Exchange rates and interest arbitrage

43 Pages·2013·1.14 MB·English

Exchange rates and interest arbitrage∗
Anella Munro
Reserve Bank of New Zealand
[email protected]
February 1, 2013
Rather than asking whether uncovered interest parity (UIP) holds, this paper
seeks to measure the contribution of interest arbitrage to exchange rate
fluctuations. Forecasts of future relative interest returns are constructed
based on AR1 models and on forecasts embedded in interest rate swap
contracts. Those forecasts are used to construct UIP-consistent exchange
rates (a forward sum of relative returns), and to decompose exchange rate
fluctuations into forecast revisions, reversion towards UIP and residual shocks
(unrelated to risk-neutral interest arbitrage). The decomposition associate
interestarbitragewiththeformertwowhichaccountfor27to40%ofexchange
rate fluctuations for a set of advanced country USD exchange rates. The bulk
ofthatcontributioncomesfromthewithin-period”Dornbuschjump”response
rather than the 1-period return that has been a focus of the parity literature.
Negative covariance of the forecast revision and the shock implies severe
downward bias in estimation of that contribution using standard techniques.
The paper discusses the potential role of capital flows in generating that
covariance and the continued appreciation of high interest rate currencies.
JEL classification: F31, G12
∗This paper has benefitted from comments from Charles Engel, Leo Krippner and
Konstantinos Theodoris.
1
1 Introduction
Avastliteratureconsiderstheempiricalfailureoftheuncoveredinterestparity
(UIP) hypothesis for developed country currencies (see, for example, Fama
(1984); and literature surveys in Engel (1996), Thornton (2009) and Engel
(2012)). UIP implies that risk-neutral investors should be indifferent between
holding domestic and foreign currency assets with similar risk characteristics.
Excess profits should have been arbitraged away. Empirically, high interest
rate currencies tend to yield excess short term returns: not only is the higher
interest return not fully offset by currency depreciation, but the high interest
currency often appreciates in the short term.
The rejection of UIP has been explored in many dimensions. The role of risk
premia has been a prominent explanation (Fama (1984), Froot and Frankel
(1989)), but is difficult to reconcile with two stylised facts. First, a high
interest currency tends to be a strong currency consistent with UIP (Engel
2012) while a high risk premium implies a high interest rate, but a weak
currency. Second, in a dynamic sense, it is unclear why a high risk premium
currency should continue to appreciate. Burnside (2011) argues that it fits
well with tail events.
Another prominent explanation is expectations bias. A key assumption in
the one-period test of UIP is rational expectations: expectations about future
spot rates are not always right, but on average they should be. According to
UIP, the expected future spot rate is the forward rate (the spot rate net of the
one-period relative interest return). In contrast, Meese and Rogoff (1983)’s
finding that a random walk forecast is hard to beat suggests that a no-change
exchange rate expectation may be rational and the forward premium may be
a biased forecast. If the exchange rate is a random walk, then the forward
premium must be inversely related to wither the expectations bias or the
shock (Engel and West (2005), van Wincoop and Bacchetta (2007),Engel
(2012)). Chinn and Frankel (1994) find greater support for UIP when using
ex-ante market expectations of future spot rates.
Finally, interaction of interest arbitrage with other factors can explain the
failure of the UIP condition. Froot and Thaler (1990) argue that partial ad-
justment to shocks can generate the UIP puzzle. van Wincoop and Bacchetta
(2007) generate the UIP puzzle in a model in which interest arbitrage is active
but expectations are subject to infrequent adjustment.
Central to the approach in this paper is the (Dornbusch 1976) asset price
view of exchange rates. In the presence of sticky prices, if the home country
2
interest rate rises, the home currency should immediately appreciate relative
to a future equilibrium rate, so that it can subsequently depreciate to equalise
returns from the two currencies each period. As such the exchange rate is a
sum of expected future relative returns.
Engel and West (2005) construct forecasts of future fundamentals based on
AR1 models and VARs for a range of exchange rate models for advanced
country USD exchange rates. They show that the forecasts are near-random
walks, suggesting the rational expectations assumption in the Fama equation
may be problematic. They find correlation between the forecasts and the
exchange rate and some evidence that the exchange rate leads fundamen-
tals, somewhat counterbalancing the usual bleak view of the role of interest
arbitrage.
