Coronavirus: Impact on Stock Prices and Growth Expectations (NBER Working Paper 27387, June 2020)

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NBER Working Paper No. 27387, Coronavirus: Impact on Stock Prices and Growth Expectations, by Niels J. Gormsen and Ralph S. J. Koijen, dated June 2020. The authors use data from the aggregate stock market and dividend futures to quantify how investors' expectations about economic growth evolve in response to the coronavirus outbreak and subsequent policy responses until June 2020. As of June 8, the authors' forecast of annual growth in dividends is down 9% in the US and 14% in the EU compared to January 1, and the forecast of GDP growth is down by 2.0% in the US and 3.1% in the EU. The paper closes by developing and estimating a simple model of the crisis.

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NBER WORKING PAPER SERIES

CORONAVIRUS: IMPACT ON STOCK PRICES AND GROWTH EXPECTATIONS

Niels J. Gormsen
Ralph S. J. Koijen

Working Paper 27387
http://www.nber.org/papers/w27387

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
June 2020

We are grateful to Nicola Benigni, Xavier Gabaix, Elisa Maffioli, Lasse Pedersen, Amir Sufi,
Stijn Van Nieuwerburgh, and Motohiro Yogo as well as seminar participants at HEC Paris,
NBIM, UNSW Sydney, and the Virtual Finance Workshop (VirtualFinance.org) for discussions
and suggestions. We thank Dong Ryeol Lee for excellent research assistance. Gormsen and
Koijen acknowledges financial support from the Center for Research in Security Prices at the
University of Chicago Booth School of Business. The views expressed herein are those of the
authors and do not necessarily reflect the views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-
reviewed or been subject to the review by the NBER Board of Directors that accompanies official
NBER publications.

© 2020 by Niels J. Gormsen and Ralph S. J. Koijen. All rights reserved. Short sections of text, not
to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including
© notice, is given to the source.
Coronavirus: Impact on Stock Prices and Growth Expectations
Niels J. Gormsen and Ralph S. J. Koijen
NBER Working Paper No. 27387
June 2020
JEL No. E44,F3,G1,G4

ABSTRACT

We use data from the aggregate stock market and dividend futures to quantify how investors’
expectations about economic growth evolve across horizons in response to the coronavirus
outbreak and subsequent policy responses until June 2020. Dividend futures, which are claims to
dividends on the aggregate stock market in a particular year, can be used to directly compute a
lower bound on growth expectations across maturities or to estimate expected growth using a
forecasting model. We show how the actual forecast and the bound evolve over time. As of June
8, our forecast of annual growth in dividends is down 9% in the US and 14% in the EU compared
to January 1, and our forecast of GDP growth is down by 2.0% in the US and 3.1% in the EU.
The lower bound on the change in expected dividends is -18% in the US and -25% in the EU at
the 2-year horizon. News about fiscal stimulus around March 24 boosts the stock market and
long-term growth but did little to increase short-term growth expectations. Expected dividend
growth has improved since April 1 in both the US and the EU. We conclude by developing and
estimating a simple model of the crisis to understand the joint dynamics of short-term dividend
futures, stock markets, and bond markets.

Niels J. Gormsen
University of Chicago
5807 South Woodlawn Avenue
Office 401
Chicago, IL 60637
niels.gormsen@chicagobooth.edu

Ralph S. J. Koijen
University of Chicago
Booth School of Business
5807 S Woodlawn Ave
Chicago, IL 60637
and NBER
Ralph.koijen@chicagobooth.edu
1 Introduction
The outbreak of the new coronavirus has caused a pandemic of respiratory disease (COVID-
19) for which vaccines and targeted therapeutics for treatment are unavailable as of May 2020
(Wang et al. (2020)). The outbreak has caused major concerns about public health around
the world. At the same time, there are growing concerns about the economic consequences
as households are required to stay home to slow the spread of the virus. The impact that
“pausing” the economy may have on supply chains and the financial stability of firms, the
financial sector, and households is largely unknown. As a result, policymakers, businesses,
and market participants try to estimate growth expectations for the years to come and assess
the shape of the recovery.
As the current situation is unprecedented, and evolving rapidly, models that use macro-
economic fundamentals to form expectations may miss some of the key forces and may be too
slow to update given the frequency with which macro-economic data become available. It has
long been recognized that asset prices may be particularly useful as they reflect investors’
expectations about future payoffs. A natural starting point may be stock markets, bond
markets (Harvey (1989)), and credit markets (Gilchrist and Zakrajsek (2012)). Indeed, much
of the media commentary has evolved around these markets. In particular the movements
in the stock market have received a lot of attention. In this paper, we provide a perspective
on how to interpret movements in the stock market and what they tells us about growth
expectations by combining it with asset pricing data from other markets.1
Equity markets in the US and the EU dropped by as much as 30%. This is an extraor-
dinary amount. To interpret this decline, it is useful to recall that the value of the stock
market, St , is equal to the discounted value of all future dividends
∞
X Et [Dt+n ]
St = (n)
, (1)
n=1 1 + µt

where Et Dt+n is the expected dividend in n years from today, conditional on today’s infor-
(n)
mation, and µt the cumulative discount rate for that cash flow. If the stock market falls,
1
Ramelli and Wagner (2020) look at the cross-section of stock price reactions to COVID-19 events to
understand the factors that impacted investors’ demand during the onset of the crisis.

2
Figure 1: Expected Dividend and GDP Growth from Dividend Futures

This figure shows the change in expected dividend and GDP growth relative to expected value at
January 1, 2020. The figure shows expected growth in the US in blue and in the EU in red. Key
events are indicated by the vertical dashed lines. The expected dividend growth is revised slowly
in response to the outbreak, particularly in the US where it was revised down by less than 5% at
March 11. By June 8, expected dividend growth is down by 9% in the US and 14% in the EU.
Expected GDP growth over the next year is down by 2.0% in the US and 3.1% in the EU. We
emphasize that these numbers are based on historical relations between growth and asset prices and
come with uncertainty. Details of the estimation are in Section 5.2.

3
Figure 2: Lower bound on revisions in expected growth at different horizons

This figure shows a lower bound on revision in expected dividend growth at different horizons. The
revisions are measured relative to expectations on January 1. The figure shows the bound for the
S&P 500 in blue and the bound for Euro Stoxx 50 in red. The lower bound bottoms out between 1
and 2 years into the future, with expected dividends being revised down by as much as 18% in the
US and 29% in the EU. The lower bound increases from years 1 to 7 in EU, which is consistent with
investors expecting catch-up growth after the recession. We emphasize that the estimates represent
lower bounds and that actual expected growth is likely higher.

