THE QUARTERLY JOURNAL OF ECONOMICS · THE QUARTERLY JOURNAL OF ECONOMICS Vol. 131 February 2016 Issue 1 RARE DISASTERS AND EXCHANGE RATES* Emmanuel Farhi and Xavier Gabaix We propose

THE

QUARTERLY JOURNALOF ECONOMICS

Vol. 131 February 2016 Issue 1

RARE DISASTERS AND EXCHANGE RATES*

Emmanuel Farhi and Xavier Gabaix

We propose a new model of exchange rates, based on the hypothesis that thepossibility of rare but extreme disasters is an important determinant of riskpremia in asset markets. The probability of world disasters as well as each coun-try’s exposure to these events is time-varying. This creates joint fluctuations inexchange rates, interest rates, options, and stock markets. The model accounts fora series of major puzzles in exchange rates: excess volatility and exchange ratedisconnect, forward premium puzzle and large excess returns of the carry trade,and comovements between stocks and exchange rates. It also makes empiricallysuccessful signature predictions regarding the link between exchange rates andtelltale signs of disaster risk in currency options. JEL Codes: G12, G15.

I. Introduction

We propose a new model of exchange rates, based on the hy-pothesis of Rietz (1988) and Barro (2006) that the possibility ofrare but extreme disasters is an important determinant of riskpremia in asset markets. The model accounts for a series of majorpuzzles in exchange rates. It also makes signature predictionsabout the link between exchange rates and currency options,which are broadly supported empirically. Overall, the modelexplains classic exchange rate puzzles and more novel links be-tween options, exchange rates, and stock market movements.

*We are indebted to the editor and two referees for detailed and helpfulsuggestions. Mohsan Bilal, Igor Cesarec, Alex Chinco, Sam Fraiberger, AadityaIyer, and Cheng Luo provided excellent research assistance. For helpful com-ments, we thank participants at various seminars and conferences andFernando Alvarez, Robert Barro, Nicolas Coeurdacier, Daniel Cohen, MarianoCroce, Alex Edmans, Francois Gourio, Stephane Guibaud, Hanno Lustig,Matteo Maggiori, Anna Pavlova, Ken Rogoff, Jose Scheinkman, John Shea,Hyun Shin, Andreas Stathopoulos, Adrien Verdelhan, and Jessica Wachter. Wethank the NSF (SES-0820517) for support.

! The Author(s) 2015. Published by Oxford University Press, on behalf of Presidentand Fellows of Harvard College. All rights reserved. For Permissions, please email:[email protected] Quarterly Journal of Economics (2016), 1–52. doi:10.1093/qje/qjv040.Advance Access publication on October 29, 2015.

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In the model, at any point in time, a world disaster mightoccur. Disasters correspond to bad times—they therefore matterdisproportionately for asset prices despite the fact that they occurwith a low probability. Countries differ by their riskiness, that isby how much their exchange rate would depreciate if a worlddisaster were to occur (something that we endogenize in the ar-ticle, relating it to the productivity of the export sector). Becausethe exchange rate is an asset price whose future risk affects itscurrent value, relatively riskier countries have more depreciatedexchange rates.

The probability of a world disaster as well as each country’sexposure to these events is time-varying. This creates large fluc-tuations in exchange rates, which rationalize their apparent‘‘excess volatility.’’ To the extent that perceptions of disasterrisk are not perfectly correlated with conventional macroeco-nomic fundamentals, our disaster economy exhibits an ‘‘exchangerate disconnect’’ (Meese and Rogoff 1983).

Relatively risky countries also feature high interest rates be-cause investors need to be compensated for the risk of an exchangerate depreciation in a potential world disaster. This allows themodel to account for the forward premium puzzle. This is trueboth in samples with no disasters and in full samples with a rep-resentative number of disasters, but the intuition is easier tograsp in the case of samples with no disasters.1 Indeed, supposethat a country is temporarily risky: it has high interest rates, andits exchange rate is depreciated. As its riskiness reverts to themean, its exchange rate appreciates. Therefore, the currenciesof high interest rate countries appreciate on average, conditionalon no disaster occurring. We also offer a detailed intuition interms of time-varying disaster risk premia for the case of fullsamples with a representative number of disasters.

The disaster hypothesis also makes specific predictions aboutoption prices. This article works them out, and finds that those

1. According to the uncovered interest rate parity (UIP) equation, the expecteddepreciation of a currency should be equal to the interest rate differential betweenthat country and the reference region. A regression of exchange rate changes oninterest rate differentials should yield a coefficient of 1. However, empirical studiesstarting with Tryon (1979), Hansen and Hodrick (1980), Fama (1984), and thosesurveyed by Lewis (2011) consistently produce a regression coefficient that is lessthan 1 and often negative. This invalidation of UIP has been called the forwardpremium puzzle: currencies with high interest rates tend to appreciate. In otherwords, currencies with high interest rates feature positive predictable excessreturns.

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signature predictions are reasonably well borne out in the data.We view this as encouraging support for the disaster hypothesis.

The starting point is that in our theory, the exchange rate ofa risky country commands high put premia in option markets—asmeasured by high ‘‘risk reversals’’ (a risk reversal is the differ-ence in implied volatility between an out-of-the-money put and asymmetric out-of-the-money call). Indeed, investors are willing topay a high premium to insure themselves against the risk thatthe exchange rate depreciates in the event of a world disaster. Acountry’s risk reversal is therefore a reflection of its riskiness.

Accordingly, the model makes four signature predictions re-garding these put premia (‘‘risk reversals’’). First, investing incountries with high risk reversals should have high returns onaverage. Second, countries with high risk reversals should havehigh interest rates. Third, when the risk reversal of a countrygoes up, its currency contemporaneously depreciates. These pre-dictions, and a fourth one detailed below, are broadly consistentwith the data.

The model is very tractable, and we obtain simple and intui-tive closed-form expressions for the major objects of interest, suchas exchange rates, interest rates, carry trade returns, yield curves,forward premium puzzle coefficients, option prices, and stocks. Toachieve this, we build on the closed-economy model with a stochas-tic intensity of disasters proposed in Gabaix (2012) (Rietz 1988;Barro 2006 assume a constant intensity of disasters), and usethe ‘‘linearity-generating’’ processes developed in Gabaix (2009).Our framework is also very flexible. We show that it is easy toextend the basic model to incorporate several factors and inflation.

We calibrate a version of the model and obtain quantitativelyrealistic values for the quantities of interest, such as the volatilityof the exchange rate, the interest rate, the forward premium, thereturn of the carry trade, as well as the size and volatility of riskreversals and their link with exchange rate movements and in-terest rates. The underlying disaster numbers largely rely onBarro and Ursua (2008)’s empirical numbers, which imply thatrare disasters matter five times as much as they would if agentswere risk neutral. As a result, changes in beliefs about disasterstranslate into meaningful volatility. This is why the model yieldssubstantial volatility, which is difficult to obtain with more tra-ditional models (e.g., Obstfeld and Rogoff 1995).

In addition, our calibration matches the somewhat puzzlinglink between stock market and exchange rate returns.

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Empirically, there is no correlation between movements in thestock market and the currency of a country. However, the mostrisky currencies have a positive correlation with world stockmarket returns, whereas the least risky currencies have a nega-tive correlation. Our calibration replicates these facts.

Finally, recent research (Lustig, Roussanov, and Verdelhan2011) has documented a one-factor structure of currency returns(they call this new factor HMLFX). Our proposed calibrationmatches this pattern. Our model also delivers the new predictionthat risk reversals of the most risky countries (respectively leastrisky) should covary negatively (respectively positively) with thiscommon factor. This prediction holds empirically.

To sum up, our model delivers the following patterns.Classic puzzles

(i) Excess volatility of exchange rates.(ii) Failure of uncovered interest rate parity. The coefficient

in the Fama regression is less than 1, and sometimesnegative.

Link between options and exchange rates

(iii) High interest rate countries have high put premia (asmeasured by ‘‘risk reversals’’).

(iv) Investing in countries with high (respectively low) riskreversals delivers high (respectively low) returns.

(v) When the risk reversal of a country’s exchange rate in-creases (which indicates that the currency becomes risk-ier), the exchange rate contemporaneously depreciates.

Link between stock markets and exchange rates

(vi) On average, the correlation between a country’s ex-change rate returns and stock market returns is zero.

(vii) However, high (respectively low) interest rate countrieshave a positive (respectively negative) correlation oftheir currency with the world stock market: their cur-rency appreciates (respectively depreciates) when worldstock markets have high returns.

Comovement structure in exchange rates

(viii) There is a broad one-factor structure in the excess cur-rency returns (the HMLFX factor of Lustig, Roussanov,and Verdelhan 2011): high interest rate currencies tend


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to comove, and comove negatively with low interest ratecurrencies.

(ix) There is a broad one-factor structure of stock marketreturns: stock market returns tend to be positively cor-related across countries.

(x) There is a positive covariance between the above twofactors.

At the same time, we match potentially challenging domesticmoments, for example,

(xi) High-equity premium.(xii) Excess volatility of stocks.

Hence, we obtain a parsimonious model of exchange rates, inter-est rates, options, and stocks that matches the main features ofthe data. It delivers novel predictions borne out in the data, no-tably the link between movements in option prices (risk rever-sals), currency returns, and stock returns.

I.A. Relation to the Literature

Our article is part of a broader research movement usingmodern asset pricing models to understand exchange rates, espe-cially the aforementioned puzzles.

In the closed-economy literature, there are three main para-digms for representative agent rational expectations models toexplain the level and the volatility of risk premia (somethingthat the plain consumption CAPM with low risk aversion failsto generate):2 habits (Abel 1990; Campbell and Cochrane 1999),long-run risks (Epstein and Zin 1989; Bansal and Yaron 2004),and rare disasters (Rietz 1988; Barro 2006; and for time-varyingdisasters, Gabaix 2012; Gourio 2012; Wachter 2013).

Economists have extended these closed-economy paradigmsto open-economy setups to understand exchange rates. Habitmodels were used by Verdelhan (2010), Heyerdahl-Larsen(2014), and Stathopoulos (2012) to generate risk premia in cur-rency markets. Long-run risks models were applied by Colacitoand Croce (2011, 2013) and Bansal and Shaliastovich (2013),using a two-country setting. Given that our model features

2. Pavlova and Rigobon (2007, 2008) provide an elegant and tractable frame-work for analyzing the joint behavior of bonds, stocks, and exchange rates whichsucceeds in accounting for comovements among international assets. However,their model is based on a traditional consumption CAPM, and therefore generateslow risk premia and small departures from UIP.


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some form of long-lasting shocks, it would be interesting to extendour framework to an Epstein-Zin setting, in particular with a moreusual production function and capital accumulation, for example,along the lines of Colacito et al. (2014). One specific feature of thedisaster approach is that it allows to think naturally about the riskreversals, which makes the four signature predictions outlinedabove. In addition, the present disaster model is particularly trac-table, so that closed forms obtain and we can think about an arbi-trary number of countries rather than just two.

To the best of our knowledge, we are the first to adapt thedisaster paradigm to exchange rates. After the present article wascirculated, Gourio, Siemer, and Verdelhan (2013) and Guo (2010)studied related and complementary models numerically in anRBC and a monetary context, respectively, while Du (2013) ex-plores quantitatively a related model, with a different focus. Hisresults are mostly numerical (as they apply to a more complexEpstein-Zin world, where closed forms are hard to obtain), applyto two countries (which makes it impossible to address an inher-ently multicountry set of issues, like Lustig, Roussanov, andVerdelhan [2011]’s HMLFX) and do not touch on the cross-moments between stocks and exchange rates and betweenstocks and currency options. Martin (2013) presents a two-countrymodel with i.i.d. shocks and characterizes the impact of deviationsfrom log normality using cumulants, and generates a Fama coef-ficient equal to 0; however, he does not investigate a number ofissues that we explore, such as currency options and stocks.Because of his conceptual focus, he only offers a limited numericalillustration rather than a full-blown quantitative analysis.3

On the empirical front, several recent papers investigate thehypothesis that disaster risk accounts for the forward premiumpuzzle. This debate is active and ongoing. Brunnermeier, Nagel,and Pedersen (2009) find evidence of a strong link between cur-rency carry trade premium and currency crash risk. Burnsideet al. (2011) compare hedged and unhedged carry trade returnsand conclude that the carry trade premium can be explained byrare events, reflecting high values of the stochastic discount

3. Another strand of the literature departs from the assumption of frictionlessmarkets. Alvarez, Atkeson, and Kehoe (2002) rely on a model with endogenouslysegmented markets to qualitatively generate the forward premium anomaly.PavlovaandRigobon(2012) studytheimportanceof incompletemarkets forexternaladjustment. Gabaix and Maggiori (2015) present a model of exchange rate determi-nation and carry trade based on limited risk-bearing capacity of the financial sector.


