The Time-Varying Risk of Macroeconomic Disasters∗

Roberto Marf`e† Julien P´enasse‡

This draft: November 18, 2018

Abstract The rare disasters model relates the equity premium to the probability of a large decline in consumption. A critical prediction of the model is that future macroeconomic disasters are forecastable. This paper tests this prediction and estimates the probability of a disaster using a dataset of 42 countries over more than a century. We ﬁnd that disasters are indeed forecastable by a wide array of variables. The most robust predictor of future disasters is the dividend-price ratio, in line with theory. Our probability estimate strongly correlates with the dividend-price ratio and forecasts stock returns. A variable disaster model, calibrated from macroeconomic data alone, further conﬁrms the link between disaster risk and the equity premium. JEL: E44, G12, G17 Keywords: rare disasters, equity premium, return predictability

∗We thank Daniel Andrei, Patrick Augustin, Jules van Binsbergen, Pierre Collin-Dufresne, George Constan- tinides, Max Croce, Magnus Dahlquist, Leland Farmer, Xavier Gabaix, Anisha Ghosh, Eric Ghysels, Fran¸cois Gourio, Daniel Greenwald, Robin Greenwood, Benjamin Holcblat, Hendrik Hülsbusch, Christian Julliard, Leonid Kogan, Peter Kondor, Christos Koulovatianos, Hening Liu, AndréLucas, Sydney Ludvingson, Rajnish Mehra, Christoph Meinerding, Alan Moreira, Tyler Muir, Urszula Szczerbowicz, Adrien Verdelhan, Tan Wang, and Michael Weber for helpful comments. We also benefitted from helpful comments by conference and seminar participants at the Columbia University Macro Lunch, U. of Luxembourg, U. of Chile, Fundao Getlio Vargas (Rio), McGill University, and conference participants at the CEPR ESSFM Gerzensee 2017 meeting, SAFE Asset Pricing Workshop 2017, NFA 2017, Paris December 2017 Finance Meeting, and the MFA 2018. We thank Robert Barro, JoséUrsúa,Asaf Manela, and Alan Moreira for making their data publicly available. †Collegio Carlo Alberto, Via Real Collegio, 30, 10024 Moncalieri (Torino), Italy; +39 (011) 670-5229; [email protected], http://robertomarfe.altervista.org ‡University of Luxembourg, 4 rue Albert Borschette, 1246 Luxembourg, Luxembourg; +352 466644 5824; [email protected], http://jpenasse.wordpress.com Many puzzles in finance arise from the inability of models to reconcile, quantitatively, asset returns and macroeconomic risks. A classic example is the equity premium puzzle: average stock returns are far too high to be explained by the observed risk in consumption (Mehra and Prescott, 1985). Another well-known example is that stock returns can be predicted by valuation ratios such as the dividend-price ratio (Campbell and Shiller 1988b). Likewise, stock market volatility is too high to reflect forecasts of future dividends (Shiller, 1981). These puzzles might be resolved if one assumes that financial markets compensate for the risk of infrequent but severe macroeconomic disasters. Barro(2006) uses international data from the last 100 years to document several episodes of large declines in consumption growth. These macroeconomic disasters occur with an annual frequency of about 3% and are strong enough to generate a sizable equity premium, as hypothesized by Rietz(1988). A first generation of models assumes that disasters arise with a constant probability (see e.g., Barro 2009; Barro and Jin 2011; Nakamura et al. 2013). The next generation of papers shows that allowing the risk of disaster to vary over time generates excess volatility and predictability (Gabaix, 2012; Gourio, 2012; Wachter, 2013) and also solves puzzles related to the markets for bonds, options, and currencies (Gabaix, 2012; Gourio, 2013; Farhi and Gabaix, 2016; Hasler and Marfe, 2016).1 Variable disasters also offer the possibility of connecting macroeconomic aggregates with asset prices in production economies (Gabaix, 2011; Gourio, 2012; Kilic and Wachter, 2015; Isoréand Szczerbowicz, 2016). This paper estimates the time-varying probability of a macroeconomic disaster. Our approach follows closely the seminal work of Barro(2006), who finds that the average number of disasters across countries and across time in the past century—in other words, the unconditional probability of a macroeconomic disaster—is high enough to rationalize the unconditional equity premium. We go one step further and propose a simple way to estimate the time-varying probability of a macroeconomic disaster. We document that the conditional probability of a disaster varies over time and is strongly correlated with the conditional equity premium. This step is critical for assessing how well disaster risk can rationalize why asset prices are volatile and why valuation ratios forecast future returns. Formally, we propose a regression-based approach to infer the probability of a macroeconomic disaster over the next year. A sufficient condition for disaster risk to vary over time is to find that rare disasters are forecastable. We thus set up a predictive panel in which the left-hand side variable is an indicator equal to one when a given country i experiences a disaster

1Another strand of research maintains a constant probability of disasters but assumes that the representative agent learns about the model parameters or states (Weitzman 2007, Koulovatianos and Wieland 2011, Du and Elkamhi 2012, Lu and Siemer 2016, Johannes et al. 2016). There is also a burgeoning literature on the implication of disagreements about the likelihood of rare disasters (see, e.g. Chen et al. 2012; Piatti 2015).

1 in t+1, and zero otherwise. In our baseline specification, a disaster is defined as a two standard deviation drop in consumption growth in a given year. On the right-hand side, we consider a wide range of time-t macroeconomic, political, and financial variables, such as consumption growth, various crisis indicators, and asset prices. The fitted values in this regression gives us

πî,t, the time-t probability of a disaster in country i at time t + 1. We find that macroeconomic crises are forecastable, which indicates that disaster risk varies over time. Among our range of right-hand side variables, dividend-price ratios on broad stock indices emerge as robust predictors of future disasters. This is particularly interesting because the DP ratio is a reasonable proxy for the conditional equity premium (Blanchard, 1993; van Binsbergen and Koijen, 2010). To get a sense of magnitudes, a 10% increase in the S&P 500 dividend-price ratio raises the likelihood of a crisis from an unconditional probability of 3.1 percent to 18.3 percent. This is precisely what one would expect if investors where shunning stocks when disaster risk is high. In other words, we find a strong relation between the global equity premium and the likelihood of a crisis, as predicted by rare-disaster models. Interestingly, variation in disaster risk mostly comes from its global component, which we noteπ ˆt and define as the average of country-estimatesπ î,t. This global component captures

62% of the variation of individual country probabilities. Figure1 shows ˆπt and the S&P 500 dividend-price ratio, our main proxy for the world equity premium. Disaster risk typically varies between 0% and 5%, with sharper increases during each of the two world wars, the Great Depression, and the Korean War. Consistently with the evidence that the dividend-price ratio forecasts individual disasters, we observe a tight connection betweenπ ˆt and the DP ratio, our proxy for the global equity premium (correlation of 0.66). Both series spike during the Great Depression and the two world wars, yet also around less prominent events such as the 1907 Knickerbocker Crisis and the 1936–1939 Spanish Civil War. In the postwar period, both are relatively low during the sixties, then surge after the ﬁrst oil crisis of 1973 and remain high thereafter, before declining again during the “Great Moderation” period. Finally, disaster risk and the equity premium rise during the 2008-2009 ﬁnancial crisis, both in modest proportion in comparison to pre-WW2 crises. When constructingπ ˆ, we exploit the fact that the DP ratio forecasts future disasters, so that part of this correlation is of course mechanical. (One can actually achieve a 100% correlation betweenπ ˆ and the DP ratio, by excluding all predicting variables except the DP ratio.) Yet, in fact, we obtain a quite similar π-estimate when we exclude the DP ratio from our panel regression (the correlation is then 0.44). Of course, forecasting power declines substantially when we do so, which is consistent with the idea that equity prices contain incremental information.

2 We next extend our methodology to calibrate a variable disaster model in the spirit of Wachter(2013). Namely, we consider a discrete-time economy, populated by a representative agent with recursive preferences, in which asset prices are derived from the dynamics of aggregate consumption. Consumption growth is subject to rare disasters, which occur with a time-varying probability. The model is of exponential affine type so it can be solved with standard techniques and yields simple, closed-form solutions. We use this model as a laboratory to assess both the qualitative and quantitative effects of our measure of time-varying disaster probability—that is, our only state variable. We emphasize that our calibration does not rely on asset price data. Consumption dynamics parameters can be estimated using our international panel of disasters. To do so, we treat the global disaster probability as a latent variable. The econometrician observes the number of disasters occurring over a given year t + 1, which is a noisy signal of the true time-t disaster probability. While assuming that this probability follows an exogenously specified time-series process, we infer the time-varying probability via a (non-Gaussian) Kalman filter. Although we only use macroeconomic data to calibrate our model, we find that it can generate a large and volatile equity premium—along with a low risk-free rate—under conservative preferences. We find that time-varying disaster risk is crucial in generating a quantitatively large and volatile equity premium as well as volatile returns. Our model can therefore rationalize both the high predictability of stock returns and the low predictability of consumption growth in actual data. This paper contributes to the growing literature on disaster risk. Our approach extends the work of Barro(2006), Barro and Jin(2011), Nakamura et al.(2013), and Barro and Jin(2016), who use international consumption data to compute the (constant) probability of a disaster. Our paper is also related in spirit to Berkman et al.(2011), who use the number and severity of international political crises to proxy for disaster risk, and Colacito et al.(2016), who use survey data to construct a measure of time-varying skewness. Chen et al.(2017) studies the structural sources of the DP ratio’s variation and find that long-run risks and habit leave considerable room (80% of variance) for additional factors; that finding is consistent with results derived using our measure of disaster risk. Another suitable vehicle for estimating disaster risk is option prices. Backus et al.(2011) extract market crash risk premia from index options, but they assume a constant disaster probability. Kelly and Jiang(2014), Farhi et al.(2015), Gao and Song (2015), Seo and Wachter(2015), and Siriwardane(2015) all assume time-varying disaster risk and infer tail risk premia from option prices and from the cross-section of stock returns. Manela and Moreira(2017) construct a proxy for the VIX using front-page articles of the Wall Street

3 Journal, which they relate to disaster concerns. While we find that stock volatility significantly forecasts disasters in univariate regressions, its explanatory power is small, and it disappears when controlling for the dividend-price ratio. To the best of our knowledge, this paper is the first that uses consumption data in constructing a measure of time-varying disaster risk. Our approach is crucial for assessing quantitatively the ability of consumption-based models to rationalize asset pricing puzzles. Another benefit of macroeconomic data is their availability over long time periods, which enables us to identify a source of risk that is more persistent than the risks present in options. In particular, we offer empirical support to the idea (previously advanced in Lettau et al. 2008) that the slow decline in the dividend-price ratio can be explained by a persistent decrease in macroeconomic risk. The rest of this paper is organized as follows. SectionI introduces the theoretical framework and our main research question. In SectionII we present the data and document that disasters are forecastable. SectionIII is concerned with the asset pricing implications of our findings. We conclude in SectionIV.

I. Theoretical Framework

A. Consumption Dynamics

We are interested in the dynamics of consumption Ci,t in a given country i = 1,...,N. The rare disasters model supposes that consumption is hit by large and infrequent shocks occuring with probability πi,t. Formally, we assume log consumption growth in country i, ∆ci,t+1 ≡ log(Ci,t+1/Ci,t) exhibits the following dynamics:

∆ci,t+1 = µi + σiεi,t+1 + vi,t+1, (1)

where εi,t+1 and vi,t+1 are two mutually independent shocks. The ﬁrst shock is a standard normal random variable, and the second shock captures rare consumption disasters. We model vi,t+1 as a compound Poisson shock: vi,t+1 = Ji,t+11∆ni,t+1>0, where ni,t+1 is a Poisson counting process such that ∆ni,t+1 > 0 describes a disaster event occurring at time t+1. The probability that the economy i encounters a disaster in in t + 1 equals2

2 We commit a slight abuse of notation since πi,t is only an approximation of the conditional probability of a disaster on the unit time interval (i.e., yearly). The Poisson counting process ni,t+1 has intensity πi,t. The exact conditional probability of a single disaster occurring over the horizon τ is therefore πi,tτ exp(−πi,tτ), while the probability that a disaster does not occur is exp(−πi,tτ). Hence the residual probability that more than a single disaster occurs is 1 − exp(−πi,tτ)(1 + πi,tτ). According to our estimates, the latter value is about 0.05% when πi,t is at its steady state and less than 1% when πi,t is at its 99th percentile. So to facilitate terminology and ease P P the notation, we shall use πi,t for the conditional disaster probability and i ∆ni,t+1 (instead of i 1∆ni,t+1>0) for the number of disasters.

4 1 X π = % π , where % > 0 for all i and % = 1. (2) i,t i t i N i i

We assume that πt follows a discretized square-root process:

√ πt+1 − π¯ = ρ(πt − π¯) + ν πtut+1, (3)

whereπ ¯ > 0, 0 < ρ < 1, and ν > 0 are constants and where ut is a standard normal random variable uncorrelated with εi,t and vi,t. The probability of a disaster in country i is the product of a country-speciﬁc component %i and a time-varying world component πt. By construction, country-speciﬁc probabilities πi,t are perfectly correlated with πt. Empirically, we document that disaster risk indeed has an approximate one-factor structure. We concentrate on this world component which we refer to as “disaster probability” (or simply “disaster risk”) in the rest of the paper.

Finally, disaster size Ji,t+1 follows a shifted gamma distribution with moment-generating function given by ϕ(u) = e−uθ(1 + uβ)−α, (4) where disasters have support on (−∞, −θ) and where the mean and variance are equal to −(θ + αβ) and αβ2, respectively. These model dynamics are fairly standard and have been shown to be capable of capturing a number of asset pricing regularities (Gabaix, 2012; Wachter, 2013). Our model is most closely related to the continuous-time model of Wachter(2013). Gabaix(2012) calibrates a richer model that allows for movements in the disaster probability and in the expected disaster size, and alternative speciﬁcations have also been proposed in the literature. For example, Barro and Jin(2011) consider diﬀerent laws for disaster size, such as single and double power laws. Gabaix(2011) and Gourio(2012) introduce time-varying disaster risk in production economies. More complex disaster paths have been considered by Gourio(2008), Nakamura et al.(2013), and Hasler and Marfe(2016) such as unfolding consumption declines and subsequent recovery as well as consumption declines leading to economic regime changes (Branger et al., 2015). We intentionally keep the model simple, tractable, and parsimonious in order to focus on the time-series relationship between disaster probability and equilibrium asset prices—the core of our empirical analysis.

5 B. Asset Prices and Disaster Risk

We next discuss how disaster risk matters for asset prices. It is natural to express the log dividend-price ratio using Campbell and Shiller’s (1988a) approximation:

∞ X j dt − pt = κ0 + Et κ1[rt+j − ∆dt+j]; (5) j=1 here dt is log of the economy-wide dividend, pt is the log of the asset price, κ0 and κ1 are log- linearizing constants, and rt is the continuously compounded return on the market portfolio. To simplify the notation, we reason at the level of a representative country and assume that

πi,t = πt. We establish in Section III.B that the log equity premium is aﬃne in the conditional probability of a disaster; that is,

  CCAPM Disaster size Disaster probability rd,t+1 log Et[e ] − rf,t = + πt ×  +  . (6) premium premium premium

The ﬁrst term is due to consumption volatility in normal times and is proportional to the relative risk aversion of the representative agent; the second term compensates for a disaster in consumption and is proportional to the current level of πt. Each of these terms obtains under expected utility (e.g., with CRRA preferences). The third term compensates for the time-varying nature of disaster risk and obtains under non-expected utility; such compensation is positive and proportional to πt when the representative investor prefers an early resolution of uncertainty (Epstein and Zin, 1989). In other words, investors fear uncertainty in their future wealth due to ﬂuctuations in πt. Under both expected and non-expected utility, Eq. (6) implies that if πt changes over time then it should forecast excess returns. Equivalently, we show in

Section III.B that (by Eq. (5)) the log dividend-price ratio is aﬃne in πt:

dt − pt = A0 + Aππt; (7) it follows that the log dividend-price ratio should be perfectly correlated with disaster risk. Of course, we do not expect Eq. (6) to hold exactly in the data and so (7) should include an error term. These circumstances motivate our regressions of the dividend-price ratio on disaster risk: a leading empirical question is the economic strength of that relation, and answering that question is our paper’s main contribution.