This paper extends Engel and West (2005) by (i) constructing forecasts that
employ long-term interest rate swaps,1 that provide a market-based forecast
of short term benchmark interest rate paths, and (ii) using the forecasts to
decompose exchange rate fluctuations into factors associated with interest
arbitrage and shocks that are not. Interest arbitrage is reflected in the
Dornbusch jump and the 1-period return (that together make up the forecast
revision), and in the reversion to a UIP-consistent level. The decomposition
provides a useful basis to consider sources of estimation bias.
While the Dornbusch jump (a response to news) should be unpredictable
ex-ante, it is of interest when examining interest arbitrage empirically for at
least three reasons. First, for a floating currency, the jump reflects interest
arbitrage, so it should be included a measure of interest arbitrage.
Second, the change in fundamentals, and so change in the forecast interest
path, can be largely observed by the time the exchange rate is set at the
end of the period. Bjørnland (2009) and Bjørnland and Halvorsen (2010)
show empirical support for a Dornbusch jump for a range of advanced small
open economy currencies using a VAR identification that allows (as does our
decomposition) for a contemporaneous interaction between interest rates and
the exchange rate.
Third, there is good reason to believe that the within-period jump is endoge-
1Chinn and Meredith (2004) and Chinn and Quayyum (2012) relate the multi-period
exchange rate changes to multi-period bond differentials and find some support for UIP
over those longer horizons. The approach here is conceptually different to those papers in
two ways: (i) the swap rate is used here to reflect a long sequence of short-term returns
rather than the long-period return on a bond; and (ii) that sequence of short-term returns
is related to 1-period exchange rate movements.
3
nously related to the shock (Fama 1984), and we know from the standard UIP
test result that the jump and/or the shock is correlated with the one-period
return.
The results show some evidence of cointegration between the forecasts and
exchange rates for a range of advanced country USD exchange rates. That co-
movementappearstobeincreasingovertime. Forecastrevisionsarepositively
correlated with changes in exchange rates consistent with interest arbitrage
being a significant part of the data generating process. The contribution of
interest arbitrage comes from mainly from the Dornbusch jump component
which has a variance two orders of magnitude larger than the one-period
relative return in the standard parity test. In the decomposition, interest
arbitrage (forecast revisions plus the error correction term) accounts for an
estimated 27 to 40% of the variance of a range of advanced country USD
exchange rates. Covariances among the shock and the interest arbitrage
components suggest severe bias in using standard estimation techniques to
examine the role of interest arbitrage in exchange rate fluctuations.
I argue that capital flows help to explain two stylised facts: the negative
covariance between forecast revisions and shocks; and the continued appre-
ciation of high interest rate currencies. Capital flows unrelated to interest
arbitrage create interest arbitrage opportunities, so influence exchange rates
in a way that covaries negatively with the effect of interest arbitrage if the
latter is incomplete ((Froot and Thaler 1990)). Second, capital should flow
to a high return to capital to equalise returns to capital over the medium
term. That flow is consistent with continued appreciation of a high interest
currency because the equalisation of returns to capital may take decades
unless countries run very large current account deficits or have relatively high
savings rates over an extended period. Capital flows are driven by a range
of factors unrelated to risk-neutral interest arbitrage (eg. risk premia, carry
trade, portfolioshifts, tradeflows, andcentralbankintervention). Empirically
financial flows are large and volatile and have explanatory power for exchange
rates (Evans and Lyons 2002).
In the next section, forecasts of fundamentals are derived analytically. Section
3, examines UIP-consistent exchange rates based on those forecasts and coin-
tegration with observed developed country US dollar exchange rates. Section
4 uses the decomposition of exchange rate changes to examine covariance
among the factors and sources of estimation bias. Section 5 concludes.
4
2 Interest arbitrage and real interest returns
UIP is derived as an optimality condition that equalises the expected returns
on domestic and foreign currency assets. Risk-neutral investors (who care
only about the mean of the risk premium) should be indifferent between
holding domestic and foreign currency assets with similar risk characteristics.