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factor and negative carry trade returns—consistent with the raredisaster hypothesis. Using a similar methodology, Jurek (2014)reaches a more skeptical conclusion about the disaster hypothesisand argues that it accounts for at most one third of the carry tradereturns. Using a different methodology that makes direct use ofoption prices at various degrees of moneyness, Farhi et al. (2015)find that global disaster risk accounts for a large fraction of thecarry trade risk premium in advanced countries in the 1996–2014sample, and that global disaster risk is an important factor in thecross-sectional and time-series variation of exchange rates, inter-est rates, and equity tail risk. These and our papers are alsorelated to an older literature on so-called peso problems (Lewis2011). Under the ‘‘pure peso’’ view, there are no risk premia andthe forward premium puzzle is simply due to a small sample bias.By contrast, under the ‘‘rare disasters’’ view, there are risk premia,and the forward premium puzzle also holds in full samples.

The rest of the article is organized as follows. In Section II, weset up the basic model and in Section III derive its implications forthe major puzzles. Section IV contains extensions to options, stocks,and the nominal yield curve. Section V shows the calibration of themodel. Section VI concludes. Most proofs are in the Appendix.

II. Model Setup

II.A. Macroeconomic Environment

We consider a stochastic infinite horizon open economymodel. There are n countries indexed by i ¼ 1; 2; . . .;n. Thereare two goods in each country i: a traded good, called T, and anontraded good, called NT. The traded good is common to allcountries, the nontraded good is country-specific.

1. Preferences. In country i, agents value consumption oftraded good CT

it and nontraded good CNTit according to

E0

X1t¼0

exp ð��tÞu CTit;C

NTit ; �it

� �" #;ð1Þ

where �it is a preference shock. We choose the following speci-fication of utility

u CTit;C

NTit ; �it

� �¼�it CT

it

� ��1� þ 1

�itCNT

it

� ��1�

h i ��1 1��ð Þ

1� �;ð2Þ


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where � is the coefficient of relative risk aversion, � is theelasticity of substitution between tradables and nontradables,and the preference shock is a vector �it ¼ �it; �itð Þ with a shifter�it for the marginal utility of wealth and a shifter �it for therelative expenditure on traded versus nontraded goods.4 Mostof our theoretical results do not in fact require this specificstructure for preferences. We adopt it only for concretenessand because we make use of it in our calibration.

2. Endowments and Technology. At each date t, each countryreceives a random endowment of the tradable good, �T

it and of thenontradable good, �NT

it . The endowment of the nontradable goodcan be used in one of two ways. It can be either consumed (CNT

it ) orinvested in the production of the tradable good (Tit). Investing oneunit of the nontraded good at time t yields exp ð�lsÞ!i;tþs units ofthe traded good in all future periods tþ s � t. Here l is the de-preciation rate of the initial investment, and !it is the yield of theexport technology. This gives the following feasibility constraints

�NTit ¼

Tit þ CNT

it ;ð3Þ

YTit ¼ �

Tit þ

X1s¼0

exp ð�lsÞ!itTi;t�s;ð4Þ

where YTit is the total output of tradable goods in country i in

period t.The consumption of tradable goods is subject to the world-

wide feasibility constraint:Xi

YTit ¼

Xi

CTit :ð5Þ

3. Complete Markets. Markets are complete: there existsa complete set of state- and date-contingent securities. As aresult, the welfare theorems apply and we can study the com-petitive equilibrium as the solution of a planning problem. Theplanner chooses a sequence of Tit; CT

it, and CNTit to maximize a

4. The shifters are not essential. Their only role is to help capture the relativemovements in the consumption of traded goods, nontraded goods, and the exchangerate (Backus and Smith 1993).


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weighted sum of welfare (equation (1)) across countries withPareto weights i subject to equations (3)–(5). The Lagrangian is:

L¼X

i

E0

X1t¼0

iexp ð��tÞu CTit;C

NTit ;�it

� �" #

þX

i

X1t¼0

E0 M�tX1s¼0

exp ð�lsÞ!it �NTi;t�s�CNT

i;t�s

� �þ�T

it�CTit

!" #;ð6Þ

where M�t is the Lagrange multiplier on the economy-wide re-source constraint for tradable goods in period t. The first-order

conditions @L@CT

it

¼ 0 and @L@CNT

it

¼ 0 deliver, respectively:

iexp ��tð ÞuCTit�M�t ¼ 0;

iexp ��tð ÞuCNTit� Et

X1s¼0

M�tþsexp ð�lsÞ!i;tþs ¼ 0:

The relative price eit ¼u

CNTit

uCT

it

of nontradable to tradable goods

within each country is obtained as

eit ¼ Et

X1s¼0

exp ð�lsÞ!i;tþsM�tþs

M�t

" #:

4. World Numeraire and Pricing Kernel. We choose the tradedgood as the world numeraire. As a result, M�t is the pricing kernelin the world numeraire. The price at time t of an asset with astochastic stream of cash flows Dtþsð Þs�0 (expressed in units of

the world numeraire) is given byEt

P1s¼0 M�tþsDtþs

� �M�t

.

5. Exchange Rate. Recall that eit is the relative price ofnontradables to tradables in country i. Since the tradable goodis the world numeraire, eit is the exchange rate of country i vis-a-vis the world numeraire if the domestic numeraire in country i isthe nontradable good. We sometimes refer to it as the ‘‘absolute’’exchange rate of country i. Our convention is such that when eit

increases, the exchange rate appreciates. We also define the bi-lateral exchange rate between country i and country j to be eit

ejt: an

exchange rate appreciation of i with respect to j corresponds to anincrease of eit

ejt.


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If the domestic numeraire is a basket of nontradable andtradable goods, then eit and eit

ejtcorrespond only approximately to

the corresponding traditional notions of real exchange rates. Thiscorrespondence becomes exact in the limit where nontradablesrepresent a large fraction of the consumption basket (�it!0).Section VII.A quantitatively discusses the issues further.

We collect these results in a proposition.

PROPOSITION 1. (Value of the exchange rate). The bilateral ex-change rate between country i and country j is eit

ejt, where the

absolute exchange rate eit of country i is the present value ofits future export productivity:

eit ¼

Et

X1s¼0

M�tþsexp ð�lsÞ!i;tþs

" #M�t

;ð7Þ

with the convention that an increase in eit means an appre-ciation of country i’s currency.

Equation (7) expresses the exchange rate directly as the netpresent value of future fundamentals. The nontradable good is anasset that produces dividends Di;tþs ¼ exp ð�lsÞ!i;tþs, and ispriced accordingly.

6. Existence of Equilibrium. To fully specify the model, we findit convenient to proceed in the following way. Take a process forproductivity !it. We posit a specific world pricing kernel M�t . Ourspecification will be chosen to be realistic yet deliver closed-formsolutions for exchange rates and interest rates. The following

lemma shows that endowment processes �Tit and �NT

it can alwaysbe found to rationalize this pricing kernel. In addition, this pro-cedure allows us to match any process for net exports

nxit ¼ YTit � CT

it.5

It would be desirable to uncover empirical evidence for theendowment and preference shocks processes postulated in thelemma.

5. We could have introduced Epstein-Zin preferences. Given processes for M�tand !it, this would only change the implied processes for CT

it and CNTit , as well as the

endowment processes �Tit and �NT

it . But it would not change our main results,namely, the characterization of exchange rates, interest rates, stocks, and options.


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LEMMA 1. (Existence of equilibrium).

(i) Take as given a process for the world pricing kernel M�t ,productivity !it, preference shocks �it, and net exportsnxit with the restriction that

Pi nxit ¼ 0 for all t. There

exist endowment processes for traded goods �Tit and non-

traded goods �NTit as well as Pareto weights i for each

country i such that the equilibrium world pricing kerneland net exports are indeed given by M�t and nxit,respectively.

(ii) Take as given a process for the world pricing kernel M�t ,productivity !it, consumptions CT

it and CNTit , and net ex-

ports nxit with the restriction thatP

i nxit ¼ 0 for all t.There exist endowment processes for traded goods �T

itand nontraded goods �NT

it ; preference shocks �it as wellas Pareto weights i for each country i such that theequilibrium world pricing kernel, consumptions and netexports are indeed given by M�t ; CT

it and CNTit , and nxit,

respectively.

II.B. Disaster Risk

We now specialize the model to incorporate variable disasterrisk. We study equilibria where the world consumption of thetradable good M�t follows the following stochastic process. Inline with Rietz (1988) and Barro (2006), we assume that in eachperiod t + 1 a disaster may happen with probability pt. If no di-

saster happens,M�tþ1

M�t¼ expð�RÞ for some R. If a disaster happens,

thenM�tþ1

M�t¼ expð�RÞB��tþ1, with Btþ1 > 0. To sum up, the pricing

kernel M�t evolves according to6

M�tþ1

M�t¼ expð�RÞ �

1 if there is no disaster at tþ 1;

B��tþ1 if there is a disaster at tþ 1:

(ð8Þ

In the calibration, we relate R to the discount factor � and thegrowth rate gc of the world endowment of tradable goods. We

6. In a more complex variant, disasters could be followed by partial recoveries(e.g., Gourio 2008; Nakamura et al. 2013). For a given �, that lowers the risk premiacoming from disaster risk. However, a slight increase in � could counteract thateffect. All in all, we find it simpler and more transparent to keep the simplest di-saster formulation, at fairly little cost to the economics.


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use the Barro-Ursua (2008) numbers to discipline the choice ofBtþ1 by relating it to the recovery rate of consumption in disas-ters. Using the first-order conditions of the planning problem,we have:

CNTit ¼ CT

it �iteitð Þ��

ð9Þ

and

CTit ¼

M�tie��t�it

� �1�

½1þ ��it e1��it �

��1

1��1� �

�1� :ð10Þ

If there was just one good, the fact that the pricing kernel M�t ismultiplied by B��tþ1 in a disaster would mean that consumptionis multiplied by Btþ1 in that disaster. In our situation withmultiple goods, the economics is broadly similar, though a bitmore complex in its details.

Consider first the case where eit, �it; and �it do not jump if adisaster occurs at t + 1. Then our assumption (8) implies that both

CTit and CNT

it are multiplied by Btþ1 between t and t + 1, and so is

the consumption basket ½ CTit

� ��1� þ 1

�itCNT

it

� ��1� �

��1.

Of course we are interested in cases where eit, �it, and �it dojump if a disaster occurs at t + 1, potentially leading to jumps inthe relative consumption of traded versus nontraded goods in theevent of a disaster. This is easiest to see in the case where �� ¼ 1.Consider for example the situation where the exchange eit is mul-tiplied by a factor Fi;tþ1 but �it; and �it do not jump if a disaster

occurs at t + 1. Then, equation (10) reduces to CTit ¼

M�tie��t�it

� ��1�

and

is independent of eit so that CTit is multiplied by Btþ1 if a disaster

occurs at t + 1 while equation (9) indicates CNTit is multiplied by

Btþ1F��i;tþ1.

1. Productivity. We assume that productivity of country ifollows:

!i;tþ1

!it¼ expðg!i

Þ�1 if there is no disaster at tþ1;

Fi;tþ1 if there is a disaster at tþ1:

(ð11Þ

During a disaster, the relative productivity of the nontradedgood is multiplied by Fi;tþ1. For instance, if productivity fallsby 20%, then Fi;tþ1 ¼ 0:8.


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In the model, a sufficient statistic for many quantities of in-terest is the ‘‘resilience’’ of a country i, defined as:

Hit ¼ ptEDt B��tþ1Fi;tþ1 � 1� �

;ð12Þ

where EDt (resp. END

t ) is the expected value conditional on adisaster happening at t + 1 (resp. conditional on no disaster hap-pening). A relatively safe country has a high resilience Hit, as ithas a high recovery rate Fi;tþ1. Conversely, a relatively riskycountry has low resilience. In equation (12), the probability pt

and the world intensity of disasters Btþ1 are common to all coun-tries, but the recovery rate Fi;tþ1 is country-specific. The changesin prospective recovery rates could be correlated across countries.

Rather than separately specifying laws of motion for its com-ponents ( pt, Btþ1, and Fi;tþ1), we directly model the law of motionfor Hit and assume that it takes a particular, convenient func-tional form. This will allow us to compute exchange rates, interestrates and options in closed form.

