Implied Dividend Volatility and Expected Growth

By Niels J. Gormsen, Ralph S.J. Koijen, and Ian Martin∗

A large literature is concerned with mea- started to fall. suring economic uncertainty and quantify- In addition to introducing this market, ing its impact on real decisions, such as in- we also provide new theoretical results that vestment, hiring, and R&D, and ultimately show how these data can be used to de- economic growth (Bloom, 2009; Jurado rive lower bounds on expected dividend re- et al., 2015). The COVID-19 pandemic turns and on expected growth rates, by ma- underscores the importance of timely mea- turity, by exploiting the insights of Mar- sures of uncertainty and expected growth tin (2017). This provides an alternative to across horizons. methods used in the literature using vec- Asset prices, such as dividend futures tor autoregressions or survey expectations, (van Binsbergen et al., 2013; Gormsen and and sharpens alternative bounds in the lit- Koijen, 2020) and index options (Gao and erature. Martin, 2020), provide particularly useful measures as they are forward looking and I. Pricing and Riskiness of Dividends available at high frequencies. Dividend futures are claims on the dividends of the ag- We denote dividends on the aggregate gregate stock market in a particular year. stock market by Dt and St denotes the value As dividend futures are differentiated by of the aggregate stock market. The present- maturity, just like nominal and real bonds, value identity implies we can use these prices to obtain growth ∞ expectations by maturity. (1) St = Et [Dt+τ Mt:t+τ ] , We extend this literature by using new τ=1 X data on the prices of options on index- τ level dividends, from which we can compute where Mt:t+τ = s=1 Mt+s and Mt denotes the stochastic discount factor. We de- implied dividend volatility. These implied Q volatilities differ from the VIX which mea- fine the τ-period dividend strip as Pt(τ) = sures uncertainty about stock prices, not Et [Dt+τ Mt:t+τ ] and the dividend futures only uncertainty about dividends. price as Ft(τ) = Pt(τ)/Et [Mt:t+τ ]. We construct a term structure of im- A. Data plied dividend volatilities that characterizes how uncertainty varies across horizons. We We use data on dividend futures for the study how this term structure developed S&P 500 index in the US (SPX) and for over the COVID-19 crisis, documenting a the Euro Stoxx 50 index in Europe (SX5E). substantial increase in the volatility of near- We source these data from Bloomberg. We future dividends that lingers even as the also use data on two ETFs; one ETF tracks volatility of the overall market portfolio has long-term Treasuries (with ticker TLT) and the other ETF tracks the investment-grade ∗ Gormsen: University of Chicago, Booth corporate bond market in the US (with School of Business, 5807 South Wood- ticker LQD). These data are from the Cen- lawn Avenue, 60637 Chicago, Illinois, USA, [email protected]. Koijen: Univer- ter for Research in Security Prices. sity of Chicago, Booth School of Business, 5807 South We use data on Euro Stoxx 50 divi- Woodlawn Avenue, 60637 Chicago, Illinois, USA, dend options trading on the Eurex Ex- [email protected]. Martin: London change. These are European options on School of Economics, Houghton Street, London WC2A 2AE, United Kingdom, [email protected]. We thank index dividends. The ten nearest succes- Lasse Pedersen for comments and suggestions. sive annual contracts of the December cy- 1 2 PAPERS AND PROCEEDINGS MONTH YEAR cle are available for trading at any point market prices substantial losses in the near in time. The Eurex Exchange records the term. daily settlement prices on the options and The left panel of Figure I.B shows the computes the ATM implied volatilities us- implied