield curve. The expectations hypothesis is rejected given that term premia exhibit cyclical variation. However, by studying the expectations hypothesis we can learn how the counter-cyclical equity term premium influences the relation between the equity yield curve and the expected development in yields, and we can learn how term premia are related to the equity yield curve. 4.1 Defining Equity Yields and the Expectations Hypothesis I define the time tequity yield en tfor maturity nas the di↵erence between log-dividends dt at time tand the log-forward price, fn t, of the time t+n dividends: en t=1 n(dtfn t) (1.22) 15Of course, a persistent process always looks less persistent in a subsample than in the full sample (Kendall,1954). However, the dividend price ratio is less persistent in this subsample even when compared to the full sample mean. 22 where nis the maturity of the dividend claim. To understand the information content in equity yields, note that equity yields can be written as the average of future returns and future growth rates: en t=1 n(dtfn t)=1 n n X i=1 rn+1i t+i1 n n X i=1 gt+i(1.23) where gt+1 is the log growth rate on dividends between period tand t+1. Idonotempirically decompose the equity yields into expected growth rates and returns. It is possible to test the expectations hypothesis and study its implications for equity term premia without decomposing yields into growth rates and returns, and I prefer to do so to avoid the uncertainty arising from such an empirical decomposition. To motivate the expectations hypothesis, note that the yield of an nmaturity claim can be decomposed into future short yields and future term premia by rewriting (1.23): en t=1 n n1 X i=0 Et⇥e1 t+i⇤+1 n n1 X i=0 Et⇥rni t+1+ir1 t+1+i⇤(1.24) The expression in (1.24) underlines the intuition in the expectations hypothesis: if term premia are constant, the variation in the long yield only comes from variation in the expected future short yields, and the long yield therefore summarizes these expectations. Before presenting the next Proposition that summarizes the testable implications of the expectations hypothesis, I define the equity yield spread sn,m t=en tem t. Proposition 3 (The expectations hypothesis). If equity term premia are constant, i.e. there exist constants cn,m such that Et[rn t+1] Et[rm t+1]=cn,m for all m, n, then the following holds: (a) The regression coefficient is n,m 1=1in enm t+men t=n,m 0+n,m 1sn,m t m nm+✏t+m(1.25) (b) The regression coefficient is n,m 1=1in k1 X i=1 ✓1i n◆(em t+im em t+(i1)m)=n,m 0+n,m 1sn,m t+⌘t+n(1.26) 23 6 Testing Asset Pricing Models: Theory vs. Stylized Facts In this section, I relate my empirical findings to canonical asset pricing models individually. Icalculate,eitheranalyticallyorthroughsimulations,theparametersfromtheregressions in the empirical analysis and compare these theoretical parameters to those observed empirically. The results are summarized in Table 10. The main challenge for the models is, as mentioned, that they cannot produce an equity term premium that is both negative on average and counter-cyclical. Rather, if the equity term premium is positive on average, it is counter-cyclical; if the term premium is negative on average, it is pro-cyclical. In addition, none of the models are able to get the test of the expectations hypothesis right: none of them have parameter estimates that are higher than one. The parameter estimates are too low because the equity term premia are positively related to the yield spread, not negatively as in the data. While much of the paper has focused on the qualitative results – that is, whether the sign on the cyclicality is correct – the results in Table 10 highlight another problem for the models: none of the models are quantitatively close to matching the observed time variation. Indeed, the Campbell and Cochrane (1999)modelandtheBansal and Yaron (2004) model both have counter-cyclical slopes, but the cyclicality is not sufficiently strong. Empirically, the parameter estimates for the regression of the (Mkt,2) premium on the dividend price ratio, Mkt,2 1, is around 0.35. However, in the habit and the long-run risks model, the estimate of Mkt,2 1is only around 0.1and0.03. In the end of this section, I propose a model that addresses the main challenge to existing models, but before doing so I address the canonical models individually. 6.1 The Habit Model by Campbell and Cochrane (1999) In the habit model by Campbell and Cochrane (1999), the term structure arises because the shortand long-maturity claims are di↵erently exposed to discount rate risk. In the habit model, discount rate risk requires a premium because discount rate shocks are conditionally perfectly negatively correlated with consumption. The negative correlation arises because a negative shock to consumption increases risk aversion and therefore the required rate of return. 30 The dynamics of the habit model are largely captured by setting a= 0 in the model from the theoretical section, meaning that Proposition 1.a applies to the model. Given the fact that the term structure is upward sloping on average, it is therefore also counter-cyclical. The economic intuition is that, in bad times, the higher price of risk causes investors to increase the required compensation for the discount rate risk inherent in long-maturity dividends, thereby making the term structure more upward sloping. This economic intuition is confirmed in simulation studies. As can be seen in Table 10, the paramamter estimate for Mkt,2in simulation studies is 0.12. This estimate is positive, as is the empirically observed value. While the parameter estimate is well below the empirical estimate, it is still large in absolute terms. Indeed, the model is calibrated to have a standard deviation of the dividend price ratio of 0.26, which means that a one standard deviation change in the dividend price ratio changes the term premium by around 3 percentage points. Finally, the model has the wrong sign on the parameter estimates in the expectations hypothesis. Both and are negative, which is evidence that equity yields spread positively predicts equity term premia. 6.2 The Long-Run Risk Model by Bansal and Yaron (2004) I analyze the long-run risk model by Bansal and Yaron (2004). Alternatively, using Bansal, Kiku, and Yaron (2012) does not fundamentally change the results. The long-run risk model has a non-degenerate treasury term structure, and I therefore subtract the corresponding bond return to get forward returns (see Binsbergen and Koijen (2017)foradecomposition of spot returns into forward and bond returns). The Bansal and Yaron model has an upward sloping equity term structure. The term structure arises because long-maturity dividends are more exposed to the long-run dividend growth risk and discount rate risk. Investors are averse towards both shocks, and longmaturity claims therefore require a premium to compensate for the additional discount rate and dividend growth rate risk. The long-run risk model is captured by setting a<0 in the model from the theory section, meaning that Proposition 1.b applies to the long-run risk model. As such, the model could in principle have a positive and pro-cyclical equity term premium, but the model’s parameters imply that it has a counter-cyclical equity term premium. In the model, periods with a high dividend price ratio are generally periods with a high price of risk, and this high 31 price of risk causes investors to require a higher premium on the long-run risk inherent in long-maturity dividends. Again, the counter-cyclical term structure is confirmed in simulations. Table 10 shows that the parameter estimate for Mkt,2is 0.03. The parameter estimate is again well below the empirical estimate, but it is economically large. The reason is lower for long-run risk than for the habit model is not that the term structure dynamics are less volatile for the long-run risk model, but rather that the dividend price ratio of the market portfolio is not determined solely by the price of risk; rather, in the long-run risk model, the dividend price ratio may be low because long-run growth rates are high, and these growth rates do not influence the term structure of expected returns.17 Finally, the long-run risk model also has the wrong signs for and . The negative signs suggest that the equity yield spread positively predicts equity premia, not negatively as in the data. 6.3 The Model by Lettau and Wachter (2007) The model by Lettau and Wachter (2007)hasadownwardslopingtermstructureofexpected returns and as such it has so far been the most successful model in explaining the equity term structure. The model has a downward sloping term structure because a negative shock to dividends causes a positive shock to dividend growth rates. This long-run insurance makes long-maturity dividends less risky than short-maturity dividends, because a negative shock to dividends over time is canceled out by a higher growth rate. The Lettau and Wachter model is captured by setting a= 0 in the theory section, meaning that Proposition 1.a applies to the model. Given that the equity term structure is downward sloping on average, it is therefore also pro-cyclical. When the dividend price ratio is low, the price of risk is on average high and investors therefore require a higher premium for holding the risky short-maturity dividends. The term premium thus increases in absolute size which is to say that it becomes more negative and the slope becomes more downward sloping. Consistent with this intuition, the value of Mkt,2is 0.08 in the model. With regards to and from the test of the expectations hypothesis, the model by Lettau and Wachter (2007) does better than the habit model and the long-run risk model. The reason is that, unlike for the two latter, the term premium and the expected changes in the yields have the same impact on the yield curve in the Lettau and Wachter (2007) 17See Beeler and Capmbell (2012) for a discussion of the relationship between the dividend price ratio and future dividend growth in the long-run risk model. 32 model. Accordingly, both and are positive. 6.4 The Disaster Model by Gabaix (2012) The spot term structure of equity returns is flat and constant in the disaster model by Gabaix (2012). The equity term structure is flat because a disaster hits all equity claims similarly. The treasury term structure is, however, upward sloping because long-maturity bonds are exposed to inflation risk. Accordingly, the forward term premium on equity is downward sloping and the slope varies over time as the slope of the bond term structure varies. For the Gabaix model, I report in Table 10 the results for the spot equity term structure instead of the results for the forward equity term structure. I do so because the spot dynamics are more representative of the risk-dynamics in the Gabaix model and because the empirical results for ,,andqualitatively are the same irrespectively of whether I consider spot or forward prices on dividend strips. Under the spot dynamics, the term structure of equities is constant and flat, which implies that the expectations hypothesis holds and and are equal to 1. 6.5 Reconciling the Facts: A Model with Negative and Counter-Cyclical Term Premia In this section, I calibrate the earlier model such that it can explain the two most important stylized facts, i.e. that equity term premia are negative on average and counter-cyclical. The model does so by having two competing drivers of term premia: long-run risk in dividend growth and a demand for hedging investment opportunities. Investors dislike long-maturity stocks because they are exposed to long-run risk in dividend growth, but they also like them because they hedge deteriorations in investment opportunities. The strength of these two competing forces varies over time in such a way that term premia are negative on average but counter-cyclical. More concretely, I assume that a>0, ⇢dx =0,and⇢dz >0. The assumptions imply that price-of-risk shocks are uncorrelated with dividend shocks and enter the stochastic discount factor negatively. In addition, the positive correlation between shock to dividend and growth rates imply that there is long-run risk in dividends. The positive value of aimplies a high demand for hedging deteriorations in investment 33 opportunities. Indeed, the high ameans that states where the price of risk drops are bad states where the stochastic discount factor is high. Intuitively, investors consider the drop in the price of risk a bad thing because their investment opportunities deteriorate. Investors are therefore willing to accept lower return on assets that hedge such deteriorations in investment opportunities. Long-maturity assets are such assets, because these realize a large capital gain when the price of risk drops. A demand for hedging deteriorations in investment opportunities features in many models in financial economics.18 The idea in these models is exactly that investors find losses that occur due to discount rate shocks less unpleasant than losses that occur due to cash flow shocks. They do so because losses that occur due to discount rate shocks are partly o↵set by higher future expected returns. However, these models rarely have a positive value of a.Indeed,whilelossesthatoccurduetodiscountrateshocksarelessunpleasantthan losses that occur due to cash flow shocks, they are nonetheless still unpleasant. Accordingly, ais negative, but less so than the coefficient in front of cash flow shocks. Indeed, Campbell (1993) shows that Epstein-Zin investors generally dislike discount rate shocks, although they dislikes them less than cash flow shocks if they have a risk aversion paramter >1. Consistent with this insight, shocks to the price of risk (or, equivalently, the conditional variance of dividends), enter the stochastic discount factor of Bansal and Yaron (2004)scaledby a constant, but the constant is positive which is to say that a<0 in the long-run risk model. The positive ais also difficult to align with the habit model because its price of risk is conditionally perfectly negatively correlated with consumption. Indeed, Santos and Veronesi (2010) show that habit models imply a high premium on discount rate risk. Apositiveacould, however, arise from underfunded pension funds. Indeed, if an underfunded pension fund plans to hold its assets to maturity, the fund is not worried about losses that occur due to discount rate shocks. However, when the discount rate goes up, the underfunded pension fund needs less additional funding to become fully funded because the additional funding o↵ers a higher return. Accordingly, states where the price of risk goes up are considered good states for underfunded pension funds, which means that acould be positive. Given the positive value of a, and the additional assumptions above, the term premia 18See Merton (1973); Campbell and Vuolteenaho (2004) for the original ICAPM and Bansal, Kiku, Shaliastovich, and Yaron (2014); Campbell, Giglio, Polk, and Turley (2017) for an ICAPM with stochastic volatility. 