6 II. Measuring Disaster Risk

A. Data

Our primary dataset consists in an updated version of the international panel on per capita consumption expenditures and GDP that was constructed by Robert Barro and JoséUrsúa and described in Barro and Ursúa(2010). The sample covers 42 countries, for many of which data is available since the early nineteenth century. We extend the data set to the years 2010– 2015—and thereby include the Great Recession—following by using the World Bank’s World Development Indicators and merge it with data on asset prices, wars and other crises. (Details on data sources and on the construction of all variables used in the analysis are in Appendix A.) Our data spans 1872 to 2015 and comprises 25 OECD countries, 14 countries from Latin America and Asia, as well as Egypt, Russia, and South Africa. We define disaster events as large declines in consumption during a given year. A country i experiences a disaster in year t if log consumption falls by 2 standard deviations (SD) from its long-term growth path:

 1, if ∆ci,t < mean(∆ci,t) − 2 × SD(∆ci,t). Disasteri,t = (8) 0, otherwise.

This choice conforms with prior research in the literature, which defines disasters as peak- to-trough declines in consumption (or GDP) that exceed a fixed cutoff value of 10% or 15%. Barro(2006), for example, considers a panel of 35 countries and documents 60 episodes of GDP contractions exceeding 15% in the twentieth century. We instead focus on one-year windows to identify disasters. Although it is natural to think of macroeconomic crises as events that unfold over several years, we need to measure disaster risk in calendar time. Crises that unfold over several years are only observed at the end of the trough; in contrast, our approach requires the identification of a disaster in real time. Measuring disasters as one-year events is also consistent with our modeling assumption that disasters occur instantaneously (Constantinides, 2008; Donaldson and Mehra, 2008; Julliard and Ghosh, 2012). Focusing one one-year event forces us to revise the cutoff that identifies disasters. We choose two standard deviations as our baseline cutoff because doing so identifies disasters that are about as rare as in the prior literature. However, in Section II.F we document that our results are not sensitive to that cutoff value and present disaster probabilities based on cutoffs of 1.5, 2.5, and 3 standard deviations. Figure2 gives an overview of disasters over our sample. 3 Small black dots indicate data

3We report the list of disasters in Table A.I.

7 unavailability and consumption disasters are marked with blue dots. Data coverage begin at various times across countries, but are generally continuous once they begin, with the exception of Austria, Singapore, and Malaysia. We choose not to interpolate missing observations, although our results are unaffected when we do so. There is a clear clustering of disasters, in particular around the two world wars (highlighted by shaded areas), and disasters are less frequent in the post-1945 period. Interestingly, the Great Recession is associated with only two disasters (both in Iceland). For example, consumption contracted by 6% in Spain in 2009, which is not a non-normal event given the volatility (6.8%) of Spain’s consumption growth over the century. Of course, adjusting disaster cutoff produces more disasters in that period. We explore changes to the way we construct disasters in Section II.F. TableI reports basic statistics of macroeconomic and non-macroeconomic crises. The top panel compares one-year disasters and peak-to-trough disasters listed in Barro and Ursúa(2008). We find that our approach produces macroeconomic disasters that are economically similar to disasters measured as peak-to-trough contractions. Both type of events are about equally likely. We obtain 147 crises, which corresponds to an average disaster probability of 3.1%. This is comparable to the 3.6% annual probability to enter a disaster reported in Barro and Ursúa(2008). 4 Although we define disasters as one-year events, they tend to be quite large. Consumption drops 15.7% on average, which is close to the 21.9% contraction for the typical 3.6-year disaster reported in Barro and Ursúa(2008). Finally, our disasters coincide, by and large, with those identified by Barro and Ursúa.For the period during which the two samples overlap, we find 128 disasters, among which 100 are identified as disasters in Barro and Ursúa. Put differently: conditional on being in a one-year disaster, our approach has a 100/128 ≈ 78.1% probability of identifying a peak-to-trough disaster. The bottom panel of TableI compares macroeconomic disasters with other crises. We collect data on wars (Sarkees and Wayman, 2010), political crises (Center for Systemic Peace), as well as, sovereign defaults, hyperinflation crises, currency crashes, and financial crises from Reinhart and Rogoff(2009). (See the AppendixA.) With the exception of civil wars, these crises tend to coincide with disasters. For instance, wars represent 6.3% of the total sample and 25.9% of the sample that includes disasters. Macroeconomic crises come in a variety of forms. Besides wars, sovereign default, hyperinflation, and currency crises are the main markers of macroeconomic disasters with conditional probabilities above 20%. In contrast, quiet periods with no crises make up 62.8% of our sample, but we find only 53 disasters (out of 147, i.e., 36.1%) without any identified crisis, and they typically correspond to the close aftermath of crises. Although

4Note that this probability is distinct from the unconditional disaster frequency reported in TableI.

8 a disaster in a given country typically coincides with an economic or geopolitical crisis of some form, the global occurrence of these various crises obey diﬀerent cycles. Figure3 depicts the share of countries entering any of the eight form of crises described earlier. While macroeconomic disasters exhibit signs of time variation, this does not seem a uniform properties of other rare events. For example, political crises, albeit infrequent, are spread about evenly over our sample. In contrast, ﬁnancial crises have been almost completely absent between 1940 and the early 1970s. Consequently, the frequencies of crises are poorly correlated (see Figure A.III in the Appendix).

B. Methodology: Regression-based Approach

Our primary goal is to examine if disaster risk varies at all and, if yes, to measure its variation. We propose two frameworks. The first one is regression-based, and exploits covariates such as past disasters and asset prices. The second one treats the global disaster probability as a latent variable that can be filtered out from past realized disasters. While the two methods differ, both rest on the idea that if macroeconomic risk is predictable, then it has to vary over time. In other words, to prove that disaster risk varies over time, it is sufficient to show that realized crises are statistically predictable. When this is the case, we can construct estimates of this disaster probability by projecting future disasters on past predictor variables. The intuition is best understood in the context of our baseline regression-based approach, which is the subject of this section. We return to the filtered-based estimates in Section III.C. Our regression-based approach proceeds in two steps. First, we use our long-run annual data for 42 countries, and estimate a probabilistic model of a macroeconomic crisis event in country i, in year t + 1, as a function of a information available at year t:

Disasteri,t+1 = ai + Xi,tb + ui,t+1, (9)

where Disasteri,t equals to one when a country experiences a 2-SD drop in consumption and zero otherwise. Disaster probabilities obtain simply in a second step, using the ﬁtted values of this regression:

πˆi,t ≡ EtDisasteri,t+1 =a ˆi + Xi,tˆb. (10)

This delivers country-level disaster probabilities. The global disaster risk estimateπ ˆt, as plotted in Figure1, follows from regressing the country estimates on time fixed effects. Our baseline specification relies on OLS estimates, but we also consider logit and probit specifications. The latter ensure that fitted estimates are well behaved probabilities. Our OLS estimates are some-

9 times negative, in which case we ﬂoor them at zero. Our simplest speciﬁcation for Eq. (9) forecasts future disasters at time t + 1 with the share of country experiencing a disaster at time t,

N 1 X F ≡ Disaster , (11) t N i,t i=1 where N denotes the number of countries in our sample. (In practice, to correct for missing observations, we construct Ft as the number of disasters divided by the number of countries for which we have data.) Ft is mechanically related to individual disaster indicators. In fact, 5 projecting these indicators on Ft (without fixed effects) yields a slope of unity. It follows that any global variable that forecasts Ft+1 also, and equivalently, forecasts individual disasters. Such variables proxy for global disaster risk. In particular, if disaster risk is persistent, the coefficient in a panel regression of a future disaster in country i on this global share (without

fixed effects) exactly recovers the first-order autocorrelation for Ft.

C. Empirical Results

TableII presents regression results of the forecasting regressions (9) on the panel of 42 countries over the period 1872-2015. The dependent variable is a dummy equal to one when there is a macroeconomic disaster in country i at time t + 1, and otherwise zero. Our panel consists of 4,734 observations of which 143 are disasters. Each regression is estimated with various predictor variables: Panel A presents results based on macroeconomic predictors; Panel B predicts disasters with six of the seven crisis dummies introduced earlier; Panel C shows estimates based on asset price predictors (including the currency crisis dummy that we classify as a financial predictor for convenience). All specifications include country fixed effects; standard errors are double-clustered on country and year. Each panel reports point estimates, standard errors, and “within” R2 (i.e., R-squareds excluding the contribution of the country fixed effects).

The ﬁrst column in Panel A forecasts macroeconomic crises with disaster frequency Ft with no controls; the second column uses a dummy equal to one if a disaster occurred in country i at time t; the third and fourth columns include lagged consumption growth in country i and GDP-weighted world consumption growth; the last column groups all macroeconomic predictors together. In Column 1, the coeﬃcient associated with lagged disaster frequency is

5 To see this, start with Disasteri,t = bFt + errori,t; summing across countries and dividing by N yields

N 1 X 1 X Disaster = bF + error . N i,t t N i,t i=1 i so that b equals unity (and errors cancel out).

10 0.597, meaning that a 10 percentage point increase in the world share of disasters Ft raises the time t + 1 probability of a disaster in country i by about 6%. The coefficient is not only economically but also statisically significant (t-stat = 5.7), meaning that disaster risk is predictable, and therefore varies over time. As noted earlier, this coefficient can be interpreted as the first-order autocorrelation for the share Ft, in that a 10 percentage points increase in Ft on average equivalently raises Ft+1 by 6%. Of course, this persistence has by itself little economic content. It says that crises in, e.g., Canada and Japan statistically predict more disasters in Turkey the following year. We argue that this predictability reveals variation in global disaster risk, which itself is a reduced form for a plurality of crises that are often international by nature.

Indeed, the influence Ft weakens when we include more predictor variables such as our crisis dummies (see TableIV). We also find evidence of persistence at the country level: a disaster in time t increases the likelihood to observe another disaster the following year by 15.5% (Column 2). The predictive power of past disasters is complemented by world consumption growth rates (Column 5). In Panel B, we report results for regressing our disaster indicator variable on lagged dummies equal to unity when a country enters into one the various crises described in Section II.A. All coefficients are positive, which indicates that the occurrence of a crisis increases the likelihood of observing a disaster. However, significance varies and only wars and political crises appear as strong predictors of macroeconomic disasters (we discuss currency crashes in Panel C). Coun- tries at war and countries that take authoritarian turns see their probability to experience a macroeconomic crisis increase by about 10%. We next forecast disasters with financial variables. We consider the S&P 500 dividend- price ratio and the S&P 500 stock market volatility, the U.S. 3-month bill rate and a dummy equal to one when a country experiences a currency crash, defined as a 15% depreciation of the exchange rate against the relevant anchor (see the Data AppendixA for details). With the exception of currency crashes, all financial predictors are meant to proxy for global variables. The main reason we focus on aggregate series is data coverage, since most disasters occur before individual financial data is available. Using U.S. data, the S&P 500 in particular, allows us to start our analysis in 1872 and to use our full sample of countries. (We return to this point below.) The key predictor here is the dividend-price ratio, which proxies for the conditional equity premium (e.g., Blanchard 1993; van Binsbergen and Koijen 2010). In the simple model that we introduce in SectionI, there is a unit correlation between the dividend-price ratio, the equity premium (in logs), and disaster risk. While the main focus of the present paper is the equity premium, note that our model also predicts that the equity premium, the risk-free

11 rate, and stock market volatility are perfectly correlated. This is a consequence of the model’s simplicity (i.e., it features a single state variable), we expect these auxiliary predictions to hold qualitatively. Namely, stock volatility should go up with disaster risk, while a precautionary motive should induce agent to save more when disaster risk rises. While our model features a closed economy country, in Farhi and Gabaix(2016)’s international setting, an increase in disaster risk causes a depreciation of the domestic currency. We thus expect our currency crash dummy to positively correlate with disaster risk. To summarize, we expect the dividend-price ratio, the risk-free rate, stock volatility, and our currency crash dummy to positively forecast disasters. Panel C shows that the dividend-price ratio is a strong predictor of future disasters. A 10 percentage point increase in the DP ratio raises the likelihood of a disaster by 17.8% (t-stat = 5.7). Increases in stock volatility and currency crashes are also followed by more disasters, although the magnitudes are smaller (e.g., a currency crisis raises the likelihood of a disaster by 2.8%). We also find that the short rate significantly predicts future disasters, albeit with a negative sign. Historically, nominal U.S. interest rates have been low during the Second World War, which coincided with a large number of disasters, and relatively high during the Volker period, when few disasters occurred. We end up with a statistically negative association between short rates and disasters, but we note that the fit is economically tiny (R2 = 0.003). We likewise find small R-squareds for stock volatility and the currency dummy. In contrast, the DP ratio forecasts disasters with an R-squared that is an order of magnitude larger (R2 = 0.027). How large is a 3% R-squared? Suppose we were to regress future realized disasters on the true disaster probability. In population, the R-squared equals

Var(π ) R2 = i,t . Var(Disasteri,t+1)

As long as πi,t is not too volatile, a good approximation of the denominator keeps πi,t constant,

Var(Disasteri,t+1) ≈ π¯(1−π¯), whereπ ¯ is the average probability of a disaster. In our sample, we haveπ ¯ = 3.1%; this yields a 0.030 approximate variance, which is close to the average variance of country disasters in our sample (0.031). Keeping this variance constant, the R-squared is pinned down by the numerator, namely the variance of the disaster probability. A 3% R- squared obtains assuming that the volatility for the disaster probability is about 3%. Because the numerator is the volatility squared, a volatility twice larger would yield a 12% R-squared. The bottom line is that, given the low frequency of realized disasters, R-squareds should be low on an absolute basis. Far from being anecdotal, our results indicate that, on its own, the dividend-price ratio likely accounts for a large share of disaster risk variability.

12 Our main specification forecasts local disasters with the S&P 500 dividend-price ratio, but local dividend-price ratio could have similar predictive power. In our model, local and global valuation ratios are indeed perfectly correlated. In the smaller sample where data is available, we actually find that local DP ratios have little predictive power. The first block of two columns of TableIII compares the predictive power of global and local dividend-price ratios. While the sample is smaller and only includes 36 disasters, we find similar evidence of predictability by the S&P 500 dividend-price ratio as in TableII. In contrast, while the coefficient for local dividend-price ratios is positive, it is small and insignificant. We then verify that other global indices predict macroeconomic crises. The next groups of columns compare the predictive power of the S&P 500 DP ratio with the Dow Jones Industrial index, GFD’s World DP ratio (both available since the 1920s), and the S&P 500 earnings-price ratio, and find similar results. Why do we find that global indices forecast disasters while local indices do not? It seems that local indices are simply noisier. Local DP ratios are twice to thrice more volatile than global indices. TableVIII finds that they do not reliably predict local stock returns, while global indices do. Likewise, in the Online Appendix, we also show that the S&P 500 dividend-price ratio forecasts political crises, defaults, and hyperinflation, while local indices do not. It seems likely that a small number of index constituents combined with changes in datasources and methodologies makes local indices encode less information than global ones in our long sample. TableIV explores the robustness of our results in subsamples, showing “kitchen sink” regressions where we include all of the predictor variables simultaneously. The first column reports the point estimates for the entire sample, which we later use to construct our baseline π-estimates. Throughout this table we continue to estimate the model in subsamples including the periods before and after 1945, and separating OECD and non-OECD countries. We also consider alternative definitions of consumption disasters, including 1.5, 2.5 and 3 standard-deviation cutoffs to define disasters, and a specification with a 2-SD cutoff that we revise after 1945 to account for a potential structural break in consumption volatility around WW2. Finally, we report results for disasters defined as 2-SD drop in GDP growth per capita. In all instances, we find that the dividend-yield coefficients have similar magnitudes regardless of the subsamples and specifications analyzed. In many configurations, the DP ratio actually subsumes most other predictors that are significant in isolation, such as disaster frequency and most of our crisis indicators.