dividend volatility of the Decem- ing a Black (76) model. We source these ber 2021 option and the implied volatil- volatilities from Bloomberg. We note that ity of the aggregate stock market. In the liquidity in the dividend options market right panel, we plot the term structure of is a concern. We view the current paper implied dividend volatilities for the 2020, mostly as a proof of concept of what can 2021, and 2022 contracts, alongside the im- be learned from these markets that may be plied volatility of the market, in January, more widely traded in the future. March, and October of 2020. For dividend options, the maturity coin- Implied volatilities before the COVID-19 cides with the year in which the dividends crisis increase with maturity, and are par- are paid. This implies that we simultane- ticularly low for the 2021 contract. The ously vary the timing of the dividend and level of volatility is comparable to the his- the maturity of the option. We use the De- torical annual dividend volatility. cember 2021, December 2022, and Decem- During the crisis, the implied dividend 1 ber 2023 contracts in our analysis. For the volatility increases sharply and, in case of Euro Stoxx 50, we choose the 12-month and the 2021 contract, rises above the implied 24-month implied volatilities (VSTOXX), volatility of the market. This increase which we get from Bloomberg. We linearly shows that short-term dividend growth is interpolate both series to target the Decem- strongly heteroskedastic. The volatility of ber 2021 maturity for the market as well. the short-term dividends remains high at All volatilities are annualized. the end of our sample, with the volatility of We sample our data weekly and use a the short-term claims approximately at the sample from January 2020 until October same level as the market. 2020. As such, the relative increase in volatility during the crisis is much stronger for B. Empirical results the short-term dividends than for the mar- We study how prices and implied volatil- ket and the increase is more persistent. A ities of both indexes and dividends changed key takeaway from this section is that finan- during the COVID-19 crisis. cial markets price the pandemic via lower Figure B in the Online Appendix shows dividend prices and high uncertainty about that aggregate stock markets in Europe and short-term cash flows. As such, while the the US fell by 20-30%, while short-term market indexes have largely recovered, the dividends fell even more for both indexes. pandemic is still reflected in the pricing of During the same period, Treasuries rallied near-future cash flows. and investment-grade corporate bonds fell by about 10%. II. Expected Returns and Growth Financial markets recovered since then, with the US stock market and the The price of a dividend claim reflects a investment-grade corporate bond market combination of the expected return on the recovering fully and the European stock claim and the expected dividend. We will market recovering about half of its losses. derive a lower bound on the expected re- However, short-term dividends have not ex- turn, and hence on the expected dividend. perienced the same recovery. Prices are still down by almost 20% in the US and more A. Methodology than 30% in Europe, suggesting that the We define Rτ as the spot-return on 1 t+τ While the December 2020 contract is also available, the τ-period dividend claim, Rτ = Dt+τ . its implied volatility mechanically dwindles during our t+τ Pt(τ) sample as more dividends are announced. Our starting point is the following identity, VOL. VOL NO. ISSUE SHORT TITLE FOR RUNNING HEAD 3 40 40 30 30 20 20 Volatility (%) Volatility (%) 10 10 0 0 2020m1 2020m4 2020m7 2020m10 January March October Date 2021 2022 Market '21 Dividend '21 2023 Market

