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Chapter Objective:

International Parity Relationships and Forecasting Foreign Exchange Rates. 6. Chapter Six. Chapter Objective: This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity. 6- 0. Chapter Outline. Interest Rate Parity

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Chapter Objective:

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  1. International Parity Relationships and Forecasting Foreign Exchange Rates 6 Chapter Six Chapter Objective: This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity. 6-0

  2. Chapter Outline • Interest Rate Parity • Covered Interest Arbitrage • IRP and Exchange Rate Determination • Reasons for Deviations from IRP • Purchasing Power Parity • The Fisher Effects • Forecasting Exchange Rates • Interest Rate Parity • Purchasing Power Parity • PPP Deviations and the Real Exchange Rate • Evidence on Purchasing Power Parity • The Fisher Effects • Forecasting Exchange Rates • Interest Rate Parity • Purchasing Power Parity • The Fisher Effects • Forecasting Exchange Rates • Interest Rate Parity • Purchasing Power Parity • The Fisher Effects • Forecasting Exchange Rates • Efficient Market Approach • Fundamental Approach • Technical Approach • Performance of the Forecasters • Interest Rate Parity • Purchasing Power Parity • The Fisher Effects • Forecasting Exchange Rates • Interest Rate Parity • Purchasing Power Parity • The Fisher Effects • Forecasting Exchange Rates 6-1

  3. Interest Rate Parity • Interest Rate Parity Defined • Covered Interest Arbitrage • Interest Rate Parity & Exchange Rate Determination • Reasons for Deviations from Interest Rate Parity 6-2

  4. Interest Rate Parity Defined • IRP is an “noarbitrage” condition. • If IRP did not hold, then it would be possible for an astute trader to make unlimited amounts of money exploiting the arbitrage opportunity. • Since we don’t typically observe persistent arbitrage conditions, we can safely assume that IRP holds. …almost all of the time! 6-3

  5. F$/£ Future value = $100,000(1 + i£)× S$/£ F$/£ (1 + i$) (1 + i£) × = (1 + i$) F$/£ = S$/£ × S$/£ (1 + i£) Interest Rate Parity Carefully Defined Consider alternative one-year investments for $100,000: • Invest in the U.S. at i$. Future value = $100,000 × (1 + i$) • Trade your $ for £ at the spot rate, invest $100,000/S$/£ in Britain at i£ while eliminating any exchange rate risk by selling the future value of the British investment forward. Since these investments have the same risk, they must have the same future value (otherwise an arbitrage would exist) 6-4

  6. $1,000 S$/£ $1,000 $1,000  (1+ i£)  (1+ i£) × F$/£ S$/£ S$/£ IRP Alternative 2: Send your $ on a round trip to Britain IRP Step 2: Invest those pounds at i£ $1,000 Future Value = Step 3: repatriate future value to the U.S.A. Alternative 1: invest $1,000 at i$ $1,000×(1 + i$) = Since both of these investments have the same risk, they must have the same future value—otherwise an arbitrage would exist 6-5

  7. $1,000  (1+ i£) × F$/£ = $1,000×(1 + i$) S$/£ F$/£ × (1+ i£) = (1 + i$) S$/£ Interest Rate Parity Defined • The scale of the project is unimportant 6-6

  8. 1 + i$ F$/¥ = 1 + i¥ S$/¥ F – S i$ – i¥ ≈ S Interest Rate Parity Defined Formally, IRP is sometimes approximated as 6-7

  9. 1 + i¥ F¥/$ = 1 + i$ S¥/$ 1 + i$ F$/¥ = 1 + i¥ S$/¥ Interest Rate Parity Carefully Defined • Depending upon how you quote the exchange rate (as $ per ¥ or ¥ per $) we have: or …so be a bit careful about that. 6-8

  10. 1 + i$ F$/¥ = S$/¥ × 1 + i¥ 1 + i¥ or F¥/$= S¥/$× 1 + i$ Interest Rate Parity Carefully Defined • No matter how you quote the exchange rate ($ per ¥ or ¥ per $) to find a forward rate, increase the dollars by the dollar rate and the foreign currency by the foreign currency rate: …be careful—it’s easy to get this wrong. 6-9

