Klaus Desmet
Notes on Uncovered Interest Parity
1. Theory
(a) The basic equation of uncovered interest parity
The uncovered interest parity (UIP) condition says that the dollar return on a
dollar investment should be the same as the dollar return on a euro investment.
Take the following information: the dollar interest rate is 5%, the euro interest rate is 3%, the spot dollar/euro exchange rate is 1.2 dollars/euro, and the
expected spot exchange rate 12 months from now is 1.25 dollars/euro.
• Option 1. Invest 1 dollar in a dollar account. One year from now I will get
1.05 dollars, that is, a return of 5%.
• Option 2. I change 1 dollar today into euros, invest it in a euro account, and
change it back into dollars 12 months from now. What I will get in expected
terms is (1/1.2)(1.03)(1.25) = 1.073 dollars, that is, a return of 7.3%.
Note that option 2 can be approximated by a much easier formula. The expected
dollar return on a euro deposit can be approximated by the sum of the euro
interest rate and the expected depreciation of the dollar vs the euro:
REU R +
EUe SD/EU R − EU SD/EU R
EU SD/EU R
Applying that to our problem, the dollar return on the euro deposit would be
0.03+(1.25-1.2)/1.2 = 0.072, that is, a return of 7.2%. As can be seen, the
approximation is good enough — it gives us 7.2%, rather than the mathematically
correct 7.3% — and a lot more intuitive. Note, however, that in countries with
very high interest rates and very high currency fluctuations (i.e., countries with
very high inflation), the approximation would not be very good.
What UIP says is that the dollar returns on dollar deposits should be the same
as the dollar returns on euro deposits. Therefore,
RU SD = REU R +
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EUe SD/EU R − EU SD/EU R
EU SD/EU R
If not, there is money to be made. In the above example, the dollar returns on
dollar deposits is 5%, whereas the dollar returns on euro deposits is 7.2%. In that
case investors will readjust their portfolio: they move out of dollar deposits into
euro deposits. This increases the value of the euro TODAY. This appreciation
continues until UIP holds, i.e., the appreciation of the euro will continue until
the expected appreciation of the euro goes down from 4.2% to 2%. At that point,
the two sides of the UIP are the same, and no investor has any incentive anymore
to readjust their portfolios. SEE GRAPHICAL ANALYSIS IN CLASS AND IN
THE BOOK.
(b) Some applications
i. Application 1: effect of changes in interest rates on the spot exchange rate
UIP tells you what should happen to the spot exchange rate, when either
the dollar interest rate or the euro interest rate goes up. For instance, if the
dollar interest rate goes up by 1%, then the right hand side of UIP should
also go up by 1%, which implies that we should now expect the dollar to
depreciate by 1% more than before (to continue with our previous example,
this would mean by 3% rather than by 2%). For a given future expected
exchange rate, this implies that you should get an immediate appreciation
of the dollar by 1%. This is consistent with our intuition: when the dollar
interest rate goes up, investors find the dollar more attractive, they move out
of the euro into the dollar, and the dollar becomes more expensive. Once
buying dollars becomes expensive enough, they cease to be interested in
adjusting their portfolios. This happens when the dollar has appreciated by
1%.
ii. Application 2: determining the expected future exchange rate
The UIP condition is very often used the other way around: instead of using
it to determine the spot exchange rate (for a given future expected exchange
rate), it is used to determine the future expected exchange rate (for a given
spot exchange rate). Again, we have one equation in one unknown and we
can solve out for the unknown. For instance, if the interest differential on
the dollar is 2% (as in our original example), then you should expect the
dollar to depreciate by 2% versus the euro. If you know the current spot
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exchange rate, you can calculate the expected future exchange rate. In fact,
the interest rate differential is what banks tend to use when they buy or sell
currency with a forward contract.
iii. Application 3: effect of expected future exchange rate on current exchange
rate
You can also study what happens when expectations about the future exchange rate change. This is very straightforward. Assume that UIP holds.