Figure 1. Volatility dynamics during the COVID-19 crisis. The years in the right panel correspond to the maturities of the implied dividend volatilities.

τ which holds for any returns Rt+τ and Rt+τ : to the asset return (as is the case in models that generate constant price-dividend τ cov∗(R ,Rt+τ ) ratios) then this condition reduces to the E [Rτ ] Rf = t t+τ t t+τ t f NCC of Martin (2017). This applies, for − Rt cov (M R ,Rτ ). example, to models such as Barro (2006), − t t+τ t+τ t+τ in which the NCC holds. Accordingly, the We use asterisks to denote risk-neutral mo- risk-neutral variance of Rt+τ (τ) constitutes f a lower bound on expected returns, giving ments and write Rt for the gross risk- free rate between t and t + τ. This rise to the following bound on expected div- relationship—a variant of the identity in idends: Martin (2017)—was exploited in the con- τ vart∗(Rt+τ ) f text of currency returns by Kremens and Et[Dt+τ ] Pt(τ) + R ≥ f t Martin (2019). Rt We explore two sets of assumptions to use this identity to derive a lower bound In Appendix A, we show that the covari- for expected returns and expected dividend ance is negative in bad states of the world in growth rates. We will refer to these ap- the Campbell and Cochrane (1999) external proaches as Method 1 and Method 2. habit model, but is positive in the Bansal τ Method 1: We pick Rt+τ = Rt+τ , which and Yaron (2004) long-run risks model for means that the first term on the RHS is the short maturities. Also, the covariance term risk-neutral variance on the dividend and is potentially positive in models in which thus observable. We assume that the sec- returns on dividends are not strongly neg- ond covariance term is non-positive. If the atively correlated with the SDF. We there- return on the dividend claim is proportional fore also consider an alternative assump- 4 PAPERS AND PROCEEDINGS MONTH YEAR tion. on the 2021 dividend varies from 0.2% be- M Method 2: We (i) pick Rt = Rt , which fore the crisis to 12.4% during the down- is the return on the aggregate stock mar- turn, and 5.0% at the end of our sample. ket, and (ii) assume that returns, dividends, This points to high expected excess re- and the SDF are jointly log-normally dis- turns on short-term claims during the cri- tributed.2 As we show in Appendix A, this sis, consistent with van Binsbergen et al. implies (2012) and van Binsbergen et al. (2013). However, it is inconsistent with the models τ M ρtσ∗(R )σ∗(R ) that motivate the covariance constraint in E [Rτ ] Rf = t t+τ t t+τ t t+τ t f the first place. Hence, we are less comfort- − Rt M τ able using this bound, and these results are cov (M R ,R ), − t t+τ t+τ t+τ best viewed as a rejection of the models in τ M which the conditional covariance condition where ρt = corrt(Rt+τ ,Rt+τ ) and σt∗ denotes risk-neutral volatility. We further as- is satisfied as the implications for dividend sume that the second covariance term is volatility under the null of the model lead non-positive. (If one adopts the perspective to too much volatility in discount rates on of an investor with log utility who chooses short-term claims. Method 2 instead relies on a covariance to invest fully in the market, then Mt+τ = M restriction that mimics Martin (2017). As 1/Rt+τ so that the covariance term is ex- actly zero. In this case, the bound holds estimating the conditional correlation is with equality.) The lower bound above challenging, we consider two values that we again gives rise to a lower bound on ex- consider plausible during times of stress, pected dividends: ρt = 50% or 75%. For these values, we plot the lower bound τ M for expected excess returns in annualized ρtσt∗(Rt+τ )σt∗(Rt+τ ) f Et[Dt+τ ] Pt(τ) f + Rt .terms in the left panel of Figure II.B. The ≥ R t expected excess return peaks at 4.5%, when Method 2 requires an estimate of ρt = 50%, or 6.7%, when ρt = 75%. As τ M before, this suggests substantial expected corrt(R ,R ). As estimating a t+τ t+τ 3 correlation model is beyond the scope excess returns on dividend claims. of this paper, we will present results for In the right panel of Figure II.B, we a range of values. We remark that the use these estimates to compute a lower assumption of log-normality is likely vio- bound on the expected dividend in Decem- lated, as discussed by Martin (2017), but ber 2021. To simplify the interpretation, the hope is that this violation has limited we scale this expectation by the December impact on the final results. We leave it 2019 dividend. This bound sharpens the for future research to derive bounds under lower bound in Gormsen and Koijen (2020), more general distributional assumptions. which corresponds to ρt = 0, and provides an alternative to VAR methods and survey B. Empirical Results expectations. We draw two conclusions from this fig- f ure. First, even though we tighten the We approximate Rt 1 during our sample. Method 1 implies that' the lower bound lower bound—as the lower bound for ex- on the annualized expected excess returns pected excess returns is well above zero— most of the variation in dividend futures is due to growth expectations. This under- 2We need the latter assumption because derivatives whose prices would reveal the risk-neutral covariance scores the usefulness of dividend futures for between the market return and dividend growth are not widely traded. By contrast, Kremens and Martin (2019) 3As the volatilities of all dividend claims and the were able to exploit the fact that index quanto contracts, market move significantly during the crisis, so do the which reveal the corresponding risk-neutral covariance lower bounds on the expected excess returns. We refer between the market return and currency appreciation, to Gormsen (2020) for estimates of variation in expected are traded. excess returns across maturities. VOL. VOL NO. ISSUE SHORT TITLE FOR RUNNING HEAD 5

