Currency Risk and Central Banks

Andrew Lilley and Gianluca Rinaldi∗

July 9, 2019

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Abstract

The correlation of currency carry returns with global equity markets increased

sharply after the recent financial crisis. Popular carry trades returned an average

4 percent per year both before and after the crisis, yet these strategies were uncor-

related with equity returns prior to 2008. To rationalize this change, we describe a

simple framework in which central banks adjust interest rate spreads to absorb

changes in currency risk. Before the financial crisis, time variation in spreads

dampened the comovement of currencies and equity markets. Afterward, as the

zero lower bound and macroeconomic concerns constrained central bank policy, the

correlation increased. We document changes in the relationship of both interest

rate spreads and foreign exchange rates with equity markets which are consistent

with this framework.

∗Harvard University: andrewlilley@g.harvard.edu, rinaldi@g.harvard.edu. We are grateful for com-

ments and suggestions from Manuel Amador, John Campbell, Tiago Florido, Xavier Gabaix, Gita

Gopinath, Robin Greenwood, Samuel Hanson, Eben Lazarus, Matteo Maggiori, Ian Martin, Hélène

Rey, Kenneth Rogoff, David Scharfstein, Andrei Shleifer, Jesse Schreger, Adi Sunderam, Erik Stafford,

Jeremy Stein, Argyris Tsiaras, Rosen Valchev, Luis Viceira, and the participants at the Harvard Finance

and International Economics lunches. We thank Refet S. Gürkaynak for providing us with high frequency

data on asset prices around Federal Reserve meeting announcements.

1

1 Introduction

Interest rate spreads and exchange rates are jointly determined. Central banks can affect

interest rates directly, and so their behavior influences the dynamics of foreign exchange

rates. In this paper, we document a sharp increase in the correlation of foreign exchange

and equity markets after the recent financial crisis and show how the conduct of monetary

policy can rationalize this change. In the two decades prior to the zero lower bound

regime, central banks of risky currencies like the Australian Dollar would increase their

rates relative to the US dollar in response to an increase in the price of risk. The opposite

was true for central banks of safe currencies, like the Japanese Yen. We argue that this

helps explain the low measured beta on each of these currencies before the recent crisis,

as central banks were absorbing changes in the price of risk by providing risk premia via

spread. Once central banks become constrained at the zero lower bound, spreads became

invariant to changes in the price of risk, and large betas emerged.

We document a new fact: conditional market betas of all developed market currencies1

display a structural break around the recent financial crisis, as shown in figure 1. An

interpretation of this finding is that currencies have become more risky and therefore

carry larger expected returns. An alternative, in line with the framework we outline, is

that the riskiness of currencies has not changed substantially, but rather we are observing

a period in which central banks are less responsive to changes in the riskiness of their

currencies. In particular, they might have been constrained by other objectives in the

way they set interest rates, or even more starkly, have been stuck at the zero lower bound

and hence unable to respond to changes in currency risk.

This interpretation has several testable implications. Firstly, the pre-crisis returns on

a currency should predict its post-crisis market beta, which we show to be true empirically.

Secondly, since many of the interest rate spreads have been substantially smaller in

the period following the financial crisis, most of the expected returns to carry trades1For each of the G10 currencies versus the US dollar, we compute their yearly rolling beta with respect

to the S&P500, a broad US equity index, which we use as a proxy for the market portfolio

2

1982 1986 1990 1994 1998 2002 2006 2010 2014 2018

−0.4

−0.2

0.0

0.2

0.4

0.6AUDNZDSEKNOKCADEURGBPCHFJPY

Figure 1: Betas estimated from a regression of the daily log appreciation of each G10 currencywith respect to the US dollar against the daily log return of the S&P 500 in US dollars.

∆ log eit = αi,t + βi log(Rmt ) + εi,t

A positive value for ∆ log eit reflects an appreciation of the foreign currency. Each beta isestimated using one year (252 trading days) of historical data, with one coefficient estimatedper currency per month. Data is from Jan 1981 to Dec 2017, collected from Bloomberg.

should come from expected currency appreciation. We show that before the financial

crisis (1981-2008), a regression of currency returns on contemporaneous market return

interacted with the conditional beta of each currency has little explanatory power, in line

with the well known Meese and Rogoff (1983) puzzle. However, in the post-crisis (2009-

2017) sample, around 24% of the variation in monthly currency returns can be explained

by interacting the conditional beta of each currency with the contemporaneous US stock

market return.

Another important building block of our framework is that central bank policy rates

respond to changes in risk premia, at least before the financial crisis. In order to test this,

we regress changes in interest rate spreads for the G10 currencies against the returns on

the S&P 500. In the period in which central banks were less constrained, we find that

spreads on risky currencies tend to increase in periods of negative equity returns, while the

3

spreads associated to safer currencies are either uncorrelated with stock market changes

or exhibit tightenings in equity downturns. We perform a simple back of the envelope

assessment of the quantitative importance of our results and find that we can account for

33% of the time variation in carry trade expected returns due to changes in the price of

risk.

We now turn to a simple explanation of this framework, and then develop a basic

model of joint interest and exchange rate determination to formalize it in section 4. In

section 2, we place our findings in the context of the existing literature. In section 3 we

test the implications of our framework, while section 5 concludes. We report additional

empirical results as well as an extended version of the model in the appendix.

1.1 Descriptive framework

Consider investing in a risky currency and funding the trade by borrowing in a safe

currency. The expected return, or risk premium, on this carry trade can be decomposed

into interest rate spread and expected relative currency appreciation. If a central bank

responds to changes in the risk premium of its currency by adjusting the interest rate

that it pays on deposits, investors are compensated for holding the currency through its

spread, and will accept a relatively higher exchange rate. If instead interest rate spreads

are unresponsive to risk changes, then the spot exchange rate adjusts to compensate

investors through expected appreciation.

As an example, consider the Australian dollar in two important historical episodes:

the stock market crash of 1987 and Lehman’s 2008 bankruptcy.

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Figure 2: AUDUSD and interest rate spread, and the S&P 500 in two equity market crashes.The left panel covers two months around Black Monday 1987 and the right covers the 2008Lehman bankruptcy. Left axes are the AUDUSD and S&P 500 cumulative returns in percent-ages, and the right axes measure the change in the AUD 3M - US 3M spread in basis points.

The stock market declined substantially in both periods, increasing the required com-

pensation per unit of risk,2 and therefore the risk premia for all risky payoffs, including

those of the Australian dollar carry trade. If central banks are not expected to offset those

changes by moving interest rates, exchange rates contemporaneously adjust in order to

allow for expected appreciation. After the crash of 1987, the US central bank immediately

eased monetary conditions, lowering the effective federal funds rate by more than 100bps

over the subsequent two days, while the Australian central bank raised the interbank rate

by 150bps. The market expected this change to be sustained - the three month interest

spread between the two currencies widened by as much as 300 basis points, while the ten

year spread increased by 100bps. During this panic episode, the Australian dollar did

not depreciate materially, and did not move together with the S&P 500. Conversely, as

evidenced by the muted move in the spread in the fall of 2008, central banks were not

expected to respond to this risk premium shock: the Australian dollar suffered a dramatic

depreciation of around 20% in this two month period, mirroring the return on the S&P2This would be the case in many standard asset pricing models, such as external habit formation

(Campbell and Cochrane (1999), Verdelhan (2010)) or time varying disaster risk (Barro (2006), Gabaix(2012), Wachter (2013)). The fact that the price of risk increased in those two periods is also confirmedempirically by standard valuation measures (Campbell and Shiller (1988), Lettau and Ludvigson (2001))and by more recent predictors based on option prices (Martin, 2017).

5

500.

If the expected return associated with a unit of risk in a given currency changes with

the value of the market portfolio, a regression of currency returns on market returns

delivers non zero betas. Even though the riskiness of currencies might not be due to

their market beta, as would be the case if the CAPM held, to the extent that central

banks do not completely offset changes in risk attitudes, exchange rates will still move

with the market portfolio. In particular, risky currencies display positive betas while safe

currencies have negative ones.

Therefore, when comparing two currencies, using the interest rate spread as a proxy

for their risk difference can be misleading if the two central banks behave differently.

For instance, suppose two central banks set interest rates only based on their internal

macroeconomic conditions and are both stuck at the zero lower bound. The interest rate

differential is zero but this does not imply that the two currencies are exposed to the

same amount of risk.

