# Chapter Objective: This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity. 5 Chapter.

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Chapter Objective: This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity. 5 Chapter Five International Parity Relationships and Forecasting Foreign Exchange Rates 5-0

Chapter Outline Interest Rate Parity Purchasing Power Parity X The Fisher Effects X 5-1

Interest Rate Parity – applied in two ways: Arbitrage & Money Market Hedge Interest Rate Parity Defined Covered Interest Arbitrage Interest Rate Parity & Exchange Rate Determination Reasons for Deviations from Interest Rate Parity 5-2

…almost all of the time! Interest Rate Parity Defined IRP is an “no arbitrage” 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. 5-3

S \$/£ × F \$/£ = (1 + i £ ) (1 + i \$ ) Interest Rate Parity Carefully Defined Consider alternative one-year investments for \$100,000: 1. Invest in the U.S. at i \$. Future value = \$100,000 × (1 + i \$ ) 2. 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. S \$/£ F \$/£ Future value = \$100,000(1 + i £ )× S \$/£ F \$/£ (1 + i £ ) ×= (1 + i \$ ) Since both of these investments have the same risk, they must have the same future value (otherwise an arbitrage would exist) 5-4

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

Interest Rate Parity Defined (1 + i \$ ) F \$/£ S \$/£ × (1+ i £ ) = \$1,000×(1 + i \$ ) \$1,000 S \$/£  (1+ i £ ) × F \$/£ = 5-6

Interest Rate Parity Defined Formally, IRP is sometimes approximated as i \$ – i £ ≈ S F – S 1 + i \$ 1 + i £ S \$/£ F \$/£ = 5-7

Interest Rate Parity Carefully Defined Depending upon how you quote the exchange rate (as \$ per £ or £ per \$) we have: 1 + i \$ 1 + i £ S £/\$ F £/\$ = or …so be a bit careful about that. 5-8

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. 1 + i \$ 1 + i £ F \$/£ = S \$/£ × or 1 + i \$ 1 + i £ F £/\$ = S £/\$ × 5-9

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. Spot exchange rateS(\$/£)=\$2.0000/£ 360-day forward rateF 360 (\$/£)=\$2.0100/£ U.S. discount ratei\$i\$ =3.00% British discount rate i £ =2.49% 5-10

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 1. Exchange \$1,000 for £500 at the prevailing spot rate, 2. Invest £500 for one year at i £ = 2.49%; earn £512.45 3. Translate £512.45 back into dollars at the forward rate F 360 (\$/£) = \$2.01/£, the £512.45 will be worth \$1,030. 5-11

Arbitrage I Invest £500 at i £ = 2.49% \$1,000 £500 £500 = \$1,000× \$2.00 £1 In one year £500 will be worth £512.45 = £500  (1+ i £ ) \$1,030 =£512.45 × £1 F £ (360) Step 3: repatriate to the U.S.A. at F 360 (\$/£) = \$2.01/£ Alternative 1: invest \$1,000 at 3% FV = \$1,030 Alternative 2: buy pounds Step 2: £512.45 \$1,030 5-12

Interest Rate Parity & Exchange Rate Determination According to IRP only one 360-day forward rate, F 360 (\$/£), can exist. It must be the case that F 360 (\$/£) = \$2.01/£ Why? If F 360 (\$/£)  \$2.01/£, an astute trader could make money with one of the following strategies: 5-13

Arbitrage Strategy I If F 360 (\$/£) > \$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 F 360 (\$/£) > \$2.01/£, then £512.45 will be more than enough to repay your debt of \$1,030. 5-14

Arbitrage I Invest £500 at i £ = 2.49% \$1,000 £500 £500 = \$1,000× \$2.00 £1 In one year £500 will be worth £512.45 = £500  (1+ i £ ) \$1,030 <£512.45 × £1 F £ (360) Step 4: repatriate to the U.S.A. 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. Step 1: borrow \$1,000 Step 2: buy pounds Step 3: Step 5: Repay your dollar loan with \$1,030. £512.45 More than \$1,030 5-15

Arbitrage Strategy II If F 360 (\$/£) < \$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 F 360 (\$/£) < \$2.01/£, then \$1,030 will be more than enough to repay your debt of £512.45. 5-16

