# International Parity Relationships and Forecasting Foreign Exchange Rates

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1 International Parity Relationships and Forecasting Foreign Exchange Rates 6 Chapter Six Chapter Objective: This chapter examines several key international parity relationships, such as interest rate parity and purchasing power parity. 6-0 Chapter Outline Interest Rate Parity Purchasing Covered Interest Power Arbitrage Parity IRP and Exchange Rate Determination Fisher The PPP Fisher Effects Deviations Effects and the Real Exchange Rate Reasons for Deviations from IRP Evidence on Purchasing Power Parity International Forecasting Exchange Fisher Effects Rates Purchasing The Fisher Effects Power Parity Efficient Market Approach The Forecasting Fundamental Fisher Effects Exchange Approach Rates Forecasting Technical Approach Exchange Rates Performance of the Forecasters 6-1 1

2 Interest Rate Parity Interest Rate Parity Defined Covered Interest Arbitrage Interest Rate Parity & Exchange Rate Determination Reasons for Deviations from Interest Rate Parity 6-2 Interest Rate Parity Defined 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 IRP holds. almost all of the time! 6-3 2

3 Variable Definitions i H : Interest Rate in the home country i F : Interest Rate in the foreign country S = Current spot rate for the foreign currency (in direct quote) F = 1 year forward rate for the foreign currency (in direct quote) FP H = one year forward premium from the home country s viewpoint = (F-S) / S FP F = one year forward premium from the foreign country s viewpoint = (S- F)/For(1/FP F FP H 1) i CH : Covered rate of interest, from the home country s viewpoint i CF : Covered rate of interest, from the foreign country s viewpoint (1 + i F \$/ = S \$/ \$ ) (1 + i ) 6-4 Interest Rate Parity Carefully Defined Consider two alternative one-year investments for \$1 1. You could invest in the US at i H. Future value of this investment in \$ will be: \$1 (1 + i H ) = (1 + i H ) 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 i F whose future value will be: [1/S (1 + i H )]. 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 \$: Fx[1/S (1 + i H )] Since these investments have the same risk, they must have the same future value (otherwise an arbitrage would exist) F (1 + i F \$/ = S \$/ \$ ) (1 + i F ) = (1 + i (1 + i ) S H ) 6-5 3

4 Interest Rate Parity Defined Formally, 1 +i H F = 1 + i F S 1 + i H Or -1 = F S 1 + i F S IRP is sometimes approximated as i H F S i F S = FP 6-6 Interest Rate Parity Carefully Defined Depending upon how you quote the exchange rates, direct (S, F) or indirect (S I, F I ), we have: 1 + i F F I = 1 + i S H I or 1 + i H F = 1 + i F S so be a bit careful about that

5 Interest Rate Parity Carefully Defined 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: F I = S I 1 + i F 1 + i H or F = S 1 + i H 1 + i F 6-8 be careful it s easy to get this wrong. Covered Rate of Interest Home Country s viewpoint (i CH ) = CH (1 + i F ) x (1 + FP H ) - 1 Foreign Country s viewpoint i (i CF ) = 6-9 (1 + i H ) x (1 + FP F \$/ ) -= 1S \$/ (1 + i \$ ) (1 + i ) 5

6 IRP & Covered Interest Arbitrage (CIA) CIA is possible when: i CH > i H i CF > i F When CIA is possible, i H, i F, and FP will have to adjust to eliminate arbitrage. IRP holds when CIA is not possible: 6-10 i CH = i H i CF = i F F \$/ = S \$/ (1 + i \$ ) (1 + i ) IRP and Covered Interest Arbitrage If IRP failed to hold, an arbitrage would exist. It is 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 for GBP S = \$ day forward rate for GBP F = \$ US interest rate i H = 5.00% British interest rate i F = 8.00%

7 IRP and Covered Interest Arbitrage If IRP failed to hold, an arbitrage would exist. It is 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 for GBP S = \$ day forward rate for GBP F = \$ US interest rate i H = 5.00% British interest rate i F = 8.00% 6-12 IRP and Covered Interest Arbitrage

