Term Structure of Consumption Risk Premia in the Cross Section of Currency Returns

• Author: IRINA ZVIADADZE
• Date: 01 August 2017
• DOI: http://doi.org/10.1111/jofi.12501
• Published date: 01 August 2017

THE JOURNAL OF FINANCE •VOL. LXXII, NO. 4 •AUGUST 2017

Term Structure of Consumption Risk Premia

in the Cross Section of Currency Returns

IRINA ZVIADADZE∗

ABSTRACT

I relate the downward-sloping term structure of currency carry returns to compensa-

tion for currency exposures to macroeconomic risk embedded in the joint dynamics of

U.S. consumption, inflation, nominal interest rate, and their stochastic variance. The

interest rate and inflation shocks play a prominent role. Higher yield currencies ex-

hibit higher multiperiod exposures to these shocks. The prices of these risk exposures

are positive and sizeable across all investment horizons. The interest rate shock is

qualitatively similar to the long-run risk of Bansal and Yaron.

IN THIS PAPER,IQUANTIFY THE RISK-RETURN relationship in the foreign exchange

(FX) market across different countries and investment horizons. My starting

point is new evidence about returns on a currency carry trade, a zero-cost

investment strategy that entails buying bonds in high-yield currencies and

short-selling bonds in low-yield currencies, across alternative investment hori-

zons. The term structure of average carry log returns is slightly downward

sloping: the average log returns are 1.13% and 0.80% per quarter at the three-

month and 10-year investment horizon, respectively. I ask which sources of

macroeconomic risk are compensated in the term structure of carry returns.

Shocks to consumption growth, inflation, interest rates, and their second-

order moments reflect different aspects of currency risk. My focus is on iden-

tifying such shocks, measuring their impact on exchange rates in the cross

∗Zviadadze is at the Stockholm School of Economics. I am grateful to the Editor,Kenneth Single-

ton, the anonymous Associate Editor, and two referees for constructive feedback. I am indebted to

my advisors, Mikhail Chernov and Francisco Gomes, for their unconditional support, stimulating

discussions, and helpful suggestions. I am also grateful to David Backus, Tomas Bjork, Jaroslav

Boroviˇ

cka, Laurent Calvet, Joao Cocco, Riccardo Colacito, Mariano M. Croce, Magnus Dahlquist,

David Edgerton, Vito Gala, Jeremy Graveline, Lars Hansen, Christian Heyerdahl-Larsen, Chris-

tian Julliard, Leonid Kogan, Lars Lochstoer, Igor Makarov, Michael Moore, Ivan Shaliastovich,

Paolo Surico, Stijn Van Nieuwerburgh, Christian Wagner, and Stanley Zin for advice and detailed

feedback. I further thank many participants of conferences and seminars, including Early Career

Women in Finance, Frontiers of Finance Warwick, SED, SoFiE, Young Scholars Nordic Finance

Workshop,WFA, World Congress of Econometric Society,Bocconi, HEC Paris, LBS, LSE, Lund Uni-

versity, New Economic School, Rutgers Business School, Stockholm School ofEconomics, Sveriges

Riksbank, UNC Kenan-Flagler Business School, and UCLA Anderson School of Management, for

many helpful discussions and comments. I gratefully acknowledge financial support from the AXA

research fund, Jan Wallander and Tom Hedelius Foundation, and Swedish House of Finance. The

author has no conflicts of interest, as identified in the Disclosure Policy.

DOI: 10.1111/jofi.12501

1529

1530 The Journal of Finance R

section and at alternative horizons, and understanding their relative impor-

tance for carry trade profitability across alternative investment horizons. I

find that interest rate and inflation shocks play a prominent role. Higher yield

currencies exhibit higher multiperiod exposures to these shocks. The prices of

these risk exposures are positive and sizeable across all investment horizons.

The interest rate shock contributes most to cross-sectional differences in cur-

rency returns across multiple investment horizons and operates qualitatively

similarly to the long-run risk of Bansal and Yaron (2004).

My analysis proceeds in two stages. In the first stage, I describe the empir-

ical properties of currency risk related to macroeconomic fluctuations. To this

end, I estimate the joint dynamics of U.S. consumption growth, inflation, the

nominal interest rate, and the cross section of exchange rate growth as a vec-

tor autoregression (VAR) with stochastic variance. I compute shock exposure

elasticities (Boroviˇ

cka et al. (2011), Boroviˇ

cka and Hansen (2014), Zviadadze

(2016)) of exchange rates to measure the importance of macroeconomic shocks

identified from the VAR. Shock exposure elasticities are marginal multiperiod

sensitivities of exchange rates to alternative current-period shocks. Intuitively,

a source of risk is economically interesting if the shock exposure elasticities of

low and high interest rate currencies exhibit high dispersion at alternative

horizons.

