The Share of Systematic Variation in Bilateral Exchange Rates

• DOI: http://doi.org/10.1111/jofi.12587
• Date: 01 February 2018
• Published date: 01 February 2018
• Author: ADRIEN VERDELHAN

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 1 •FEBRUARY 2018

The Share of Systematic Variation in Bilateral

Exchange Rates

ADRIEN VERDELHAN∗

ABSTRACT

Sorting countries by their dollar currency betas produces a novel cross section of

average currency excess returns. A slope factor (long in high beta currencies and

short in low beta currencies) accounts for this cross section of currency risk premia.

This slope factor is orthogonal to the high-minus-low carry trade factor built from

portfolios of countries sorted by their interest rates. The two high-minus-low risk

factors account for 18% to 80% of the monthly exchange rate movements. The two

risk factors suggest that stochastic discount factors in complete markets’ models

should feature at least two global shocks to describe exchange rates.

THE CORRELATION STRUCTURE OF BILATERAL EXCHANGE RATES can be summarized

by a small number of principal components, but those principal components

offer a purely statistical description of exchange rates and are difficult to in-

terpret in any micro- or macrofinance model. In this paper, in contrast, I report

that two currency risk factors account for a substantial share of individual

exchange rate time series. These factors are priced in currency markets and

the shares of systematic currency risk have implications for any no-arbitrage

model in international finance.

Two risk factors, namely, carry and dollar, are constructed from portfolios

of currencies. The carry factor corresponds to the change in exchange rates

between baskets of high and low interest rate currencies, while the dollar

factor corresponds to the average change in the exchange rate between the

U.S. dollar and all other currencies. All exchange rates are defined here with

respect to the U.S. dollar. I regress changes in exchange rates on the carry

factor, the same carry factor multiplied by the country-specific interest rate

difference (the latter is referred to as “conditional carry”), and the dollar factor.

The change in bilateral exchange rate on the left-hand side of these regressions

is measured between tand t+1; on the right-hand side, the carry and dollar

∗Adrien Verdelhan is with MIT Sloan and NBER. The author thanks the Editor, Ken Sin-

gleton, two anonymous referees, as well as Andrew Atkeson, Nittai Bergman, John Campbell,

Francesca Carrieri, Mike Chernov, Martin Evans, Emmanuel Farhi, Tarek Hassan, Chris Jones,

Pab Jotikasthira, Thomas Knox, Leonid Kogan, Ralph Koijen, David Laibson, Hanno Lustig, Mat-

teo Maggiori, Nelson Mark, Philippe Mueller, Jun Pan, Tarun Ramadorai, Steve Ross, Barbara

Rossi, Nick Roussanov, Lucio Sarno, Piet Sercu, Chris Telmer, Jiang Wang, Ken West, and partic-

ipants at many seminars and conferences for helpful comments and discussions. I have read the

Journal of Finance’s disclosure policy and have no conflicts of interest to disclose.

DOI: 10.1111/jofi.12587

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factors also correspond to changes between tand t+1, while the domestic and

foreign interest rates are known at date t. The carry and dollar factors do not

include the bilateral exchange rate that is the dependent variable.

The factor regressions offer a novel picture of bilateral exchange rate move-

ments. Both factors appear highly significant. With the carry factors, the ad-

justed R2s range from 0% to 23% among developed countries at the monthly

frequency over the 1983 to 2010 sample. While a lot of research focuses on the

carry trade, the dollar factor appears to be a more important driver of exchange

rates. When the two factors are combined, the adjusted R2s range from around

20% to 90% in developed countries and from 10% to 75% among developing

countries with floating currencies. The distribution of R2s on the factor regres-

sions is quite stable across frequencies; similar distributions appear at daily,

monthly, quarterly, and annual frequencies. The substantial R2s of the factor

regressions do not imply that bilateral exchange rates are easy to forecast: the

corresponding regressions use contemporaneous variables, not predictive ones.

Large R2s can naturally be obtained by using three or more principal compo-

nents. The dollar factor is actually close to the first principal component, while

the carry factor is different from any of them—the information contained in

short-term interest rates matters. Both factors deliver a more stable descrip-

tion of exchange rates than the principal components. More crucially, principal

components do not imply that risks are priced. In contrast, both the carry and

the dollar factors are priced in currency markets. They are risk factors in the

asset pricing sense, consistent with the logic of an Euler equation.

