2. Chapter 6 International Parity Relationships and Forecasting Exchange Rates

3. Introduction
It is important to have a firm understanding of the forces driving exchange rate changes.
This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity.
An understanding of these parity relationships provides insights into:
how foreign exchange rates are determined.
how to forecast foreign exchange rates.

4. Arbitrage definition
Arbitrage : the act of simultaneously buying and selling equivalent assets for the purpose of making guaranteed profits.

5. Interest rate parity (IRP)
IRP is an arbitrage condition that must hold when international financial markets are in equilibrium. 
If IRP did not hold, then it would be possible for an trader to make unlimited amounts of money exploiting the arbitrage opportunity.

6. Interest rate parity (IRP)
Suppose that you have $1 to invest over a 1 year period. Consider two alternative ways:
invest domestically at the U.S. interest rate.
invest in a foreign country (e.g. UK) at the foreign interest rate and use a forward contract to eliminate the exchange risk.
your U.K. investment coupled with forward hedging is a perfect substitute for the domestic U.S. investment
Both investments yield the same cash flow.
It is assumed here that you want to consider only default-free investments.

7. Interest rate parity (IRP)
US investment:
The value of $1 would be $1 1+ i $
UK investment:
Exchange $1 for a pound amount, that is, £(1/S). (where, S=S($/£))
Invest the pound amount at the U.K. interest rate (i£), with the maturity value of £(1/S)(1 + i£)
Sell the maturity value of the U.K. investment forward in exchange for a predetermined dollar amount, that is, $[(1/S)(1 + i£)]F. (where, F =F($/£))
Arbitrage equilibrium would dictate that:
1+ 𝑖 $ = 𝐹( $ £ ) 𝑆( $ £ ) 1+ 𝑖 £
𝐹( $ £ )=𝑆( $ £ ) 1+ 𝑖 $ 1+ 𝑖 £

8. Interest rate parity (IRP)
An arbitrage equilibrium condition holding that the interest rate differential between two countries should be equal to the forward exchange premium or discount. Violation of IRP gives rise to profitable arbitrage opportunities.
𝐹( $ £ ) 𝑆( $ £ ) = 1+ 𝑖 $ 1+ 𝑖 £
If i$ > i£ , then the £ Forward is at premium to spot.
If i$ < i£ , then the £ Forward is at discount to spot.

9. Covered interest arbitrage
A situation that occurs when IRP does not hold, thereby allowing certain arbitrage profits to be made without the arbitrageur investing any money out of pocket or bearing any risk.

10. Example of CIA Suppose that i$=5%, i£=8%, S($/£)=1.8, F12($/£)=1.78
Suppose the arbitrager can borrow $1,000,000 or £555,556 (its equivalent in £).
Check if IRP holds:
𝐹( $ £ ) 𝑆( $ £ ) 1+ 𝑖 £ = =1.068
1+ 𝑖 $ =1+0.05=1.05
IRP does not hold implying that profitable arbitrage exists..

11. Example of CIA
The arbitrager can carry out the following transactions:
In the United States, borrow $1,000,000. Repayment in one year will be $1,050,000 = $1,000,000 × 1.05.
Buy £555,556 spot using $1,000,000.
Invest £555,556 in the U.K.; the maturity value will be £600,000 = £555,556 × 1.08.
Sell £600,000 forward in exchange for $1,068,000 = (£600,000)($1.78/£).
In one year, the arbitrager will
Receive £600,000 from UK investment.
Deliver £600,000 to the counterparty of the forward contract and receive $1,068,000.
Repay $1,050,000 for his US loan.
Arbitrage profit= $1,068,000 − $1,050,000= $18,000

12. Market adjustment How long will this arbitrage opportunity last?
Only for a short while. As soon as deviations from IRP are detected, informed traders will immediately carry out CIA transactions.
The following adjustments restore IRP:
The pound will appreciate in the spot market (S↑).
The pound will depreciate in the forward market (F↓).
The interest rate will rise in the United States (i$↑).
The interest rate will fall in the U.K. (i£↓).