4
then either expected future dividends fall or investors discount future dividends at a higher
(n)
rate, that is, µt rises.
For the stock market to decline by 30% only due to revised growth expectations, the
shock to future dividends needs to be large and highly persistent. It would for instance be
inconsistent with a V-shaped recovery. To see this, we can sum the dividend prices over the
first 10 years and find that this accounts for about 20% of the value of the stock market.
This implies that if discount rates do not move and the economic impact on dividends lasts
no more than 10 years, a 30% decline in the stock market would mean that firms pay no
dividends in the next 10 years - seemingly a rather extreme scenario. It would correspond
to an L-shaped recovery in which dividends permanently drop by 30%, with no catch-up
growth.
However, focusing on fundamentals only is typically not the right way to interpret move-
ments in the stock market. The seminal work by Shiller (1981) and Campbell and Shiller
(1988) shows that most of the variation in the value of the stock market is due to changes
(n)
in expected returns, µt , not revisions in expected future growth rates. See COCHRANE
(2011) for an excellent review. This insight brings good and bad news. The good news is
that investors’ expectations did not decline as dramatically as in the earlier calculation. The
bad news, however, is that we learn little about growth expectations by taking cues from
the stock market. Instead, we learn about investors’ changes in discount rates that may be
driven by shifts in risk aversion, sentiment, or uncertainty about long-run growth.
Our main point is that data from a related market, namely dividend futures, are useful
to obtain estimates of growth expectations by maturity. Dividend futures are contracts that
only pay the dividends of the aggregate stock market in a given year.2 We can convert these
prices to make each of the components of (1),

(n) Et [Dt+n ]
Pt = (n)
, (2)
1 + µt

directly observable.
(n)
We refer to Pt as the price of the n−year dividend strip at time t. If we sum all dividend
2
See van Binsbergen et al. (2012), van Binsbergen et al. (2013), van Binsbergen and Koijen (2017), and
Gormsen (2020) for earlier work on dividend strips and dividend futures.

5
strip prices, they add to the market, St = ∞
P (n)
n=1 Pt . There are two important reasons that
data on dividend strip prices are informative. First, van Binsbergen et al. (2013) show that
prices of dividend strips provide good forecasts of dividend growth and economic growth
more broadly. Second, and particularly relevant during this period, dividend strips are
differentiated by maturity, just like nominal, real, and corporate bonds. We use this feature
of the data to provide an estimate of expected growth over the next year and to obtain a
lower bound on the term structure of growth expectations by maturity.
Figure 1 shows the dynamics of expected dividend and GDP growth expectations in
the US and in the EU until June 8. Growth expectations did not respond much to the
Wuhan lockdown. Following the lockdown in Italy, growth expectations start to deteriorate.
The travel restrictions on visitors to the US from the EU leads to a sharp deterioration of
growth expectations. This is occurs once again following the declaration of the national
emergency and the subsequent actions by the Federal Reserve on March 15. Following the
US fiscal stimulus program, GDP growth has stabilized somewhat in the US but continued to
deteriorate in the EU. By June 8, expected dividend growth over the next year is down by 9%
for the S&P 500 index and 14% for the Euro Stoxx 50 index. The estimate of GDP growth
over the next year is down by 2.0% in the US and 3.1% in the EU.3 As a word of caution, we
emphasize that these estimates are based on a forecasting model estimated using historical
data. In these unprecedented times, there is a risk that the historical relation between growth
and asset prices changes, meaning these estimates come with uncertainty.4 Nevertheless, in
discussing what asset markets may tell about investors’ growth expectations, we argue that
dividend futures should play a central role.
We also derive a lower bound on expected dividend growth by horizon, which can be
computed directly using observed prices. The lower bound is forward looking and requires
neither a forecasting model nor historical data, which makes it useful in our setting, and
only relies on the assumption that expected excess returns have not decreased.
The lower bound is plotted in Figure 2. The figure displays the lower bound on the
3
The chief economist of Goldman Sachs, Jan Hatzius, revised his forecast for GDP growth in 2020 down
to 0.4%, compared with a prior growth estimate of 1.2% on March 15.
4
An additional reason for the changing link between dividend futures and future GDP growth is that
governments and regulators may impose restrictions on firms’ payout policies in return for financial support
or to safeguard the financial system in case of banks and insurance companies.

6
change in expected dividends on the vertical axis and the horizon on the horizontal axis.
As of June 8, the lowest value of the lower bound is 18% in the US and 29% in the EU,
relative to January 1. There are signs of catch-up growth from year 1 to year 7 in the EU
as the bound is substantially higher at longer horizons. We study how the bound evolves
during the crisis in response to news and policy decisions, which provides a narrative as to
how investors interpreted these events.
We compare the lower bound observed during the coronavirus crisis to the lower bound
observed during the November 2008 of the global financial crisis (GFC). On March 23 2020,
the day with the lowest price of S&P 500 during the coronavirus crisis, the lower bound
on dividend growth is as low, or lower, than what we observed during the financial crisis.
The lower bound does, however, show stronger signs of catch-up growth than during the
GFC. The comparison to the financial crisis is useful as we show that the lower bound
was quite tight during the previous recession. The lower bound on the change in growth
rates was almost 30% at the 2-year horizon, and dividends indeed fell short of the pre-crisis
trend by almost 30% after two years. These results suggest that even in a stressed financial
system, dividend futures are closely related to future fundamentals and therefore contain
useful information.5
We also use the dividend futures to better understand the overall movement in the stock
market. During the onset of the crisis, the stock market drops substantially more than the
1- to 7-year dividend strips. This finding implies that the value of distant-future dividends
– dividends paid out more than 7 years from today – must have dropped by more than
the value of the near-future dividends. As we find it unlikely that long-run dividends, in
levels, are hit harder than near-term dividends, the drop must come from discount rates.6
Hence, prices on the market and the futures jointly suggest that discount rates initially
increased substantially on long-maturity claims such as the market portfolio.7 We formalize
this analysis at the end of the paper.
5
A related concern is that dividend futures market may not be as liquid as other equity markets. However,
van Binsbergen et al. (2013) show that dividend futures forecast economic growth better than other price-
based forecasts such as bond yields.
6
We refer to Eichenbaum et al. (2020) for a macroeconomic model of epidemics that is consistent with
this assumption.
7
The importance of long-horizon discount rate variation to understand movements in the aggregate stock
market is consistent with (Gormsen, 2020).

7
As of June 8, the expected return on the market has returned to the pre-crisis level. On
June 8, the S&P 500 trades at $3232, which is $64 lower than the average price between
January 1 and February 19. This drop can largely be explained by the first 7 years of
dividends, as they are down by a total of $72. As such, the distant-future dividends, the
dividends beyond year 7, must have approximately the same value as before the crisis. If
expected long-run dividends are the same as before the crisis, expected returns on the long-
run dividends must therefore also be the same as before the crisis. However, interest rates
have dropped substantially, which means the expected return in excess of the interest rates
is higher than before the crisis.
Our results have implications for asset pricing theories. It is well known that it is of-
tentimes difficult to identify the economic shocks that caused asset prices to move (Cutler
et al. (1989)). The unique feature of the ongoing events is that the nature of the shock is
clear, and we have a prior regarding the temporal structure. We discuss this in more detail
in Section 9. We conclude by developing a simple asset pricing model of pandemics that we
can easily calibrate and estimate in Section 10. The model allows us to understand the joint
dynamics of short-term dividend prices, the aggregate stock market, and bond prices during
the crisis.

