International Parity Relationships and Forecasting Exchange Rates Chapter

International Parity Relationships and Forecasting Exchange Rates Chapter Six Copyright © 2012 by the Mc. Graw-Hill Companies, Inc. All rights reserved.

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

Interest Rate Parity Defined § IRP is a “no arbitrage” condition. § If IRP did not hold, then it would be possible for an astute trader to make unlimited amounts of money by 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

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 and invest $100, 000/S$/£ in Britain at i£ while eliminating any exchange rate risk by selling the future value of the F$/£ British investment forward. Future value = $100, 000(1 + i )× £ S$/£ Since these investments have the same risk, they must have the same future value (otherwise an arbitrage would exist). F$/£ (1 + i£) × = (1 +F i$) = S × (1 + i$) S$/£ $/£ (1 + i£) 6 -4

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

Interest Rate Parity Defined § The scale of the project is unimportant. $1, 000 (1+ i ) × F $1, 000×(1 + i$) = £ $/£ S$/£ F$/£ × (1+ i£) (1 + i$) = S$/£ § IRP is sometimes approximated as: i$ – i£ ≈ F – S S 6 -6

Interest Rate Parity Carefully Defined § No matter how you quote the exchange rate ($ per ¥ or ¥ per $) to find a forward rate, increase the dollars by the dollar rate and the foreign currency by the foreign currency rate: 1 + i¥ F¥/$ = S¥/$ × 1 + i$ or 1 + i$ F$/¥ = S$/¥ × 1 + i ¥ …be careful—it’s easy to get this wrong. 6 -7

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 rate 360 -day forward rate S($/£) = $2. 0000/£ F 360($/£) = $2. 0100/£ U. S. discount rate i$ = 3. 00% British discount rate i£ = 2. 49% 6 -8

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. 6 -9

Other Choice: Buy £ 500 at $2/£. £ 1 £ 500 = $1, 000× $2. 00 $1, 000 One Choice: Invest $1, 000 at 3%. FV = $1, 030 Arbitrage I £ 500 Step 2: Invest £ 500 at i£ = 2. 49%. £ 512. 45 In one year £ 500 will be worth Step 3: Repatriate £ 512. 45 = to the U. S. at £ 500 (1+ i£) F 360($/£) = $2. 01/£ $1, 030 F£(360) $1, 030 = £ 512. 45 × £ 1 6 -10

IRP& 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. 6 -11

Arbitrage Strategy I § If F 360($/£) > $2. 01/£: 1. Borrow $1, 000 at t = 0 at i$ = 3%. 2. Exchange $1, 000 for £ 500 at the prevailing spot rate (note that £ 500 = $1, 000 ÷ $2/£. ); invest £ 500 at 2. 49% (i£) for one year to achieve £ 512. 45. 3. Translate £ 512. 45 back into dollars; if F 360($/£) > $2. 01/£, then £ 512. 45 will be more than enough to repay your debt of $1, 030. 6 -12

Step 2: Buy pounds £ 1 £ 500 = $1, 000× $2. 00 $1, 000 £ 500 Arbitrage I Step 3: Invest £ 500 at i£ = 2. 49%. £ 512. 45 In one year £ 500 will be worth £ 512. 45 = £ 500 (1+ i£) Step 4: Repatriate to the U. S. Step 1: Borrow $1, 000. More F£(360) $1, 030 < £ 512. 45 × Step 5: Repay than $1, 030 £ 1 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 -13

Arbitrage Strategy II § If F 360($/£) < $2. 01/£: 1. Borrow £ 500 at t = 0 at i£= 2. 49%. 2. Exchange £ 500 for $1, 000 at the prevailing spot rate; invest $1, 000 at 3% for one year to achieve $1, 030. 3. 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. 6 -14

Step 2: Buy dollars $2. 00 $1, 000 = £ 500× £ 1 $1, 000 £ 500 Arbitrage II Step 1: Borrow £ 500. Step 5: Repay More Step 3: your pound loan than Invest $1, 000 with £ 512. 45 at i$ = 3%. Step 4: Repatriate to the U. K. In one year $1, 000 F£(360) will be worth $1, 030 > £ 512. 45 × $1, 030 £ 1 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 -15

