f55 er ch06 irp ppp and ife

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6Chapter Six

International Parity Relationships and Forecasting Foreign Exchange Rates

Chapter Objective:

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

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Chapter Outline

Interest Rate Parity Interest Rate Parity Interest Rate Parity Interest Rate Parity Interest Rate Parity Interest Rate Parity

Purchasing Power Parity

The Fisher Effects

Forecasting Exchange Rates

Covered Interest Arbitrage

IRP and Exchange Rate Determination

Reasons for Deviations from IRP


The Fisher Effects

Purchasing Power Parity PPP Deviations and the Real Exchange Rate

Evidence on Purchasing Power Parity

The Fisher Effects

F i E h R


The Fisher Effects

Forecasting Exchange Rates


The Fisher Effects

Forecasting Exchange Rates Efficient Market Approach


Fisher Effects

International Fisher Effects

The Fisher Effects

Forecasting Exchange Rates Forecasting Exchange Rates Fundamental Approach

Technical Approach

Performance of the Forecasters

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Interest Rate Parity

Interest Rate Parity Defined

Covered Interest Arbitrage

Interest Rate Parity & Exchange Rate Determination

Reasons for Deviations from Interest Rate Parity

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IRP is an “no arbitrage” condition.

If IRP did not hold, then it would be possible for a 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

…almost all of the time!arbitrage conditions, we can safely assume that IRP holds.

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Variable DefinitionsiH : Interest Rate in the home country

iF : Interest Rate in the foreign country

S = Current spot rate for the foreign currency (in direct quote)S = Current spot rate for the foreign currency (in direct quote)

F = 1 year forward rate for the foreign currency (in direct quote)

FPH = one year forward premium from the home country’s viewpoint = (F-S) / S

FPF = one year forward premium from the foreign country’s viewpoint = (S- F) / F or (1/ FP – 1)

S$/£ ×F$/£ = (1 + i£)(1 + i$)

viewpoint (S- F) / F or (1/ FPH 1)

iCH : Covered rate of interest, from the home country’s viewpoint

iCF : Covered rate of interest, from the foreign country’s viewpoint

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Interest Rate Parity Carefully Defined

Consider two alternative one-year investments for $1

1. You could invest in the US at iH. Future value of this investment in $ will be: $1 × (1 + iH) = (1 + iH)

2. Or you could convert $1 into the foreign currency at the going spot rate (S) and invest 1/S in the foreign country at iF whose future value will be: [1/S × (1 + iH)]. In order to eliminate any exchange rate risk, you will have to sell this amount at forward rate (F) to get you money back in $: F x [1/S × (1 + i )]

S$/£ ×F$/£ = (1 + i£)(1 + i$)

rate (F) to get you money back in $: F x [1/S × (1 + iH)]

SF

(1 + iF) × = (1 + iH)

Since these investments have the same risk, they must have the same future value (otherwise an arbitrage would exist)

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Formally, 1 + i Fy,

IRP is sometimes approximated as

1 + iH

1 + iF SF

=

Or FP 1 + iH

1 + iF

- 1 =F – S

S=

IRP is sometimes approximated as

iH – iF ≈S

F – S

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Depending upon how you quote the exchange rates, direct (S, F) or indirect (SI, FI), we have:

1 + iH

1 + iF

SI

FI=1 + iH

1 + iF SF

=or

so be a bit caref l abo t that…so be a bit careful about that.

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No matter how you quote the exchange rate (direct or indirect) to find a forward rate, increase the dollars by the dollar rate and the foreign currency by the foreign currency rate:

1 + iH1 + iF

…be careful—it’s easy to get this wrong.