2 The Stock and Bond Market Response to COVID-19
Figure 3 shows the cumulative return on the stock markets in the US and in the EU in the
top panels. We use the S&P500 index as the representative stock index in the US and the
Euro Stoxx 50 index in the EU. The bottom panels show the cumulative return on 30-year
nominal bonds in the US and in Germany. Neither of the stock markets responded strongly
to the outbreak in China or the lockdown of Wuhan, China, on January 23. However, once
it is apparent that the outbreak spread to Italy, South Korea, and Iran, around February 20,
stock markets declined sharply.
In response to the US’ decision on March 12 to severely restrict travel from the EU, with
the exception of the UK, and decisions by governments in the EU to impose lockdowns to
various degrees, stock markets around the world declined by 10% or more. By March 18,
stock markets have dropped more than 30% from their peak. On March 24, S&P 500 rallies

8
Figure 3: The response of the stock and nominal bond markets in the US and EU

This figure shows the cumulative return on the S&P 500, the Euro Stoxx 50 index, 30-year US
Treasuries, and 30-year German bunds. We depict using dashed vertical lines the following five
events: The lockdown of Wuhan, China on January 23, the announcement of the quarantine in
Italy on February 22, the announcement by the US government that it would ban travel from the
EU on March 11, the declaration of national emergency in the US on March 13, and the news that
congress is close to passing a stimulus bill on March 24.

9
almost 10% following news of fiscal stimuli.
In search of safety, investors’ demand for long-term government bonds issued by the US
and Germany increased. Over the same period, the yield on 30-year US Treasuries decreases
by almost a percentage point, driving prices of 30-year bonds up by approximately 30%. We
see a similar rally in German Bunds, which are the safe assets in the Euro area.
Stock returns are often measured in excess of the return on bonds. When measured in
excess of 30-year bonds, the aggregate stock market falls by almost 60% at the bottom.
This is a lower excess return than observed in any calendar month in modern US history.
A central question for policymakers and market participants is how to read this decline in
the stock market. That is, what does the decline tell us about the expected trajectory of
future growth or changes in expected excess returns. In the remainder of this paper, we
show that we can make progress on this question by using data on dividend futures and the
stock market jointly.

3 The Temporal Nature of COVID-19 and Past Pan-
demics
To interpret the evolution of asset prices, it is useful to place the ongoing pandemic into
historical context. The Centers for Disease and Control and Prevention (CDC) provides
an overview of past pandemics.8 The key takeaway is that pandemics tend to be relatively
short-lived. For instance, the H1N1 virus spread in 1918 and 1919, the H2N2 virus in 1957
and 1958, the H3N2 virus in 1968, and the H1N1pdm09 virus in 2009. While the pandemic
may spread more easily in today’s interconnected world, the expectation is that a vaccine
can be available within 2 years. So while the economic contraction may be very sharp, and
potentially have long-lasting effects due to defaults of households, firms, parts of the financial
sector, and even governments, we believe at the time of writing that it is reasonable to assume
that the economic consequences are most severe in the next one or two years. Indeed, this
reasoning has prompted policy proposals to flatten not only the pandemic curve, but also
8
https://www.cdc.gov/flu/pandemic-resources/basics/past-pandemics.html.

10
the recession curve (Gourinchas (2020)). We will interpret the dynamics of equity and bond
markets through this lens.

4 The Response of Dividend Futures to COVID-19
To better understand the expected impact of COVID-19 on the economy over the next few
years, we turn to the term structure of dividend prices. The equity term structure are prices
of claims to the dividends of all firms in an index in a given year. To interpret the dividend
strip price, we can write (2) as
(n)
(n) Gt
Pt = Dt (n)
,
1 + µt
h i
(n)
where Gt = Et DDt+n t
is the expected growth rate between years t and t+n. In practice,
we do not directly observe the dividend strip price, but instead observe the dividend futures
(n)
price, which we denote by Ft . The two prices are linked by the no-arbitrage relationship
(n) (n) (n)
Ft = Pt (1 + yt ), which implies

(n)
(n) Gt
Ft = Dt (n)
,
1 + θt
(n)
(n) (n) 1+µ
where yt is the cumulative n−year risk-free interest rate and θt = (n) −1 is the expected
t
1+yt
excess n-period return on n−period dividend risk.
(n)
We directly observe the futures price, Ft , which informs us about the market’s expecta-
(n)
tion of the growth rate by maturity and the expected excess return, θt , again, by maturity.
The unique feature is that we can get information about growth expectations by maturity,
while the stock market is informative about growth rates and expected returns across all
maturities combined.
Dividend futures are exchange-traded products, traded on the Chicago Mercantile Ex-
change in the US and on the Eurex Exchange in EU, and also to a large extent on over-the-
counter markets. Because the contracts expire in December, the maturity of the available
contracts varies over the calendar year. We therefore interpolate prices across the different
contracts to obtain constant maturity prices. We use the mid-quotes at close as pricing data

11
in the US and settlement prices, which is the volume-weighted average price during the day,
in the EU. We address the liquidity of the dividend futures market later in section 8. The
conclusion is that trading frictions have a negligible impact on the conclusions in this paper.
Figure 4 shows how the prices on dividend futures evolve between January 1 and June 8.
The figure shows the change in prices of dividend futures relative to the price of the same-
maturity claim on January 1. The top left corner shows the cumulative change in prices on
March 5. Prices drop only modestly during the initial spread of the virus from January 1
to March 5. In contrast, equity markets drop by more than 10% between January and early
March. Since near-future dividends do not drop in value, the initial drop in the stock market
must come from a drop in the value of distant-future dividends.
Dividend prices drop substantially between March 5 and March 20. The top right corner
shows the change in priced from January 1 to March 20. Prices are down by more than
30% for the S&P 500 and more than 40% for the Euro Stoxx 50. The drop is biggest on
the 2-year horizon. Between March 5 and March 20, stock markets drop substantially in
both the US and the EU, with the S&P 500 experiencing its biggest daily loss since 1987.
Important dates are March 11, when the US limits travel from the EU, and March 13, when
the US declares a state of emergency. On March 13, stock markets soar after the declaration
of the national emergency. Dividend prices also increase but only at the long end.
The bottom left corner shows the change in prices from January 1 to March 26. On
March 25, Congress comes close to passing a 1.8 trillion dollar fiscal stimulus bill. Stock
markets soar already on the March 24 following news of the bill. Overall, stock markets
increase by around 10% from March 20 to March 26, presumably driven by news about
stimulus.9 However, the short-term dividend futures actually decrease slightly over this
period. This finding implies that fiscal stimulus lifted the stock market by lifting the value
of distant-future dividends, not by improving prices of near-term cash flows.
Finally, the bottom right corner shows the change in price from January 1 to June 8.
Dividend prices increases overall between March 26 and June 8. Stock prices also increase
substantially over this period, with the S&P 500 almost returning to the level of January 1
9
Congress is also rumored to include a ban on paying dividends until September 2020 for firms receiving
financial support, something that is likely to decrease the value of the 2020 dividend claim but has less of
an impact on the 2-year claim.