Expected excess return Expected dividend 8 r=50% 0 r=50% r=75% r=75% r=0% -10 6 -20 4 -30 Expected dividend (%) 2 -40 Expected excess return dividend claim (%) -50 0

2020m1 2020m4 2020m7 2020m1 2020m4 2020m7 2020m10 2020m10 Date Date

Figure 2. A lower bound on expected excess returns on the short-term dividend claim (left panel )and on the expected dividend in 2021, scaled by the 2019 dividend (right panel). The different lines correspond to conditional correlations between market returns and short-term dividend returns. forecasting economic growth. Second, while Quarterly Journal of Economics, 121(3): the market recovers, in the second part of 823–866, 2006. 2020, the near-term growth expectations improve only slightly and, in fact, deteri- Nicholas Bloom. The impact of uncer- orate towards the year’s end. Paired with tainty shocks. Econometrica, 77(3):623– the high level of implied dividend volatili- 685, 2009. ties, the short-term economic outlook is un- certain and not expected to recover in the Tim Bollerslev, George Tauchen, and Hao near term in Europe. Zhou. Expected stock returns and variance risk premia. Review of Financial REFERENCES Studies, 22:4463–4493, 2009.

Ravi Bansal and Amir Yaron. Risks for the John Y. Campbell and John H. Cochrane. long run: A potential resolution of asset By force of habit: A consumption-based pricing puzzles. Journal of Finance, 59 explanation of aggregate stock market (4):1481–1509, 2004. behavior. Journal of Political Economy, 107(2):205–251, 1999. Ravi Bansal, Dana Kiku, and Amir Yaron. An empirical evaluation of the long-run Ian Dew-Becker, Stefano Giglio, Anh risks model for asset prices. Critical Fi- Le, and Marius Rodriguez. The nance Review, 1:183–221, 2012. price of variance risk. Journal of Financial Economics, 123(2):225 – Robert J Barro. Rare disasters and asset 250, 2017. ISSN 0304-405X. doi: markets in the twentieth century. The https://doi.org/10.1016/j.jfineco.2016.04.003. 6 PAPERS AND PROCEEDINGS MONTH YEAR

Itamar Drechsler and Amir Yaron. What’s vol got to do with it. Review of Financial Studies, 24:1–45, 2011. Can Gao and Ian Martin. Volatility, Valu- ation Ratios, and Bubbles: An Empirical Measure of Market Sentiment. working paper, 2020. Niels Joachim Gormsen. Time variation of the equity term structure. Journal of Fi- nance, forthcoming, 2020. Niels Joachim Gormsen and Ralph S J Koi- jen. Coronavirus: Impact on Stock Prices and Growth Expectations. The Review of Asset Pricing Studies, 10(4):574–597, 2020. Kyle Jurado, Sydney C. Ludvigson, and Serena Ng. Measuring uncertainty. American Economic Review, 105(3): 1177–1216, March 2015. Lukas Kremens and Ian Martin. The Quanto Theory of Exchange Rates. American Economic Review, 109(3):810– 843, 2019. Ian Martin. What is the Expected Return on the Market? The Quarterly Journal of Economics, 132(1):367–433, 2017. Jules van Binsbergen, Michael Brandt, and Ralph Koijen. On the timing and pricing of dividends. American Economic Re- view, 102(4):1596–1618, 2012. Jules H. van Binsbergen, Wouter Hueskes, Ralph S.J. Koijen, and Evert B. Vrugt. Equity yields. Journal of Financial Eco- nomics, 110(3):503 – 519, 2013. Jessica A. Wachter. Can time-varying risk of rare disasters explain aggregate stock market volatility? Journal of Finance, 68:987–1035, 2013. VOL. VOL NO. ISSUE SHORT TITLE FOR RUNNING HEAD 7

Online appendix

A1. Method 1

We evaluate whether the covariance condition holds such that the bound is indeed a lower bound in several asset pricing models.