34 are given by: ✓n,1 t=aBn1 xx | {z } negative (demand for hedging) +Bn1 z⇢dzz | {z }xt positive (long-run risk in dividends) (1.30) The first term in the term premium reflects that investors are willing to accept a lower return on long-maturity equity because it hedges deteriorations in investment opportunities. The second term reflects that investors want a higher return on long-maturity equity because it is more exposed to long-run risk. The net e↵ect can be either positive or negative depending on the specification of the model. In the term premium, the e↵ect of the demand for hedging is constant but the e↵ect of long-run risk depends on the price of risk: when the price of risk increases, the compensation demanded for long-run risk increases as well. Accordingly, the equity term premium is counter-cyclical. I calibrate the model to fit the 1996-2016 sample where I have data on dividend prices. In this sample, the annual dividend growth rate is 3.57 percent per year with an annualized quarterly standard deviation of 0.05. I use those inputs and specify the following quarterly variables to fit the data: x=0.022, z=0.001, ⇢dz =0.16, 'x='z=0.65, ¯x=0.85, and a=0.2. The moments of the model are summarized in Table 11. Qualitatively, the model has the right regression coefficients in the three tests covered in this paper: it has a counter-cyclical term structure and it has regression coefficients in the tests of the expectations hypothesis above 1. In addition, the equity term premium is negative on average. The results of the model is summarized in Figure 7. 7 Conclusion I document a new stylized fact about the equity term structure, namely that the equity term premium is counter-cyclical. The result is robust: (1) it holds in the post-2005 sample across four di↵erent indexes when measuring the equity term premia using dividend futures; (2) it holds when excluding the financial crisis from the sample; (3) it holds in the U.S. sample from 1996 when using equity term premia implied from options prices; and (4) it holds in the U.S. from 1963 when using the cross-section of equities to measure equity term premia. In addition, the variation in the equity term premia is large: in the post 2005 sample, the 35 equity term premium varies with nine percentage points between good and bad times. A series of recent studies documents the importance of short-maturity risks in understanding average risk premia in equities, bonds, variance-swaps, housing markets, and currencies (Binsbergen and Koijen,2017;Du↵ee,2011;Dew-Becker, Giglio, Le, and Rodriguez, 2017;Giglio, Maggiori, and Stroebel,2015;Giglio, Maggiori, Stroebel, and Weber,2015; Lustig, Stathopoulos, and Verdelhan,2013). Given these previous studies, the countercyclical equity term structure is surprising because it suggests that long-maturity risks are the main drivers of variation in risk premia. Accordingly, the short-maturity risk appears important for explaining average risk premia, but long-maturity risk appears the most important for explaining the variation in these risk premia, at least for equities. This pattern is puzzling. Indeed, none of the standard asset pricing models that I study can generate this pattern, i.e. a term premium that is negative on average and counter-cyclical: the recent downward sloping models have pro-cyclical term premia; the traditional upward sloping models have counter-cyclical term premia. I present a new model as a potential solution to the puzzle. In my model, investors trade o↵a demand for hedging investment opportunities with an aversion towards long-run risk in dividend growth. In good times, investors require a lower return on long-maturity dividends than on short-maturity dividends because they want to hedge deteriorations in investment opportunities. In bad times, investors require a higher return on long-maturity dividends than on short-maturity dividends because they are averse to the increased long-run risk in dividend growth. The model is, however, not grounded in a utility function, and doing so remains an interesting avenue for future research. In addition to having strong implications for macro-finance models, the counter-cyclical term structure also has real e↵ects. The equity term structure influences the di↵erence in cost of capital between firms with di↵erent cash-flow maturity. In bad times, when the equity term premia are higher, long-maturity firms find capital relatively more expensive than short-maturity firms and therefore use relatively more labor and less capital. In this sense, the equity term structure has important consequences for individual firms and for workers in di↵erent industries. 8Proofs Proof of proposition 1.a 36 Note first that the price-dividend ratio of the market portfolio is given by the following sum: Pt Dt = 1 X i=1 Pi t Dt = 1 X i=1 exp Ai+Bi zzt+Bi xxt(1.31) where Ptis the price of the market portfolio at time t. We first establish that the price dividend ratio and the negative of the price of risk, xt, are positively quadrant dependent (Lehmann,1966)andhavepositivecovariance. Tosee this, define first the random variable yt=xtand note that the dividend price ratio Pt/Dt is an increasing, monotone function of ztand yt,i.e. Pt Dt =f(zt,y t) where f:R2!Ris nondecreasing in both ztand yt. Similarly, define the function g(yt)= xtwhere g:R!Ris nondecreasing in yt.Becauseztand ytare independent, f(zt,y t) and g(yt)havepositivecovarianceandthetwoarepositivelyquadrantdependent. The positive quadrant dependence means that the positive covariance between the two functions caries over through monotonic transformations of the two variables (see e.g. Oliveira,2012). Accordingly, we can write cov(P/Dt;xt)=cov ✓f(zt,y t); g(yt)◆>0 (1.32) and cov(ptdt;xt)=cov ✓ln f(zt,y t);g(yt)◆>0 (1.33) The sign of the covariance between the dividend price ratio and the term premium is therefore determined as sign cov(dtpt;✓n,1 t)= sign (Bn1 x⇢dxx+Bn1 z⇢dzz)(1.34) Next, the average term premium is given by: E[✓n,1 t]=(Bn1 x⇢dxx+Bn1 z⇢dzz)¯x(1.35) 37 Because ¯xis positive, the sign is given by sign E[✓n,1 t]= sign (Bn1 x⇢dxx+Bn1 z⇢dzz)(1.36) which proves the proposition. Proof of proposition 1.b The asset pricing model in this paper is an example where a6= 0 and sign E[✓n,1 t]6= sign cov(dtpt;✓n,1 t)which proves the proposition. Proof of proposition 2.a Note first that taking the natural logarithm of (1.16)and(1.17) and subtracting (1.17) from (1.16) gives the following expression for the capital to labor ratio: ln(kn t)=ln(Et[Rn t+1]) + ln w+lnln (1.37) Subtracting (1.37)foranm-maturity claim from (1.37)forann-maturity claim gives ln(kn t)ln(km t)=ln(Et[Rn t+1]Et[Rm t+1]) Proof of proposition 2.b Note first that EtMt+1 Pn1 t+1 Dt+1 Ft(Kn t,L n t)= exp( ˜ An+(Bn x+d)xt+Bn zzt)Ft(Kn t,L n t) (1.38) where ˜ Anis a constant. Inserting this expression in (1.16)and(1.17), taking the natural logarithm, and solving for Kn tgives (supressing constants): ln(Kn t)= 1 1↵✓Bn zzt+(Bn x+d)xt+(↵1) ln Et[Rn t+1]◆(1.39) Subtracting a one-period claim from the expression in (1.39) gives the expression in proposition 2.b. 38 39 Table 1 The Equity Term Structure: Stylized Facts versus Theory The equity term premium 𝐸[𝑇𝑃]=𝐸[𝑟  −𝑟 ] is the conditional expected annual return to long-maturity equity minus the annual return to short maturity equity. The cyclicality of the equity term premium is measured by a linear projection of the realized term premium on the ex ante dividend price ratio of the market portfolio. The expectations hypothesis refers to the relation between the equity yield spread and future equity yields. The hypothesis is evaluated in a regression of future yield changes on the yields spread multiplied by a maturity modification. A parameter estimate of one implies that the yields develop exactly as the yield curve suggests under the expectations hypothesis. The habit model refers to the Campbell & Cochrane (1999) model. The long-run risk model refers to the Bansal & Yaron (2004) model. Average slope Cyclicality Expectations Hypothesis Paper (van Binsbergen, Brandt, Koijen, 2012) (this paper) (this paper) Data Measured as 𝐸 [ 𝑇𝑃 ] = 𝐸 [ 𝑟     − 𝑟       ] 𝑇 𝑃  = 𝛽  + 𝛽  𝐷  / 𝑃  + 𝑒  ∆ yield    = 𝛾  + 𝛾  yield spread  + ϵ    Result Downward sloping 𝐸 [ 𝑇𝑃 ] < 0 Counter-cyclical 𝛽  > 0 Yield spread predicts change in yield 𝛾  > 1 Theories Habit Upward Counter-cyclical 𝛾  < 0 Long-run risk Upward Counter-cyclical 𝛾  < 0 Lettau Wacther (2007) Downward Pro-cyclical 𝛾  ∈ ( 0 ; 1 ) Gabaix (2012) Flat Constant 𝛾  = 1 Hasler Marfe (2016) Downward Pro-cyclical 𝛾  ∈ ( 0 ; 1 ) My model Downward Counter-cyclical 𝛾  > 1 46 Panel D: FTSE 100 m=1 4.02 4.32 4.64 4.60 4.43 4.12 3.20 (3.81) (3.80) (3.81) (3.88) (4.03) (3.83) (3.85) m=2 2.17 2.11 1.75 1.57 1.25 0.88 (1.47) (1.27) (1.01) (0.99) (0.78) (0.74) m=3 0.57 0.17 -0.05 -0.30 0.22 (0.38) (0.11) (-0.04) (-0.22) (0.24) m=mean(1-7) 0.77 (0.99) 47 Table 6 Counter-Cyclical Equity Term Premia: Alternative Measures This table shows the relation between term premia and the dividend price ratio of the market portfolio. The term premia are measured in two different ways: (1) as the implied term premium from equity options (van Binsbergen, Brandt, and Koijen, 2012) and (2) as the return to a term premium mimicking portfolio from the cross-section of equities. The term premium mimicking portfolio is long a portfolio with long duration firms and short a portfolio with short duration firms. To construct the long and short duration portfolios, I run a regression of the excess return to 30 portfolios sorted on book-to-market, profitability, and investment onto the realized return difference between longand short-maturity equity claims between 1996 and 2015. For each style, I rank portfolios in ascending order based on their beta with respect to longminus short-maturity return difference. I assign the two portfolios with highest (lowest) beta to the long (short) duration portfolio. Within the long (short) duration portfolio I equal weight the excess return. I include as independent variables the ex ante dividend price ratio of the market portfolio, the five Fama and French (20015) factors, the ex ante one-year treasury yield, and the ex ante treasury yield spread (five-year yield in excess of one-year yield). The yields are from Fama and Bliss (1987). All returns are measured in rolling one-year log returns. I report t-statistics below the parameter estimates. The t-statistics are based on Newey and West (1987) standard errors corrected for 18 lags. The results are from the U.S. Panel A: Alternative measures of the equity term premium Option implied term premium Cross-sectional term premium Period 1996-2009 1996-2009 1963-2015 1963-2015 1996-2015 1963-1996 d t - p t 1.04 0.90 0.04 0.07 0.13 0.07 (2.55) (5.93) (0.92) (3.68) (1.54) (1.68) Mkt 0.30 0.10 -0.03 0.11 (0.91) (3.23) (-0.51) (3.09) SMB -0.65 0.25 0.23 0.25 (-2.41) (7.09) (2.85) (4.83) HML -0.45 -0.34 -0.20 -0.38 (-1.32) (-7.41) (-3.14) (-7.03) RMW -0.79 -0.59 -0.89 -0.43 (-1.44) (-9.24) (-9.46) (-5.01) CMA 0.26 -0.09 -0.03 -0.05 (0.55) (-1.13) (-0.20) (-0.51) Bond yield 0.05 -0.01 0.00 -0.01 (1.31) (-2.92) (-0.19) (-3.76) Bond yield spread 0.10 -0.02 -0.01 -0.02 (1.28) (-3.40) (-0.86) (-3.00) 48 Panel B: Alternative measures of cyclicality Option implied term premium Cross-sectional term premium Period 1996-2009 1996-2009 1996-2009 1996-2009 1963-2015 1963-2015 CAPEt 0.59 0.96 0.05 (1.46) (5.17) (2.42) cay t 3.07 -2.78 0.81 (0.61) (-0.61) (1.99) Mkt 0.30 0.98 0.10 0.10 (0.88) (1.78) (3.02) (3.28) SMB -0.73 -1.14 0.26 0.18 (-2.51) (-1.64) (7.09) (2.37) HML -0.64 -0.81 -0.32 -0.13 (-2.35) (-1.50) (-6.48) (-1.12) RMW -0.77 0.22 -0.62 -0.08 (-1.43) (0.48) (-9.66) (-0.55) CMA 0.15 0.69 -0.15 0.13 (0.37) (0.55) (-1.74) (0.72) Bond yield 0.11 -0.03 -0.01 0.00 (2.11) (-0.52) (-2.01) (-0.96) Bond yield spread 0.13 0.04 -0.02 -0.02 (1.39) (0.33) (-3.09) (-1.74) 49 Table 7 The Expectations Hypothesis: The Yield Spread and Long-Yield Changes This table shows the relation between long-yield changes and the equity yield spread. The table reports the parameter estimate from the following regression: 𝑒  −𝑒 =𝛾 , +𝛾 ,𝑠 , 𝑚 𝑛−𝑚+𝜖 The expectations hypothesis is that 𝛾 , =1 for all maturity pairs n,m. The t-statistics for this hypothesis are reported below the parameter estimates. The t-statistics are based on Newey and West (1987) standard errors corrected for 1.5m lags. The maturities n and m are both measured in years. The sample is from 2005 to 2016. Maturity of long-maturity claim (n) 2 3 4 5 6 7 Panel A: S&P 500 m=1 1.00 1.08 1.18 1.27 1.36 1.38 (-0.00) (0.17) (0.46) (0.70) (0.93) (1.07) m=2 1.61 1.77 1.87 1.95 2.04 (2.08) (2.16) (2.51) (2.91) (3.32) m=3 1.73 1.94 2.09 2.15 (3.04) (3.25) (3.09) (3.29) Panel B: Nikkei 225 m=1 1.62 1.59 1.30 1.17 1.07 0.99 (1.02) (0.92) (0.53) (0.32) (0.14) (-0.02) m=2 1.05 0.97 0.85 0.79 0.74 (0.11) (-0.07) (-0.33) (-0.46) (-0.59) m=3 1.23 1.29 1.19 1.14 (0.34) (0.37) (0.26) (0.19) Panel C: EuroStoxx 50 m=1 2.72 3.86 3.10 2.79 2.62 2.53 (2.03) (5.37) (5.91) (5.92) (5.76) (5.58) m=2 1.61 1.58 1.46 1.36 1.33 (2.80) (2.05) (1.78) (1.59) (1.52) m=3 1.04 1.13 1.01 0.93 (0.09) (0.25) (0.03) (-0.19) Panel D: FTSE 100 m=1 -1.20 1.71 2.04 2.03 1.96 1.88 (-3.50) (0.85) (1.55) (2.05) (2.41) (2.56) m=2 1.56 1.71 1.59 1.52 1.47 (1.90) (1.74) (1.70) (1.68) (1.66) m=3 1.26 1.19 1.03 0.95 (1.06) (0.58) (0.10) (-0.17) 50 Table 8 The Expectations Hypothesis: The Yield Spread and Short-Yield Changes This table shows the relation between short-yield changes and the equity yield spread. The table reports the parameter estimate from the following regression: 1−𝑖 𝑛𝑒 −𝑒()    =𝜙 , +𝜙 ,𝑠 , +𝜂 The expectations hypothesis is that 𝜙 , =1 for all maturity pairs n,m. The t-statistics for this hypothesis are reported below the parameter estimates. The t-statistics are based on Newey and West (1987) standard errors corrected for 1.5m lags. The maturities n and m are both measured in years. The sample is from 2005 to 2016. Maturity of long-maturity claim (n) 2 3 4 5 6 Panel A: S&P 500 m=1 1.00 1.23 1.23 1.25 1.26 (-0.00) (1.41) (1.98) (2.58) (2.02) m=2 1.38 1.37 (2.24) (3.60) Panel B: Nikkei 225 m=1 1.31 1.15 0.98 1.06 1.07 (0.92) (0.70) (-0.10) (0.35) (0.37) m=2 0.98 1.13 (-0.07) (0.63) Panel C: EuroStoxx 50 m=1 1.86 2.07 1.49 1.30 0.72 (3.07) (6.18) (3.81) (3.55) (-3.82) m=2 1.29 1.12 (2.66) (2.09) Panel D: FTSE 100 m=1 -0.10 1.43 1.37 1.34 1.17 (-4.58) (1.45) (1.84) (2.68) (1.69) m=2 1.36 1.20 (1.79) (2.07) 51 Table 9 Cyclicality of the Investment Duration This table reports the results from a regression of investment characteristics on the dividend price ratio of the market portfolio plus controls. I consider the investment rates for