13 D. Properties of Disaster Risk Estimates

Figure4 shows disaster probabilities πi,t (solid lines) and realized disasters (shaded areas) for the 42 countries in our sample. Disaster probabilities are constructed using fitted values in the regression (9). Figure1 shows the global probability πt. Our specification utilizes the full range of predictor variables and the left column in TableIV shows point estimates. Disaster risk is clearly volatile and persistent. It typically ranges between 0% and 10%, and occasionally spikes higher during macroeconomic crises, reflecting the autocorrelation of realized disasters. While country experiences vary, country disaster probabilities have a marked single-factor structure.

The global disaster probability πt, obtained by regressing individual disaster probabilities on year fixed effects, explains 62% of their variance. Interestingly, this result is consistent with Lewis and Liu(2016), who show that a a high degree of common disaster risk is necessary to reproduce the fact that international asset return correlations are greater than consumption growth correlations. TableV further confirms the importance of the global component of disaster risk. We regress local disasters at of 1-, 3-, 5-, and 10-year ahead cumulative horizons, on local and global disaster riskπ î,t andπ ˆt, and on dividend-price ratios. To distinguish the predictive effect of the (global) dividend-price ratio from other macroeconomic variables, we also present regression for disaster probabilities estimated without the DP ratio, denoted asπ ˆ\DP. We report slope estimates, standard errors, and R2 statistics for the full sample (1872–2015) as well as individually for the pre-war (1872–1945) and post-war (1945–2015) periods. In essentially all specifications, higher disaster risk is followed by more disasters, with coefficients significant at the 1% level. The slopes are about one at the 1-year horizon. (They don’t exactly equal unity because disaster probabilities are bounded at zero.) In the full sample, R-squareds forπ î,t,π ˆt ratio are close (they are, respectively, 0.070 and 0.051), in line with our earlier result that the global component explains the largest share of local disaster risk. The predictive slopes and, to a lesser extent, R-squareds tend to rise as the forecasting horizon increase. Interestingly, R- squareds increase (or decrease less) for global disaster risk than for local estimates. This means that the global factor dominates further at longer horizons. Finally, the results are robust to forecasting with the dividend-price ratio instead of withπ î,t orπ ˆt.

E. Predicting Macroeconomic Crises Out of Sample

So far we have only considered in-sample regressions, which means our forecasts may be aﬀected by look-ahead bias, even though we only use lagged data. In this section, we ask if our methodology to forecast disasters could have provided early-warning signals of upcoming disasters in

14 real time. To this end, we start forecasting in 1922 rather than 1872, which gives us a 50-year training period (3,621 country-observations and 92 disasters) to estimate our forecasting model. Each year t, we estimate Eq. (9) using data available up to t and use coefficient estimates to forecast disasters in t + 1. These forecasts (with a floor at zero) constitute our out-of-sample estimates of the local probabilitiesπ î,t, while the global probabilityπ ˆt obtains as the average of the local estimates. As in the previous section, we construct our baseline estimates using the full range of predictors (see TableV), but we also consider a more parsimonious specification based on the dividend-price ratio only. Although we start with a 50-year training period, for many countries the starting samples are often much shorter, which implies noisy fixed effects. We thus do not include country fixed effects, to ensure a fair comparison across all specifications. (The out-of-sample performance of local probabilities deteriorates sensibly when we do include them, although it remains statistically significant). A concern with our approach is that we still use in-sample consumption moments to date consumption disasters—our dependent variable—so that our results may still suffer from look- ahead bias. The reason we do so is to maximize sample coverage. Identifying disasters in real time requires to use a training period to compute consumption moments in each countries, which costs us more observations. We nevertheless repeat our analysis in a smaller sample where we require 50 years of data per country (2,634 country-observations and 31 disasters) so that the dependent variable is also constructed in real time. We compare our out-of-sample probabilities with in-sample probabilities based on the same data. We do so in two ways. First, we compute the t-stat and R2 on a regression of future disaster outcomes on probability forecasts with not constant. We next calculate Receiver Operating Characteristic (ROC) curves, a common tool used in binary classification problems, recently used to assess early-warning signals of financial crises (see, e.g., Schularick and Taylor 2012). Figure5 shows the curve corresponding to in-sample local disaster probabilities (point estimates are shown in TableIV, Column 1). In this type of setting, one is interested in converting probabilities into binary forecasts, which requires choosing a threshold over which a value of one is assigned. The ROC curve shows the true positive rate (i.e., of all the disasters that did happen, what fraction did the model predict?) against the false positive rate (i.e., how often the model wrongly predict a macroeconomic crisis in t + 1?). The curve obtains by varying all possible threshold values. As the threshold increases, the number of disaster signals drops, so fewer disasters are correctly identified and incorrectly signaled, while for a lower threshold more disasters are correctly identified, the cost being that the frequency of false signals also increases. A model with no forecasting power results in a 45-degree line whereas a model with

15 a perfect fit would have an elbow-shaped ROC running from (0,0) to (0,1) to (1,1). Goodness of fit is measured by the area under the ROC curve (AUROC), with 0.5 corresponding to no explanatory power and 1 to a perfect fit. TableVI reports the results of our out-of-sample forecast experiment. We report R-squareds and AUROC statistics, both evaluated in- and out-of-sample. The top panel shows results for disasters identified as 2-SD drop in consumption, based on full-sample consumption moments. The bottom panel reports statistics based on the smaller sample of macroeconomic crises identified with real-time moments. In each case, we forecast disasters with our π-estimates. As earlier, we consider multiple variants:π ˆ andπ î are local and global probabilities in a regression that exploits the full range of predictors (see Column 1 in TableIV). We also consider a specification that excludes the dividend-price ratio,π ˆ\DP, and another that only excludes all variables but the DP ratio, denoted asπ ˆDP. In all instances, forecasts are significant at the 1% level. Logically, in sample, we find that predictability is larger the richer the information set (e.g., R-squareds are larger for local probabilities than for their global counterparts). While R-squareds are lower out of sample,6 this hierarchy is preserved, indicating that, even if our forecasting model is fairly rich, the extra forecasting power brought by including additional predictors remains out-of-sample. In the baseline specification, R-squareds range from 0.020 to 0.044, which is slightly lower than in-sample statistics. The results are similar when disasters are identified in real time, although both in-sample and out-of-sample R2 are lower than when disasters are dated ex post. The reason is that disasters are less frequent in the latter, which mechanically reduces the R2. In contrast, AUROCs are similar in both specifications, ranging from 0.718 to 0.867 in-sample, and 0.722 to 0.809 out-of-sample. In all cases, the statistics are statistically larger than the non-informative level of 0.5.

F. Robustness

Figures A.I and A.II illustrate that our π-estimates are not very sensitive to the sample or specification choices. Figures A.I shows the fitted global disaster probabilities based on the regression shown in TableIV. Panel A plots estimates for the pre- and post-war samples. Panel B compares results for OECD and non-OECD countries. Panel C show results for GDP- disasters and consumption disasters constructed with a revised cutoff after 1945. Panel D assumes different disaster cutoffs: 1.5, 2.5 and 3 standard deviations from average consumption growth rate, instead of the 2-SD cutoff in Eq. (8). In all cases, our estimates exhibit very similar

6An exception is the dividend-price ratio, which has similar, if not better, out-of-sample forecasting power. Our probability-estimates are sometimes truncated at zero (i.e., when OLS would forecast a negative disaster probability). For the dividend-price ratio, out-of-sample π-estimates hit the zero-lower-bound in the quiet 15-year period that precede the Great Recession. This ends up improving the out-of-sample performance.

16 dynamics. In Figure A.II, Panel A shows real-time estimates constructed in Section II.E. Panel B finds that GDP-weighting individual country probabilities yields a very similar global probability than equally-weighting, as we do in our baseline specification. Panel C compares fitted probabilities with probit and logit models, instead of the linear specification (9). Regression results, reported in Tables A.III and A.IV are qualitatively similar. Non-linear π-estimates tend to be more extreme during the two world wars and the Great Depression, but exhibit similar dynamics otherwise. Finally, Panel D of the figure examines filtered probability estimates (discussed later in Section III.C): one with square-root latent dynamics, the other with log-autoregressive dynamics. While the filter only uses realized disasters to infer the latent global probability, the estimated probabilities capture similar events than our richer regression-based estimates.

III. Asset Pricing Implications

A. Predicting Consumption and Stock Returns

Our main hypothesis is that the risk of a disaster occurring varies over time. Changes in the forecasts of future disasters induce fluctuations not only in expected future cash flows but also in the premium required to hold stocks. The second of these effects dominates and so stock prices appear to be cheaper under a high risk of disaster, which leads to predictable returns. This dynamic holds especially in the absence of disasters, when all variation in the dividend- price ratio and in expected returns seems unrelated to future cash flows. Therefore, in this section we ask whether πt forecasts consumption growth and excess returns. And since the only source of dividend-price ratio variability in our model is predictable disaster risk, we repeat this forecasting exercise while using the DP ratio. Later, in Section III.E, we shall compare these empirical results with the simulated moments generated by our model. Although this paper finds consumption disaster risk to be predictable, this needs not be true for the consumption growth rate. Indeed, the literature has emphasized the dividend-price ratio’s limited power to forecast future consumption (see, e.g., Beeler and Campbell(2012) for US evidence), meaning that most of the DP’s variation stems from changes in either risk magnitude or risk premia. TableVII establish that, in line with prior evidence, neither π-estimates nor the dividend-price ratio have strong predictive power for international consumption growth. We report, in the same format as TableV, regression results of consumption growth at 1-, 3-, 5-, and 10-year ahead cumulative horizons, on local and global disaster riskπ î,t andπ ˆt, and on dividend-price ratios. An increase in disaster risk predicts a decline in consumption growth,

17 but the effect is short-lived and reverses at horizons larger than one year. Subsample results indicate that most of the negative predictability in consumption growth can be attributed to the pre-war period. After 1945, consumption appears somehow predictable, but with a small positive sign, meaning that an increase in disaster risk forecasts higher growth. We find similarly mixed evidence when forecasting with DP ratios, consistently with Beeler and Campbell(2012). The main message of the paper is that disaster risk can rationalize variation in the equity premium. Indeed, our estimated probability of a disaster strongly correlates with the world dividend-price ratio, which itself forecasts stock returns. TableVIII presents more direct evidence: namely, disaster risk predicts future excess returns. We report predictive regression results with similar predictor variables as in TablesV andVII. First observe that the global DP ratio strongly predicts future returns, with magnitudes that increase with the forecast horizon. A one percent increase in the dividend-price ratio forecasts returns 3.6 and 20.5 percent higher at one and ten year horizons, respectively. As noted in Cochrane(2011), this implies a highly volatile equity premium: in our sample, the volatility of the dividend-price ratio is about 1.6%, so our estimates imply that the world equity premium varies by about 5.7% per year. This is very similar to results in the prior literature about time- varing risk premia. For instance, Cochrane(2011) estimates that the US equity premium varies by about 5.4% per year. In contrast, local dividend-price ratio have essentially no forecasting power (bottom panel); this echoes our previous finding in TableIII that only global valuation ratios forecast future disasters. Next, observe that in all instances π-estimates also significantly forecast returns. This is not simply a restatement of the fact that the dividend-price ratio forecasts future disasters. Examine, for instance, the point estimates forπ ˆ\DP. This probability is constructed using the same predictor variable as our baseline estimateπ ˆ, with the exclusion of the DP ratio. A one percent increase inπ ˆ\DP forecasts a 2.1 percentage point larger return the next year. The forecasts add up to 6.8 percentage increase over a ten-year period. Given that our π-estimates are about twice more variable than the dividend-price ratio, this implies a volatility for the 1-year equity premium of about 7%. In other words, variation in disaster risk can entirely rationalize the magnitude of stock market predictability.

B. The Model

This section presents a simple equilibrium asset pricing model with time-varying disaster risk. The model is a discrete-time version of the one oﬀered by Wachter(2013) and belongs to the discrete-time aﬃne class proposed in Drechsler and Yaron(2011). After describing the economy and the dynamics of aggregate consumption, we derive equilibrium asset prices. Solution details

18 are provided in AppendixC. We consider a pure exchange economy, `ala Lucas(1978), populated by a representative investor with recursive preferences (Epstein and Zin, 1989):

1−1/ψ 1−1/ψ 1/(1−1/ψ) 1−γ 1−γ Vt = (1 − δ)Ct + δ(Et[Vt+1 ]) .

In this expression, δ is the time discount factor, γ 6= 1 is the relative risk aversion, and ψ is the elasticity of intertemporal substitution. To ensure tractability, we focus on the case of ψ equal to one. Similarly to Collin-Dufresne et al.(2016), we normalize utility V by consumption level C such that the log value function vct ≡ log Vt/Ct is given by

δ vc = log [e(1−γ)(∆ct+1+vct+1)]. (12) t 1 − γ Et

The aggregate dividend paid by the equity claim is modeled in a parsimonious way (after Abel

1999, Campbell 2003, and Wachter 2013): ∆dt+1 = φ∆ct+1. Although this model is simplistic, when φ > 1 dividends fall by more than consumption in the event of a disaster—which is consistent with US data (Longstaﬀ and Piazzesi, 2004). We solve for asset prices by expressing the stochastic discount factor in terms of the investor’s value function, which is aﬃne in the disaster probability πt. We can then solve for the return on the equity claim via the investor’s Euler equation up to the usual Campbell and Shiller (1988a) log linearization. As in Section I.B, we reason at the level of a typical country and so the country disaster probability is πi,t = πt. The stochastic discount factor is given by

e(1−γ)vct+1 M = δe−γ∆ct+1 × (13) t+1 (1−γ)(∆c +vc ) | {z } Et[e t+1 t+1 ] discounting of | {z } expected utility discounting of continuation utility

(Collin-Dufresne et al., 2016) and the value function satisﬁes

vct = v0 + vππt, (14) where

δ γσ2 v = µ +π ¯(1 − ρ)v − , 0 1 − δ π 2 δρ − 1 − p1 + δ(2δν2 + ρ2 − 2ρ − 2δν2ϕ(1 − γ)) v = . π δ(γ − 1)ν2

19 Note that the stochastic discount factor variance (i.e., the price of risk in the economy) increases with the disaster probability:

2 2 2 2 2 2 2 vart(log Mt+1) = γ σ + (γ − 1) vπν πt + γ Et[Jt+1]πt,

2 where Et[Jt+1] is given by the second-order derivative of the moment-generating function evaluated at 0.

Let Wt be the present value of the future aggregate consumption stream. If the elasticity of intertemporal substitution is equal to 1 then investor wealth is proportional to consumption, δ Wt = Ct 1−δ , in which case the log return on wealth satisﬁes rc,t+1 = − log δ + ∆ct+1.

The log risk-free rate, rf,t = − log Et[Mt+1], is aﬃne in the disaster probability:

2 rf,t = − log δ + µ − γσ + πt(ϕ(1 − γ) − ϕ(−γ)). (15)

This risk-free rate is stationary and decreases linearly with disaster probability π. An increased likelihood of disaster increases the consumption risk, which the investor can hedge with risk-free investments. The resulting increase in holdings of the risk-free asset entails an increased price for that asset and hence a reduction in the risk-free rate. This eﬀect increases in magnitude with relative risk aversion. To solve for the equity claim price, we log-linearize returns around the unconditional mean of the log DP ratio dp ≡ E[dt − pt] with dt − pt ≡ log Dt/Pt:

−dt+1+pt+1 rd,t+1 = log(e + 1) + dt − pt + ∆dt+1

≈ k0 − k1(dt+1 − pt+1) + dt − pt + ∆dt+1;

where the endogenous constants k0 and k1 satisfy

−dp dp k0 = −k1 log(k1) − (1 − k1) log(1 − k1) and k1 = e /(1 + e ).

Campbell et al.(1997) and Bansal et al.(2012) document the high accuracy of such a log linearization, which we hereafter assume to be exact. We can use the Euler equation 1 = r Et[Mt+1e d,t+1 ] to recover that the log DP ratio is aﬃne in the disaster probability:

dt − pt = A0 + Aππt, (16)

20 where

A0 = log(1 − k1) − log(k1) − Aππ¯ and q 1 2 2 2 Aπ = 2 2 Ω + 2k1ν (ϕ(1 − γ) − ϕ(φ − γ)) − Ω k1ν

2 with Ω = 1 − k1(ρ + (1 − γ)vπν ).