Lucas model

Consider a Lucas model with power utility and similar technology processes as in Camp- bell and Cochrane (1999)

Δct+1 = g + vt+1,

Δdt+1 = g + wt+1, where v N(0, σ2), w N(0, σ2 ), and corr(v , w ) = ρ. The dividend price is t+1 ∼ t+1 ∼ w t+1 t+1

Et (Mt+1Dt+1) = δDt Et (exp (1 γ)g γvt+1 + wt+1 ) { − − } 1 1 2 = D R− exp g + σ γσσ ρ , t f 2 w − w 1 1 2 2 where Rf = δ− exp γg γ σ . The dividend return, Rd,t+1 = Dt+1/ Et (Mt+1Dt+1), is { − 2 } 1 R = R exp γσσ ρ σ2 + w , d,t+1 f w − 2 w t+1 2 so E Rd,t+1 = Rf exp γσσwρ and var log Rd,t+1 = σw. As Mt+1 and Rd,t+1 are conditionally lognormal, the NCC{ holds for} the dividend strip return if and only if

log Et Rd,t+1 log Rf − σt(log Rd,t+1) . σt(log Rd,t+1) ≥

The quantity on the left-hand side of the inequality is, essentially, the Sharpe ratio of the dividend strip. Hence the NCC condition holds if and only if

γρσ σ . ≥ w However, we can also compute the covariance term and avoid bounds altogether

2 cov (Mt+1Rd,t+1,Rd,t+1) = E Mt+1Rd,t+1 E (Rd,t+1) . − The first term is given by

2 2 2 E Mt+1Rd,t+1 = E Rf exp 2γσσwρ σw + 2wt+1 γg γvt+1 − − − 1 = R2 exp 2γσσ ρ σ2 + 2σ2 γg + γ2σ2 2 σσ ργ f w − w w − 2 − w 2 = Rf exp σw Putting it all together implies that

var∗ (Rd,t+1) = Rf [E Rd,t+1 Rf + cov (Mt+1Rd,t+1,Rd,t+1)] − = R2 exp σ2 1 . f { w} − 8 PAPERS AND PROCEEDINGS MONTH YEAR

We observe the left-hand side and notice that it is highly volatile, which rejects the model. More broadly, in any model of the form

1 1 2 2 M = R− exp λ σ λ v , t+1 f {−2 t − t t+1} 1 R = R exp μ σ + w , d,t+1 f { t − 2 w t+1} for any risk price λt and μt = ρλtσσw, it holds

2 2 var∗ (R ) = R exp σ 1 . d,t+1 f { w} − These conditions are satisfied in the Campbell and Cochrane (1999) model to whichwe turn next.

Campbell and Cochrane (1999) In this subsection, we use the notation from Campbell and Cochrane (1999) without further comment. The price of a claim to the first dividend is

Dt+1 Et (Mt+1Dt+1) = Dt Et Mt+1 D t = δDt Et (exp γg γ (φ 1)(st s) + [1 + λ(st)]vt+1 + g + wt+1 ) {− − { − − } } 2 2 γ σ 1 2 γρσσw = δDt exp g(1 γ) + γ(1 φ)(st s) + (1 2(st s)) + σ 1 2(st s) . − − − 2 − − 2 w − − − 2S S p So the return on this dividend strip is

2 2 Dt+1 Dt+1 1 γ σ 1 2 γρσσw Rd,t+1 = = δ− exp g(γ 1) γ(1 φ)(st s) (1 2(st s)) σ + 1 2(st s) . (M D ) D − − − − − 2 − − − 2 w − − Et t+1 t+1 t 2S S p Hence the expected return is

