Chapter 11
Interest Parity
The Investment and Borrowing Criteria
Determining the Currency of Investment
An Example: Comparing of Investments
Determining the Currency in Which to Borrow
Borrowing and Investing for Arbitrage Profit
The Covered Interest Parity Condition
Mathematical Statement of Interest Parity
Market Forces Resulting in Covered Interest Parity: A Graphical Presentation
Combining PPP and Interest Parity
The Uncovered Interest Parity Condition
The Expectations Form of PPP
The Interrelationship of the Parity Conditions
Why Covered Interest Differences Persist
Transaction Costs and Interest Parity
Political Risk and Interest Parity
Taxes and Interest Parity
Liquidity Differences and Interest Parity
Effect of the Reasons for Interest Disparity on Investment and borrowing Decisions
1. The Covered Interest Parity Condition
(1) Definition: It states that when steps have been taken to avoid foreign exchange risk, costs of borrowing and rates of return on financial investment will be equal irrespective of the currency investment or the currency borrowed.
(2) Rationale: How shopping around for the highest investment yield or lowest borrowing cost pushes yields and costs in different currencies towards equality.
2. Some Useful Terms
(1) Money Market: This is the market in which short-term securities such as treasury bills and commercial paper are traded.
(2) Foreign Exchange Risk and Exposure: It is the result of uncertainty in asset or liability values, or in income flows, due to unexpected changes in exchange rates.
(3) Aviva Corporation
◎is a manufacturing company
◎is headquartered in the United States
◎has uneven cash flows; on some occasions it has surplus funds to invest, and on other occasions it needs to borrow
The Investment and Borrowing Criteria -- Determining the Currency of Investment
1. Background
(1) Aviva Corporation has some funds to place in the money market for 3 months
(2) Choices:
◎invest in securities denominated in its own domestic currency at an interest rate on locally traded commercial paper or treasury bills
◎invest in foreign currency-denominated securities
2. Securities
(1) Many countries have money markets in which financial securities denominated in the countries’ own currencies are actively traded. For example: Canadian dollar securities in Canada
(2) In the large international financial centers such as London and New York, there are active markets in securities denominated in a variety of different currencies. For example: in London there are active markets in U.S. dollar securities, Japanese yen securities, and so on.
3. How an exchange-risk-free investment decision is made
r$ is the annualized U.S. dollar interest rate
r￡is the annualized interest rate on 3-month British pound bank deposits
S($/￡) is the spot exchange rate; the number of dollars per pound
F1/4($/￡) is the 3-month forward rate at the time of investment
(1) Investing in a dollar-denominated (domestic currency) bank deposit
each dollar will provide $ (1+r$/4)
(2) Investing in a pound-denominated (foreign currency) bank deposit
◎$1 will obtain in 1/S($/￡) British pounds, assuming that there are no transaction costs
◎each dollar will receive ￡[1/S($/￡)] (1+r￡/4) after 3 months
This certain number of pounds represents an uncertain number of dollars, but a forward contract can offer a complete hedge and guarantee the number of dollars that will finally be received.
◎at the time of buying the 3-month pound-denominated deposit, Aviva sells forward the amount of pounds to be received at maturity
the number of dollars received $ [F1/4($/￡)/S($/￡)] (1+r￡/4)
◎The purchase of the spot pounds, the investment in the pound-denominated deposit, and the forward sale of pounds all take place at the same time, and there is no doubt about the number of dollars the will be received.
(3) The rule for deciding the currency in which to invest
U.S. dollar deposit / Pound deposit1+r$/4 / [F1/4($/￡)/S($/￡)] (1+r￡/4) / ˇ
= / indifferent /
Indifferent
ˇ◎a general form (1+rd) = [F(d/f)/S(d/f)] (1+rf)
(4) From the equality above, with cancelation and rearrangement we get
(11.4)r$ = r￡+ 4 {[F1/4($/￡) – S($/￡)] / S($/￡)} (1+r￡/4)
by dropping the term {[F1/4($/￡) – S($/￡)] / S($/￡)} (r￡/4); which is a small number
(11.5)r$ = r￡+ 4 [F1/4($/￡) – S($/￡)] / S($/￡)
◎the first term on the RHS is the annualized pound interest rate
◎the second term on the RHS is the annualized forward premium on pounds
◎Investors should be indifferent between home- and foreign-currency denominated securities if the home-currency interest rate equals the foreign-currency rate plus the annualized forward exchange premium/discount on the foreign currency.
An Example: Comparing Investments
1. Table 11.1. Exchange Rates and Interest Rates on Different Currency-Denominated 3-Month Bank Deposits
(1) The Covered (Hedged) Yield: The yield on foreign currency-denominated deposits when the proceeds are sold forward.
￡Covered yield = r￡+ 4 {[F1/4($/￡) – S($/￡)] / S($/￡)} (1+r￡/4)
◎This yield is slightly higher than the yield on U.S. dollar deposits.
(2) The highest yield shown in this Table is the yield on British pound deposits.