three portfolios: a portfolio with short-duration firms, a portfolio with long-duration firms, and a long/short portfolio. To construct the longand short-duration portfolio, I run a regression of the excess return to 30 portfolios sorted on book-to-market, profitability, and investment onto the realized return difference between longand short-maturity equity claims between 1996 and 2015. For each style, I rank portfolios in ascending order based on their beta with respect to longminus short-maturity return difference. I assign the two portfolios with highest (lowest) beta to the long (short) duration portfolio. Within the long (short) duration portfolio I equal-weight the characteristics. Capital to labor is measured as capital expenditures to total salary. Investment is measured as capital expenditure to plant, property, and equipment. Change in Capex is the change in capital expenditure. R&D is the annual change in salary spend on R&D activities. Total salary expenses is the annual change in total salaries. All characteristics are measured in cross-sectional percentiles. In the regressions, I use as the independent variables the ex ante dividend price ratio of the market portfolio, the one-year treasury yield, the treasury yield spread (five-year in excess of one-year), and a dummy variable for NBER recessions. I report tstatistics below the parameter estimates. The t-statistics are based on Newey and West (1987) standard errors corrected for 18 lags. The sample is U.S. equities from 1963-2015. Panel A: Capital to labor Portfolio Short duration Long duration Longminus short duration Period 1963-2015 1963-2015 1963-2015 1963-2015 1996-2015 19631996 dt - pt 2.85 -4.03 -6.88 -6.86 -0.54 -4.08 (10.93) (-9.09) (-10.56) (-5.50) (-2.64) (-1.76) Bond yieldt 0.01 -0.31 0.38 (0.04) (-8.99) (1.96) Bond yield spreadt 0.91 -0.57 1.38 (1.52) (-4.27) (2.92) Recession dummyt 1.35 0.22 0.20 (1.70) (1.74) (0.23) Panel B: Detrended capital to labor Capital to labor (HP filter) Capital to labor (first difference) Portfolio Long-short Long-short Long-short Long-short Long-short Long-short Period 1963-2015 1996-2015 1963-2015 1963-2015 1996-2015 1963-1996 d t - p t (HP filter) -2.00 -0.45 -3.78 (-2.02) (-1.70) (-1.92) d t - p t (first differenced) -1.14 -0.97 -1.33 (-0.95) (-2.18) (-0.60) Bond yield t 0.02 -0.08 0.03 0.05 -0.04 0.14 (0.67) (-1.58) (0.39) (1.36) (-0.62) (1.47) Bond yield spread t -0.16 -0.26 -0.26 0.10 -0.24 0.21 (-0.93) (-1.63) (-1.12) (0.49) (-1.60) (0.73) Recession dummy t 0.35 -0.01 0.51 0.21 0.21 0.11 (1.13) (-0.03) (1.16) (0.44) (0.83) (0.16) 52 Panel C: Investment Investment measure Investment (Capex to PPE) Change in CapEx Portfolio Short duration Long duration Longshort Short duration Long duration Longshort Period 1963-2015 1963-2015 1963-2015 1963-2015 1963-2015 19632015 d t - p t 1.84 -3.48 -9.13 2.54 -4.17 -10.14 (3.27) (-4.13) (-4.16) (4.86) (-4.85) (-4.51) Bond yieldt 0.74 0.66 (2.12) (1.91) Bond yield spreadt 1.68 1.10 (1.87) (1.11) Recession dummyt 2.58 1.90 (2.64) (1.57) Panel D: R&D and Salary Characteristic: R&D Total Salary Expenses Portfolio Short duration Long duration Longshort Short duration Long duration Longshort Period 1963-2015 1963-2015 1963-2015 1963-2015 1963-2015 1963-2015 d t - p t 2.74 -5.85 -9.26 0.07 0.83 0.82 (6.37) (-9.96) (-6.12) (0.22) (2.15) (1.21) Bond yieldt 0.10 0.06 (0.40) (0.53) Bond yield spreadt 0.67 0.77 (1.01) (2.60) Recession dummyt 1.88 -0.38 (1.77) (-0.75) 53 Table 10 Asset Pricing Theories versus Stylized Facts This table shows the result of simulations of different asset pricing models. I simulate the models and estimate the following regressions: 𝑟; −𝑟; =𝛽 , +𝛽 ,(𝑑−𝑝)+𝜖, 𝑒  −𝑒 =𝛾 , +𝛾 ,𝑠 , 𝑚 𝑛−𝑚+𝜖 1−𝑖 𝑛𝑒 −𝑒()    =𝜙 , +𝜙 ,𝑠 , +𝜂 where 𝑟  is the forward log-return on the n maturity claim between month t and t+12, 𝑑−𝑝 is the log dividend price ratio of the market portfolio, 𝑒  is the yield on the n maturity dividend at time t, and 𝑦, is the yield spread between the n and the m maturity dividend at time t. Maturities n and m are measured in years. The habit model is the Campbell and Cochrane (1999) model. The model is simulated using the series method of Wachter (2005). The long-run risk model is the model by Bansal and Yaron (2004) that features stochastic volatility. 𝐸 [ 𝑟     − 𝑟     ] 𝛽   ,  𝛾   ,  𝜙   ,  Data Empirical observation -0.035 0.35 1.27 1.25 Theories Habit 0.038 0.12 -1.32 -0.80 Long-run risk 0.029 0.03 -0.95 -0.03 Lettau & Wacther (2007) -0.067 -0.08 0.32 0.72 Disaster (Gabaix, 2012) 0 0 1 1 My model -0.01 0.10 2.4 1.6 54 Table 11 Reconciling Theory with the Empirical Facts: Simulated Results in an Asset Pricing Model This table shows results of simulations in my model. The results are based on 100,000 years of artificial data. The model is simulated to fit the U.S. data in the 1996-2016 period. The model is simulated on the quarterly horizon. Expected returns and standard deviations are annualized (multiplied by 4 and 2). Data Model Stock market moments 𝐸 [ 𝑅     − 𝑅  ] 0.07 0.07 𝜎 [ 𝑅     − 𝑅  ] 0.18 0.23 𝐸 [ 𝑃  / 𝐷  ] 57.3 49.7 𝜎 ( 𝑝  − 𝑑  ) 0.22 0.28 AR1( 𝑝  − 𝑑  ) 0.92 0.90 Term structure of equity results 𝐸 [ 𝑟     − 𝑟     ] -0.035 -0.01 𝛽   ,  0.35 0.10 𝛾   ,  1.27 2.4 𝜙   ,  1.25 1.6 55 Figure 1 The Term Structure of One-Year Equity Returns This figure shows the term structure of holding-period equity returns for the S&P 500. The figure shows the unconditional average return (solid line), the average return in bad times (dashed line), and the average return in good times (dash-dotted line). Good and bad times are defined by the ex ante dividend price ratio. Short-maturity equity claims is the average return to dividend futures of 1 to 7 years maturity. The long-maturity claim is the average return to the market portfolio. Returns are annual spot returns, 2005 – 2016. -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Short-maturity claims Long-maturity claims Annual return Average Bad times Good times 62 Table A1 Summary Statistics on Dividend Futures This table provides summary statistics for dividend futures. Returns and standard deviations are for annualized claims. The loading on the dividend price ratio refers to the regression coefficient in a regression of realized excess returns on the ex ante log dividend price ratio of the market portfolio. Good and bad times are based on the median dividend price ratio. Maturity of dividend claim (years) 1 2 3 4 5 6 7 Mkt Panel A: S&P 500 Average futures returns 0.015 0.012 0.021 0.021 0.024 0.028 0.037 -0.024 Variance annual returns 0.002 0.017 0.027 0.031 0.038 0.044 0.048 0.077 Variance quarterly returns 0.006 0.023 0.025 0.030 0.037 0.041 0.046 0.080 Loading on d  − p  0.059 0.275 0.328 0.404 0.489 0.548 0.575 0.677 Average good times 0.017 -0.006 0.010 0.002 0.000 0.000 0.012 -0.080 Average bad times 0.013 0.029 0.034 0.045 0.053 0.062 0.067 0.041 Average yields -0.038 -0.044 -0.044 -0.042 -0.041 -0.038 -0.035 Standard dev. of yield 0.115 0.077 0.052 0.042 0.036 0.033 0.029 Panel B: Nikkei 225 Average futures returns 0.109 0.071 0.048 0.077 0.081 0.084 0.089 -0.040 Variance annual returns 0.006 0.071 0.117 0.135 0.150 0.161 0.170 0.121 Variance quarterly returns 0.018 0.084 0.107 0.113 0.119 0.121 0.123 0.117 Loading on d  − p  0.088 0.611 0.855 0.953 1.015 1.055 1.086 1.002 Average good times 0.055 -0.051 -0.083 -0.011 -0.004 0.004 0.017 -0.097 Average bad times 0.13 0.15 0.16 0.16 0.17 0.16 0.16 0.02 Average yields 0.020 0.004 -0.013 -0.017 -0.016 -0.014 -0.012 Standard dev. of yield 0.160 0.131 0.097 0.077 0.065 0.056 0.048 Panel C: Euro Stoxx 50 Average futures returns 0.033 0.044 0.019 0.004 -0.003 -0.007 -0.007 -0.062 Variance annual returns 0.005 0.044 0.091 0.100 0.100 0.099 0.099 0.099 Variance quarterly returns 0.010 0.063 0.080 0.083 0.084 0.082 0.084 0.090 Loading on d  − p  0.096 0.500 0.605 0.604 0.587 0.570 0.549 0.560 Average good times -0.004 -0.015 -0.035 -0.047 -0.050 -0.050 -0.045 -0.120 Average bad times 0.033 0.126 0.110 0.089 0.075 0.064 0.056 0.035 Average yields 0.056 0.046 0.031 0.025 0.020 0.015 0.012 Standard dev. of yield 0.128 0.124 0.086 0.066 0.053 0.045 0.038 Panel D: FTSE 100 Average futures returns 0.018 0.038 0.027 0.020 0.017 0.017 0.020 -0.035 Variance annual returns 0.005 0.030 0.064 0.072 0.073 0.071 0.070 0.063 Variance quarterly returns 0.009 0.043 0.054 0.058 0.059 0.060 0.061 0.062 Loading on d  − p  0.151 0.763 0.853 0.876 0.854 0.823 0.787 0.917 Average good times -0.012 -0.018 -0.013 -0.023 -0.026 -0.023 -0.018 -0.089 Average bad times 0.069 0.089 0.071 0.069 0.065 0.063 0.064 0.025 Average yields 0.000 0.005 0.001 0.000 -0.002 -0.002 -0.002 Standard dev. of yield 0.098 0.099 0.069 0.053 0.042 0.035 0.030 63 Table A2 Counter-Cyclical Equity Term Premia: Quarterly Returns This table shows the relation between term premia and the dividend price ratio of the market portfolio. The table reports the parameter estimate from the following regression: 𝑟; −𝑟; =𝛽 , +𝛽 ,(𝑑−𝑝)+𝜖, where 𝑑−𝑝 is the dividend price ratio of the market portfolio and 𝑟,  is the three-month forward return to the dividend claim with n year maturity. The regression is based on monthly rolling regressions. The t-statistics are based on Newey and West (1987) standard errors corrected for 8 lags. The maturities n and m are both measured in years. The row m=mean(1-7) refers to the average return to the onethrough seven-year maturity dividend claim. The sample is from 2005 to 2016. Maturity of long-maturity claim (n) 2 3 4 5 6 7 Mkt Panel A: S&P 500 m=1 0.03 0.06 0.09 0.12 0.13 0.15 0.23 (0.50) (1.07) (1.63) (1.91) (2.00) (2.09) (3.23) m=2 0.01 0.04 0.06 0.07 0.07 0.08 (1.33) (2.66) (2.57) (2.29) (2.06) (1.03) m=3 0.02 0.02 0.03 0.04 0.08 (1.13) (0.84) (1.15) (1.03) (1.22) m=mean(1-7) 0.08 (1.18) Panel B: Nikkei 225 m=1 0.13 0.17 0.16 0.18 0.19 0.20 0.23 (2.33) (2.27) (2.07) (2.17) (2.15) (2.30) (1.31) m=2 0.01 0.02 0.04 0.04 0.05 0.16 (0.49) (0.72) (0.96) (0.95) (0.97) (1.39) m=3 0.00 0.01 0.01 0.00 0.06 (0.53) (0.56) (0.30) (-0.05) (0.68) m=mean(1-7) 1.20 (0.19) Panel C: EuroStoxx 50 m=1 0.09 0.07 0.06 0.04 0.03 0.03 0.04 (1.08) (0.75) (0.65) (0.49) (0.41) (0.31) (0.37) m=2 -0.03 -0.04 -0.05 -0.06 -0.06 -0.02 (-1.33) (-1.50) (-1.65) (-2.18) (-2.36) (-0.59) m=3 -0.01 -0.02 -0.03 -0.03 0.02 (-1.69) (-1.38) (-1.55) (-1.63) (0.38) m=mean(1-7) 0.03 (0.90) continued… 64 Panel D: FTSE 100 m=1 0.18 0.16 0.17 0.17 0.17 0.16 0.19 (1.62) (1.29) (1.48) (1.51) (1.65) (1.59) (2.02) m=2 -0.06 -0.07 -0.08 -0.08 -0.08 0.00 (-1.48) (-1.42) (-1.50) (-1.31) (-1.33) (-0.04) m=3 -0.01 -0.02 -0.02 -0.03 0.06 (-0.84) (-0.99) (-0.63) (-0.66) (0.66) m=mean(1-7) 0.08 (1.17) 65 Table A3 Counter-Cyclical Difference in Sharpe Ratios This table shows the relation between the difference in Sharpe ratios for equity claims with different maturity and the dividend price ratio of the market portfolio. The table reports the parameter estimate from the following regression: 𝑆𝑅, −𝑆𝑅, =𝛽 , +𝛽 ,(𝑑−𝑝)+𝜖, where 𝑑−𝑝 is the dividend price ratio of the market portfolio and 𝑆𝑅,  is the one-year log-Sharpe ratio of the log-return to the n maturity claim. The regression is based on monthly rolling regressions. The t-statistics are based on Newey and West (1987) standard errors corrected for 18 lags. The maturities n and m are both measured in years. The sample is from 2005 to 2016. Maturity of long-maturity claim (n) 2 3 4 5 6 7 Mkt Panel A: S&P 500 m=1 1.47 2.07 2.64 3.31 3.64 3.04 3.51 (2.63) (4.11) (5.56) (5.85) (5.65) (6.15) (12.89) m=2 0.42 0.98 1.55 1.69 1.18 1.18 (2.24) (2.83) (2.97) (2.45) (2.70) (1.29) m=3 0.63 0.95 0.87 0.51 1.08 (3.68) (2.85) (1.54) (1.26) (1.43) m=mean(1-7) 0.93 (1.47) Panel B: Nikkei 225 m=1 0.63 1.24 1.59 1.74 1.47 1.23 2.27 (0.64) (1.11) (1.22) (1.20) (0.90) (0.69) (1.48) m=2 0.49 0.94 1.19 1.15 1.02 0.99 (3.64) (2.57) (2.24) (1.80) (1.43) (1.09) m=3 0.35 0.47 0.35 0.03 0.12 (1.79) (1.39) (0.83) (0.07) (0.19) m=mean(1-7) 0.25 (0.51) Panel C: EuroStoxx 50 m=1 1.45 1.83 2.01 2.12 2.29 2.22 2.00 (5.73) (3.85) (3.71) (4.04) (4.75) (4.42) (2.43) m=2 0.19 0.44 0.49 0.65 0.59 0.72 (0.77) (1.53) (1.75) (2.66) (2.52) (1.17) m=3 0.17 0.31 0.44 0.44 0.69 (2.04) (4.56) (5.40) (3.77) (1.42) m=mean(1-7) 0.61 (1.23) Continued… 66 Panel D: FTSE 100 m=1 2.76 3.52 4.00 4.17 4.10 3.87 3.62 (4.32) (3.94) (3.64) (3.60) (3.58) (3.25) (3.31) m=2 0.23 0.65 0.71 0.77 0.63 0.70 (0.56) (1.17) (1.12) (1.39) (1.14) (1.09) m=3 0.49 0.42 0.45 0.24 0.29 (2.52) (1.79) (2.13) (0.90) (0.62) m=mean(1-7) 0.47 (1.19) 67 Table A4 Counter-Cyclical Equity Term Premia: Stambaugh Correction This table shows the relation between term premia and the dividend price ratio of the market portfolio. The table reports the parameter estimate from the following regression: 𝑟; −𝑟; =𝛽 , +𝛽 ,(𝑑−𝑝)+𝜖, where 𝑑−𝑝 is the dividend price ratio of the market portfolio and 𝑟,  is the twelve-month forward return to the dividend claim with n year maturity. The regression is based on monthly rolling regressions. The parameter estimate is corrected for the Stambaugh (1999) bias. The t-statistics are based on Newey and West (1987) standard errors corrected for 18 lags. The maturities n and m are both measured in years. The row m=mean(1-7) refers to the average return to the onethrough seven-year maturity dividend claim. The sample is from 2005 to 2016. Maturity of long-maturity claim (n) 2 3 4 5 6 7 Mkt Panel A: S&P 500 m=1 0.10 0.13 0.20 0.28 0.35 0.38 0.49 (1.59) (1.97) (2.75) (3.41) (3.71) (3.94) (3.63) m=2 0.02 0.10 0.18 0.23 0.24 0.27 (1.26) (2.94) (3.67) (3.66) (3.44) (1.83) m=3 0.05 0.07 0.09 0.10 0.21 (1.84) (1.15) (1.00) (1.02) (1.54) m=mean(1-7) 0.20 (1.46) Panel B: S&P 500 (using cay) m=1 1.30 2.26 3.51 4.72 5.66 5.63 6.54 (1.10) (1.61) (2.44) (3.23) (3.80) (4.07) (3.47) m=2 0.77 2.09 3.32 4.01 3.95 3.64 (1.74) (4.05) (4.82) (4.81) (3.45) (1.94) m=3 1.22 2.31 2.90 2.82 2.93 (4.72) (3.56) (3.31) (2.41) (1.65) m=mean(1-7) 1.93 (1.01) Panel C: Nikkei 225 m=1 0.39 0.63 0.69 0.73 0.74 0.77 0.65 (4.91) (3.23) (2.89) (2.64) (2.42) (2.31) (2.29) m=2 0.17 0.25 0.30 0.33 0.35 0.32 (2.42) (2.17) (2.05) (1.91) (1.80) (1.52) m=3 0.05 0.06 0.06 0.03 0.00 (1.08) (0.86) (0.57) (0.21) (-0.02) m=mean(1-7) 0.05 (0.38) continued… 68 Panel D: EuroStoxx 50 m=1 0.20 0.31 0.31 0.31 0.29 0.27 0.26 (1.70) (2.02) (2.05) (2.09) (2.02) (1.75) (1.65) m=2 0.01 0.00 -0.02 -0.04 -0.06 -0.05 (0.19) (-0.03) (-0.23) (-0.41) (-0.60) (-0.31) m=3 -0.05 -0.08 -0.10 -0.13 -0.07 (-1.52) (-1.83) (-2.17) (-2.61) (-0.67) m=mean(1-7) 0.02 (0.23) Panel E: FTSE 100 m=1 0.26 0.32 0.34 0.33 0.30 0.27 0.43 (2.29) (2.52) (2.81) (2.53) (2.14) (1.75) (1.89) m=2 -0.01 0.00 -0.02 -0.05 -0.08 0.06 (-0.14) (-0.02) (-0.20) (-0.40) (-0.59) (0.23) m=3 -0.01 -0.04 -0.07 -0.11 0.01 (-0.16) (-0.55) (-0.80) (-0.99) (0.07) m=mean(1-7) 0.09 (0.56) 69 Table A5 Predictability of Term Premia: R2 This table shows the adjusted R2 from the following regression of realized term premia on the