We decompose

Hit ¼ Hi� þbHit;ð13Þ

where Hi� and bHit are the constant and variable parts of resil-ience, respectively. For tractability, we posit that the law ofmotion for bHit follows a linearity-generating process:

bHi;tþ1 ¼1þHi�

1þHitexpð��Hi

ÞbHit þ "Hi;tþ1;ð14Þ

where �Hidenotes the speed of mean reversion of resilience and

the innovations "Hi;tþ1 have mean zero, both unconditionally and

conditional on a disaster (Et½"Hi;tþ1� ¼ E

Dt ½"

Hi;tþ1� ¼ 0).

The economic meaning of equation (14) is that bHit mean-re-verts toward 0, but is subject to shocks. Because Hit hovers around

Hi�;1þHi�

1þHitis close to 1 and the process behaves like a regular AR(1)

up to second-order terms in bHit: bHi;tþ1 ’ exp ð��HiÞbHitþ "

Hi;tþ1. The

‘‘twist’’ term 1þHi�

1þHitis innocuous from an economic perspective but

provides analytical tractability (see the technical appendix inGabaix 2009 for a discussion). Linearity-generating processesallow the derivation of the equilibrium exchange rate in closedform.7

7. For the process to be well behaved, we need to impose that bHit is alwaysabove a negative lower bound bH

i¼ 1þHi�ð Þ exp ð��Hi

Þ � 1� �

. To ensure this, the


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2. Resilience as a Sufficient Statistic. As we shall see shortly,resilience Hit is a sufficient statistic for the impact of pt;Btþ1; andFi;tþ1 on exchange rates, interest rates, as well as consumption.We leverage this insight and do not independently specifythe processes for pt;Btþ1; and Fi;tþ1. We simply assume thatthey lead to the process for resilience Hit that we have positedabove.

III. Main Results

III.A. Exchange Rates and Interest Rates

1. Exchange Rate. We start by deriving the value of the ex-change rate. We define hi� :¼ ln 1þHi�ð Þ and

rei :¼ Rþ l� g!i� hi�:ð15Þ

As we shall see below, rei � l is the interest rate when thetemporary component of resilience, bHit, is 0.

PROPOSITION 2. (Level of the exchange rate). The bilateral ex-change rate between country i and country j is eit

ejt, where eit

is the exchange rate of country i in terms of the world numer-aire and is equal to

eit ¼!it

rei1þ

bHit

rei þ �Hi

!ð16Þ

in the limit of small time intervals.

Equation (16) implies that the exchange rate eit increases(appreciates) with hi� and bHit: risky (i.e., low resilience) countrieshave a low (depreciated) exchange rate. Safer (i.e., high resil-ience) currencies have a high (appreciated) exchange rate.Risky countries are those whose currency value (and, more prim-itively, whose relative price of nontradables) is expected to dropduring disasters.

The exchange rate fluctuates with the resilience bHit. As weshall see in the calibration, these fluctuations are plausibly largeand can therefore generate ‘‘excess volatility’’ of the exchange rate(see the Online Appendix for a generalization with several factors).

variance of the innovations "Hi;tþ1 goes to 0 when bHit is close to bH

i. The Online

Appendix details this.


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http://qje.oxfordjournals.org/lookup/suppl/doi:10.1093/qje/qjv040/-/DC1




To the extent that they are imperfectly correlated with tra-ditional macroeconomic fundamentals, the fluctuations in resil-ience can generate an ‘‘exchange rate disconnect.’’

2. Derivation of Proposition 2. The appendix provides a rigor-ous proof. Here we guess and verify a solution of the form

eit

!it¼ aþ bbHitð17Þ

with coefficients a and b to be determined. Equation (7) gives:

eit

!it¼ 1þ e�lEt

M�tþ1

M�t

!i;tþ1

!it

ei;tþ1

!i;tþ1

�:

Using the process for M�t , !it (equations (8) and (11)) we have:

EtM�tþ1

M�t

!i;tþ1

!it

�¼ e�Rþg!i 1� ptð Þ þ ptE

Dt B��tþ1Fi;tþ1

� �� Et

M�tþ1

M�t

!i;tþ1

!it

�¼ e�Rþg!i 1þHitð Þ;ð18Þ

which shows the origin of the definition of resilience. Using the

process for bHi;tþ1 in equation (14), and the fact that innovation

"Hi;tþ1 has mean 0 independently of the realization of the disaster

(i.e., EDt ½"

Hi;tþ1� ¼ E

NDt ½"

Hi;tþ1� ¼ 0), we get:

eit

!it� 1 ¼ e�lEt

M�tþ1

M�t

!i;tþ1

!it

�Et

ei;tþ1

!i;tþ1

�¼ e�l�Rþg!i 1þHitð ÞEt

ei;tþ1

!i;tþ1

�¼ e�l�Rþg!i 1þHitð ÞEt aþ bbHi;tþ1

h i; using ð17Þ

¼ e�l�Rþg!i a 1þHitð Þ þ b 1þHitð ÞEtbHi;tþ1

� �:

Using equations (13) and (14),eit

!it�1¼ e�l�Rþg!i a 1þHi� þ

bHit

� �þb 1þHi�ð Þexp ð��Hi

ÞbHit

� �¼ e�l�Rþg!i ehi� aþ ae�hi� þbexp ð��Hi

Þ� � bHit

� �; using 1þHi� ¼ ehi�

¼ e�rei aþ ae�hi� þbexp ð��HiÞ

� � bHit

� �:


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By (17) we have eit

!it�1¼a�1þbbHit, hence:

a� 1 ¼ e�reia;

b ¼ ae�rei�hi� þ be�rei��Hi :

Solving for a and b gives:

eit ¼!it

1� exp �reið Þ1þ

exp �rei � hi�ð Þ

1� exp �rei � �Hi

� � bHit

!:ð19Þ

Taking the limit of small time intervals (i.e., small rei and �Hi)

gives expression (16). w

3. Domestic Pricing Kernel. We can use this proposition to derivean expression for the domestic pricing kernel, Mit ¼ ie

��tui;CNTit

.We get

Mit ¼M�t eit ¼M�t!it

rei1þ

bHit

rei þ �Hi

!:ð20Þ

If there is a world disaster on the tradable good (so that M�t ismultiplied by B��tþ1Þ, there is also a disaster on the domestic,nontradable good (so that Mit is multiplied by B��tþ1Fi;tþ1, as!it is multiplied by Fi;tþ1). Disasters affect marginal utilitywith respect to both tradable and nontradable goods.

The decomposition also shows how the domestic pricingkernel Mit is subject to both common disaster shocks with coun-try-specific loadings (B��tþ1Fi;tþ1) that command an internationalrisk premium and nondisaster shocks that do not command aninternational risk premium: the innovations to bHit (and also theinnovations to !it that are orthogonal to disasters, if we had in-troduced them), which can be correlated across countries, but stilldo not command a disaster premium.

4. Interest Rate. We first derive the interest rate in termsof the pricing kernel, before giving its concrete value. The argu-ment is standard, but for completeness we spell it out here.One unit of the domestic currency is worth eit at time t. Onecan invest it in the domestic bond and get 1þ ritð Þ in the domesticcurrency at time t + 1, hence 1þ ritð Þei;tþ1 of the world numeraire


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at time t + 1. The value of the payoff is EtM�tþ1

M�t1þ ritð Þei;tþ1

h iat time

t. Hence

eit ¼ EtM�tþ1

M�t1þ ritð Þei;tþ1

�;

so that the interest rate is given by:

1

1þ rit¼ Et

M�tþ1ei;tþ1

M�t eit

�:ð21Þ

The following proposition calculates its value in the disastermodel. Recall that rei :¼ Rþ l� g!i

� hi�.

PROPOSITION 3. (Interest rate). The value of the interest rate incountry i is

rit ¼ rei � l�reibHit

rei þ �Hiþ bHit

ð22Þ

in the limit of small time intervals.

Proof. In this proof, it is useful to define xit ¼ e�hi� bHit. Then,equation (18) gives:

EtM�tþ1!i;tþ1

M�t !it

�¼ exp �Rþg!i

� �1þHitð Þ ¼ exp �reiþlð Þ 1þxitð Þ:

Equation (14) implies Et xi;tþ1

� �¼ exp ��Hi

� �xit

1þxit, and equation (19)

yields eit ¼ !itA 1þ Bxitð Þ with A ¼ 11�exp �reið Þ

; B ¼ exp �reið Þ

1�exp �rei��Hi

� �.Thus:

1þ rit ¼eit

EtM�tþ1

M�tei;tþ1

� ¼ !itA 1þ Bxitð Þ

EtM�tþ1

M�t!i;tþ1A 1þ Bxi;tþ1

� � �¼

1þ Bxit

EtM�tþ1!i;tþ1

M�t !it

�Et 1þ Bxi;tþ1

� �¼

1þ Bxit

exp �rei þ lð Þ 1þ xitð Þ 1þ Bexp ��Hi

� �xit

1þ xit

� ¼ exp rei � lð Þ

1þ Bxit

1þ xit 1þ Bexp ��Hi

� ��


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¼ exp rei � lð Þ

1þexp �reið Þ


� � xit

1þ1


� � xit

¼ exp rei � lð Þ 1�1� exp �reið Þð Þexp �hi�ð ÞHit


� �þ exp �hi�ð ÞHit

" #:

Hence,

rit ¼ exp rei � lð Þ 1�1� exp �reið Þð Þexp �hi�ð ÞbHit


� �þ exp �hi�ð ÞbHit

" #� 1:

ð23Þ

Taking the limit of small time intervals gives equation (22). wWhen a country is ‘‘risky’’ (low hi� or bHit), its interest rate is

high according to equation (22) because its currency has a highrisk of depreciating in bad states of the world. Note that this is areal risk of depreciation, not a default risk. Safe countries canborrow at a lower interest rate, which may explain why histori-cally the dollar or Swiss franc interest rates were low (Gourinchasand Rey 2007).

5. Safe Haven Countries Can Borrow at Low Interest Rates andHave an Appreciated Currency. Consider two countries, one withlow risk/high average resilience hi� (Switzerland, the safehaven), and one with high risk/lower average resilience hi�

(Brazil). Equations (15)–(22) imply that, on average (i.e.,when bHit ¼ 0), Switzerland has low interest rates (equation(22) with bHit ¼ 0), while Brazil has high interest rates. Thisis a compensation for disaster risk, not default: investors arewilling to lend to Switzerland at low interest rates because theSwiss exchange rate will appreciate relative to Brazil’s in adisaster.

At the same time, the exchange rates are eit ¼!it

reiwhen resil-

iences are at their central value (equation (16) with bHit ¼ 0).Hence, the Swiss exchange rate is on average appreciated(‘‘strong’’) compared with the Brazilian exchange rate.

Switzerland (the safe haven) therefore benefits from the ‘‘ex-orbitant privilege’’ of borrowing at low interest rates. This under-lying mechanism is different from those of Gourinchas, Govillot,and Rey (2010), who emphasize differences in risk aversion,


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and Maggiori (2013), who emphasizes differences in financial de-velopment. A distinctive feature of our model is that the exchangerates of safe haven countries appreciate in the times of crisis.Colacito et al. (2015) generate similar facts using a model withheterogeneous exposures to a long-run risk, rather than a disas-ter, factor.

III.B. Forward Premium Puzzle and Carry Trade

1 Carry Trade. Consider the following carry trade from theperspective of an international investor. The trade borrows oneunit of international numeraire in currency j at interest rate rjt

and invests it in currency i at interest rate rit for one period. Thereturn on this carry trade is

Rij;tþ1 ¼ei;tþ1

eit1þ ritð Þ �

ej;tþ1

ejt1þ rjt

� �:

It is important to characterize the expected return on this carrytrade. Because disasters are rare events, it is useful to computethis expected return both in samples with no disasters and infull samples with representative disasters. Formally, the formercorresponds to the expectation conditional on no disasters END

t

Rij;tþ1

� �and the latter corresponds to the unconditional expec-

tation Et Rij;tþ1

� �.

PROPOSITION 4. (Carry trade return). Consider two countries i andj with rek = re, Hk� ¼ H�; �Hk

¼ �H for k ¼ i; j, and consider thelimit of small time intervals. The expected returns of thecarry trade (unconditionally, and conditionally on no disas-ters in the sample) are equal to:

Et Rij;tþ1

� �¼ 1�B

�� bHjt�

bHit

� �¼ 1�B

��

1þ�H

re

� rit� rjt

� �:ð24Þ

ENDt Rij;tþ1

� �¼ bHjt �

bHit ¼ 1þ�H

re

� rit � rjt

� �:ð25Þ

The first expression requires the additional assumptionthat Btþ1 is constant with value B. The two right-handsides expressed with rit � rjt hold up to second-orderterms in bHit and bHjt.