A higher real interest return in the home country r relative to the foreign
t
country r∗ should be offset by an expected depreciation (rise) of the home
t
currency so returns are equal and there is no excess return to holding the
home or foreign currency: 2
E (q )−q = (r −r∗)+λ (1)
t t+1 t t t t
where q is the log of the nominal exchange rate, defined as the home price
t
of a unit of foreign currency (a fall in q represents an appreciation of the
t
home currency), E (q ) is the expected spot exchange rate at t+k based
t t+k
on information at time t, and λ is the excess home currency return (Engel
t
2012).3
2.1 The standard one-period test
Assuming rational expectations (on average, the expected future spot rate
should equal the observed future spot rate), the standard test for UIP written
in real terms is: 4
∆q = α+β(r −r∗)+(cid:15) (2)
t+1 t t t+1
H : β = 1
0
2UIP can be expressed in real or nominal terms. Assuming stationarity of the real exchange
rate, it is convenient to express UIP in real terms so that the equilibrium exchange rate is
constant. If the nominal interest differential in part reflects an inflation differential, then
the nominal equilibrium must adjust to satisfy purchasing power parity (PPP) in the long
term. See Appendix A
3λ mayreflectavarietyoffactorsincludingcapitalflowsthatarenotdrivenbyrisk-neutral
t
interestarbitrage, relativedefaultrisk, liquidityrisk, measurementerror, forecasterroretc.
4The same hypothesis can be (and was originally) expressed as a test of the ”unbiasdenss”
of the forward exchange rate f as a predictor of the future nominal spot rate, s :
t,t+1 t+1
∆s = α +β(f −s )+(cid:15) ; H : α = 0, β = 1. The forward rate f is
t+1 t,t+1 t t+1 0 t,t+1
the certainty-equivalent expected future spot rate according to covered interest parity
((f −s )∼=i −i∗ ) whereby the entire transaction is contracted ex-ante, so there is
t,t+1 t t t
no uncertainty and arbitrage tends to hold in normal times. See Baba and Packer (2009)
on deviations from covered interest parity during the global financial crisis.
5
The hypothesis H : β = 15 has been widely rejected in both nominal and
0
real terms. The overwhelming evidence is that, not only is β not equal to
one, but it is often of the opposite sign with unity outside the 90% confidence
band (despite large standard errors). The high interest rate currency offers
not only a higher yield, but also tends to appreciate over the higher yield
period.
2.2 The exchange rate as an asset price
Equation 1 can be substituted forward to express the exchange rate as a
relative asset price that is an infinite forward sum of future relative returns:
∞ ∞
(cid:88) (cid:88)
q = −E (r −r∗ )−E λ + lim E (q ) (3)
t t t+k t+k t t+k t t+k
k→∞
k=0 k=0
Assuming that the final term in 3, is a constant purchasing power parity
(PPP) equilibrium, q¯, and using the notation of (Engel 2012),
∞ ∞
(cid:88) (cid:88)
q −q¯ = −E (r −r∗ )−E λ (4)
t t t+k t+k t t+k
k=0 k=0
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
EtRt Λt
The deviation of the real exchange rate from PPP equilibrium is the expected
sum of real relative returns E R consistent with risk-neutral UIP, plus the
t t
expected sum of excess returns Λ that drives a wedge between the exchange
t
rate and its risk-neutral UIP-consistent level. At t+1,
q −q¯ = −E R −Λ (5)
t+1 t+1 t+1 t+1
Subtracting 4 from 5,
∆q = −[E R −E R ]−[Λ −Λ ]
t+1 t+1 t+1 t t t+1 t
= (r −r∗)−[E R −E R ]+(1−ρ )Λ −λ (6)
t t t+1 t+1 t t+1 Λ t t+1
The term [E R −E R ] is the forecast revision. It comprises the one-
t+1 t+1 t t
period relative return (r − r∗) plus the change in the expected path of
t t
subsequent relative returns (the Dornbusch jump).6
5The hypothesis is sometimes stated as H : α = 0, β = 1 but in practice only β = 0
0
matters (McCallum (1994)).
6See Appendix C for a graphical exposition.
6
While the revision of the UIP-implied exchange rate at t + 1 should not
be forecastable ex-ante, it is important for understanding the contribution
of interest arbitrage in three respects. First, the within-period response to
news-driven forecast revisions is an integral part of interest arbitrage. Second,
empirically, it is much larger than the subsequent adjustment embodied
in the 1-period return. Finally, there is reason to believe that it covaries
endogenously with the shock. For example, a capital inflow that appreciates
the home currency and depresses the path of home interest rates creates
an interest arbitrage opportunity. Arbitrage should respond, leading to an
offsetting capital outflow and/or a downward adjustment in the value of the
home currency (repricing may occur without flows).