dividend price ratio: 𝑟; −𝑟; =𝛽 , +𝛽 ,(𝑑−𝑝)+𝜖, where 𝑑−𝑝 is the dividend price ratio of the market portfolio and 𝑟,  is the twelve-month forward return to the dividend claim with n year maturity. The regression is based on monthly rolling regressions. The maturities n and m are both measured in years. The row m=mean(1-7) refers to the average return to the onethrough seven-year maturity dividend claim. The sample is from 2005 to 2016. Maturity of long-maturity claim (n) 2 3 4 5 6 7 Mkt Panel A: S&P 500 m=1 0.12 0.10 0.14 0.17 0.20 0.20 0.17 m=2 0.02 0.11 0.19 0.21 0.20 0.07 m=3 0.10 0.06 0.05 0.06 0.06 m=mean(1-7) 0.05 Panel B: Nikkei 225 m=1 0.25 0.23 0.23 0.23 0.22 0.23 0.21 m=2 0.14 0.18 0.19 0.19 0.19 0.09 m=3 0.12 0.09 0.06 0.02 -0.01 m=mean(1-7) 0.01 Panel C: Euro Stoxx 50 m=1 0.40 0.21 0.20 0.19 0.18 0.17 0.25 m=2 0.00 0.00 0.00 -0.01 -0.01 0.00 m=3 -0.01 0.00 0.01 0.03 -0.01 m=mean(1-7) 0.02 Panel D: FTSE 100 m=1 0.41 0.27 0.26 0.26 0.26 0.25 0.21 m=2 0.01 0.00 0.00 -0.01 -0.01 -0.01 m=3 0.04 0.06 0.09 0.12 0.00 m=mean(1-7) 0.01 70 Chapter 2 Conditional Risk with Christian Skov Jensen Abstract: We present a new direct methodology to study conditional risk, that is, the extra return compensation for time-variation in risk. We show theoretically that the conditional part of the CAPM can be captured by augmenting the standard market model with a conditionalrisk factor, which is a specific market timing strategy. Both in the U.S. and global sample covering 23 countries, all major equity risk factors load on our conditional-risk factor, implying that each factor has a higher conditional market beta when the market risk premium is high or the market variance is low. Accordingly, these factor returns can be partly explained by conditional risk. Studying the economic drivers of these results, we find evidence that conditional risk arises from variation in discount rate betas (not cash flow betas) due to the endogenous e↵ects of arbitrage trading. Keywords: asset pricing, conditional CAPM, factor models, time-varying discount rates. JEL classification: G10, G12. We are grateful for helpful comments from Malcolm Baker, Peter Christo↵ersen, Robin Greenwood, Sam Hanson, Eben Lazerus, Dong Lou, Stefan Nagel, Tobias Moskowitz, Lasse Heje Pedersen, Andrei Shleifer, Adi Sunderam, Christian Wagner, and Paul Whelan, as well as from seminar participants at Copenhagen Business School and Harvard Business School. Both authors gratefully acknowledge support from the FRIC Center for Financial Frictions (grant no. DNRF102) and the European Research Council (ERC grant no. 312417). 71 Example continued: The SDF Approach We can arrive at the results above easily if we use the stochastic discount factor language instead of the beta language. The stochastic discount factor approach is also useful when generalizing the results to a multi-factor model. The stochastic discount factor of the CAPM2is mt+1 =1 Rf t 1 Rf t bt˜rm t+1 (2.9) which can be written as mt+1 =1 Rf t 1 Rf t b˜rm t+1 1 Rf t (btb)˜rm t+1 (2.10) The law of one prices implies that 0=Et[mt+1ri t+1]=Et[Rf tmt+1ri t+1] (2.11) By the law of iterated expectations we have 0=E[Rf tmt+1ri t+1)] (2.12) =E[ri t+1]+cov(ri t+1;Rf tmt+1) (2.13) meaning that E[ri t+1]=cov(ri t+1;Rf tmt+1) (2.14) =˜ E[rm t+1]+cov(ri t+1;ct+1) | {z } Conditional Risk (2.15) which is the same expression as in (2.8). In the following section we use the stochastic 2The notation for the stochastic discount factor for the CAPM in expression (2.9) di↵ers slightly from the one usually used. Cochrane (2001)uses mt+1 =At+BtRM t+1 where At=1/Rf tBtEtRM t+1 and Bt=bt/Rf t. But this expression is of course the same as ours: mt+1 =At+BtRM t+1 =1 Rf t +Bt(RM t+1 Et[RM t+1]) = 1 Rf t 1 Rf t bt˜rm t+1 78 discount factor language to more formally derive a multi factor model with conditional risk. 1.2 Conditional Risk in Factor Models We now derive a general statement for conditional risk in factor models. Consider the class of factor models captured by the following stochastic discount factor for k=1,...,K traded risk factors: mt+1 =1 Rf t 1 Rf t K X k=1 bk t˜rk t+1 (2.16) where ˜rk t+1 =rk t+1 Et[rk t+1] (2.17) and bk t=Et[rk t+1] vart(˜rk r+1) is the time tshock and price of risk for factor k.Theexpressionin(2.16)canberewritten as mt+1 =1 Rf t 1 Rf t K X k=1 bk˜rk t+1 1 Rf t K X k=1 (bk tbk)˜rk t+1 (2.18) where bkis the unconditional price of risk for factor k bk=E[rk t+1] var(˜rk r+1) By applying the law of one price and taking unconditional expectations, we can state an unconditional model that incorporates conditional risk. Before doing so, we define the conditional risk factors ck t+1 =˜rk r+1(bk tbk). Proposition 1 (conditional risk in factor models) The unconditional expected excess return on an asset i is given by E[ri t+1]= K X k=1 ˜ kk+ K X k=1 k ck c(2.19) 79 where ˜ k=cov(ri t+1;˜rk t+1) var(˜rk t+1), k=E[rk t+1],(2.20) k c=cov(ri t+1;ck t+1) var(ck t+1), k c=var(ck t+1) (2.21) In the factor model above, each factor kis represented by two betas: one for its unconditional risk and the other for its conditional risk. These two orthogonal factors capture all of the unconditional implications of the stochastic discount factor in (2.16). The following proposition summarizes the properties of the two factors and their betas. Proposition 2 (properties of conditional risk factors and betas) 2.a (zero mean factors):The means of all factors are zero: E[˜rk t+1]=E[ck t+1] = 0 (2.22) 2.b (uncorrelated factors):For each factor k, the return and shock to the risk factor is uncorrelated with the conditional risk factor: cov(rk t+1;ck t+1)=cov(˜rk t+1;ck t+1) = 0 (2.23) 2.c (shock betas for the factors):The factor khas a loading of one on its own shock: cov(rk t+1;˜rk t+1) var(˜rk t+1)= 1 (2.24) 2.d (constant-beta equivalence):If an asset j has a constant conditional beta, the expected return is given by the usual unconditional beta. That is, if k t=covt(rj t+1;rk t+1) vart(rk t+1)=c(2.25) then ˜ k=kand k c= 0 (2.26) 80 While Proposition 1 allows for the estimation of a kfactor model, we will focus on the one factor CAPM model in the empirical section. We do so because conditional risk with respect to the market portfolio has the most tangible interpretation and because the market factor is the most widely used factor. 1.3 Conditional Cash Flow and Discount Rate Risk Conditional market risk arises because conditional market betas are higher when the price of risk is higher. As shown by Campbell and Vuolteenaho (2004), conditional market betas are the sum of the given asset’s conditional cash flow and discount rate betas. Accordingly, the conditional risk must come from either conditional cash flow or discount rate betas being high when the price of risk is high. In this section, we show how to estimate these two sources of conditional risk by decomposing the conditional risk factor into two. First note that shocks to the market portfolio, ˜rt+1,aregivenbycashflownewsand discount rate news (Campbell and Shiller,1988): ˜rm t+1 =NCF,t+1 +NDR,t+1 (2.27) The beta of an individual stock can then be expressed as: t=covt(ri t+1;NCF,t+1) vart(˜rt+1)+covt(ri t+1;NDR,t+1) vart(˜rt+1)(2.28) t⌘CF t+DR t(2.29) Similarly, the market’s conditional risk factor can be decomposed into two parts: cm t+1 =˜rm r+1(bm tbm) (2.30) =NCF,t+1(bm tbm)+NDR,t+1(bm tbm) (2.31) ⌘cCF t+1 +cDR t+1 (2.32) where cCF t+1 is the conditional cash-flow-risk factor and cDR t+1 is the conditional discount-raterisk factor. Loading on conditional cash flow risk and conditional discount rate risk has a tangible economic interpretation. The unconditional covariance with the two risk factors summarizes the covariance of cash flowand discount rate betas with the expected return 81 and variance: cov(ri t+1,c CF t+1)=covCF t;Et[rm t+1]bvart(˜rm t+1)(2.33) and cov(ri t+1,c DR t+1)=covDR t;Et[rm t+1]bvart(˜rm t+1)(2.34) 2 Methodology 2.1 Identifying Conditional Moments In order to estimate our factor model, we must estimate the conditional mean and variance of the factors. In this section, we outline the identifying assumptions we rely on in doing so. To estimate the conditional market risk premium, we use the three pass estimator suggested by Kelly and Pruitt (2013). The estimator uses the cross-section of valuation ratios to estimate the expected return. By using the cross-section of valuation ratios rather than just the valuation ratio for the market, it is possible to separate the e↵ect of expected growth rates and expected discount rates. Accordingly, the methodology consistently recovers the conditional market risk premium based on two simple identifying assumptions: (1) the expected log return and log growth rates are linear in a set of latent factors, and (2) these factors evolve according to a first-order vector autoregression. We rely on the Kelly and Pruitt estimator for multiple reasons. Most importantly, the method is proven to predict the one-month expected market return well both inand outof-sample, and it is proven to work in both the U.S. and internationally. Indeed, Kelly and Pruitt (2013) show that the estimator predicts the one-month expected return on the U.S. market portfolio with an R2of 2.38 in-sample and 0.93 out-of-sample; and it predicts the global market portfolio with an R2of 1.5 out-of-sample. In addition, the estimator consistently recovers the market risk premium under assumptions that are consistent with the null-hypothesis we test against when we are testing for conditional risk. With respect to the variance, we similarly assume that the market variance evolves according to a first-order autoregression. We rely on this assumption because it is transparent and in line with recently published papers revolving around time-varying variance, such as 82 Campbell, Giglio, Polk, and Turley (2017). Our results in the empirical section are highly robust to other measures of expected return and variance. We verify in the Appendix that the results are robust to estimating expected returns based on the measures of Campbell and Thompson (2008). We also verify that the results are robust to using the variance estimated in Bollerslev, Tauchen, and Zhou (2009) or calculating variance based on SVIX. In order to estimate conditional cash flow and discount rate risk, we need to decompose returns into cash flow news and discount rate news. For simplicity, we rely on the quarterly time series estimated by Campbell, Giglio, Polk, and Turley (2017). The authors make the time series available online. 2.2 Data Our sample consists of equities from 23 di↵erent countries between August 1963 and December 2016. The 23 markets in our sample correspond to the countries belonging to the MSCI World Developed Index as of December 31, 2016. Stock returns are from the union of the CRSP tape and the XpressFeed Global Database. All returns are in USD and do not include any currency hedging. All excess returns are measured as excess returns above the U.S. Treasury bill rate. We report summary statistics in Table 1. We study conditional risk in each country in our sample and in a broad global sample. Our broad sample of global equities contains all available common stocks on the union of the CRSP tape and the XpressFeed Global database. For companies traded in multiple markets we use the primary trading vehicle identified by XpressFeed. Our global sample runs from January 1986 to December 2016 because XpressFeed’s Global coverage starts in 1986 for most countries (see Table 1). The Kelly and Pruitt (2013) estimator takes as input portfolios sorted on size and bookto-market. In the U.S., we use 100 portfolios sorted unconditionally on size and book-tomarket from Ken French’s website. In the global sample, we similarly create 100 portfolios sorted unconditionally on size and book-to-market. In the individual international countries, we create 25 portfolio sorted first on size and then conditionally on book-to-market. We use only 25 portfolios and conditional sorts because some of the countries have few firms in the beginning of the sample and the conditional sorts into 25 portfolios helps ensure an adequate number of firms in each portfolio. 83 We calculate monthly variance as the sum of squared daily residuals over the month with a degree of freedom adjustment for the estimation of the mean. cvart(˜rm t+1)= n n1 n X i=1 (rm i¯rm)2(2.35) where nis the number of trading days in the month. The estimation assumes that the expected return is constant during each month. The expected time tvariance is then calculated as: vart(˜rm t+1)=ˆ ✓0+ˆ ✓1cvart1(˜rm t) (2.36) where ˆ ✓0and ˆ ✓1are parameter estimates from the following regression: cvart(˜rm t+1)=✓0+✓1cvart1(˜rm t) (2.37) We rely on in-sample estimations for the expected variance, but the results are robust to using out-of-sample estimates of the variance as in Bollerslev, Tauchen, and Zhou (2009). 