The DP ratio is stationary and increases (resp., decreases) with the disaster probability πt when γ is (resp., is not) greater than 1. These dynamics reflect a preference for the early (resp., late) resolution of uncertainty about time variation in πt. An increase in disaster probability makes it more likely that disasters will affect future consumption. An investor who prefers early resolution of uncertainty (γ > 1) is worried about current disaster risk and also about uncertainty in future disaster risk. Hence prices are low, relative to dividends, when πt is high (and vice versa). Note that the substitution effect and the income effect offset each other when the elasticity of intertemporal substitution is equal to one, as we assume here. The log equity premium is given by

rd,t+1 2 2 log Et[e ] − rf,t = γφσ + (γ − 1)k1Aπvπν πt | {z } | {z } non-disaster risk disaster probability risk

+ (ϕ(φ) + ϕ(−γ) − ϕ(φ − γ) − 1)πt, (17) | {z } disaster size risk and the return variance is

2 2 2 2 2 2 ∂2 vart(rd,t+1) = φ σ + k1Aπν πt + φ πt ∂u2 ϕ(u) u=0 . (18) | {z } | {z } | {z } non-disaster risk disaster probability risk disaster size risk

Both the equity premium and the return variance are given by three terms. The ﬁrst of these terms concerns non-disaster risk and gives rise to the usual consumption-CAPM compensation (Lucas, 1978). Variation in disaster probability gives rise to the second component of the equity premium and return variance. This term increases (resp., decreases) with current disaster probability and also with its persistence and volatility for γ > 1 (resp., γ < 1), whereas it disappears for γ = 1; this term also captures the excess volatility of returns over dividends. The third term is associated with disaster realizations and increases with both disaster size variance and current disaster probability.

21 Finally, the risk-neutral return variance (i.e., the model-implied VIX) has a similar form:

Q 2 2 2 2 2 2 ∂2 vart (rd,t+1) = φ σ + k1Aπν πt + φ πt ∂u2 ϕ(u) u=−γ, (19) | {z } | {z } non-disaster risk disaster probability risk | {z } disaster size risk where only the third term diﬀers from the physical return variance. Hence, the variance risk premium is proportional to πt.

C. Methodology: Filtered-based Approach

We now describe the econometric framework. Our goal is to estimate πt, the annual probability of a macroeconomic disaster over the next year in a typical country, as well as the parameters governing that probability’s dynamics. The remaining parameters, which correspond to the size distribution of disasters and to consumption in normal times, are estimated in a straightforward way using maximum likelihood techniques. Here we provide a general overview of the estimation procedure; readers are referred to AppendixB for a detailed description.

We assume that πt follows the square-root process given by Eq. (3). Although this probability πt is latent, we observe disasters ∆ni,t as they occur across countries and time. In light of our assumption that individual disaster probabilities are perfectly correlated, we focus on 7 the number of disasters occurring in a given year t, or ∆nt. Because country-speciﬁc disasters follow Poisson distributions, the sum of observed disasters in t + 1 also follows a Poisson distribution: X X ∆nt+1 ≡ ∆ni,t+1 ∼ Poisson πi,t = Poisson(Nπt). (20) i i

According to Eq. (20), ∆nt follows a Poisson distribution with intensity Nπt. Eqs. (3) and (20) together define a filtering problem: the true disaster probability is unobserved but we can still measure it, albeit with noise. Hence our procedure aims to remove this noise in order to recover an estimate of πt and of the parameters in Eq. (3). Standard filtering techniques require Gaussian observations and that the latent variable is represented as an autoregressive process. Because our observations instead obey a Poisson distribution, we follow Durbin and Koopman(1997) and approximate the non-Gaussian model by a traditional linear Gaussian model. In their approach, Nπt is assumed to follow a log- autoregressive process. That differs from our assumption of a square-root process, but it does ensure a positive probability of disaster. In order to estimate a square-root process, we proceed

7Our econometric model does not account for the increase, over time, in the number of countries for which we have data. To correct for that shortcoming, we divide the number of disasters at time t by the number of countries for which data is available in t and then multiply that quotient by the total number of countries N = 42. The result is then rounded to the nearest integer.

22 in two steps. First, we estimate a log-autoregressive disaster probability using the Durbin and Koopman(1997) methodology. In the second step, we recover the square-root–process probability and parameters by way of a simple OLS regression. The two probabilities are close, in practice, so little information is lost in this process (see Figure A.II, Panel D., in the Appendix). We simulate 1,000 bootstrap samples in order to assess statistical signiﬁcance. Finally, we assume that our disaster-related parameters (i.e.,π ¯, ρ, µ, θ, α, and β) are common across countries and also time invariant; at the same time, we allow the parameters that are unrelated to disasters (µi and σi) to vary across countries. Because we focus primarily on the US economy, we estimate these parameters with US data.

D. Estimation Results

TableIX presents the point estimates, standard errors, and 95% confidence intervals for the consumption process parameters defined by Eqs. (1)–(4). Disaster parameters are estimated by pooling information across countries, although estimates for consumption growth in “normal times” are based on US data. Our first key result is that disaster risk is both persistent (ρ = 0.78) and volatile, with a scale parameter ν of about 0.07. These two parameters contribute to the unconditional volatility of disaster risk, which is pν2π/¯ (1 − ρ2) = 2.1%. Interestingly, the point estimates are close to the corresponding (continuous-time) parameters in Wachter(2013). Wachter chooses a more persistent ρ = 0.92 (using our notation), which is set to match the autocorrelation of the price-dividend ratio.8 The scale parameter is lower than in our setup (ν = 0.067). Altogether, these parameters imply a very similar volatility of disaster risk (3.2%). The mean disaster size is θ+αβ = 15.7% with a 7.3% standard deviation (the minimum disaster size is θ = 3.4%). We plot the distribution of consumption declines, along with the fitted density, in Figure6. Consumption growth parameters in “normal times” belong to the usual range of values: the long-term consumption growth rate is 1.9%, and consumption volatility is 3.1%.

E. Asset Pricing Results

In this section we discuss the calibration of unobserved parameters and spell out the model’s main predictions about consumption growth and asset prices. We show that our estimated parameters governing the dynamics of πt oﬀer quantitative support for the asset pricing predictions of our variable disaster model. In view of the given consumption parameters, we set the dividends’ leverage parameter

8The persistence of scaled price variables varies according to how they are deﬁned. In our sample, the autocorrelation of the dividend-price ratio is 0.80, while the autocorrelation of the price-dividend ratio is 0.93. When measured in logs, the autocorrelations are both equal to 0.89.

23 to φ = 2.6. This value implies that dividend growth and consumption growth are perfectly correlated (the empirical correlation does, in fact, exceed 60%) and that dividends are more volatile than consumption. Evidence from the Depression era suggests that disasters have a much larger effect on dividends than on consumption (Longstaff and Piazzesi, 2004). Wachter (2013) argues that a value of 2.6 is conservative and captures well the extra risk of dividends relative to consumption in both normal times and disaster periods. We complete the calibration by setting the preference parameters δ and γ to fit the main asset pricing moments. We find that low values (between 6 and 8) of relative risk aversion suffice to generate a high equity premium that conforms with the data. Hereafter we consider the pair (δ = 97.5%, γ = 7.5) as our baseline calibration. TableX shows that this calibration leads to a good fit of the most important unconditional moments of asset prices. That outcome is hardly unexpected because our estimated disaster risk parameters are relatively close to those used by Wachter(2013), which are calibrated to match actual asset prices. The main difference concerns the persistence of disaster probability which is lower for our estimates: this requires a higher relative risk aversion. Panel A of TableX reports historical asset pricing moments and their model counterparts for several preference settings. The risk-free rate is about 1.3% with 2.2% unconditional volatility. The equity premium is approximately 7.2%, and the return volatility is about 13.4%; hence the Sharpe ratio is 53.8%. The DP ratio is about 4.2% with 0.5% unconditional volatility—values that are quite close to the data. In Panel B of the table, we study the effect of a change in either the persistence ρ or the disaster scale ν (while holding all else constant). Raising either persistence or scale parameters increases both the equity premium and return volatility, notwithstanding the common level of risk aversion (γ = 7.5). Conversely, a decrease in persistence or scale parameters significantly lowers the first two moments of stock returns. We note that variation in disaster risk is important for generating a large equity premium. Agents who prefer the early resolution of uncertainty dislike bad news. They also dislike the possibility of more bad news in the future. Thus agents price not only disasters but also the uncertainty associated with variation in the probability of disaster. In the baseline calibration, about 17% of the equity premium reflects compensation for disaster probability risk at the steady state (see Eq. (17)), whereas non-disaster risk and disaster size risk contribute for 26% and 57% respectively. Likewise, disaster probability risk accounts for about 28% of return variance in the steady state (Eq. (18)), whereas both non-disaster risk and disaster size risk contribute for 36%. These results indicate that a preference for the early resolution of uncertainty is important for quantitatively reconciling disaster risk and asset prices, and they are in line with our previous

24 ﬁnding that disaster risk is correlated with the US DP ratio in post-war data (i.e., in the absence of disasters). Now we investigate the extent of predictability that is implied by our model. We show in

TableXI that πt is a good predictor of returns but not of consumption growth. For the sake of consistency with Section III.A’s empirical analysis, we simulate 1,000 paths of the economy with 116 yearly observations for each simulation. To avoid generating a negative probability of disaster, we simulate monthly series and then convert them to annual frequency.9 We then run regressions using a horizon of 1, 3, 5, and ﬁnally 10 years. The leftmost four data columns of Ta- bleXI report average regression estimates for each horizon. We can see that disaster probability does not forecast consumption growth. This lack of predictability is not surprising given the assumed consumption dynamics of Eq. (1); it also accords with actual data, as documented in the middle (pre-war) panel of TableVII. Disaster probability is a reasonably good predictor of excess returns, with a positive slope and explanatory power increasing with the horizon. Although the explanatory power is due to our model’s simplifying assumptions, that model captures the positive intertemporal relationship—between disaster probability and excess returns—found in the data. Results reported in the center and the right side of TableXI allow for a more accurate comparison between model-implied predictability and the empirical evidence. The center four columns report predictability results from simulations of the economy using a sample of 46 years in which four disasters occur; in the rightmost four columns we consider the case of a 70-year economy in which no disasters take place. In this way we characterize scenarios comparable to the US experience during the pre- and post-war periods, respectively. In neither period does disaster probability forecast consumption growth, yet it forecasts excess returns in both periods. We ﬁnd that excess returns are predicted more reliably in the absence of realized disasters, which is also consistent with empirical evidence from the post-war United States. Overall, our measure of disaster probability enables a quantitative assessment of variable disaster models and strongly supports the notion that asset prices depend heavily on the time- varying nature of disaster risk.

IV. Conclusion

The rare disaster model is one of the few workhorses of empirical asset pricing. The model is based on the idea that equity markets compensate investors’ fear of rare but large macroe-

9 4 4 √ Simulations of πt at monthly frequency (i.e., πt+4 = (1 − ρ )¯π + ρ πt + ν πt4ut for 4 = 1/12) have but a negligible likelihood of realizations lower than 0.1%, which we replace by a small positive threshold.

25 conomic events. In this paper we use consumption data to assess such a model by directly estimating its unique and all-important state variable: the time-varying probability of a macroeconomic disaster. We use several approaches to estimate the time-varying expected probability of a macroeconomic disaster; for this purpose, we use a data set comprising 42 countries and covering the period from 1900 to 2015. The results reported here constitute strong evidence in support of the rare disaster hypothesis. Disaster risk is volatile, persistent, and strongly correlated with the US DP ratio. Our model generates a large and volatile equity premium whose coeﬃcient of relative risk aversion is 7.5 and whose elasticity of intertemporal substitution is 1.

26 Table I: Taxonomy of crises

Event Av. size Av. duration Frequency Conditional frequency

Comparison with peak-to-through disasters: Disaster 15.7% 1 3.1% 100.0% Peak-to-through disastera 21.9% 3.6 12.0% 78.1%

Comparison with various crises: War 2.8 6.3% 25.9% Civil war 4.6 7.6% 6.8% Political crisis 2.0 2.1% 10.2% Sovereign default 7.6 9.5% 22.4% Hyperinflation 2.9 9.9% 22.4% Currency crash 1.7 14.5% 27.9% Financial crisis 2.8 8.9% 15.0% Quiet period 6.6 62.8% 36.1% Notes. We report the average disaster size (when appropriate), crises duration (in years), and crises unconditional and conditional frequencies. Frequency is defined as the number of crisis years divided by total years. The conditional frequency column reports the frequency of crises that coincide with macroeconomic disasters. The top panel compares one-year disasters and peak-to-trough disasters listed in Barro and Ursúa(2008). A one-year disaster is defined as a 2 standard-deviation drop in consumption growth over one year. The bottom panel compares macroeconomic disasters with other crises. See Data AppendixA for further details. Observations are over the sample of 42 countries, 1872-2015.

27 Table II: Predicting macroeconomic disasters

Dependent variable: Indicator = 1 if local disaster in year t + 1 (1) (2) (3) (4) (5) Panel A: Macroeconomic Predictors Disaster frequency† 0.597∗∗∗ – – – 0.386∗∗∗ (0.10) – – – (0.11) Disaster – 0.155∗∗∗ – – 0.107∗∗∗ – (0.03) – – (0.03) Consumption growth – – −0.231∗∗∗ – 0.016 – – (0.08) – (0.07) Consumption growth† ––– −1.261∗∗∗ −0.512∗∗ – – – (0.26) (0.21) Adj. R2 0.037 0.024 0.008 0.023 0.050 Panel B: Crisis Predictors War 0.101∗∗∗ – – – 0.095∗∗∗ (0.02) – – – (0.02) Civil war 0.005 – – – −0.003 (0.01) – – – (0.01) Political crisis – 0.097∗∗∗ – – 0.077∗∗ – (0.03) – – (0.03) Sovereign default – – 0.033∗ – 0.022 – – (0.02) – (0.02) Hyperinflation – – 0.022 – 0.017 – – (0.02) – (0.01) Financial crisis – – – 0.017 0.019∗ – – – (0.01) (0.01) Adj. R2 0.019 0.007 0.004 0.001 0.027 Panel C: Asset Price Predictors Dividend-price ratio† 1.782∗∗∗ – – – 1.749∗∗∗ (0.31) – – – (0.34) Stock volatility† – 0.204∗∗ – – 0.049 – (0.09) – – (0.08) Short rate† –– −0.306∗∗ – −0.391∗∗∗ – – (0.15) – (0.14) Currency crash – – – 0.028∗∗∗ 0.027∗∗∗ – – – (0.01) (0.01) Adj. R2 0.027 0.004 0.003 0.003 0.034 Notes. This table shows estimates from the linear probability model specified in equation (9). Observa- tions are over the sample of 42 countries, 1872-2015. Our panel consists of 4,734 observations of which 143 are disasters. The dependent variable is a dummy equal to one when there is a macroeconomic disaster in country i at time t + 1, defined as a 2 standard-deviation drop in consumption growth, and 0 otherwise. The disaster indicator is regressed on macroeconomic, crisis, and asset price predictors. Global variables are indicated with a †. Disaster frequency denotes the share of countries experiencing a macroeconomic disaster. Crisis variables are indicators equal to unity when a country enters in a given crisis. With the exception of currency crash indicators, asset prices utilize US data. The dividend-price ratio, stock volatility are based on the S&P 500 and the short rate is the 3-month bill rate. See Data AppendixA for further details. Standard errors in parenthesis are computed from standard errors dually clustered on country and year. ∗, ∗∗, and ∗∗∗ indicate statistical significance at 10%, 5%, and 1% levels, respectively.