1 iH

1 + iFF = S ×or

1 + iH

1 iFFI = SI ×

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Covered Rate of Interest

Home Country’s viewpoint (iCH) = CH

(1 + iF) x (1 + FPH ) - 1

F i C ’ i i ( )

S$/£ ×F$/£ = (1 + i£)(1 + i$)

Foreign Country’s viewpoint (iCF) =

(1 + iH) x (1 + FPF ) - 16-9

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IRP & Covered Interest Arbitrage (CIA)

CIA is possible when:

iCH > iHCH H

iCF > iF

When CIA is possible, iH, iF , and FP will have

to adjust to eliminate arbitrage.

S$/£ ×F$/£ = (1 + i£)(1 + i$)

IRP holds when CIA is not possible:

iCH = iH

iCF = iF6-10

IRP and Covered Interest Arbitrage

If IRP failed to hold, an arbitrage would exist. It is , geasiest 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 for GBP S = $2 0000Spot exchange rate for GBP S $2.0000

360-day forward rate for GBP F = $1.9700

US interest rate iH = 5.00%

British interest rate iF = 8.00%

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Spot exchange rate for GBP S = $2 0000Spot exchange rate for GBP S $2.0000

360-day forward rate for GBP F = $1.9100



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Spot exchange rate for BP S = $2 0000Spot exchange rate for BP S $2.0000

360-day forward rate for BP F = $2.0400



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Spot exchange rate for BP S = $2 000Spot exchange rate for BP S $2.000

360-day forward rate for BP F = $2.090



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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

Capital Controls Governments sometimes restrict import and export of Governments sometimes restrict import and export of

money through taxes or outright bans.

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The concept of Absolute and Relative Purchasing Power Parity (PPP)

PPP and Exchange Rate Determination

PPP Deviations and the Real Exchange Rate

Consequences of PPP Violations

Evidence on PPP

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Absolute Purchasing Power Parity

A dollar should buy the same quantities of goods and services in all countries

According to absolute PPP, in the long run, currencies should move towards the rate which equalizes the prices of an identical basket of goods and services in each country

The exchange rate (direct quote) between two (S)

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currencies should equal the ratio of the countries’ price levels in the home (PH) and foreign (PF) country: S = (PH / (PF)

Examples

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Absolute Purchasing Power Parity and Exchange Rate Determination

S =P

PH

PF

S$1200

= $1 50/ €

For example, if an ounce of gold costs $1200 in the U.S. and € 800 in Europe, then the price of one euro in terms of dollars should be:

S = € 800= $1.50/ €

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What happens if S = 1.25 or S = 1.75?

Does PPP Hold? More Examples

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Evidence on Absolute PPP

Absolute PPP probably doesn’t hold precisely in h l ld f i fthe real world for a variety of reasons: Tradable and non-tradable factors of production 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.

Relative PPP-determined exchange rates can provide a more valuable benchmark.

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PPP: Evidence

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Does PPP Hold? The Case of Big Mac

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Relative Purchasing Power Parity

Even if the dollar does not buy the same basket of goods in other countries, the purchasing power of h d ll i h i ld i blthe dollar in these countries could remain stable

over time.

We can show that according to Relative PPP: If two countries have different inflation rates, then the

exchange rates between the two countries will adjust to

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exchange rates between the two countries will adjust to maintain equality of relative purchasing power for the citizens of both countries.

The “real” exchange rate will remain constant

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Variable DefinitionS= Current spot rate (price of foreign currency) in direct quote

S1 = Actual spot rate, 1 year from nowF = 1-year forward rateFP = the forward premium = [(F-S) / S] = [(F/S) - 1]H = Inflation rate in the home countryF = Inflation rate in the foreign countryE(S1) = Expected spot rate, 1 year from now, based on PPPE(e) = [E(S1)/S] – 1 = The expected percentage change, or 1rate of change, in the spot rate, based on PPPe = (S1/S) – 1 = The actual percentage change, or rate of change, in the spot rateSr= real spot rate

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Absolute Purchasing Power Parity and Exchange Rate Determination

S =P

PH

PF

S$1200

= $1 50/ €

For example, if an ounce of gold costs $1200 in the U.S. and € 800 in Europe, then the price of one euro in terms of dollars should be:

S = € 800= $1.50/ €

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What happens if S = 1.25 or S = 1.75?