3 Conditional Risk in Stock Returns Table 1 o↵ers summary statistics of the 24 exchanges in our sample. The first three columns show the time-series median market capitalization of the firms listed in a given country, the time-series median number of firms, and the time-series average weight of the given country in the global portfolio. The U.S. has a high average weight in the global portfolio, but this is largely driven by the early years 1986-1990 where the U.S. constitutes the most of the sample. The weight of the U.S. market is downward trending throughout the sample and towards the end of the sample the weight of the U.S. is around .2. The fifth and the sixth columns in Table 1 show the average standard deviation and market risk premium in annualized terms. The last three columns of Table 1 shows the R2of the expected variance and return to the market portfolio. Regarding the variance, the R2is generally around 30% to 50%, with the U.S. and the global portfolio being in the low end. This high R2corresponds to previous studies on predicting variance (Bollerslev, Tauchen, and Zhou,2009;Bollerslev, Hood, Huss, 84 and Pedersen,2016), suggesting that our simple method for predicting variance works well. The two last columns of Table 1 summarize the R2of the expected return on the market portfolio. The first column shows the R2of the expected log return to the market portfolio, which is what the Kelly Pruitt estimator extracts. The last column shows the expected excess returns, which is calculated under the assumption of log-normally distributed returns by adding one-half the conditional log-variance to the log-return, taking the exponential, and subtracting the risk-free rate. The table shows that the R2for the log-returns in the U.S. and the global sample is 1.6% and 2.4%, which is around the same as reported by Kelly and Pruitt (2013). Internationally, the R2vary between 0.5% to 3.3%, with the median being 2.2%. The results reported by Kelly and Pruitt for the U.S. and global sample thus appear to extend to most individual exchanges. The R2for the expected excess return are similar to those for the log-return. The expected variance and market return is used to calculate the relative price of risk btb, which is an important input for the conditional-risk factor. Figure 1 visually inspects this relative price of risk in the U.S. (Panel A) and the global sample (Panel B). The price of risk varies substantially on both the short and long horizon. The substantial short horizon variation in the price of risk underlines the importance of using a forward looking measure of the price of risk. Indeed, an alternative to our approach is to implement the conditional CAPM over short horizons for which the price of risk is assumed to be constant. If daily data are available, the horizon is often around three to six months, and if daily data are not available, the horizon is substantially longer. The price of risk in Figure 1 exhibits substantial variation over these horizons, which, if statistically significant and not driven by forecast errors, invalidates this unparemetric approach. The price of risk in Figure 1 also shows substantial long-run variation that appears closely linked to economic conditions. In the U.S. in particular, the price of risk tends to increase in the years following economic recessions: the price of risk increases in the years following the recessions in 1973-1975, 1981-1982, 1990-1991, 2001, and 2007-2009. On the other hand, the price of risk is lowest during the tech bubble. The price of risk is also low during the onset of the financial crisis. This result is similar to the findings in Moreira and Muir (2017). Moreira and Muir argue that in the beginning of the financial crisis, and crises more generally, the variance increases by more than the market risk premium which causes the price of risk to go down. 85 Another way to visualize conditional risk is by looking at the realizations of the conditionalrisk factor. Doing so gives us a rough idea of which kinds of assets that have positive conditional risk: an assets with a return that net of the market mimics the conditional-risk factor would has a high level of conditional risk. Panel A in Figure 2 plots the two-year cumulative realization of the conditional-risk factor in the U.S.. The two-year realization shows distinct patterns. In particular, the cumulative value decreases during the tech bubble, indicating that an asset that performed poorly during the tech bubble has a high level of conditional risk. The value factor (HML) is a prominent example of such an asset: value stocks lost heavily to growth stocks during the tech bubble. Accordingly, we would expected HML to be an asset with high conditional risk. Consistent with this logic, and previous research3, we find empirically that HML has substantial conditional risk. More generally, Panel B in Figure 2 plots, along with the two-year realization of the conditional-risk factor, the two-year realization to an average cross-sectional risk factor which we call the composite risk factor. The composite risk factor is the average return to the factors value (HML), profitability (RMW), investment (CMA), momentum (UMD), and betting against beta (BAB). The figure shows that the two-year realizations of the composite risk factor and the conditional-risk factor are correlated. For instance, both factors earn high return during the 1980s, lose substantially during the tech bubble, earn high returns again during the 2001-2003 stock market contraction, and lose substantially during the market rebound after the financial crisis. This visual evidence suggests that the composite risk factor might load on conditional risk. We next address formally whether the risk factors load on conditional risk through factor analysis. 3.1 Conditional Risk in the Cross-Section of Stock Returns In this section, we analyze conditional risk in the cross-sectional of equities by implementing the conditional CAPM as an unconditional two-factor model following Proposition 1. Table 2 summarizes the results for seven cross-sectional risk factors: size (SMB), value (HML), profitability (RMW), investment (CMA), momentum (UMD), and betting against beta (BAB), and a composite factor (COMP) which is the average return to the last five major risk factors. Panel A shows the results in the U.S. The first row shows the monthly alpha in percent. The alpha is statistically significant for all the strategies except the size factor. 3For previous research on conditional risk in HML, see Lettau and Ludvigson (2001b); Lewellen and Nagel (2006). 86 The positive two-factor alphas mean that the well-documented unconditional CAPM alphas of these factors cannot be explained by conditional risk. For HML and UMD, this result is similar to those found by Lewellen and Nagel (2006)andBoguth, Carlson, Fisher, and Simutin (2011), but for betting against beta our results di↵er from previous findings in Cederburg and O’Doherty (2016). We compare our results more closely to the literature later in the paper. The third row of Panel A, which shows the loading on the conditional risk factor, reveals a striking relationship between alpha and loading on conditional risk: all the risk factors that have positive alpha are positively exposed to conditional risk. Indeed, value, profitability, investment, momentum, and betting against beta all have positive loadings on conditional risk. The only factor that does not have a positive loading is size, for which the loading is statistically insignificant. The fourth row summarizes how large a compensation the conditional risk loadings warrant. As mentioned, conditional risk cannot explain the full alpha of the strategies, but it does explain a meaningful amount. Indeed, conditional risk explains between between .03 and .12 percentage point of monthly return, equivalent to 0.39 to 1.41 percentage point of annual return. This is a large amount in absolute terms, considering it arises simply from a failure to implement the CAPM correctly in the first place. It is also a large amount relative to the unconditional CAPM alpha, as can be seen in the sixth row. Indeed, conditional risk explains 8% to 15% of the unconditional CAPM alpha for these strategies; for the average factor COMP, conditional risk explains 11% of CAPM alpha. Panel B reports the results from the global sample. Qualitatively, the results are similar: value, profit, investment, momentum, betting against beta, and the composite factor are all positively exposed to conditional risk; but the e↵ect of conditional risk is not large enough to render the two-factor alphas insignificant. Furthermore, the size factor is negatively exposed to conditional risk but the exposure is close to zero and statistically insignificant, as in the U.S. The economic magnitude of conditional risk is larger in the global sample than in the U.S.. Indeed, conditional risk explains as much as 0.17 percentage points of monthly return, equivalent to 2.03 percentage point of annual return. This larger absolute e↵ect of conditional risk, combined with the fact that the average risk factor has lower alpha in the global sample, means that conditional risk explains a larger fraction of the unconditional CAPM 87 have the second and third most conditional risk in the late sample. Finally, it is worth providing a simple example of how the arbitrage trading actually creates time-variation in the conditional betas. Consider an economy with two states: a high and a low price-of-risk state. In the high price-of-risk state, the arbitrageurs are long the market and in the low price-of-risk state they are short the market. In both states, they hold the composite risk factor. In the good state, funding shocks to the arbitrageurs create a positive correlation between the composite risk factor and the market factor because they are long both factors, which means that the conditional beta of the composite factor is high in this high state. Similarly, funding shocks create negative correlation between the composite risk factor and the market when the price of risk is low because the arbitrageur is short the market. Accordingly, the arbitrage trading creates low conditional betas for the composite risk factor in this low state. Taken together, the funding shocks and the resulting arbitrage trading thus create high conditional betas when the price of risk is high and low conditional betas when the price of risk is low, which is to say that it creates conditional risk. In conclusion, we find strong evidence that conditional risk is driven in part by arbitrage activity. If arbitrageurs trade both the cross-sectional risk factors and the conditionalrisk factor, these become correlated and the cross-sectional risk factors therefore load on conditional risk. This e↵ect is stronger the more the arbitrageurs are present in the factors. Consistent with this, we find that the conditional risk in the cross sectional risk factors is stronger when: (1) the factors have higher expected return and arbitrageurs therefore are trading them more; (2) there is more leverage in the economy and arbitrageurs therefore are likely to hold larger positions; and (3) there is more arbitrage capital. In addition, using the net speculative position in S&P 500 futures, we find evidence that arbitrageurs trade the conditional-risk factor. 5 Robustness In this section we consider how robust our results are to using other measures of the conditional market risk premium and variance. Tables A1 to A10 report a range of robustness checks where we redo our analysis using di↵erent measures of these conditional estimates. Tables A1 through A4 show the results of using the measures of expected return reported in Campbell and Thompson (2008). Campbell and Thompson show how the dividend price 94 ratio, earnings price ratio, and book-to-market are linked to the expected market return through accounting identities. Using these identities, we calculate the expected market risk premium and estimate the conditional risk factor and the market shock factor. The results are robust to using these measures of expected returns. The composite risk factors load on the conditional risk factor in both the U.S. and global sample for all three measures of expected return. In addition, the composite risk factor loads on conditional risk in at least 18 of the 22 other samples. The economic e↵ect of conditional risk is, however, smaller than when using the Kelly-Pruitt measure of expected return. Tables A5 and A6 report results where the market risk premium is assumed to be constant, such that only the expected variance varies. The results are also robust to this specification: the composite risk factor loads on conditional risk in all samples. However, the economic e↵ect is smaller, with conditional risk only explaining about 4% of the alpha. Tables A7 and A8 report results where the variance is assumed constant, such that only the expected market risk premium varies. We again extract the conditional market risk premium using the Kelly-Pruitt method. The results are robust, with all risk factors loading on conditional risk in the U.S. sample (except SMB), and with the composite risk factor loading on conditional risk in all but two of the 24 samples. In addition, the economic e↵ect is close to as large as when using time-varying variance. Finally, Table A9 reports results in the U.S. sample when using more sophisticated methods for estimating expected variance. Panel A reports results of using the expected variance in Bollerslev, Tauchen, and Zhou (2009). The results are robust. Similarly to our baseline case, all the major risk factors load on conditional risk. The economic e↵ect is slightly larger than in the baseline case. Panel B reports the results of using the risk neutral variance SVIX. These results are also robust, with all risk factors loading on conditional risk. Finally, in Table A10 we report results based on using an index of multiple estimators for the conditional market risk premium. We calculate the index of expected returns as the average of the expected return in Campbell and Thompson (2008); Kelly and Pruitt (2013); Lettau and Ludvigson (2001a) and Martin (2016). The results are robust to using this index of expected returns: both in the U.S. and globally, the factors have the same conditional-risk loadings as when using just the Kelly and Pruitt (2013)estimateforthe conditional market risk premium. 95 In conclusion, we find that our results are robust to various di↵erent methods for identifying and estimating the conditional market risk premium and variance. The results are stronger than the baseline results if we use more sophisticated measures of conditional variance, but weaker if we assume constant variance. The results are robust to, but weaker, when using simpler measures of the conditional market risk premium. 