Week 3
The Parities
The Parities
• There are three fundamental parity conditions that,
in equilibrium, are supposed to hold across
international markets.
– Covered Interest Rate Parity
– Purchasing Power Parity (also called the “Law
of One Price”)
– Uncovered Interest Rate Parity (also called the
“International Fisher effect”).
Covered Interest Rate Parity
• The covered interest rate parity is simply the
relation between the forward price, the spot, and
interest rates (that we have already seen): F = S(1
+ r n/36)/(1 + r*n/360).
• In other words, the forward premium or discount
for one currency relative to another should be
proportional to the difference between nominal
interest rates.
• If this relation does not hold, then there is an
arbitrage. In equilibrium, there should not be
arbitrage opportunities. Therefore, this relation
should always hold.
The Other Two Parities
• Unlike the covered interest rate parity, the PPP and
the uncovered interest rate parities are not
arbitrage relationships. They are relationships that
we expect to hold in equilibrium – but if they do
not hold, it may not be possible to arbitrage the
violation of the parity.
• Because we cannot arbitrage them, we only expect
these parities to hold over the long run.
• In the long run, what factors would we expect to
determine the exchange rate?
• In general, we will see that one might expect
inflation, interest rates, and exchange rates (across
the two countries) to be determined
simultaneously.
• We will express the relation between exchange
rates and inflations in terms of a “Purchasing
Power Parity” (or the Law of One Price).
• And we will term the relation between exchange
rates and interest rates as the “International Fisher
Effect” or the “Uncovered interest rate parity”.
Purchasing Power Parity (PPP)
• We use money to buy goods. Given a certain
amount of money, should we, in the long run, be
able to buy the same quantity of goods in either
country?
• If we can, then given the existing price levels (or
inflation) in each country, we should be able to
figure out the exchange rate.
• P(DC) = S(DC/FC) x P*(FC)
• Here, P is the price-level in the domestic
country,and P* is the price level in the foreign
country (FC). S=exchange rate in direct terms.
• This equation is called the “Law of One Price”.
• We can use the PPP to compute the real exchange
rate.
Real Exchange Rate
• The real exchange rate is defined as the ratio of
the actual exchange rate and the exchange rate
implied by PPP.
• Real S = (Actual S)/(PPP Implied Exchange Rate)
• If there was PPP, then the real exchange rate
would be equal to 1. If it is greater than 1, then the
foreign currency is overvalued relative to the
domestic currency (and, of course, the domestic
currency is undervalued.)
• If it is less than 1, the foreign currency is
undervalued relative to the domestic currency.
An Illustration
• The Economist constructs the Big Mac Index as a
(light-hearted) way of computing PPP using the
prices of Big Macs across different countries.
• http://www.economist.com/markets/bigmac/displa
yStory.cfm?story_id=4065603
“Italians like their coffee strong and their
currencies weak. That, at least, is the conclusion
one can draw from their latest round of grumbles
about Europe's single currency. But are the
Italians right to moan? Is the euro overvalued?
Our annual Big Mac index (see table) suggests
they have a case: the euro is overvalued by 17%
against the dollar. How come? The euro is worth
about $1.22 on the foreign-exchange markets. A
Big Mac costs €2.92, on average, in the euro zone
and $3.06 in the United States. The rate needed
to equalise the burger's price in the two regions is
just $1.05. To patrons of McDonald's, at least,
the single currency is overpriced.”
Calculation of PPP and Real Exchange
Rate
• How does one compute the real over- or under-valuation?
• One Big Mac cost $3.06 in the US, and Euro 2.92, on
average, in the Euro Zone. The current exchange rate is
1.22 $/Euro.
• The Big Mac PPP implies an absolute PPP implied
exchange rate of 3.06/2.92=1.05 $/Euro.
• Thus, the real exchange rate (in direct terms for the US):
• Real Rate = Actual/PPP = (1.22)/(1.05)=1.1642 .
• So the Euro is 16.42% stronger in real terms compared to
the US$.
• In indirect terms, the real rate is 1/1.1642. Therefore, the
US$ is (1-1/1.1642)=(1 – 0.86) = 14% weaker in real
terms relative to the Euro.
• Here is another way of computing the real rate, which is
equivalent to the earlier methodology (please verify).
• To figure out if the FC is under or over-valued, take the
money that buys you 1 unit in the foreign country, convert
it at the current exchange rate to units of DC, and ask how
much of that same good you can buy in the domestic
country. The amount of goods you can buy in the domestic
country is the real value of the FC.
• The cost of a Big Mac in US is $3.06.
• The cost of a Big Mac in China is $1.27.