Currency Carry Trade § Currency carry trade involves buying a currency that has a high rate of interest and funding the purchase by borrowing in a currency with low rates of interest, without any hedging. § The carry trade is profitable as long as the interest rate differential is greater than the appreciation of the funding currency against the investment currency. 6 -16

Currency Carry Trade Example Suppose the 1 -year borrowing rate in dollars is 1%. The 1 -year lending rate in pounds is 2½%. The direct spot ask exchange rate is $1. 60/£. A trader who borrows $1 m will owe $1, 010, 000 in one year. Trading $1 m for pounds today at the spot generates £ 625, 000 invested for one year at 2½% yields £ 640, 625. The currency carry trade will be profitable if the spot bid rate prevailing in one year is high enough that his £ 640, 625 will sell for at least $1, 010, 000 (enough to repay his debt). § No less expensive than: § § § § $1, 010, 000 S 360($/£) = £ 640, 625 b = $1. 5766 £ 1. 00 6 -17

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 -18

Transactions Costs Example § Will an arbitrageur facing the following prices be able to make money? Borrowing Lending $ 5. 0% 4. 50% € 5. 5% 5. 0% Spot F($/ €) = S($/ €) × Bid $1. 42 = € 1. 00 (1 + i$) (1 + i€) Ask $1. 45 = € 1, 00 Forward $1. 415 = € 1. 00 $1. 445 = € 1. 00 F 1 b($/€) = a b S (1+i ($/€) $) 0 (1+i€l ) F 1 a($/€) = b l S (1+i ($/€) $) 0 (1+i€b) 6 -19

$1 m 0 Borrow $1 m at i$b Step 1 $1 m×(1+ib$) IRP 1 Step 2 1 $1 m × a ×(1+il€)×Fb 1($/€) = $1 m×(1+ib$) Buy € at S 0($/€) spot ask No arbitrage forward bid price (for customer): b b a (1+i ) S (1+i ($/€) $ 0 $) Step 4 b F 1($/€) = = 1 l l Sell € at ×(1+i ) € € S 0 a($/€) forward bid = $1. 4431/€ 1 1 l Step 3 invest € at i€ $1 m × a ×(1+il€) $1 m × a S 0($/€) 6 -20 (All transactions at retail prices. )

b € 1 m × S 0($/€) 0 € 1 m × Sb 0($/€) × (1+il$) l lend at i$ Step 3: IRP 1 € 1 m × Sb 0($/€) × (1+il$) ÷ F 1 a($/€) = € 1 m×(1+ib€) No arbitrage forward ask price: Step 2: sell € 1 m at spot bid € 1 m F 1 a($/€) = l b S 0 (1+i ($/€) $) (1+i€b) = $1. 4065/€ Step 1: borrow € 1 m at i€b Step 4 buy € at forward ask € 1 m×(1+i€b) (All transactions at retail prices. ) 6 -21

Why This May Seem Confusing § On the last two slides we found “no arbitrage. ” – Forward bid prices of $1. 4431/€. – 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 -22

Setting Dealer Forward Bid and Ask § Dealer stands ready to be on the 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 posted borrowing rates. il$ = 4. 5% and i€l = 5. 0% Borrowing Lending $ 5. 0% 4. 50% € 5. 5% 5. 0% i$b = 5. 0%, ib€ = 5. 5%. Spot Bid $1. 42 = € 1. 00 Ask $1. 45 = € 1. 00 Forward $1. 415 = € 1. 00 $1. 445 = € 1. 00 6 -23

Setting Dealer Forward Bid Price Our dealer is indifferent between buying euros today at the spot bid price and buying euros in 1 year at the forward bid price. $1 m × $1 m×(1+ib$) Invest at i$b He is willing to spend He is also willing to buy at $1 m today and receive b 1 $1 m × b S 0($/€) 1 S 0 b($/€) Invest at i€b F 1 b($/€) = b S 0 (1+i ($/€) $) (1+i€b) forward bid spot bid $1 m 1 b $1 m × b ×(1+i€) 6 -24 S 0($/€)