6 Relation to the Literature Our results relate to and extend a long strand of literature on the conditional CAPM. Jagannathan and Wang (1996) show that the conditional CAPM helps explain asset returns when using instruments to measure betas. Lettau and Ludvigson (2001b)showthatusingthecay variable variable as instrument explains the returns to size and value sorted portfolio, but Lewellen and Nagel (2006) argue that the e↵ect is overestimated and that the conditional CAPM cannot explain the cross-section of stocks. Lewellen and Nagel further advocate the use of short-horizon regressions as an instrument-free way of testing the conditional CAPM. However, Boguth, Carlson, Fisher, and Simutin (2011) argue that the short-horizon regressions has certain small-sample issues, and instead advocate the use of an instrumental approach that uses past betas and state variables as instruments. Using this approach, they show that momentum portfolios load on conditional risk. More recently, Cederburg and O’Doherty (2016) argue that the conditional CAPM explains the low risk anomaly documented by Black, Jensen, and Scholes (1972)andFrazzini and Pedersen (2014). Going beyond unconditional expected returns, Nagel and Singleton (2011) test the additional implication that conditional expected returns must be consistent with the conditional factor models. In the following, we address more closely the two papers most closely related to ours. Lettau and Ludvigson (2001b): Our approach is most closely related to the study by Lettau and Ludvigson. Similarly to Lettau and Ludgivson, we estimate the price of risk rather than time-variation in betas and use this estimate of the price of risk to implement the conditional CAPM. The di↵erence in our approaches is that we explicitly estimate the price of risk by estimating the expected return and variance, whereas Lettau and Ludvigson assume that it is a function of the cay variable. To estimate the risk premium on their conditional risk factor, they must estimate the variance of the price of risk, which they do in a second-stage regression in the cross-section of stocks. However, this second-stage 96 regression may estimate unrealistically high variance of the price of risk, leading to an overestimation of the ability of the conditional CAPM to explain the cross-section of stock. Indeed, Lewellen and Nagel (2006) argue that the results do not hold under reasonable estimates of the variance of expected return and variance. We avoid this problem by instead estimating the price of risk in first-stage regressions. This approach ensures that we rely on reasonable estimates of the variance of expected return and variance. Indeed, as can be seen in Table 1, the R2of the expected return is a few percent, which is generally believed to be a meaningful variability in expected returns on short horizons (see e.g. Ross (2015)). Cederburg and O’Doherty (2016): Our results on the low-risk e↵ect di↵er substantially from those by Cederburg and O’Doherty, as they find that the low-risk e↵ect is statistically insignificant once controlling for conditional risk. There are two potential reasons for this discrepancy. First, Cederburg and O’Doherty use instruments to pick up variation in betas. These instruments are unlikely to pick up all the variation in betas. The instruments may therefore miss variation in betas that is either negatively correlated with the expected return or positively correlated with volatility. If this is the case, the estimate of conditional risk in Cederburg and O’Doherty (2016) too high. Alternatively, the instruments may have picked up variation in expected return or variance that was not expected ex ante, in which case the estimate of conditional risk may also be too high. Second, we study the returns to the monthly betting against beta factor and not quarterly beta sorted portfolios as Cederburg and O’Doherty do. The advantage of studying the betting against beta factor is that the factor is hedged ex ante to have a conditional beta of zero, mitigating the risk of missing variation in conditional betas. In addition, the fact that the factor is hedged conditionally to have a beta of zero, and has an alpha of 10 percentage points per year, means that it is unlikely that the conditional CAPM can explain its average return in the first place: it would require the estimated conditional betas to be far from the true betas. 7 Conclusion We document a global pattern of conditional risk in stock returns: across 23 developed countries, the five largest risk factors are exposed to conditional risk. The conditional risk 97 explains around 20% of the alpha to these strategies. In addition, conditional risk explains all of the alpha of time-series strategies such as volatility-managed portfolios. This conditional risk has broad economic implications. For instance, a CFO of a value firm who discounts cash flows using the unconditional CAPM would use the company’s beta of, say, 1 times the global market risk premium, which gives an annual discount rate of around 5 percent. However, given the conditional risk in global value firms, the CFO should in fact use an annual discount rate of 6.5 percent to also reflect the conditional-risk premium. In addition to NPV analysis, the conditional risk is also important for judging the economic importance of di↵erent anomalies, understanding market efficiency, evaluating the performance of asset managers, and in financial analysis more generally. We document the global pattern in conditional risk by using a new, simple method for estimating conditional risk. The method augments the unconditional CAPM with a conditional-risk factor, which is a precisely defined market timing factor. This conditionalrisk factor is sufficient to capture all of the implications of the conditional CAPM and can easily be applied in future factor analysis, performance analysis of money managers, or by CFOs to determine their company cost of capital. We also address the economic source of the conditional risk. Previous explanations of why conditional risk arises are often unique to a single factor. However, the fact that all the major risk factors load on conditional risk hints at a common explanation for all the factors. We find evidence of such a common explanation, namely that the conditional risk in all the factors arises from arbitrage trading. Consistent with this arbitrage trading hypothesis, we find that there is more conditional risk in the the cross-sectional risk factors when arbitrageurs trade these more intensely and when the abitrageurs are more levered. 98 8 Appendix Proof of (2.3). Note that we can write the conditional beta as t=E[]+⌘t.Wecan then write the unconditional covariance between the excess return to asset iand the shock to the market portfolio as cov(ri t+1;˜rt+1)=cov(E[]˜rt+1 +⌘t˜rt+1;˜rt+1) =E[]var(˜rt+1)+cov(⌘t;var(˜rt+1)) given that cov(⌘t˜rt+1;˜rt+1)=E[⌘t˜r2 t+1]=cov(⌘t;˜r2 t+1) and using that ˜r2 t+1 =Et[˜r2 t+1]+✏t+1 = vart(˜r2 t+1)+✏t+1 where cov(⌘t;✏t+1) = 0. By dividing both sides by the unconditional variance of ˜rt+1 we obtain the expression in (3). Proof of (2.8). Note that the covariance term in (2.5)canbewrittenas cov t;Et[rm t+1]bvart(˜rm t+1)=E⇥t(Et[rm t+1]bvart(˜rt+1))⇤ E[t]E⇥(Et[rm t+1]bvart(˜rt+1))⇤ where the first term is equal to Eri t+1˜rt+1 vart(˜rt+1)(Et[rm t+1]bvart˜rt+1)=E⇥ri t+1⇤E[˜rt+1(btb)] + cov(ri t+1;˜rt+1(btb)) =cov(ri t+1;ct+1), given that E[ct+1]=0(shownlater),andthesecondtermisequaltozero: E[t]E⇥(Et[rm t+1]bvart˜rt+1)⇤=E[t]E[rm t+1]bvar(˜rt+1)=0 Finally, to see that E[ct+1]=0,notethat E[ct+1]=E[˜rt+1]E[btb]+cov(˜r;btb) which is equal to zero because the shock to the market portfolio has a zero mean and is uncorrelated with (unpredictable by) the ex ante price of risk, bt. 99 Proof of Proposition 2.b. The covariance between the conditional-risk factor and the shock to the market is cov(˜rt+1;ct+1)=E[˜rt+1ct+1]=E⇥˜r2 t+1(btb)⇤=E⇥rm t+1⇤E⇥rm t+1⇤=0 100 101 Table 1 Summary Statistics This table reports summary statistics for the 24 exchanges in our sample. Our sample consist of the union of all U.S. common stocks on CRSP tape (“shrcd” equal to 10 or 11) and all global stocks in the Xpressfeed global databse (“tcpi” equal to 0). The expected market risk premium is calculated using the Kelly and Pruitt (2013) method. Expected variance is calculated using an AR(1) regression. All returns are in USD. The standard deviation of the market risk premium is annualized. The market risk premium is in annual percent. The R2 is based on monthly regressions. Returns are total log returns. Excess returns are simple returns in excess of the risk-free rate. Exchange Starting year Median number of firms Mean weight in global portfolio Market risk premium R2 in predictive regressions St. dev Average Variance Returns Excess returns AUS 1990 1178.5 0.020 0.195 7.9% 0.487 0.005 0.005 AUT 1992 88 0.002 0.199 2.3% 0.510 0.029 0.029 BEL 1991 164 0.009 0.181 7.7% 0.444 0.015 0.019 CAN 1986 574.5 0.025 0.160 6.3% 0.571 0.019 0.019 CHE 1991 267 0.027 0.171 7.5% 0.337 0.031 0.030 DEU 1991 1002.5 0.080 0.181 3.0% 0.379 0.036 0.034 DNK 1991 175 0.005 0.183 8.8% 0.426 0.021 0.022 ESP 1991 151 0.017 0.208 3.9% 0.391 0.019 0.018 FIN 1991 135.5 0.005 0.254 9.2% 0.540 0.041 0.038 FRA 1991 739.5 0.050 0.194 5.0% 0.460 0.014 0.014 GBR 1987 2045.5 0.113 0.170 5.8% 0.454 0.019 0.019 HKG 1991 932 0.035 0.217 10.9% 0.429 0.022 0.021 IRL 1995 52 0.003 0.237 5.7% 0.344 0.032 0.025 ISR 1996 246 0.003 0.217 6.8% 0.350 0.026 0.023 ITA 1992 273 0.019 0.218 2.9% 0.422 0.020 0.022 JPN 1989 3578 0.169 0.204 1.9% 0.214 0.014 0.021 NLD 1991 181.5 0.022 0.176 6.3% 0.452 0.034 0.028 NOR 1991 179.5 0.004 0.233 7.0% 0.496 0.037 0.037 NZL 1991 126 0.003 0.204 7.6% 0.466 0.014 0.018 PRT 1997 57 0.003 0.216 4.1% 0.472 0.042 0.039 SGP 1991 480.5 0.012 0.184 7.9% 0.367 0.035 0.038 SWE 1991 328 0.014 0.223 9.0% 0.444 0.015 0.015 USA 1964 4789 0.450 0.173 5.8% 0.305 0.016 0.018 WOR 1986 18366 NA 0.147 5.0% 0.259 0.024 0.023 102 Table 2 Conditional Risk in Equity Factors This table reports results from evaluation of different equity strategies in the conditional CAPM. We regress the monthly excess returns of different factors on the shock to the market factor and the conditional risk factor and evaluate returns in the following two-factor model from Proposition 1: 𝐸𝑟  = 𝛼+ 𝛽𝜆+ 𝛽 𝜆  where 𝑟  is the excess return to the risk factor i, 𝛽 and 𝛽  are the beta for the marketand conditional-risk factor, and 𝜆 and 𝜆  are their risk premia. The composite factor COMP is the average return to HML, RMW, CMA, UMD, and BAB. “Compensation for conditional risk” is the product 𝛽 𝜆 . All alphas are in monthly percent. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. Standard errors are bootstrapped to account for generated regressors. The U.S. and global samples run from 1964-2015 and 1986-2015. SMB HML RMW CMA UMD BAB COMP Panel A: U.S. Sample Alpha 0.13 0.36 0.26 0.29 0.71 0.90 0.50 (1.00) (3.24) (2.45) (3.63) (3.19) (4.71) (5.12) Market beta 0.21 -0.17 -0.12 -0.15 -0.13 -0.05 -0.12 (6.36) (-5.05) (-4.88) (-7.86) (-2.15) (-0.97) (-5.37) Conditional risk beta 0.00 0.02 0.03 0.02 0.05 0.07 0.04 (-0.23) (2.08) (1.94) (2.66) (1.46) (2.36) (2.23) Compensation for conditional risk 0.00 0.03 0.05 0.03 0.09 0.12 0.06 Fraction of alpha explained by conditional risk -0.03 0.08 0.15 0.09 0.11 0.12 0.11 Observations 623 623 623 623 623 623 623 Adjusted R2 0.09 0.08 0.09 0.13 0.04 0.08 0.19 Panel B: Global Sample Alpha 0.08 0.24 0.27 0.21 0.66 0.65 0.39 (0.64) (1.81) (2.88) (2.58) (2.80) (3.82) (4.32) Market beta 0.06 -0.09 -0.18 -0.08 -0.22 -0.13 -0.14 (2.01) (-3.17) (-9.53) (-4.50) (-3.27) (-2.34) (-6.51) Conditional risk beta 0.00 0.03 0.01 0.02 0.02 0.04 0.02 (-0.25) (2.29) (1.43) (3.36) (1.19) (2.40) (3.01) Compensation for conditional risk -0.01 0.13 0.03 0.08 0.09 0.17 0.10 Fraction of alpha explained by conditional risk -0.09 0.36 0.10 0.28 0.12 0.21 0.20 Observations 354 354 354 354 354 306 354 Adjusted R2 0.01 0.12 0.21 0.13 0.07 0.12 0.25 103 Table 3 Conditional Risk in Portfolios Sorted on Aggregate Characteristics This table reports results from evaluation of characteristics sorted portfolios in the conditional CAPM. We regress the monthly excess returns of different portfolios on the shock to the market factor and the conditional risk factor and evaluate returns in the following two-factor model from Proposition 1: 𝐸𝑟  = 𝛼+ 𝛽𝜆+ 𝛽 𝜆  where 𝑟  is the excess return to the portfolio i, 𝛽 and 𝛽  are the beta for the marketand conditional-risk factor, and 𝜆 and 𝜆  are their risk premia. Each month, we sort stocks into ten portfolios based on the aggregate size, book-to-market, profitability, investment, momentum, and beta characteristic. We measure characteristics in cross-sectional percentiles and chose signs such that higher characteristics are associated with higher returns. In the U.S., we use NYSE breakpoints for the portfolios sorts. In the international sample, we use calculate breakpoints based on the 20% largest firms. Portfolios are value-weighted, refreshed, and rebalanced monthly. “Compensation for conditional risk” is the product 𝛽 𝜆 . All alphas are in monthly percent. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. Standard errors are bootstrapped to account for generated regressors. The U.S. and global samples run from 1964-2015 and 1986-2015. Portfolios sorted on aggregate characteristics 1 (low) 2 3 4 5 6 7 8 9 10 (high) 11 (10-1) Panel A: U.S. Sample Alpha -0.56 -0.04 0.00 0.10 0.20 0.20 0.22 0.22 0.38 0.50 1.06 (-2.13) (-0.18) (-0.01) (0.58) (1.25) (1.33) (1.38) (1.47) (2.36) (2.87) (4.02) Market beta 1.36 1.13 1.01 0.98 0.90 0.85 0.84 0.79 0.75 0.77 -0.59 (32.10) (39.81) (41.88) (37.33) (37.99) (35.18) (31.65) (27.83) (24.39) (21.56) (-9.38) Conditional risk beta -0.03 -0.01 0.01 0.02 0.03 0.03 0.03 0.05 0.04 0.05 0.08 (-1.83) (-1.02) (1.76) (2.33) (2.35) (2.27) (2.36) (2.72) (2.24) (2.23) (2.15) Compensation for conditional risk -0.06 -0.01 0.02 0.04 0.04 0.04 0.05 0.07 0.07 0.08 0.13 Fraction of alpha explained by conditional risk 0.09 0.26 1.15 0.27 0.18 0.18 0.20 0.25 0.16 0.14 0.11 Observations 623 623 623 623 623 623 623 623 623 623 623 Adjusted R 2 0.86 0.89 0.90 0.88 0.86 0.85 0.82 0.79 0.71 0.70 0.33 110 Table 9 Arbitrage Trading and Conditional Risk Panel A reports the results of a regression of the price of risk (bt) on the net speculative demand (NSDt). Net speculative demand is calculated as the difference in commercial long and short positions in S&P 500 futures dividend by open interest at time t. Panel B reports the results of a regression of the returns to the composite risk factor on the shock to the market portfolio and the product of the shock and the ex ante NSD. Panel C reports the results of a regression of the composite risk factor on the conditional risk factor and the product of the conditional risk factor and the ex ante intermediary leverage as defined by He, Kelly, and Manela (2017). t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. The t-statistics in Panel A are based on Newey and West (1987) standard errors. The table considers both the U.S. and global (WOR) sample. Panel A: Correlation between the price of risk (bt) and net speculative demand (NSD) Dependent variable Intercept 𝑁𝑆 𝐷  Obs R2 Correlation U.S. 𝑏  2.80 6.42 358 0.04 0.21 (8.27) (1.96) WOR 𝑏  3.06 13.46 354 0.07 0.26 (5.49) (2.06) Panel B: The composite factor has higher market beta when NSD is higher Dependent variable Intercept 𝑟      𝑟      × 𝑁𝑆 𝑃  Obs R2 U.S. 𝐶𝑂𝑀 𝑃    0.46 -0.16 1.05 358 0.20 (4.98) (-7.80) (5.04) WOR 𝐶𝑂𝑀 𝑃    0.42 -0.14 0.71 354 0.21 (5.72) (-8.88) (4.79) Panel C: The composite factor has higher beta to the conditional risk factor when leverage is higher Dependent variable Intercept 𝑐    𝑐    × 𝐿𝐸 𝑉  Obs R2 U.S. 𝐶𝑂𝑀 𝑃    0.48 -0.04 1.17 550 0.17 (6.94) (-3.32) (7.33) WOR 𝐶𝑂𝑀 𝑃    0.36 -0.03 0.56 354 0.13 (4.56) (-2.37) (4.60) 111 Table 10 Expected Factor Return and Conditional Risk Loading This table reports results from the following regression of the composite risk factor on the conditional risk factor and the conditional risk factor interacted with the value spread of the composite risk factor: 𝐶𝑂𝑀𝑃 = 𝛾+ 𝛾𝑐 + 𝛾𝑐 ×𝑉𝑆 + 𝜖 where 𝑉𝑆 is the average value spread at time t of the five factors in 𝐶𝑂𝑀𝑃 . The composite factor 𝐶𝑂𝑀𝑃 is the average return to HML, RMW, CMA, UMD, and BAB. