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/displayStory.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.