Chapter Objective:
This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity.
6Chapter Six
International Parity Relationships and Forecasting Foreign Exchange Rates
6-1
Chapter Outline Interest Rate Parity Purchasing Power Parity The Fisher Effects Forecasting Exchange Rates
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
6-2
Interest Rate Parity Interest Rate Parity Defined Covered Interest Arbitrage Interest Rate Parity & Exchange Rate
Determination Reasons for Deviations from Interest Rate Parity
6-3
…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.
6-4
S$/£ ×F$/£ = (1 + i£)(1 + i$)
Interest Rate Parity Carefully DefinedConsider 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 these investments have the same risk, they must have the same future value (otherwise an arbitrage would exist)
6-5
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:
$1,000S$/£
(1+ i£) × F$/£
$1,000S$/£
(1+ i£)
=
IRP
6-6
Interest Rate Parity Defined The scale of the project is unimportant
(1 + i$) F$/£
S$/£
× (1+ i£)=
$1,000×(1 + i$) $1,000
S$/£
(1+ i£) × F$/£=
6-7
Interest Rate Parity Defined
Formally,
IRP is sometimes approximated as
i$ – i¥ ≈S
F – S
1 + i$
1 + i¥ S$/¥
F$/¥=
6-8
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¥/$ =
1 + i$
1 + i¥ S$/¥
F$/¥=or
…so be a bit careful about that.
6-9
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¥/$ ×
6-10
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%
6-11
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 $1,000 for £500 at the prevailing spot rate, 2. Invest £500 for one year at i£ = 2.49%; earn £512.453. Translate £512.45 back into dollars at the forward rate
F360($/£) = $2.01/£, the £512.45 will be worth $1,030.
6-12
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
$1,030
6-13
Interest Rate Parity & Exchange Rate Determination
According to IRP only one 360-day forward rate, F360($/£), can exist. It must be the case that
F360($/£) = $2.01/£
Why?
If F360($/£) $2.01/£, an astute trader could make money with one of the following strategies:
6-14
Arbitrage Strategy I
If F360($/£) > $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 ÷ $2/£) invest £500 at 2.49% (i£) for one year to achieve £512.45iii. Translate £512.45 back into dollars, if F360($/£) > $2.01/£, then £512.45 will be more than enough to repay your debt of $1,030.
6-15
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 $1,030
6-16
Arbitrage Strategy II
If F360($/£) < $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 £512.45.6-17
Arbitrage II
$1,000
£500
$1,000 = £500×£1
$2.00
In one year $1,000 will be worth $1,030 > £512.45 ×
£1F£(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
6-18
IRP and Hedging Currency RiskYou 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%
6-19
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 $195,140,989.36 into pounds at the spot
rate S($/£) = $2/£ to receive £97,570,494.68 3. Invest £97,570,494.68 in the UK at i£ = 2.49% for
one year.4. In one year your investment will be worth £100
million—exactly enough to pay your supplier.6-20
Money Market HedgeWhere do the numbers come from? We owe our supplier £100
million in one year—so we know that we need to have an investment with a future value of £100 million. Since i£ = 2.49% we need to invest £97,570,494.68 at the start of the year.
How many dollars will it take to acquire £97,570,494.68at the start of the year if S($/£) = $2/£?
£97,570,494.68 = £100,000,000
1.0249
$195,140,989.36 = £97,570,494.68 × $2.00£1.00
6-21
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.
6-22
Money Market Hedge: an Example
Step 6Pay supplier £100 million
Step 1Order Inventory; agree to pay supplier £100 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/£)
6-23
Another Money Market HedgeA 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)
6-24
Another Money Market Hedge
$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($/€)×
6-25
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.
$x = S($/£)× £y
(1+ i£)T
0 T
$x –$x(1 + i$)TRepay the loan in T years
6-26
Generic Money Market Hedge: Step Two
at the prevailing spot rate.
£y(1+ i£)Tii. 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.6-27
Generic Money Market Hedge
1. Calculate the present value of £y at i£
£y(1+ i£)T
2. Borrow the U.S. dollar value of receivable 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 receivable.
6. Repay your dollar-denominated loan with $x(1 + i$)T.6-28
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:
F180($/€) – S($/€) S($/€) ×
360180f180,€v$ = =
$1.30 – $1.25$1.25 × 2 = 0.08
6-29
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.