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. Standard errors are bootstrapped to account for generated regressors. The U.S. and global samples runs from 1964-2015 and 1986-2015. Exchange 𝛾   𝛾   𝛾   Observations Adjusted R2 AUS 0.70 (5.74) -0.04 (-1.23) 0.19 (1.95) 306 0.05 AUT 0.34 (2.22) -0.01 (-0.83) 0.11 (1.65) 282 0.00 BEL 0.34 (2.59) 0.01 (0.43) 0.00 (0.04) 294 0.00 CAN 0.78 (5.23) -0.03 (-0.80) 0.09 (0.99) 354 0.01 CHE 0.40 (3.21) 0.00 (0.25) 0.00 (0.04) 294 -0.01 DEU 0.62 (3.44) -0.01 (-0.28) 0.15 (0.74) 294 0.09 DNK 0.35 (2.26) 0.02 (2.58) 0.07 (2.00) 294 0.06 ESP 0.31 (2.18) 0.00 (0.05) 0.07 (0.63) 294 0.02 FIN 0.07 (0.24) -0.01 (-0.21) 0.10 (0.82) 294 0.10 FRA 0.44 (3.10) -0.06 (-1.19) 0.24 (1.19) 294 0.04 GBR 0.48 (5.22) -0.01 (-0.63) 0.12 (1.29) 342 0.09 HKG 0.40 (2.38) 0.00 (-0.00) -0.01 (-0.36) 294 0.00 IRL 0.76 (2.61) 0.00 (-0.15) 0.13 (1.45) 246 0.01 ISR 0.61 (3.51) -0.01 (-0.42) 0.07 (0.71) 234 0.00 ITA 0.17 (1.29) -0.01 (-0.62) 0.12 (1.06) 282 0.05 JPN 0.21 (2.42) -0.01 (-0.31) 0.13 (0.82) 318 0.00 NLD 0.43 (3.08) 0.00 (0.38) 0.06 (0.82) 294 0.02 NOR 0.54 (3.07) -0.01 (-0.72) 0.05 (1.15) 294 0.01 NZL 0.39 (2.88) 0.00 (-0.23) 0.12 (1.87) 294 0.03 PRT 0.16 (0.88) 0.01 (0.30) -0.02 (-0.14) 222 0.00 SGP 0.26 (1.37) 0.00 (0.00) 0.01 (0.06) 294 -0.01 SWE 0.41 (2.41) -0.16 (-1.31) 0.49 (1.85) 294 0.09 USA 0.45 (4.78) -0.01 (-0.88) 0.16 (2.05) 623 0.13 WOR 0.37 (4.08) -0.03 (-1.50) 0.14 (1.78) 354 0.15 112 Table 11 Conditional Risk in Equity Factors Pre and Post 1994 This table reports results from evaluation of different equity strategies in the conditional CAPM. We regress the monthly excess returns of different factors on the shock to the market factor and the conditional risk factor and evaluate returns in the following two-factor model from Proposition 1: 𝐸𝑟  = 𝛼+ 𝛽𝜆+ 𝛽 𝜆  where 𝑟  is the excess return to the risk factor i, 𝛽 and 𝛽  are the beta for the marketand conditional-risk factor, and 𝜆 and 𝜆  are their risk premia. The composite factor COMP is the average return to HML, RMW, CMA, UMD, and BAB. “Compensation for conditional risk” is the product 𝛽 𝜆 . All alphas are in monthly percent. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. Standard errors are bootstrapped to account for generated regressors. Panel A shows the U.S. sample from 1994-2015 and Panel B shows the U.S. sample from 1964-1993. SMB HML RMW CMA UMD BAB COMP Panel A: U.S. Sample Post 1994 Alpha 0.00 0.13 0.55 0.25 0.73 0.97 0.53 (-0.02) (0.86) (3.03) (2.45) (2.43) (4.02) (3.83) Market beta 0.19 -0.12 -0.28 -0.11 -0.34 -0.29 -0.23 (4.74) (-2.79) (-8.74) (-4.51) (-4.32) (-4.13) (-7.60) Conditional risk beta 0.01 0.05 0.04 0.03 0.07 0.11 0.06 (0.47) (1.99) (2.07) (2.05) (1.60) (2.57) (2.52) Compensation for conditional risk 0.01 0.07 0.06 0.05 0.10 0.15 0.09 Fraction of alpha explained by conditional risk 1.53 0.36 0.09 0.16 0.12 0.14 0.14 Observations 262 262 262 262 262 262 262 Adjusted R2 0.05 0.08 0.22 0.11 0.11 0.18 0.32 Panel B: U.S. Sample Pre 1994 Alpha 0.25 0.50 0.18 0.30 0.84 1.03 0.57 (1.71) (3.82) (2.50) (2.88) (4.34) (8.19) (8.82) Market beta 0.22 -0.19 0.00 -0.17 0.02 0.13 -0.04 (6.37) (-6.15) (-0.25) (-9.56) (0.43) (3.28) (-2.62) Conditional risk beta -0.02 0.01 -0.01 0.02 0.01 0.01 0.01 (-1.57) (1.36) (-1.57) (2.52) (0.30) (0.89) (0.98) Compensation for conditional risk -0.02 0.01 -0.01 0.02 0.01 0.01 0.01 Fraction of alpha explained by conditional risk -0.10 0.03 -0.08 0.08 0.01 0.01 0.02 Observations 361 361 361 361 361 361 361 Adjusted R2 0.12 0.11 0.00 0.17 0.00 0.04 0.03 113 Figure 1 The Conditional Price of Risk This figure plots the time series of the conditional price of risk minus the unconditional price of risk. Panel A plots the price of risk in the U.S. sample and Panel B plots the price of risk in the global sample. 0 0 0 0 0 1 1 1 1 1 1 -15.000 -10.000 -5.000 0.000 5.000 10.000 15.000 06-1963 04-1965 02-1967 12-1968 10-1970 08-1972 06-1974 04-1976 02-1978 12-1979 10-1981 08-1983 06-1985 04-1987 02-1989 12-1990 10-1992 08-1994 06-1996 04-1998 02-2000 12-2001 10-2003 08-2005 06-2007 04-2009 02-2011 12-2012 10-2014 Panel A: U.S. conditional price of risk (bt-b) 0 0 0 0 0 1 1 1 1 1 1 -25.000 -20.000 -15.000 -10.000 -5.000 0.000 5.000 10.000 15.000 20.000 06-1986 07-1987 08-1988 09-1989 10-1990 11-1991 12-1992 01-1994 02-1995 03-1996 04-1997 05-1998 06-1999 07-2000 08-2001 09-2002 10-2003 11-2004 12-2005 01-2007 02-2008 03-2009 04-2010 05-2011 06-2012 07-2013 08-2014 09-2015 Panel B: Global conditional price of risk (bt-b) 114 Figure 2 Rolling Realizations of the Conditional-Risk Factor This figure plots the two-year rolling realizations of the conditional-risk factor together with the two-year rolling CAPM alpha to the composite risk factor (Panel B). Shaded bars indicate NBER recessions. The conditional-risk factor is the unexpected return to the market portfolio multiplied by the difference between the conditional and unconditional price of risk. The composite risk factor is the average return to HML, RMW, CMA, UMD, and BAB. The figure shows the U.S. sample. 115 Figure 3 Compensation for Conditional Risk in the Cross-Section of Equity Returns This figure shows how many percentage point of return conditional risk justifies for portfolios sorted on aggregate characteristics. Each month, we sort stocks into ten portfolios based on the aggregate size, book-to-market, profitability, investment, momentum, and beta characteristic. We measure characteristics in cross-sectional percentiles and chose signs such that higher characteristics are associated with higher returns. In the U.S., we use NYSE breakpoints for the portfolios sorts. In the international sample, we use calculate breakpoints based on the 20% largest firms. Portfolios are value-weighted, refreshed, and rebalanced monthly. The figure shows “compensation for conditional risk”, which is the conditional risk beta multiplied by the risk premium on the conditional risk factor. The U.S. sample runs from 1964-2015. The global sample runs from 1986-2015. -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 Compensation for conditional risk (monthly percent) Portfolios sorted on expected return (high U.S. sample Global sample 116 Figure 4 Conditional Risk in the Cross-Section of Equity Returns around the World This figure plots how many percent of the unconditional CAPM alpha to the composite risk factor that can be explained by conditional risk. The composite risk factor is the average return to HML, RMW, CMA, UMD, and BAB. -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Compensation for conditional risk (monthly %) Fraction of alpha explained by conditional risk 117 Figure 5 The Composite Factor has Larger Conditional Risk When it has Higher Expected Return This figure plots 𝛾  from the following regression of the composite risk factor on the conditional risk factor and the conditional risk factor interacted with the value spread of the composite risk factor: 𝐶𝑂𝑀𝑃 = 𝛾 + 𝛾 𝑐 + 𝛾 (𝑐 ×𝑉𝑆 )+ 𝜖  where k denotes country and 𝑉𝑆  is the average value spread in country k at time t of the five factors in 𝐶𝑂𝑀𝑃. The composite factor 𝐶𝑂𝑀𝑃is the average return in country k to HML, RMW, CMA, UMD, and BAB. -0.10 0.00 0.10 0.20 0.30 0.40 0.50 118 Table A1 Robustness: Conditional Risk Using Campbell and Thompson (2008) This reports robustness analysis where the market risk premium used in the conditional-risk factor is measured using the dividend to price ratio of the market portfolio as in Campbell and Thompson (2008). We regress the monthly excess returns of different factors on the shock to the market factor and the conditional risk factor and evaluate returns in the following two-factor model from Proposition 1: 𝐸𝑟  = 𝛼+ 𝛽𝜆+ 𝛽 𝜆  where 𝑟  is the excess return to the risk factor i, 𝛽 and 𝛽  are the beta for the marketand conditional-risk factor, and 𝜆 and 𝜆  are their risk premia. The composite factor COMP is the average return to HML, RMW, CMA, UMD, and BAB. “Compensation for conditional risk” is the product 𝛽 𝜆 . All alphas are in monthly percent. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. The U.S. and global samples run from 1964-2015 and 1986-2015. SMB HML RMW CMA UMD BAB COMP Panel A: U.S. Sample Alpha 0.13 0.35 0.31 0.30 0.79 1.01 0.55 (1.08) (3.39) (3.66) (4.31) (4.76) (7.46) (8.69) Market beta 0.21 -0.17 -0.12 -0.15 -0.13 -0.05 -0.12 (7.80) (-7.35) (-6.47) (-9.71) (-3.38) (-1.67) (-8.71) Conditional risk beta 0.00 0.05 0.00 0.04 0.00 0.02 0.02 (0.30) (4.43) (0.19) (4.39) (0.18) (1.34) (3.11) Compensation for conditional risk 0.00 0.04 0.00 0.03 0.00 0.02 0.02 Fraction of alpha explained by conditional risk 0.02 0.10 0.00 0.08 0.00 0.01 0.03 Observations 624 624 624 624 624 624 624 Adjusted R2 0.09 0.10 0.06 0.15 0.01 0.00 0.12 Panel B: Global Sample Alpha 0.08 0.31 0.32 0.28 0.79 0.77 0.48 (0.73) (2.85) (3.86) (3.82) (4.04) (4.85) (6.50) Market beta 0.05 -0.10 -0.18 -0.08 -0.21 -0.15 -0.14 (2.12) (-4.10) (-9.99) (-5.31) (-4.98) (-4.27) (-8.68) Conditional risk beta -0.01 0.07 0.01 0.03 -0.03 0.04 0.02 (-1.07) (6.01) (1.29) (3.70) (-1.27) (2.14) (2.78) Compensation for conditional risk -0.01 0.06 0.01 0.02 -0.02 0.03 0.02 Fraction of alpha explained by conditional risk -0.16 0.16 0.03 0.08 -0.03 0.04 0.04 Observations 359 359 359 359 359 306 359 Adjusted R2 0.01 0.12 0.22 0.10 0.06 0.06 0.18 119 Table A2 Robustness: Conditional Risk Using Dividend to Price Ratio from Campbell and Thompson (2008) This table reports robustness analysis where the market risk premium used in the conditional-risk factor is measured using the dividend to price ratio of the market portfolio as in Campbell and Thompson (2008). For each exchange, we regress the monthly excess returns COMP on the shock to the market factor and the conditional risk factor for the given exchange. For each exchange, we evaluate returns in the following two-factor model from Proposition 1: 𝐸[𝑟 ]= 𝛼+ 𝛽𝜆+ 𝛽 𝜆  where 𝑟  is the excess return to COMP, 𝛽 and 𝛽  are the beta for the marketand conditional-risk factor, and 𝜆 and 𝜆  are their risk premia. The composite factor COMP is the average return to HML, RMW, CMA, UMD, and BAB. “Compensation for conditional risk” is the product 𝛽 𝜆 . All alphas are in monthly percent. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. The U.S. and global samples runs from 1964-2015 and 1986-2015. Exchange Alpha Market Beta Conditional Risk Beta Compensation for conditional risk Fraction of alpha explained by conditional risk AUS 0.74 (6.32) -0.06 (-3.16) 0.01 (0.97) 0.01 0.02 AUT 0.39 (2.74) -0.07 (-2.88) 0.06 (1.18) 0.01 0.01 BEL 0.43 (3.74) -0.11 (-5.26) 0.10 (4.57) 0.03 0.06 CAN 0.86 (6.82) -0.13 (-5.50) 0.01 (1.39) 0.03 0.04 CHE 0.49 (4.23) -0.10 (-4.19) 0.01 (0.74) 0.01 0.01 DEU 0.76 (5.46) -0.20 (-7.58) 0.11 (4.03) 0.03 0.04 DNK 0.52 (3.91) -0.11 (-4.50) 0.05 (3.24) 0.04 0.07 ESP 0.44 (3.78) -0.10 (-6.07) 0.04 (1.80) 0.01 0.03 FIN 0.47 (2.16) -0.21 (-8.18) 0.03 (2.88) 0.13 0.22 FRA 0.52 (4.03) -0.13 (-5.64) 0.07 (3.80) 0.03 0.06 GBR 0.52 (5.80) -0.05 (-2.88) 0.02 (1.67) 0.01 0.02 HKG 0.46 (3.82) -0.10 (-6.02) 0.01 (0.74) 0.01 0.02 IRL 0.84 (3.15) -0.14 (-4.05) 0.03 (0.64) 0.01 0.01 ISR 0.65 (3.92) -0.05 (-1.81) 0.00 (-0.18) 0.00 0.00 ITA 0.24 (1.90) -0.02 (-0.87) -0.03 (-0.54) 0.00 -0.01 JPN 0.21 (2.57) -0.05 (-3.54) -0.01 (-0.78) 0.00 -0.02 NLD 0.53 (4.48) -0.09 (-3.98) 0.03 (2.08) 0.02 0.03 NOR 0.59 (3.75) -0.05 (-2.17) -0.01 (-0.66) -0.01 -0.02 NZL 0.47 (4.05) -0.02 (-0.78) 0.01 (0.41) 0.00 0.01 PRT 0.25 (1.62) -0.09 (-4.33) 0.05 (0.95) 0.00 0.02 SGP 0.40 (3.10) -0.14 (-7.93) -0.01 (-0.76) -0.01 -0.03 SWE 0.52 (3.23) -0.08 (-3.27) 0.03 (1.15) 0.01 0.02 USA 0.55 (8.69) -0.12 (-8.71) 0.02 (3.11) 0.02 0.03 WOR 0.48 (6.50) -0.14 (-8.68) 0.02 (2.78) 0.02 0.04 126 Table A9 Robustness: Conditional Risk Using Bollerslev et al (2009) and SVIX This reports robustness analysis where the market variance used in the conditional-risk factor is measured using either Bollerslev, Tauchen, Zhou (2009) or the SVIX measure of risk neutral variance (Martin, 2017). For each exchange, we regress the monthly excess returns COMP on the shock to the market factor and the conditional risk factor for the given exchange. We evaluate returns in the following two-factor model from Proposition 1: 𝐸[𝑟 ]= 𝛼+ 𝛽𝜆+ 𝛽 𝜆  where 𝑟  is the excess return to COMP, 𝛽 and 𝛽  are the beta for the marketand conditional-risk factor, and 𝜆 and 𝜆  are their risk premia. The composite factor COMP is the average return to HML, RMW, CMA, UMD, and BAB. “Compensation for conditional risk” is the product 𝛽 𝜆 . All alphas are in monthly percent. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. The sample is the U.S. sample from 1990-2015 (Panel A) and 1996-2015 (Panel B). SMB HML RMW CMA UMD BAB COMP Panel A: Bollerslev et. al (2009) Alpha 0.00 0.16 0.48 0.25 0.79 0.99 0.53 (-0.01) (1.05) (3.47) (2.42) (2.95) (4.58) (5.17) Market beta 0.18 -0.13 -0.25 -0.13 -0.30 -0.26 -0.21 (4.43) (-3.72) (-8.22) (-5.77) (-4.91) (-5.44) (-9.28) Conditional risk beta 0.00 0.02 0.02 0.02 0.04 0.05 0.03 (0.44) (2.86) (2.88) (3.55) (2.48) (4.64) (5.55) Compensation for conditional risk 0.01 0.08 0.07 0.06 0.11 0.17 0.10 Fraction of alpha explained by conditional risk 1.17 0.32 0.13 0.20 0.13 0.15 0.16 Observations 311 311 311 311 311 311 311 Adjusted R2 0.05 0.06 0.19 0.12 0.08 0.14 0.27 Panel B: SVIX Alpha 0.05 0.15 0.56 0.25 0.76 1.04 0.55 (0.21) (0.80) (3.33) (2.12) (2.27) (4.00) (4.42) Market beta 0.20 -0.12 -0.29 -0.12 -0.35 -0.31 -0.24 (4.05) (-3.19) (-7.94) (-4.55) (-4.92) (-5.50) (-8.87) Conditional risk beta 0.02 0.07 0.05 0.04 0.06 0.11 0.07 (0.95) (3.68) (3.00) (3.77) (1.71) (4.28) (5.31) Compensation for conditional risk 0.02 0.06 0.05 0.04 0.05 0.11 0.06 Fraction of alpha explained by conditional risk 0.30 0.31 0.08 0.15 0.07 0.09 0.10 Observations 238 238 238 238 238 238 238 Adjusted R2 0.06 0.08 0.23 0.12 0.10 0.17 0.31 127 Table A10 Robustness: Index of Expected Return This reports robustness analysis where the market risk premium used in the conditional-risk factor is measured as the average of estimators by: the Campbell and Thompson (2008), Kelly and Pruitt (2013), Lettau and Ludvigson (2001), and Martin (2017). We regress the monthly excess returns of different factors on the shock to the market factor and the conditional risk factor and evaluate returns in the following two-factor model from Proposition 1: 𝐸𝑟  = 𝛼+ 𝛽𝜆+ 𝛽 𝜆  where 𝑟  is the excess return to the risk factor i, 𝛽 and 𝛽  are the beta for the marketand conditional-risk factor, and 𝜆 and 𝜆  are their risk premia. The composite factor COMP is the average return to HML, RMW, CMA, UMD, and BAB. “Compensation for conditional risk” is the product 𝛽 𝜆 . All alphas are in monthly percent. t-statistics are reported below the parameter estimates and statistical significance at the 5% level is indicated in bold. The U.S. and global samples run from 1964-2015 and 1986-2015. SMB HML RMW CMA UMD BAB COMP Panel A: Bollerslev et. al (2009) Alpha 0.17 0.36 0.26 0.29 0.69 0.93 0.51 (1.46) (3.50) (3.17) (4.20) (4.25) (7.13) (8.32) Market beta 0.21 -0.17 -0.12 -0.15 -0.13 -0.05 -0.12 (7.87) (-7.32) (-6.60) (-9.57) (-3.53) (-1.68) (-9.07) Conditional risk beta -0.01 0.00 0.03 0.01 0.10 0.09 0.05 (-0.89) (0.44) (3.91) (1.36) (5.67) (6.69) (7.40) Compensation for conditional risk -0.01 0.00 0.03 0.01 0.08 0.08 0.04 Fraction of alpha explained by conditional risk -0.06 0.01 0.10 0.03 0.11 0.08 0.08 Observations 623 623 623 623 623 623 623 Adjusted R2 0.09 0.08 0.08 0.13 0.06 0.07 