Arbitrage II \$1,000 £500 \$1,000 = £500× £1 \$2.00 In one year \$1,000 will be worth \$1,030 >£512.45 × £1 F £ (360) Step 4: repatriate to the U.K. 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. Step 1: borrow £500 Step 2: buy dollars Invest \$1,000 at i \$ = 3% Step 3: Step 5: Repay your pound loan with £512.45. \$1,030 More than £512.45 5-17

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. Spot exchange rateS(\$/£)=\$2.00/£ 360-day forward rateF 360 (\$/£)=\$2.01/£ U.S. discount ratei\$i\$ =3.00% British discount rate i £ =2.49% 5-18

IRP and a Money Market Hedge To form a money market hedge: 1. Borrow \$195,140,989.36 in the U.S. (in one year you will owe \$200,995,219.05). 2. Translate \$195,140,989.36 into pounds at the spot rate S(\$/£) = \$2/£ to receive £97,570,494.68 3. Invest £97,570,494.68 in the UK at i £ = 2.49% for one year. 4. In one year your investment will be worth £100 million—exactly enough to pay your supplier. 5-19

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 £97,570,494.68 at 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/£? £97,570,494.68 = £100,000,000 1.0249 \$195,140,989.36 = £97,570,494.68 × \$2.00 £1.00 5-20

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. 5-21

A Similar Example – outlined as 7 steps X Step 6 Pay supplier £100 million Step 1 Order Inventory; agree to pay supplier £100 million in 1 year. 01 Step 5 Redeem £-denominated investment receive £100 million Suppose that the spot dollar-pound exchange rate is \$2.00/£ and i \$ = 1% i £ = 4% Step 4 Invest £96,153,846 at i £ = 4% Step 3 Buy £96,153,846 = at spot exchange rate. £100,000,000 1.04 Step 2 Borrow \$192,307,692 at i \$ = 1% Step 7 Repay dollar loan with \$194,230,769 (\$192,307,692 = £96,153,846×\$2/£) 5-22

Another Money Market Hedge (For students’ practicing) X 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% \$1.50 €1.00 Dollar cost today = \$144,230.77 = €96,153.85 × €100,000 1.04 €96,153.85 =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) 5-23

Another Money Market Hedge (For students’ practicing) X \$148,557.69 = \$ 144,230,77 × (1.03) 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 = €100,000 (1+ i € ) T (1+ i \$ ) T ×S(\$/€)× 5-24

Generic Money Market Hedge (maturity of T) – only for reference X 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. \$x = S(\$/£)× £y (1+ i £ ) T 0T \$x\$x–\$x(1 + i \$ ) T Repay the loan in T years 5-25

Generic Money Market Hedge (maturity of T) – only for reference X at the prevailing spot rate. £y (1+ i £ ) T ii. Exchange the borrowed \$x for Invest at i £ for the maturity of the payable. £y (1+ i £ ) T 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. 5-26

Generic Money Market Hedge Summary X 1. Calculate the present value of £y at i £ £y (1+ i £ ) T 2. Borrow the U.S. dollar value x of 1.’s value at the spot rate. \$x = S(\$/£)× £y (1+ i £ ) T 3. Exchange for £y (1+ i £ ) T 4. Invest at i £ for T years. £y (1+ i £ ) T 5. At maturity your pound sterling investment pays your payable. 6. Repay your dollar-denominated loan with \$x(1 + i \$ ) T. 5-27

Reasons for Deviations from IRP Transactions Costs The interest rate available to an arbitrageur for borrowing, i b may (normally) exceed the rate he can lend at, i l. In addition, the bid-ask spreads of currency exchange should also be taken into account. 5-28

Transactions Costs Example Will an arbitrageur facing the following prices be able to make money? BorrowingLending \$5.0%4.50% €5.5%5.0% BidAsk Spot\$1.42 = €1.00\$1.45 = €1,00 Forward\$1.415 = €1.00\$1.445 = €1.00 (1 + i \$ ) (1 + i € ) F(\$/ €) = S(\$/ €) × (1+i \$ ) b (1+i € ) l S 0 (\$/€) a F 1 (\$/€) = b (1+i \$ ) l (1+i € ) b S 0 (\$/€) b F 1 (\$/€) = a 5-29