If you think the dollar is going to depreciate by 1% more than previously
expected over the next 12 months, the spot exchange rate will go down by
1% today. Therefore, lots of the volatility of the nominal exchange rate can
be due to changes in expectations about the future rate (without any happening today!). As is usual in these markets, there is lots of noise, and any
little new piece of information will cause fluctuations.
(c) Further comments on uncovered interest parity
In this section we make a couple of additional remarks about uncovered interest
parity
i. The UIP condition of course includes some risk, because it is phrased in terms
of expected value of future exchange rates. We therefore cannot talk about
real arbitrage. In contrast, the covered interest parity condition involves no
such risk, because we fix today how many dollars we will get for a euro 12
months from now:
RU SD = REU R +
FU SD/EU R − EU SD/EU R
EU SD/EU R
where FU SD/EU R is the forward exchange rate. If the right hand side is
larger than the left hand side, you are sure to make a profit by switching
your money from dollars to euros. Given that there is a pure arbitrage
opportunity, you can be sure that the covered interest parity condition holds
at all times. (This assumes away default risk, more about this later.)
ii. Of course risk may play a role. Assets denominated in certain currency may
be perceived as being more risky than others: default risk, high indebtedness
of the government, history of financial instability, lack of liquidity, country
risk, etc. In that case, assuming agents are risk averse, we would need to
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introduce a risk premium into the UIP condition:
RU SD = REU R +
EUe SD/EU R − EU SD/EU R
EU SD/EU R
+ρ
where ρ would represent the risk premium of investing in dollars, rather
than in euros. Indeed, if ρ > 0 it implies that the dollar returns on euro
investments are lower than the dollar returns on dollar investments. This
means that the euro gets away with giving a lower return than the dollar. By
the way, this risk premium would also appear on the covered interest parity
condition. (Note: study the case of the dollar as a “safe haven” during
the global financial crisis! The dollar strengthened in spite of interest rates
dropping dramatically.)
iii. The inability of forecasting exchange rates is often seen as discrediting the
UIP condition, which says that the expected depreciation of the currency
should be equal to the interest differential. However, this expectation does
not say anything about the variance in that expectation. As long as the UIP
leads to an unbiased estimator of the future exchange rate, we should not
criticize the UIP condition. Indeed, errors could be small or could be large,
as long as on average the error we make is zero, then the UIP condition is
a good description of reality. As long as all available information today is
included into our expectation of the future exchange rate, then we are fine.
If that is so, then any additional fluctuation in the future exchange rate,
beyond what UIP predicts, is simply due to future unexpected shocks. More
on this when we discuss the paper by Froot and Thaler.
2. Froot and Thaler, JEP 1990
(a) Introduction
The uncovered interest parity condition tells you that in expected terms, it does
not make a difference whether you invest your money in dollars or in euros.
Therefore, flipping a coin is the right solution. As already pointed out, the idea
is that UIP incorporates all the information available today. It should therefore
be an unbiased estimator: although we are of course likely to end up making a
mistake ex post, the ex ante expected error is zero. In other words, the error we
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make on average may be small or large, but the average error we make should
be zero. However, in reality it turns out that returns are systematically higher
on currencies that give higher interest rates. For instance, if the dollar gives a
higher interest than the euro, you should invest in the dollar. This is somewhat
of a puzzle.
(b) Empirical evidence
The UIP equation in the Krugman/Obstfeld book is written as:
RU SD = REU R +
EUe SD/EU R − EU SD/EU R
EU SD/EU R
This can be re-written as:
EUe SD/EU R − EU SD/EU R
EU SD/EU R
= RU SD − REU R
This implies that the interest rate differential is an unbiased estimator of the future spot exchange rate. Again, this does not mean that it predicts the exchange
well or not, it simply says that on average the error we make should be zero. We
can therefore run the regression on page 182 of Froot/Thaler:
∆s = α + β(RU SD − REU R ) + ε
where ∆s is the log difference in the spot exchange rate. Under the null hypothesis, α = 0 and β = 1.