0.18 Panel B: SVIX Alpha 0.08 0.29 0.27 0.25 0.64 0.68 0.42 (0.72) (2.66) (3.23) (3.39) (3.31) (4.48) (5.90) Market beta 0.06 -0.10 -0.18 -0.08 -0.22 -0.12 -0.14 (2.26) (-4.06) (-10.05) (-5.37) (-5.26) (-3.47) (-9.36) Conditional risk beta 0.00 0.05 0.01 0.03 0.06 0.08 0.04 (-0.43) (5.44) (2.04) (4.97) (3.59) (6.04) (7.48) Compensation for conditional risk -0.01 0.07 0.02 0.04 0.08 0.12 0.06 Fraction of alpha explained by conditional risk -0.08 0.20 0.07 0.15 0.12 0.14 0.13 Observations 354 354 354 354 354 306 354 Adjusted R2 0.01 0.11 0.23 0.13 0.10 0.15 0.28 128 Figure A1 Conditional Risk in Volatility Managed Portfolios around the World This figure the loading of volatility managed portfolios on the conditional risk factor around the world. -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 129 Figure A2 Conditional Risk in Time Series Momentum around the World This figure the loading of time-series momentum portfolios on the conditional risk factor around the world. -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 130 Figure A3 Conditional Risk in Cross-Sectional Risk Factors around the World This figure shows the compensation for conditional risk is in different cross-sectional strategies in different countries. The composite risk factor is the average return to HML, RMW, CMA, UMD, and BAB. -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 HML RMW CMA UMD BAB Chapter 3 Bettin Against Correlation: Testing Theories of the Low-Risk E↵ect with Cli↵Asness, Andrea Frazzini, and Lasse Heje Pedersen Abstract: We test whether the low-risk e↵ect is driven by (a) leverage constraints and thus risk should be measured using beta vs. (b) behavioral e↵ects and thus risk should be measured by idiosyncratic risk. Beta depends on volatility and correlation, where only volatility is related to idiosyncratic risk. Hence, the new factor betting against correlation (BAC) is particularly suited to di↵erentiating between leverage constraints vs. lottery explanations. BAC produces strong performance in the US and internationally, supporting leverage constraint theories. Similarly, we construct the new factor SMAX to isolate lottery demand, which also produces positive returns. Consistent with both leverage and lottery theories contributing to the low-risk e↵ect, we find that BAC is related to margin debt while idiosyncratic risk factors are related to sentiment and casino profits. Keywords: asset pricing, leverage constraints, lottery demand, margin, sentiment. JEL classification: G02, G12, G14, G15. Asness and Frazzini are at AQR Capital Management, Two Greenwich Plaza, Greenwich, CT 06830, email: [email protected]; web: www.econ.yale.edu/ af227/. Pedersen is at AQR Capital Management, Copenhagen Business School, NYU, and CEPR; e-mail: [email protected]; web: www.lhpedersen.com. Gormsen is at Copenhagen Business School; e-mail [email protected]. We are grateful for helpful comments from Robin Greenwood, Toby Moskowitz, and Tatyana Marchuk (discussant) as well as from seminar participants at AQR, BI-SHoF conference on Asset Pricing and Financial Econometrics 2017, the Q-group spring seminar 2017, Instinet Annual Global Quant Research Conference, Bernstein Research conference on Controversies in Quantitative Finance, and Copenhagen Business School. Gormsen and Pedersen gratefully acknowledge support from FRIC Center for Financial Frictions (grant no. DNRF102). 131 1 Introduction The relation between risk and expected return is a central issue in finance with broad implications for investment behavior, corporate finance, and market efficiency. One of the major stylized facts on the risk-return relation, indeed in empirical asset pricing more broadly, is the observation that assets with low risk have high alpha, the so-called “low-risk e↵ect” (Black, Jensen, and Scholes, 1972).1However, the literature o↵ers di↵erent views on the underlying economic drivers of the low-risk e↵ect and the best empirical measures. In short, the debate is whether (a) the low-risk e↵ect is driven by leverage constraints and risk should be measured using systematic risk vs. (b) the low-risk e↵ect is driven by behavioral e↵ects and risk should be measured using idiosyncratic risk.2This paper seeks to test these theories using broad global data, controlling for more existing factors, using measures of the economic drivers, and using new factors that we call betting against correlation (BAC) and scaled MAX (SMAX) that help solve the problem that the existing low-risk factors are highly correlated. The theory of leverage constraints for the low-risk e↵ect was proposed by Black (1972) and extended by Frazzini and Pedersen (2011, 2014) who study an extensive set of global stocks, bonds, credits, and derivatives based on their betting against beta (BAB) factor. Hence, the systematic low-risk e↵ect is based on a rigorous economic theory and has survived more than 40 years of out of sample evidence. Further, a number of papers document evidence consistent with the underlying economic mechanism of leverage constraints: Jylh¨a (2018) finds that exogenous changes in margin requirements influence the slope of the security market line, Boguth and Simutin (2017) show that funding constraints as proxied by mutual fund beta predict BAB, Malkhozov, Mueller, Vedolin, and Venter (2016) show that international illiquidity predict BAB, and Adrian, Etula, and Muir (2014) document a strong link between the return to BAB and financial intermediary leverage.3 The alternative view is that the low-risk e↵ect stems from behavioral biases leading to 1We use the standard term “low-risk e↵ect” to refer to the (risk-adjusted) return spread between lowand high-risk stocks (i.e., it does not just refer to low-risk stocks). 2A related but distinct debate is whether other factors subsume low-risk factors or vice versa (see, for instance, Novy-Marx, 2014 and Fama and French, 2016) and we also address this debate herein as discussed below. We note however, that BAB and BAC are based on equilibrium theories of asset pricing while the other factors are ad hoc empirical specifications. 3See also the related evidence on corporate finance and banking (Baker and Wurgler, 2015, 2016), benchmark constraints (Brennan, 1993; Baker, Bradley, and Wurgler, 2011) and leverage constraints and di↵erences of opinion (Hong and Sraer, 2015). 132 a preference for lottery-like returns (Barberis and Huang, 2008; Brunnermeier, Gollier, and Parker, 2007) and therefore the focus should be on idiosyncratic risk. Indeed, Ang, Hodrick, Xing, and Zhang (2006, 2009) find that stocks with low idiosyncratic volatility (IVOL) have high risk-adjusted returns in the U.S. and internationally. In a similar vein, Bali, Cakici, and Whitelaw (2011) consider stocks sorted on the maximum return (MAX) over the past month, finding that low MAX is associated with high risk-adjusted returns,4and Bali, Brown, Murray, and Tang (2017) argue that the low-risk e↵ect is driven by idiosyncratic risk rather than systematic risk. Also, Liu, Stambaugh, and Yuan (2017) argue that the low-risk e↵ect is driven by idiosyncratic risk and only appears among over-priced stocks. The challenge with the existing literature is that it seeks to run a horse race between factors that are, by construction, highly correlated since risky stocks are usually risky in many ways. Indeed, the reason that all these factors are known under the umbrella term “the low-risk e↵ect” is that they are so closely related. Hence, the most powerful way to credibly distinguish these theories is to construct a new factor that captures one theory while at the same being relatively unrelated to factors capturing the alternative theory. To accomplish this, we decompose BAB into two factors: betting against correlation (BAC) and betting against volatility (BAV). BAC goes long stocks that have low correlation to the market and shorts those with high correlation, while seeking to match the volatility of the stocks that are bought and sold. Likewise, BAV goes long and short based on volatility, while seeking to match correlation. This decomposition of BAB creates a component that is relatively unrelated to the behavioral factors (BAC) and a closely related component (BAV). To see that BAC is relatively unrelated to the behavioral-based factors, we note that the long and short sides of BAC have similar average volatility, skewness, and MAX.5At the same time, sorting on ex ante market correlation successfully creates a BAC factor that is long stocks with low ex post market correlations (and short stocks with high ones). Since stocks with low market correlation have low market betas, the theory of leverage constraints implies that BAC has positive risk-adjusted returns, just like BAB. Empirically, we find that BAC is about as profitable as the BAB factor and BAC has a highly significant CAPM alpha as predicted by the theory of leverage constraints. This evidence thus supports the theory of leverage constraints and is clearly separate from the behavioral factors. To address the findings of Liu, Stambaugh, and Yuan (2017), we double-sort on their measure 4See also the measure related to idiosyncratic skewness studied by Boyer, Mitton, and Vorkink (2009). 5See also the measure related to idiosyncratic skewness studied by Boyer, Mitton, and Vorkink (2009). 133 of each stock’s “mispricing” and our measure of each stock’s correlation with the market, finding that low-correlation stocks deliver higher risk-adjusted returns in each quintile of mispricing, providing further evidence that the low-risk e↵ect is not just about idiosyncratic risk or its interaction with mispricing. Another challenge to the low-risk e↵ect, both with systematic and idiosyncratic risk, is posed by Fama and French (2016) who argue that that a five-factor model of the market (MKT), size (SMB), value (HML), profitability (RMW), and investment (CMA) explains the low-risk e↵ect (and the majority of the cross-section of returns more broadly, except for momentum). While they don’t test BAB explicitly, they suggest that there is no relationship between alpha and systematic risk once controlling for the five factors. We study this question explicitly and, further, we also control for short-term reversal (REV), which is particularly relevant for the idiosyncratic risk factors (due to their high turnover as discussed below). We find significant alpha for BAB and BAC for a variety of combinations of control factors in the US and globally. For example, BAC has a five-factor alpha of 0.62 Turning to the behavioral theory, we next consider the factors that go long stocks with low MAX return (LMAX) or low idiosyncratic volatility (IVOL). We sign all factors such that they are long low-risk stocks (even though the literature is not always consistent in this regard).6Since IVOL is already based on decomposing volatility into its systematic and idiosyncratic parts, we do not further decompose IVOL. For LMAX, however, we can again create a new factor that helps di↵erentiate alternative hypotheses by removing the common component (namely, volatility). Just like we created BAC to remove the e↵ect of volatility from beta (which left us with correlation), we can remove the e↵ect of volatility from MAX: We construct a scaled-MAX (SMAX) factor that goes long stocks with low MAX return divided by ex ante volatility and shorts stocks with the opposite characteristic. This factor captures lottery demand in a way that is not as mechanically related to volatility as it is more purely about the shape of the return distribution. Behavioral theories imply that these idiosyncratic risk factors should have positive alphas, which we confirm in the data. In the U.S., SMAX, LMAX, and IVOL all produce significant alphas with respect to the Fama-French five-factor model, but SMAX performs stronger than both LMAX and IVOL. In the global sample, however, none of the factors are robust to controlling for the five Fama-French factors and short-term reversal. 6For example, LMAX is the negative of the FMAX factor considered by Bali, Cakici, and Whitelaw (2011). 134 To go beyond studying the risk-adjusted returns, we study additional predictions arising from the di↵erent economic theories for the low-risk e↵ect. To capture the idea underlying the theory of leverage constraints, we consider the margin debt held by customers at NYSE member organizations (broker-dealers). To capture the behavioral e↵ects, we consider investor sentiment as suggested by Liu, Stambaugh, and Yuan (2017). We find that BAB and BAC are predicted by measures of leverage constraints, while these factors are not predicted by investor sentiment. In contrast, MAX and IVOL are related to sentiment, but not measures of leverage constraints. This evidence is consistent with both of the alternative theories playing a role and that the alternative factors may, to some extent, capture di↵erent e↵ects.7 The result that sentiment predicts the idiosyncraticrisk factors supports the role of behavioral biases, but it does not tell exactly which behavioral bias is at play. To study behavioral lottery demand more specifically, we consider two new measures of lottery demand: profits earned by casinos in the U.S. and sales of lottery tickets in the UK. We find that casino profits predict the returns to LMAX and IVOL, consistent with theories of lottery demand. We find no evidence, however, that higher sales of lottery tickets predict higher return to any of the lottery factors. Finally, we find that neither BAB nor BAC load on any of the lottery demand measures, consistent with these factors being driven by leverage constraints and not lottery demand. Having tested the specific predictions arising from the competing theories of the leverage e↵ect, we next run “horseraces” between the di↵erent low-risk factors to judge their relative importance. We regress each type of low-risk factor (systematic/idiosyncratic) on the alternate type of low-risk factor as well as several controls (the Fama-French factors and short-term reversal). We find that BAB and BAC are robust to controlling for LMAX in the US and globally. Turning things around, we find that SMAX is robust to controlling for BAB in the US, but LMAX and IVOL both have insignificant alphas when we control for BAB (recall that the behavioral factors were insignificant globally even before we control for BAB). These insignificant alphas of the idiosyncratic risk factors arise because their returns are 7We also consider other alternative theories of the low-risk e↵ect. In particular, the literature also includes so-called Money Illusion as suggested by Modigliani and Cohn (1979) and studied by Cohen, Polk, and Vuolteenaho (2005). However, we find no evidence that inflation predicts either BAB or BAC. This result holds despite the fact that we include the 70’s and 80’s, time periods that included large shocks to inflation. 135