8 IRP and Covered Interest Arbitrage If IRP failed to hold, an arbitrage would exist. It is 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 for BP S = \$ day forward rate for BP F = \$ US interest rate i H = 8.00% British interest rate i F = 4.00% 6-14 IRP and Covered Interest Arbitrage If IRP failed to hold, an arbitrage would exist. It is 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 for BP S = \$ day forward rate for BP F = \$2.090 US interest rate i H = 8.00% British interest rate i F = 4.00%

9 Reasons for Deviations from IRP Transactions Costs The interest rate available to an arbitrageur for borrowing, i b may exceed the rate he can lend at, i l. There may be bid-ask spreads to overcome, F b /S a < F/S Capital Controls Governments sometimes restrict import and export of money through taxes or outright bans Reasons for Deviations from IRP

10 Purchasing Power Parity 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 6-18 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) currencies should equal the ratio of the countries price levels in the home (P H ) and foreign (P F ) country: S = (P H / (P F ) Examples

11 Absolute Purchasing Power Parity and Exchange Rate Determination P H S = P F 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: \$1200 S = = \$1.50/ 800 What happens if S = 1.25 or S = 1.75? 6-20 Does PPP Hold? More Examples

12 Evidence on Absolute PPP Absolute PPP probably doesn t hold precisely in the real world ldfor 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 PPP: Evidence

13 Does PPP Hold? The Case of Big Mac 6-24 Relative Purchasing Power Parity Even if the dollar does not buy the same basket of goods in other countries, the purchasing power of the 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 maintain equality of relative purchasing power for the citizens of both countries. The real exchange rate will remain constant

14 Variable Definition S= Current spot rate (price of foreign currency) in direct quote S 1 = Actual spot rate, 1 year from now F = 1-year forward rate FP = the forward premium = [(F-S) / S] = [(F/S) - 1] H = Inflation rate in the home country F = Inflation rate in the foreign country E(S 1 ) = Expected spot rate, 1 year from now, based on PPP E(e) = [E(S 1)/S] 1 = The expected percentage change, or rate of change, in the spot rate, based on PPP e = (S 1 /S) 1 = The actual percentage change, or rate of change, in the spot rate S r = real spot rate 6-26 Absolute Purchasing Power Parity and Exchange Rate Determination P H S = P F 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: \$1200 S = = \$1.50/ 800 What happens if S = 1.25 or S = 1.75?

15 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) E(S 05) 1 ) = \$1.50 (1.05) \$1.575 = = \$ (1.03) 1.03 E(e) = [E(S 1 )/S 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 (1.05) E(S 1 ) 1.00 (1.03) = = H = S \$ F 1.00 Relative 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(S 1 ) = F So that: expected rate of change in the exchange rate = forward premium, or E(e) = FP

16 Relative Purchasing Power Parity According to Relative PPP: Approximately: E(e) E(S 1 ) 1 + = H S 1 + F 1 + E(e) = H F E(S 1 ) = = H - F S [1 + E(e)] 6-30 Real Exchange Rate Real exchange rate is the spot rate adjusted for inflation, let us call it S r. It is supposed to tell us if a foreign currency has appreciated or depreciated, after adjusting for inflation. S r = S 1 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 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

17 PPP & Competitiveness We can also use PPP to determine the competitiveness of the home country s currency q = = = E(S 1 )/S 1 q = 1: Competitiveness of home country is unchanged q < 1: Competitiveness of home country has improved q > 1: Competitiveness of home country has deteriorated 6-32 PPP Conditions Summarized PPP is Violated PPP Holds Foreign currency has Foreign currency has appreciated (USD has depreciated) in real terms depreciated (USD has appreciated) in real terms No Change US exports more competitive US exports less competitive S 1 =E(S 1 ) S 1 > E(S 1 ) S 1 < E(S 1 ) e= E(e) e> E(e) e< E(e) S = S r S > S r S< S r q = 1 q < 1 q >