The intuition for expression (24) is as follows. First, if there

are no disasters (B ¼ 1), the expected carry trade return is zero,


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Et Rij;tþ1

� �¼ 0. Second, if there are disasters (B < 1), then the ex-

pected carry trade return is positive if the investment currency i

has lower resilience than the funding currency j (bHjt �bHit > 0),

that is, if currency i is riskier than currency j, so that this carrytrade will have highly negative returns in a disaster. At the same

time, we have bHjt �bHit ¼ � 1þ �H

re

� �rjt � rit

� �: if currency i is risk-

ier than currency j, its (real) interest rate rit is greater than coun-try j’s interest rate rjt. In other words, investing in currencieswith high interest rates yields positive expected returns, whichis simply a compensation for bearing disaster risk.

A related intuition can be provided for expression (25).Suppose again that currency i is riskier than currency

j (bHjt �bHit > 0; rit � rjt > 0). If no disaster occurs, the investor

pockets the interest differential rit � rjt. Moreover, becausebHj;tþ1 �bHi;tþ1 mean-reverts toward 0, the exchange rate of coun-

try i appreciates in expectation (conditional on no disasters)against that of country j. As a result, the investor also collectsan expected capital gain, and the expected return of the carry

trade (conditional on no disasters) 1þ �H

re

� �rit � rjt

� �is greater

than rit � rjt.8

2. Fama Regression. These results can be understood usingthe related language of Fama (1984) regressions:

Fama regression:ei;tþ1

eit�

ej;tþ1

ejt¼ �þ ðrjt � ritÞ þ "ij;tþ1;ð26Þ

where "ij;tþ1 is a random variable with mean 0. As for theexpected return on the carry trade, it is useful to evaluatethis regression both in samples with no disasters (conditionalon no disasters) and in full samples with representative disas-ters (unconditionally). We denote the corresponding coefficientsby ND and Full.

8. We can derive related expressions that are valid whether or not there aredifferences in steady state resiliences (Hi� 6¼ Hj�). Indeed, we have Et Rij;tþ1

� �¼ 1� B

�Þ Hjt �Hit

� ��and END

t Rij;tþ1

� �¼ Hjt �Hit. This applies in particular with

constant resiliences Hjt �Hit ¼ Hj� �Hi� ¼ ri� � rj� (assuming g!i¼ g!j

). In thatcase, there is a positive return to the carry trade equal to the constant interest ratedifferential (as argued by Hassan and Mano 2015).


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We have ei;tþ1

eit�

ej;tþ1

ejt¼ Rij;t � ðrit � rjtÞ in the limit of small

time intervals, so that the coefficients in the Fama regressions

are simply Full ¼ 1� bFull and ND ¼ 1� bND, where bFull ¼ ð1�

B�Þð1þ �H

reÞ and bND ¼ 1þ �H

reare the coefficients on interest rate

differentials in Proposition 4. Hence, we obtain the followingproposition.

PROPOSITION 5. (Fama coefficients). Consider two countries i and jwith rek = re, �Hk

¼ �H for k ¼ i; j, and consider the limit of

small time intervals as well as small bHit and bHjt. In theFama regression (26), if in addition Bt is constant with

value B, then in a full sample the coefficient Full is:

Full ¼ ��H

reþ 1þ

�H

re

� B�� 1;ð27Þ

while in a sample with no disasters the coefficient is:

ND ¼ ��H

re:ð28Þ

The uncovered interest rate parity (UIP) condition can berestated as requiring = 1. The ‘‘forward premium puzzle’’ isthe observation that empirically, the coefficient in the Fama re-gression is < 1. Our results can be seen as a rationalization ofthe forward premium puzzle.

When there are no disasters (B ¼ 1) we have Full ¼ 1.

But when there are disasters (B < 1), we have Full < 1. The in-tuition is similar to the one given for the expected return on thecarry trade and can be traced back to the existence of a time-varying disaster risk premium. Note that we can even have

Full < 0.In addition, we always have ND < 0. There again, the intu-

ition is similar to the one given for the expected return on thecarry trade (conditional on no disasters), and derives from the

mean reversion in relative resilience bHj;tþ1 �bHi;tþ1. In this sim-

plest model with one factor (resilience), ND is always negative; inricher models with more factors (resilience and inflation, seebelow), ND can have both signs depending on the relative impor-tance of the different factors.


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IV. Extensions

IV.A. Options and Risk Reversals

Disaster risk is inherently hard to measure, but options offera powerful way to assess its importance. Here, we characterizethe way disasters are incorporated into option prices. We discussthe empirical validity of the model’s predictions.

Consider two countries i and j. Call j the domestic currencyand i the foreign currency. A put gives the investor the right atdate 1 to receive K units of currency j in exchange for ej0

ei0units of

currency i, where K is the strike. As those ej0

ei0units of currency i

have a market value of ei1

ej1

ej0

ei0units of currency j at time 1, the put

pays off x1 ¼ K � ei1

ej1

ej0

ei0

� �þin units of currency j at date 1. This cor-

responds to ej1x1 units of the international numeraire, so the date

0 price of the put is E0M�1M�

0ej1x1

h iin the international numeraire,

and VP Kð Þ ¼ 1ej0E0

M�1M�

0ej1x1

h iin currency j. Hence, the put price in

currency j is:

VP Kð Þ ¼ E0M�1M�0

ej1

ej0K �

ei1

ei0

� þ �:ð29Þ

Likewise, the currency j price at date 0 of a call with strike K is

VC Kð Þ ¼ E0M�1M�

0

ei1

ei0�

ej1

ej0K

� �þ �.

1. Option Prices without Disasters. The Black-Scholes for-mula for equity options was adapted by Garman and Kohlhagen(1983) to currency options. We call VP

BSðS; �; �; r; r�;TÞ and

VCBSðS; �; �; r; r

�;TÞ the Black-Scholes prices for a put and a call,respectively, when the exchange rate is S, the strike is �, theexchange rate volatility is �, the home interest rate is r, the for-eign interest rate is r�, and the time to maturity is T. The pricingformulas in that case are well known.9

2. Option Prices in the Model. We use the index k for a state-ment applying to both countries i, j. For tractability, we make twosimplifying assumptions as in Gabaix (2012). First, we assume

that if a disaster occurs in period 1, "Hk1 is equal to 0. Second,

9. Calling � the Gaussian cumulative distribution function, we have:


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we assume that the distribution of ek1 conditional on date 0 infor-mation and no disaster occurring in period 1 is log-normalwith drift k and volatility �k, where k indexes countries:ek1

ek0¼ exp k þ "k �

�2k

2

� �, where "k � Nð0; �2

kÞ, and k :¼ R� ln

1þ rkð Þ 1þHkð Þð Þ is the expected exchange rate appreciationconditional on no disasters.10 This enables us to derive optionprices in closed form. The standard deviation of the bilateral log

exchange rate (conditional on no disaster) is �ijj :¼ �2i þ �

2j�

�2�ij�i�jÞ

12, where �ij is the correlation between "i and "j.

PROPOSITION 6. (Option prices). The price of a put with strike Kand maturity 1 is:

VP Kð Þ ¼ VP;ND Kð Þ þ VP;D Kð Þ;ð30Þ

where VP;ND Kð Þ and VP;D Kð Þ are the part of the price cor-responding respectively to the no-disaster and disasterstates:

VP;ND Kð Þ¼ exp �Rþið Þ 1�p0ð ÞVPBS Kexp j�i

� �;�ijj

� �;ð31Þ

VP;D Kð Þ ¼ exp �Rþið Þp0ED0 B��1 Kexp j�i

� �Fj1�Fi1

� �þh i;

ð32Þ

where VPBS K;�ð Þ :¼VP

BSð1;K;�;0;0;1Þ is the Black-Scholesput value when the strike is K, the volatility �, the interestrates 0, the maturity 1, the spot price 1.

VCBSðS; �; �; r; r

�;TÞ ¼ Se�r�T�ðd1Þ � �e�rT�ðd2Þ;

VPBSðS; �; �; r; r

�;TÞ ¼ �e�rT�ð�d2Þ � Se�r�T�ð�d1Þ;

d1 ¼

ln

�S

�

þ

�r� r� þ

�2

2

T

��

ffiffiffiffiTp ; d2 ¼ d1 � �

ffiffiffiffiTp

:

10. This can be ensured as in Gabaix (2012). We assume that if there is nodisaster, then !tþ1

!t¼ eg 1þ "!tþ1

� �, with Et "

!tþ1

� �¼ Et "

!tþ1"

Htþ1

� �¼ 0. This does not

change any of the formulas for the exchange rate and the interest rate. The distur-bance term "!tþ1 can be designed to ensure that

eNDi1

ei0has the log-normal noise de-

scribed above. The exact expression for i comes from the Euler equation1 ¼ E 1þ rið Þe�Rþi 1þHið Þ

� �:


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The price of a call is given by the put-call parity equation:

VC Kð Þ ¼ VP Kð Þ þ1

1þ ri0�

K

1þ rj0:ð33Þ

The option price (30) is the sum of two terms. The first one isa familiar Black-Scholes term. The second is a pure disaster term.

If the foreign currency i is riskier than the home currency j,then out-of-the-money put prices on the currency pair (home, for-eign) should be higher than out-of-the-money call prices as theprice of protection against a devaluation of the foreign currencyshould be high. We next present a simple metric—risk rever-sals—to measure the gap between out-of-the-money puts andout-of-the-money calls.

3. Implied Volatility Smirk and Risk Reversals. We start bysurveying well-known notions in option theory. Given a calloption with strike K and price v, the implied volatility of theoption is the volatility b� Kð Þ that needs to be assumed in the

Black-Scholes formula to match the price: VCBS K;b� Kð Þð Þ ¼ v.

Implied volatilities on puts are defined similarly. For instance,if a currency has a lot of disaster risk, its put price will be high(Proposition 6) and its implied volatility will be high.

In particular, consider the implied volatility curve (i.e., thegraph of the implied volatilityb� Kð Þ as a function of the strike K) ofa pair of currencies: a risky currency and a safe currency. Out-of-the-money puts protect against the crash of the exchange rate of iversus j and out-of-the-money calls protect against the crash ofthe exchange rate of j versus i. Imagine that i is riskier than j.Then the implied volatility of deep out-of-the-money puts ishigher than the implied volatility of out-of-the-money calls—apattern referred to as a ‘‘smirk.’’

A popular way to quantify the smirk is the risk reversal (RR).Intuitively, it is the difference in implied volatility of an out-of-the-money put and a symmetric out-of-the-money call. Hence, avery risky currency will have a high RR.

To formulate a more precise definition of the RR, we need todefine the delta of an option. It is the derivative with respect tothe time 0 currency price in the Black-Scholes formula. Formally,if the call price (in the Black-Scholes/Garman-Kohlhagen model)

is VCBS S; �; r; r�; �;Tð Þ, the delta is � :¼

@VCBS

@S ). The delta of a calloption decreases monotonically from 1 to 0 as �

S increases.


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Symmetrically, the delta of a put option decreases monotonicallyfrom 0 to –1 as �

S increases. Given a value � 2 0; 1ð Þ, we define the

� risk reversal to be the difference in implied volatilities betweenan out-of-the-money put and an out-of-the-money call with thefollowing properties. The strike of the put is chosen such that theBlack-Scholes/Garman-Kohlhagen delta is ��. Symmetrically,the strike of the call is chosen such that the Black-Scholes/Garman-Kohlhagen delta is �. In practice we work with� ¼ 0:25, corresponding to a ‘‘25 delta’’ RR.

We state a lemma to better understand the risk reversal. It isdrawn from Farhi et al. (2015, Proposition 5), and the OnlineAppendix gives a self-contained proof.11

LEMMA 2. In the limit of small time intervals, the risk reversal canbe expressed as:

RRijt ¼ k� Hjt �Hit

� �:ð34Þ

where k� :¼ 1�2�� 1 �ð Þð Þ

is a numerical constant, with � and �

the density and cumulative distribution functions of a stan-dard Gaussian.

Hence, if country i has more disaster risk than country j(Hjt �Hit > 0), then the risk reversal is positive: put prices oncurrency i are very expensive and have a high implied volatility(compared to symmetric call prices).

4. Four Signature Predictions of Disasters. The model makesfour broad predictions regarding option prices. The first threewere seen already, and the fourth one is detailed in Section V.C.

(i) Countries with high risk reversals have high interestrates.

(ii) Investing in countries with high risk reversals generateshigh expected returns.

11. Formula (34) holds for a ‘‘one-period’’ option, and Hit is expressed per period.

Suppose that ‘‘one period’’ is � years (e.g., � ¼ 112 if a period is one month), and implied

volatilities are expressed in annual units, and the maturity of the option is T years.