2 2 1 γ σ γρσσw Et Rd,t+1 = δ− exp gγ γ(1 φ)(st s) 2 (1 2(st s)) + 1 2(st s) − − − − 2S − − S − − q γρσσw = Rf exp 1 2(st s) , S − − q and 2 vart log Rd,t+1 = σw . From the above equations, the NCC holds for the dividend strip return if and only if γρσ 1 2(st s) σw. S − − ≥ q This condition mimics the Lucas model above when s s and γLucas = γ . t S Figure A.A1 shows that the NCC holds in sufficiently− bad states of the world, butnot in good states of the world and not at the steady state level of habit, S.

Bansal and Yaron (2004)

In this subsection, we use the notation from Bansal and Yaron (2004) without further comment. We focus on the Case II calibration that features stochastic volatility. Single-period calculations. A claim to the first dividend earns zero risk premium because dividend growth is conditionally uncorrelated with the log stochastic discount fac- VOL. VOL NO. ISSUE SHORT TITLE FOR RUNNING HEAD 9

Sharpe ratio 0.4 covt Mt 1Rd,t 1,Rd,t 1 0.3 0.01

0.2 0.00 Habit,S 0.02 0.04 0.06 0.08 0.10 0.01 0.1 0.02 0.0 Habit,S 0.00 0.02 0.04 0.06 0.08 0.10 0.03

Figure A1. The Sharpe ratio, and the covariance term, for the one-period dividend strip in Campbell and Cochrane (1999). The dotted line in the left panel indicates the volatility of the dividend strip

return, σw. The solid vertical line indicates the steady state level of habit. The dashed vertical line indicates the maximum attainable level of habit.

Note: Figure notes without optional leading.

tor, covt(gd,t+1, mt+1) = 0. (We are following Bansal and Yaron by treating the model as conditionally lognormal, relying on loglinearizations.) Hence Et Rd,t+1 = Rf,t+1. Again exploiting lognormality, the risk-neutral variance takes the form above,

2 var log g 2 ϕ2 σ2 var∗ R = R e t d,t+1 1 = R e d t 1 , t d,t+1 f,t+1 − f,t+1 − and

1 2 2 ϕdσt covt (Mt+1Rd,t+1,Rd,t+1) = vart∗ Rd,t+1 (Et Rd,t+1 Rf,t+1) = Rf,t+1 e 1 . Rf,t+1 − − − Hence the conditional covariance is positive (but very small) in Bansal and Yaron (2004). Bansal et al. (2012) make dividends and consumption growth correlated in the short run. 2 2 γπσt As a result, covt(gd,t+1, mt+1) = γπσt , and hence Et Rd,t+1 = Rf,t+1e , where π is a new parameter introduced in equation− (3) of Bansal et al. (2012). Risk-neutral variance changes slightly: 2 2 2 2 (π +ϕ )σt var∗ R = R e 1 t d,t+1 f,t+1 − and π2+ϕ2 σ2 γπσ2 cov (M R ,R ) = R e( ) t e t . t t+1 d,t+1 d,t+1 f,t+1 − This is positive in their calibration, in which π2 + ϕ2 = 42.3 > γπ = 26. Multi-period calculations. Bansal and Yaron (2004) calibrate the model to a monthly frequency, while we use 1- to 3-year dividend claims. We therefore compute the risk-neutral variance and the covariance for longer horizons for completeness. The model implies that

2 Pt(τ) = Dt exp(Aτ + Bτ xt + Cτ σt ), where the coefficients follow from

2 Pt(τ) = Dt Et Mt+1 exp gd,t+1 + Aτ 1 + Bτ 1xt+1 + Cτ 1σt+1 − − − 2 2 = Dt exp μd + φxt + Aτ 1 + Bτ 1ρxt + Cτ 1σ (1 ν1) + Cτ 1ν1σt − − − − − × 1 1 2 2 2 2 1 2 2 2 2 Rft− exp ϕd + Bτ 1ϕe σt + Cτ 1σw λm,eBτ 1ϕeσt λm,wCτ 1σw 2 − 2 − − − − − 10 PAPERS AND PROCEEDINGS MONTH YEAR where R = exp s + s x + s σ2 and thus ft { 0 1 t 2 t }