It turns out that in ALL empirical tests β < 1. Even worse, very often we get
β < 0. Note that this validates the view of always investing in that currency
which gives the higher interest rates. It obviously also invalidates the simple UIP
view. An example can help us see this. Assume you start off with the following
case: RU SD = REU R = 5%. In that case no matter what β is, the expected
depreciation of the dollar is zero. However, assume that the dollar interest rate
goes up by 1%, so that the interest differential on the dollar is now 1%. If β = 1,
we now expect a future depreciation of the dollar by 1%. However, if β < 1, for
instance if β = 0.5, you would only expect a depreciation of the dollar of 0.5%.
This clearly violates UIP: the expected return on dollar deposits is now 5.5% and
the expected return on euro deposits is 5%. In other words, UIP is not correct,
and the view of investing in the currency which gives the higher interest rate
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seems the superior strategy. The rest of the paper is then dedicated to proposing
some reasons as to why UIP does not hold.
(c) Solution 1: Risk premia.
If the risk premium on currencies is an increasing function of the interest rate,
then β would end up being less than 1. Including a risk premium in the UIP
expression gives you:
RU SD = REU R +
EUe SD/EU R − EU SD/EU R
EU SD/EU R
+ρ
where ρ is the risk premium on the USD. Now go back to our previous example
with β = 0.5. In the case of RU SD = REU R = 5%, UIP holds with ρ = 0. In
the case of RU SD = 6% and REU R = 5%, UIP holds with ρ = 0.5. In other
words, if an increasing interest rate leads to a higher risk premium, then we
have a story consistent with β < 1 and consistent with an augmented version of
the UIP condition. At first sight, this actually may seem sensible. Higher risk
premia (for instance, due to high level of government debt) may actually feed into
higher interest rates, giving a positive correlation between interest rates and risk
premia. How could we go about testing this? The most straightforward way is
to collect survey data on exchange rate expectations, plug it into our augmented
UIP condition, and then back out exchange rate risk. However, it turns out that
the risk premia are uncorrelated with the interest differential.
(d) Solution 2: Expectational errors.
In this case, we need to explain not just that we make errors (that is to be
expected!), but that the errors are systematic. In other words, we always end up
making errors in the same direction, so that the error does not have mean zero.
Consider what they refer to as the peso problem. In fact, what β < 1 suggests is
that a positive interest differential is associated with less expected depreciation.
For instance, a positive interest differential of 5% on the peso would be associated
with an expected depreciation of less than 5%, at least that’s what the data
seem to tell. However, this β < 1 may be due to a problem with the estimation
procedure. The following example should clarify things. Assume Mexico has a
fixed exchange rate. Each years investors expect with a probability of 90% that
the fixed exchange will be maintained, and with probability 10% they think there
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will be a depreciation of 50%. Expected depreciation is therefore 5%. For the
simple UIP condition to hold, investors will require a 5% interest differential on
the Mexican peso. Now, given the low probability of the devaluation, you would
need a very long time series to test the UIP. For instance, with a short time series
of 10 years, it is quite likely that you won’t observe any devaluation. In that
case, what you are observing is a persistent interest differential on the Mexican
peso, but no devaluation. This would give you a low β, in fact it would give you
a β = 0. Note that the reason why you are getting this low β has nothing to do
with theory, it only has to do with using a wrong data set (i.e., a too short time
series) to test the theory.
(e) Solution 3: investors are slow to react.
Here the idea is the following. Again, start off with the same example as before:
RU SD = REU R = 5%. This means that the expected depreciation of the exchange
rate is zero. Then increase the interest rate on the dollar by 1%. What does UIP
predict? You get an instantaneous appreciation of the dollar by 1%, as investors
adjust their portfolios, and you then get a subsequent depreciation of the dollar
by 1% over the next 12 months. Therefore, increases in interest rates are followed
by short run periods of depreciation. However, assume the initial instantaneous
effect of a 1% appreciation takes time to take effect. In this case, you might see
an increase in interest rate followed by a short run appreciation of the dollar.
This may lead to a lower β. This point of view is of course criticizable, as there
may be no reason why we should expect financial markets to react slowly. In
fact, financial markets are the paradigm of fast reactions.
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