  11. IRP and Covered Interest Arbitrage If IRP failed to hold, an arbitrage would exist. It’s easiest to see this in the form of an example. Consider the following set of foreign and domestic interest rates and spot and forward exchange rates. 6-10

  12. IRP and Covered Interest Arbitrage A trader with $1,000 could invest in the U.S. at 3.00%, in one year his investment will be worth $1,030 = $1,000  (1+ i$) = $1,000  (1.03) Alternatively, this trader could • Exchange $1,000 for £500 at the prevailing spot rate, • Invest £500 for one year at i£ = 2.49%; earn £512.45 • Translate £512.45 back into dollars at the forward rate F360($/£) = $2.01/£, the £512.45 will be worth $1,030. 6-11

  13. £500 £1 £500 = $1,000× $2.00 $1,000 F£(360) $1,030 = £512.45 × £1 Alternative 2: buy pounds Arbitrage I Step 2: Invest £500 at i£ = 2.49% In one year £500 will be worth £512.45 = £512.45 Step 3: repatriate to the U.S.A. at F360($/£) = $2.01/£ £500  (1+ i£) Alternative 1: invest $1,000 at 3% FV = $1,030 $1,030 6-12

  14. Interest Rate Parity & Exchange Rate Determination According to IRP only one 360-day forward rate, F360($/£), can exist. It must be the case that F360($/£) = $2.01/£ Why? If F360($/£)  $2.01/£, an astute trader could make money with one of the following strategies: 6-13

  15. Arbitrage Strategy I If F360($/£) > $2.01/£ i. Borrow $1,000 at t = 0 at i$ = 3%. ii. Exchange $1,000 for £500 at the prevailing spot rate, (note that £500 = $1,000 ÷ $2/£) invest £500 at 2.49% (i£) for one year to achieve £512.45 iii. Translate £512.45 back into dollars, if F360($/£) > $2.01/£, then £512.45 will be more than enough to repay your debt of $1,030. 6-14

  16. £500 £1 £500 = $1,000× $2.00 $1,000 F£(360) $1,030 < £512.45 × £1 Step 2: buy pounds Arbitrage I Step 3: Invest £500 at i£ = 2.49% In one year £500 will be worth £512.45 = £512.45 £500 (1+ i£) Step 4: repatriate to the U.S.A. Step 1: borrow $1,000 More than $1,030 Step 5: Repay your dollar loan with $1,030. If F£(360) > $2.01/£ , £512.45 will be more than enough to repay your dollar obligation of $1,030. The excess is your profit. 6-15

  17. Arbitrage Strategy II If F360($/£) < $2.01/£ i. Borrow £500 at t = 0 at i£= 2.49% . ii. Exchange £500 for $1,000 at the prevailing spot rate, invest $1,000 at 3% for one year to achieve $1,030. iii. Translate $1,030 back into pounds, if F360($/£) < $2.01/£, then $1,030 will be more than enough to repay your debt of £512.45. 6-16

  18. Step 2: buy dollars $2.00 $1,000 = £500× £1 Step 3: Invest $1,000 at i$ = 3% Step 4: repatriate to the U.K. F£(360) $1,030 > £512.45 × £1 Arbitrage II £500 Step 1: borrow £500 $1,000 Step 5: Repay your pound loan with £512.45 . More than £512.45 In one year $1,000 will be worth $1,030 If F£(360) < $2.01/£ , $1,030 will be more than enough to repay your dollar obligation of £512.45. Keep the rest as profit. 6-17

  19. IRP and Hedging Currency Risk You are a U.S. importer of British woolens and have just ordered next year’s inventory. Payment of £100M is due in one year. IRP implies that there are two ways that you fix the cash outflow to a certain U.S. dollar amount: a) Put yourself in a position that delivers £100M in one year—a long forward contract on the pound. You will pay (£100M)($2.01/£) = $201M in one year. b) Form a money market hedge as shown below. 6-18

