Chapter Objective:

This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity.

5Chapter Five

International Parity Relationships and Forecasting Foreign Exchange Rates

5-1

Chapter Outline

Interest Rate ParityPurchasing Power ParityX The Fisher EffectsX

5-2

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

5-3

…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.

5-4

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)

5-5

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

5-6

Interest Rate Parity Defined

(1 + i$) F$/£

S$/£

× (1+ i£)=

$1,000×(1 + i$) $1,000

S$/£

(1+ i£) × F$/£=

5-7

Interest Rate Parity Defined

Formally,

IRP is sometimes approximated as

i$ – i£ ≈S

F – S

1 + i$

1 + i£ S$/£

F$/£=

5-8

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.

5-9

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$/£ = S$/£ × or1 + i$

1 + i£F£/$ = S£/$ ×

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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 S($/£) = $2.0000/£

360-day forward rate F360($/£) = $2.0100/£

U.S. discount rate i$ = 3.00%

British discount rate i£ = 2.49%

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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 F360($/£) = $2.01/£, the £512.45 will be worth $1,030.

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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

F360($/£) = $2.01/£ Alternative 1:

invest $1,000 at 3%FV = $1,030

Alternative 2:buy pounds

Step 2:

£512.45

$1,030

5-13

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:

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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.

5-15

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

5-16

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.

5-17

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

5-18

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 rate S($/£) = $2.00/£

360-day forward rate F360($/£) = $2.01/£

U.S. discount rate i$ = 3.00%

British discount rate i£ = 2.49%

5-19

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.

5-20

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 £97,570,494.68at 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

£1.005-21

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.

5-22

A Similar Example – outlined as 7 steps X

Step 6Pay supplier £100 million

Step 1Order Inventory; agree to pay supplier £100 million in 1 year.

0 1Step 5Redeem £-denominated investment receive £100 million

Suppose that the spot dollar-pound exchange rate is $2.00/£ andi$ = 1%i£ = 4%

Step 4 Invest £96,153,846 at i£ = 4%

Step 3Buy £96,153,846 =at spot exchange rate.

£100,000,0001.04

Step 2Borrow $192,307,692 at i$ = 1%

Step 7Repay dollar loan with $194,230,769

($192,307,692 = £96,153,846×$2/£)

5-23

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€1.00Dollar cost today = $144,230.77 = €96,153.85 ×

€100,0001.04€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)

5-24

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($/€)×

5-25

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

0 T

$x –$x(1 + i$)TRepay the loan in T years

5-26

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.

5-27

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.5-28

Reasons for Deviations from IRP

Transactions Costs The interest rate available to an arbitrageur for borrowing,

ib may (normally) exceed the rate he can lend at, il. In addition, the bid-ask spreads of currency exchange

should also be taken into account.

5-29

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%  Bid Ask

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 S0($/€)a

F1($/€) =b (1+i$)l

(1+i€)b S0($/€)b

F1($/€) =a

5-30

0 1IRP

No arbitrage forward bid price (for customer):Buy € at spot ask

$1m ×S0($/€)a

1

Step 2

Sell € at forward

bid

Step 4

$1m ×S0($/€)a

1 ×(1+i€)×l F1($/€) =b $1m×(1+i$)b

$1m $1m×(1+i$)bBorrow $1m at i$

Step 1

b

invest € at i€l $1m ×S0($/€)a

1 ×(1+i€)lStep 3

(All transactions at retail prices.)

F1($/€) =b (1+i$)b

S0($/€)a

1 ×(1+i€)l

(1+i$)b

(1+i€)l

S0($/€)a=

= $1.4431/€

5-31

0 1

buy € at forward

ask

Step 4

sell €1m at spot bid

Step 2

€1m × S0($/€)b lend at i$

Step 3

l

IRP€1m×(1+i€)b€1m × S0($/€) × (1+i$) ÷ F1($/€) =b l a

€1m×(1+i€)b€1m Step 1borrow €1m at i€b

(All transactions at retail prices.)

No arbitrage forward ask price:

F1($/€) =a (1+i$)l

(1+i€)b

S0($/€)b

= $1.4065/€

€1m × S0($/€) × (1+i$) b l

5-32

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.

5-33

Purchasing Power ParityX

Purchasing Power Parity and Exchange Rate Determination

*PPP-determined exchange rates also provide a

valuable benchmark.

5-34

Purchasing Power Parity and Exchange Rate DeterminationX

The exchange rate between two currencies should equal the ratio of the countries’ price levels:

S($/£) = P£

P$

S($/£) = 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:

5-35

Purchasing Power Parity and Exchange Rate DeterminationX

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

=

5-36

Purchasing Power Parity and Exchange Rate DeterminationX

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

1.051 + $

1 + €

= =

Relative PPP states that the rate of change in the exchange rate is equal to differences in the rates of inflation—roughly 2%5-37

Purchasing Power Parity and Interest Rate ParityX

Notice that our two big equations today equal each other:

= = F($/€)S($/€)

1 + $

1 + €

PPP

1 + i€

1 + i$ =F($/€)S($/€)

IRP

5-38

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 + €

–

1 + €

1 + €

= F($/€)S($/€)

1 + $

1 + €

= F($/€) – S($/€)

S($/€)$ – €

1 + €

E(e) = ≈ $ – €

5-39

Expected Rate of Change E(e) in Exchange Rate as Interest Rate DifferentialX

= F($/€) – S($/€)

S($/€)i$ – i€

1 + i€E(e) = ≈ i$ – i€

5-40

Quick Short CutX

Given the difficulty in measuring expected inflation, managers often use

≈ i$ – i€$ – €

4-41

The Exact Fisher EffectsX

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

5-42

International Fisher EffectX

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¥ =

5-43

International Fisher EffectX

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¥/$

5-44