Then, formula (34) becomes RRijt ¼1�2�

� ��1 �ð Þð ÞHjt �Hit

� � ffiffiffiffiTp

, where RRijt ¼RRijtffiffi�p and

Hit ¼Hit

� are the RR and the resilience expressed in annual units. In addition, for 25

delta options (� ¼ 0:25Þ; 1�2�� 1 �ð Þð Þ

’ 1:57.


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(iii) When risk reversals go up, the exchange rate contempo-raneously depreciates.

(iv) The risk reversal of risky (i.e., high risk reversal, highinterest rate) countries should covary negatively withHMLFX (which is the payoff of a portfolio going longhigh interest rate currencies and short low interestrate currencies), while the risk reversal of less riskycountries should covary positively with it.

Empirical support for these predictions can be found in variouspapers: Carr and Wu (2007, prediction iii), Brunnermeier, Nagel,and Pedersen (2009, predictions i—iii), Du (2013, prediction i),Farhi et al. (2015, predictions i—ii). Section V.C finds support forprediction iv.

These four signature predictions of the disaster hypothesisare therefore qualitatively borne out in the data. The calibrationwill show that the correspondence between empirics and theorycan be made quantitative as well.

5. Illustration: Impact of a Change in the World DisasterProbability. An important object is the probability of a world di-saster, pt. Its movements have a number of signature effects thatwe now study. Consider two countries, again one safe (high Fjt),Switzerland, and one risky (low Fit<Fjt), Brazil. The difference intheir resiliences is

Hjt �Hit ¼ ptEDt B��tþ1 Fj;tþ1 � Fi;tþ1

� �� :

Suppose that pt increases, while EDt B��tþ1 Fj;tþ1 � Fi;tþ1

� �� re-

mains the same. Then, Hjt �Hit increases: Switzerland becomesrelatively more resilient than Brazil. As a result, Switzerland’scurrency appreciates relative to Brazil’s.

IV.B. Stocks

Our model allows us to think in a tractable way aboutthe joint determination of exchange rate and equity values. Weconsider the case of a stock of a generic firm in country i, that


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produces the traded good.12 Its dividend is Dit in units of thetraded good (i.e., the international numeraire), and dit ¼

Dit

eit

when expressed in the domestic currency. It follows the followingprocess

Di;tþ1

Dit¼ exp gDð Þð1þ "

Di;tþ1Þ �

1 if there is no disaster;

FDi;tþ1 if there is a disaster;

(where "D

i;tþ1 is an idiosyncratic shock uncorrelated with the pric-ing kernel.

We define the resilience HDit of the dividend Dit of stock i as

HDit ¼ ptEDt B��tþ1FDi;tþ1 � 1� �

¼ HDi� þbHDit:

As before, we posit that the law of motion for bHDit is an LGprocess:

bHDi;tþ1 ¼1þHDi�

1þHDitexp ��HDi

� � bHDit þ "HD

i;tþ1;ð35Þ

where �HDiis the speed of mean reversion of the resilience of

the stock. We also define hDi� :¼ ln 1þHDi�ð Þ.

PROPOSITION 7. (Price of stocks). The domestic price of the stockPDit is, in the continuous time limit

PDit ¼ dit

1þ HbDit

rDiþ�HDi

rDi;ð36Þ

where dit is the dividend expressed in the domestic cur-rency and rDi :¼ R� gD � hDi�.

A more resilient stock (high bHDit) has a higher price-dividendratio and lower future returns. The next lemma states the equitypremium.

LEMMA 3. (Equity premium). The expected excess return of stocks(in the domestic currency, over the domestic risk-free rate) is,in the limit of small time intervals, and neglecting second-order Ito terms: ptEt B��tþ1 Fi;tþ1 � FDi;tþ1

� �� .

12. The dividend of this firm may not be equal to total exports, as only a segmentof firms are listed in the stock market. The NBER working paper version of thisarticle also develops the case of a firm producing the nontraded good.


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IV.C. Yield Curve, Forward Rates, and Nominal ExchangeRates

Until recently, forward real interest rates were not available.Only their nominal counterparts were actively traded. Eventoday, most bonds are nominal bonds. To model nominal bonds,

we build on the real model developed above. Let Pit ¼Pi0Qt�1

s¼0

1��isð Þ

be

the price level in country i, where �it is inflation at time t (thisformulation will prove tractable). The nominal exchange rate is:

~eit ¼eit

Pit;ð37Þ

where we denote nominal variables with a tilde. The nominal

interest rate ~rt satisfies 1 ¼ EtM�tþ1

~ei;tþ1

M�t ~eit1þ ~ritð Þ

h i, so that in the

continuous-time limit

~rit ¼ rit þ �it;ð38Þ

that is, the nominal interest rate, is the real interest rate plusinflation. Because there is no burst of inflation during disasters,Fisher neutrality applies. With a risk of a burst of inflation,even short-term bonds would command a risk premium.

We posit that inflation hovers around �i�, roughly accordingto an AR(1) process. More specifically, to ensure tractability of themodel, we posit the linearity-generating process:

�i;tþ1 ¼ �i� þ1� �i�

1� �itexp ��ið Þ �it � �i�ð Þ þ "�i;tþ1;ð39Þ

where "�i;tþ1 has mean 0 and is uncorrelated with innovations inM�tþ1, in particular with disasters. This means, to the leadingorder, that �i;tþ1 � �i� ’ exp ��ið Þ �it � �i�ð Þ þ "�i;tþ1, that is, theprocess is a (twisted) AR (1). One could allow for nonzero cor-relation, but the analysis would become a bit more complicated.

PROPOSITION 8. (Forward rates). In the continuous-time limit, thedomestic nominal forward rate is, up to second-order terms inbHit and �it � �i�:

fit Tð Þ ¼ rei�l�rei

reiþ�Hi

exp ��HiT

� � bHitþ�i� þexp ��iTð Þ �it��i�ð Þ:ð40Þ


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The nominal forward rate in equation (40) depends on realand nominal factors. The real factor is the resilience of theeconomy bHit. The nominal factor is inflation �it.

Each term is multiplied by a term of the type exp ��HiT

� �.

For small speeds of mean reversion �, the forward curve is fairlyflat.

We can derive the implications of our model for a Fama re-gression in nominal terms:

~ei;tþ1

~eit�

~ej;tþ1

~ejt¼ ~� þ ~ ð ~rjt � ~ritÞ þ ~"ij;tþ1;ð41Þ

where ~rit and ~rjt are the nominal interest rates in countries iand j. Our model’s prediction is in the next proposition.

PROPOSITION 9. (Value of the coefficient in the Fama regression innominal terms). In the nominal Fama regression (41) withforward rates, the coefficients are:

~ ND¼ ~� ND þ 1� ~� ; ~

Full¼ ~� Full þ 1� ~�;ð42Þ

where ND and Full are the coefficients in the Fama regres-sion defined in Proposition 5 and

~� ¼1

1þ Varð�it��jtÞ

reireiþ�Hi

� �2

Var Hbit�Hbjt

� �ð43Þ

is the share of variance in the forward rate due to bHt.

In this simple model, the higher the variance of inflation,the closer ~

NDis to 1. Hence, countries with very variable infla-

tion (typically countries with high average inflation) approxi-mately satisfy the uncovered interest rate parity condition.Bansal and Dahlquist (2000) provide empirical support for thisphenomenon. When disaster risks are very variable—and thereal exchange rate is very variable—then ~

NDis smaller and

can turn negative.


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V. Quantitative Analysis

V.A. Calibration

We present a simple calibration of the model to work out itsbasic quantitative properties. We leave a full-blown estimation ofthe model for future work. As our data is nominal, use the exten-sion to a nominal setup presented in Section IV.C.13 We usemonthly data from JP Morgan presented in Farhi et al. (2015)for the January 2009–May 2013 period.14 We take the same pa-rameters for all countries.

1. Key and Ancillary Parameters. Table I summarizes themain inputs of the calibration, which are sufficient statistics forthe output of the model, summarized in Tables III–V. Thesesufficient statistics can in turn be rationalized in terms of‘‘ancillary’’ parameters shown in Table II. We call these parame-ters ‘‘ancillary’’ because they matter only via their impact on theaforementioned sufficient statistics listed in Table I. For instance,take rei ¼ Rþ l� g!i

� hi� (see equation (15)). Table I calibratesit to be 6%. At the same time, many combinations of R; l;�g!i

,and �hi� can add up to rei ¼ 6%. We present one such combina-tion of ancillary parameters in Table II.

Similarly, we directly calibrate the process (14) for resilience

Hit ¼ ptEDt B��tþ1Fi;tþ1 � 1� �

. In the ancillary Table II, we show oneexample of underlying processes of the disaster probability pt,severity Btþ1, and country-specific factors Fi;tþ1 that generatethe sufficient statistic Hit. This allows for a more parsimoniouscalibration that focuses on the key drivers of exchange rates.

2. Calibrating the Key Parameters. Cross-country correla-tions: In the baseline calibration, we take the innovations to

"Hit ; "

HD

it , and "Dit to be uncorrelated across countries. This is just

for clarity and simplicity. We introduce cross-country correlationslater in Section V.C. The Online Appendix details the exact spe-

cification that we adopt for the mean-zero innovations "Hit ; "

HD

it ,

and "Dit .

13. The countries we use are Australia, Canada, euro area, Japan, Norway,New Zealand, Sweden, Switzerland, United Kingdom, United States.

14. There was a large crash of carry trade in fall 2008. The post-2008 sample ismore economically homogeneous than the full sample.


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Exchange rate and interest rate: We call � the time-differ-ence operator, �xt ¼ xt � xt�1, and �x ¼ stdev �xtð Þ the volatility ofa variable xt. For two countries, we define the volatility of the

bilateral exchange rate as �bile ¼ stdev �ln eit

ejt

� �and the volatility

of the difference in interest rates �bilr ¼ stdev � rit � rjt

� �� .

Equations (16) and (22) give �bilr ¼ rei�

bile . The above equation

guides our calibration. We set rei ¼ 6%. This implies �bile ’ � ~e

bil

’ 11% and a reasonable value of the bilateral real interest rate

volatility, �bilr ¼ 0:7%.

For the speed of mean reversion of resilience, we take�Hi¼ 18%, which gives a half-life of ln 2=�Hi

¼ 3:8 years, in linewith estimates from the exchange rate predictability literature

TABLE I

KEY PARAMETER INPUTS

Exchange rate discount rate rei ¼ 6%

Resilience: volatility and speed ofmean-reversion

�Hi¼ 1:87%; �Hi

¼ 18%

Inflation: volatility and speed ofmean-reversion

��i¼ 0:6%; ��i

¼ 23%

Stocks: discount rate, volatility andgrowth of dividends

rDi ¼ 4:4%; �Di¼ 8:4%; gDi

¼ 2:5%

Stocks resilience: volatility and speedof mean-reversion

�HDi¼ 3:1%; �HDi

¼ 13%

Notes. This table reports the coefficients used in the model. �X is the average volatility, and �X is thespeed of mean reversion. The time unit is a year (the model is simulated at a monthly frequency, but forreadability the numbers are all annualized).

TABLE II

ANCILLARY PARAMETERS

Coefficient of risk aversion, IES � ¼ 4; � ¼ 1�

Rate of time preference � ¼ 5:9%

Probability and severity of disasters p ¼ 3:6%; B ¼ 0:66Recovery rate of export productivity F ¼ 1; �Fi

¼ 9:8%

Growth rate of consumption, export productivity gci¼ 2:5%; g!i

¼ 0Depreciation rate l ¼ 5:5%

Recovery rate of stock dividend FDi¼ B

Notes. This table reports the coefficients used in the model to generate the parameters of Table I. �X

is the average volatility, and �X is the speed of mean reversion. The time unit is a year (the model issimulated at a monthly frequency, but for readability the numbers are all annualized). The parametersalso generate: Hi* = 15.37%, R = 15.9%, �FDi

= 1.5%.