12
2020.

5 What Do Dividend Futures Tell About Growth Expec-
tations?
5.1 A Lower Bound on Dividend Growth
We provide a simple lower bound on the expected growth rate in dividends that can be
computed using market prices only. If we consider a change in the price of a dividend future
over a short period of time from t to t0 , t0 > t, we have

(n)
(n) ∆Gt0
∆Ft0 = (n)
,
∆Θt0

x (n) (n)
where ∆xt0 = xtt0 and Θt = 1 + θt .
To obtain a lower bound on the change in growth expectations, our key assumption
is that the expected excess return, which for instance reflects investors’ risk aversion, did
(n)
not decline since the outbreak, ∆Θt0 ≥ 1. This implies that we can bound the change in
expected growth from below by

(n) (n)
∆Gt0 − 1 ≥ ∆Ft0 − 1,

which depends only on market prices on the right-hand side that are readily available. Hence,
(n) (n)
the change in expected growth over the next n years, ∆Gt0 , is bounded from below by ∆Ft0 .
We provide more details on the necessary technical assumptions in the Appendix.
The lower bound is shown in Figure 2 in the introduction. The lower bound on dividend
growth expectations is revised down by as much as 18% in the US for the 2- to 7-year horizon.
In the EU, the 1-year growth is revised down by as much as 29%. It is important to keep
in mind that the lower bound represents the revision in expected growth rates relative to
previous expectations, not a lower bound on the actual growth rate. If investors expected
a nominal growth rate of 6% annually prior to the outbreak, the expected growth on the

13
Figure 4: The Development of the Dividend Term Structure over the COVID-19 Outbreak

This figure shows the relative price of dividend of dividend futures with different maturity.
We consider the percentage change in prices since January 1. The dividend futures are claims
on the dividend paid out on the index in a given year. We consider the S&P 500 index and
the Euro Stoxx 50 index. Maturity measured on the horizontal axis is expressed in years. For
instance, the top left figure shows that, between January 1 and March 5, dividend futures
prices fall by only a few percent for the 2-year claim but by as much as 10% for the 7-year
claim.

14
2-year horizon would be more than 12%. Revising the 2-year growth expectations down by
18% would thus imply a negative growth of “only” 6% over a 2-year horizon. In addition,
we measure a lower bound that is equal to the actual expectations only when expected
excess returns did not go up. We next discuss a methodology, which requires additional
assumptions, that we use to compute an estimate of expected growth.

5.2 Estimating Dividend Growth Expectations
We estimate growth expectations directly using out-of-sample forecasting. These estimates
are plotted in Figure 1. Here we explain how we estimate these. We first define the equity
yields on index i as:

!
(n) 1 Dt
eit = ln (n)
,
n Ft

where n is measured in years. Using a training sample from 2006 to 2017, we run a pooled
regression of realized dividend growth rates on the S&P 500 and the Euro Stoxx 50 onto the
2-year equity yield on the associated index:

D (2)
∆1 Di,t = β0i + β1D eit + i,t+4 , (3)

where t is measured in quarters, i refers to either S&P 500 or Euro Stoxx 50, and ∆n xt ≡
xt+4n
xt
− 1. We then use the parameter estimates in this regression to estimate expected
dividend growth at every trading day since January 1 2020. Dividends on the left hand side
are measured in nominal terms. The R2 in this forecasting regression is 0.65. We report
regression details in Table 2 in the Appendix B.

15
6 Mapping Dividend Growth Expectations to GDP Growth
Expectations
6.1 Dividend and GDP Growth
We can use dividend futures to compute a lower bound and point estimate for GDP growth
expectations. Indeed, dividend expectations are related to GDP expectations as dividends
summarize the profits and production of listed firms, which in turn is part of the GDP.
However, there is obviously also independent variation in both series.
To illustrate the relation between the two series at the business-cycle frequency, we extract
the cyclical component of real dividends and real GDP using the methodology developed in
Hamilton (2018)
X11
zt = d0 + di zt−j + ct ,
j=8

where zt corresponds to either log real dividends or log real GDP. The residual, ct , corre-
sponds to the cyclical component.
Figure 5 presents the results from the first quarter in 1985 to the fourth quarter in
2019.10 We standardize each of the series. The two move strongly together with a time-
series correlation of 54%. We note that the two series are not perfectly synchronized and
that the series appear more strongly related during economic downturns, that is, when the
series are below average.This is precisely what we care about in the current environment,
which makes dividend futures particularly relevant to estimate investors’ expectation of GDP
growth as well.

6.2 A Lower Bound on GDP Growth
To calculate the lower bound on changes in expected GDP growth, we multiply our lower
bound on dividend growth by a country-specific constant bi that maps dividend growth into
GDP growth:
10
We adjust the aggregate dividend series for Microsoft’s special dividend in November 2004, which oth-
erwise would show as a substantial outlier.

(n)
Et [∆n Yit ] − 1 ≥ ∆Fi,t0 − 1 × bi , (4)

where Et [∆n Yit ] is the change in expected GDP growth at horizon n for country i. The
constant bi measures how much GDP changes when dividends change. One way to estimate
bi is to regress GDP growth on dividend growth. However, this estimate is likely to be
biased downwards due to asynchronicities between the series shown in Figure 5 and other
independent variation in dividend growth. A downward bias in bi would be problematic as it
leads to an upward bias in our lower bound. To ensure that our lower bound is conservative,
we instead run the regression

∆1 Dit = a0i + a1i ∆1 Yit + εt+4 , (5)

and use bi = a11i . In this way, asynchronicities between GDP and dividends, and other
independent variation in GDP, leads to a lower estimate of ai and a higher estimate of bi .
This results in a more conservative lower bound.
We run separate regressions in the US and EU. In the US, we use the 1985 to 2019 sample
of real growth in GDP and dividends. In the EU, we use a shorter 2003 to 2019 sample of
real growth in GDP and dividends on the Euro Stoxx 50. We use real series to avoid putting
too much weight on the early US sample with high inflation.11 The resulting estimates of bi
are 0.67 in the US and 0.33 in the EU. The lower bound is plotted in Figure 6.