28 Table III: Only global valuation ratios forecast disasters

Dependent variable: Indicator = 1 if local disaster in year t + 1 (1) (2) (3) (4) DP ratio 1.107∗∗∗ – 1.576∗∗∗ – 1.638∗∗∗ – 1.782∗∗∗ – (0.33) – (0.39) – (0.37) – (0.31) – DP ratio (local) – 0.058 – – – – – – – (0.07) – – – – – – DP ratio (GFD World) – – – 1.524∗∗∗ –––– – – – (0.37) – – – – DP ratio (GFD DJ) – – – – – 1.464∗∗∗ –– – – – – – (0.36) – – EP ratio – – – – – – – 0.742∗∗∗ – – – – – – – (0.20) Adj. R2 0.014 0.000 0.024 0.014 0.026 0.019 0.027 0.016 Sample start 1873 1873 1926 1926 1921 1921 1873 1873 Sample end 2015 2015 2015 2015 2015 2015 2015 2015 Disasters 36 36 92 92 101 101 143 143 N 2,168 2,168 3,541 3,541 3,697 3,697 4,734 4,734 Notes. This table compares the forecasting power of global and local DP ratios in the linear probability model specified in equation (9). A disaster is defined as a 2 standard-deviation drop in consumption growth in a given country over a given year. The disaster indicator is regressed on global and local valuation ratios. DP ratio indicates our main predictor, based on the S&P 500, whereas GFD World and DJ indicate the world and Dow Jones Industrial indices constructed by Global Financial Data. EP indicates the earnings-price ratio. Standard errors in parenthesis are computed from standard errors dually clustered on country and year. ∗, ∗∗, and ∗∗∗ indicate statistical significance at 10%, 5%, and 1% levels, respectively.

29 Table IV: Predicting disasters: full speciﬁcation results

Dependent variable: Indicator = 1 if local disaster in year t + 1 Full Pre-1945 Post-1945 OECD Non-OECD 1945 break GDP 1.5 SD 2.5 SD 3 SD Disaster frequency† 0.169 0.213 0.065 0.159 0.196 0.157∗∗ 0.231∗∗ 0.188 0.152∗∗∗ 0.099∗ (0.10) (0.15) (0.18) (0.12) (0.18) (0.07) (0.11) (0.13) (0.06) (0.06) Disaster 0.096∗∗∗ 0.078∗∗ 0.128∗∗ 0.074∗∗ 0.121∗∗ 0.082∗∗ 0.116∗∗∗ 0.107∗∗∗ 0.064∗∗ 0.034 (0.03) (0.04) (0.06) (0.04) (0.05) (0.03) (0.04) (0.03) (0.03) (0.02) Consumption growth 0.027 0.086 −0.070 −0.035 0.103 0.036 −0.060 0.040 −0.030 −0.043 (0.07) (0.11) (0.08) (0.09) (0.12) (0.09) (0.05) (0.07) (0.06) (0.04) Consumption growth† −0.419∗∗ −0.162 −0.360 −0.591∗∗∗ 0.063 0.079 −0.336 −0.738∗∗∗ −0.014 −0.112 (0.17) (0.29) (0.32) (0.22) (0.51) (0.13) (0.27) (0.26) (0.14) (0.09) War 0.060∗∗∗ 0.102∗∗∗ 0.029∗ 0.054∗∗∗ 0.071∗ 0.042∗∗∗ 0.030∗ 0.081∗∗∗ 0.043∗∗∗ 0.033∗∗∗ (0.02) (0.03) (0.02) (0.02) (0.04) (0.01) (0.02) (0.03) (0.01) (0.01) Civil war 0.004 −0.025 0.017 −0.010 0.017 0.002 0.017∗ 0.011 0.005 0.005 (0.01) (0.02) (0.01) (0.02) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) Political crisis 0.045 0.023 0.060 0.053 0.006 0.019 0.086∗∗∗ 0.071∗∗ 0.033 0.008 (0.03) (0.04) (0.05) (0.04) (0.05) (0.03) (0.03) (0.03) (0.03) (0.02) Sovereign default 0.004 −0.001 0.000 0.007 0.001 −0.017 −0.002 0.009 −0.002 −0.001 30 (0.01) (0.03) (0.01) (0.03) (0.01) (0.01) (0.01) (0.02) (0.01) (0.01) Hyperinflation 0.004 0.016 0.004 0.028 −0.019 0.014 0.016 −0.002 0.009 0.006 (0.02) (0.05) (0.02) (0.03) (0.01) (0.02) (0.02) (0.02) (0.01) (0.01) Financial crisis 0.017 −0.006 0.025∗ 0.008 0.035∗ 0.031∗∗∗ 0.012 0.021 0.003 −0.001 (0.01) (0.02) (0.01) (0.01) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) Dividend-price ratio† 0.941∗∗∗ 1.197∗ 0.773∗∗ 0.821∗∗∗ 1.126∗∗ 0.476∗∗ 0.396∗ 1.264∗∗∗ 0.446∗∗∗ 0.357∗∗ (0.29) (0.64) (0.31) (0.26) (0.48) (0.23) (0.20) (0.35) (0.17) (0.15) Stock volatility† 0.031 −0.121 0.043 0.013 0.084 0.006 0.096 0.124 0.076 −0.008 (0.08) (0.11) (0.07) (0.09) (0.13) (0.05) (0.08) (0.10) (0.05) (0.05) Short rate† −0.225∗∗ −0.359 −0.194∗ −0.231∗ −0.212 −0.096 −0.127 −0.282∗ −0.036 −0.094 (0.11) (0.44) (0.10) (0.13) (0.15) (0.10) (0.10) (0.15) (0.08) (0.06) Currency crash 0.019∗ 0.043 0.010 0.022 0.014 0.020 0.011 0.032∗∗ 0.010 −0.001 (0.01) (0.03) (0.01) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) Adj. R2 0.067 0.062 0.059 0.081 0.053 0.036 0.074 0.079 0.048 0.042 Disasters 143 106 45 93 50 118 133 229 80 43 N 4,734 1,865 2,901 3,158 1,576 4,734 4,734 4,734 4,734 4,734 Notes. This table shows estimates from the linear probability model specified in equation (9). Observations are over the sample of 42 countries, 1872-2015. The dependent variable is a dummy equal to one when there is a macroeconomic disaster in country i at time t + 1, defined as a 2 standard-deviation drop in consumption growth, and 0 otherwise, unless indicated otherwise. The disaster indicator is regressed on macroeconomic, crisis, and asset price predictors (see TableII). Results are shown for the full sample, as well as subsamples comparing pre- and post-WW2 data, OECD and non-OECD countries. The next five columns report results with different disaster definitions. “1945 break” indicates consumption disasters constructed with a revised cutoff after 1945; “GDP” correspond to GDP-disasters; finally, we consider different disaster cutoffs: 1.5, 2.5 and 3 standard deviations from average consumption growth rate, instead of the 2-SD cutoff in Eq. (8). Analogous results based on probit and logit models are given in Tables A.III and A.IV. Standard errors in parenthesis are computed from standard errors dually clustered on country and year. ∗, ∗∗, and ∗∗∗ indicate statistical significance at 10%, 5%, and 1% levels, respectively. Table V: Predicting disasters with πî, πˆ, and DP ratios

Full sample (1873-2015) Prewar (1873-1945) Postwar (1945-2015) K 1 3 5 10 1 3 5 10 1 3 5 10 PK k=1 Disasteri,t+k = a + bπˆi,t + ui,t+k b 1.07∗∗∗ 2.42∗∗∗ 3.03∗∗∗ 3.75∗∗∗ 1.07∗∗∗ 2.24∗∗∗ 2.34∗∗∗ 1.42∗ 0.99∗∗∗ 1.45∗∗∗ 1.58∗∗∗ 1.97∗∗∗ (0.14) (0.30) (0.40) (0.47) (0.15) (0.36) (0.60) (0.83) (0.32) (0.40) (0.40) (0.52) Adj. R2 0.070 0.095 0.084 0.062 0.054 0.068 0.040 0.008 0.055 0.036 0.026 0.025 N 4,734 4,642 4,550 4,325 1,865 1,793 1,721 1,549 2,901 2,817 2,733 2,523 PK k=1 Disasteri,t+k = a + bπˆt + ui,t+k b 1.14∗∗∗ 2.87∗∗∗ 3.84∗∗∗ 5.01∗∗∗ 1.11∗∗∗ 2.61∗∗∗ 3.12∗∗∗ 2.89∗∗∗ 0.93∗∗ 1.61∗∗∗ 1.88∗∗∗ 2.47∗∗∗ (0.16) (0.34) (0.49) (0.61) (0.18) (0.41) (0.71) (0.97) (0.41) (0.48) (0.49) (0.70) Adj. R2 0.051 0.086 0.086 0.071 0.043 0.069 0.051 0.024 0.023 0.021 0.018 0.019 N 4,767 4,675 4,583 4,357 1,888 1,816 1,744 1,570 2,911 2,827 2,743 2,533 PK \DP k=1 Disasteri,t+k = a + bπˆt + ui,t+k b 1.08∗∗∗ 2.73∗∗∗ 3.48∗∗∗ 4.17∗∗∗ 1.05∗∗∗ 2.52∗∗∗ 2.97∗∗∗ 2.68∗∗ 0.83 1.16∗∗ 1.18∗∗ 1.79∗∗

31 (0.17) (0.37) (0.54) (0.68) (0.18) (0.36) (0.66) (1.04) (0.52) (0.57) (0.57) (0.85) Adj. R2 0.044 0.074 0.068 0.047 0.041 0.067 0.050 0.021 0.017 0.010 0.006 0.009 N 4,734 4,642 4,550 4,325 1,865 1,793 1,721 1,549 2,901 2,817 2,733 2,523 PK k=1 Disasteri,t+k = a + bDPt + ui,t+k b 1.78∗∗∗ 4.47∗∗∗ 6.66∗∗∗ 9.78∗∗∗ 2.50∗∗∗ 5.04∗∗∗ 5.59∗∗∗ 3.58 0.84∗∗∗ 1.78∗∗∗ 2.35∗∗∗ 2.74∗∗∗ (0.31) (0.66) (0.98) (1.35) (0.59) (1.30) (2.05) (2.50) (0.30) (0.55) (0.61) (0.59) Adj. R2 0.027 0.044 0.054 0.053 0.020 0.025 0.017 0.004 0.009 0.012 0.013 0.010 N 4,734 4,642 4,550 4,325 1,865 1,793 1,721 1,549 2,901 2,817 2,733 2,523 PK k=1 Disasteri,t+k = a + bDPi,t + ui,t+k b 0.06 0.14 0.19 0.34 −1.14 −1.95 −2.71∗ −7.01∗∗∗ −0.01 −0.09 −0.17 −0.34 (0.07) (0.13) (0.17) (0.25) (1.10) (1.61) (1.55) (2.48) (0.04) (0.09) (0.13) (0.22) Adj. R2 0.000 0.000 0.000 0.000 0.007 0.006 0.006 0.027 0.000 0.001 0.001 0.005 N 2,168 2,096 2,024 1,835 504 484 464 405 1,674 1,602 1,530 1,346 Notes. This table presents slope coefficients, standard errors, and adjusted R2 statistics for predictive panel regressions of cumulated future disasters on disaster probabilities and the dividend-price ratio (DP). A disaster is defined as a 2 standard-deviation drop in consumption growth in a given country over a given year. An index i indicates a local variable, e.g., DP is the world dividend-price ratio (proxied with the S&P 500 dividend-price ratio), while DPi indicates a local dividend-price ratio. Disaster probabilities are denoted byπ ˆ andπ ˆ\DP. The latter indicates a disaster probability constructed without the global DP. Observations are over the sample of 42 countries, 1872-2015. Results are reported for the full sample as well as for subsamples covering (respectively) the pre- and post-WW2 periods. Standard errors are dually clustered on country and time. ∗, ∗∗, and ∗∗∗ indicate statistical significance at 10%, 5%, and 1% levels, respectively. Table VI: Predicting disasters out of sample

R2 AUROC In sample Out of sample In sample Out of sample Baseline disasters (N = 3,621, 92 disasters) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ πî,t 0.075 0.044 0.863 0.782 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ πˆt 0.044 0.035 0.772 0.766 πˆ\DP 0.036∗∗∗ 0.032∗∗∗ 0.742∗∗∗ 0.762∗∗∗ DP ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ πˆt 0.021 0.020 0.733 0.735 Real-time disasters (N = 2,634, 31 disasters) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ πî,t 0.054 0.034 0.867 0.809 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ πˆt 0.027 0.022 0.744 0.754 πˆ\DP 0.024∗∗∗ 0.021∗∗∗ 0.737∗∗∗ 0.722∗∗∗ DP ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ πˆt 0.010 0.011 0.718 0.737 Notes. This table reports R2 and Area under the Receiver Operating Characteristic (AUROC) statistics for in-sample and out-of-sample probability estimates. In-sample and out-of-sample estimates are compared over the 1922-2015 forecasting period. In-sampleπ ˆ are estimated over the full sample period (1872-2015). Out-of-sampleπ ˆ are constructed by fitting Eq. (9) on a expanding window after a 50-year training period (1872-1921). The top panel shows forecasting results with disasters identified in-sample. The bottom panel reports results where disasters are identified out-of-sample, based on consumption means and standard deviations estimated over the same expanding window as out-of-sample forecasts. Disaster probabilities are denoted byπ ˆ,π ˆ\DP, andπ ˆDP. The superscript \DP indicates a disaster probability constructed without the global DP, while DP denotes a probability constructed with DP, excluding other predictor variables. R2 are calculated by regressing π-estimates on disaster outcomes (with no constant). We assess statistical significance by testing the slope estimate against zero, with standard errors double clustered on country and year. The AUROC equals the area under the ROC curve, which measures the accuracy of a predictive signal by comparing the true positive rate (signal ratio) against the false positive rate (noise ratio) for every possible threshold value (see Figure5), with 0.5 corresponding to no explanatory power and 1 to a perfect fit. We report statistical significance based on bootstrapped confidence bands for the AUROC. ∗, ∗∗, and ∗∗∗ indicate statistical significance at 10%, 5%, and 1% levels, respectively.