(3) If Aviva were to invest its $10 million for 3 months in the British pound deposits, covered in the forward market, it would receive back
$10,000,000 (1+0.046167/4) = $10,115,415
Determining the Currency in Which to Borrow
1. The rule for deciding the currency in which to borrow
(1)
Costs of Borrowing / U.S. dollars / Pounds1+r$/4 / [F1/4($/￡)/S($/￡)] (1+r￡/4) / ˇ
= / indifferent /
Indifferent
ˇ(2) Aviva should borrow in dollars: the amount to be repaid is lower from a dollar loan than from a pound loan.
(3) The borrowing decision criterion is seen to be the same as the investment criterion with the inequality reversed.
Borrowing and Investing for Arbitrage Profit
1. Covered Interest Arbitrage
(1) Definition: Borrowing and investing with foreign exchange exposure hedged in order to profit from differences in yields/borrowing costs on securities denominated in different currencies.
(2) The company has begun with no funds of its own, and has taken no risk. Borrowing in dollars and simultaneously investing in pounds will result in a profit if the number of dollars received from the hedged pound investment exceeds the repayment on the dollar loan, that is, if
1+r$/4 < [F1/4($/￡)/S($/￡)] (1+r￡/4)
(3) The reverse activity, borrowing in pounds and investing in dollars, will be profitable if the reverse inequality holds.
The Covered Interest Parity Condition -- Mathematical Statement of Interest Parity
1. The Covered Interest Parity Condition
(1) If we allow for compound interest, investors and borrowers will be indifferent between the dollar and pound for investing and borrowing when
(11.10)(1+r$)n = [Fn($/￡) / S($/￡)] (1+r￡)n
(2) When equation (11.10) holds, no covered interest arbitrage is profitable.
Market Forces Resulting in Covered Interest Parity: A Graphical Presentation
1. The Covered Interest Parity Diagram
(1) Covered Interest Parity: From (11.4)
(11.11)r$ - r￡= 4 {[F1/4($/￡) – S($/￡)] / S($/￡)} (1+r￡/4)
(2) Figure 11.1. The Covered Interest Parity Diagram
◎The vertical axes: The annualized 3-month forward premium on the pound
◎The horizontal axes: The annualized interest advantage of the dollar versus the pound
A pound forward premiumA dollar interest disadvantage / A pound forward premium
A dollar interest advantage
A pound forward discount
A dollar interest disadvantage / A pound forward discount
A dollar interest advantage
◎The parity condition is represented by a 45-degree line in which traces the points where the two sides of equation (11.11) are equal.
2. Inequality
(1) Suppose we have the following inequality
(11.12)r$ - r￡< 4 {[F1/4($/￡) – S($/￡)] / S($/￡)} (1+r￡/4)
It is profitable for an interest arbitrager to borrow dollars and make a covered investment in pounds.
(2) The actions taken by arbitragers would lead to the following activities
◎Borrow dollars, perhaps by issuing and selling a security—and thus tend to increase r$
◎Buy spot pounds with the borrowed dollars—and thus increase S($/￡)
◎Buy a pound security—and thus reduce r￡
◎Sell the pound proceeds forward for U.S. dollars—and thus reduce F1/4($/￡)
All four steps of arbitrage occur simultaneously, all the forces shown by the arrows occur simultaneously.
(3) Inequality (11.12) can be represented in Figure 11.1 by points such as A, B, and C that are above and to the left of the 45-degree line. We find that above the covered interest parity line, flows of funds from dollars to pounds push us back toward it, and below the line, flows of funds from pounds to dollars also push us back toward it.
(4) Table 11.2. Points off the Interest Parity Line
◎The covered margin: The advantage, if any, from engaging in covered interest arbitrage.
Combining PPP and Interest Parity -- The Uncovered Interest Parity Condition
1. The Approximation of the Forward Exchange Rate
(1) Definition: the expected future spot exchange rate
Sn*($/￡) is the expected spot exchange rate between the dollar and the pound in n years time.
(2) Speculation will make the forward exchange rate approximately equal to the expected future spot rate.
(11.13)Sn*($/￡) = Fn($/￡)
2. The Uncovered Interest Parity Condition
(1) Substituting equation (11.13) into equation (11.10),
(11.14)(1+r$)n = [Sn*($/￡) / S($/￡)] (1+r￡)n
(2) By definition
(11.15)Sn*($/￡) ≡S($/￡) (1+S*-dot)n
where S*-dot is the average annual expected rate of change of the spot exchange rate
(11.16)(1+r$)n = (1+S*-dot)n (1+r￡)n
(11.17)1 + r$ = 1 + S*-dot + r￡+ S*-dot × r￡
(11.18)r$ - r￡= S*-dot
That is, the interest differential should approximately equal the expected rate of change of the spot exchange rate.
The Expectations Form of PPP
1. The PPP Condition in terms of Expextations
(1) From Chapter 10
(10.12)PUS*-dot – PUK*-dot = S*-dot
*: expectations; dot: the rate of change
e.g., PUS*-dot is the expected dollar return from holding commodities in the United States; the expected U.S. rate of inflation
◎The expected return in terms of dollar from buying and holding the basket of commodities in the United States will be equal to the expected return in terms of dollars from buying and holding the same basket of commodities in Britain.