  20. IRP and a Money Market Hedge To form a money market hedge: • Borrow $195,140,989.36 in the U.S. (in one year you will owe $200,995,219.05). • Translate $195,140,989.36 into pounds at the spot rate S($/£) = $2/£ to receive £97,570,494.68 • Invest £97,570,494.68 in the UK at i£ = 2.49% for one year. • In one year your investment will be worth £100 million—exactly enough to pay your supplier. 6-19

  21. £100,000,000 £97,570,494.68 = 1.0249 $2.00 $195,140,989.36 = £97,570,494.68 × £1.00 Money Market Hedge Where do the numbers come from? We owe our supplier £100 million in one year—so we know that we need to have an investment with a future value of £100 million. Since i£ = 2.49% we need to invest £97,570,494.68at the start of the year. How many dollars will it take to acquire £97,570,494.68 at the start of the year if S($/£) = $2/£? 6-20

  22. Money Market Hedge • This is the same idea as covered interest arbitrage. • To hedge a foreign currency payable, buy a bunch of that foreign currency today and sit on it. • Buy the present value of the foreign currency payable today. • Invest that amount at the foreign rate. • At maturity your investment will have grown enough to cover your foreign currency payable. 6-21

  23. 0 1 £100,000,000 Step 3 Buy £96,153,846 = at spot exchange rate. 1.04 Money Market Hedge: an Example Suppose that the spot dollar-pound exchange rate is $2.00/£ and i$ = 1% i£ = 4% Step 1 Order Inventory; agree to pay supplier £100 in 1 year. Step 5 Redeem £-denominated investment receive £100 million Step 2 Borrow $192,307,692 at i$ = 1% Step 6 Pay supplier £100 million ($192,307,692 = £96,153,846×$2/£) Step 7 Repay dollar loan with $194,230,769 Step 4 Invest £96,153,846 at i£ = 4% 6-22

  24. Can hedge this payable by buying today and investing €96,153.85 at 4% in Italy for one year. At maturity, he will have €100,000 = €96,153.85 × (1.04) €100,000 €96,153.85 = 1.04 $1.50 Dollar cost today = $144,230.77 = €96,153.85 × €1.00 Another Money Market Hedge A U.S.–based importer of Italian bicycles • In one year owes €100,000 to an Italian supplier. • The spot exchange rate is $1.50 = €1.00 • The one-year interest rate in Italy is i€ = 4% 6-23

  25. €100,000 $148,557.69 = S($/€)× × (1+ i$)T (1+ i€)T Another Money Market Hedge • With this money market hedge, we have redenominated a one-year €100,000 payable into a $144,230,77 payable due today. • If the U.S. interest rate is i$ = 3% we could borrow the $144,230,77 today and owe in one year $148,557.69 = $ 144,230,77 × (1.03) 6-24

  26. £y Repay the loan in T years $x = S($/£)× $x –$x(1 + i$)T (1+ i£)T 0 T Generic Money Market Hedge: Step One Suppose you want to hedge a payable in the amount of £y with a maturity of T: i. Borrow $x at t = 0 on a loan at a rate of i$per year. 6-25

  27. ii. Exchange the borrowed $x for £y £y (1+ i£)T (1+ i£)T Invest at i£ for the maturity of the payable. Generic Money Market Hedge: Step Two at the prevailing spot rate. At maturity, you will owe a $x(1 + i$)T. Your British investments will have grown to £y. This amount will service your payable and you will have no exposure to the pound. 6-26

  28. 1. Calculate the present value of £y at i£ £y $x = S($/£)× 3. Exchange for (1+ i£)T £y £y (1+ i£)T (1+ i£)T 4. Invest at i£ for T years. £y (1+ i£)T Generic Money Market Hedge 2. Borrow the U.S. dollar value of receivable at the spot rate. 5. At maturity your pound sterling investment pays your receivable. 6. Repay your dollar-denominated loan with $x(1 + i$)T. 6-27