2 1 2 2 2 Aτ = s0 + μd + Aτ 1 + Cτ 1σ (1 ν1) + Cτ 1σw λm,wCτ 1σw, − − − − 2 − − − Bτ = s1 + φ + Bτ 1ρ, − − 1 2 2 2 Cτ = s2 + Cτ 1ν1 + ϕd + Bτ 1ϕe λm,eBτ 1ϕe. − − 2 − − − Returns are given by

2 2 Rd,t+1(τ) = exp gd,t+1 + Aτ 1 Aτ + Bτ 1xt+1 Bτ xt + Cτ 1σt+1 Cτ σt − − − 2 − − − = ft(τ, xt, σt ) exp ϕdσtut+1 + Bτ 1ϕeσtet+1 + Cτ 1σwwt+1 , { − − } where

2 2 2 ft(τ, xt, σt ) = exp μd + φxt + Aτ 1 Aτ + (Bτ 1ρ Bτ ) xt + Cτ 1σ (1 ν1) + (Cτ 1ν1 Cτ ) σt − − − − − − − − 1 2 2 1 2 2 2 2 2 2 = Rft exp Cτ 1σw ϕd + Bτ 1ϕe σt + λm,wCτ 1σw + λm,eBτ 1ϕeσt . −2 − − 2 − − − Exploiting lognormality,

2 2 2 2 2 2 2 2 vart∗ Rd,t+1 = Rft (exp vart log Rd,t+1(τ) 1) = Rft exp ϕd + Bτ 1ϕe σt + Cτ 1σw 1 . { } − − − − For the risk premium, we have

2 2 Et (Rd,t+1(τ)) Rft = Rft exp λm,wCτ 1σw + λm,eBτ 1ϕeσt 1 . − { − − } − This implies for the covariance

2 2 2 2 2 2 covt (Mt+1Rd,t+1(τ),Rd,t+1(τ)) = Rft(exp ϕd + Bτ 1ϕe σt + Cτ 1σw − − { 2 2 } exp λm,wCτ 1σw + λm,eBτ 1ϕeσt ), − { − − } which we linearize (and approximating Rft = 1 as it does not affect the sign and is the relevant empirical case)

2 2 2 2 covt (Mt+1Rd,t+1(τ),Rd,t+1(τ)) ϕdσt + Bτ 1ϕe (Bτ 1ϕe λm,e) σt + Cτ 1 (Cτ 1 λm,w) σw. ' − − − − − −

Note that Bτ ,Bτ0 > 0, Cτ ,Cτ0 < 0, for τ > 0, and λm,e > 0 and λm,w < 0. At longer horizons, the NCC will be satisfied, but the coefficients (Bτ ,Cτ ) change only slowly with maturity due to the persistence of the processes. We therefore conclude that the NCC condition of method 1 is likely not satisfied in Bansal and Yaron (2004) when calibrated to our sample period.