32 Table VII: Disaster risk and the dividend-price ratio weakly forecast consumption growth

Full sample (1873-2015) Prewar (1873-1945) Postwar (1945-2015) K 1 3 5 10 1 3 5 10 1 3 5 10 PK k=1 ∆ci,t+k = a + bπˆi,t + ui,t+k b −0.21∗∗∗ −0.14 0.18 0.56∗∗ −0.24∗∗∗ −0.15 0.28 0.97∗∗∗ 0.01 0.41∗ 0.56∗ 0.67∗ (0.07) (0.17) (0.20) (0.23) (0.07) (0.16) (0.20) (0.22) (0.14) (0.24) (0.29) (0.37) Adj. R2 0.019 0.003 0.003 0.014 0.023 0.003 0.007 0.054 0.000 0.016 0.019 0.014 N 4,734 4,642 4,550 4,325 1,865 1,793 1,721 1,549 2,901 2,817 2,733 2,523 PK k=1 ∆ci,t+k = a + bπˆt + ui,t+k b −0.20∗∗ −0.16 0.17 0.50∗ −0.23∗∗∗ −0.22 0.14 0.67∗∗ 0.24 0.93∗∗∗ 1.19∗∗∗ 1.37∗∗∗ (0.08) (0.20) (0.25) (0.28) (0.08) (0.18) (0.25) (0.29) (0.16) (0.33) (0.35) (0.41) Adj. R2 0.011 0.002 0.001 0.007 0.015 0.005 0.001 0.018 0.008 0.040 0.041 0.028 N 4,767 4,675 4,583 4,357 1,888 1,816 1,744 1,570 2,911 2,827 2,743 2,533 PK \DP k=1 ∆ci,t+k = a + bπˆt + ui,t+k b −0.18∗ −0.09 0.33 0.70∗∗ −0.20∗∗∗ −0.20 0.19 0.77∗∗∗ 0.24 1.04∗∗∗ 1.34∗∗∗ 1.40∗∗∗

33 (0.10) (0.24) (0.30) (0.29) (0.08) (0.18) (0.27) (0.25) (0.20) (0.27) (0.31) (0.40) Adj. R2 0.009 0.001 0.005 0.013 0.012 0.004 0.002 0.025 0.008 0.046 0.047 0.027 N 4,734 4,642 4,550 4,325 1,865 1,793 1,721 1,549 2,901 2,817 2,733 2,523 PK k=1 ∆ci,t+k = a + bDPt + ui,t+k b −0.37∗∗∗ −0.56∗∗ −0.52 −0.58 −0.66∗∗∗ −0.24 0.47 1.70∗ 0.12 0.42 0.53 0.63 (0.13) (0.24) (0.33) (0.49) (0.23) (0.45) (0.74) (0.89) (0.12) (0.28) (0.33) (0.45) Adj. R2 0.009 0.006 0.003 0.002 0.011 0.001 0.001 0.013 0.001 0.004 0.004 0.003 N 4,734 4,642 4,550 4,325 1,865 1,793 1,721 1,549 2,901 2,817 2,733 2,523 PK k=1 ∆ci,t+k = a + bDPi,t + ui,t+k b −0.10∗∗ −0.15 −0.15 0.02 0.23 0.44 0.81∗ 2.07∗∗∗ −0.07∗∗ 0.03 0.17∗∗ 0.58∗∗∗ (0.04) (0.11) (0.12) (0.15) (0.21) (0.35) (0.47) (0.68) (0.03) (0.06) (0.08) (0.18) Adj. R2 0.006 0.003 0.002 0.000 0.004 0.004 0.009 0.041 0.003 0.000 0.004 0.022 N 2,168 2,096 2,024 1,835 504 484 464 405 1,674 1,602 1,530 1,346 Notes. This table presents slope coeﬃcients, standard errors, and adjusted R2 statistics for predictive panel regressions of cumulated consumption growth on disaster probabilities and the dividend-price ratio (DP). An index i indicates a local variable, e.g., DP is the world dividend-price ratio (proxied with the S&P \DP 500 dividend-price ratio), while DPi indicates a local dividend-price ratio. Disaster probabilities are denoted byπ ˆ andπ ˆ . The latter indicates a disaster probability constructed without the global DP. Observations are over the sample of 42 countries, 1872-2015. Results are reported for the full sample as well as for subsamples covering (respectively) the pre- and post-WW2 periods. Standard errors are dually clustered on country and time. ∗, ∗∗, and ∗∗∗ indicate statistical signiﬁcance at 10%, 5%, and 1% levels, respectively. Table VIII: Disaster risk and the dividend-price ratio forecast excess returns

Full sample (1873-2015) Prewar (1873-1945) Postwar (1945-2015) K 1 3 5 10 1 3 5 10 1 3 5 10 PK f k=1 ri,t+k − ri,t+k = a + bπˆi,t + ui,t+k b 1.37∗∗∗ 2.41∗∗∗ 3.68∗∗∗ 5.45∗∗∗ 1.01∗∗∗ 1.85∗∗∗ 2.52∗∗∗ 2.68∗∗∗ 2.08∗∗∗ 2.63∗∗∗ 4.00∗∗∗ 6.55∗∗∗ (0.31) (0.43) (0.42) (0.50) (0.33) (0.58) (0.58) (0.64) (0.63) (0.67) (0.65) (0.75) Adj. R2 0.014 0.022 0.037 0.064 0.070 0.079 0.096 0.114 0.014 0.011 0.018 0.039 N 1,572 1,487 1,406 1,217 170 158 146 119 1,408 1,323 1,242 1,053 PK f k=1 ri,t+k − ri,t+k = a + bπˆt + ui,t+k b 2.23∗∗∗ 3.96∗∗∗ 6.28∗∗∗ 9.41∗∗∗ 1.26∗∗∗ 2.55∗∗∗ 3.61∗∗∗ 4.49∗∗∗ 4.46∗∗∗ 6.13∗∗∗ 9.46∗∗∗ 15.00∗∗∗ (0.53) (0.70) (0.77) (0.89) (0.38) (0.71) (0.77) (0.92) (1.47) (1.58) (1.96) (1.83) Adj. R2 0.019 0.028 0.053 0.092 0.055 0.078 0.103 0.168 0.027 0.025 0.045 0.083 N 1,572 1,487 1,406 1,217 170 158 146 119 1,408 1,323 1,242 1,053 PK f \DP k=1 ri,t+k − ri,t+k = a + bπˆt + ui,t+k b 2.10∗∗∗ 3.12∗∗∗ 4.72∗∗∗ 6.79∗∗∗ 1.14∗∗∗ 2.17∗∗∗ 2.72∗∗∗ 4.02∗∗∗ 3.80∗∗ 3.62∗∗ 5.38∗∗ 7.36∗∗∗

34 (0.58) (0.78) (0.90) (1.10) (0.36) (0.68) (0.89) (1.05) (1.49) (1.82) (2.17) (2.76) Adj. R2 0.016 0.017 0.028 0.045 0.049 0.061 0.064 0.131 0.019 0.009 0.014 0.020 N 1,572 1,487 1,406 1,217 170 158 146 119 1,408 1,323 1,242 1,053 PK f k=1 ri,t+k − ri,t+k = a + bDPt + ui,t+k b 3.59∗∗∗ 7.54∗∗∗ 13.95∗∗∗ 20.54∗∗∗ 2.70∗∗ 4.80∗ 9.88∗∗∗ 8.18∗∗ 4.27∗∗ 8.62∗∗∗ 15.25∗∗∗ 22.74∗∗∗ (1.35) (2.18) (2.18) (2.26) (1.09) (2.50) (2.35) (3.35) (1.76) (2.72) (2.70) (2.71) Adj. R2 0.015 0.032 0.079 0.124 0.024 0.027 0.080 0.065 0.017 0.033 0.078 0.130 N 1,572 1,487 1,406 1,217 170 158 146 119 1,408 1,323 1,242 1,053 PK f k=1 ri,t+k − ri,t+k = a + bDPi,t + ui,t+k b −0.23 1.40 2.63 2.57 1.01 4.93∗ 10.47∗∗∗ 6.86∗ −0.21 1.43 2.60 2.46 (0.56) (1.39) (2.28) (2.60) (1.68) (2.94) (2.65) (3.70) (0.58) (1.45) (2.33) (2.62) Adj. R2 0.000 0.008 0.021 0.016 0.002 0.017 0.056 0.036 0.000 0.009 0.023 0.017 N 1,480 1,400 1,326 1,149 145 133 121 94 1,341 1,261 1,187 1,010 Notes. This table presents slope coeﬃcients, standard errors, and adjusted R2 statistics for predictive panel regressions of cumulated excess returns on disaster probabilities and the dividend-price ratio (DP). An index i indicates a local variable, e.g., DP is the world dividend-price ratio (proxied with the S&P 500 \DP dividend-price ratio), while DPi indicates a local dividend-price ratio. Disaster probabilities are denoted byπ ˆ andπ ˆ . The latter indicates a disaster probability constructed without the global DP. Observations are over the sample of 42 countries, 1872-2015. Results are reported for the full sample as well as for subsamples covering (respectively) the pre- and post-WW2 periods. Standard errors are dually clustered on country and time. ∗, ∗∗, and ∗∗∗ indicate statistical signiﬁcance at 10%, 5%, and 1% levels, respectively. Table IX: Estimation results

Full sample 95% Conﬁdence interval Estimate S.E. Normal Times: Average growth in consumption µ 0.019 0.003 (0.013, 0.024) Volatility of consumption growth σ 0.031 0.002 (0.027, 0.035)

Disaster probability: Long-term disaster probabilityπ ¯ 0.032 0.010 (0.016, 0.059) Persistence ρ 0.781 0.084 (0.743, 0.880) Volatility parameter ν 0.075 0.060 (0.056, 0.317)

Disaster size: Shape parameter α 2.804 0.492 (1.564, 4.736) Scale parameter β 0.044 0.007 (0.028, 0.068) Shifting parameter θ 0.034 0.006 (0.019, 0.053) Notes. This table presents maximum likelihood estimates of the time-varying disaster probability model. Log consumption growth evolves according to

∆ci,t = µi + σiεi,t + vi,t,

where µi and σi are constants, εi,t is a standard normal random variable, and vi,t = Ji,t1∆ni,t>0. Here ∆ni,t follows a Poisson distribution with time-varying probability πt: √ πt − π¯ = ρ(πt−1 − π¯) + ν πt−1ut.

Finally, Ji,t follows a shifted gamma distribution that takes a minimum value of θ and gamma parameters α and β. The log consumption growth mean µi and variance σi are estimated from US data while excluding four consumption growth disasters (in 1920, 1921, 1930, and 1932). Disasters are deﬁned according to Eq. (8). The remaining parameters are estimated via maximum likelihood from an international data set of 42 countries for the period 1900 to 2015. The likelihood for the non-Gaussian state-space model is computed by simulation (see Section III.C).

35 Table X: Asset Pricing Moments

Panel A Data Data γ = 6 γ = 7.5 γ = 9 % 1871-2015 1946-2015 δ = 98.5% δ = 97.5% δ = 96.5%

Risk-free rate level 2.4 1.1 1.2 1.3 0.9 Risk-free rate volatility 4.7 3.1 1.5 2.2 3.4

Equity premium 5.8 5.8 4.5 7.2 12.5 Return volatility 18.6 16.9 12.3 13.4 16.4 Sharpe ratio 31.1 41.5 36.6 53.8 76.2

Dividend yield level 4.2 3.4 1.5 4.2 9.0 Dividend yield volatility 1.6 1.4 0.1 0.5 1.9

Panel B % ρ = 0.76 ρ = 0.60 ν = 0.16 ν = 0.08

Risk-free rate level 1.3 1.3 1.3 1.3 Risk-free rate volatility 2.5 2.0 3.1 1.5

Equity premium 10.0 6.7 10.5 6.4 Return volatility 17.5 12.5 18.4 12.0 Sharpe ratio 57.1 53.6 57.1 53.3

Dividend yield level 6.5 3.8 6.8 3.6 Dividend yield volatility 1.9 0.3 2.1 0.2 Notes. Panel A reports unconditional moment statistics from S&P 500 real returns and three-month US Treasury real rates—in addition to model-implied steady-state moments based on the Table I parameters—for several preference settings. Panel B reports model-implied steady-state moments using all but one of the parameters from Table I together with the preference parameters γ = 7.5 and δ = 97.5%.

36 Table XI: Model-Implied Long-Horizon Regressions

Unrestricted (116 years) Four disasters (46 years) No disasters (70 years) K 1 3 5 10 1 3 5 10 1 3 5 10

Consumption growth PK k=1 ∆ct+k = α + βπt + t+k β 0.11 0.21 0.24 0.22 0.12 0.25 0.30 0.32 -0.00 -0.01 -0.01 -0.01 s.e. 0.13 0.22 0.28 0.39 0.13 0.23 0.29 0.41 0.10 0.18 0.22 0.31 R2 0.02 0.03 0.03 0.04 0.02 0.03 0.04 0.04 0.01 0.03 0.04 0.06

Excess returns PK k=1 rm,t+k − rf,t+k = α + βπt + t+k β 1.47 3.04 3.71 4.09 1.48 3.03 3.66 3.95 1.91 3.82 4.51 4.70 s.e. 0.49 0.80 1.02 1.42 0.50 0.82 1.05 1.46 0.51 0.81 1.05 1.53 R2 0.09 0.14 0.14 0.11 0.08 0.13 0.13 0.10 0.18 0.27 0.26 0.18

Notes. This table presents slope coeﬃcients, standard errors, and R2 statistics for predictive regressions of consumption growth and excess returns on disaster probability. The economy has been simulated 1,000 times at monthly frequency with 116 yearly observations for each simulation. Estimates reported in the left group of columns are obtained by pooling regression results from all simulations. Regression results reported in the center columns are based on 1,000 simulations, each of 46 years’ length, and four realized disasters. Results reported in the rightmost four columns are based on 1,000 simulations, each of 70 years’ length, and no realized disasters.

37 Figure 1: Disaster risk and the DP ratio This graph shows the global disaster probability πt, and the S&P 500 dividend-price ratio. The correlation between the two series is 0.66.

38 Figure 2: Timeline of macroeconomic disasters Macroeconomic disasters are deﬁned as two-standard-deviation declines in consumption growth below their long-term growth rate. Small black dots indicate data unavailability.

39 Figure 3: Share of new crises

40 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000

1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000

1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 41

1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000 1900 1950 2000

Figure 4: Local disaster probabilities, πˆi Figure 5: Receiver operating characteristic (ROC) curve This curve assess the accuracy of binary forecasts (in this case in-sampleπ ˆi estimates). Each dot correspond to a given threshold rule, with darker dots representing higher thresholds. The x-axis indicates how often the model wrongly predict a macroeconomic crisis in t + 1; the y-axis gives the fraction of predicted disaster among all disasters that did happen.

42 12 Empirical density Estimated density (shifted gamma distribution) 10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 6: Size distribution of macroeconomic disasters The ﬁgure plots the empirical density of consumption disasters alongside the density estimated from the shifted gamma distribution. The estimated parameters of the ﬁtted distribution are given in TableIX.

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46 Appendices – For Online Publication

Appendix A. Data Appendix

Financial data The data come from Global Financial Data (GFD). We compute international excess return by subtracting the continuously compounded 3-month short-term interest rate from the total equity return (in logs). The former is GFD’s country Treasury Bill Yields. Our world variables are mostly based on US series, which have the largest coverage. Our baseline global dividend-price ratio is the Standard & Poor’s 500 DP ratio computed by GFD, which starts in 1871. We also consider GFD’s World dividend-price ratio, the Dow Jones Industrials Average Dividend Yield, and the Standard & Poor’s 500 earnings-price ratio (see TableIII). Our proxy for annual world stock volatility also uses S&P 500 data, which is available monthly until Jan 1918, weekly from 1918 To 1927, and daily from Jan 1928 onwards. Finally, we use US 3-month Treasury Bill yields to proxy for the short-term risk-free rate.

Wars. The data for war and civil wars come from Sarkees and Wayman(2010), which extends from 1816-2007. Wars correspond to the events listed as inter-state wars, while civil war correspond to the events classiﬁed as either intra-state or non-state wars. We use Wikipedia pages listing wars involving each country in our sample for wars from 2008 to 2015.10

Political Crises. We use data from the Center for Systemic Peace (CSP): Integrated Network for Societal Conﬂict Research. The constraint on the executive variable is constructed by the Polity IV project by coding the authority characteristics of governments across the globe from 1800 to 2016. The scale ranges from 1 (weak constraints on executive power) to 7 (strong constraints on executive power). The variable measures the extent of institutional constraints on executive power. We deﬁne political crises as a 4 or more decline in constraints on executive, or a collapse or interruption in authority (coded separately as “-66” and “-77” in the Polity IV dataset).

Sovereign Defaults. We utilize external default dummies assembled by Carmen Reinhart and Kenneth Rogoﬀ (RR, see Reinhart and Rogoﬀ(2009)). The data covers the period 1800- 2014. We use Wikipedia’s list of sovereign debt crises to complete our data.11

Hyperinflation. Our hyperinflation dates primarily come from RR, who define inflation crises as periods with annual inflation rate exceeding 20%. The data covers the period 1800-2010. We extend the sample to 2015 using inflation data from the World Development Indicators as well as GFD.

Currency crises. We use currency crises dates from RR, which are deﬁned as annual depreciation against the US dollar (or the relevant anchor currency) of 15 percent or more. The data covers the period 1800-2010; we use exchange rate data from WDI and the Federal Reserve Bank of St. Louis to calculate crisis dates until 2015.

Financial Crises. We use banking crises dates constructed by RR, which covers the period 1800-2010. Banking crises correspond to events characterized by substantial bank runs

10E.g., https://en.wikipedia.org/wiki/List_of_wars_involving_Argentina (as of November 28, 2017). This yields a single new civil war: the Sinai Insurgency (Egypt, 2011-2015), and extended periods for two other civil wars, namely the Second Sri Lanka Tamil (Sri Lanka, 2006-2009), as well as unrest in Colombia (1989-2015). 11Accessed on November 28, 2017 at https://en.wikipedia.org/wiki/List_of_sovereign_debt_crises. In our sample of countries, we ﬁnd one default in 2015 (Greece).

