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International Parity Relationships and Forecasting FX Rates

Chapter Objective: • This chapter examines several key international parity

relationships, such as interest rate parity and purchasing power parity.

Chapter Outline Interest Rate Parity Purchasing Power Parity The Fisher Effects Forecasting Exchange Rates

5Chapter five

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Interest Rate Parity

IRP: Relationship between nominal interest rates and forward/spot exchange rates or forward premiums

IRP is an zero arbitrage condition. Arbitrage = the act of buying and selling simultaneously the same

or equivalent assets to make a guaranteed (riskless) profit.

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.

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Interest Rate Parity

Suppose you have $100,000 to invest for one year.1) invest in Canada at i$ : Future value = $100,000(1 + i$), or

2) trade your dollars for pound at the spot rate, invest in UK at i£ and hedge your exchange rate risk by selling the future value of the British investment forward. Future value = $100,000(F/S)(1 + i£)

Since both investments have the same risk, they must have the same future value—otherwise an arbitrage would exist. Therefore,

$100,000(F/S)(1 + i£) = $100,000(1 + i$) (F/S)(1 + i£) = (1 + i$) (F/S) = (1 + i$) / (1 + i£)

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Interest Rate Parity

Formally, (F/S)(1 + i£) = (1 + i$)

or if you prefer,

IRP is sometimes approximated as

1 + i£

1 + i$ =SF

i$ – i£ =S

F – S

If i$ > i £ then F>S

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Interest Rate Parity Carefully Defined

Depending upon how you quote the exchange rate:

(1) direct quote is $ per £

(2) indirect quote is £ per $ we have:

(2) (1)

1 + i$

1 + i£

S£/$

F£/$ = 1 + i$

1 + i£ S$/£

F$/£=or

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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($/£) = $1.25/£

360-day forward rate F360($/£) = $1.20/£

Canadian interest rate i$ = 7.10%

British interest rate i£ = 11.56%

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Interest Rate Parity

$1,000 $1,071

$1,0711. Trade $1000 for £800

2. Invest £800 at 11.56% = i£

3. One year later, trade £892.48 for $ at F360($/£) = $1.20/£

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IRP and Covered Interest Arbitrage (CIA)

(1) $1,000 invested in Canada:

One-year result: $1,000(1+ i$) = $1,000(1.071) = $1,071

(2) - Exchange $1,000 for £800 at the prevailing spot rate, (£800 = $1,000 ÷ $1.25/£)

- Invest £800 at i£ = 11.56% for one year.

One-year result: £800 (1+ i£)= £800 (1.1156) = £892.48

-Translate £ back into dollars at F360($/£) = $1.20/£, $1.20/£ £892.48 = $1,071.

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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($/£) = $1.20/£

Why?

If F360($/£) $1.20/£, an astute trader could make money with

one of the following strategies:

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Arbitrage Strategy I

If F360($/£) > $1.20/£

i. Borrow $1,000 at t = 0 at i$ = 7.1%.

ii. Exchange $1,000 for £800 at the prevailing spot rate, (note that £800 = $1,000÷$1.25/£) invest £800 at 11.56% (i£) for one year to achieve £892.48

iii. Translate £892.48 back into dollars, if

F360($/£) > $1.20/£ , £892.48 will be more than enough to repay your dollar obligation of $1,071.

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Arbitrage Strategy II

If F360($/£) < $1.20/£

i. Borrow £800 at t = 0 at i£= 11.56% .

ii. Exchange £800 for $1,000 at the prevailing spot rate, invest $1,000 at 7.1% for one year to achieve $1,071.

iii. Translate $1,071 back into pounds, if

F360($/£) < $1.20/£ , $1,071 will be more than enough to repay your £ obligation of £892.48.

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Covered Interest Arbitrage (CIA)

Because the spot and forward markets are not always in a state of equilibrium opportunity for arbitrage exists.

The arbitrager who recognizes this imbalance can invest in the currency that offers the higher return on a covered basis, i.e., exchange rate back to $, for example, is guaranteed at the current forward rate.

CIA opportunities continue until IRP is reestablished.

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180 days = 6 monthsS =¥ 106.00/$

¥ 106,000,000

Dollar rate = 8.00 % per year

Dollar money market

$1,000,000 $1,040,000 1.04

Start End

Yen money market

¥ 108,120,000 1.02

Yen rate = 4.00 % per year

CIA Example - 2

F180 = ¥ 103.50/$

ArbitragePotential$1,044,638

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IRP and Exchange Rate Determination

According to IRP:

Given F, S depends on relative interest rates All else equal, an increase in i$ will lead to a

higher foreign exchange value of the dollar. i.e. when i$ => S

xFi

iS

)1(

)1(

$

£

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IRP and Exchange Rate Determination

Under certain conditions F can be viewed as the expected future S conditional on all relevant information being available.