From this perspective, the quantitative empirical failure to explain carry trade returns

with measured currency betas is also not surprising. During times in which central banks

are concerned about other objectives than currency stability, large betas will emerge and

currencies will display more expected appreciation: the expected return to holding risky

currencies comes through expected appreciation rather than the interest rate differential.

Conversely if central banks try to offset risk variation, currency betas are small, and

currency appreciations are both unpredictable and unexplainable by a traditional capital

asset pricing framework. This also implies that we can observe large variation in interest

rate spreads and in the conditional market betas of currencies even if their intrinsic

riskiness has not changed: a change in the behavior of central banks is sufficient.

A large literature has sought to explain why non-US central banks use interest rate

policy partially to stabilize their currencies. The more open the economy, the more likely

it is that the inclusion of their exchange rate in their Taylor rule can reduce inflation

volatility (Taylor, 2001). Moreover, separately from the direct effects on the inflation

6

targeting mandate, exchange rate misalignments which deviate prices from the law of one

price result in inefficient allocations, creating a trade-off between inflation and exchange

rate targeting (Engel, 2011); depreciations cause adverse balance sheet effects due to

the prevalence of dollar liabilities (Calvo and Reinhart, 2002). Thomas Mertens et al.

(2017) show that a fiscal policy which appreciates one’s own currency in bad times will

raise the capital-labor ratio of a country by lowering its risk premium mechanically. In a

similar vein, we consider the dollar as the global unit of account, and introduce a simple

model where non-US central banks attempt to smooth movements in their exchange rates,

driving the covariance of exchange rates and risk factors toward zero.

7

2 Literature Review

This paper bridges the literature on currency risk premia with the literature on central

banks’ de facto management of exchange rates against the US dollar.

Given the widely documented failure of uncovered interest rate parity (Fama (1984),

Engel (1996), Chernov and Creal (2018), Valchev (2019)), researchers have attempted to

link the returns to the carry trade with traditional risk factors.3 A classic approach has

been to sort currencies into portfolios by their interest rate level in order to capture the

conditional risk within currencies (Lustig and Verdelhan, 2007). The returns on those

portfolios have been linked to their CAPM beta, which showed that high interest rate

currencies displayed a positive beta, but their magnitudes were too low to justify the

expected returns on the carry trade (Lustig and Verdelhan, 2007).

Carry trade returns have been better explained using conditional models of risk: carry

trade returns display higher comovements with the market during periods of bad market

returns (Lettau et al., 2014); they are more vulnerable to crashes, and particularly so

when the price of protection against stock market crashes is high (Brunnermeier et al.,

2008; Fan et al., 2019), or when currency options show a higher cost of protecting against

depreciation risk than appreciation risk (Farhi et al., 2015); the risk premium in the

dollar, vis-à-vis the currencies of the rest of the world, is lower in U.S. recessions, when

the price of risk is high (Lustig et al., 2014); a currency’s average covariance with a

broad dollar index has predictive power for its return, even after accounting for its carry

factor loading (Verdelhan, 2018); as is the case with the equity market, currencies which

depreciate during periods of low cross-sectional foreign exchange correlations have positive

excess returns (Mueller et al., 2017); currencies with a high cost of insuring against

volatility appreciate relative to those with a low cost, and their predicted returns are

largely not spanned by interest rate differentials (Della Corte et al., 2016); the carry factor3A recent theoretical literature has focused on the time variation in currency risk premia and hence

on justifying violations of uncovered interest parity. For instance, see Verdelhan (2010) for a treatmentusing habit formation preferences, Colacito and Croce (2011) for a justification based on time variationin long-run risk, and Farhi and Gabaix (2016) for an approach using time-varying country resilience torare disasters.

8

is subsumed by trailing economic momentum (Dahlquist and Hasseltoft, 2019); countries

with more cyclical budget surpluses have currency returns which are more predictable by

the carry factor (Jiang, 2019).

More recently, Kremens and Martin (2018) construct a measure of expected currency

appreciation from derivative prices and verify that it strongly predicts currency returns

in their 2009-2015 sample of developed country currencies. Their results are consistent

with our view that, after the financial crisis, compensation for currency risk mostly came

from expected currency appreciation. Calomiris and Mamaysky (2019) demonstrate this

even more strongly – in this post-crisis sample, the carry factor has predicted negative,

rather than positive, returns.

We also contribute to the literature on foreign exchange stability as a de facto objective

of monetary policy, broadly reviewed in Ilzetzki et al. (2017). In particular, we focus on

the impact of central bank behavior on traditional measures of currency risk and the

expected appreciation of currencies. Central banks have an objective of smoothing their

exchange rates, and tend to lean against foreign currency flows using their own foreign

exchange reserves - a fact documented in Fratzscher et al. (2018), and motivated by

Amador et al. (2017). We consider the parallel role of using their policy rate to this end,

as described in Taylor (2001). In related empirical work, Inoue and Rossi (2018) show

that central banks can influence their currencies through monetary policy shocks which

depreciate their currencies via expectations of future policy spreads. Also in this domain,

? considers the interplay of central bank and fiscal policy in determining carry trade

returns.

We add to a nascent literature on the impact of the zero lower bound on asset prices.

With respect to currencies, Ferrari et al. (2017) document that monetary policy shocks

have had larger impacts in the era of low rates, while Inoue and Rossi (2018) show the

impact of unconventional policy is mostly associated with changes in expectations of

future monetary policy spreads. In other asset classes, recent work by Datta et al. (2018)

links the constraint of the zero lower bound to a significant change in the correlation of

9

US equities and oil prices; Ngo and Gourio (2016) find a similar sign reversal between

US equities and inflation swaps; and Bilal (2017) documents the decrease in correlation

between stock and nominal bond returns. In related theoretical work, Bilal (2017) and

Campbell et al. (2018) associate these changes to shifts in central bank policy.

Our work complements a growing literature on the specialness of the US dollar, to

which our contribution is in documenting foreign central banks’ preference for currency

stability against the US dollar specifically. Various authors have documented this special

role in the form of a lower return on dollar denominated assets, including work by Ca-

ballero et al. (2008), Mendoza et al. (2009), Gourinchas et al. (2010), Maggiori (2017),

Farhi and Maggiori (2018). An assumption underlying our empirical exercises is that the

price of risk can be related to US dollar asset prices. Previous work has demonstrated

the dollar’s role as a global unit of account, as in Chahrour and Valchev (2018), and as

a store of value and unit of invoicing Gopinath and Stein (2018), alongside work which

documents the dominance of the dollar in the denomination of financial assets (Maggiori

et al., 2018).

We also contribute to the long-standing literature on the explicability of exchange rate

movements. Meese and Rogoff (1983) show that exchange rates are unforecastable, but

also that they are hard to explain even including contemporaneous information on asset

returns, a broad literature which is reviewed in (Rossi, 2013). We argue that prior to

2008, beta provided a poor estimate of currency risk and so this result is not surprising.

On the other hand, when most central banks are constrained by objectives other than

currency stabilization, the return on the stock market explains a substantial proportion of

currency movements at monthly frequencies. We concord with the assessment of Itskhoki

and Mukhin (2017) that only shocks to the price of risk can successfully explain most of

exchange rate variation.

Other related work includes the measurement of correlations between foreign curren-

cies with global equity markets for the purpose of optimal portfolio construction (Camp-

bell et al., 2010). Our framework also provides an explanation for the puzzle documented

10

by Shah (2018), wherein high interest rate currencies tend to depreciate the most under

contractionary Fed shocks, in tandem with the largest increase in yields. Shah shows this

joint reaction is a puzzle in standard macroeconomic models of exchange and interest

rates; we extend upon this result in section 3.4.

11

3 Empirical evidence

We present empirical evidence consistent with our framework in four parts. We begin

by describing the data. We then document a structural break in measured betas around

the start of the great recession, and show that the betas of these currencies during the

post-crisis regime are predicted by their carry trade returns prior to it. Next, we turn

to showing that spreads on risky currencies increased in response to changes in the price

of risk in the period before the recent financial crisis. We also examine the responses of

currencies and interest rates to changes in the price of risk using high frequency shocks

around Federal Reserve Open Market announcements. Finally, we reexamine the Meese

and Rogoff (1983) puzzle in light of our findings by showing that measured betas together

with contemporaneous market returns fail to explain currency changes when central banks

have flexibility in interest rate setting, but provide significant explanatory power during

the zero lower bound regime.