A2. Method 2

Additional calculations under lognormality

1 2 1 2 rf,t σ σ1,tZ1,t+1 μd,t σ +σ2,tZ2,t+1 If Mt+1 = e− − 2 1,t− and Rd,t+1 = e − 2 2,t and Rt+1 = μ 1 σ2 +σ Z e t− 2 3,t 3,t 3,t+1 then we must have μ r = ρ σ σ and μ r = ρ σ σ so d,t − f,t 12,t 1,t 2,t t − f,t 13,t 1,t 3,t that E MR = 1 holds, where we are writing ρij,t for corrt(Zi,t+1,Zj,t+1. Straightforward cal- μd,t+μt ρ23,tσ2,tσ3,t culations show that covt(Rd,t+1,Rt+1) = e (e 1) and covt∗(Rd,t+1,Rt+1) = 2 2 2rf,t ρ23,tσ2,tσ3,t 2μd,t σ − 2μt σ e (e 1); similarly, vart Rd,t+1 = e (e 2,t 1) and vart Rt+1 = e (e 3,t 1), − 2r σ2 − 2r σ2 − while risk-neutral variances are var∗ R = e f,t (e 2,t 1) and var∗ R = e f,t (e 3,t 1). t d,t+1 − t t+1 − VOL. VOL NO. ISSUE SHORT TITLE FOR RUNNING HEAD 11

It follows that the true and risk-neutral correlations are equal:

eρ23,tσ2,tσ3,t 1 corrt (Rd,t+1,Rt+1) = corrt∗ (Rd,t+1,Rt+1) = − ρ23,t . σ2 σ2 ≈ e 2,t 1 e 3,t 1 − − r The covariance condition

Although we have focussed on dividend volatility, consumption-based models also have difficulty matching the time series behavior of price volatility. Martin (2017, TableIV) reports time series of various statistics (mean, median, standard deviation, min, max, skewness, kurtosis, and autocorrelation) of sample paths of the VIX and SVIX indices gen- erated in the model economies of Campbell and Cochrane (1999), Bansal and Yaron (2004), Bansal et al. (2012), Bollerslev et al. (2009), Drechsler and Yaron (2011), and Wachter (2013). None of these is able to generate sample paths that resemble those observed empirically. All the models apart from Drechsler and Yaron (2011) generate volatility series that are more persistent than the data. The empirically observed mean gap between VIX and SVIX—a measure of the average importance of extreme left-tail events—is outside the support of the million sample paths in every model: Wachter (2013) overstates the importance of such events, and all other models understate it. All the models apart from Wachter (2013) fail, on 99% of sample paths, to generate the maximum levels of VIX that have been observed in reality. All models apart from Drechsler and Yaron (2011) fail, on 99% of sample paths, to match the kurtosis of VIX and SVIX. We also refer to Dew-Becker et al. (2017) for the a set of related challenges of models to match the term structure of variance swaps and variance risk premia. Dew-Becker et al. show that the monthly risk premium on short-term variance ( 3 months) is negative and large whereas the risk premium on longer-term variance (> 3≤ months) is essentially zero. The fact that very-short run variance is priced but longer-term variance is not implies the existence of a transitory element in realized volatility that investors are highly averse towards. The results also imply that investors are not averse towards changes in long- term expected volatility – something that is hard to reconcile with long-run risk models (Drechsler and Yaron, 2011) and disaster models with Epstein-Zin preferences (Wachter, 2013) as investors in such models are averse towards increases in expected volatility, which is modeled to be persistent, meaning that claims on longer-term variance should be priced. 12 PAPERS AND PROCEEDINGS MONTH YEAR

additional figures

We summarize the broad patterns in the data as opposed to high-frequency event studies.4 We therefore sample the data at three moments in time: the pre-pandemic peak of the market, the bottom of the market, and at the end of our sample. For both Europe and the US, we determine the peak of the index level before the start of the pandemic and compute the average prices and volatilities of each asset during the three week period before the peak. Similarly, we determine the bottom of the market indexes and average the prices and implied volatilities in the three weeks surrounding the bottom. We also average the prices during the last three weeks of our sample. To succinctly present the results, we present the returns averaged for the 2021, 2022, and 2023 dividend futures prices. The dividend prices are indicated by “ST” in the legend of the figure. 10 0 -10 % change -20 -30 -40 Downturn Recovery

SPX SPX ST LT Treasuries IG bonds SX5E SX5E ST

Figure B1. The dynamics of asset prices during the COVID-19 crisis.

4Gormsen and Koijen (2020) analyze the dynamics of the index and dividend futures around some of the key events during the crisis for the European, Japanese, and US market.