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TABLE III

MOMENTS: EMPIRICAL AND IN THE MODEL

Moments Data Calibration

Std. dev. (�ln ~eijtÞ 12.35% 11.9%Carry trade return 3.44% 4.9%Fama regression coefficient –1.29 –0.61Mean(jRRijtjÞ 1.31% 1.25%Std. dev. (RRijt) 1.24% 1.28%Std. dev. (�RRijt) 2.60% 1.10%Std. dev. ( ~rit) 1.38% 1.28%Std. dev. (�ð ~rit � ~rjtÞ) 0.71% 1.18%Corr(�ln ~eijt;�RRijtÞ –0.58 –0.90Corr(ln ~eij;tþ1; ln ~eijt) 0.88 0.98Corr(�ln ~eij;tþ1;�ln ~eijt) –0.13 0.00Corr( ~rit � ~rjt;RRijtÞ 0.55 0.65

Notes. The table reports the moments generated by the model, using the inputs from Table I. The riskreversal RRijt is defined as the implied volatility of an out-of-the-money put minus the implied volatility ofan out-of-the-money call, all at 25-delta. A high RRijt means that the price of protection from depreciationof currency i (against country j) is high. ~rit is the nominal interest rate. We define ~eijt :¼ ~e it

~e jt, the nominal

bilateral exchange rate between countries i and j: a high ~eijt means that currency i appreciates. Carrytrade returns are the returns from a long-short portfolio going $1 long (resp. short) an equally sizedportfolio of high (resp. low) RR countries. The time unit is a year (the model is estimated and simulatedat a monthly frequency, but for readability the numbers are all annualized).

TABLE IV

MOMENTS RELATED TO STOCKS: EMPIRICAL AND IN THE MODEL


Equity premium: mean (rStockit � rit) 6% 7.5%

Volatility of stock returns: std. dev. (rStockit ) 18% 18%

Mean Pit

Dit23 18.7

Standard deviation of ln Pit

Dit0.33 0.37

Volatility of stock returns in foreign currency:std. dev. (rStock;$

it )20% 21%

Corr(�ln ~eijt; rStockit � rStock

jt Þ 0.00 0.03

Corr(�ln ~eijt; rStock;$it � rStock;$

jt Þ 0.75 0.55

Corr(�RRijt; rStockit � rStock

jt Þ –0.05 –0.01

Corr(�RRijt; rStock;$it � rStock;$

jt Þ –0.46 –0.49

Notes. The table reports the moments generated by the model, using inputs from Table I. rstockit (resp.

rstock;$it ) is the stock return of country i expressed in country i’s domestic currency (resp. in US$). The risk

reversal RRijt is defined as the implied volatility of an out-of-the-money put minus the implied volatility ofan out-of-the-money call, all at 25-delta. A high RRijt means that the price of protection from depreciationof currency i (against country j) is high. We define ~eijt :¼ ~e it

~e jt, the nominal bilateral exchange rate between

countries i and j: an increase in ~eijt means that currency i appreciates. Here we take country j to be theUnited States. The time unit is a year (the model is estimated and simulated at a monthly frequency, butfor readability the numbers are all annualized).


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(Rogoff 1996). The standard deviation of the innovations to resil-ience Hit are chosen to roughly match the level and volatility ofthe risk reversals, as well as the volatility of the bilateral ex-change rate, as reported in Table III. This leads us to set�Hi¼ 1:87%.Inflation: Data (e.g., on currency options) are nominal, and

the essence of our model is real. Inflation contains a substantialhigh-frequency and transitory component, which is in part due tomeasurement error. Call ~�it ¼ �it þ �it the measured inflation(which can be thought of as trend inflation plus mean-zeronoise), while �it is the trend inflation. We estimated inflationusing the Kalman filter, with �i;tþ1 ¼ C1 þ C2�i;t þ "

0i;tþ1for the

trend inflation and ~�it ¼ �it þ �it for the noisy measurement ofinflation, in the sample 1985–2013 (so that we capture the post-Volcker disinflation regime). This led to a volatility of inflation of��i¼ 0:6% (by the way the estimate is quite unstable and sample-

dependent) and speed of mean reversion of ��i¼ 23%.

Carry trade returns: A simple and important statistic thatcaptures risk premia in exchange rates is the average carrytrade return. We operationalize this strategy in the followingmanner (see, e.g., Farhi et al. 2015). To better capture disasterrisk, we sort on risk reversals rather than interest rates. Wedivide countries into two equally sized bins of resilience: therisky countries are those in the bottom half of resilience andthe less risky countries are in the top half. We define the carrytrade as going long $1 in the equally weighted portfolio of risky

TABLE V

MOMENTS RELATED TO HMLFX: EMPIRICAL AND IN THE MODEL


Corr(HMLFX;t; rStockat Þ 0.39 0.33

Test of the one factor-structure: �i 0* 0*in rCurrency

it ¼ �i þ i;HMLHMLFXtþ "it

(for H,M,L portfolio) (for H,M,L portfolio)Corr(HMLFX;t;�ln ~eit

~eatÞ 0:60;�0:01;�0:51ð Þ ð0:50; 0:04;�0:43Þ

Corr(HMLFX;t;�RRitÞ �0:43;�0:05; 0:36ð Þ ð�0:47;�0:02; 0:43Þ

Notes. Here HMLFX,t is the return of a portfolio going long high interest rate currencies and short lowinterest rate currencies. rStock

at is the average of stock market returns across countries. rCurrencyit is the

currency return (capital gains plus interest rate) when going long a basket i (high / medium / low interestrate currencies) and short an equally weighted basket of all currencies. 0* means that the value is notstatistically different from 0. ~e it

~e atis the nominal exchange rate of country i vis-a-vis an equally weighted

basket of all currencies ( ~eat is the average exchange rate across countries, ln ~eat ¼1n

Pnj¼1 ln ~ejtÞ. Countries

are sorted by interest rates and are divided into three groups of high, medium, and low interest rates(H, M, L).


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countries (low bHit) and going short $1 in the equally weightedportfolio of safe countries (high bHit).

Stocks: Empirically, domestic stock returns and the ex-change rate are uncorrelated on average (see Table IV). We spe-cify the correlation between innovations to HDit and Hit to matchthis (the procedure is detailed in the Online Appendix). We matchthe volatility of dividends dit of 11%, as in Campbell andCochrane (1999); this leads to �Di

¼ 8:4% (remember that Dit isin the world numeraire, and dit is expressed in the domestic cur-rency). We use the stock calibration in Gabaix (2012). We take thespeed of mean reversion of dividend resilience to be �HDi

¼ 13%,which mimics the speed of mean reversion of the stock’s price-dividend ratio and rDi ¼ 4:4%, to match the average price-divi-dend ratio. We choose �HDi

to match the volatility of stock returns.

2. Interpreting Key Parameters in Terms of Specific AncillaryParameters. Resiliences are sufficient statistics for the calibra-tion. To be more precise, our calibration only requires specifyingthe joint laws of motion for the resiliences Hit ¼ ptEt B��tþ1

�Fi;tþ1 � 1�. The results of the calibration do not depend onwhether the shocks that drive the changes in resiliences comefrom movements in pt, B��tþ1, or Fi;tþ1. However, we discuss nowhow their variations are related to deeper disaster parameters.

One simple specification to generate the volatility of resil-ience in Table I is to assume that resilience innovations comeentirely from idiosyncratic movements of Fit. We keep pt and

Btþ1 constant at p ¼ 3:6% and B ¼ 0:66, which as recommendedby Barro and Ursua (2008), is based on a certainty equivalent of

historical disasters (B :¼ E B��½ ��1� ¼ 0:66, averaging over histor-

ical episodes; see also Barro and Jin 2011). We use a coefficient ofrelative risk aversion � = 4.

Then we calibrate �Fi¼

�Hi

pB�� ¼ 9:8%.15 This generates a dis-

persion in prospective recovery rates of var Fitð Þ1=2’

�Fiffiffiffiffiffiffiffiffi2�Hi

p , so that

the dispersion in expected bilateral recovery rates is

15. An interesting alternative is as follows. Suppose that innovations in differ-

ential resilience come entirely from movements in pt (keeping Et

B��tþ1 Fi;tþ1� Fj;tþ1

� �� constant). With fixed values of Fit, for example, jFi;tþ1� Fj;tþ1j

¼ 0:23 (similar to the numbers above) we have �pt ¼ 1:5%. This is of the sameorder of magnitude as the calibration in Wachter (2013), which uses �pt ’ 1:1%.


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var ln Fit � ln Fjt

� �1=2’

�Fiffiffiffiffiffiffi�Hi

p ¼ 23%. Hence, if the theory is cor-

rect, a weak consequence is that the realized dispersion of cur-rency jumps during disasters should be at least 23%.16

Empirically, the dispersion of actual currency movements isindeed larger, at 53% (the interquartile range of 41%) duringWorld Wars I and II.17 Hence, by that metric, the volatility inresilience is reasonable.

Exports and productivity: In the calibration, we take � ¼ 1� for

parsimony (so that the utility becomes separable) and alsoassume that �it and �it do not jump in a disaster. In a disaster,

M�tþ1 is multiplied by B��tþ1. So, since CTit ¼

M�tie��t�it

� ��1�

(from equa-

tion (10)), we have that CTit is multiplied by Btþ1 in a disaster.

Given CNTit ¼ CT

it �iteitð Þ�1� (from equation (9)), CNT

it is multiplied

by Btþ1F�1�

i;tþ1 in a disaster. For simplicity in the choice of ancillary

parameters, we will take the case where the typical value of Fi;tþ1

is F ¼ 1. This way, in a typical disaster, the ratio of nontradableto tradable consumption remains unchanged. In other words,both types of consumption collapse by the same factor, Btþ1.

This leads us to take Hi� ¼ pðB��

F � 1Þ ¼ 15:3%. We catego-rize Hi� as ancillary because its value (given the existing keyparameters of Table I) does not affect (up to second-orderterms) the exchange rate and interest rate moments that wereport in Tables III and IV.

The growth rate of productivity g!iis irrelevant in practice,

but for completeness we propose a specific value. Though resultsare not sensitive to the choice of this parameter, we set g!i

¼ 0; inan economy where endowments all grow at a rate gci

, the

16. Indeed, consider the very mild enrichment of the model, where !i;tþ1

!itis the

process assumed in the article (equation (11)), but multiplied by ð1þ "!i;tþ1Þ where

"!i;tþ1 is 0 if there is no disaster, and is a mean-0 random variable is there is a

disaster. Then, none of the pricing expressions in the article change (includingthe value of the exchange rate and interest rate). If there is a disaster at t,

�ln eit

ejt¼ �ln Fit

Fjtþ�ln

"!it"!

jt, so varDð�ln eit

ejtÞ � varDð�ln Fit

FjtÞ.

17. The data come from Global Financial Data, and comprises Australia,Austria, Belgium, Brazil, Canada, Denmark, Finland (only World War II),France, Germany, India, Ireland (only World War II), Italy, Japan, theNetherlands, New Zealand (only World War II), Norway, Portugal (only WorldWar II), Russia (only World War II), South Africa, Spain, Sweden, Switzerland,and the United Kingdom.


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consumption of tradables and nontradables then both grow at gci,

as there is no differential advantage to the production of trad-ables. We take l ¼ 5:5% which leads to an average real interestrate of rei � l ¼ 0:5%:

Given R ¼ rei � lþ g!iþ hi�, we obtain R ¼ 15:9%. Given our

assumptions, we have R ¼ �þ �gci, where gci

is the growth rate oftradable consumption. We choose the growth rates so that innormal times consumption of nontradables grows at a rategci¼ 2:5%. This in turn pins down the rate of time preference,

� ¼ R� �gci¼ 5:9%.

Stocks: We take the recovery of the stock dividend to be thatof the economy FDi

¼ B. By the same reasoning as before, we have�FDi¼

�HDi

pB�� ¼ 1:5%.

V.B. Main Quantitative Findings

Tables III and IV present results from the calibration.18

As Table III shows, the model hits the volatility of the bilat-eral exchange rate, that is, generates the right amount of ‘‘excessvolatility’’ in exchange rates. The model also roughly matches(and slightly undershoots) the size of disaster risk as measuredby the average size of risk reversals. At the same time, the modelgenerates a moderate volatility of the interest rate, as in the data.

We showed earlier that in the model countries with high riskreversals have high interest rates, and that increases in risk re-versals are associated with depreciations of the exchange rate. Thecalibration confirms this. Table III reports the calibrated values ofCov ~rit � ~rjt;RRt

� �, which broadly matches up with its empirical

counterpart. Corr(�ln ~eijt;�RRijtÞ is a bit far from its empiricalvalue. The reason is that we have a one-factor model, which (es-sentially) tends to generate a –1 correlation. If we have a two-factormodel (as in the Online Appendix), then the fit becomes quite good.

The carry trade generated by the model gives average re-turns in line with the empirical evidence (see Farhi et al. 2015for more variants of the carry trade). Investing in countries withhigh risk reversals generates high expected returns. Indeed, theexpected return of the carry trade (given positive RR) is about

18. The Online Appendix presents a table that verifies that our calibration ofstocks also matches the pattern of predictability of the stock market with the price-dividend ratio, with a slope and R2 rising with the horizon (at least for a while). Theresults in Lettau, Maggiori and Weber (2014) could help constrain the calibration ofthe model’s higher moments.