I document economically sizeable and statistically significant gaps between

the shock exposure elasticities in the cross section of currency baskets. Higher

yield currencies exhibit higher exposures to the interest rate and inflation

shocks across multiple investment horizons. To the best of my knowledge, this

is the first study to explicitly relate interest rate and inflation shocks, identified

from U.S. macroeconomic data, to the cross section of currency baskets, and to

explore the term structure of currency exposures to macroeconomic risk. These

novel facts complement the influential evidence of Lustig and Verdelhan (2007)

in a nontrivial way.

The dispersion in multiperiod currency exposures to the sources of risk is

a necessary but not sufficient condition for relating the cross-sectional spread

in currency returns across alternative investment horizons to macroeconomic

risk. An open question is whether currency sensitivities to risk have an eco-

nomically sizeable price. This is where the second stage of my analysis comes

in. To measure price of risk, I introduce a U.S. representative agent with recur-

sive utility and an aversion to consumption risk, and I interpret the estimated

VAR as a model of consumption growth with the explicitly specified dynamics

of expected consumption growth and stochastic variance. Consequently, the

macroeconomic shocks represent multiple sources of consumption risk.

Combining the VAR with recursive preferences comes with its challenges.

The nominal yield plays a dual role. On the one hand, the pricing kernel de-

pends on the nominal yield, as it is one of the state variables in the model.

On the other hand, the nominal yield is the (log) conditional expectation of the

pricing kernel and hence an affine function of the model’s states. The first-order

condition implies a set of pricing restrictions on the structural parameters of

the model. The structural restrictions imposed on the parameters of the VAR

Term Structure of Consumption Risk Premia 1531

lead to nontrivial complications in estimation that I resolve by using Bayesian

Markov Chain Monte Carlo (MCMC) methods.

Armed with the estimated dynamics of the pricing kernel, I characterize

how currency sensitivities to macroeconomic risk are compensated in the mar-

ketplace at horizons from one quarter to 10 years. Shock price elasticities

(Boroviˇ

cka et al. (2011), Boroviˇ

cka and Hansen (2014), Zviadadze (2016)) as-

sign prices to shock exposure elasticities. Put simply, shock price elasticities

constitute a term structure of marginal prices of risk, or a term structure of

marginal Sharpe ratios.

The analysis of the shock elasticities reveals that interest rate risk plays the

most prominent role in the FX market. Two important properties lead to this

conclusion. First, I confirm in the context of the structural VARthat higher yield

currencies exhibit larger sensitivities to the shock at horizons from one quarter

to 16 quarters.1This fact means that, upon realization of the positive shock,

high-yield currencies are expected to appreciate relative to low-yield currencies

over a four-year period. Second, the marginal price for bearing interest rate risk

is the largest at all investment horizons. Taken together, the evidence suggests

that the term structure of currency risk premia reflects the interest rate shock

at short- and medium-term horizons.

An examination of the empirical properties of the interest rate shock sug-

gests that this risk is qualitatively similar to the long-run risk of Bansal and

Yaron (2004). First, the shock affects consumption growth only with a lag of

one quarter. Second, the shock originates via a persistent variable and has the

highest price of risk. Finally, it is the dominant source of variation in the real

risk-free rate. These empirical results complement the argument of Colacito

and Croce (2013) about the special role of long-run risk in currency markets.

Their conclusion is based on a symmetric two-country setting that, by construc-

tion, does not accommodate the term structure of the cross-sectional spreads

in currency risk premia. I leave identification of the economic origins of the

long-run risk shock operating in the FX market for future research.

Estimation of the structural VAR reveals that, similar to the case of the in-

terest rate shock, higher yield currencies have higher exposures to the inflation

shock across multiple investment horizons. The marginal price of the shock is

less sizeable than that of the interest rate shock but is economically mean-

ingful. Given the quantitative prominence of the inflation shock in the term

structure of currency carry premia, future models of FX risk should explore not

only real (as has been the main focus so far) but also nominal sources of risk.

The variance shock contributes to the term structure of currency risk premia

in a conceptually different way. Currencies are not sensitive to the variance

shock at any investment horizon, yet it is the only source of time-variation

in the risk premium. As a result, the log expected return on the one-period

carry trade strategy, for example, has fluctuated between 0.43% and 3.54% per

quarter.

1Henceforth, I use the structural VARand the VAR augmented with cross-equation restrictions

interchangeably.