6.3 GDP Growth Expectations
To estimate GDP expectations, we can in principle follow the procedure of section 5.2 and use
the equity yields to forecast GDP growth. However, to obtain the most accurate estimate, we
want to account for (i) the small asynchronicities between GDP and dividends documented
in Figure 5 and (ii) the potentially stronger relation between the two series when growth is
below average, which most closely mimics the current situation.
11
The lower bound is nonetheless still a lower bound on nominal growth.

17
The sample with dividend futures is too short to effectively deal with these issues and
we therefore prefer a slightly different approach. We first use the long US sample to map
dividend growth into GDP growth, and we then use this mapping to transform our dividend
growth expectations from section 5.2 into expectations about GDP growth. We note that
one could make other reasonable modeling assumptions that may lead to somewhat different
estimates.
We first map real GDP growth to real dividend growth using the following regression in
the 1985-2019 US sample:
∆n Yt = An + Bn ∆n Dt + et+4n ,

only using data when ∆n Dt < ∆n Dt , with x̄ ≡ T1 t xt , and Yt denoting real GDP. There
P

are two important features of this regression. First, by only considering observations where
realized growth is below average, we estimate the downside relation between the two series,
which is what is relevant in our context. Second, by using longer horizons, n, we can mitigate
the effect of small asynchronicities.
As our benchmark case, we estimate Bn using 2-year growth (n = 2) in the 1985 to 2019
US sample. The baseline estimate is 0.22, meaning that dividends move approximately four
times as much as GDP in downturns (see Table 3 in the Appendix for details). The baseline
estimate is robust to using a longer sample period and to using 3- instead of 2-year growth,
as reported in the Appendix B. The estimate is lower if we consider 1-year growth, reflecting
that GDP and dividends are not perfectly synchronized. If we consider the unconditional
relation between GDP and dividends, the slope coefficient is only half as large, suggesting a
weaker upside relation between GDP and dividend growth.
Having estimated the relation between GDP and dividends, we forecast GDP growth as:

(2)
Et [∆1 Yit ] = Ai + B2 β1D ei,t ,

where Ai can be chosen such that our forecast is correct on average in the 2006 to 2017
sample (the constant is irrelevant as we forecast the change in expectations since January
2020, which only depends on the slope coefficients and the dividend yield). We use the same
estimate of Bn , which is based on US data, for both the US and the EU, as we have a longer

18
Figure 5: Cyclical components of log real dividends and log real GDP.
2
Dividends
GDP
1

-1

-2

-3

-4
1985 1990 1995 2000 2005 2010 2015 2020

sample available for the US. We emphasize once more that these estimates are based on a
forecasting model estimated using historical data. In unprecedented times, there is a risk
that historical relations change, implying that these estimates come with uncertainty.
For comparison, we also show a series of contemporary GDP forecasts in Table 1 until the
middle of March. We also refer to the CEPR book on “Economics in the Time of COVID-
19” for further analysis of the economic effects, and in particular the chapter by Wren-Lewis
(2020) who estimates a decline of GDP of around 1% to 2%, and at most 5%.

7 Comparison to the Global Financial Crisis of 2008
We compare the market’s response to COVID-19 to the GFC of 2008. On March 23, the
VIX is at a similar level as the one observed during the GFC. Stock prices have also dropped
as much as in the fall of 2008, at least when measured in excess of 30-year treasuries. These
observations underline the severity of the impact of the COVID-19 outbreak on financial
markets.
Figure 7 shows the lower bound during the GFC. The blue line plots the lower bound on
revisions in expected dividends between July 31 2008 and November 31 2008. The red line

19
Figure 6: A lower bound on GDP growth expectations by maturity as implied by dividend
markets

The figure shows a lower bound on changes in expected GDP at the 1 to 7-year horizon. The
changes are measured relative to expectations on January 1. The figure shows that expected GDP
may have been revised down by as much as 31% in the US and 19% in the EU. It is revised down
the most on the 1- to 2-year horizon. This estimate is a lower bound meaning actual expectations
are likely be higher (see text for description).

20
Table 1: GDP growth forecasts

Organization company Period Change Level Region Date
Office for Budget Responsibility 2020 -0.30% Britain 12-Mar
Rabobank 2020.H1 Below last year’s level EU 12-Mar
EUan Commission 2020 -2.30% EU 13-Mar
Rabobank 2020 -1% Eurozone 12-Mar
Rabobank 2020 -0.60% France 12-Mar
Berenberg Bank 2020.Q1 -0.10% Germany 19-Feb
Rabobank 2020 -0.90% Germany 12-Mar
OECD 2020 -0.50% Global 3-Feb
IMF 2020 -0.10% Global 22-Feb
IMF 2020 Below last year’s level Global 4-Mar
Goldman Sachs 2020 -1% Global 6-Mar
Moody’s 2020 -0.30% Global 10-Mar
Rabobank 2020 -1.30% Global 12-Mar
Rabobank 2020 -1.70% Italy 12-Mar
Rabobank 2020 -0.60% Netherlands 12-Mar
Rabobank 2020 -0.50% Spain 12-Mar
Rabobank 2020 -0.50% UK 12-Mar
Goldman 2020.Q1 -1.10% US 2-Mar
OECD 2020.Q2 Below last year’s level US 6-Mar
WSJ Survey among economists 2020 -0.70% US 12-Mar
Rabobank 2020 -0.50% US 12-Mar
Capital Economist 2020.Q2 -1% US 13-Mar
Bank of America 2020 -0.40% US 14-Mar
Bruce Kasman (JPMorgan) 2020.H1 Below last year’s level US & Global 13-Mar

21
plots the subsequent realized dividends measured relative to a pre-crisis trend of around 4%
growth.12 For the S&P 500, the lower bound on changes in expected dividends lines up well
with the realized dividends at the short-end. For Euro Stoxx 50, realizations are slightly
below the bound at the short end, above the bound in the middle, and around the bound at
the long end.13 It is comforting that even during a period of high financial turbulence, the
future prices appear well linked to fundamentals and align well with realizations.
Figure 7 also plots the lower bound following the outbreak of COVID-19. On March 23,
which is the day the stock market reaches the bottom, the bound is lower than observed
during the financial crisis, but the curve indicates more catch-up growth, particularly in
Europe.

8 Liquidity of the Dividend Futures Market
Dividend futures for the Euro Stoxx 50 have traded on the Eurex exchange since 2008. The
size of the market has increased steadily since its inception as shown in Figure 8. At the
end of 2019, there are around 1 million contracts outstanding. Each contract is for a 100
dividend points and trades at around EUR 12,000 depending on the maturity. This gives a
total notional outstanding of around EUR 12 billion. The total number of contracts increases
from around 800,000 to 1,200,000 during the spring of 2020, but the notional measured in
EUR drops because the value of the futures decreases.
Figure 9 shows the number of contracts traded daily. The trading volume increases during
the coronavirus crisis, peaking at 100,000 daily contracts. As a comparison, the average daily
traded contracts in 2019 is around 20,000. This heightened volume alleviates concerns that
the market dried up during the crisis.
The average bid-ask spread varies with the maturity of the claim. Between January 1
and May 19, the bid-ask spread for the 2021 claim is on average 0.27% in the middle of the
day. The bid-ask spread increases steadily in the maturity of the contract to around 1.2%
12
We measure the growth rates as the real-growth in dividends observed between 1947 and 2007, which is
close to 2% plus 2% for expected long-run inflation in 2007.
13
We note, however, that the low realized dividends on the long end could reflect the European sovereign
debt crisis of 2011, which was probably unexpected in 2008.