13. CIA example 2:
Suppose that the market condition is summarized as follows:
Three-month interest rate in the United States: 8.0% per annum.
Three-month interest rate in Germany: 5.0% per annum.
Current spot exchange rate: €0.800/$.
Three-month forward exchange rate: €0.7994/$.
Does a CIA opportunity exists? If yes, calculate the arbitrage profit, assuming that the arbitrager can borrow up to $1,000,000 or the equivalent € amount.

14. Interest Rate Parity and Exchange Rate Determination
𝑆( $ £ )= 1+ 𝑖 £ 1+ 𝑖 $ 𝐹( $ £ )
All else equal, an increase in the U.K. interest rate will lead to a higher foreign exchange value of the £.
 A higher U.K. interest rate will attract capital to the United States, increasing the demand for dollars.  
All else equal, an increase in the U.S. interest rate will lead to a higher foreign exchange value of the dollar.
 A higher U.S. interest rate will attract capital to the United States, increasing the demand for dollars.
A decrease in the U.S. interest rate will lower the foreign exchange value of the dollar.

15. Foreign exchange expectations
Some forecasters believe that for the major floating currencies, foreign exchange markets are “efficient” and forward exchange rate are unbiased predictors of future exchange rates.
The forward expectations hypothesis states that the forward exchange rate Ft is equal to the expected value of the spot exchange rate at time t.
© Jon Moulton

16. Foreign exchange expectations
The forward expectations hypothesis can be written as:
𝐹 𝑡 =𝐸 𝑆 𝑡
An unbiased predictor doesn't mean the future spot rate will actually be equal to the forward rate. It will overestimate and underestimate in equal frequency.
Academic evidence is mixed.
© Jon Moulton

17. Uncovered interest arbitrage (carry trade)
Borrowing in “cheap” rate country, exchanging to desired country for investment, and investing in desired securities.
Without using forward contracts  FX risk exists
Works when exchange rate is relatively stable and interest rate differential exists between countries.

18. Uncovered interest arbitrage (carry trade)
Suppose Japan’s 1-year rate is 0.4% per annum, S(Yen/$)=80, F1(Yen/$)= , and US rate is 0.65%.
Is it possible to do CIA? No
But if you can borrow more cheaply in Japan , and the spot rate does not vary within the year, then uncovered re- exchange investment would be profitable.
© Jon Moulton

19. Reasons for Deviations from Interest Rate Parity
IRP tends to hold well, but it may not hold precisely all the time for two reasons: transaction costs and capital controls.
The interest rate at which the arbitrager borrows, ia, tends to be higher than the rate at which he lends, ib, reflecting the bid-ask spread.
Governments sometimes restrict capital flows, inbound and/or outbound.

20. Japan adopted the new Foreign Exchange and Foreign Trade Control Law, which generally liberalized foreign exchange transactions
Capital restrictions were removed in 1979
New capital controls introduced

21. Purchasing Power Parity (PPP)
The PPP theory states that the exchange rate between currencies of two countries should be equal to the ratio of the countries' price levels. 
It is an application of the law of one price internationally to a standard consumption basket.

22. What is the law of one price?
The law of one price states that two identical assets should sell for the same price regardless of where they are sold (assuming no market frictions).

23. What is a consumption basket?
It is a list of the most common consumption items bought by people all around the world.
Provides an insight on the cost of living and is used to calculate inflation.

24. Absolute PPP
Formally, PPP states that the exchange rate between the dollar and the pound should be:
𝑆 𝑃𝑃𝑃 ( $ £)= 𝑃 $ 𝑃 £
 P$ :dollar price of the standard consumption basket in the US.
 P£ :pound price of the same basket in the United Kingdom. 
This equation is called the absolute version of PPP.

25. Big Mac PPP
The Economist each year compiles local prices of Big Macs around the world and computes the so-called “Big Mac PPP.”
They calculate the exchange rate that would equalize the hamburger prices between America and elsewhere. 
It is a proxy for under/over-valuation of currencies.

26. Example
A Big Mac costs $4.33 in America and ¥ in China. Thus, the Big Mac PPP would imply the exchange rate would be
SPPP(¥/$)= ¥15.65 /$4.33 = 3.62¥/$.
The actual exchange rate, however, is 6.39¥/$, implying that
the yuan is substantially undervalued.