The revision to the UIP-consistent path can be estimated at time t+1 when
we measure ∆q using data available at t+1−(cid:15), (cid:15) → ∞ (interest rates
t+1
are persistent). By the time the markets have set q , arbitrageurs have
t+1
revised expectations about the forward path of relative returns based on data
available up to t+1−(cid:15), and their response to that revised forecast should
influence ∆q .
t+1
The final term in equation 6 is the change in expected excess returns. If Λ ,
t+1
the deviation of the real exchange rate from the UIP-consistent equilibrium,
is stationary, then the final term includes an error correction term and a
shock since Λ = ρ Λ +λ .
t+1 Λ t t+1
(cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125)
EtΛt+1 shock
The forecast revision and the ex-ante deviation from parity create incentives
for interest arbitrage opportunities. If shocks were small relative to interest
arbitrage, then changes in the forecast plus the error correction term would
keep the exchange rate near parity; the forecasts would be cointegrated
with the observed exchange rate; and forecast revisions would be positively
correlated with exchange rates. As shocks become large relative to the
resources devoted to interest arbitrage, the exchange rate deviates further
from parity, is less cointegrated with the forecast path, and forecast revisions
become less correlated with exchange rate fluctuations.
In the rest of this section, specific expressions for E R are derived, based on
t t
forecasts of future interest rate paths. These are based on AR1 models (as in
(Engel and West 2005)) and on the nominal interest rate forecasts embedded
in the 10 year interest rate swap.
7
2.2.1 Expectations based on an AR1 model
Suppose agents have a simple AR1 forecasting model for interest rates,
inflation rates and λ :
t
x = ρ x +(cid:15) , where 1 > ρ > 0 and (cid:15) ∼ N(0,σ2) (7)
t+1 x t t x t+k
E (x ) = ρ x
t t+1 x t
forx ∈ [r ,r∗,π ,π∗,λ ] (8)
t t t t t t
Using the geometric sum approximation 1+a+a2+a3+... (cid:39) 1/(1−a), the
expected sum of the future path of x is:
t
∞
(cid:88) x
t
E x (cid:39)
t t+k
1−ρ
x
k=0
Substituting AR1 forecasts, based on variables observed at time t, into
equation 6, the level of the real exchange rate consistent with UIP is:
i i∗ ρ2π ρ2 π∗
−E R (cid:39) − t + t + π t−1 − π∗ t−1 (9)
t t
1−ρ 1−ρ 1−ρ 1−ρ
i i∗ π π∗
Small but persistent changes in relative returns may have large effects when
summed over a long horizon. Short-term interest differentials are persistent
(AR1 coefficients for monthly rates are typically about 0.98) so the coefficient
ρ is expected to be large ( ρ = 49 for ρ = 0.98).
1−ρ 1−ρ
Similarly, at t+1, the level of the exchange rate consistent with UIP is:
i i∗ ρ2π ρ2 π∗
−E R (cid:39) (cid:39) − t+1 + t+1 + π t − π∗ t (10)
t+1 t+1
1−ρ 1−ρ 1−ρ 1−ρ
i i∗ π π∗
Substituting (9) and (10) into 6, the change in the exchange rate consistent
with UIP between t and t+1 can be written:
∆i ∆i∗ ρ2∆π ρ2 ∆π∗
∆q = − t+1 + t+1 + π t − π∗ t +(1−ρ )Λ −λ
t+1 λ t t+1
1−ρ 1−ρ 1−ρ 1−ρ
i i∗ π π∗
Since interest rates are persistent, and ρ is large so the forecast revision is
1−ρ
expected to be large. Moreover, the Dornbusch jump (∆R −(r −r∗)) is
t+1 t t
potentially correlated with (r −r∗) if R is near a random walk (Engel and
t t t
West 2005).
8
2.2.2 Expectations embedded in long-term swap rates
Long-term swap rates provide an alternative, and probably considerably more
accurate, proxy for the expected sum of future nominal interest rates than an
AR1model. Theswaprateisafixedinterestrateequaltotheex-anteexpected
average 90-day (or 180 day) rate over the 10-year duration. Therefore, the
10-year swap rate differential provides a measure of the expected sum of
relative short-term interest returns over ten years.