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Purchasing Power Parity and Exchange Rate Determination

Suppose the spot exchange rate (S) is $1.50 = €1.00

If the inflation rate in the U.S. (H) is expected to be 5% in the next year and 3% in the euro zone(F),

Then the expected exchange rate in one year E(S) should be such that $1.50×(1.05) = €1.00×(1.03)

$1 50 (1 05) $1 575E(S1)$1.50×(1.05)€1.00×(1.03)

$1.575€1.03

=

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=

E(e) = [E(S1)/S – 1] $1.5291$1.50

=

= $1.5291

- 1 = .019 = 1.94%

Purchasing Power Parity and Exchange Rate Determination Because of the inflation differential, the euro is expected to

appreciate by 1.94% in the spot market by the end of the year:

$1.50€1.00

= E(S1)

S

$1.50×(1.05)€1.00×(1.03) 1.05

1.031 + H

1 + F= =

Relative PPP states that the rate of change in the exchange rateRelative PPP states that the rate of change in the exchange rate is equal to differences in the rates of inflation—roughly 2%Also remember that E(S1) = FSo that: expected rate of change in the exchange rate = forward premium, or E(e) = FP

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Relative Purchasing Power Parity

According to Relative PPP:

=E(S1)

S1 + H

1

S 1 + F

= E(e)1 + H

1 + F

- 1

= Approximately: E(e) H - F

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= E(S1) S [1 + E(e)]

pp y ( )

“Real” Exchange RateReal exchange rate is the spot rate adjusted for inflation, let us call it Sr . It is supposed to tell us if a foreign currency has appreciated or depreciated, after adjusting for inflation.

S S= Sr S1

Under PPP, real exchange rates should remain constant

Suppose the US the current spot rate for € is 1.50 and US inflation rate is 5% while the inflation rate is 3% in the euro zone If the spot rate next year turns out to be 1 52 the real

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zone. If the spot rate next year turns out to be 1.52, the real exchange rate is: 1.52*(1.03/1.05) = $1.491• We can say that although the spot rate for € appreciated in “nominal”

terms from $1.50 to $1.54, it actually depreciated in “real” terms from $1.50 to $1.491

• This would weaken the US’s competitive position against Europe

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PPP & CompetitivenessWe can also use PPP to determine the competitiveness of the home country’s currency

=q q

=

E(S1)/S1=

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q = 1: Competitiveness of home country is unchangedq < 1: Competitiveness of home country has improvedq > 1: Competitiveness of home country has deteriorated

PPP Conditions Summarized

PPP is Violated

PPP Holds Foreign currency has Foreign currency has g yappreciated (USD has depreciated) in “real” terms

g ydepreciated (USD has appreciated) in “real” terms

No Change US exports more competitive US exports less competitive

S1 = E(S1) S1 > E(S1) S1 < E(S1)

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e = E(e) e > E(e) e < E(e)

S = Sr S > Sr S< Sr

q = 1 q < 1 q > 1

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PPP: EXAMPLE 1

Inflation rate in the US is 5%; H = 0.05 Inflation rate in the Europe is 3%; F = 0.03 Current spot rate for € is $1.50; S = 1.50 To maintain relative PPP the expected percentage change in the spot To maintain relative PPP, the expected percentage change in the spot

exchange rate for €, E(e) = (1.05) / (1.03) - 1 = 1.9417 % To maintain relative PPP, the expected spot exchange rate for €, at the

end of the year, E(S1) = $1.50 ( 1 + 0.019417)= $1.5291 per €

If, 1 year latter the actual spot rate, S1 for € turns out to be $1.54 $1.52

Compared to E(S1) of $1.5291, S1 is higher lower

Actual % change in S: e = (S /S) 1 2 027 % 1 333 %

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Actual % change in S: e = (S1/S) -1 2.027 % 1.333 %