3.1 Data

The three core pieces of data for this analysis are exchange rates, short term interest rates,

and the S&P 500 index. S&P 500 and currency data are collected at daily frequencies,

while the data on interest rates is collected at a monthly frequency. For the section on

high frequency FOMC shocks, we also collect intra-day exchange rate data as detailed in

appendix A.5.

We focus on the most traded currencies, according to the Bank of International Set-

tlements Triennial Surveys, commonly referred to as the G10. From 1995 to 2016, the

US dollar, Euro4, Japanese Yen, British Pound, Swiss Franc, Australian Dollar, Cana-

dian Dollar, Norwegian Krona, Swedish Krona, and New Zealand Dollar represented an

outsized share of global foreign exchange turnover, accounting for 95 percent of annual

foreign exchange turnover, whereas every other currency occupied less than an average 14Prior to the introduction of the Euro in 1999, we use the Deutsche Mark in its place.

12

percent of turnover over this horizon. Our analysis focuses on these currencies since they

should most reliably respond to changes in the price of risk at high frequencies.

For each of the aforementioned currencies, we collect daily exchange rate data from

Bloomberg, measured at the foreign exchange market closing time of 5pm EST. We collect

data on the S&P 500 from Yahoo Finance, measured at the market close of 4pm EST.

We collect 2 year government bond yields from Global Financial Data.5

For the section on high frequency FOMC announcement shocks, we use data on

changes in currencies and the S&P 500 collected from 15 minutes before, and 45 minutes

after, Federal Reserve Open Market Committee announcements. Since government bond

data is not available at such a high frequency, we use 2 day changes in government bond

yields collected from Bloomberg. We are grateful to Refet S. Gürkaynak for sharing

with us data on the S&P 500 index and Fed Fund futures around FOMC announcement

windows, hand collected from the Chicago Mercantile Exchange. We compile high fre-

quency exchange rate data using tick data sourced from HistData.com, and minute level

exchange rate data sourced from Forexite.com.

Descriptive statistics and further details regarding data availability and sample con-

struction are provided in the appendix sections A.1 and A.5 respectively.

3.2 Currency betas

We document a sharp increase in the betas of major currencies with global equities during

the period in which interest rates have been constrained by the zero lower bound. To

establish this fact, we estimate the conditional CAPM beta of each currency using daily

data, in a rolling regression.

We estimate this beta for every currency pair with the US dollar at the end of every

month from January 1982 to December 2017, and display the time series of estimated

betas for each currency pair graphically in figure 1. The clear break in the magnitude5No other source provides data on bond or swap yields for the majority of these currencies earlier

than 1993, and currency forwards data for most of these currencies begins between 1993 and 1995.

13

of betas is apparent at the start of 2008, after which each of these currencies measures

its largest beta over the entire sample. The break is equally stark when measured as

correlations, shown in the appendix.

In our framework, the increase in beta of each currency does not have the traditional

CAPM interpretation of an increase in riskiness, as the measured beta is dependent on

the quantity of risk in each currency, the price of risk, and the behavior of its central bank.

Particularly in this instance, the stark increase in betas, which occurs for all currencies

at the onset of the crisis, can be explained by central banks being unwilling or unable

to offset changes in their currencies’ risk premia by adjusting interest rates. For some,

their exchange rate stabilization objective was overshadowed by macroeconomic concerns

during this phase. For others, even more starkly, their optimal interest rate was well

below zero, hence the zero lower bound prevented any adjustment of spreads to changes

in the price of risk.

1987 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2017

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AUDNZDSEKNOKCADEURGBPCHFJPY

Figure 3: Spread over US Treasuries for each of the G10 government bonds, at the 2 yeartenor. Data is from Apr 1987 through to Dec 2017.

Seen this way, the time series of currency betas is an opposite counterpart to the

familiar pattern in yield spreads to the Federal Funds rate over this time horizon. As

14

shown in figure 3, spreads were large and time varying prior to the crisis, but have

converged towards zero. More importantly, the time variation in spreads is now much

smaller - these have become fairly constant since the onset of ultra-low interest rates.

Prior to 2008 the average absolute change in spreads was 28bps per month, whereas it

has fallen to an average of 13bps per month since 2009.

In figure 4, we show that the pre-crisis return on each currency predicts its post-crisis

market beta. Prior to the crisis, investors were predominantly compensated by spread,

but in the last decade, this compensation has switched to expected currency appreciation.

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Figure 4: The horizontal axis shows the average annual return to the carry trade versus theUS dollar in the pre-crisis sample (Jan 1986 to Dec 2007). The vertical axis shows the averageestimated beta for each currency, in the pre-crisis and post-crisis (Jan 2008 to Dec 2017) samples.

The zero lower bound is not the only mechanism by which central bank policy can

become constrained, but it is a particularly clean one. In support of our interpretation,

we show this increase in betas is only apparent for developed economy currencies whose

central bank policies were constrained. Emerging market economies’ monetary policies

were not constrained by the zero lower bound, as they have higher natural nominal

15

interest rates. In the appendix figure 9, we show that the betas of currencies of Brazil,

India, Mexico, Turkey and South Korea have either increased gradually or stayed flat

over the last two decades, but do not display a similar structural break in 2008.

3.3 Spreads and the price of risk

We now show that changes in the price of risk were reflected in changes in bond spreads,

prior to this period. A key component of our model is that central banks act in part to

dampen changes in their currencies which occur in response to changes in the price of risk.

Take for example, the exchange rate between the Australian dollar and the US dollar,

two currencies underlying a popular carry trade position. Suppose the risk premium in

Australian dollars rises because of an increase in the price of risk. The Australian central

bank is faced with a choice: they can allow the exchange rate to depreciate, so that it

has room to appreciate in expectation, or they can increase the cash rate to offset that

risk. If they increase interest rates, or markets anticipate they will, then investors can be

compensated by spread, such that the currency does not need to depreciate by as much

as it otherwise would have if the central bank were not expected to take any action.

According to our framework, following an increase in the price of risk, investors would

anticipate that the Australian central bank will set interest rates higher to compensate

investors via spread. The fact that central banks react at a much lower frequency than

changes in the price of risk creates an empirical challenge. Notwithstanding this, if

carry trade positions are relatively sticky and investors anticipate that central banks will

react in the future, or if carry trade investors hold short-dated bonds, then they can be

compensated by future policy rate changes.

We measure investor expectations of central bank intentions by the yields on 2 year

government bonds for each currency, and construct spreads to the corresponding US

Treasury yield. For each currency, we regress changes in the 2 year spread on changes in

the price of risk, proxied by S&P 500 returns. We use monthly changes in bond yields

for two reasons. Firstly, daily data is not available for most of these currencies prior to

16

the early 1990s. Secondly, we cannot observe these prices at the same cutoff time, as

the yields on the bonds are measured with respect to local market closing times.6 We

perform this exercise both for periods in which central banks were constrained and for

periods in which they were not, reporting the results in figure 5: the dark bars refer to

the unconstrained period and the light bars to the constrained.

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NZD AUD SEK NOK CAD EUR GBP CHF JPY

Bet

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Unconstrained Constrained

Figure 5: Regression coefficients of monthly currency appreciations against the US dollar onthe monthly return on the S&P 500, by period:

∆(rit − r$t ) = αi + βi,unc logRmt + βi,con logRmt + εi,t

where rit is the yield on the 2 year government bond of country i, r$t is the yield on the USD 2

year government bond yield, logRmt is the log appreciation of the S&P 500 over the month. Wedefine a month to be constrained if it is either after 2008, or if the central bank was operating atthe zero lower bound before 2008, as has been the case for Japan (from 1998) and Switzerland(from 2003 to 2004). The dark bars correspond to estimates of βi,unc and the light to βi,con.The sample is from Jan 1987 to Dec 2017.

During the period in which central banks were unconstrained, we observe opposing

behavior between central banks whose currencies are bought on the long side of the carry6Using monthly changes in spreads, the difference in cut-times, of up to 16 hours, is minimized.

Currency forward data, which does not suffer a cut-time problem, cannot be used for our sample, due tothe time series length.

17

trade (such as the Reserve Banks of Australia and New Zealand), and those on the short

side (such as the Bank of Japan and the Swiss National Bank). In months where we

observe declines in the value of equities, we see yields rise on Australian government

bonds by more than US government bonds at the 2 year tenor, while bonds in Japanese

yen decline by the most.