Then

using the approximation formula:

where E(e) is the expected rate of change in the exchange rate that is [E(St+1)-St]/St

)( 1 tt ISEF

)()1(

)1(1

$

£tt ISEx

i

iS

)()( £$ eEii

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Reasons for Deviations from IRP

Arbitrage Profit exists when (F/S)(1 + i£) (1 + i$ )> 0

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

exceed the rate he can lend at, ib. There may be bid-ask spreads to overcome, Fb/Sa < F/S Thus (Fb/Sa)(1 + i£

b) (1 + i$ a) 0 because

(Fb/Sa)< (Fb/Sa) ; (1 + i£b)< (1 + i£); (1 + i$

a)> (1 + i$ )

Capital Controls Governments sometimes restrict import and export of money

through taxes or outright bans.

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Purchasing Power Parity (PPP)

Absolute PPP : 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 the standard commodity basket costs $300 in the U.S. and £150 in the U.K., then the price of one pound sterling in terms of dollars should be:

• Spot exchange rate is determined by relative prices of similar basket of goods

PPP : Law of one price applied internationally to a standard commodity basket.

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Relative Purchasing Power Parity

If the assumptions of absolute PPP theory are relaxed, we observe relative purchasing power parity This idea is that PPP is not particularly helpful in

determining what the spot rate is today, but that the relative change in prices between countries over a period of time determines the change in exchange rates.

Moreover, if the spot rate rate between 2 countries starts in equilibrium, any change in the differential rate of inflation between them tends to be offset over the long run by an equal but opposite change in the spot rate.

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Relative Purchasing Power Parity and Exchange Rate Determination

Relative PPP states that the rate of change in the exchange rate is equal to the differences in the rates of inflation:

e = (1 + £ )

($ – £) ≈ $ – £

If Canadian inflation is 5% and U.K. inflation is 3.5%, the dollar should depreciate by 1.45% or around 1.5%

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PPP Deviations and the Real Exchange Rate

The real exchange rate is q =

(1 + e)(1+ £)

(1 + $)

If PPP holds, (1 + e) = (1 + £ )

(1 + $ )then q = 1.

If q < 1 competitiveness of domestic country improves with currency depreciations (if dollar depreciates by more than is warranted by PPP)If q > 1 competitiveness of domestic country deteriorates with currency depreciations (if dollar depreciates by less than the inflation differential)

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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. Ipod, 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 long-run benchmark.

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The 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 Canada, the Fisher effect is written as:

1 + i$ = (1+ $)(1 + E[$]) = 1 + $ + E[$] + $E[$]

i$ = $ + E($) + $E[$] ≈ $ + E[$]Where

$ is the equilibrium expected “real” Canadian interest rateE[$] is the expected rate of Canadian inflationi$ is the equilibrium expected nominal Canadian interest rate

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Expected Inflation

The Fisher effect implies that the expected inflation rate is approximated as the difference between the nominal and real interest rates in each country, i.e.

E[$] = (1 + $)

(i$ – $) ≈ i$ – $

i$ = $ + (1 + $)E[$] ≈ $ + E[$]

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International Fisher Effect

If the Fisher effect holds in Canada

i$ = (1+ $)(1 + E[$]) ≈ $ + E[$]

and the Fisher effect holds in Japan,

i¥ = (1+ ¥ )(1 + E[¥]) ≈ ¥ + E[¥]

and if the real rates are the same in each country (i.e. $ = ¥ ), then we get the International Fisher Effect

E(e) = (1 + i¥)

(i$ – i¥) ≈ i$ – i¥

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International Fisher Effect

If the International Fisher Effect holds,

then Forward Expectations parity holds.

E(e) = (1 + i¥)

(i$ – i¥)

and if IRP also holds

= (1 + i¥)

(i$ – i¥)

S

F – S

S

F – SE(e) =

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Approximate Equilibrium Exchange Rate Relationships

E($ – £)

≈ IRP≈ PPP

≈ FE ≈ FPPP

≈ IFE ≈ FEP

S

F – S

E(e)

(i$ – i¥)

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Forecasting Exchange rates

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] and

Ft = E[St+1| It] Predicting exchange rates using the efficient markets

approach is affordable and is hard to beat.

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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.

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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

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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.”