22
Figure 7: Comparing the Lower bound to the lower bound observed during the Global
Financial Crisis

The blue line shows the lower bound on changes in expected dividend growth between between July
31 and November 31 in 2008. The line shows that dividend growth was revised down with up to 25%
on the 2-year horizon for the S&P 500. The red line shows the realized dividends x years into the
future (relative to a pre-crisis trend of 4% nominal). The realized dividends where approximately
30% below the pre-crisis trend after 2 years for the S&P 500. The dotted yellow line shows the
lower bound on changes in expected dividend growth between January 1 and June 8 2020. The
lower bound is as lower than observed during the financial crisis. The figure shows results for the
S&P 500 to the left and for Euro Stoxx 50 to the right.

23
Figure 8: Notional Outstanding for Euro Stoxx 50 Dividend Futures

This figure shows the total notional outstanding on Euro Stoxx 50 dividend futures on the Eurex
exchange.

for the 2027 claim. We note that bid-ask spreads vary over day and tend to be larger in the
morning, before the cash market on European exchanges starts trading, and in the evening,
after the cash markets closes (the futures themselves trade from 8:30 to 22:00). Figure 10
shows the bid and ask prices for the 2021 claim. We measure the average bid and ask prices
over each 15-min interval during opening hours of the cash market in Frankfurt (that is,
from 9:00 to 17:30). The bid and ask prices are close at all points in the sample.

24
Figure 9: Daily Volume in the Market for Euro Stoxx 50 Dividend Futures

The left side of this figure shows the daily volume traded in the market for Euro Stoxx 50 dividend
futures. Each contract is for 100 dividend points. The right side of the figure shows the total value
of the contracts traded on a given day. Sample is Jan 1 to May 19 2020.

25
Figure 10: Bid and Ask Prices for the 2021 Euro Stoxx 50 Dividend Claim

This figure shows the bid and ask prices of the 2021 Euro Stoxx 50 dividend futures. The figure
shows the average price over each 15-minute interval. We only consider the prices during the time
of the day where the Frankfurt Stock Exchange is open for trading in the cash equity market
(9:00-17:30). Sample is Jan 1 to May 19 2020.

26
9 Reconciling the Price of the Stock Market with Divi-
dend Strip Prices
Given the modest decline in growth expectations in the first weeks of the outbreak, we can
learn something about how expected excess returns changed during this period. To illustrate
this, we fit a simple model for dividend prices that we require to simultaneously price the
dividend futures as well as the aggregate stock market. Starting from (1), we observe the
dividend prices for the first 7 years and we observe St . We model the expected growth minus
the expected excess return, gtn − θtn , as a function of maturity. The functional form that we
fit in each period follows Nelson and Siegel (1987).
The results are presented in Figure 11. As the short-term dividend prices in the US
did not move much for the first 7 years, even though the market fell, from February 20 to
March 5, the long-term dividend prices fell. The expected value of short-term dividends is
likely to fall more than the expected value of long-term dividends as some catch-up growth
is expected. The large drop in the value of long-term dividend must therefore come from an
increase in discount rates. Increases in expected returns on long-term claims is commonly
observed during times of stress (Gormsen (2020)). In the EU, both short- and long-term
dividend prices dropped during the same period, which is more consistent with a shock to
both growth expectations and expected excess returns. During the period from March 5 to
12, in which growth expectations changed sharply, we see that both short- and long term
dividend prices fell sharply in both geographies.
These results have implications for asset pricing theories. It is well known that it is
oftentimes difficult to identify the economic shocks that caused asset prices to move (Cutler
et al. (1989)). The unique feature of the ongoing events is that the nature of the shock
is clear, as well as the temporal structure. Although there is uncertainty about the long-
term consequences, it seems reasonable to assume that the short-term economic growth
consequences are more severe than the consequences after, say, five years. Moreover, the
initial decline in the aggregate stock market in the US, with a small response to short-term
dividend prices, suggests that modest shocks to short-term expectations can trigger large
and persistent changes in expected excess returns.

27
Figure 11: Reconciling the stock market and dividend price responses

This figure shows estimated prices of dividends for different maturity. For both S&P 500 and Euro
Stoxx 50, we fit the term structure of dividend prices to the functional form on Nelson and Siegel
(1987) under the restriction that the price of all the dividends sum to the market. We estimate the
prices separately on February 20, March 5, and March 12. Maturity measured on the horizontal
axis is in years.

28
10 A Simple of Theory of Pandemics and Asset Prices

We propose a simple model to provide a more structural interpretation of the key facts that

we document empirically.

10.1 Model Specification

Time is indexed by ¢ and all shocks in the model have a standard normal distribution. We
distinguish three phases in the model: (i) the period before the pandemic, (ii) the initial
outbreak and containment period, and (iii) the period after the virus. During the first period,

t < tg, log dividend growth, Ad;, has a constant mean and variance,

1
Ad; = pn — BON + one.

We summarize investors’ preferences via the stochastic discount factor (SDF), M;, that is

given by
1
M, = exp (-» — 5) — Ae) .

In this simple environment, the prices of dividend futures, normalized by the current level

of dividends, F Dw”, are given by
FD,” = exp (np),

with w* = uw —oA. We assume that investors did not anticipate the arrival of a pandemic
during this period. As such, we cannot learn about the risk, or pricing of pandemic risks,
during this period.