6-30
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 = €1,00Forward $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
6-31
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 1b
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€)lS0($/€)a
=
= $1.4431/€
6-32
0 1
buy € at forward
ask
Step 4
sell €1m at spot bid
Step 2
€1m × S0($/€)b lend at i$
Step 3l
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$)l
(1+i€)bS0($/€)b
= $1.4065/€
€1m × S0($/€) × (1+i$) b l
6-33
At these forward bid and ask prices the customer is indifferent
between a forward market hedge and a money market hedge.
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.
6-34
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
Borrowing Lending
$ 5.0% 4.50%€ 5.5% 5.0% Bid Ask
Spot $1.42 = €1.00 $1.45 = €1.00Forward $1.415 = €1.00 $1.445 = €1.00
l li$ = 4.5% and i€ = 5.0% b bi$ = 5.0%, i€ = 5.5%.
6-35
Setting Dealer Forward Bid PriceOur dealer is indifferent between buying euro today at spot bid price and buying euro in 1 year at forward bid price.
spot
bid He is willing to spend
$1m today and receive
$1m ×S0($/€)b
1
$1m
$1m ×S0($/€)b
1
Invest at i$b $1m×(1+i$)b
Invest at i€b
×(1+i€)b$1m ×S0($/€)b
1
forw
ard
bid
F1($/€) =b (1+i$)b
(1+i€)bS0($/€)b
He is also willing to buy at
6-36
Setting Dealer Forward Ask PriceOur dealer is indifferent between selling euro today at spot ask price and selling euro in 1 year at forward ask price.
Invest at i€b €1m×(1+i€)b
forw
ard
ask
F1($/€) =a (1+i$)b
(1+i€)bS0($/€)a
He is also willing to buy at
spot
ask He is willing to spend
€1m today and receive
€1m × S0($/€)b
€1m
€1m × S0($/€)b
Invest at i$b ×(1+i$)b€1m × S0($/€)b
6-37
Purchasing Power Parity Purchasing Power Parity and Exchange Rate
Determination PPP Deviations and the Real Exchange Rate Evidence on PPP
6-38
Purchasing Power Parity and Exchange Rate Determination The exchange rate between two currencies should
equal the ratio of the countries’ price levels:
S($/£) = 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:
6-39
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 $1.25×(1.03) = €1.00×(1.05)
F($/€) = $1.25×(1.03)€1.00×(1.05)
$1.23€1.00
=
6-40
Purchasing Power Parity and Exchange Rate Determination
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%6-41
Purchasing Power Parity and Interest Rate Parity Notice that our two big equations today equal
each other:
= = F($/€)S($/€)
1 + $
1 + €
PPP
1 + i€
1 + i$ =F($/€)S($/€)
IRP
6-42
Expected Rate of Change in Exchange Rate as Inflation Differential 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) = ≈ $ – €
6-43
Expected Rate of Change in Exchange Rate as Interest Rate Differential
= F($/€) – S($/€)
S($/€)i$ – i€
1 + i€E(e) = ≈ i$ – i€
6-44
Quick and Dirty Short Cut Given the difficulty in measuring expected
inflation, managers often use
≈ i$ – i€$ – €
6-45
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.
6-46
Approximate Equilibrium Exchange Rate Relationships
E($ – £)
≈ IRP≈ PPP
≈ FE ≈ FRPPP
≈ IFE ≈ FEP
SF – S
E(e)
(i$ – i¥)
6-47
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 rateE($) is the expected rate of U.S. inflationi$ is the equilibrium expected nominal U.S. interest rate
6-48
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 + ¥)E(1 + $)1 + i$
1 + i¥ =6-49
International Fisher Effect
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¥/$
6-50
PPP
FRPPPFE
FEPIFE
Exact Equilibrium Exchange Rate Relationships
$/
$/ )(
¥
¥
SSE
$/
$/
¥
¥
SF
IRP
E(1 + ¥)E(1 + $)
1 + i$
1 + i¥
6-51
Forecasting Exchange Rates Efficient Markets Approach Fundamental Approach Technical Approach Performance of the Forecasters
6-52
Efficient Markets Approach Financial Markets are efficient if prices reflect all
available and relevant information. If this is so, exchange rates will only change when
new information arrives, thus:St = E[St+1]
andFt = E[St+1| It]
Predicting exchange rates using the efficient markets approach is affordable and is hard to beat.
6-53
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.
6-54
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
6-55
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 forward rate.
The founder of Forbes Magazine once said: “You can make more money selling financial advice than following it.”
6-56
End Chapter Six
6-57