  29. F180($/€) – S($/€) 360 $1.30 – $1.25 f180,€v$ = × = × 2 = 0.08 S($/€) 180 $1.25 Forward Premium • It’s just the interest rate differential implied by forward premium or discount. • For example, suppose the € is appreciating from S($/€) = 1.25 to F180($/€) = 1.30 • The forward premium is given by: 6-28

  30. Reasons for Deviations from IRP • Transactions Costs • The interest rate available to an arbitrageur for borrowing, ib may exceed the rate he can lend at, il. • There may be bid-ask spreads to overcome, Fb/Sa < F/S • Thus (Fb/Sa)(1 + i¥l)  (1 + i¥ b)  0 • Capital Controls • Governments sometimes restrict import and export of money through taxes or outright bans. 6-29

  31. (1+i€) l S0($/€) (1+i$) a b F1($/€) = b S0($/€) (1+i$) b l F1($/€) = a (1+i€) b Transactions Costs Example • Will an arbitrageur facing the following prices be able to make money? (1 + i$) F($/ €) = S($/ €) × (1 + i€) 6-30

  32. Step 1 Borrow $1m at i$ b $1m 0 1 Step 2 1 $1m × ×(1+i€)× l Buy € at spot ask Step 4 F1($/€) = F1($/€) = = b b invest € at i€ 1 Sell € at forward bid l (1+i$) $1m×(1+i$) $1m×(1+i$) (1+i$) b b b b ×(1+i€) ×(1+i€) (1+i€) = $1.4431/€ l l l S0($/€) S0($/€) S0($/€) S0($/€) S0($/€) 1 a a a a a 1 $1m × Step 3 $1m × (All transactions at retail prices.) IRP No arbitrage forward bid price (for customer): 6-31

  33. Step 3 lend at i$ l €1m × S0($/€) × (1+i$) l b 0 1 IRP €1m × S0($/€) × (1+i$) ÷ F1($/€) = l a b No arbitrage forward ask price: Step 4 F1($/€) = Step 2 a buy € at forward ask (1+i$) €1m×(1+i€) €1m×(1+i€) b b l sell €1m at spot bid = $1.4065/€ (1+i€) b S0($/€) b €1m × S0($/€) €1m Step 1borrow €1m at i€ b b (All transactions at retail prices.) 6-32

  34. Why This Seems Confusing • On the last two slides we found “no arbitrage” • Forward bid prices of $1.4431/€ and • Forward ask prices of $1.4065/€ • Normally the dealer sets the ask price above the bid—recall that this difference is his expected profit. • But the prices on the last two slides are the prices of indifference for the customer NOT the dealer. At these forward bid and ask prices the customer is indifferent between a forward market hedge and a money market hedge. 6-33

  35. i$ = 4.5% and i€ = 5.0% l l i$ = 5.0%, i€ = 5.5%. b b Setting Dealer Forward Bid and Ask • Dealer stands ready to be on opposite side of every trade • Dealer buys foreign currency at the bid price • Dealer sells foreign currency at the ask price • Dealer borrows (from customer) at the lending rates • Dealer lends to his customer at the borrowing rate 6-34

  36. $1m $1m×(1+i$) b Invest at i$ Invest at i€ b b He is willing to spend $1m today and receive He is also willing to buy at (1+i$) b spot bid forward bid F1($/€) = (1+i€) b b S0($/€) S0($/€) S0($/€) S0($/€) 1 1 1 b b b b $1m × $1m × $1m × ×(1+i€) b Setting Dealer Forward Bid Price Our dealer is indifferent between buying euro today at spot bid price and buying euro in 1 year at forward bid price. 6-35

  37. ×(1+i$) b Invest at i€ Invest at i$ b b F1($/€) = He is willing to spend €1m today and receive a He is also willing to buy at (1+i$) b spot ask forward ask (1+i€) b S0($/€) a €1m €1m×(1+i€) b €1m × S0($/€) €1m × S0($/€) €1m × S0($/€) b b b Setting Dealer Forward Ask Price Our dealer is indifferent between selling euro today at spot ask price and selling euro in 1 year at forward ask price. 6-36

  38. Purchasing Power Parity • Purchasing Power Parity and Exchange Rate Determination • PPP Deviations and the Real Exchange Rate • Evidence on PPP 6-37