27. Example 2
The price of Big Mac is Switzerland is CHF 6.5 and $4.33 in the USA. Calculate the Big Mac PPP implied CHF/$exchange rate . If S(CHF/$)=0.99, would you the say the CHF is over or undervalued?
the Big Mac PPP for Switzerland is 1.52 Swiss francs per dollar, compared with the actual exchange rate of 0.99 francs per dollar. This implies that the Swiss franc is very much overvalued.

29. Testing Big Mac PPP
Pakko and Pollard (2003) conclude that Big Mac PPP holds in the long run, but currencies can deviate from it for lengthy periods.
The absolute version of PPP assumes no barriers to trade. High tariffs and transportation costs on beef and lettuce.
Prices are affected by taxes and competition.
Prices of non-tradeable goods (real estate, utilities, labor) are also inputs that affect production costs.
© Jon Moulton

30. Relative form of PPP
The relative version of the PPP states exchange rate movement should offset any inflation differential between countries:
𝑒≈ 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 $ − 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 £
𝑤ℎ𝑒𝑟𝑒, 𝑒=𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒 𝑟𝑎𝑡𝑒
For example, if the inflation rate is 6% per year in the United States and 4% in the U.K., then the pound should appreciate against the dollar by about 2 percent, that is, e ≈ 2 percent.

31. Relative form of PPP Relative PPP can be written as:
𝑆 𝑡+1 𝑃𝑃𝑃 ( 𝐴 𝐵) 𝑆 𝑡 ( 𝐴 𝐵) = 1+𝑖𝑛𝑓 𝐴 1+𝑖𝑛𝑓 𝐵 or
𝑆 𝑡+1 𝑃𝑃𝑃 ( 𝐴 𝐵) = 𝑆 𝑡 ( 𝐴 𝐵) × 1+𝑖𝑛𝑓 𝐴 1+𝑖𝑛𝑓 𝐵

32. Relative PPP If 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝐵 < 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝐴
 PPP predicts country B’s currency will appreciate.
If 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝐵 > 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝐴
 PPP predicts country B’s currency will depreciate.
In a longer term view, relative PPP predicts where foreign exchange rate are headed.
© Jon Moulton

33. Evidence on Purchasing Power Parity
Even if PPP may not hold in reality, it can still play a useful role in economic analysis.
PPP-determined exchange rate can be used as a benchmark in deciding if a country's currency is undervalued or overvalued against other currencies.
PPP may hold over the very long term but not in the short term.
More meaningful international comparisons of economic data can be made using PPP-determined rather than market-determined exchange rates.

34. Evidence on Purchasing Power Parity
Reasons for failure of PPP:
Baskets of goods might be different between countries.
Substantial barriers to international commodity arbitrage exist. 
Deviations from PPP can result from transportation costs or tariffs and quotas imposed on international trade.
As long as there are nontradables in the consumption basket, PPP will not hold in its absolute version. 

35. International Fisher Effect
The Fisher effect is a theory stating that the nominal interest rate is the sum of the real interest rate and the expected inflation rate.
The domestic Fisher Effect says:
1+𝑖 = 1+𝑟 1+𝑖𝑛𝑓 =1+𝑖𝑛𝑓+𝑟+𝑟×𝑖𝑛𝑓
𝑖=𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
𝑟=𝑟𝑒𝑎𝑙 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛
𝑖𝑛𝑓=𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒
© Jon Moulton

36. International Fisher Effect
International fisher effect takes fisher effect in two countries:
1+ 𝑖 $ = 1+ 𝑟 $ 1+ 𝑖𝑛𝑓 $
1+ 𝑖 £ = 1+ 𝑟 £ 1+ 𝑖𝑛𝑓 £
Dividing the two equations we get..
1+ 𝑖 $ 1+ 𝑖 £ = 1+ 𝑟 $ 1+ 𝑟 £ 𝑖𝑛𝑓 $ 1+ 𝑖𝑛𝑓 £
Assuming real rates of return are equal =1
© Jon Moulton

