# International Parity Conditions

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International Parity Conditions

Law of One Price In the absence of shipping costs, tariffs, and other frictions, identical goods should trade for the same real price in different economies: Pi = s P*i The Law of One Price holds perfectly for homogeneous goods with low transaction costs. Why? Examples: precious metals, wheat, oil

Purchasing Power Parity is simply the extension of the Law of One Price to all products in two economies. It says that the overall real price levels should be identical: P = s P* Example: Costs \$1400 to purchase a certain basket of U.S. consumption goods. If Swiss Franc trades at 2 (\$ per Franc), how many Swiss Francs will the same basket cost in Geneva?

Because overall economy price levels consist of different goods in different countries, a more appropriate form of PPP is the relative form. Relative Purchasing Power Parity asserts that relative changes in price levels will be offset by changes in exchange rates: % DP - % DP* = % Ds Or denoting inflation (%DP) as   - D * = %Ds RPPP asserts that differences in inflation rates will be offset by changes in the exchange rate.

Example: A year ago, the Brazilian Real traded at \$0.917/Real. For 2003, Brazil’s inflation was 4.1% and the U.S. inflation was 1.7%. What should be the value of the Real today?

Exchange Rates and Asset Prices
Exchange rates are determined by the relative supplies and demands for currencies. Since buyers and sellers are ultimately interested in purchasing something with the currency - goods, services, or investments - their prices and returns must indirectly influence the demand for a given currency. So, prices, exchange rates, and interest rates must be linked….

Forward Market Basics Forward Contract involves contracting today for the future purchase or sale of foreign exchange.

90 - day Swiss franc contract
Forward Market Basics 90 - day Swiss franc contract You buy Swiss Francs (long position) S90(\$/SF)

90 - day Swiss franc contract
Forward Market Basics 90 - day Swiss franc contract F90(\$/SF) = .8446 S90(\$/SF)

90 - day Swiss franc contract
Forward Market Basics Profit \$ 90 - day Swiss franc contract Y -axes measures profits or losses in \$. S90(\$/SF) Forward price a buyer will pay in dollars for Swiss franc in 90 days X- axes shows the spot price on maturity date of the forward contract

90 - day Swiss franc contract
Forward Market Basics 90 - day Swiss franc contract Long Contract Profit \$ S90(\$/SF) F90(\$/SF) = .8446 If price drops to 0 then the buyer will pay \$.8446 while he could pay \$0. His loss then is

90 - day Swiss franc contract
Forward Market Basics 90 - day Swiss franc contract Long Contract Profit \$ If price is .8446 then his profit is then 0. S90(\$/SF) F90(\$/SF) = .8446 -F90(\$/SF)

90 - day Swiss franc contract
Forward Market Basics Profit \$ 90 - day Swiss franc contract Long position S90(\$/SF) F90(\$/SF) = .8446 -F90(\$/SF)

90 - day Swiss franc contract
Forward Market Basics Profit \$ 90 - day Swiss franc contract F90(\$/SF) S90(\$/SF) F90(\$/SF) = .8446 Short position

Law of One Price for Assets
Absent frictions, identical goods must trade for identical prices in different countries when converted into a common currency. The same condition should hold for assets. One important difference between goods and assets: Price is not paid immediately - it is paid over time in the form of returns. This introduces the primary friction for exchanging assets - a friction not found in goods. Risk.

Law of One Price for Assets
Hence, there must exist a corresponding version of LOP for assets which requires returns to be identical across countries once this friction has been removed: Covered Interest Parity Exactly like the Law of One Price, Covered Interest Parity requires frictionless markets to offer identical rates of returns for identical assets. How do make assets in two countries identical? Eliminate risk: 1. Eliminate exchange rate risk with forward contracts. 2. Compare assets whose other risks are minimal (i.e. default).

Law of One Price for Assets
Arbitrageurs will guarantee that the following two strategies will generate the exact same common-currency return: 1. a. Purchasing \$1 worth of U.S. short-term treasuries (lend money).

Law of One Price for Assets
Arbitrageurs will guarantee that the following two strategies will generate the exact same common-currency return: 1. a. Purchasing \$1 worth of U.S. short-term treasuries. b. Obtain an n-period return of 1+Rt,t+n.

Law of One Price for Assets
Arbitrageurs will guarantee that the following two strategies will generate the exact same common-currency return: 1. a. Purchasing \$1 worth of U.S. short-term treasuries. b. Obtain an n-period return of 1+Rt,t+n. 2. a. Convert \$1 into foreign currency at rate 1/st (FC/\$).

Law of One Price for Assets
Arbitrageurs will guarantee that the following two strategies will generate the exact same common-currency return: 1. a. Purchasing \$1 worth of U.S. short-term treasuries. b. Obtain an n-period return of 1+Rt,t+n. 2. a. Convert \$1 into foreign currency at rate 1/st (FC/\$). b. Purchase corresponding foreign short-term treasuries (borrow money in foreign currency).