18 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 exchange rate for, E(e) = (1.05) / (1.03) - 1 = % To maintain relative PPP, the expected spot exchange rate for, at the end of the year, E(S 1 ) = \$1.50 ( )= \$ per If, 1 year latter the actual spot rate, S 1 for turns out to be \$1.54 \$1.52 Compared to E(S 1 ) of \$1.5291, S 1 is higher lower Actual % change in S: e = (S 1 /S) % % The real rate for, S r = S 1 *[1.03/1.05] \$ \$ q is equal to: [1+E(e)] / [1 + e ] = E(S 1 )/S The real rate (S r ) has: increased decreased In real terms, has: appreciated depreciated US s competitiveness has: improved deteriorated 6-34 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 eexpectedexpected percentage change in the spot exchange rate for SF, E(e) = (1.05) / (1.08) - 1 = % To maintain relative PPP, the expected spot exchange rate for SF, at the end of the year, E(S 1 ) = \$0.90 ( )= \$0.875 per SF If, 1 year latter the actual spot rate, S 1 for SF turns out to be \$0.86 \$0.88 Compared to E(S 1 ) of \$0.875, S 1 is lower higher Actual % change in S: e = (S 1 /S) % % The real rate for SF, S r =S 1 *[1.08/1.05] \$ \$ q is equal to: [1+E(e)] / [1 + e ] = E(S)/S The real rate (S r ) has: decreased increased In real terms, SF has: depreciated appreciated US s competitiveness has: deteriorated improved

19 Purchasing Power Parity and Interest Rate Parity Notice that the PPP & IRP equations are equal because E(S) = F or E(e) = FP: PPP IRP E(S) 1 + = H 1 + i H F = = S 1 + F 1 + i F S E(e) = 1 + H 1 + i -1 H = -1 = FP 1 + F 1 + i F 6-36 PPP: Evidence

20 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 + F(\$/ ) S(\$/ ) = 1 + \$ 1 = 1 + \$ 1 + S(\$/ ) E(e) = F(\$/ ) S(\$/ ) = \$ S(\$/ ) 1 + \$ Fisher Effect The 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 + + ρ Approximately: i = ρ + If real rates are equal across countries, or: ρ H = ρ F Then: (1 + i H ) / (1 + i F ) = (1 + H ) / (1 + F ) Approximately : i H -i F = H - F

21 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 6-40 International Fisher Effect (IFE) 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

22 International Fisher Effect (IFE) 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 IFE: Terminology i H = Nominal interest rate for the home country i F = Nominal interest rate for the foreign country S = Current spot rate (direct quote) for the foreign currency (in home currency units) S 1 = Next year s spot rate (direct quote) for the foreign currency

23 Uncovered Rate: Home County s View point The uncovered rate from the home county s point of view (i UH ) is the rate earned by the holders of dollars by: 1. Converting DOLLARS into FOREIGN CURRENCY today at the current spot exchange rate (S), and 2. Investing the FOREIGN CURRENCY at the FOREIGN INTEREST RATE (i F ), and 3. Converting FOREIGN CURRENCY back into DOLLARS at maturity using the future spot exchange rate (S 1 ) This return is affected by two factors: whether the foreign currency appreciates or depreciates against the dollar = % change in direct quote (DQ) = (S 1 -S)/ S The rate of interest you earn in the foreign country = i F You calculate it as: i UH = (1 + % change in DQ)*(1 + i F ) 1 Profit making Strategy: If i UH > i H then borrow in dollars and invest in foreign currency If i UH < i H then borrow in foreign currency and invest in dollars If i UH = i H then you cannot make any profit 6-44 Uncovered Rate: Foreign County s View point The uncovered rate from the foreign county s point of view (i UF ) 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), and 2. Investing the DOLLARS at the US INTEREST RATE (i H ), and 3. Converting DOLLARS back into FOREIGN CURRENCY at maturity using the future spot exchange rate (S 1 ) This return is affected by two factors: whether the US Dollars appreciates or depreciates against the foreign currency = % change in indirect quote (IQ) = (S 0 -S 1 ) / S 1 The rate of interest you earn in the home country (US) = i H You calculate it as: i UF = (1 + % change in IQ)*(1 + i H ) 1 Profit making strategy : If i UF > i F then borrow in foreign currency and invest in dollars If i UF < i F then borrow in dollars and invest in foreign currency If i UF = i F then you cannot make any profit

24 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 UH = i H or i UF = i F According to IFE the above conditions will hold only when the expected percentage change in the spot rate, E(e): E(e)= (1 + i H ) / (1 + i F ) 1 Approximately: E(e)= i H -i F According to IFE, the expected spot rate 1 year from now, E(S 1 ), should be: E(S 1 ) = S [1 + E(e)] 6-46 Uncovered Rate and IFE: Summarized If i UH > i H or i UF < i F 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 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), i H, i F, until i UH = i H or i UF = i H, or IFE holds If i UF > i F or i UH < i H 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 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), i H, i F, until i UH = i H or i UF = i H, or IFE holds

