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 H¨ulsbusch, Christian Julliard, Leonid Kogan, Peter Kondor, Christos Koulovatianos, Hening Liu, Andr´e Lucas, 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, Jos´e Urs´ua, 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;
roberto.marfe@carloalberto.org, http://robertomarfe.altervista.org
‡University of Luxembourg, 4 rue Albert Borschette, 1246 Luxembourg, Luxembourg; +352 466644 5824;
julien.penasse@uni.lu, 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;Isor´e and Szczerbowicz,2016).
This paper estimates the time-varying probability of a macroeconomic disaster. Our ap- proach 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 uncondi- tional 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 macroeco- nomic 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).
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 ˆ
πi,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 note ˆπt and define as the average of country-estimates ˆπi,t. This global component captures 62% of the variation of individual country probabilities. Figure 1 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 between ˆπt 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 constructing ˆπ, 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 between ˆπ 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.
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 gen- erate 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 ofBarro(2006),Barro and Jin(2011),Nakamura et al.(2013), andBarro and Jin(2016), who use international consumption data to compute the (constant) probability of a disaster.
Our paper is also related in spirit toBerkman et al.(2011), who use the number and severity of international political crises to proxy for disaster risk, andColacito 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
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 construct- ing 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 inLettau 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. SectionIintroduces the theoretical framework and our main research question. In SectionIIwe present the data and document that disasters are forecastable. SectionIIIis concerned with the asset pricing implications of our findings. We conclude in Section IV.
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+1as a compound Poisson shock: vi,t+1= Ji,t+11∆ni,t+1>0, where ni,t+1is 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
2We commit a slight abuse of notation since πi,tis only an approximation of the conditional probability of a disaster on the unit time interval (i.e., yearly). The Poisson counting process ni,t+1has 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,tis at its steady state and less than 1% when πi,tis at its 99th percentile. So to facilitate terminology and ease the notation, we shall use πi,tfor the conditional disaster probability andP
i∆ni,t+1(instead ofP
i1∆ni,t+1>0) for the number of disasters.
πi,t = %iπt, where %i > 0 for all i and 1 N
X
i
%i = 1. (2)
We assume that πtfollows a discretized square-root process:
πt+1− ¯π = ρ(πt− ¯π) + ν√
πtut+1, (3)
where ¯π > 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) andGourio(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.
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:
dt− pt= κ0+ Et
∞
X
j=1
κj1[rt+j− ∆dt+j]; (5)
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 SectionIII.B that the log equity premium is affine in the conditional probability of a disaster; that is,
log Et[erd,t+1] − rf,t= CCAPM
premium
+ πt ×
Disaster size
premium
+ Disaster probability premium
. (6)
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 πtwhen 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 SectionIII.Bthat (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.
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 Jos´e Urs´ua and described in Barro and Urs´ua (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:
Disasteri,t =
1, if ∆ci,t < mean(∆ci,t) − 2 × SD(∆ci,t).
0, otherwise.
(8)
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 SectionII.Fwe 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.
Figure 2 gives an overview of disasters over our sample.3 Small black dots indicate data
3We report the list of disasters in TableA.I.
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 excep- tion 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.
Table I reports basic statistics of macroeconomic and non-macroeconomic crises. The top panel compares one-year disasters and peak-to-trough disasters listed inBarro and Urs´ua(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 Urs´ua (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 Urs´ua (2008). Finally, our disasters coincide, by and large, with those identified by Barro and Urs´ua. For the period during which the two samples overlap, we find 128 disasters, among which 100 are identified as disasters in Barro and Urs´ua.
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 TableIcompares 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 fromReinhart 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.
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. Figure 3 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 SectionIII.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:
ˆ
πi,t≡ EtDisasteri,t+1= ˆai+ Xi,tˆb. (10) This delivers country-level disaster probabilities. The global disaster risk estimate ˆπt, as plotted in Figure 1, 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-
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,
Ft≡ 1 N
N
X
i=1
Disasteri,t, (11)
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, projecting these indicators on Ft (without fixed effects) yields a slope of unity.5 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
TableIIpresents 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 predic- tor 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
5To see this, start with Disasteri,t= bFt+ errori,t; summing across countries and dividing by N yields 1
N
N
X
i=1
Disasteri,t= bFt+ 1 N
X
i
errori,t.
so that b equals unity (and errors cancel out).
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 Fton 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 Section I, 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
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
R2 = Var(πi,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), where ¯π is the average probability of a disaster. In our sample, we have ¯π = 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.
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 Table IIIcompares 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 Table II. 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.
Table VIII 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.
TableIVexplores the robustness of our results in subsamples, showing “kitchen sink” regres- sions 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 alter- native 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.
D. Properties of Disaster Risk Estimates
Figure 4 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). Figure1shows the global probability πt. Our specification utilizes the full range of predictor variables and the left column in Table IV 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.
Table V 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 risk ˆπi,t and ˆπt, 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 as ˆπ\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 for ˆπi,t, ˆπt 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 with ˆπi,t or ˆπt.
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 method- ology to forecast disasters could have provided early-warning signals of upcoming disasters in
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 probabilities ˆπi,t, while the global probability ˆπtobtains 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 R2on a regression of future disas- ter 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).