Law of One Price for Assets
Arbitrageurs will guarantee that the following two strategies will generate the exact same common-currency return: 1. a. Purchasing \$1 worth of U.S. short-term treasuries. b. Obtain an n-period return of 1+Rt,t+n. 2. a. Convert \$1 into foreign currency at rate 1/st (FC/\$). b. Purchase corresponding foreign short-term treasuries. c. Receive an n-period foreign currency return of 1+R*t,t+n.

Law of One Price for Assets
Arbitrageurs will guarantee that the following two strategies will generate the exact same common-currency return: 1. a. Purchasing \$1 worth of U.S. short-term treasuries. b. Obtain an n-period return of 1+Rt,t+n. 2. a. Convert \$1 into foreign currency at rate 1/st (FC/\$). b. Purchase corresponding foreign short-term treasuries. c. Receive an n-period foreign currency return of 1+R*t,t+n. d. Eliminate the currency risk of the foreign return by locking in an exchange rate of Ft,t+n (\$/FC).

Law of One Price for Assets
Arbitrageurs will guarantee that the following two strategies will generate the exact same common-currency return: 1. a. Purchasing \$1 worth of U.S. short-term treasuries. b. Obtain an n-period return of 1+Rt,t+n. 2. a. Convert \$1 into foreign currency at rate 1/st (FC/\$). b. Purchase corresponding foreign short-term treasuries. c. Receive an n-period foreign currency return of 1+R*t,t+n. d. Eliminate the currency risk of the foreign return by locking in an exchange rate of Ft,t+n (\$/FC). e. Obtain an overall n-period return of: Ft,t+n (1+R*t,t+n) / st

Synthetic Forward Contract
Another way to derive the forward price of FC is replicate it synthetically: 1. Borrow \$ 2. Convert to FC (at St) 3. Lend the FC. I now effectively have a forward contract. I have committed to pay a certain quantity of \$ in the future in return for receiving a certain quantity of FC in the future. Through exchange rate and money markets, we can synthetically deposit, lend, exchange currency spot, or exchange currency forward. We just need to keep proper track of differences between bid and ask prices and borrowing and lending rates.

Spot, Forward, and Money Market Relationships
Time Dimension t t+n Borrow at \$ loan rate \$ A D Currency Dimension Buy FC Spot at ask Sell FC Forward at bid FC B C Lend at FC deposit rate

Spot, Forward, and Money Market Relationships
Time Dimension t t+n \$ A D Lend at \$ deposit rate Currency Dimension Buy FC Forward at ask Sell FC Spot at bid Borrow at FC loan rate FC B C

Spot, Forward, and Money Market Relationships
Time Dimension t t+n Borrow at \$ loan rate \$ A D Lend at \$ deposit rate Currency Dimension Buy FC Spot at ask Buy FC Forward at ask Sell FC Forward at bid Sell FC Spot at bid Borrow at FC loan rate FC B C Lend at FC deposit rate

Spot, Forward, and Money Market Relationships
Time Dimension t t+n 1/(1+Rt,t+n) L \$ A D Currency Dimension A 1/st Ft,t+n B FC B C (1+R*t,t+n) D

Spot, Forward, and Money Market Relationships
Time Dimension t t+n \$ A D (1+Rt,t+n) D Currency Dimension B st 1/Ft,t+n A 1/(1+R*t,t+n) L FC B C

Spot, Forward, and Money Market Relationships
Time Dimension t t+n 1/(1+Rt,t+n) L \$ A D (1+Rt,t+n) D Currency Dimension A B 1/st st 1/Ft,t+n Ft,t+n A B 1/(1+R*t,t+n) L FC B C (1+R*t,t+n) D

An arrow from FC to \$, can be thought of
as SELLING FC or BUYING \$. (2) The reverse arrow from \$ to FC represents the reverse transaction, SELLING \$ or BUYING FC. (3) An arrow from right to left (from the future to the present), can be thought of as borrowing - taking cash from the future and bringing it to the present. (4) The reverse arrow from left to right (from the present to the future), can be thought of as investing - taking cash that you have now and putting it away until the future.

Exchange Rate Risk Covered Interest Parity says that if we lock in the forward rate to eliminate exchange rate risk, the common-currency return to otherwise riskless deposits in two currencies will be identical: 1+Rt,t+n = Ft,t+n (1+R*t,t+n) / st What happens if we don’t lock in the forward rate? How will the returns compare if we use an unhedged or “uncovered” version and just convert returns at the future spot rate? 1+Rt,t+n vs. st+n (1+R*t,t+n) / st

Exchange Rate Risk If exchange rate risk is not priced (if investors do not require compensation for bearing exchange rate risk) then expected returns are equal: 1+Rt,t+n vs. E [ st+n ] (1+R*t,t+n) / st and, if those expectations are rational, on average they are right: 1+Rt,t+n = st+n (1+R*t,t+n) / st Alternatively, this says that on average the forward rate equals the future spot rate: Ft,t+n = st+n . This is known as the unbiased forward hypothesis.