The risk-based interpretation of the carry factor is well known. Previous

research on currency portfolios shows that the carry factor accounts for the

cross section of currency excess returns sorted by interest rates: covariances of

the carry factor with currency returns align with the cross section of average

excess returns. A consistent result appears here on individual currencies: the

higher the interest rate, the larger the loading on the carry risk factor. This is

the risk-based explanation of the classic currency carry trade.

This paper shows that the dollar factor also has a risk-based interpretation.

The price of dollar risk cannot be estimated precisely from portfolios of countries

sorted by interest rates because these portfolios all load in the same way on the

dollar factor. Instead, I build portfolios of countries sorted by their time-varying

exposures to the dollar factor (i.e., dollar betas). After transaction costs, the low

dollar beta portfolio offers an average log excess return of just 0.6% per year

for investors who go long foreign currencies when the average forward discount

(average foreign minus U.S. interest rate) is positive and short otherwise. The

high dollar beta portfolio offers an average log excess return of 5.8% for similar

investments, implying a large Sharpe ratio of almost 0.6. Conditioning on the

average forward discount, covariances of the dollar factor with portfolio returns

account for this new cross section of average excess returns, while covariances

with the carry factor do not. The estimated price of dollar risk is significant

and close to the mean of the factor, as implied by the absence of arbitrage.

The key pricing information is contained in a long-minus-short risk factor,

built from the same set of portfolios used as test assets as the difference in

The Share of Systematic Variation in FX 377

exchange rates between high and low dollar beta portfolios. The loadings of

dollar beta portfolios on this long-minus-short risk factor are significant and

vary monotonically across portfolios. I refer to the difference in exchange rates

between high and low dollar beta portfolios as the global component of the

dollar factor. It is highly correlated (0.85) with the dollar factor itself but its

correlation with the carry factor is statistically insignificant.

Overall, by building portfolios of currency returns, one can extract two risk

factors defined as differences in exchange rate changes, and thus immune to

U.S.-specific shocks: the carry factor and the global component of the dollar

factor. These two variables describe bilateral exchange rates expressed in U.S.

dollars or in different units. Regressions of changes in bilateral exchange rates

on the two high-minus-low risk factors deliver R2s between 18% and 83% for

exchange rates defined in U.S. dollars. This is the key result of the paper,

as only “dollar-neutral” explanatory variables are used in these regressions.

The same variables describe exchange rates defined in other currencies, for

example, Japanese yen and U.K. pounds. The R2s are lower but still range

from 18% to 46% for yen-based exchange rates and from 6% to 48% for pound-

based exchange rates. The loadings on the global component of the dollar factor

are significant in 20 out of 26 regressions. If the factors were capturing country-

specific shocks, they would not describe currencies in other units. The empirical

findings in this paper suggest that global shocks are key to describing exchange

rates.

The economic source of those global shocks is an open question. At the annual

frequency, the global component of the dollar tends to be low when developed

countries are close to the troughs of their business cycles, as measured by

the OECD turning points. The dollar risk could thus be interpreted as global

macroeconomic level risk, while carry shocks appear more related to volatil-

ity and uncertainty.1Of course, this tentative interpretation does not exclude

others based on shocks to, for example, liquidity, intermediaries’ wealth, mon-

etary policies, international trade, or international capital flows. The different

potential sources and interpretations of those global shocks, as well as the un-

derlying economic sources of the differences in country betas, are potentially

fruitful research avenues. As an initial step in that research agenda, I show

in a qualitative reduced-form model that two kinds of global shocks to the

stochastic discount factors (SDFs) are necessary to describe exchange rates.

In the tradition of Frachot (1996), Backus, Foresi, and Telmer (2001), and

Hodrick and Vassalou (2002), I start from the law of motion of lognormal SDFs.

In the model, each log SDF is driven by country-specific and two world shocks.

When markets are complete, log changes in exchange rates correspond to the

differences between domestic and foreign log SDFs. The volatilities of the SDF

shocks are time-varying to account for the forward premium puzzle. For the

sake of clarity, the main text focuses on a special case of the model, while the

1The carry factor can be proxied by measures of global volatility on equity (Lustig, Roussanov,

and Verdelhan,2011) oron currency markets (Menkhoff et al., 2012a), or by measuresof downside

equity risk (Lettau, Maggiori, and Weber,2014).