47 and closure, merging, takeover, or large-scale government assistance of an important ﬁnancial institution. We use Wikipedia’s list of banking crises to complete our data to 2015.12

Appendix B. Estimation Procedure

This appendix details the procedure used to estimate the latent disaster probability πt. We ∗ proceed in two steps: the ﬁrst estimates an approximation πt , which is assumed to follow a log-autoregressive process; the second step recovers πt. ∗ Let λt ≡ log(Nπt ), and assume that λt follows a ﬁrst-order autoregressive or AR(1) process

∗ ∗ λt+1 − λ¯ = ρ (λt − λ¯) + ν et. (B1)

According to Eq. (20), the number of countries experiencing a disaster at time t (∆nt) follows a ∗ Poisson distribution parameterized by Nπt. With λt so deﬁned, if we assume that πt ≈ πt then

p(∆nt+1|λt) = exp[∆nt+1λt − exp(λt) − log(∆nt+1!)], (B2) from which it follows that the expected number of disasters in t + 1 is exp(λt). Our state-space model is given by Eqs. (B1) and (B2). The latter equation expresses an observed value and is our measurement equation; the former equation expresses an unobserved value and will be estimated using the Kalman filter. It would be ideal theoretically for πt to replace λt as the state variable, in which case the law of motion for πt would be given by (3). From an econometric perspective, however, the log-autoregressive specification is preferable ∗ because it ensures that πt remains positive. We remark that, at annual frequency, the discretized square-root process has a non-negligible probability of generating negative values. Our model consists of three parameters: ψ = (λ,¯ ρ∗, ν∗). Analytical methods cannot be used because the observations are not normally distributed, so we rely on simulation techniques. Thus we adopt a method, developed in Durbin and Koopman(1997), that consists of estimating the model with maximum likelihood techniques and then using a traditional linear Gaussian model to approximate the non-Gaussian one. The true likelihood can then be computed by adjusting the likelihood—which is available in closed form—of the approximating Gaussian model. The adjustment term, which we obtain via simulation, corrects for the departure of the true likelihood from the Gaussian likelihood. Formally, the likelihood is defined as L(ψ) = p(∆nt+1|ψ) and corresponds to the joint density of the model defined by Eqs. (B1) and (B2). Dropping time subscripts, we can write Z L(ψ) = p(∆n|ψ) = p(∆n, λ|ψ) dλ λ Z = p(∆n|λ, ψ)p(λ|ψ) dλ. (B3) λ The first line of Eq. (B3) expresses the likelihood as the joint distribution of the observed disasters ∆n and the unobserved state variables λ, given ψ. In the second line, the joint distribution of ∆n and λ is expressed as the conditional density of ∆n (given λ) multiplied by the marginal density of λ. A closed-form solution for the integral in (B3) does not exist if p(∆n, λ|ψ) follows a Poisson distribution. In principle, we could simulate trajectories of λ from the density p(λ|ψ) and then evaluate Eq. (B3) by the sample mean of the corresponding values of p(∆n|λ, ψ). Yet we choose to rely instead on the importance sampling technique (e.g., Ripley(1987)), which has been shown to be much more efficient in this context. Importance sampling involves choosing a

12Accessed on November 28, 2017 at https://en.wikipedia.org/wiki/List_of_banking_crises. In our sample of countries, we find one default in 2015 (Greece). We find no evidence of sovereign defaults for Iceland and Switzerland over our sample period. We extend the Spanish and Icelandic financial crises from 2008 to 2012.

48 density g(λ|∆n, ψ) that is as close to p(λ|∆n, ψ) as possible, after which one can evaluate the likelihood (B3) while making an appropriate adjustment to correct for the difference between the true density and its approximation. Observe that, provided g(λ|∆n, ψ) is positive everywhere, Eq. (B3) can be written as Z p(λ|ψ) L(ψ) = p(∆n|λ, ψ) g(λ|∆n, ψ) dλ λ g(λ|∆n, ψ) p(λ|ψ) = p(∆n|λ, ψ) , (B4) Eg g(λ|∆n, ψ) where Eg denotes expectation with respect to the importance density g(λ|∆n, ψ). Although g(λ|∆n, ψ) can be any conditional density a priori, we choose a Gaussian distribution—one parametrized to be as close as possible to p(λ|∆n, ψ)—in order to maximize the estimation’s efficiency. The importance density is selected using the mode estimation method described in Chapter 10 of Durbin and Koopman(2012). Next we point out that, although the conditional densities differ between the exact and approximate models, the marginal densities must be the same in both models. As a result, g(λ|∆n, ψ) 6= p(λ|∆n, ψ) but g(λ|ψ) = p(λ|ψ). We can therefore rewrite the importance density as g(∆n, λ|ψ) g(∆n|λ, ψ)g(λ|ψ) g(∆n|λ, ψ)p(λ|ψ) g(λ|∆n, ψ) = = = . g(∆n|ψ) g(∆n|ψ) g(∆n|ψ) This expression implies, in turn, that p(λ|ψ) g(∆n|ψ) = . (B5) g(λ|∆n, ψ) g(∆n|λ, ψ) Observe that g(∆n|ψ) is the model’s likelihood function under the approximating Gaussian distribution g; hence we write Lg(ψ) = g(∆n|ψ). Substituting Eq. (B5) into Eq. (B4) now yields (ψ) p(∆n|λ, ψ) (ψ) = p(∆n|λ, ψ) Lg = (ψ) . (B6) L Eg g(λ|∆n, ψ) Lg Eg g(∆n|λ, ψ) Equation (B6) expresses the non-Gaussian likelihood L(ψ) as an adjustment to the linear Gaus- sian likelihood Lg(ψ), which is easily computed with the Kalman filter. An important advantage of this expression is that it requires simulation only of the second term and not of the likelihood itself. To summarize: our first step filters out λt by maximizing the log of (B6), which is given as the sum of a standard Gaussian log likelihood and a correction term. Our second step recovers πt, ρ, and ν by running the OLS regression ∗ ∗ πt − π¯ πt−1 − π¯ ∗ p ∗ = ρ p ∗ + νut , (B7) πt−1 πt−1 ∗ where πt = exp(λt)/N andπ ¯ is taken to equal the sample average of disasters in the data. We find that this approximation produces only small errors: the root-mean-square error is 0.0049, ∗ and the correlation between the filtered series πt and the series for πt obtained in the second stage is greater than 0.99 (see Panel D of Figure A.II).

Appendix C. Solving the Model

Recall that the dynamics of consumption belong to the aﬃne class and are given by

∆ct = µ + σεt + vt, √ πt = (1 − ρ)¯π + ρπt−1 + ν πt−1ut.

49 Therefore, the following expectation has exponential aﬃne solution (Drechsler and Yaron, 2011):

0 0 u1∆ct+1+u2πt+1 g0(u)+g1(u) [∆ct,πt] 0 2 Et[e ] = e , u = (u1, u2) ∈ R , where

σ2 1 2 2 g0(u) = µ − 2 u1 +π ¯(1 − ρ)u2 + 2 σ u1, 1 2 2 0 g1(u) = 0, 2 ν u2 + ρu2 + ϕ(u1) − 1 . The representative agent has recursive utility of the form

1−1/ψ 1−1/ψ 1/(1−1/ψ) 1−γ 1−γ Vt = (1 − δ)Ct + δ(Et[Vt+1 ]) .

Normalized utility obtains if we take the limit as ψ → 1, divide by Ct, rearrange, and then take the logarithm: δ vc = log( [e(1−γ)(∆ct+1+vct+1)]). t 1 − γ Et

Hypothesizing an aﬃne form for vct,

vct = v0 + vc∆ct + vππt, we can substitute to obtain δ v + v ∆c + v π = log( [e(1−γ)(∆ct+1+v0+vc∆ct+1+vππt+1)]) 0 c t π t 1 − γ Et and then compute the expectation on the right-hand side: δ v + v ∆c + v π = (1 − γ)v + g ([(1 − γ)(1 + v ), (1 − γ)(v )]0) 0 c t π t 1 − γ 0 0 c π 0 0 0 + g1([(1 − γ)(1 + vc), (1 − γ)vπ] ) [∆ct, πt] .

Finally, we solve for the coeﬃcients:

vc : vc = 0, 2 2 δ vπ : vπ = vπδρ + δ(1 − γ)vπν /2 + 1−γ (ϕ(1 − γ) − 1), 2 v0 : v0 = v0δ + µδ + vπδ(1 − ρ)¯π − δγσ /2, where we take the negative root to solve for vπ. In order to derive asset prices, we ﬁrst solve for the stochastic discount factor:

e−γ∆ct+1+(1−γ)vct+1 M = δ t+1 (1−γ)(∆c +vc ) Et[e t+1 t+1 ] 0 0 0 = δe−γ∆ct+1+(1−γ)vct+1−(1−γ)v0−g0([1−γ,(1−γ)vπ] )−g1([1−γ,(1−γ)vπ] )[∆ct,πt] .

−r Since the risk-free rate satisﬁes e f,t = Et[Mt+1], we obtain

0 0 0 e−rf,t = δe(1−γ)v0+g0([−γ,(1−γ)vπ] )+g1([−γ,(1−γ)vπ] )[∆ct,πt] 0 0 0 × e−(1−γ)v0−g0([1−γ,(1−γ)vπ] )−g1([1−γ,(1−γ)vπ] )[∆ct,πt] and therefore 2 rf,t = − log δ + µ − γσ + πt(ϕ(1 − γ) − ϕ(−γ)).

50 Recall that dividends can be expressed as ∆dt = φ∆ct. Stock returns are therefore given by r = log Pt+1+Dt+1 = log Pt+1/Dt+1+1 + log Dt+1 d,t+1 Pt Pt/Dt Dt −dt+1+pt+1 = log(e + 1) + dt − pt + ∆dt+1

≈ k0 − k1(dt+1 − pt+1) + dt − pt + ∆dt+1 for some endogenous constants k0 and k1 to be derived later. We posit an aﬃne form for the logarithm of the DP ratio: dt − pt = A0 + Ac∆ct + Aππt. r The Euler equation for the stock (i.e., the claim asset on Dt) is 1 = Et[Mt+1e d,t+1 ]. We now plug in Mt+1, the log-linearized rd,t+1, and our aﬃne guesses for dt − pt and dt+1 − pt+1:

0 0 0 −γ∆ct+1+(1−γ)vct+1−(1−γ)v0−g0([1−γ,(1−γ)vπ] )−g1([1−γ,(1−γ)vπ] )[∆ct,πt] 1 = Et[δe × ek0−k1(A0+Ac∆ct+1+Aππt+1)+(A0+Ac∆ct+Aππt)+φ∆ct+1 ]. Then we rearrange terms and solve the expectation as follows:

0 0 0 1 = δe−g0([1−γ,(1−γ)vπ] )−g1([1−γ,(1−γ)vπ] )[∆ct,πt] +k0−k1A0+(A0+Ac∆ct+Aππt)

−γ∆ct+1+(1−γ)vππt+1−k1(Ac∆ct+1+Aππt+1)+φ∆ct+1 × Et[e ] 0 0 0 = δe−g0([1−γ,(1−γ)vπ] )−g1([1−γ,(1−γ)vπ] )[∆ct,πt] +k0−k1A0+(A0+Ac∆ct+Aππt) 0 0 0 × eg0([φ−γ−k1Ac,(1−γ)vπ−k1Aπ] )+g1([φ−γ−k1Ac,(1−γ)vπ−k1Aπ] )[∆ct,πt] . Finally, we solve for the coeﬃcients of the DP ratio and the log linearization constants:

k0 = −k1 log(k1) − (1 − k1) log(1 − k1), log k1 = log(1 − k1) − A0 − AcE[∆ct] − AπE[πt]. The coeﬃcients satisfy

Ac : Ac = 0, 0 0 0 0 Aπ : Aπ = g1([1 − γ, (1 − γ)vπ] )[0, 1] − g1([φ − γ − k1Ac, (1 − γ)vπ − k1Aπ] )[0, 1] , 0 0 A0 : A0 = − log δ − k0 + k1A0 − g0([φ − γ − k1Ac, (1 − γ)vπ − k1Aπ] ) + g0([1 − γ, (1 − γ)vπ] ).

Note that one must solve simultaneously for k1 and Aπ (we take the positive root) and then for k0 and A0. Hence the equity premium is given by

rd,t+1 log Et[e ] − rf,t rd,t+1 = log(Et[e ]Et[Mt+1]) J J rd,t+1 J rd,t+1 J C C = log Et[e ] + log Et[Mt+1] − log Et[e Mt+1] − covt[rd,t+1, mt+1] 2 σ 0 0 = [φ, −k1Aπ] 2 [γ, (γ − 1)vπ] + πt[ϕ(φ) + ϕ(−γ) − ϕ(φ − γ) − 1], 0 ν πt where the superscripts C and J denote (respectively) the normal and non-normal components. The return variance can be expressed as

2 σ 0 0 2 ∂2 vart[rd,t+1] = [φ, −k1Aπ] 2 [φ, −k1Aπ] + φ 2 ϕ(u) πt. 0 ν πt ∂u u=0

Finally, the risk-neutral return variance satisﬁes

2 Q σ 0 0 2 ∂2 var [rd,t+1] = [φ, −k1Aπ] 2 [φ, −k1Aπ] + φ 2 ϕ(u) πt t 0 ν πt ∂u u=−γ

(Drechsler and Yaron, 2011).

51 Appendix D. Additional Tables and Figures referenced in the text

52 Table A.I: List of Disasters

Year Disasters 1876 Canada (0.09) 1942 Austria (0.14), Colombia (0.12), France (0.21), Germany (0.10), Greece (0.22), Netherlands (0.32), Russia (0.42), Taiwan (0.14) 1877 Switzerland (0.15) 1943 Australia (0.09), Colombia (0.11), France (0.14), Italy (0.13), Korea (0.14) 1878 Norway (0.06) 1944 Austria (0.17), Japan (0.19), Korea (0.11), Taiwan (0.28), Turkey (0.16) 1879 Turkey (0.23) 1945 Germany (0.18), Italy (0.09), Japan (0.35), Korea (0.18), New Zealand (0.11), Spain (0.13), Sweden (0.09), Taiwan (0.40) 1888 Switzerland (0.16) 1946 India (0.10) 1896 Argentina (0.17), Spain (0.13) 1947 Denmark (0.14) 1897 New Zealand (0.25) 1948 India (0.08) 1900 Argentina (0.19) 1950 Iceland (0.12), India (0.08), Korea (0.18), South Africa (0.08) 1905 Taiwan (0.16) 1951 Korea (0.12), Singapore (0.10) 1907 Brazil (0.15) 1952 Korea (0.13), Malaysia (0.12), New Zealand (0.10) 1908 Canada (0.11) 1953 Egypt (0.12) 1914 Austria (0.27), Belgium (0.20), Canada (0.11), 1959 China (0.09) Germany (0.18) 1915 Belgium (0.24), Chile (0.24), France (0.13), Por- 1960 China (0.05) tugal (0.07), Turkey (0.30) 1916 Germany (0.10), Portugal (0.08), United Kingdom 1961 China (0.06), Sri Lanka (0.09) (0.09) 1917 Australia (0.14), Belgium (0.20), Finland (0.13), 1963 Sri Lanka (0.08) Germany (0.12), Norway (0.10), Russia (0.21), Turkey (0.17), United Kingdom (0.08) 1918 Austria (0.23), Finland (0.23), Malaysia (0.13), 1972 Sri Lanka (0.08) Netherlands (0.38), Norway (0.08), Russia (0.31) 1919 Canada (0.12), Malaysia (0.18), Singapore (0.10) 1974 Chile (0.21) 1920 Egypt (0.15), Malaysia (0.19), Russia (0.24) 1975 Chile (0.17), Portugal (0.09) 1921 Brazil (0.15), Canada (0.17), Denmark (0.18), 1978 Peru (0.10) Norway (0.16), Sweden (0.13), United States (0.07) 1922 Chile (0.18), New Zealand (0.15) 1982 Chile (0.20) 1923 Germany (0.13) 1983 Uruguay (0.16) 1927 Chile (0.20) 1985 South Africa (0.06) 1930 Colombia (0.17), Malaysia (0.12), Peru (0.11), 1986 Malaysia (0.12) United States (0.06) 1931 Australia (0.20), Chile (0.27), Malaysia (0.16), 1988 Indonesia (0.09), Peru (0.10) New Zealand (0.12), Venezuela (0.21) 1932 Canada (0.09), Mexico (0.15), United States (0.10) 1989 Peru (0.17) 1933 Venezuela (0.22) 1995 Mexico (0.11) 1936 Portugal (0.09), Spain (0.43) 1998 Indonesia (0.08), Korea (0.14), Malaysia (0.12) 1939 Australia (0.10), Greece (0.27) 2002 Argentina (0.15), Uruguay (0.17) 1940 Belgium (0.25), Colombia (0.12), Denmark (0.23), 2008 Iceland (0.11) Egypt (0.17), Finland (0.16), Greece (0.28), Japan (0.11), Portugal (0.08), United Kingdom (0.10) 1941 Belgium (0.33), France (0.31), United Kingdom 2009 Iceland (0.15) (0.04) Notes. This table lists disasters based on consumption data in our panel of 42 countries from 1900 to 2015. Disaster sizes are indicated in parentheses.