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4.9% a year. Finally, the model generates Fama coefficients ND ¼ �0:61 in line with estimates of the literature cited above.In the table, we report Verdelhan (2010)’s estimate, which isrepresentative.

Stocks. Table IV shows the moments related to stocks. We call

rStockit ¼

PitþDit

Pi;t�1� 1 the return of the stock in country i, in the do-

mestic currency. We also call rStock;$it ¼

eijt

eij;t�1

PitþDit

Pi;t�1� 1 ’ rStock

it þ�

lneijt the return in a foreign currency, which we take to be the

dollar; ~eijt ¼~eit~ejt

the bilateral exchange rate between countries i

and j. The equity premium is about 6%, in line with the usualempirical estimates. We report the correlations between changesin the exchange rate of two countries and changes in the relative

stock returns: corrð�ln ~eijt; rStockit � rStock

jt Þ. Empirically, this corre-

lation is close to 0; movements in the stock market and the ex-change rate are uncorrelated on average. The model reproducesthat fact. Likewise, changes in risk reversals and relative stockreturns are essentially uncorrelated in the data and in the model

(corrð�RRijt; rStockit � rStock

jt Þ is close to 0).

We also study the correlation between stock returns in acommon currency and the change in the exchange rate:

corrð�ln ~eijt; rStock;$it � rStock;$

jt Þ. Empirically, this correlation is

high, about 0.75. The model is roughly in line with this.Likewise, there is a negative correlation between changes in

risk reversals and relative stock returns: corrð�RRijt; rStock;$it �

rStock;$jt Þ < 0 in the data and the model. Economically, when coun-

try i becomes less risky, the risk reversal RRijt goes down, theexchange rate ~eijt appreciates and its realized stock return differ-

ential rStock;$it � rStock;$

jt is higher.

We conclude that the disaster model can be made quantita-tively broadly congruent with the salient empirical facts.

V.C. Additional Findings: Covariance Structure in Currenciesand HMLFX

In their influential work, Lustig, Roussanov, and Verdelhan(2011) find a one-factor structure in currency returns. Namely,they form a portfolio, HMLFX, going long high interest rate cur-rencies and short low interest rate currencies. They also find that


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currency excess returns are accounted for by the HMLFX factor.

They find, in essence, that regressing currency return rCurrencyit ¼

�i þ iHMLFXtþ "it yields �i ¼ 0. Here, rCurrency

it is the currencyreturn (capital gains plus interest rate) when going long abasket i (e.g., high interest rate currencies) and short a diversi-fied basket of currencies.

In this subsection, we keep the previous calibration but giveit the additional structure of a one-factor model in currencies andstocks, respectively. We find that we can match the salient factsrelated to HMLFX and make a new, successful prediction linkingit to risk reversals.

1. Factor Structure. We define the normalized resiliences ofstocks and exchange rates as:

hDit :¼1þ HDit

rDiþ�HDi

1þ H it

reiþ�Hi

� 1;heit :¼bHit

rei þ �Hi

:

Using this notation, we can reexpress equations (16) and (36) asfollows:

eit ¼!it

rei1þ heitð Þ;PDit ¼

rei

rDi

Dit

!it1þ hDitð Þ:

The innovation to the exchange rate and the stock price (and re-turn) are captured by heit and hDit, respectively. We call "Xt

theinnovation to a random variable Xt ("Xt

¼ Xt � Et�1 Xt½ �).We posit that the innovation to normalized resilience follows

a one-factor structure: "heit¼ ei;t�1fet þ �eit where fet and �eit are

mean-0 innovations. We also specify eit ¼ b Hat �Hitð Þwith b>0,and Hat is the average of Hit over all other countries. This meansthat when fet is positive, the spread in the resilience betweenrisky and less risky countries shrinks, that is, risky currenciesappreciate over less risky currencies. Hence, fet is proportionalto HMLFX;t. When fet is positive, HMLFX;t is positive and therisk reversal of risky countries goes down while their exchangerate appreciates. The key free parameter is b, which we set to0.114.

Empirically, international stock markets tend to covary. Thisnaturally suggests a one-factor structure of stock resilience:


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"hDit¼ Di;t�1fDt þ �Dit. We set Dit ¼ 1 and corr fet; fDtð Þ ¼ 0:65,

which allows us to match the empirical correlation between theHMLFX factor and the average of international stock market re-turns (the Online Appendix details the process, including linear-ity-generating terms.) The factors �eit and �Dit are uncorrelatedwith other variables.

2. Results. Table V shows the results.19 As expected, if we sortcountries by interest rates (high, medium, and low interest rates:H,M,L; recall that in our model, risky countries have high realinterest rates), we observe that risky countries have a positivecorrelation with HMLFX and the least risky countries have a neg-

ative correlation with it (see the row Corr(HMLFX;t;�ln ~eit~eat

)). The

model produces a good quantitative fit with this fact.We also verify that the one-factor structure shown in Lustig,

Roussanov, and Verdelhan (2011) is replicated in our model (see

the row on rCurrencyit ¼ �i þ i;HMLHMLFXt

þ "it).20 In addition, the

model replicates the positive correlation between HMLFX returnsand average stock market returns.

The new prediction of the model is that risk reversals shouldcovary with HMLFX: the risk reversal of risky countries shouldcovary negatively with HMLFX, whereas the risk reversal of lessrisky countries should covary positively with it. This is indeed thecase in the data, as indicated in Table V. We view this as anadditional comforting, previously undocumented disaster-likefeature of the data.

In conclusion, a parsimonious calibration of the model canreplicate the major moments of the link between currencies, in-terest rates, stocks, and options, including the factor structuredocumented in stocks and currencies.

19. This table is computed over the whole sample to maximize representative-ness. The numbers are broadly the same when restricting to the post-2009 sample,except Corr(HMLFX;t; rStock

at Þ, which is smaller in that sample. We suspect that thisnumber is not representative of typical samples.

20. If we computed the returns of portfolios short a given currency (say, thedollar), then we would need to add a second factor, namely, the return of thatcurrency.


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VI. Conclusion

We have proposed a disaster-based tractable frameworkfor exchange rates. Our framework accounts qualitatively andquantitatively for both classic exchange rate puzzles (e.g.,excess volatility of exchange rates, forward premium puzzle,excess return of the carry trade) and links between currency op-tions, exchange rates, and interest rates—signature predictionsof the disaster hypothesis.

The model is fully solved in closed form. It can readilybe extended in several ways. The Online Appendix worksout various extensions, including a detailed model of theterm structure and the incorporation of business cyclemovements.21

The model offers a unified, tractable, and calibrated treat-ment of the major assets and their links: exchange rates, bonds,stocks, and options. Hence, we hope it may be a useful point ofdeparture to think about issues in international macro-finance.In particular, studying more specifically the dynamics of produc-tion and consumption in the disaster environment seems like afruitful direction for research.22 We speculate that this might in-volve modeling adjustment costs in investment, imperfect risksharing, and price setting imperfections leading to pricing tomarket and incomplete cost-to-price pass-through. Pursuingthis direction could lead to a unified international macro modelto think jointly about prices and quantities.

21. These extensions rationalize additional empirical facts uncovered byBoudoukh, Richardson, and Whitelaw (2012) on forward rates and deviationsfrom UIP, and Lustig, Stathopoulos, and Verdelhan (2014) on the term premiumin the bond and currency riskiness. Boudoukh, Richardson, and Whitelaw (2012)find that the carry trades based on long-dated forward rates exhibit small devia-tions from UIP: in our model this is because at a long horizon, other factors (e.g.,business cycle or inflation) are more important, and they tend to generate no devi-ation from UIP. Lustig, Stathopoulos, and Verdelhan (2014) find that ‘‘risky’’ cur-rencies have a lower term premium. In our model this holds even if all countrieshave the same inflation dynamics, where inflation goes up in disasters (as in thehistorical experience on average), which creates a term premium. Because riskycurrencies will depreciate in a disaster, a portfolio long their short-term bond andshort their long-term bond will have little value in a disaster, whereas the equiva-lent portfolio for safe countries will have a positive value: hence, the term premiumis high in safe countries and small or zero in risky countries.

22. To keep a tractable model of production with disasters, the ‘‘disasterization’’procedure may be useful. It was proposed in Gabaix (2011), and later used in Gourio(2012).


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Appendix: Complements and Proofs

A. Different Notions of the Exchange Rate

In the article, we define the ‘‘absolute’’ exchange rate eit to bethe price of the nontraded good in country i in terms of the worldnumeraire. The more traditional definition would be Eit, the priceof the consumption basket in country i in terms of the worldnumeraire.23 Using the usual algebra of CES price indexes, thelink between the two is:

Eit ¼ ��it þ eit1��

� � 11��:ð44Þ

The share of traded goods in consumption is

��ite1��it ��it þ eit

1�� 1

. In the data, this share is small, so that �itis close to 0. The consequence is that Eit ’ eit and Eit

Ejt’

eit

ejt, so that

the two notions are quantitatively close. This approximation isexact up to a term O ��it

� �. It is analytically simpler to characterize

the behavior of eit. In any case, it is possible to go back and forthbetween the two notions using equation (44).

B. Proofs

We present the proofs of the main results. Additional proofsare in the Online Appendix. For simplicity, we drop the countryindex i in most proofs.

The Limit of Small Time Intervals. We often take the limit ofsmall time intervals. We formalize this procedure here. Take forinstance the interest rate ‘‘per period’’ rit. Let us call �t the phy-sical length of a time period (e.g., �t ¼ 1

12 of a year if the timeperiod is one month), and rit the interest rate in continuoustime notation. Then, rit ¼ rit�t. Likewise,bHit ¼

bH it�t;pt ¼ pt�t; � ¼ q�t; l ¼ k�t:ð45Þ

However, the exchange rate is not in ‘‘per period units,’’ so thateit ¼ eit. Likewise, Bt ¼ Bt; Fit ¼ Fit. On the other hand, t ¼ t

�t,where t is the physical time (in years). The limit of small timeintervals corresponds to �t!0.

As an example, let us detail how equation (19) becomes equa-tion (16) in the limit of small time intervals.

23. The bilateral exchange rate between country i and country j is then Eit

Ejt.


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eit ¼!it




� � bHit

!

¼vit�t

1� exp �rei�tð Þ1þ

exp �rei�t� hi��tð Þ

1� exp �rei�t�fHi�t

� � bH it�t

!

¼vit�t

rei�tþO �tð Þ2� � 1þ

1þO �tð Þ

rei�tþfHi�tþO �tð Þ2

� � bH it�t

!

¼vit

rei þO �tð Þ1þ

1þO �tð Þ

rei þfHiþO �tð Þ

bH it

�

¼vit

rei1þ

bH it

rei þfHi

!þO �tð Þ ¼

!it

rei1þ

bHit

rei þ �Hi

!þO �tð Þ:

Complement to the Proof of Proposition 2. We now present arigorous proof. Let Dt ¼ exp �ltð Þ!t and Xt ¼

bHt (for simplicity,we drop the subscript i in this proof). By Proposition 1, we have

e0 ¼

E0

X1t¼0

M�t Dt

" #M�0

:ð46Þ

We calculate the moments:

EtM�tþ1Dtþ1

M�t Dt

�¼ exp �R�lþg!ð Þ ð1�ptÞþptE

Dt B��tþ1Ftþ1

� � �¼ exp �R�lþg!ð Þ 1þHtð Þ ¼ exp �R�lþg!ð Þð1þH�Þ

þexp �R�lþg!ð ÞbHt

¼ exp �R�lþg!ð Þð1þH�Þþexp �R�lþg!ð ÞXt

¼ exp �reð Þþexp ð�re�h�ÞXt;

using re¼Rþl�g!�h�. Also:


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EtM�tþ1Dtþ1

M�t DtXtþ1

�¼ Et

M�tþ1Dtþ1

M�t Dt

�Et Xtþ1½ �

¼ e�R�lþg! 1þHtð Þ1þH�1þHt

e��H bHt

¼ e�R�lþg!��H 1þH�ð ÞbHt ¼ e�re��H Xt:

There are two ways to conclude. The first way uses thenotation of Gabaix (2009) 24: the above two moment calcu-lations show that Yt ¼M�t Dt 1;Xtð Þ is an LG process withgenerator �:

� ¼exp �reð Þ exp �re � h�ð Þ

0 exp �re � �Hð Þ

!:

Using Theorem 2 in Gabaix (2009), or equation 3 of the‘‘Precis,’’ we find ei0 ¼ !i0 1; 0ð Þ I ��ð Þ

�1 1;Xtð Þ0,

ei0 ¼!i0




� � bHi0

!:ð47Þ

More generally, where !it is the current productivity of thecountry, we get equation (19). The second way is the moreheuristic proof in the main text.