This result is particularly surprising considering the confounding effect of changes in

global growth. Whilst changes in the price of equities convey information about global

growth alongside the price of risk, the component relating to growth prospects works

against the result - we would anticipate the central banks of the commodity currencies,

Australia and New Zealand, to ease monetary conditions the most when equity prices are

falling. Rather, we find the goal of exchange rate smoothing takes precedence, and they

do the opposite.

Table 1: Regression of monthly changes in the market expected return lower bound oncontemporaneous returns in the S&P 500:

EPt+1 − EPt = α+ β ·Rmt+1 + εt+1

The data on the equity premium lower bound is obtained from the online supplemental materialfor Martin (2017) and the sample is from Jan 1996 to Dec 2012. ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

Equity PremiumS&P 500 −0.141∗∗∗

(0.012)

Constant 0.001(0.001)

N 192R2 0.402

While the magnitudes of the coefficients in figure 5 might seem small, one should

compare them to changes in risk premia associated with equity price changes. Of course,

since changes in risk premia are hard to measure, this is not a simple task. Nevertheless,

we can use the option measure introduced by Martin (2017). As shown in table 1, a 1%

decrease in the S&P 500 is associated with a 14 basis points increase in the risk premium

18

of the market.7

This estimate should be rescaled by an estimate of the relative “quantity of risk" of

carry trades as compared to the equity market. Over the whole sample, shorting the

USD to buy AUD returned around 2% a year on average, while estimates of the equity

premium are around 6% a year.

Taking each of these numbers at face value, a 1% fall in the S&P 500 is associated

with an estimated increase of around 4.5 basis points in the expected return of this carry

trade.8 Our estimate of the change in 2 year spread is closer to 2 basis points. Given

the omitted variable concerns discussed above and the fact that the magnitudes of the

yield responses are larger in the high frequency exercise discussed in the next section, we

consider this back of the envelope calculation to be consistent with our proposed view: the

variation in expected central bank policy rates was responsible for a substantial fraction

of carry trade risk premia changes before the recent crisis.

3.4 High frequency FOMC shocks

We use changes in the price of risk over high frequency windows around FOMC an-

nouncements as a means to test these hypotheses with a higher degree of power. For

the empirical test in the previous section, we used monthly changes in yields and in the

price of risk. One drawback of this approach is that currencies, interest rate spreads, and

the price of risk are all reacting to other news. While we cannot account for all such

news, we can instead use changes in these variables around FOMC announcements, as

these windows have been shown to be associated to large changes in the price of risk, at

a time when the impact of other macroeconomic news is small to nonexistent (Lucca and

Moench (2015), Bernanke and Kuttner (2005)).

We confirm our results for the reaction of currencies and equities to the price of risk7The measure is actually a lower bound on the equity premium, but Martin (2017) argues the bound

is tight and that the time series can be a useful proxy for the time variation in risk premia8To obtain this, we simply multiply our estimate of the change in equity risk premium by the ratio

of the unconditional returns: 2%6% × 16bps ≈ 4.5bps

19

in those windows. The first regression specification pertains to currency reactions; we

regress the log appreciation of the foreign currency, measured in dollars, against the

log appreciation of the S&P 500, over the 15 minutes before and 45 minutes after the

FOMC announcements, run separately for each 9 currency pairs. The second measures

the inferred reaction of foreign central banks to the price of risk; we use the change in

the 2 year bond yield as the dependent variable.9 For both specifications, we control for

the direct effect on foreign currencies and yields stemming from changes in the expected

path of monetary policy in the US. These controls are the implied basis points change

to the effective federal funds rate in the current and the next three FOMC meetings,

derived from federal funds rate futures changes over these windows. We describe the

FOMC meeting coverage in the appendix, and show that the results are similar when we

do not control for changes in the expected path of the federal funds rate.

Here we focus here on the market responses of interest and exchange rates in both

periods. In line with our earlier results, we show in figure 6 that currency reactions

are indeed significantly larger during the zero lower bound regime, given the inability

of central banks to dampen their changes. Moreover, the market’s inferred reaction of

central banks to the shock is confirmed to be smaller, as documented in figure 7. The

betas for each currency are ordered along the horizontal axis according to their quantity

of risk, as measured by their average carry trade return, as in figure 4.

We note that the market’s inferred responsiveness of central bank policy to the change

in the S&P500 is largest for the high risk currencies, such as the NZD and AUD, and

insignificant for the currencies which have the lowest quantity of risk and have provided

little carry trade return against the US dollar - the CHF and JPY. That is, the curren-

cies which depreciate most against declines in the S&P500 invoke the most aggressive

responses of their central banks to protect the currency. The exception to this rule is

the CAD, which displays a small responsiveness to changes in the S&P500, given its9Since many of these bond markets are not open during the FOMC announcement window, we use

the 2 day change in bond yields as the dependent variable, while using the one hour change in the S&P500 to ensure we are still using a high frequency shock free of other macroeconomic news as our sourceof variation.

20

−0.3

0.0

0.3

0.6

0.9

NZD AUD CAD SEK NOK EUR GBP CHF JPY

Bet

a

Unconstrained Constrained

Figure 6: Regression coefficients of currency appreciations against the US dollar on the returnon the S&P 500 over one hour windows around FOMC announcements. The return on theS&P 500 is interacted with a variable indicating whether this meeting occurred after January2009, resulting in pre-crisis and post-crisis coefficients. The regression specification is:

∆ log eit = αi + βi,unc logRmt + βi,con logRmt + γiXt + εi,t

where ∆ log eit is the log appreciation of currency i in US dollars, logRmt refers to the logappreciation of the S&P 500 equity index in the hour surrounding the FOMC announcement,and Xt are controls for the direct change in Fed monetary policy expectations. Currencies areordered along the horizontal axis by decreasing risk, as measured by their average pre-crisiscarry trade return. Further details on data construction and sample coverage are provided inthe appendix.

comparatively high quantity of risk in this set of currencies.

Our framework also provides a simple way to understand the puzzling fact documented

by Shah (2018). He shows that high interest rate currencies tend to depreciate the most

under contractionary Fed shocks whereas their 10 year yields display the largest increase.

These facts are puzzling in standard complete markets models of international finance,

since the currency reaction suggest stochastic discount factors rise the most in low-rate

countries, while bond market reactions suggest the opposite. We consider this fact a

natural consequence of our framework. FOMC shocks also change the price of risk, with

21

−12

−8

−4

0

4

NZD AUD CAD SEK EUR GBP CHF JPY

Bet

a

Unconstrained Constrained

Figure 7: Regression coefficients of changes in the 2 year yields of each bond in a FOMCannouncement day on the return on the S&P 500 over an hour window around the FOMCannouncement. The return on the S&P 500 is interacted with a variable indicating whetherthis meeting occurred after January 2009, resulting in pre-crisis and post-crisis coefficients. Theregression specification is:

∆rit = αi + βi,unc logRmt + βi,con logRmt + γiXt + εi,t

where ∆ log rit is the yield of the government bond of currency i, logRmt refers to the log appreci-ation of the S&P 500 equity index in the hourly window surrounding the FOMC announcement,and Xt are controls for the direct change in Fed monetary policy expectations. Currencies areordered along the horizontal axis by decreasing risk, as measured by their average pre-crisiscarry trade return. Further details on data construction and sample coverage are provided inthe appendix, alongside robustness checks with further controls.

a 25bp contractionary shock lowering equity valuations by 100 to 250bps according to

prior research (Bernanke and Kuttner (2005), Chuliá et al. (2010)). The historically high

rate currencies (AUD and NZD) are risky, while the low rate currencies (JPY and CHF)

are safe. Following a contractionary shock, the price of risk rises and currencies depreciate

in accordance with their quantity of risk. Foreign central banks act to offset that risk,

and given identical preferences, the Australian and New Zealand central banks would be

expected to increase interest rates by the most to offset the depreciation. Thus the joint

puzzle is rationalized by the reaction of foreign central banks to Federal Reserve policy.

22

3.5 Relation to the Meese-Rogoff puzzle

The second testable implication of our framework is that for two currencies constrained

at the zero lower bound, return must come through currency appreciation. Since central

banks are constrained from compensating investors for return via spread, currencies will

bear the full brunt of changes in the price of risk. Over this period, changes in the price

of risk will show up as beta for currencies, whether or not CAPM risk is the fundamental

which drives currency returns.