The “real” rate for €, Sr = S1 *[1.03/1.05] $1.5107 $1.4910

q is equal to: [1+E(e)] / [1 + e ] = E(S1)/S1 0.9929 1.0060

The “real” rate (Sr) has: increased decreased

In real terms, € has: appreciated depreciated

US’s competitiveness has: improved deteriorated

PPP: EXAMPLE 2 Inflation rate in the US is 5%; H = 0.05 Inflation rate in the Switzerland is 2%; F = 0.08 Current spot rate for SF is $0.90; S = 0.90 To maintain relative PPP, the expected percentage change in the spot o e ve , e expected pe ce tage c a ge i t e spot

exchange rate for SF, E(e) = (1.05) / (1.08) - 1 = - 2.778 % To maintain relative PPP, the expected spot exchange rate for SF, at the

end of the year, E(S1) = $0.90 ( 1 + 0.02941)= $0.875 per SF

If, 1 year latter the actual spot rate, S1 for SF turns out to be $0.86 $0.88

Compared to E(S1) of $0.875, S1 is lower higher

Actual % change in S: e = (S /S) 1 4 444 % 2 222 %

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Actual % change in S: e = (S1/S) -1 - 4.444 % - 2.222 %

The “real” rate for SF, Sr =S1*[1.08/1.05] $0.8846 $0.9051

q is equal to: [1+E(e)] / [1 + e ] = E(S)/S1 1.017 0.994

The “real” rate (Sr) has: decreased increased

In real terms, SF has: depreciated appreciated

US’s competitiveness has: deteriorated improved

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Purchasing Power Parity and Interest Rate Parity

Notice that the PPP & IRP equations are equal because E(S) = F or E(e) = FP:

= E(S)

S1 + H

1 + F1 + iF

1 + iH =FS=

PPP IRP

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= E(e) 1 + H

1 + F 1 + iF

1 + iH -1 = FP= - 1

PPP: Evidence

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Expected Rate of Change in Exchange Rate as Inflation Differential

We could also reformulate our equations as inflation or interest rate differentials:

=F($/€) – S($/€) 1 + $ 1 =

1 + $ 1 + €

= F($/€)S($/€)

1 + $

1 + €

= S($/€) 1 + €

– 1 = 1 + €

–1 + €

= F($/€) – S($/€)

S($/€)$ – €

1 + €

E(e) = ≈ $ – €

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Fisher EffectThe nominal interest rate is composed of a real interest rate and an expected inflation rate. Nominal interest rate: i; Real rate: ρ; Expected inflation:

(1 + i) = (1 + ρ) (1 + )

i = ρ + + ρApproximately: i = ρ +

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If real rates are equal across countries, or: ρH = ρF

Then: (1 + iH) / (1 + iF) = (1 + H) / (1 + F)

Approximately : iH - iF = H - F

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International Fisher Effect (IFE)

The concept of IEF

IFE Conditions

Deviations of from IFE: uncovered rates of interest: from the home and foreign country’s view point

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In an integrated global money and capital markets:

(1) Domestic fisher effect holds in each country.

(2) All investors have the same real rate of return worldwide.

(3) Therefore all nominal interest rate differences must be due to inflation differences

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The exchange rate of a country with a higher (lower) interest rate than its trading partner should depreciate (appreciate) by the amount of the interest rate difference to maintain equality of real rates of return.