During the second period, t € |to,t,), the virus spreads and behavioral or regulatory

social distancing measures are put in place. As a result, dividends decline,
dD = Di, exp (—y;) ’

with y, > 0 for t € [to, ti). We refer to vy: as the cost of social distancing. These costs are

29
uncertain and evolve as
∆ϕt = σϕ ϕt ,

which is the first source of pandemic-specific risk.
At t = t1 , the pandemic ends and we assume for simplicity that dividends recover to
pre-crisis levels, Dt1 = Dt0 .14 However, there is uncertainty about the long-run damage
done during the economy’s shutdown. Defaults among firms and households can weaken the
balance sheets of the financial sector and the recession may escalate into a financial crisis.
This financial disaster, if it were to happen, occurs at t = t1 .
If the disaster does not happen, relative dividend futures prices restore to pre-crisis levels

(n)
F Dt = exp (nµ? ) ,

for t ≥ t1 . However, if a financial crisis does happen, relative dividend futures prices are
permanently lowered for t ≥ t1 ,

(n)
F Dt = exp (nµ? ρ) ,

with ρ ∈ (0, 1) . As µ? = µ − σλ < 0, a lower risk-neutral growth rate is consistent with
a lower rate of economic growth, higher uncertainty or higher risk prices, for instance, due
to an increase in investors’ risk aversion. The possibility of a financial crisis is the second
source of pandemic-specific risk.
The probability of a disaster at t1 , given the information at time t1 − 1, is given by

πt1 −1 = π0 exp (gt1 −1 + ϕt1 −1 ) ,

which implies that the probability is increasing in the cost of social distancing, ϕt1 −1 . The
disaster probability is also increasing in gt1 −1 , which are additional factors that increase the
probability of a financial crisis. For instance, gt may decrease (or become negative) as a result
of fiscal or monetary policy actions. Alternatively, gt may increase due to concerns about
defaults among households and small firms that weaken (shadow) banks’ balance sheets but
14
In reality, dividends tend to recover more gradually. Adding this feature to the model does not affect
the basic economics at work and we omit it to keep the model as stylized as possible.

30
are not perfectly correlated with y;. We refer to gq as financial crisis concerns.
As y, and g are both equal to zero for t < to, 7 reflects the unconditional probability

of a disaster once the pandemic starts. The dynamics of g is
Ag = O got + Og; (6)

where the first term, o,,€7, captures the correlation between the cost of social distancing and
financial crisis concerns. The shock ef captures the independent variation, and we assume
Ey [ety iefi1] = 0.

Uncertainty about government interventions is the third and final source of pandemic-
specific risk. During the outbreak of the pandemic, which is the period covered by our paper,
asset prices are driven by just two shocks: ¢? and ef. Short-term dividend prices are less
impacted by government policy actions than the market itself, or long-term dividend prices,
as a financial disaster has a larger impact on longer-term claims.

To complete the model, we specify how investors price risk when t € |to, t1). The SDF is
given by

Mz = exp (—y — ? + Ae? + Ae?) ,

where we assume for simplicity that the risk prices on all shocks are identical. Given the
short sample, we do not have enough power to separately estimate risk prices on different
sources of risk. Moreover, we are primarily interested in the dynamics of 9, y:, and 7, for

which the risk prices are not important as we will show.

10.2. Model Solution

We solve for the relative dividend futures prices, F’ Dd”, for the second period when t €

[to, ti). The dividend prices for n < t; — t are given by
FD” = exp (Con)
and for n > t, — t, we have

FD,” = exp (don + dine) — EXP (bon + Din Pr + bane) 5 (7)

ol
and we derive the expressions for the coefficients in Appendix

10.3. Model Implications

For the short-term dividend prices, we consider the case where n < t; — t. This implies that

short-term dividend futures directly reveal y; as

where we ignore the declining maturity over a short period of time in calculating the constant.
Equity prices depend on y; and g, which implies that short-term dividend futures prices
and the market together allow us to uncover both (g, y,). The value of the stock market,

S;, relative to the current level of dividends, D;, is denoted by SD, and given by

SD= S- exp (—x1"n) FD”
n=1

t;-t-1

= 5° exp (<n + Con) -

n=1
lore)

S- exp (-n + Q0,n + aint) — exp (<n + bon + bi nYt + band) :

n=t1-t

As before, we consider

In S; —In Sy

=InSD,—In SD,- + In D, — In Di-

=InSD,—In SD,- — Yt.
This shows that based on movements in stock prices, bond yields, and dividend futures prices
(which allow us to measure y;), we can recover g,. Based on dividend futures prices alone,

we can recover (Y;, see (8). To simplify the calculations, we linearize the price-dividend ratio

around the price-dividend ratio right before the pandemic. In Appendix |C| we show that

32
this implies
yi & constant — (in FE” —In Fe”) ; (9)
0

ge & constant — (in S,;—In S,.) — S- Orn Ce _ y) — Yt; (10)
n=1

where the expression for g” is given in equation (13),
0

With both series in hand, we can also compute the relative change in the disaster prob-

ability
TM
— = exp (9 + 1). (11)
to
10.4 Results
In estimating vy, and g, we set w* = —2%. We report all series relative to their values on

February 20 as in Figure The results are reported in Figure [12] The left panel displays
the cost of social distancing, which is directly linked to the decline in dividend futures, see
(10). To estimate the series, we use the 2-year dividend futures contract. The middle panel
shows our estimates of financial crisis concerns, g,. The right panel compute the relative
disaster probability, see (a As the government bond market experienced substantial stress
in March, which leads to large swings in bond yields, we report the 7-day moving average as
the red line|!>]

We find that the cost of social distancing rises sharply in March and peaks at approxi-
mately 55% in the beginning of April. The cost sharply declines early April and stabilizes
until the middle of May. As of then, the cost falls sharply once again to 25%. The finan-
cial crisis concerns, in the middle panel, increase sharply from the middle of February, even
though the expected cost of social distancing are still low. This illustrates the rapidly ris-
ing concern of a potential downturn, by 20% in a 2-week period, but not in the immediate
future. Such sentiment has a larger impact on the stock market compared to dividend fu-

tures. As the cost of social distancing rises, g; declines. This presumably reflects investors’

15We smooth the series 3 days backwards and 3 days forward, leading to a 7-day window including the
current day.

33
Figure 12: Dynamics of the cost of social distancing, financial crisis concerns, and the relative
probability of a financial crisis.

34
expectations about fiscal and monetary policy actions.
The right panel reports the relative disaster probability, where the probability is relative
to its value on February 20. We find that the disaster probability increases by 55%, and
subsequently steadily declines. However, by the end of our sample, it is still 10% higher
compared to February 20. This is even though the stock market recovered to pre-crisis
levels. The reason is that long-term bond yields declined during this period. As a result,
stock prices should have increased substantially more if the disaster probability would have
reduced to February 20 levels. This implies that even though expected returns may be
the same as before the crisis, the expected return in excess of risk-free bonds is likely still
elevated.

11 Conclusion
In periods of economic and financial distress, getting frequently-updated and forward-looking
measures of the expected path of the economy is key for policy makers and market partici-
pants. We show that dividend futures can constitute a useful tool in this regard.

35
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37
A Technical Details of the Lower Bounds

In this appendix, we detail the assumptions we use to derive a lower bound on dividend (in

Section |A.1) and GDP growth rates (in Section |A.2).