In light of this, we can reinterpret the Meese-Rogoff puzzle in the context of our

framework. The Meese-Rogoff puzzle is not merely that exchange rates are unforecastable

conditional on current information, but that exchange rates remain unaccountable even

including contemporaneous information. Perhaps most surprisingly, currencies remain

unexplainable even after including the contemporaneous return on the market alongside

broad array of financial indicators - a surprising result for risky assets. We argue that prior

to 2008, beta provided a poor estimate of currency risk, even measured at a reasonably

high freqency, and so this result is not surprising. We take our estimates of beta from

the method underlying figure 1, which are estimated using 1 year of historical daily

information, and use them to explain the next 1 month exchange rate appreciation by

interacting this measure of risk with the future return on the market.

As reported in table 2, in the regime prior to the zero lower bound, this specification

has little explanatory power, with an R-squared of 1 percent. In the sample where

currencies are constrained with spreads of zero, compensation for risk comes through

expected appreciation, and we find an R-squared of 24 percent, outperforming models

which include a broad array of financial and economic indicators (Rossi, 2013).

23

Table 2: Panel regression of monthly exchange rate appreciations on a constant and theconditional beta estimated from the year before the start of the month interacted with the S&P500 appreciation in the same month. The regression specification is:

∆ log ei,t+1 = α1 + α2 · βit logRmt+1 + ui,t+1

where βit is the generated regressor of the beta of foreign currency i using daily market returndata over the year prior. ∆ log eit+1 and logRmt+1 are the log appreciation of currency i in USdollars and the return on the S&P 500, in the month subsequent to the window of estimationof the generated regressor βit. The left column reports results for the sample from Jan 1981 toDec 2008 while the right column is for the period from Jan 2009 to Dec 2017. The standarderrors, given in parentheses below, are estimated by a block bootstrap with block size 500 daysto account for the generated regressors. ∗: p<0.1; ∗∗: p<0.05; ∗∗∗: p<0.01.

Pre-Zero Lower Bound Zero Lower BoundConstant 0.0002 −0.002

(0.002) (0.002)

β ×Rm 0.823 1.280∗∗∗(0.841) (0.345)

N 2,888 972R2 0.018 0.243

In table 4 of the appendix, we show this result is not aided by the shorter window of

the post-zero lower bound sample: dividing the pre-zero lower bound sample into three

windows of the same length as the post-zero lower bound sample does not improve the

explicability of exchange rate changes in the pre-period.

24

4 Model

We adapt the variable disasters risk model of Gabaix (2012) to develop a stylized frame-

work for exchange rates. As opposed to the model of Farhi and Gabaix (2016), our model

does not fully specify a macroeconomic Context, but rather takes a reduced form short-

cut to model exchange rates as prices of domestic assets. While lacking an explicit micro

foundation, this reduced form approach allows us to tractably analyze foreign interest

rates that vary in response to the overall resilience of the economy, which is a way of

capturing the central bank behavior described in the empirical part of this paper.

Time is discrete and runs from t = 0 to infinity. The representative agent has CRRA

utility with risk aversion parameter γ > 1.

Ut =∞∑i=0

e−ρ·iC1−γt+i

1− γ (1)

In this endowment economy, at each period, consumption grows at a constant rate

gC unless the disaster state is realized, which happens with constant probability p. If

the disaster realizes at time t + 1, consumption growth is lowered by a random factor

Bt+1 < 1.

Ct+1

Ct= egC

1 if no disaster occurs at t+1

Bt+1 if a disaster occurs at t+1(2)

The stochastic discount factor is, therefore, given by

Mt+1

Mt

= e−ρ−γgC

1 if no disaster occurs at t+1

B−γt+1 if a disaster occurs at t+1(3)

We model each currency as a carry trade asset. The representative agent prices

currencies by computing the net present value of future interest payments obtained by

owning a currency, converted back to dollars. The agent believes these dollar payoffs

evolve according to

Ri,t+1

Ri,t

=

fi(1 + εRi,t+1

) (1 + εEi,t+1

)if no disaster occurs at t+1

fiBt+1 if a disaster occurs at t+1(4)

25

In normal times, the dividend stream is perceived to grow or shrink proportionally

by a factor (1 + εRi,t+1)(1 + εEi,t+1), where the two shocks are independent and mean zero.

In the disaster state, the foreign exchange rate crashes proportionally to fiBt+1, lowering

the future effective dividend in dollars permanently. fi is a constant for each foreign

currency i and indexes the riskiness of a currency: fi < 1 implies a currency which is

more risky than the consumption stream itself.

We can think of εRi,t+1 as the change of the local interest rate of country i and of εEi,t+1

as the currency movement. The riskiness of the consumption claim varies over time as

Bt+1 changes. We introduce the resilience of the consumption claim

Hc,t = pEDt [B1−γt+1 − 1] (5)

where the superscript D indicates that the expectation is conditional on a disaster hap-

pening at time t+1. Following Gabaix (2012) we assume this resilience follows a linearity

generating process

Hc,t = H∗c + Hc,t (6)

Hc,t+1 = 1 + Hc∗

1 +Hc,t

e−φHHc,t + εHc,t+1 (7)

This process is close to an autoregressive process with persistence governed by φc. The

extent of the disaster for currency i, fiBt, is a fixed multiple of the time varying overall

disaster, Bt. Thus, each foreign currency has its own time-varying resilience, but these

are perfectly correlated across currencies:

Hi,t = pEDt [B1−γt+1 fi − 1] = fiHc,t + p(fi − 1) (8)

Using equation 8, we can split this term into a permanent and a temporary component

Hi,t = H∗c + p(fi − 1) + Hi,t (9)

Hi,t = fiHc,t (10)

As usual, Et[εHi,t+1] = Et[εEi,t+1] = Et[εRi,t+1] = 0 but we depart from Gabaix (2012) by

allowing the innovation to foreign interest rates, εRi,t+1 and εHc,t+1 to be correlated. The

covariance is currency specific and denoted by si ≡ Cov(εRi,t+1, εHc,t+1). This is the pa-

26

rameter characterizing the behavior of foreign central banks. A currency with a negative

si will be one for which the central bank tends to increase their spreads relative to the

US interest rate in bad times. We can now provide an expression for the price of the

consumption claim, Pi, defined as the asset that pays off Ct at each period t.

Result 1. The price of the consumption claim is given by

Pt = Ct1− e−δc

(1 + eδc−h

∗c

1− e−δc−φH Hc,t

)(11)

where h∗c ≡ log(1 +H∗c ) and δc = ρ− γgc − h∗c.

Proof. See Theorem 1 in Gabaix (2012).

The exchange rate Ei,t is the time t price of the stream of payoffs defined by 4. This

approach has an important weakness: it does not link back actual changes in Ei,t to εEi,t+1,

which is effectively a departure from rational expectations. In particular, assuming that

the shock εEi,t+1 is independent of εHi,t+1 in the mind of the representative agent means

that variation in the exchange rate is perceived as separate from changes in the overall

resilience of the economy, but in the model the exchange rate is actually correlated with

changes in resilience: the agent is not able to invert the equilibrium exchange rate to

understand where changes are coming from. While this is not the standard approach to

modeling exchange rates, it makes the link between central bank policy and exchange

rates transparent while keeping the model analytically tractable.

Result 2. The foreign exchange rate is given by

Ei,t = Ri,tfie−δi

1− fie−δi − f 2i si(1− p) e

−2(δi+hi∗ )

1−e−δi−φH

(1 + eh

∗i

1− e−δi−φh(Hi,t + fisi(1− p)e−δi+h

∗i

))(12)

where hi∗ ≡ log(1 +Hi∗) and δi = ρ+ γgc − h∗i .

27

Proof.