  39. For example, if an ounce of gold costs $300 in the U.S. and £150 in the U.K., then the price of one pound in terms of dollars should be: P$ P$ $300 S($/£) = P£ S($/£) = = = $2/£ P£ £150 Purchasing Power Parity and Exchange Rate Determination • The exchange rate between two currencies should equal the ratio of the countries’ price levels: 6-38

  40. $1.25×(1.03) $1.23 F($/€) = = €1.00×(1.05) €1.00 Purchasing Power Parity and Exchange Rate Determination • Suppose the spot exchange rate is $1.25 = €1.00 • If the inflation rate in the U.S. is expected to be 3% in the next year and 5% in the euro zone, • Then the expected exchange rate in one year should be $1.25×(1.03) = €1.00×(1.05) 6-39

  41. F($/€) = S($/€) $1.25×(1.03) €1.00×(1.05) Purchasing Power Parity and Exchange Rate Determination • The euro will trade at a 1.90% discount in the forward market: 1.03 1 + $ = = 1.05 1 + € $1.25 €1.00 Relative PPP states that the rate of change in the exchange rate is equal to differences in the rates of inflation—roughly 2% 6-40

  42. F($/€) PPP IRP = S($/€) 1 + $ 1 + i$ F($/€) = 1 + i€ 1 + € S($/€) Purchasing Power Parity and Interest Rate Parity • Notice that our two big equations today equal each other: = 6-41

  43. F($/€) = S($/€) F($/€) – S($/€) F($/€) – S($/€) 1 + $ 1 + $ 1 + € = = – 1 – S($/€) S($/€) 1 + € 1 + € 1 + € Expected Rate of Change in Exchange Rate as Inflation Differential • We could also reformulate our equations as inflation or interest rate differentials: 1 + $ 1 + € = $– € E(e) = ≈ $– € 1 + € 6-42

  44. F($/€) – S($/€) = S($/€) Expected Rate of Change in Exchange Rate as Interest Rate Differential i$– i€ ≈ i$ –i€ E(e) = 1 + i€ 6-43

  45. $– € ≈ i$ –i€ Quick and Dirty Short Cut • Given the difficulty in measuring expected inflation, managers often use 6-44

  46. Evidence on PPP • PPP probably doesn’t hold precisely in the real world for a variety of reasons. • Haircuts cost 10 times as much in the developed world as in the developing world. • Film, on the other hand, is a highly standardized commodity that is actively traded across borders. • Shipping costs, as well as tariffs and quotas can lead to deviations from PPP. • PPP-determined exchange rates still provide a valuable benchmark. 6-45

  47. F–S S Approximate Equilibrium Exchange Rate Relationships E(e) ≈ IFE ≈ FEP ≈ PPP (i$–i¥) ≈ IRP ≈ FE ≈ FRPPP E($– £) 6-46

  48. The Exact Fisher Effects • An increase (decrease) in the expected rate of inflation will cause a proportionate increase (decrease) in the interest rate in the country. • For the U.S., the Fisher effect is written as: 1 + i$ = (1 + $ ) × E(1 + $) Where $ is the equilibrium expected “real” U.S. interest rate E($) is the expected rate of U.S. inflation i$ is the equilibrium expected nominal U.S. interest rate 6-47

  49. E(1 + ¥) 1 + i¥ 1 + i$ E(1 + $) International Fisher Effect If the Fisher effect holds in the U.S. 1 + i$ = (1 + $ ) ×E(1 + $) and the Fisher effect holds in Japan, 1 + i¥ = (1 +¥ ) ×E(1 + ¥) and if the real rates are the same in each country $ = ¥ then we get the International Fisher Effect: = 6-48

  50. E(1 + ¥) 1 + i¥ = 1 + i$ E(1 + $) 1 + i¥ F¥/$ = 1 + i$ S¥/$ F¥/$ E(1 + ¥) = S¥/$ E(1 + $) International Fisher Effect If the International Fisher Effect holds, and if IRP also holds then forward rate PPP holds: 6-49

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