Figure 5 shows the curve corresponding to in-sample local disaster probabilities (point esti- mates 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
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.
TableVIreports 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 iden- tified with real-time moments. In each case, we forecast disasters with our π-estimates. As earlier, we consider multiple variants: ˆπ and ˆπi are local and global probabilities in a regres- sion that exploits the full range of predictors (see Column 1 in Table IV). We also consider a specification that excludes the dividend-price ratio, ˆπ\DP, and another that only excludes all variables but the DP ratio, denoted as ˆπDP. 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 Table IV. 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.
dynamics.
In FigureA.II, Panel A shows real-time estimates constructed in SectionII.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 SectionIII.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 πtforecasts 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. TableVIIestablish 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 Table V, regression results of consumption growth at 1-, 3-, 5-, and 10-year ahead cumulative horizons, on local and global disaster risk ˆπi,t and ˆπt, and on dividend-price ratios. An increase in disaster risk predicts a decline in consumption growth,
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 withBeeler 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. Table VIII presents more direct ev- idence: namely, disaster risk predicts future excess returns. We report predictive regression results with similar predictor variables as in Tables Vand VII.
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 Table IIIthat 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 for ˆπ\DP. This probability is constructed using the same predictor variable as our baseline estimate ˆπ, with the exclusion of the DP ratio. A one percent increase in ˆπ\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 inDrechsler and Yaron(2011). After describing the economy and the dynamics of aggregate consumption, we derive equilibrium asset prices. Solution details
are provided in Appendix C.
We consider a pure exchange economy, `a la Lucas (1978), populated by a representative investor with recursive preferences (Epstein and Zin,1989):
Vt=(1 − δ)Ct1−1/ψ+ δ(Et[Vt+11−γ])
1−1/ψ
1−γ 1/(1−1/ψ)
.
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 toCollin-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
vct= δ
1 − γlog Et[e(1−γ)(∆ct+1+vct+1)]. (12) The aggregate dividend paid by the equity claim is modeled in a parsimonious way (afterAbel 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
Mt+1= δe−γ∆ct+1
| {z }
discounting of expected utility
× e(1−γ)vct+1 Et[e(1−γ)(∆ct+1+vct+1)]
| {z }
discounting of continuation utility
(13)
(Collin-Dufresne et al.,2016) and the value function satisfies
vct= v0+ vππt, (14)
where
v0= δ 1 − δ
µ + ¯π(1 − ρ)vπ−γσ2 2
,
vπ = δρ − 1 −p1 + δ(2δν2+ ρ2− 2ρ − 2δν2ϕ(1 − γ))
δ(γ − 1)ν2 .
Note that the stochastic discount factor variance (i.e., the price of risk in the economy) increases with the disaster probability:
vart(log Mt+1) = γ2σ2+ (γ − 1)2vπ2ν2πt+ γ2Et[Jt+12 ]πt,
where Et[Jt+12 ] is given by the second-order derivative of the moment-generating function eval- uated 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, Wt= Ct δ
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:
rf,t = − log δ + µ − γσ2+ 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:
rd,t+1= log(e−dt+1+pt+1+ 1) + dt− pt+ ∆dt+1
≈ k0− k1(dt+1− pt+1) + dt− pt+ ∆dt+1;
where the endogenous constants k0 and k1 satisfy
k0= −k1log(k1) − (1 − k1) log(1 − k1) and k1= e−dp/(1 + edp).
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 = Et[Mt+1erd,t+1] to recover that the log DP ratio is affine in the disaster probability:
dt− pt= A0+ Aππt, (16)
where
A0 = log(1 − k1) − log(k1) − Aππ¯ and Aπ = 1
k21ν2 q
Ω2+ 2k21ν2(ϕ(1 − γ) − ϕ(φ − γ)) − Ω
with Ω = 1 − k1(ρ + (1 − γ)vπν2).
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 πtis 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
log Et[erd,t+1] − rf,t= γφσ2
| {z }
non-disaster risk
+ (γ − 1)k1Aπvπν2πt
| {z }
disaster probability risk
+ (ϕ(φ) + ϕ(−γ) − ϕ(φ − γ) − 1)πt
| {z }
disaster size risk
, (17)
and the return variance is
vart(rd,t+1) = φ2σ2
| {z }
non-disaster risk
+ k21A2πν2πt
| {z }
disaster probability risk
+ φ2πt∂u∂22ϕ(u) u=0
| {z }
disaster size risk
. (18)
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.
Finally, the risk-neutral return variance (i.e., the model-implied VIX) has a similar form:
varQt(rd,t+1) = φ2σ2
| {z }
non-disaster risk
+ k21A2πν2πt
| {z }
disaster probability risk
+ φ2πt∂u∂22ϕ(u) u=−γ
| {z }
disaster size risk
, (19)
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 AppendixBfor a detailed description.
We assume that πt follows the square-root process given by Eq. (3). Although this proba- bility 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 the number of disasters occurring in a given year t, or ∆nt.7 Because country-specific disas- ters follow Poisson distributions, the sum of observed disasters in t + 1 also follows a Poisson distribution:
∆nt+1≡X
i∆ni,t+1∼ Poisson X
iπi,t
= Poisson(N πt). (20) According to Eq. (20), ∆ntfollows a Poisson distribution with intensity N πt. 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, N πt 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.