Et[Mt+1Ri,t+1

MtRi,t

]= e−ρ−γgc

(pEDt [B1−γ

t+1 fi] + (1− p)ENDt [fi(1 + εRi,t+1)(1 + εEi,t+1)])

= e−ρ−γgcfi(1 +Hc,t)

= e−ρ−γgcfi(1 +H∗c + Hc,t)

Et[Mt+1Ri,t+1

MtRi,t

Hc,t+1

]= e−ρ−γgc

(pEDt [B1−γ

t+1 fiHc,t+1] + (1− p)ENDt [fi(1 + εRi,t+1)(1 + εEi,t+1)Hc,t+1])

= e−ρ−γgc(pEDt [B1−γ

t+1 fi]EDt [Hc,t+1] + (1− p)fi·(ENDt [Hc,t+1] + cov(Hc,t+1, ε

Ri,t+1)︸ ︷︷ ︸

=si

+ cov(Hc,t+1, εEi,t+1)︸ ︷︷ ︸

=0

+ENDt [Hc,t+1εRi,t+1ε

Ei,t+1]︸ ︷︷ ︸

=0

)

since the shock to consumption resilience εHc,t+1 is independent of whether a disaster

occurs, EDt [Hc,t+1] = ENDt [Hc,t+1] = Et[Hc,t+1] and the expression above is equal to

e−ρ−γgc(fiEt[Hc,t+1]

(pEDt [B1−γ

t+1 ]− p+ 1)

+ (1− p)sifi)

= e−ρ−γgc(

1 +H∗c1 +Hc,t

e−φHHc,t(Hc,t + 1) + (1− p)sifi)

= e−ρ−γgc(e−φH (1 +H∗c )Hc,t + (1− p)sifi

)Therefore, MtRt(1, Hc,t)t=0,1,... is a linearity generating process with parameters α, κ, γ

and Γ:

Et[Mt+1Di,t+1

MtDi,t

]= α + κHi,t

Et[Mt+1Di,t+1

MtDi,t

Hi,t

]= γ + ΓHi,t

where,

Di,t ≡ Ri,t

α ≡ fie−δi

κ ≡ fie−δi+h∗

i

γ ≡ (1− p)fisie−δi+h∗i

Γ ≡ e−δi−φH

28

and

h∗i ≡ log(1 +H∗i )

δi ≡ ρ+ γgc − h∗i

We can therefore obtain a closed form expression for the price:

ei,t = Di,t

[1

1− α− κγ1−Γ

(α + κ

1− Γ(Hi,t + γ

))]

29

5 Conclusion

In this paper, we documented a large shift in the relationship of currency movements

and risk factors after the recent financial crisis and proposed a simple framework linking

this to central banks’ behavior. Correlations between risky assets and exchange rates

increase when central banks do not adjust spreads in response to changes in risk premia.

We document significant time variation in the betas of major currencies with the S&P

500, and a structural break at the onset of the period in which interest rates have been

constrained by the zero lower bound.

We also showed that interest rate spreads in the period before the financial crisis

tended to move with risk premia in a way consistent with our framework: risky (safe)

currency spreads increased (decreased) with the price of risk. We also show that these

responses can account for a substantial part of currency risk premia variation.

Moreover, we highlighted that while currency appreciations are unexplained by con-

temporaneous equity market returns before the financial crisis, in line with the results

of Meese and Rogoff (1983), this is not the case in the recent post crisis period in which

interest rate spreads across currencies have not reacted to changes in risk premia.

Essential to our framework is the notion that certain currencies are risky while others

are safe. In particular, we assume currencies have a certain quantity of risk but we do

not explain why that is the case. Moreover, we find the quantity of risk of currencies

seems to be time varying. For example, the British pound has the highest beta in the

sample around the time of Brexit and all currencies appear to become less risky relative

to the US dollar from 2012 to 2014. We do not attempt to explain what determines the

intrinsic riskiness of currencies here, but it clearly is the next important step in this line

of work.

30

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A Appendix: data and robustness

A.1 Data descriptive statistics

In table 3 we report the mean and standard deviation of monthly currency appreciations

and spreads, for the entire sample as well as splitting before and after 2008. While FX

standard deviation slightly increased in the post sample, once we remove 2008 and the

first half of 2009 from the sample, the difference in the average standard deviations is

negligible. On the other hand, the standard deviation of spreads is much lower in the

post period.

Table 3: Summary statistics for currency moves and 2 year interest rates spreads.

AUD CAD CHF EUR GBP JPY NOK NZD SEK Mean87 - 08

FX mean -0.07 0.06 0.15 0.16 0.01 0.22 0.03 -0.01 -0.05 0.05sd 3.05 1.63 3.27 2.86 2.95 3.32 2.94 3.50 3.16 2.97

Spread mean 1.79 0.34 -2.29 -0.86 1.08 -3.80 1.14 2.45 1.13 0.11sd 1.85 1.28 1.74 1.92 1.33 1.69 2.25 2.04 2.60 1.86

08 - 17FX mean -0.09 -0.19 0.13 -0.16 -0.32 -0.01 -0.34 -0.06 -0.19 -0.14

sd 4.12 2.97 3.30 3.19 2.76 3.00 3.54 4.24 3.56 3.41Spread mean 2.09 0.14 -1.07 -0.55 -0.11 -0.96 0.64 2.12 -0.22 0.23

sd 1.21 0.50 0.71 0.89 0.62 0.60 0.88 0.92 1.10 0.83AllFX mean -0.08 -0.01 0.14 0.07 -0.08 0.15 -0.08 -0.03 -0.09 -0.00

sd 3.38 2.09 3.28 2.96 2.90 3.24 3.12 3.72 3.27 3.11Spread mean 1.89 0.28 -1.89 -0.76 0.69 -2.87 0.97 2.35 0.69 0.15

sd 1.67 1.09 1.58 1.66 1.28 1.95 1.93 1.78 2.31 1.70

37

A.2 Correlations of currencies and equities

We repeat the main beta graph in correlations, and show the correlation of each exchange

and interest rate in figure 8.

1982 1986 1990 1994 1998 2002 2006 2010 2014 2018

−0.5

0.0

0.5

AUDNZDSEKNOKCADEURGBPCHFJPY

Figure 8: Correlations between the daily log appreciation of each G10 currency against theUS dollar and the daily log return on the S&P 500 in US dollars. Each correlation is estimatedusing one year (252 trading days) of historical data, with one correlation estimated per currencyper month. Data is from Jan-1981 to Dec-2017, collected from Bloomberg.

We also construct the betas for a series of emerging market currencies that did not

set interest rates close to the zero lower bound through the recent financial crisis.

38

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8KRWMXNINRTRYBRL

Bet

a

Figure 9: Betas estimated from a regression of the daily log appreciation of currencies againstthe US dollar on the daily log return on the S&P 500 in US dollars. Each beta is estimatedusing one year (252 trading days) of historical data, with one coefficient estimated per currencyper month. Data is from Jan-1986 to Dec-2017, collected from Bloomberg.

We note from figure 9 that those betas do not display a clear structural break, contrary

to the pattern in figure 1.

39

A.3 High frequency responses around FOMC announcements

We repeat the empirical specifications underlying figures 6 and 7, without controls for

changes in the path of the effective federal funds rate, and report the results in figures

10 and 11.

0.00

0.25

0.50

0.75

NZD AUD CAD SEK NOK EUR GBP CHF JPY

Bet

a

Unconstrained Constrained

Figure 10: Regression coefficients of currency appreciations against the US dollar on thereturn on the S&P 500 over one hour windows around FOMC announcements. The returnon the S&P 500 is interacted with a variable indicating whether this meeting occurred afterJanuary 2009, resulting in pre-crisis and post-crisis coefficients. The regression specification is:

∆ log eit = αi + βi,unc logRmt + βi,con logRmt + εi,t

where ∆ log eit is the log appreciation of currency i in US dollars, logRmt refers to the logappreciation of the S&P 500 equity index in the hour surrounding the FOMC announcement.Currencies are ordered along the horizontal axis by decreasing risk, as measured by their averagepre-crisis carry trade return. Further details on data construction and sample coverage areprovided in the appendix.

40

−10

−5

0

5

NZD AUD CAD SEK EUR GBP CHF JPY

Bet

a

Unconstrained Constrained

Figure 11: Regression coefficients of the changes in the 2 year yields of each bond in a FOMCannouncement day on the return on the S&P 500 over an hour window around the FOMCannouncement. The return on the S&P 500 is interacted with a variable indicating whetherthis meeting occurred after January 2009, resulting in pre-crisis and post-crisis coefficients. Theregression specification is:

∆rit = αi + βi,unc logRmt + βi,con logRmt + εi,t

where ∆ log rit is the yield of the government bond of currency i, logRmt refers to the log appreci-ation of the S&P 500 equity index in the hourly window surrounding the FOMC announcement.Currencies are ordered along the horizontal axis by decreasing risk, as measured by their aver-age pre-crisis carry trade return. Further details on data construction and sample coverage areprovided in the appendix, alongside robustness checks with further controls.