Setting Dealer Forward Ask Price Our dealer is indifferent between selling euros today at the spot ask price and selling euros in 1 year at the forward ask price. € 1 m Invest at i$b € 1 m × S 0 b($/€) ×(1+ib$) He is willing to spend He is also willing to buy at € 1 m today and receive b € 1 m × S 0 b($/€) Invest at i€b Fa 1($/€) = a S 0 (1+i ($/€) $) (1+i€b) forward ask spot ask € 1 m × S 0 b($/€) € 1 m×(1+ib€) 6 -25

PPP and Exchange Rate Determination § The exchange rate between two currencies should equal the ratio of the countries’ price levels: P$ S($/£) = P£ For example, if an ounce of gold costs $300 in the U. S. and £ 150 in the U. K. , then the price of one pound in terms of dollars should be: S($/£) = P$ = $300 = $2/£ P£ £ 150 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 -26

PPP and Exchange Rate Determination § The euro will trade at a 1. 90% discount in the forward market: F($/€) = S($/€) $1. 25×(1. 03) € 1. 00×(1. 05) $1. 25 € 1. 00 1. 03 1 + $ = = 1. 05 1 + € Relative PPP states that the rate of change in the exchange rate is equal to differences in the rates of inflation—roughly 2%. 6 -27

PPP and IRP § Notice that our two big equations equal each other: PPP F($/€) 1 + $ = 1 + € S($/€) IRP = 1 + i$ F($/€) = 1 + i€ S($/€) 6 -28

Expected Rate of Change in Exchange Rate as Inflation Differential § We could also reformulate our equations as inflation or interest rate differentials: F($/€) 1 + $ = S($/€) 1 + € F($/€) – S($/€) 1 + $ 1 + € = – 1 = – S($/€) 1 + € F($/€) – S($/€) $ – € E(e) = ≈ $ – € = S($/€) 1 + € 6 -29

Expected Rate of Change in Exchange Rate as Interest Rate Differential F($/€) – S($/€) i$ – i€ = E(e) = S($/€) 1 + i€ ≈ i$ – i€ § Given the difficulty in measuring expected inflation, managers often use a “quick and dirty” shortcut: $ – € ≈ i$ – i€ 6 -30

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 -31

Approximate Equilibrium Exchange Rate Relationships E(e) ≈ IFE (i$ – i¥) ≈ FEP ≈ PPP F – S ≈ IRP S ≈ FE ≈ FRPPP E( $ – £) 6 -32

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 -33

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: E(1 + ¥) 1 + i¥ = 1 + i$ E(1 + $) 6 -34

International Fisher Effect If the International Fisher Effect holds, E(1 + ¥) 1 + i¥ = 1 + i$ E(1 + $) and if IRP also holds, 1 + i¥ F¥/$ = 1 + i$ S¥/$ then forward rate PPP holds: F¥/$ E(1 + ¥) = S¥/$ E(1 + $) 6 -35

Exact Equilibrium Exchange Rate Relationships FEP IFE 1 + i¥ 1 + i$ PPP IRP FE FRPPP E(1 + ¥) E(1 + $) 6 -36

Forecasting Exchange Rates: Efficient Markets Approach § Financial markets are efficient if prices reflect all available and relevant information. § If this is true, exchange rates will only change when new information arrives, thus: St = E[St+1] and Ft = E[St+1| It] § Predicting exchange rates using the efficient markets approach is affordable and is hard to beat. 6 -37

Forecasting Exchange Rates: Fundamental Approach § Involves econometrics to develop models that use a variety of explanatory variables. This involves three steps: – Step 1: Estimate the structural model. – Step 2: Estimate future parameter values. – Step 3: Use the model to develop forecasts. § The downside is that fundamental models do not work any better than the forward rate model or the random walk model. 6 -38

Forecasting Exchange Rates: Technical Approach § Technical analysis looks for patterns in the past behavior of exchange rates. § Clearly it is based upon the premise that history repeats itself. § Thus, it is at odds with the EMH. 6 -39

Performance of the Forecasters § Forecasting is difficult, especially with regard to the future. § As a whole, forecasters cannot do a better job of forecasting future exchange rates than the forecast implied by the forward rate. § The founder of Forbes Magazine once said, “You can make more money selling financial advice than following it. ” 6 -40