A.1 Dividend Growth

We derive the lower bound for the one-period dividend growth expectations and the argu-
ments directly extend to longer-term growth expectations. By no arbitrage, the price of a

one-period dividend futures, FO. is given by

FO = E[Mis1Di41]
‘ EM)”

where M;,, denotes the stochastic discount factor. We rewrite the equation as

FY 7 Ky Mest eet

D EM] ’

and using E[XY] = E|X]E[Y] + Cov(X,Y), we have

FY a) Cov (Miss, Pit )

t 4 Di
dD; t Ey | Misi]
Coa
where
Cov (Me Pt ) 7.
> Dt
oY) = jit

Ee(MiylGe?

is the (gross) risk premium on dividend growth. Our central assumption is that following

the crisis at t/ > t, investors’s risk aversion increases, implying e 1) > eM), As a result, we

38
have

which yields the lower bound

ve (12)

A.2 GDP Growth

To derive a lower bound on GDP growth, we start from a regression as in (5), where we

regress dividend growth on GDP growth

De
dD;

Vist
Y;

-1=a9 +a =1] +6.

Taking conditional expectations and rewriting gives

GY) L400 +arky Ee _ 1|

GY 4 ag tak, se — 1

Y; Y;
t t

using that ag ~ O and that ins —1~a2-y for (x,y) small. The above step assumes

that expectations about € are not updated between tand ¢’. Inserting into and using
Lte

—l1~ax-y again gives

l+y
Y;
Ey |] 1 | RS?
—l27 a7!
Ky =| 1 | F;

39
Table 2: Predictive Regressions of Dividend Growth on Dividend Yields

This table shows results from regressions similar to (3). In a pooled sample across S&P 500 and
Euro Stoxx 50, we regress realized dividend growth onto the ex-ante two-year yield and a dummy
equal to 1 for Euro Stoxx 50 observations. HAC standard errors based on (Lazarus et al.}|2019) are

presented in parenthesis. Observations are quarterly.

Intercept EU dummy ef ) R? + Obs
Ai Dj 0.046 -0.018 -0.9518 0.65 98
(0.02) (0.03) (0.19)

B~ Additional Tables

C Model Solution

First, for n < t; — t, we have F DW” = exp (Co,,) and we can solve for the coefficients using

rp Ex [Me PDE) |
fo Ey (Mi41]

1
=E, lex (-3* + se) exp (Con—1 — Oe i1)

1
= exp (ons + 50¢ — Ao]

= Exp (Con) ’

implying

1
2
Con = Coyn—1 + 9°" _ AD.

For n >t, — 1, the prices take the form as announced in (7). We first consider t = t, — 1,
for which it holds:

40
Table 3: Mapping Dividends to GDP
This table shows the slope coefficient Bn for a regression of GDP growth onto dividend growth at
different horizons (n), conditions and sample periods. The baseline coefficient is estimated in the
1985-2019 sample using using rolling 2-year growth and only considers observations where realized
dividend growth is below the time-series average. Dividend growth is real dividend growth for S&P
500 and GDP growth is real GDP growth in US. HAC standard errors based on (Lazarus et al.,
2019) are presented in parenthesis. Observations are quarterly.

Baseline: Robustness

Horizon (years) 2 1 3 2 3 2 3

Condition: Downside Downside Unconditional

Sample: 1985-2019 1985-2019 1958-2019 1985-2019

Bn 0.22 0.14 0.31 0.23 0.27 0.13 0.12
s.e. (0.03) (0.05) (0.04) (0.07) (0.11) (0.06) (0.07)
R2 0.50 0.18 0.67 0.19 0.23 0.14 0.12
Observations 62 66 62 125 131 240 236

41
h i
(n)
Et1 −1 Mt1 F Dt1 exp (∆dt1 )
(n+1)
F Dt1 −1 =
Et1 −1 [Mt1 ]
= (1 − πt1 −1 ) exp (nµ? ) exp (ϕt1 −1 )
+ πt1 −1 exp (nµ? ρ) exp (ϕt1 −1 )
= exp (nµ? ) exp (ϕt1 −1 )
− (exp (nµ? ) − exp (nµ? ρ)) π0 exp (gt1 −1 + (1 + ξ) ϕt1 −1 )
= exp (a0,n+1 + a1,n+1 ϕt1 −1 ) − exp (b0,n+1 + b1,n+1 ϕt1 −1 + b2,n+1 gt1 −1 ) ,

implying

a0,n = (n − 1) µ? ,
a1,n = 1,
b0,n = ln (exp ((n − 1) µ? ) − exp ((n − 1) µ? ρ)) + ln π0 ,
b1,n = 1 + ξ,
b2,n = 1.

For t ∈ [t0 , t1 − 1), we recursively solve for prices

42
Ey Me P DI.) exp (Adis.)|

FDMV =
Ey [Mi+1]

= FE, ste exp (aon + GinPe + (Gin _ 1) otha)

Ey [Mi+1]

— E; te exp (bo. + bi npe + b2,n9¢ + (bin — 1) Oy + ban qv) Er + bom gei.1)

Ey [Mi41]

= exp (on + QinPt + 9 (din _— 1) Oo + (din _ 1) od)

1
— exp (0 + Di nQe + b2n9t + 5 (bin —l)o, + baatrs)*)
1
x exp (S0.0% + ((Bin — 1) oy + ba nF qo + b2nFq) \) ,
implying

FD” = exp (don + Qi n¥t) — exp (bon + bi nPt + bon Jt) ’

where

Aon = Gn—1 + = (A1,n—-1 — 1)° o + (@in—1 — 1) 0%,

2
Qin = a1 n-1
=,
1 2,1 2
bon = bon—1 + 5 ((in-1 — Log + ban—18 9) + 52-19% 9 + ((O1n-1 — 1) Og + ba,n-1F 9p + b2,n-19y) A;

bin = bi n—1
=14+6
ban = bo n—1

= 1.

43
To derive a tractable expression for the price-dividend ratio, recall that

SD,= Yes (-i" n) FDI”
ty-t-1

— S- exp (<n + Con) +

n=1
oo

S- exp (<n + don + aint) — exp (<n + bon + bine + bandt)

n=t1-t

We consider a first order Taylor expansion of In SD, in (<n + Con; —yn + don + Qi nPt; —yn + bon -

(n) (n) (n)

around (-¥" n+ np*,In2— YM + ny”, yn + nui”), which implies

[© @) ty-t-1
InSD,—nSD,- ~ — yan Ca = uy”) - S- a” (Con — nyt”)
n=1 n=1
0 (hom — nut + RinYe + hom
+ 3 [ese ne + hi nYe + hang) | ;
n=t1—

where

SD,- 16 yn + nut)

gi”) — ta

to SD,

(13)

and hon = (2d0n — bon); hin = (2a1n — bin) = 1—- €, and ho, = —b2, = —1. Note that

a are weights that add to one across maturities. We notice that with t, — t equal to, say,

3. we havd"]

oe)

‘Formally, we approximate )7° ,, gy) x ~ 1,

44
This ultimately implies
∞
X
(n) (n) (n)
ln S t − ln S t−0 = constant − ξϕt − gt − θt− n yt − yt− .
0 0
n=1

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