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IFE: Terminology

iH = Nominal interest rate for the home countryiF = Nominal interest rate for the foreign countryF g yS = Current spot rate (direct quote) for the foreign

currency (in home currency units)S1 = Next year’s spot rate (direct quote) for the

foreign currency

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Uncovered Rate: Home County’s View point The uncovered rate from the home county’s point of view (iUH) is the rate earned by the

holders of dollars by:1. Converting DOLLARS into FOREIGN CURRENCY today at the current spot exchange

rate (S), andrate (S), and2. Investing the FOREIGN CURRENCY at the FOREIGN INTEREST RATE (iF), and3. Converting FOREIGN CURRENCY back into DOLLARS at maturity using the future

spot exchange rate (S1) This return is affected by two factors:

whether the foreign currency appreciates or depreciates against the dollar = % change in direct quote (DQ) = (S1 - S) / S

The rate of interest you earn in the foreign country = iF

You calculate it as:

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You calculate it as:iUH = (1 + % change in DQ)*(1 + iF) – 1

Profit making Strategy:If iUH > iH then borrow in dollars and invest in foreign currencyIf iUH < iH then borrow in foreign currency and invest in dollarsIf iUH = iH then you cannot make any profit

Uncovered Rate: Foreign County’s View point

The uncovered rate from the foreign county’s point of view (iUF) is the rate earned by the holders of foreign currency by:1. Converting FOREIGN CURRENCY into DOLLARS today at the current spot

exchange rate (S) andexchange rate (S), and2. Investing the DOLLARS at the US INTEREST RATE (iH), and3. Converting DOLLARS back into FOREIGN CURRENCY at maturity using the

future spot exchange rate (S1) This return is affected by two factors:

whether the US Dollars appreciates or depreciates against the foreign currency = % change in indirect quote (IQ) = (S0 - S1) / S1

The rate of interest you earn in the home country (US) = iH

You calculate it as:

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You calculate it as:iUF = (1 + % change in IQ)*(1 + iH) – 1

Profit making strategy :If iUF > iF then borrow in foreign currency and invest in dollarsIf iUF < iF then borrow in dollars and invest in foreign currencyIf iUF = iF then you cannot make any profit

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IFE Conditions According to IFE one should not be able to make money by

consistently borrowing in one country and investing in another

These conditions are met when: i = i or i = i These conditions are met when: iUH = iH or iUF = iF

According to IFE the above conditions will hold only whenthe expected percentage change in the spot rate, E(e):

E(e)= (1 + iH) / (1 + iF) – 1Approximately: E(e)= iH - iF

According to IFE the expected spot rate 1 year from now

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According to IFE, the expected spot rate 1 year from now,E(S1), should be:

E(S1) = S [1 + E(e)]

Uncovered Rate and IFE: SummarizedIf iUH > iH or iUF < iF then investors will profit if they:• borrow in the home country (US)• convert the $ loan amount into foreign currency• invest in the foreign capital market

h d f h b i /i i d h f i b k• at the end of the borrowing/investment period convert the foreign currency back into domestic currency ($) and pay off the domestic (US) loan

• If this continues then: S↑, E(S1)↓, iH↑, iF↓, until iUH = iH or iUF = iH, or IFE holds

If iUF > iF or iUH < iH then investors will profit if they:• borrow in the foreign country• convert the loan amount from foreign currency into domestic currency ($)

invest in the domestic (US) capital market

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• invest in the domestic (US) capital market• at the end of the borrowing/investment period convert the domestic currency ($)

back into foreign currency and pay off the foreign loan• If this continues then: S↓, E(S1)↑, iH↓, iF↑, until iUH = iH or iUF = iH, or IFE holds

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IFE : Example 1

Interest rate in US, iH = 7 % & Euro zone interest rate, iF = 9 % Current spot rate for €, S = $1.40 According to IFE, the percentage change in exchange rate, based on

direct quote for € should be:direct quote, for € should be:E(e) = (1.07) / (1.09) - 1 = - 1.83486%

According to IFE, the expected spot rate for € at the end of the year should be:E(S1) = $1.40 ( 1 - 0.0183) = $ 1.37 / €

What happens if you believe (predict) that S1 will be $1.39 ? You could make money by borrowing in $ and investing in €

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Can you show how? What happens if you believe (predict) that S1 will be $1.35 ? You could make money by borrowing in € and investing in $ Can you show how?