.

.

.

.

41

A.4 Uniform Sample Size Meese Rogoff Regressions

We repeat the Meese Rogoff regressions in table 2 for windows of equal sizes, to demon-

strate the result is not aided by the use of a shorter sample window.

Table 4: Panel regression of monthly exchange rate appreciations on a constant and theconditional beta estimated from the year before the start of the month interacted with the S&P500 appreciation in the same month. The regression specification is:

∆ log ei,t+1 = α1 + α2 · βit logRmt+1 + ui,t+1

where βit is the generated regressor of the beta of foreign currency i using daily market returndata over the year prior. ∆ log eit+1 and logRmt+1 are the log appreciation of currency i in USdollars and the return on the S&P 500, in the month subsequent to the window of estimationof the generated regressor βit. The four columns report results for the period indicated in theirrespective title row. The standard errors, given in parentheses below, are estimated by a blockbootstrap with block size of 500 days to account for generated regressors. ∗: p<0.1; ∗∗: p<0.05;∗∗∗: p<0.01.

82-90 91-09 00-08 09-17Constant −0.0002 −0.001 0.002 −0.002

(0.003) (0.003) (0.003) (0.002)β ×Rm 0.550 0.454 1.068 1.280∗∗∗

(1.503) (0.920) (1.305) (0.345)

N 992 969 927 972R2 0.004 0.005 0.051 0.243

42

A.5 High frequency sample

We focus on FOMC announcement dates from June 2000 to October 2015, which is

the sample for which we have been provided high frequency data on movements in the

S&P500. We use the meeting dates recorded by Lucca and Moench (2015) until 2011,

and and collect the remainder from Bloomberg thereafter.

Currency data: We collect tick-level exchange rate data from HistData.com where

available, and use minute-level data from Forexite.com for the remainder. The below

table summarizes our data sources and windows.

Table 5: Sources of dependent variable data, and sample sizes for regressions of currencyreactions to S&P500 movements during FOMC announcement windows. The following data isapplicable to to the regressions underlying figures 6, 7, 10, and 11.

AUDUSD EURUSD GBPUSDSource HistData HistData HistDataWindow 2001-06/2015-12 2000-06/2015-12 2000-06/2015-12N obs 112 123 121

NZDUSD USDCAD USDCHFSource Forexite HistData HistDataWindow 2003/01-2015 2001-01/2015-12 2000-06/2015-12N obs 102 116 120

USDJPY USDNOK USDSEKSource HistData Forexite ForexiteWindow 2000-06/2015-12 2005-02/2015-12 2005-02/2015-12N obs 121 85 85

Yield data: We collect daily 2-year government bond yield data from Bloomberg.

Data is not available with a constant cut time, as they are measured with respect to each

market’s own bond closing time. For the euro, we use German government bonds. In

order to take measurements over similarly timed windows, we take two day yield changes,

aligning the measurement windows such that we take the change in the yield from the

local market close prior to the FOMC announcement, to the second market close after

the FOMC announcement. For example, for an FOMC announcement which occurs at

14:00EST on a Wednesday, the change in Australian yields is measured from 02:00EST

43

on Wednesday, to 02:00EST on Friday, while the change in Canadian Treasury yields is

measured from 17:00EST on Tuesday to 17:00EST on Thursday.

We make the following sample adjustments. We exclude Norway Government Bonds

due to a paucity of available data - all yield curve points are recorded only intermittently,

and for less than half the sample. We replace the New Zealand 2 year government bond

yield with a predicted yield from a regression of the 2 year government bond yield on the

5 year government bond yield during months where no New Zealand 2 year government

bond was on issue.

Table 6: Sources of independent variable data, and sample sizes for regressions of currencyreactions to S&P500 movements during FOMC announcement windows. The following data isapplicable to the regressions underlying figures 11 and 7.

AUD EUR GBPBloomberg code GTAUD2Y GTDEM2Y GTGBP2Y

Window 2000-06/2015-12 2000-06/2015-12 2000-06/2015-12N obs 123 123 123

NZD CAD CHFBloomberg code GTNZD2Y GTCAN2Y GTCHF2Y

Window 2000-06/2015-12 2000-06/2015-12 2000-06/2015-12N obs 109 123 120

JPY NOK SEKBloomberg code GTJGB2Y - GTSEK2Y

Window 2000-06/2015-12 - 2000-06/2015-12N obs 120 0 121

44

A.6 Multi-period model

The model can be readily extended to any multi-period setting, while retaining the key

results. Here we demonstrate the extension to a three period setting.

pf0

DpfD,1

δ2p

δ1− pp

NDpfND,1

δp

11− p

1−p

Figure 12: Exchange rate determination in a three period model.

In the third period, the exchange rate has lost cumulatively δ with each disaster, and

therefore is 1 with probability (1−p)2, δ with probability 2p(1−p) and δ2 with probability

p2 in the final period. Thus the exchange rates in each state of the second period are

given by:

pfND,1 = (1 + r)(

1− pd−γ(1− δ)1− p+ pd−γ

)(13)

pfD,1 = δpfND,1 = δ(1 + r)(

1− pd−γ(1− δ)1− p+ pd−γ

)(14)

Assuming a constant interest rate across all periods, this gives rise to an exchange

rate at time zero of:

pf0 = a(1 + r)((1− p) · pfND,1 + p · d−γpfD,1)

= (1 + r)

((1+r)δpd−γ(1−p+δpd−γ)

1−p+pd−γ + (1+r)(1−p)(1−p+δpd−γ)1−p+pd−γ

)1− p+ pd−γ

=[(1 + r)

(1− pd−γ(1− δ)

1− p+ pd−γ

)]2

= (pfND,1)2 ≡ (pf )2 (15)

45

where the final equality refers to the initial exchange rate under the two period model.

More generally, the exchange rate at time zero in the n period version of the model

above will be given by

pf0 = (pf )n (16)

The exchange rate in any state in period t < n under the n period model will be given

by

pft = δ∑t

i=1 1(disaster in period i)(pf )n−(t+1) (17)

A.7 Illustrative Model

There are two periods, 0 and 1. In the second period, the good state occurs with proba-

bility 1−p, and the disaster state occurs with probability p. There is a risky asset in unit

positive supply which pays off 1 in state ND at time 1, but only B < 1 in the disaster

state D, as in figure 12.

pe

1p

B1− p

t = 1t = 0

Figure 13: Risky asset payoff

We think of this risky asset as a broad equity index, or a claim to consumption. A valid

stochastic discount factor for time 1 payoffs in this endowment economy is therefore

M = a

1 if no disaster at 1

B−γ if disaster at 1(18)

Where a < 1 is a constant. We make the simplifying assumption that the constant

a is such that E[M ] = 1, which requires that a = 1(1−p)+pB−γ . This is an assumption on

46

the rate of time preference which ensures the risk-free rate in the home (US) economy is

equal to zero, which simplifies the intuition without being essential to the framework or

the results.

There are two other assets in zero net supply: a domestic bond and a foreign bond,

both without default risk. The domestic bond pays off 1 with certainty at time 1. The

payoff of the foreign bond is 1+r units of foreign currency, but the proceeds are uncertain

when measured in domestic currency. We assume that the exchange rate at time 1 is 1 in

the good state and δ in the disaster state, capturing the riskiness of the foreign currency.

A value of δ below (above) 1 indicates a risky (safe) currency. Hence, the domestic

currency payoffs of the foreign bond are

X =

1 + r if no disaster at 1

(1 + r)δ if disaster at 1(19)

The key quantity of interest is the exchange rate at time 0. We define it to be the ratio

of the foreign bond price to the domestic bond price, e0, such that a higher e0 is a more

appreciated foreign currency. The domestic bond price is one by the definition of a, hence

the price of the foreign bond is the exchange rate:

e0 ≡ E[MX] = a(1 + r)(1− p+ pδB−γ) = (1 + r)(

1− pB−γ(1− δ)1− p+ pB−γ

)(20)

The expected foreign currency appreciation is therefore(1− p) + pδ

e0= (1− p) + pδ

a(1 + r)(1− p+ pδB−γ) (21)

Notice that in the case of risk neutral preferences: γ = 0, expected appreciation is 11+r ,

such that UIP holds in expectation.