01 IRP No arbitrage forward bid price (for customer): Buy € at spot ask \$1m × S 0 (\$/€) a 1 Step 2 Sell € at forward bid Step 4 \$1m × S 0 (\$/€) a 1 ×(1+i € )× l F 1 (\$/€) = b \$1m×(1+i \$ ) b \$1m \$1m×(1+i \$ ) b Borrow \$1m at i \$ Step 1 b invest € at i € l \$1m × S 0 (\$/€) a 1 ×(1+i € ) l Step 3 (All transactions at retail prices.) F 1 (\$/€) = b (1+i \$ ) b S 0 (\$/€) a 1 ×(1+i € ) l (1+i \$ ) b (1+i € ) l S 0 (\$/€) a = = \$1.4431/€ 5-30

01 buy € at forward ask Step 4 sell €1m at spot bid Step 2 €1m × S 0 (\$/€) b lend at i \$ Step 3 l IRP €1m×(1+i € ) b €1m × S 0 (\$/€) × (1+i \$ ) ÷ F 1 (\$/€) = b la €1m×(1+i € ) b €1mStep 1borrow €1m at i € b (All transactions at retail prices.) No arbitrage forward ask price: F 1 (\$/€) = a (1+i \$ ) l (1+i € ) b S 0 (\$/€) b = \$1.4065/€ €1m × S 0 (\$/€) × (1+i \$ ) b l 5-31

More true stories On the last two slides we found “no arbitrage” Forward bid prices of \$1.4431/€ and Forward ask prices of \$1.4065/€ Questions: For profitable arbitrage, the market forward bid price of € should be higher than \$1.4431/€, or the market forward ask price of € should be lower than \$1.4065/€. The above situations make no sense, because normally the dealer (bank) sets the ask price above the bid—recall that this difference is dealer’s expected profit. 5-32

Purchasing Power Parity X Purchasing Power Parity and Exchange Rate Determination *PPP-determined exchange rates also provide a valuable benchmark. 5-33

Purchasing Power Parity and Exchange Rate Determination X The exchange rate between two currencies should equal the ratio of the countries’ price levels: S(\$/£) = P£P£ P\$P\$ P£P£ P\$P\$ £150 \$300 == \$2/£ 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: 5-34

Purchasing Power Parity and Exchange Rate Determination X 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) F(\$/€) = \$1.25×(1.03) €1.00×(1.05) \$1.2262 €1.00 = 5-35

Purchasing Power Parity and Exchange Rate Determination X The euro will trade at a 1.90% discount in the forward market : \$1.25 €1.00 = F(\$/€) S(\$/€) \$1.25×(1.03) €1.00×(1.05) 1.03 1.05 1 +  \$ 1 +  € == Relative PPP states that the rate of change in the exchange rate is equal to differences in the rates of inflation—roughly 2% 5-36

Purchasing Power Parity and Interest Rate Parity X Notice that our two big equations today equal each other: = = F(\$/€) S(\$/€) 1 +  \$ 1 +  € PPP 1 + i € 1 + i \$ = F(\$/€) S(\$/€) IRP 5-37

Expected Rate of Change E(e) in Exchange Rate as Inflation DifferentialX We could also reformulate our equations as inflation or interest rate differentials: = F(\$/€) – S(\$/€) S(\$/€) 1 +  \$ 1 +  € – 1 = 1 +  \$ 1 +  € – = F(\$/€) S(\$/€) 1 +  \$ 1 +  € = F(\$/€) – S(\$/€) S(\$/€)  \$ –  € 1 +  € E(e) = ≈  \$ –  € 5-38

Expected Rate of Change E(e) in Exchange Rate as Interest Rate Differential X = F(\$/€) – S(\$/€) S(\$/€) i \$ – i € 1 + i € E(e) = ≈ i \$ – i € 5-39

Quick Short Cut X Given the difficulty in measuring expected inflation, managers often use ≈ i \$ – i €  \$ –  € 4-40

The Exact Fisher Effects X 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 5-41

International Fisher Effect X 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: E(1 +  ¥ ) E(1 +  \$ ) 1 + i \$ 1 + i ¥ = 5-42

International Fisher Effect X If the International Fisher Effect holds, then forward rate PPP holds: E(1 +  ¥ ) E(1 +  \$ ) 1 + i \$ 1 + i ¥ = and if IRP also holds 1 + i \$ 1 + i ¥ S ¥/\$ F ¥/\$ = E(1 +  ¥ ) E(1 +  \$ ) = S ¥/\$ F ¥/\$ 5-43

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