Proof of Proposition 4. Derivation of equation (25): Usingequation (16), we calculate:

24. See also the compact version, on Gabaix’s web page: ‘‘Precis of Results onLinearity-Generating Processes.’’ http://pages.stern.nyu.edu/�xgabaix/papers/lg-precis.pdf


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http://pages.stern.nyu.edu/~xgabaix/papers/lg-precis.pdf




ENDt

ei;tþ1

eit� 1

�¼ END

t

!i;tþ1

!it

1þbHi;tþ1

re þ �H

1þbHit

re þ �H

� 1

26643775

¼ ENDt expðg!i

Þ

1þbHi;tþ1

re þ �H

1þbHit

re þ �H

� 1

26643775

¼ g!iþ END

t

1þbHi;tþ1

re þ �H

1þbHit

re þ �H

� 1

26643775þ o �tð Þ

¼ g!iþ END

t

bHi;tþ1 �bHit

re þ �H þbHit

�þ o �tð Þ

¼ g!i�

�HbHit þ

bH2

it

re þ �H þbHit

þ o �tð Þ;

where we use the fact that equation (14) becomes, in the limitof small time intervals, bHi;tþ1 �

bHit ¼ � �H þbHit

� � bHit þ "Hi;tþ1þ

o �tð Þ2� �

.Using equation (22), rit ¼ re � l� reHbit

reþ�HþHbit

and re ¼ Rþ l�g!i� hi�,

ENDt

ei;tþ1

eit1þ ritð Þ

��1¼END

t

ei;tþ1

eit�1

�þ ritþo �tð Þ

¼ g!i��H

bHitþbH2

it

reþ�HþbHit

þ re�l�rebHit

reþ�HþbHit

þo �tð Þ

¼R�hi� �reþ�Hþ

bHit

� � bHit

reþ�HþbHit

þo �tð Þ

¼R�Hi� �bHitþo �tð Þ ¼R�Hitþo �tð Þ:

This implies:


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ENDt Rij;tþ1

� �¼ END

t

ei;tþ1

eit1þ ritð Þ

�� END

t

ej;tþ1

ejt1þ rjt

� � �

¼ Hjt �Hit þ o �tð Þ ¼ bHjt �bHit þ o �tð Þ

Next, we turn to proving equation (24).

Etei;tþ1

eit1þ ritð Þ

��1¼ 1�ptð ÞEND

t

ei;tþ1

eit1þ ritð Þ�1

�

þptEDt

ei;tþ1

eit1þ ritð Þ�1

�¼ 1�ptð Þ R�Hitð ÞþptE

Dt Fi;tþ1�1� �

þo �tð Þ:

The last step is verified as follows:

ptEDt

ei;tþ1

eit1þ ritð Þ � 1

�¼ pt�tED

t Fi;tþ1 1þ rit�tð Þ � 1� �

¼ pt�t EDt Fi;tþ1 � 1� �

þO �tð Þ� �

¼ ptEDt Fi;tþ1 � 1� �

þ o �tð Þ:

Given Hit ¼ ptEDt ½B

��Fi;tþ1 � 1� (equation (12)), as we

assumed that Btþ1 ¼ B is deterministic,

Et Rij;tþ1

� �¼ Et

ei;tþ1

eit1þ ritð Þ

�� Et

ej;tþ1

ejt1þ rjt

� � �¼ 1� ptð Þ Hjt �Hit

� �þ ptE

Dt Fi;tþ1 � Fj;tþ1

� �þ o �tð Þ

¼ 1� ptð Þ Hjt �Hit

� �þ B

�Hit �Hjt


¼ 1� B�

� �Hjt �Hit

� �þ o �tð Þ:

The last step uses equation (45), which gives pt ¼ Oð�tÞ and

Hjt �Hit ¼ Oð�tÞ, so that ptðHjt �HitÞ ¼ Oðð�tÞ2Þ ¼ oð�tÞ.Finally, we use equation (22), which gives rit � rjt ¼ �

rebHit

reþ�Hþ

OðbH2

itÞ þrebHjt

reþ�HþOðbH2

jtÞ, hence bHjt �bHit ¼ ð1þ

�HreÞðrit � rjtÞ þOðbH2

jt þbH2

itÞ.


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Proof of Proposition 5. We use equation (25), which gives (inthe limit of small time intervals and resiliences):

ENDt

ei;tþ1

eit�

ej;tþ1

ejt

�þ rit� rjt

� �¼END

t

ei;tþ1

eit1þ ritð Þ�

ej;tþ1

ejt1þ rjt

� � �þo �tð Þ

¼ 1þ�H

re

� rit� rjt

� �þo �tð Þ;

that is,

ENDt

ei;tþ1

eit�

ej;tþ1

ejt

�¼�H

rerit � rjt

� �þ o �tð Þ:

Hence, an econometrician running the Fama regression inequation (26) will find ND ¼ �

�Hre

.Likewise,

Etei;tþ1

eit�

ej;tþ1

ejt

�¼ 1� B

��

1þ�H

re

� � 1

�rit � rjt


¼ ��H

reþ 1þ

�H

re

� B�

�rjt � rit


so that Full ¼ ��H

reþ 1þ �H

re

� �B�.

Proof of Proposition 6. Put price: We start with the put price:

VP Kð Þ ¼E0M�1M�0

Kej1

ej0�

ei1

ei0

� þ �

¼ 1�p0ð ÞEND0

M�1M�0

Kej1

ej0�

ei1

ei0

� þ �þp0E

D0

M�1M�0

Kej1

ej0�

ei1

ei0

� þ �

¼ 1�p0ð Þe�RE

ND0 K

ej1

ej0�

ei1

ei0

� þ �þp0e�R

ED0 B��1 K

ej1

ej0�

ei1

ei0

� þ �;

where ND and D superscripts denote expectation conditional onno disasters and a disaster, respectively. The next calculationuses the following lemma, which is standard.25

25. To verify it, we calculate that the characteristic function of y is the charac-teristic function of distribution (48):

EQ eky� �¼E ex�E x½ ��2

x=2ekyh i

¼ exp kE y½ �þk2�2

y

2þkCov x;yð Þ

!¼ exp k E y½ �þCov x;yð Þð Þþ

k2�2y

2

" #:


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LEMMA 4. (Discrete-time Girsanov). Suppose that ðx; yÞ are jointlyGaussian under P. Consider the measure Q defined bydQdP ¼ expðx� E½x� � VarðxÞ

2 Þ. Then, under Q, y is Gaussian,with distribution

y�QN E y½ � þ Cov x; yð Þ;Var yð Þð Þ;ð48Þ

where E y½ �; Cov x; yð Þ, and Var yð Þ are calculated under P.

To perform the calculation, write for the ND case

ei1

ei0¼ exp i þ "i �

�2i

2

� ð49Þ

and the analog for j. We call � ¼ "i � "j, and calculate:

V1¼END0 K

ej1

ej0�

ei1

ei0

� þ �

¼END0 Kexp jþ"j�

�2j

2

!�exp iþ"i�

�2i

2

� !þ" #

¼ exp ið ÞEND0 exp "j�

�2j

2

!Kexp j�i

� ��exp "i�"jþ

�2j

2��2

i

2

! !þ" #:

We define dQdP¼ exp "j�

�2j

2

� �, and use Lemma 4. Under Q, with �

¼ "i� "j; y¼ �þ�2

j

2 ��2

i

2 is a Gaussian variable with variance �2�

and mean:

EQ y½ � ¼

�2j

2��2

i

2þ Cov "i � "j; "j

� �¼ �

�2j

2��2

i

2þ �i;j ¼ �

Var �ð Þ

2:

Hence,

V1 ¼ exp ið ÞEQ Kexp j � i

� �� ey

� �þh i¼ exp ið ÞV

PBS Kexp j � i

� �; �ijj

� �;

where �ijj ¼ ðVarð"j � "iÞÞ12 and VP

BSðK; �Þ ¼ E½ðK � expð�u� �2

2 ÞÞþ�

(with u a standard Gaussian) is the Black-Scholes put value


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when the interest rate is 0, the maturity 1, the strike K, thespot price 1, and the volatility �.

Next we observe that in disasters, ei1ei0¼ exp ið ÞFi1. This

implies:

ED0 B��1 K

ej1

ej0�

ei1

ei0

� þ �¼ ED

0 B��1 exp j

� �KFj1 � exp ið ÞFi1

� �þh i:

We conclude that the value of the put is equation (30).Call price: We use the put-call parity. Using the identity xþ

¼ xþ �xð Þþ and the fact that E0M�1M�

0

ei1ei0

h i¼ 1

1þri0, we have:

VC Kð Þ¼E0M�1M�0

ei1

ei0�

ej1

ej0K

� þ �¼E0

M�1M�0

ei1

ei0�

ej1

ej0K

� �þE0

M�1M�0

ej1

ej0K�

ei1

ei0

� þ �¼

1

1þri0�

K

1þrj0þVP Kð Þ:

Proof of Proposition 7. Using the same proof as in Gabaix(2012, Theorem 1), the price of the stock in the internationalnumeraire (the traded good) is:

PDit ¼ Dit

1þ HbDit

rDiþ�HDi

rDi:

Hence, expressed in the domestic currency, the price is:

PDit ¼PDit

eit¼ dit

1þHDit

rDiþ�HDi

rDi. Away from the continuous time limit,

the price of the stock is:

PDit ¼ dit

1þ exp �rDi�hDi�ð Þ

1�exp �rDi��HDi

� � bHDit

1� exp �rDið Þ:

The Model with Nominal Prices. The inflation process is as inGabaix (2012), so we can take results from that paper. Let

Qit ¼ Qi0Qt�1

s¼0 1� �isð Þ be the value of money (the inverse of theprice level) in country i. Using the LG results, the expected valueof one unit of currency T periods later is:


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EtQi;tþT

Qit

�¼ 1� �i�ð Þ

T 1�1� exp ��iTð Þ

1� exp ��ið Þ

�it � �i�

1� �i�

� ;ð50Þ

or EtQi;tþT

Qit

h i¼ exp ��i�Tð Þ 1� 1�exp ��iTð Þ

��i�it � �i�ð Þ

� �in the con-

tinuous-time limit.The time t price of a nominal bond yielding one unit of cur-

rency at time t + T is ~Zt Tð Þ ¼ EtM�

tþTetþTQi;tþT

M�t etQit

h i. Because we assume

that shocks to inflation are uncorrelated with disasters, the pre-sent value of one nominal unit of the currency is:

~Zt Tð Þ ¼ EtM�tþTetþT

M�t et

�Et

Qi;tþT

Qit

�:ð51Þ

Proof of Proposition 8. The derivation of the forward rate is asin Gabaix (2012), Theorem 2 and Lemma 2, using equation (51).

Proof of Proposition 9. We start with the case of the regressionin a sample that does not contain disasters. So up to second-orderterms in bHt and �it, using the fact that (up to o �tð Þ terms), thechange in the nominal exchange rate is the sum of the change inthe real rate plus inflation differential,

ENDt

~ei;tþ1� ~eit

~eit�

~ej;tþ1� ~ejt

~ejt

�¼��H

reþ�H

bHit�bHjt

� �� it��jt

� �þo �tð Þ

:¼a bHit�bHjt

� �þb �it��jt

� �þ cþo �tð Þ

~rit� ~rjt¼�re

reþ�H

bHit�bHjt

� �þ �it��jt

� �þo �tð Þ

:¼A bHit�bHjt

� �þB �it��jt

� �þCþo �tð Þ;

hence


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~ ND¼ �

Cov ENDt

~ei;tþ1 � ~eit

~eit�

~ej;tþ1 � ~ejt

~ejt

�; ~rit � ~rjt

� Var ~rit � ~rjt

� �¼ �

aAVar bHit �bHjt

� �þ bBVar �it � �jt

� �A2Var bHit �

bHjt

� �þ B2Var �it � �jt

� �¼ �~�

a

A� 1� ~�ð Þ

b

B¼ ~� ND þ 1� ~�;

where ~� :¼ 1

1þB2Var �it��jtð Þ

A2Var H it�H jtð Þ

:

The case of the full sample regression is proved similarly.

Harvard University

New York University

Supplementary Material

An Online Appendix for this article can be found at QJEonline (qje.oxfordjournals.org).

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THE QUARTERLY JOURNAL OF ECONOMICS · THE QUARTERLY JOURNAL OF ECONOMICS Vol. 131 February 2016 Issue 1 RARE DISASTERS AND EXCHANGE RATES* Emmanuel Farhi and Xavier Gabaix We propose