An advantage of using swap rates rather than long-term bond rates here is
that they carry mainly interest rate risk and relatively little credit risk. The
interest rate risk is desirable for our purpose: exposure to interest rate risk
means that the forecasts implied by the swap curve need to be as accurate as
possible. Also, the forecasts embedded in interest rate swaps are the basis for
a vast volume of transactions. The Bank for International Settlement reports
that the notional amount outstanding in June 2011 was over $400 trillion.7
The fact that a swap has little credit risk is also desirable. It means that
the swap differential is mainly a forecast of the difference in short term
risk-free interest rates. The credit risk embedded in a swap comes from two
main sources. One is counter-party risk (associated with the swap being
out-of-the-money due to movements in interest rates) which is commonly
mitigated by posting collateral.
The second is the credit risk embodied in the underlying 90- or 180-day
benchmark (Libor or equivalent) rate8 that tends to be small because it is
relatively short-term and because central bank liquidity operations contribute
to keeping short-term spreads modest.9 In contrast government or private
bonds can carry a large ‘‘specialness” discount or credit premium.
Limiting the forecasting horizon to 10 years (after 10 years, the interest rates
and exchange rate are assumed to be at the long-run equilibrium), the level
7BIS Quarterly Bulletin, June 2011.
8Libor is not without problems: it is a quoted (not necessarily transacted) rate based on
submissions from a panel of banks. The panel is dominated by large AA to A-rated banks.
The panel varies from currency to currency so there are credit risk differences between the
domestic and foreign interest rate series (unless quotes from a matched panel are used).
There may also be differences in liquidity risk and country-currency risk in the domestic
and foreign rates, and quotes have at times been subject to manipulation.
9During the global financial crisis the Libor spread to the OIS curve (expected average
policy rate) did rise sharply, but the rise was short lived compared to longer term bond
spreads to swap, and Libor quickly convereged toward the policy rate.
9
of the exchange rate consistent with UIP:
119 ∞
(cid:88) (cid:88)
−E R = − (i −i∗ )+E (π −π∗ )
t t t+k t+k t t+k t+k
k=0 k=1
ρ2π ρ2 π∗
≈ −120(i10S −i∗10S)+ π t − π∗ t (11)
t t 1−ρ 1−ρ
π π∗
(cid:124) (cid:123)(cid:122) (cid:125)
EtRt
where i10S and i∗10S are home and foreign 10-year swap rates (% per month,
theexpectedaveragemonthlyreturnontheshort-termrateforthesubsequent
120 months) and the expected inflation differential is based on a AR1 forecast
as before.10
Substituting equation (11), at t and t+1 into 6, the change in the exchange
rate consistent with interest parity is:
ρ2∆π ρ2 ∆π∗
∆q = −120∆(i10S −i∗10S)+ π t − π∗ t −(1−ρ )Λ −λ (12)
t+1 t+1 t+1 1−ρ 1−ρ λ t t+1
π π∗
= (r −r∗)+[∆R −(r −r∗)]+(1−ρ )Λ −λ (13)
t t t+1 t t λ t t+1
Again, ∆R is potentially large and volatile, being based on large or infinite
t+1
sums; and ∆R −(r −r∗) is potentially correlated with (r −r∗), if R
t+1 t t t t t+1
is a near-random walk, so may bias the estimate of β in equation 2.
3 UIP exchange rates and cointegration
3.1 Data and forecasts
The data set here includes eight USD currencies: the Australian dollar
(AUD), Canadian dollar (CAD), Swiss Franc (CHF), euro (EUR), British
pound (GBP), Japanese yen (JPY), New Zealand dollar (NZD) and Swedish
Krona (SEK). The sample period is January 1990 to July 2012, or the sub-
sample for which the relevant data are available (see Appendix B for details).
The data are monthly, so the sample size is about 260 for most countries,
and shorter for the euro and for currencies where long term interest swap
markets developed later. This covers the bulk of the period for which reliable
10The sum of relative inflation could be based on break-even inflation rates from inflation-
indexed government bonds, but inflation-indexed bonds are only systematically issued in a
few jurisdictions, and tend to be traded in much less liquid markets than nominal bonds.
10