The Interrelationship of the Parity Conditions
1. From equations (10.12) and (11.18), we have
(11.19)r$ - r￡= PUS*-dot – PUK*-dot
(11.20)r$- PUS*-dot = r￡– PUK*-dot
(1) The real interest rate is the nominal interest rate minus the expected rate of inflation.
(2) Equation (11.20) is called the Fisher-open condition which is the condition that the real interest rates are equal in different countries.
2. The interdependence of Exchange Rates, Interest Rates, and Inflation Rates
(1) Mathematical representations
◎Uncovered interest parity:r$ – r￡= S*-dot($/￡)
◎Expected PPP:PUS*-dot – PUK*-dot = S*-dot($/￡)
◎Fisher-open:r$ - PUS*-dot = r￡– PUK*-dot
(2) Figure 11.2.: Any one of these conditions can be derived from the other two.
Why Covered Interest Differences Persist
The failure to achieve exact covered interest parity could occur because in actual financial markets there are:
1. Transaction costs
2. Political Risks
3. Potential tax advantages to foreign exchange gains versus interest earnings
4. Liquidity differences between foreign securities and domestic securities
Transaction Costs and Interest Parity
1. Sources of transaction costs
(1) The cost of transaction in foreign exchange is reflected in the bid-ask spread in exchange rates.
(2) Covered investment or borrowing involve two foreign exchange transaction costs－one on the spot market and the other on the forward market.
(3) There are additional transaction costs of interest arbitrage due to interest-rate spreads. This is because the borrowing interest rate is likely to exceed theinvestment interest rate.
2. Situations off the interest parity line
(1) It SEEMS that there could be large deviations from interest parity by up to the extra transaction costs of investing or borrowing in foreign currency before the benefits of the foreign alternative are sufficient to compensate for the added costs.
◎interest rate spread
◎foreign exchange transaction costs
3. Reasons for the recognition that transaction costs DO NOTcontribute to deviations from interest parity
(1) One-way interest arbitrage
Figure 11.3. One-way and Round-trip interest arbitrage
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◎For round-trip interest arbitrage to be profitable it is necessary to overcome the transaction costs in the foreign exchange markets and the borrowing-lending spread on interest rates.
◎Figure 11.4. Interest Parity in the Presence of Transaction Costs, Political Risk, or Liquidity Premiums
A band around the interest parity line within which round-trip interest arbitrage is unprofitable.
◎One-way interest arbitrage
Look at equation (11.25) on p.277.
Either method of going from ＄0 to ￡n requires buying pounds, the only difference being whether they are purchased on thespot or the forward market. Therefore, transaction costs will be paid whatever the choice.
(2) The choice faced by third-country borrowers and investors
◎The Japanese or Germans pay foreign exchange costs whatever the currency of their investment or borrowing, i.e., ＄or ￡, and compare investment or borrowing rates in thetwo currencies.
(3) Summary
For the reasons described above, we can expect deviations from interest parity to be too small for round-trip arbitrage to ever occur. We conclude that transaction costs are probably NOTa cause of deviations from interest parity.
Political Risk and Interest Parity
1. Definition:
Political risk involves the uncertainty that while funds are invested in a foreign country, they may be frozen, become inconvertible into other currencies, or be confiscated.
2. Alternative situations
(1) Usually, the investment that involves the least political risk is at home.
(2) It is possible that for investors in some countries, it is politically less risky to send funds abroad. For example, people in some volatile countries have invested in Switzerland and the United States for political safety.
3. It is important to remember that political risk relates to the country, not the currency, of investment.
(1) The extra spread when political risk is faced versus when it is not faced is a measure of the political risk premium.
4. Even when covered yields are on instruments which trade in different countries, third-country investors might force interest rates and exchange rates onto the interest parity line.
◎Japanese investors view the United States and Britain as equally risky politically
◎If conditions are driven onto the interest parity line, this will encourage U.S. and British investors to keep funds at home, because neither is receiving compensation for the perceive risk of investing in the other country.
Taxes and Interest Parity
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Liquidity Differences and Interest Parity
1. Definition:
The liquidity of an asset can be judged by how quickly and cheaply it can be converted into cash.
2. Sources of Costs
Situation: a covered foreign-currency 90-day investment is sold after only 50 days
(1) The brokerage costs for selling a security
(2) Transaction costs are faced when investors convert the foreign exchange from the hedged foreign-currency security on the spot market.
(3) When a security is sold prior to maturity and the funds are converted into domestic currency, there is still thematter of honoring the original forward contract to sell the foreign exchange at the maturity of the foreign currency investment.
◎to buy spot foreign currency at the date of maturity
◎to buy a new forward exchange contract of 40 days maturity
3. The amount of extra required return and hence the potential width of the band due to liquidity preference depend on the likelihood that the funds will be needed early and on whether these funds can be borrowed by using the original covered investment to secure the loans.
◎The more uncertainty there is concerning future needs and alternative sources of short-term financing, the higher are the premiums that should be required before venturing into foreign-currency securities.
Effect of the Reasons for Interest Disparity on Investment and Borrowing Decisions
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TM11-1