53 Table A.II: Predicting Crises: Local vs. Global Dividend-Price Ratios

Dependent variable: Indicator = 1 if crisis in year t + 1 Disaster War Civil war Political crisis DP ratio 0.841∗∗∗ – −0.303 – −0.105 – 0.584∗∗∗ – (0.26) – (0.61) – (0.23) – (0.19) – DP ratio (local) – 0.066 – 0.039 – 0.036 – 0.015 – (0.07) – (0.13) – (0.06) – (0.05) Adj. R2 0.010 0.000 0.001 0.000 0.000 0.000 0.009 0.000 Sample start 1873 1873 1873 1873 1873 1873 1873 1873 Sample end 2015 2015 2015 2015 2015 2015 2015 2015 Events 31 31 59 59 33 33 16 16 N 2,168 2,168 2,168 2,168 2,168 2,168 2,168 2,168

54 (a) Disasters, Wars, and Political Crises

Dependent variable: Indicator = 1 if crisis in year t + 1 Sovereign default Hyperinﬂation Currency crash Financial crisis DP ratio 0.609∗∗ – 0.637∗∗∗ – 0.224 – 0.452 – (0.24) – (0.19) – (0.68) – (0.53) – DP ratio (local) – 0.057 – 0.264∗ – 0.114 – 0.270 – (0.07) – (0.15) – (0.22) – (0.23) Adj. R2 0.008 0.000 0.005 0.005 0.000 0.000 0.001 0.002 Sample start 1873 1873 1873 1873 1873 1873 1873 1873 Sample end 2015 2015 2015 2015 2015 2015 2015 2015 Events 21 21 33 33 179 179 95 95 N 2,168 2,168 2,168 2,168 2,168 2,168 2,168 2,168 (b) Sovereign Defaults, Inﬂation, Currency, and Financial Crises Notes. Table A.III: Predicting Disasters: Subsample Results (Probit)

Dependent variable: Indicator = 1 if local disaster in year t + 1 Full Pre-1945 Post-1945 OECD Non-OECD 1945 break GDP 1.5 SD 2.5 SD 3 SD Disaster frequency† 1.094 1.805 2.357 0.691 1.907 2.106∗∗ 1.885∗ 0.568 2.117∗∗ 1.216 (0.95) (1.16) (2.14) (1.11) (1.76) (0.88) (1.04) (0.91) (0.90) (1.06) Disaster 0.524∗∗∗ 0.386∗ 0.230 0.309 0.774∗∗ 0.500∗∗ 0.580∗∗∗ 0.503∗∗∗ 0.414∗ 0.280 (0.18) (0.20) (0.41) (0.23) (0.30) (0.23) (0.18) (0.14) (0.24) (0.27) Consumption growth 0.713 0.957 −1.119 0.110 1.397 0.518 −0.606 0.574 −0.158 −0.644 (0.67) (0.76) (1.43) (0.99) (1.13) (0.95) (0.70) (0.53) (1.01) (1.15) Consumption growth† −4.840∗∗ −0.897 −12.772 −7.768∗∗ 1.513 1.169 −3.794 −6.139∗∗ −0.255 −4.755 (2.20) (3.02) (10.72) (3.13) (5.00) (1.86) (3.64) (2.48) (2.68) (3.88) War 0.637∗∗∗ 0.667∗∗∗ 0.807∗∗∗ 0.559∗∗∗ 0.792∗∗∗ 0.530∗∗∗ 0.324∗∗ 0.622∗∗∗ 0.589∗∗∗ 0.583∗∗∗ (0.13) (0.16) (0.27) (0.14) (0.24) (0.12) (0.15) (0.13) (0.12) (0.14) Civil war 0.105 −0.186 0.482∗ −0.099 0.287 −0.017 0.229 0.160 0.109 0.331 (0.14) (0.21) (0.27) (0.27) (0.21) (0.17) (0.14) (0.13) (0.19) (0.26) Political crisis 0.436∗∗ 0.168 0.940∗∗ 0.407∗∗ 0.285 0.281 0.575∗∗∗ 0.457∗∗∗ 0.381∗ 0.060 (0.19) (0.21) (0.45) (0.20) (0.55) (0.23) (0.16) (0.15) (0.21) (0.25) Sovereign default 0.115 0.032 −0.029 0.158 0.063 −0.235 −0.022 0.088 0.033 0.167 55 (0.15) (0.20) (0.24) (0.22) (0.23) (0.16) (0.17) (0.14) (0.17) (0.25) Hyperinflation 0.046 0.156 −0.132 0.272 −0.378 0.141 0.189 0.009 0.132 0.113 (0.17) (0.24) (0.29) (0.20) (0.25) (0.19) (0.16) (0.16) (0.20) (0.24) Financial crisis 0.331∗∗ −0.018 0.835∗∗∗ 0.207 0.563∗∗∗ 0.507∗∗∗ 0.263∗ 0.264∗∗ 0.103 0.060 (0.15) (0.21) (0.21) (0.19) (0.21) (0.12) (0.14) (0.12) (0.15) (0.32) Dividend-price ratio† 15.865∗∗∗ 9.842∗∗ 28.244∗∗∗ 15.270∗∗∗ 17.443∗∗∗ 8.917∗∗ 8.673∗∗∗ 15.021∗∗∗ 12.805∗∗∗ 17.284∗∗∗ (3.46) (4.80) (7.18) (3.56) (5.79) (3.49) (3.15) (3.08) (3.26) (4.30) Stock volatility† 1.152 −0.423 1.949 0.986 1.673 0.333 2.068∗ 1.666∗ 2.339∗∗ 0.962 (0.93) (1.11) (1.89) (1.26) (1.36) (0.81) (1.09) (0.91) (0.96) (1.52) Short rate† −3.996∗∗ −3.875 −5.582∗∗ −4.438∗ −3.324 −1.742 −3.123 −3.492∗∗ −1.494 −6.164 (1.74) (3.93) (2.39) (2.59) (2.15) (1.76) (2.23) (1.72) (2.10) (4.00) Currency crash 0.245∗∗ 0.353∗∗ 0.182 0.314∗∗ 0.131 0.275∗ 0.202∗ 0.254∗∗ 0.195 0.036 (0.12) (0.14) (0.25) (0.14) (0.16) (0.14) (0.11) (0.10) (0.13) (0.16) N 4,666 1,840 1,630 3,158 1,508 4,434 4,213 4,666 4,161 3,224 Notes. This table shows estimates from a Probit model variant of the regression specified in equation (9). Baseline OLS results are reported in TableIV. Observations are over the sample of 42 countries, 1872-2015. The dependent variable is a dummy equal to one when there is a macroeconomic disaster in country i at time t + 1, defined as a 2 standard-deviation drop in consumption growth, and 0 otherwise, unless indicated otherwise. The disaster indicator is regressed on macroeconomic, crisis, and asset price predictors (see TableII). Results are shown for the full sample, as well as subsamples comparing pre- and post-WW2 data, OECD and non-OECD countries. The next five columns report results with different disaster definitions. “1945 break” indicates consumption disasters constructed with a revised cutoff after 1945; “GDP” correspond to GDP-disasters; finally, we consider different disaster cutoffs: 1.5, 2.5 and 3 standard deviations from average consumption growth rate, instead of the 2-SD cutoff in Eq. (8). Standard errors in parenthesis are computed from standard errors dually clustered on country and year. ∗, ∗∗, and ∗∗∗ indicate statistical significance at 10%, 5%, and 1% levels, respectively. Table A.IV: Predicting Disasters: Subsample Results (Logit)

Dependent variable: Indicator = 1 if local disaster in year t + 1 Full Pre-1945 Post-1945 OECD Non-OECD 1945 break GDP 1.5 SD 2.5 SD 3 SD Disaster frequency† 1.561 3.073 3.058 0.598 3.792 4.329∗∗ 3.642∗ 0.590 4.032∗ 2.341 (1.99) (2.32) (4.29) (2.21) (4.13) (2.08) (2.12) (1.78) (2.13) (2.43) Disaster 0.938∗∗ 0.671∗ 0.099 0.536 1.481∗∗ 0.990∗ 1.003∗∗∗ 0.870∗∗∗ 0.708 0.397 (0.39) (0.41) (0.98) (0.49) (0.72) (0.54) (0.35) (0.27) (0.56) (0.65) Consumption growth 1.203 1.389 −3.103 0.016 2.767 1.146 −1.549 0.987 −0.618 −1.864 (1.58) (1.75) (3.11) (2.24) (2.68) (2.56) (1.59) (1.13) (2.54) (3.00) Consumption growth† −11.665∗∗ −2.584 −41.383 −18.207∗∗ 2.634 2.834 −7.842 −13.697∗∗ −1.507 −10.395 (5.54) (7.39) (31.59) (7.30) (12.30) (4.16) (8.16) (5.67) (7.53) (10.61) War 1.250∗∗∗ 1.181∗∗∗ 1.534∗∗∗ 1.114∗∗∗ 1.529∗∗∗ 1.049∗∗∗ 0.568∗ 1.203∗∗∗ 1.183∗∗∗ 1.254∗∗∗ (0.26) (0.31) (0.58) (0.28) (0.48) (0.26) (0.33) (0.25) (0.27) (0.35) Civil war 0.268 −0.355 1.111 −0.132 0.679 0.041 0.559∗ 0.329 0.319 0.746 (0.34) (0.44) (0.69) (0.64) (0.52) (0.42) (0.32) (0.28) (0.48) (0.69) Political crisis 0.830∗∗ 0.297 1.897 0.764∗ 0.559 0.514 1.146∗∗∗ 0.841∗∗∗ 0.715 0.117 (0.42) (0.42) (1.24) (0.45) (1.29) (0.54) (0.33) (0.30) (0.49) (0.59) Sovereign default 0.250 0.073 0.022 0.319 0.178 −0.463 −0.029 0.201 0.107 0.417 56 (0.33) (0.43) (0.52) (0.49) (0.52) (0.41) (0.38) (0.29) (0.44) (0.68) Hyperinflation 0.168 0.322 −0.095 0.615 −0.811 0.340 0.434 0.084 0.312 0.153 (0.36) (0.47) (0.62) (0.43) (0.59) (0.43) (0.33) (0.32) (0.49) (0.58) Financial crisis 0.674∗∗ −0.008 1.747∗∗∗ 0.365 1.163∗∗ 1.085∗∗∗ 0.587∗ 0.490∗∗ 0.316 0.088 (0.34) (0.45) (0.52) (0.43) (0.46) (0.29) (0.32) (0.25) (0.37) (0.91) Dividend-price ratio† 32.980∗∗∗ 19.272∗∗ 63.501∗∗∗ 30.925∗∗∗ 36.672∗∗∗ 18.971∗∗ 17.396∗∗∗ 29.747∗∗∗ 28.028∗∗∗ 39.216∗∗∗ (7.31) (9.59) (16.48) (7.45) (13.54) (8.30) (6.46) (6.00) (7.30) (9.61) Stock volatility† 1.841 −1.261 3.420 1.792 2.934 0.599 3.865 2.749 4.594∗∗ 1.829 (2.06) (2.43) (4.69) (2.78) (3.23) (1.99) (2.35) (1.88) (2.28) (3.88) Short rate† −8.682∗∗ −7.842 −12.832∗∗ −9.490 −6.369 −4.053 −7.412 −7.043∗ −3.790 −14.941 (4.13) (8.35) (5.36) (6.41) (4.93) (4.39) (5.62) (3.77) (5.30) (10.72) Currency crash 0.514∗∗ 0.721∗∗ 0.354 0.665∗∗ 0.201 0.552∗ 0.383 0.508∗∗ 0.446 0.170 (0.26) (0.28) (0.63) (0.31) (0.37) (0.33) (0.25) (0.20) (0.31) (0.42) N 4,666 1,840 1,630 3,158 1,508 4,434 4,213 4,666 4,161 3,224 Notes. This table shows estimates from a Logit model variant of the regression specified in equation (9). Baseline OLS results are reported in TableIV. Observations are over the sample of 42 countries, 1872-2015. The dependent variable is a dummy equal to one when there is a macroeconomic disaster in country i at time t + 1, defined as a 2 standard-deviation drop in consumption growth, and 0 otherwise, unless indicated otherwise. The disaster indicator is regressed on macroeconomic, crisis, and asset price predictors (see TableII). Results are shown for the full sample, as well as subsamples comparing pre- and post-WW2 data, OECD and non-OECD countries. The next five columns report results with different disaster definitions. “1945 break” indicates consumption disasters constructed with a revised cutoff after 1945; “GDP” correspond to GDP-disasters; finally, we consider different disaster cutoffs: 1.5, 2.5 and 3 standard deviations from average consumption growth rate, instead of the 2-SD cutoff in Eq. (8). Standard errors in parenthesis are computed from standard errors dually clustered on country and year. ∗, ∗∗, and ∗∗∗ indicate statistical significance at 10%, 5%, and 1% levels, respectively. Table A.V: Estimation Results: Sensitivity to the Disaster Cutoff

Cutoﬀ = 2.5 s.d. Cutoﬀ = 3 s.d. Estimate S.E. Estimate S.E. Normal Times: Average growth in consumption µ 0.019 0.003 0.019 0.003 Volatility of consumption growth σ 0.031 0.002 0.031 0.002

Disaster probability: Long-term disaster probabilityπ ¯ 0.032 0.010 0.032 0.010 Persistence ρ 0.781 0.084 0.781 0.084 Volatility parameter ν 0.075 0.060 0.075 0.060

Disaster size: Shape parameter α 2.804 0.492 2.804 0.492 Scale parameter β 0.044 0.007 0.044 0.007 Shifting parameter θ 0.034 0.006 0.034 0.006 Notes. This table presents maximum likelihood estimates of the time-varying disaster probability model, under stricter conditions to treat consumption declines (deﬁned as 2.5 and 3 standard deviations from average consumption growth rate, instead of 2 as in Eq. (8)). Corresponding πt-estimates are plotted in panel D of Figure A.I.

57 Table A.VI: Estimation Results: Alternative Datasets

OECD GDP Estimate S.E. Estimate S.E. Normal Times: Average growth in consumption µ 0.019 0.003 0.019 0.003 Volatility of consumption growth σ 0.031 0.002 0.031 0.002

Disaster probability: Long-term disaster probabilityπ ¯ 0.032 0.011 0.032 0.010 Persistence ρ 0.848 0.023 0.781 0.084 Volatility parameter ν 0.084 0.043 0.075 0.060

Disaster size: Shape parameter α 2.804 0.492 2.804 0.492 Scale parameter β 0.044 0.007 0.044 0.007 Shifting parameter θ 0.034 0.006 0.034 0.006 Notes. This table gives maximum likelihood estimates of the time-varying disaster probability model based on OECD consumption data only (25 countries) and GDP data. Corresponding πt-estimates are plotted in panel B and C of Figure A.I.

58 Figure A.I: Disaster Risk Estimates: Sub-Samples See the main text, Section II.F.

59 Figure A.II: Disaster Risk Estimates: Alternative Speciﬁcations See the main text, Section II.F.

60 Figure A.III: Frequency of disasters and other events: correlation matrix

61 References

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