The price of the broad equity index at time 0 is, similarly

pe = a(1− p+ pB1−γ) = 1− pB−γ(1−B)1− p+ pB−γ

(22)

The foreign central bank sets r internalizing its impact on the exchange rate. We denote

actual and natural output of the foreign economy by yf and yf respectively. The IS curve

47

is given by:

yf = yf − φ(r − rf ) + zf (23)

where zf can be thought of as a demand shock, and φ governs the responsiveness of

demand to interest rates. The zero output gap interest rate is therefore given by zf

φ+ rf .

The central bank’s objective function trades off a zero output gap with its preference

for a stable exchange rate with a relative weight on exchange rate stability denoted by s:

Lf = (yf − yf )2 + s(e0 − E[e1])2

= (φ(r − rf )− zf )2 + s(a(1 + r)(1− p+ pδB−γ)− 1 + p− pδ

)2(24)

Therefore, the optimal interest rate is

1 + r = φ2(1 + rf ) + φzf + sa(1− p+ pδB−γ)(1− p+ pδ)φ2 + sa2(1− p+ pδB−γ)2 (25)

If we eliminate the central bank’s preference for smoothing, s = 0, we obtain r =zf

φ+ rf . Conversely as the preference for smoothing dominates macroeconomic stabil-

ity concerns, s → ∞, the interest rate is set to exactly compensate carry trade risk:

1 + r → 1−p+pδa(1−p+pδB−γ) . We want to understand how the comovement of equity prices

and foreign exchange rates is affected by changes in the objective function of the foreign

central bank. Since we are considering a two period model, we use comparative statics of

e0 and pe with respect to changes in γ at time 0. We interpret changes in γ as changes

in the price of risk.

Note that the covariance of the price of risk and the price of the equity asset does not

depend on the actions of the foreign central bank. The covariance of the foreign exchange

rate and the price of risk, or the equity asset, is highly dependent on the foreign central

bank’s preferences, as captured by the example in figure 14. Being constrained by the

zero lower bound has identical implications for the exchange rate to that of a central bank

which is dominated by macroeconomic concerns (s → 0): interest rate spreads become

unresponsive to changes in the price of risk.

48

0.0

0.3

0.6

0.9

1.2

0 1 2 3 4 5γ

s = 10

0.0

0.3

0.6

0.9

1.2

0 1 2 3 4 5

e0pe

r

s = 0

Figure 14: Effect of the price of risk (proxied by γ) on the price of the equity index pe, thetime 0 exchange rate e0 and the foreign central bank rate r for two values of s. The left panel,for s = 10 represents the actions of a central bank that is focused on stabilizing exchange rates.On the other hand, the right panel represents a central bank with no such concerns. The otherparameters for this calibration are: r = .04, B = .4, δ = .7, p = .1 and φ = .5.

B Model

There are three periods, 0, 1 and 2. In the third period, the good state occurs with

probability 1 − p, and the disaster state occurs with probability p. The probability p is

a fixed parameter. The consumption process is exogenous and simply equals 1 at time 1

and in the good state at time 2, but only B < 1 in the disaster state at time 2. There is

a risky asset in zero net supply which only pays off at time 2 and which we interpret as

the broad equity index. The disaster state payoff F of this asset in the disaster state is

unknown as of t = 0 but is revealed at t = 1. The timing of the model is represented in

the following diagram:

We think of this risky asset as a broad equity index, or a claim to consumption. The

other securities, in zero net supply, are foreign and domestic bonds. On the domestic

side, there are two short term bonds paying off at times 1 and 2 as well as a long term

bond trading at time 0 and which pays off 1 for sure at time 2. There are equivalent

49

Uncertaintyabout F

F isrevealed

NDp

D1− p

t = 2t = 1t = 0

Figure 15: Three period model timing

bonds denominated in the foreign currency. The gross interest rates paid on a one period

bond paying off at time t+1 is denoted as Rcurrt where curr is f for foreign bonds (paying

one unit of foreign currency) and d for domestic bonds. The two-period interest rates are

denoted by Rf and Rd for the foreign and domestic ones, respectively. We denote by Et

the exchange rate: the ratio of the prices of the foreign and domestic short term bonds.

While this is an unconventional way of defining exchange rates, it is natural in a context

in which there is only one representative agent. To capture the riskiness of the foreign

currency, we assume that the time 2 exchange rate, E2, is 1 in the good state ND but δ

in state B.

A representative agent has CRRA utility

for t = 1, 2. Under our notation, ψ is the intertemporal elasticity of substitution and γ

is the relative risk aversion coefficient for static gambles. Since we are considering a three

period model, utility at time 3 can not depend on future expected utility. In order to

emphasize the importance of risk aversion over time 3 payoffs, we assume CRRA utility

over time 3 consumption.

U3 = β2 c1− 1

ψ

31− 1

ψ

. (26)

Domestic interest rates, i.e. Rd1, Rd

2 and Rd are pinned down by the representative

agent Euler equation, we let interest rates on foreign bonds be set by a foreign central

bank. Foreign bonds payoffs need to be converted back into domestic consumption, so

they are risky from the perspective of the representative agent. For instance, buying one

50

unit of the foreign bond at time 1 delivers Rf2 units of domestic consumption at time 2

in the ND state and δRf2 in the disaster state.

Given our definition of the exchange rate and the fact that its time 2 value is deter-

mined exogenously, we can characterize E0 and E1.

The foreign central bank sets interest rates internalizing its impact on those exchange

rates and trades off macroeconomic concerns and exchange rate stabilization. Since our

focus is on the riskiness of exchange rates and not on the determinants of optimal mone-

tary policy in the foreign currency, we simply model the central bank problem at t = 1, 2

as maximizing

Lft = (Rft − Rf

t )2 + s(Et − E[Et+1])2 (27)

Where Rft can be interpreted as the interest rate which would be optimal if exchange

rate smoothing were not important for the foreign central bank. The parameter s is

the relative weight that the central bank assigns to exchange rate stabilization, and is

therefore key to our model. We begin analyzing the case in which s = 0, where the

central bank does not try to smooth its exchange rate. In this case risky currencies will

move together with the risky asset in the expected way: their beta is pinned down by

their "quantity of risk", parametrized by δ.

Proposition 3. If s = 0, the beta of foreign currency bonds and the risky asset is given

byCov(P f

1 , P1)Var(P1) = 1 + (B − δ)K · (1 +Rf ) (28)

where K is a positive constant independent of the riskiness of the foreign currency δ.

Proof. The price of the risky asset at time 1 is

P1 = f(p)(1− p+ pB1− 1ψ ) (29)

while the price of the foreign bond is given by

P f1 = f(p)(1 +Rf )(1− p+ pδB−

1ψ ) (30)

51

where the term

f(p) ≡(

1− 1ψ

){(1− β){(1− β) + β(1− β)[(1− p) + pB1− 1

γ ]1− 1

ψ1−γ }

ψ1−ψ

}−1

.

If s = 0, Rf doesn’t depend on p. The covariance of these two prices is hence given

by:

Cov(P f1 , P1) ≡ Cov(P1 + f(p)B−

1ψ p(δ −B), P1) · (1 +Rf )

=(Var(P1) + (δ −B)B−

1ψCov (f(p) · p, P1)

)· (1 +Rf )

=

= (Var(P1) + (B − δ)K) · (1 +Rf )

Where K = B−1ψ (1−B1− 1

ψ ) ·Var (f(p)p), a positive constant. The second equality follows

from noting that P1 = f(p) · p

The carry trade in our model consists in borrowing in domestic currency in order to

fund the purchase of foreign bonds. For instance, the carry trade from time 0 to time 1

pays off E1Rf1 − 1 at time 1, the long run carry trade pays off ....

Proposition 4. The stronger the preference for a stable exchange rate from the foreign

central bank, the lower the comovement between the foreign exchange rate and the risky

asset. The expected return to the carry trade is unaffected by the bank’s preference for

smoothing.∂

∂s

(∂e0

∂p

)< 0 (31)

Proposition 5. The higher the preference for a stable exchange rate from the foreign

central bank, the lower is the expected return to the carry trade from period 0 to 1, and

the higher the expected return to the carry trade between periods 1 and 2.

Proposition 6. The higher the preference for a stable exchange rate from the foreign

central bank, the larger the term premium on its long term bonds at time 0.

52