• Thus, the amount of money that buys 1 Big Mac in China
buys only 0.415 (1.27/3.06) Big Macs in US.
• Thus, the real value of the Chinese Yuan is 0.415.
• Equivalent, the Yuan is 58.5% (1-0.415) undervalued
against the US$.
• By the Big Mac index, the Chinese currency is the most
undervalued currency in the world (and the Swiss Franc, by
65%, is the most overvalued).
Real Impact of the Asian Currency Crisis
• The following example is instructive in understanding the
“real” impact of the Asian currency crisis. (From The
Economist (2/7/1998))
• Indonesia: 1996 GDP = $226b, 1998 Feb: $51b. However,
GDP measured in PPP = $1020b.
• S Korea: 1996 GDP =$485b, 1998 Feb=$272b, GDP using
PPP = 660 B.$
• Hong Kong: 1998: $188b, and PPP is also 190b (That is,
no change).. Singapore also had little change in GDP on a
PPP basis.
Why should PPP not hold for a single
good?
• Do we really expect PPP to hold for every single
good? Here are some good reasons why PPP
would not hold at the single good level.
• 1. Transaction costs of undertaking spatial
arbitrage.
• 2. The goods may not be equivalent, or the
perceived quality might be different.
• 3. Tariffs and other trade barriers may prevent
arbitrage.
• 4. The good may not be tradable.
• So why are Big Mac prices so different across countries?
• Big Macs are perishable, so cannot be traded.
• However, if the components of the Big Mac (labor,
material, other costs) are tradable, then we again might
expect the Big Mac prices to converge.
• However, major components of costs are not tradable –
labor, electricity, and rent. These costs comprise between
55 and 64% of the cost of the Big Mac.
– For more detail, see David Parsley and Shang-Jin Wei, “A Prism
into PPP Puzzles.”
http://www2.owen.vanderbilt.edu/david.parsley/research.htm)
Absolute PPP
• Traditionally, we do not compare PPP at the level
of a single good. We do so using price indices.
• Absolute PPP: P(US) = S ($/FC) x P*(FC)
• This is the same as what we saw earlier, except
that P, P* are now the price indices (like the CPI).
• All the reasons that suggest why PPP would not
hold for a single good also apply when we
compute PPP using price indices.
• In addition, absolute PPP may not hold for yet
another reason: The weights that construct the
price deflator may be different in different
countries, depending on the relative importance of
the good in the country.
Alternative to Absolute PPP
• In fact, even within a single country, the relative
importance of a consumption item changes. For
example, the basket used to calculate the CPI in
the US has recently (1997-98) been changed to
include, amongst others, communication costs,
and the fact that Americans eat more pasta than
meat than in 1987 (when the previous update took
place.)
• Given all these problems, it is not surprising if
absolute PPP does not hold.
• Thus, we instead ask whether PPP holds in a
relative sense.
• PPP holds in relative terms if the real exchange
rate does not change over time.
Relative PPP
• Suppose we assume that the exchange rates at some point in
time (either now or in the past) was “correct”. Then,
assuming relative PPP, we expect that the exchange rate
changes thereafter should reflect relative difference in
inflations.
• S(t) = exchange rate at time t
• S(T) = PPP implied exchange rate at time T (after t)
• I = cumulative domestic inflation between T and t
• I* = cumulative foreign inflation between T and t
• Then:
• S(T) = S(t) (1 + I)/(1+I*)
• 1 + (S(T) – S(t))/S(t) = (1 + I)/(1+I*)
• (1 + change in exchange rates) = (1 + I)/(1+I*).
• An alternative way of writing this:
• (S(T)-S(t))/S(t) = (I - I*)/(1+I)
• i.e. The appreciation or depreciation is approximately equal
to the difference in inflations.
Real Exchange Rate using Price Indices
• Consider the base year as 1973. The CPI for
US=40.3, and Japan=44. The exchange rate then
was $0.003762/Yen.
• In 1989: the CPI for US is 117.2 and Japan is
104.6. The exchange rate is $0.006971/Y.
• The PPP implied exchange rate for 1989 is:
0.003762 (117.2/40.3)/(104.6/44)=$0.004602.
• The real exchange rate in 1989: actual/PPP=
0.006971/0.004602=1.5148.
• Thus, the Yen has appreciated in real terms, and is
overvalued by 51.48%.
– PPP did not hold in relative terms because the real
exchange rate for the Yen changed over time.
Real Effective Exchange Rate
• The effective exchange rate (or the trade-weighted
exchange rate) is defined as an average of the
country’s exchange rates with its trading partners,
weighted by the proportion of trade done with it.
Thus, it is a measure of the competitiveness of the
country.