25 IFE : Example 1 Interest rate in US, i H = 7 % & Euro zone interest rate, i F = 9 % Current spot rate for, S = \$1.40 According to IFE, the percentage change in exchange rate, based on direct quote, for should be: E(e) = (1.07) / (1.09) - 1 = % According to IFE, the expected spot rate for at the end of the year should be: E(S 1 ) = \$1.40 ( ) = \$ 1.37 / What happens if you believe (predict) that S 1 will be \$1.39? You could make money by borrowing in \$ and investing in Can you show how? What happens if you believe (predict) that S 1 will be \$1.35? You could make money by borrowing in and investing in \$ Can you show how? 6-48 IFE : Example 2 Interest rate in US, i H = 7% & Interest rate in Switzerland, i F = 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: E(e) = (1.06) / (1.03) - 1 = % According to IFE, the expected spot rate for SF at the end of the year should be: E(S 1 ) = \$0.85 ( ) = \$0.883 / SF What happens if you believe (predict) that S 1 will be \$0.90? You could make money by borrowing in \$ and investing in FF Can you show how? What happens if you believe (predict) that S 1 will be \$0.87? You could make money by borrowing in SF and investing in \$ Can you show how?

26 Approximate Equilibrium Exchange Rate Relationships IFE E(e) ( ) FEP (i H i F ) IRP PPP F S S E( H F ) 6-50 Exact Equilibrium Exchange Rate Relationships IFE E ( S 1 ) S FEP 1 + i H 1 + i F IRP PPP F S E(1 + H ) E(1 + F )

27 Variable Definitions S= Current spot rate (price of foreign currency) in direct quote S 1 = Actual spot rate, 1 year from now F = 1-year forward rate FP H = the forward premium = [(F-S) / S] = [(F/S) - 1] from the home country s view point FP F = the forward premium = [(S-F) / F] = [(S/F) - 1] from the foreign country s view point H = Inflation rate in the home country F = Inflation rate in the foreign country ρ = Real rate of interest E(S 1 ) = Expected spot rate, 1 year from now, based on PPP E(e) = [E(S 1 )/S] 1 = The expected percentage change, or rate of change, in the spot rate, based on PPP e = (S 1 /S) 1 = The actual percentage change, or rate of change, in the spot rate S r = real spot rate i H = Nominal interest rate for the home country i F = Nominal interest rate for the foreign country Formula: Purchasing Power Parity (PPP) Exact relationship: E(e) =(1 + π H )/(1+π π F ) - 1 Approximate relationship: E(e) = π H - π F S 1 = S 0 * [ 1 + E(e) ] S R = S 1 *(1 + π F ) / (1 + π H ) Fisher Equation: (1 + i) = (1 + ρ) )(1 + ) ) i = ρ + + ρ Approximately: i = ρ + 27

28 Formula: International Fisher Effect (IFE) Fisher Equation: (1 + i) = (1 + ρ) (1 + ) i = ρ + + ρ Approximately: i = ρ + i uh : Uncovered rate of return, home country s viewpoint i uf : Uncovered rate of return, foreign country s viewpoint i uh = (1 + i f ) (1 + % change in DQ ) 1 i uf = (1 + i h ) (1 + % change in IQ ) 1 % change in DQ (direct quote)= (S 1 - S 0 )/S 0 % change in IQ (indirect quote) = [1 /(1+ % change in DQ] - 1 The IFE relationship holds when: E(e)= (1 + i H ) / (1 + i F ) 1 Approximately: E(e)= i H i F S 1 =S 0 * [1 + E(e)] Formula: Interest Rate Parity (IRP) Calculating the covered rate of returns (home & foreign country s view point) i ch = Covered rate of return, home country s view point i cf = Covered rate of return, foreign country s view point i ch = (1 + i f ) (1 + FP h ) 1 i cf = (1 + i h ) (1 + FP f ) 1 The IRP relationship holds when the expected forward premium from the home country s point of view (FP h ): FP h = (1 + i h )/(1 + i f ) 1 S 1 = S 0 * ( 1 + FP h ) 28

29 The Hamburger Standard: Further Discussion

30 30

31 31

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