37. International Fisher Effect
The international fisher effect says:
Relative nominal rate equate to relative inflation rate.
Evidence shows this holds in the long run.
Combined with the relative PPP, we get:
𝑆 𝑡+1 ( 𝐴 𝐵) 𝑆 𝑡 ( 𝐴 𝐵) = 1+ 𝑖 𝐴 1+ 𝑖 𝐵
The international fisher effect can be written as:
𝐸 𝑒 ≈ 𝑖 𝐴 − 𝑖 𝐵
© Jon Moulton

38. International Fisher Effect
Thus, actual results says:
Currencies with high interest rates tend to depreciate and currencies with low interest rates tend to appreciate.
Evidence indicates this also holds in the long-run but , but with significant deviations in the short-run.
© Jon Moulton

39. Forecasting foreign exchange
Three approaches to to exchange rate forecasting :
Efficient market approach: uses such market-determined prices as the current exchange rate or the forward exchange rate to forecast the future exchange rate.
Fundamental approach: uses various formal models of exchange rate determination for forecasting purposes.
Technical approach: identifies patterns from the past history of the exchange rate and projects it into the future.
The existing empirical evidence indicates that neither the fundamental nor the technical approach outperforms the efficient market approach.

40. Practice problem 1
Suppose that the treasurer of IBM has an extra cash reserve of $100,000,000 to invest for six months. The six-month interest rate is 8 percent per annum in the United States and 7 percent per annum in Germany. Currently, the spot exchange rate is €1.01 per dollar and the six-month forward exchange rate is €0.99 per dollar. The treasurer of IBM does not wish to bear any exchange risk. Where should he/she invest to maximize the return?

41. Practice problem 2
While you were visiting London, you purchased a Jaguar for £35,000, payable in three months. You have enough cash at your bank in New York City, which pays 0.35% interest per month, compounding monthly, to pay for the car. Currently, the spot exchange rate is $1.45/£ and the three- month forward exchange rate is $1.40/£. In London, the money market interest rate is 2.0% for a three-month investment. There are two alternative ways of paying for your Jaguar. (a) Keep the funds at your bank in the U.S. and buy £35,000 forward. (b) Buy a certain pound amount spot today and invest the amount in the U.K. for three months so that the maturity value becomes equal to £35,000. Evaluate each payment method. Which method would you prefer? Why?

42. Practice problem 3
Currently, the spot exchange rate is $1.50/£ and the three-month forward exchange rate is $1.52/£. The three-month interest rate is 8.0% per annum in the U.S. and 5.8% per annum in the U.K. Assume that you can borrow as much as $1,500,000 or £1,000,000. a. Determine whether the interest rate parity is currently holding. b. If the IRP is not holding, how would you carry out covered interest arbitrage? Show all the steps and determine the arbitrage profit. c. Explain how the IRP will be restored as a result of covered arbitrage activities.

43. Practice problem 5
In the issue of October 23, 1999, the Economist reports that the interest rate per annum is 5.93% in the United States and 70.0% in Turkey. Why do you think the interest rate is so high in Turkey? Based on the reported interest rates, how would you predict the change of the exchange rate between the U.S. dollar and the Turkish lira?

44. Practice problem 6
As of November 1, 1999, the exchange rate between the Brazilian real and U.S. dollar is R$1.95/$. The consensus forecast for the U.S. and Brazil inflation rates for the next 1- year period is 2.6% and 20.0%, respectively. What would you forecast the exchange rate to be at around November 1, ?

45. Practice problem 8
Suppose that the current spot exchange rate is €1.50/₤ and the one- year forward exchange rate is €1.60/₤. The one-year interest rate is 5.4% in euros and 5.2% in pounds. You can borrow at most €1,000,000 or the equivalent pound amount, i.e., ₤666,667, at the current spot exchange rate.
Show how you can realize a guaranteed profit from covered interest arbitrage. Assume that you are a euro-based investor. Also determine the size of the arbitrage profit.
Discuss how the interest rate parity may be restored as a result of the above transactions.
Suppose you are a pound-based investor. Show the covered arbitrage process and determine the pound profit amount.
answers: a)profit=€68,134 c) profit= £42.584