• Suppose the base year is 1995, and US has 60% of
its trade with Japan, and 40% of its trade with
Germany.Suppose the actual exchange rate in the
base year was 100Yen/$, and DM 1.5/$. Let the
effective exchange rate index be defined to be 100
in this base year.
• In 1996: suppose the actual exchange rate is
Yen105/$, and DM1.65/$.
• The effective real exchange rate is then:100
[(105/100)x0.6 + (1.65/1.5)x0.4]=107.
• So that the US$ has appreciated by 7%, against its
trading partners.
• The effective exchange rate is useful when we we
are trying to determine the overall effect of
exchange rate changes on the country’s trade. If
we use real exchange rates to make the above
calculations, then the real effective exchange rate
will give us an overall picture of the
competitiveness of the US economy in the global
economy.
Empirical Evidence for PPP
• First, of all it is important to understand that the
inflation series is less volatile than exchange rates
- so there is going to be considerable noise when
we compare the two series.
• Empirical evidence seems to suggest that there are
persistent deviations from PPP both in the short
and long run.
Empirical Evidence for PPP
• 1. In a simple regression of changes in nominal
exchange rate on inflation differentials, the slope
coefficient is less than 1. Thus, the exchange rate
change is less than what one would expect. (This
is not entirely surprising, as one would expect that
inflation is only one of the factors that determine
exchange rate changes.).
• 2. The slope coefficient increases with the length
of the interval.
• 3. The coefficient is stronger under hyperinflation.
Other Evidence
• The regression tests demonstrate the simple
observation that the exchange rate series is far
more volatile than the inflation series.
• Perhaps it is worth asking how long and by how
much the real exchange rate can vary from 1. This
can be tested by checking if the changes in the real
exchange rate follow a random walk. Abuaf and
Jorian (Journal of Finance, 1990) find that the real
exchange rate seems to mean-revert to 1, but may
take 3-5 years to reduce its over/under valuation
by half.
Uncovered Interest Rate Parity
• What should be the relation between exchange
rates and interest rates?
• Suppose you have a choice of investing your
money in USA at the eurodollar interest rate, r, or
in Japan at the euro-yen interest rate, r*. Which
will you choose?
• Of course, as we have seen before, that if you
invested in the foreign currency and hedged
yourself, then the covered interest rate parity (i.e.
by the definition of the forward rate) tells you
that, in either case, you should earn the same
amount.
• But what if you did not want to hedge yourself?
• In the long run, if economies were globally
integrated then one might again expect that it
should not matter.
• Suppose S(t) is the current spot rate, and S(t+1) is
the spot rate you expect to hold in the future (so to
be precise we should actually write it as E[S(
t+1)]).
• If you invest $1 today in the US, you will get
$(1+r) one year from now.
• If you invest $1 in the foreign currency, you
expect to get $(1)(1+r*)( s(t+1)/s(t) ).
• So that if in equilibrium you were indifferent
between the two, then you should expect
• E[S(t+1)] = S(t)(1 + r)/(1 + r*)
• This is the same equation as we had earlier for the
forward price except that we replaced the forward
price, F, by the expected future spot price,
E[S(t+1)].
• As previously we can rewrite this as:
• [E(S(t+1)-S(t)]/S(t) = (r-r*)/(1+r*)
• So that the expected change in spot rate is
approximately equal to the difference in interest
rates.
• This suggests that if the domestic interest rate is
higher than the foreign, then investors expect a
depreciation of the US$, and if its lower than the
foreign then they expect an appreciation of the $.
• This is the same as saying that the speculators
expect the future spot rate to be the forward rate.
Empirical Evidence
• The empirical evidence regarding the central
prediction of the interest rate parity is very poor.
• We can, as we did with the PPP, test the relation
by a linear regression: if we regress (S[t+1]S[t])/S[t] on interest rate differentials, (r-r*), then
the theory would indicate a slope coefficient of 1.
But when we do the empirical test, we rarely get
such a coefficient, and even sometimes get a
negative slope coefficient!
• For example, the Japanese interest rates have been
about consistently lower than the US for the last
few years - but the fluctuations in the Yen have
included both a 20% depreciation, and a 20%
appreciation.
International Fisher Effect
• We can combine the PPP and UIP relationships as
follows.
• PPP => S(T) = S(t) x (1 + I)/(1+I*)
• UIP => S(T) = S(t) x (1 + r)/(1+r*)
• This implies that:
• (1+I)/(1+I*) = (1+r)/(1+r*)
• (1+r*)/(1+I*)= (1+r)/(1+I)
• Thus, (r* - I*) is approximately equal to (r – I).
• In other words, the real exchange rates should be
approximately equal across countries.