IFE : Example 2

Interest rate in US, iH = 7% & Interest rate in Switzerland, iF = 3% Current spot rate for SF (S)= $0.85 According to IFE, the percentage change in exchange rate, based on

direct quote SF should be:direct quote, SF should be:E(e) = (1.06) / (1.03) - 1 = 3.8845%

According to IFE, the expected spot rate for SF at the end of the year should be:E(S1) = $0.85 ( 1 + 0.0288) = $0.883 / SF

What happens if you believe (predict) that S1 will be $0.90 ? You could make money by borrowing in $ and investing in FF

C h h ?

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Can you show how? What happens if you believe (predict) that S1 will be $0.87 ? You could make money by borrowing in SF and investing in $ Can you show how?

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Approximate Equilibrium Exchange Rate Relationships

( )

≈ IRP≈ PPP

≈ IFE ≈ FEP

S

F – S

E(e)

(iH – iF)

E(H – F)

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Exact Equilibrium Exchange Rate Relationships

SE )( 1

PPP

FEPIFES

SE )( 1

S

FIRP

1 + iF

1 + iH

E(1 + H)E(1 + F)

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Variable DefinitionsS= Current spot rate (price of foreign currency) in direct quoteS1 = Actual spot rate, 1 year from nowF = 1-year forward rateF = 1-year forward rateFPH = the forward premium = [(F-S) / S] = [(F/S) - 1] from the home country’s view pointFPF = the forward premium = [(S-F) / F] = [(S/F) - 1] from the foreign country’s view pointH = Inflation rate in the home countryF = Inflation rate in the foreign countryρ = Real rate of interestE(S ) = Expected spot rate 1 year from now based on PPPE(S1) = Expected spot rate, 1 year from now, based on PPPE(e) = [E(S1)/S] – 1 = The expected percentage change, or rate of change, in the spot rate, based on PPPe = (S1/S) – 1 = The actual percentage change, or rate of change, in the spot rateSr= real spot rateiH = Nominal interest rate for the home countryiF = Nominal interest rate for the foreign country

Formula: Purchasing Power Parity (PPP)

Exact relationship: E(e) =(1 + πH) / (1 + πF) - 1E(e) (1 + πH) / (1 + πF) - 1Approximate relationship: E(e) = πH - πF

S1 = S0 * [ 1 + E(e) ]SR = S1 *(1 + πF) / (1 + πH)

i i (1 i) (1 ) (1 )Fisher Equation: (1 + i) = (1 + ρ) (1 + )i = ρ + + ρApproximately: i = ρ +

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Formula: International Fisher Effect (IFE)

Fisher Equation: (1 + i) = (1 + ρ) (1 + )

i = ρ + + ρApproximately: i = ρ +

iuh : Uncovered rate of return, home country’s viewpointiuf : Uncovered rate of return, foreign country’s viewpoint

iuh = (1 + if) (1 + % change in DQ ) – 1iuf = (1 + ih) (1 + % change in IQ ) – 1% h i DQ (di t t ) (S S ) / S% change in DQ (direct quote)= (S1 - S0) / S0

% change in IQ (indirect quote) = [1 /(1+ % change in DQ] - 1The IFE relationship holds when:E(e)= (1 + iH) / (1 + iF) – 1Approximately: E(e)= iH – iFS1 = S0 * [1 + E(e)]

Formula: Interest Rate Parity (IRP)Calculating the covered rate of returns (home & foreign country’s view point)ich= Covered rate of return, home country’s view point icf= Covered rate of return, foreign country’s view point

ich = (1 + if) (1 + FPh) – 1icf = (1 + ih) (1 + FPf) – 1The IRP relationship holds when the expected forward premium from the

home country’s point of view (FPh ):FPh = (1 + ih)/(1 + if) – 1 S1= S0 * ( 1 + FPh)

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The Hamburger Standard: Further Discussion

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f55 er ch06 irp ppp and ife

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