Uncovered Interest Parity
Put differently, if exchange rate risk is not priced, an ‘unhedged’ version of covered interest parity should hold as well. 1+Rt,t+n = Ft,t+n (1+R*t,t+n) st

Uncovered Interest Parity
Put differently, if exchange rate risk is not priced, an ‘unhedged’ version of covered interest parity should hold as well. On average: 1+Rt,t+n = st+n (1+R*t,t+n) st

Uncovered Interest Parity
Put differently, if exchange rate risk is not priced, an ‘unhedged’ version of covered interest parity should hold as well. On average: 1+Rt,t+n = st+n (1+R*t,t+n) st Which can be closely approximated by the Uncovered Interest Parity equation: Rt,t+n - R*t,t+n = % D st,t+n.

Covered and Uncovered Interest Parity
The Intuition of Covered and Uncovered Interest Parity In CIP, if FC interest rates are low, how can we get US\$ based investors to hold FC assets? The answer is that we offer them a more favorable forward rate (higher F in terms of \$/FC) to offset the low FC interest rate. So the market is working by pricing F to offset a known low FC interest rate. (2) In UIP, if we expect the US\$ to be weaker in the future (meaning more \$ per FC) how would we get investors to willingly hold US\$ assets? The answer is, we offer them an added bonus in the form of a higher \$ interest rate - just high enough to offset the loss of a weaker US\$. So the market is working by setting a high \$ interest rate to offset an expected depreciation of the US\$.

Uncovered Interest Parity
High interest rate currencies don’t, on average, depreciate sufficiently. There are 3 possible explanations: 1. Risk Premia: The high interest rates of discount currencies are not only compensating investors for an expected decline in the exchange rate, but also for the bearing risks associated with that currency. Peso Problem: Remember, UIP holds “on average.” We may have difficulty observing the true average in the data. High interest rate currencies may include the possibility of extremely large depreciations which have not occurred during the sample period. 3. Irrational Expectations: investors systematically get the future exchange rate wrong.

Key International Relationships

Key International Relationships Relative Inflation Rates
Exchange Rate Change

Key International Relationships
RPPP: P - P* = %Ds Inflation differentials are offset by changes in spot exchange rate. Relative Inflation Rates Exchange Rate Change

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Exchange Rate Change

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change CIP: Ft,t+n / st =(1+ R) /(1+ R*) Forward differs from spot by interest rate differential Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change Unbiased Forward: Ft,t+n = E(st+n) Forward is expectation of spot Covered Interest Parity Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
Fisher Effect: 1+R = (1+r)(1+P) Interest rate equals real rate plus expected inflation Relative Inflation Rates Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
1+R = (1+r)(1+E(P)) Relative Inflation Rates R - R* = P - P* With RIP, interest rates reflect expected inflation differential. Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Fisher Effect and Real Interest Parity Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Fisher Effect and Real Interest Parity Purchasing Power Parity Relative Interest Rates Exchange Rate Change Unbiased Forward Rate Forward Exchange Rates Ft,t+n / st =(1+ R) /(1+ R*)

Key International Relationships
Relative Inflation Rates Fisher Effect and Real Interest Parity Purchasing Power Parity Relative Interest Rates Exchange Rate Change Forward Exchange Rates Ft,t+n / st =(1+ R) /(1+ R*) Ft,t+n = E(st+n)

Key International Relationships
Relative Inflation Rates Fisher Effect and Real Interest Parity Purchasing Power Parity Uncovered Interest Parity: R - R* = %Ds Exchange rate changes offset interest differentials Relative Interest Rates Exchange Rate Change Forward Exchange Rates Ft,t+n / st =(1+ R) /(1+ R*) Ft,t+n = E(st+n)

Key International Relationships
Relative Inflation Rates 1+R = (1+r)(1+P) R - R* = P - P* Purchasing Power Parity Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
Relative Inflation Rates 1+R = (1+r)(1+P) R - R* = P - P* P - P* = %Ds Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
Relative Inflation Rates 1+R = (1+r)(1+P) R - R* = P - P* P - P * = %Ds Uncovered Interest Parity: R - R* = %Ds Exchange rate changes offset interest differentials Relative Interest Rates Exchange Rate Change Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

Key International Relationships
Relative Inflation Rates Fisher Effect and Real Interest Parity Purchasing Power Parity Relative Interest Rates Exchange Rate Change Uncovered Interest Parity Covered Interest Parity Unbiased Forward Rate Forward Exchange Rates

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