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CHAPTER 6 INTERNATIONAL PARITY RELATIONSHIPS AND FORECASTING FOREIGN EXCHANGE RATES ANSWERS & SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS QUESTIONS 1. Give a full definition of arbitrage. Answer: Arbitrage can be defined as the act of simultaneously buying and selling the same or equivalent assets or commodities for the purpose of making certain, guaranteed profits. 2. Discuss the implications of the interest rate parity for the exchange rate determination. Answer: Assuming that the forward exchange rate is roughly an unbiased predictor of the future spot rate, IRP can be written as: S = [(1 + i£)/(1 + i$)]E[St+1It]. The exchange rate is thus determined by the relative interest rates, and the expected future spot rate, conditional on all the available information, It, as of the present time. One thus can say that expectation is self-fulfilling. Since the information set will be continuously updated as news hit the market, the exchange rate will exhibit a highly dynamic, random behavior. 3. Explain the conditions under which the forward exchange rate will be an unbiased predictor of the future spot exchange rate. Answer: The forward exchange rate will be an unbiased predictor of the future spot rate if (i) the forward risk is insignificant and (ii) foreign exchange markets are informationally efficient. 5. Discuss the implications of the deviations from the purchasing power parity for countries’ competitive positions in the world market. Answer: If exchange rate changes satisfy PPP, competitive positions of countries will remain unaffected following exchange rate changes. Otherwise, exchange rate changes will affect relative competitiveness of countries. If a country’s currency appreciates (depreciates) by more
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than is warranted by PPP, that will hurt (strengthen) the country’s competitive position in the world market. 6. Explain and derive the international Fisher effect. Answer: The international Fisher effect can be obtained by combining the Fisher effect and the relative version of PPP in its expectational form. Specifically, the Fisher effect holds that E($) = i$ - $, E(£) = i£ - £. Assuming that the real interest rate is the same between the two countries, i.e., $ = £, and substituting the above results into the PPP, i.e., E(e) = E($)- E(£), we obtain the international Fisher effect: E(e) = i$ - i£. The international Fisher effect holds that the expected rate of exchange rate change is equal to the interest rate differential between a pair of countries. 7.
Researchers found that it is very difficult to forecast the future exchange rates more
accurately than the forward exchange rate or the current spot exchange rate. How would you interpret this finding? Answer: This implies that exchange markets are informationally efficient. Thus, unless one has private information that is not yet reflected in the current market rates, it would be difficult to beat the market.
10. Explain the following three concepts of purchasing power parity (PPP): a. The law of one price. b
b. Absolute PPP. c. Relative PPP.
Answer: a. The law of one price (LOP) refers to the international arbitrage condition for the standard consumption basket. LOP requires that the consumption basket should be selling for the same price in a given currency across countries. b. Absolute PPP holds that the price level in a country is equal to the price level in another country times the exchange rate between the two countries.
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c. Relative PPP holds that the rate of exchange rate change between a pair of countries is about equal to the difference in inflation rates of the two countries.
PROBLEMS 1. Suppose that the treasurer of IBM has an extra cash reserve of $100,000,000 to invest for six months. The six-month interest rate is 8 percent per annum in the United States and 7 percent per annum in . Currently, the spot exchange rate is €1.01 per dollar and the six-month forward exchange rate is €0.99 per dollar. The treasurer of IBM does not wish to bear any exchange risk. Where should he/she invest to maximize the return? $105,590,909The market conditions are summarized as follows: i$ = 4%; i€ = 3.5%; S = €1.01/$; F = €0.99/$. If $100,000,000 is invested in the U.S., the maturity value in six months will be Equation: FV = PV(1+i/m): $104,000,000 = $100,000,000 (1 + .08/2 interest). Alternatively, $100,000,000 can be converted into euros and invested at the German interest rate, with the euro maturity value sold forward. In this case the dollar maturity value will be = ($100,000,000 x €1.01) invested in = €101,000,000 that will increase to FV = PV(1+i/m): €104,535,000 = €101,000,000 (1 + .07/2 interest) and which will be exchanged back to $ at €0.99: €104,535,000/.99 = =$105,590,909 Clearly, it is better to invest $100,000,000 in with exchange risk hedging. Online problem format – original question: What will the value of the investment be in six months if invested in US$? a. $100,000,000 x 1.04 = $104,000,000 b. $100,000,000 x 1.035 = $103,500,000 c. $100,000,000 x (1.04 -1.035) = $100,500,000 d. None of the above
What will the value of the investment in Euros be if invested in for 6 months?
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a. $100,000,000 x 1.04 = $104,000,000 x 0.99 = €102,960,000 b. $100,000,000 x 1.01 x 1.04 = €105,040,000 c. $100,000,000 x 1.01 x 1.035 = €104,535,000 (Correct) d. $100,000,000 x 0.99 x 1.035 = €102,465,000 If he considers to do an investment in Euros – then what forward position will he take with preference to the currency that he will exchange his investment to after 6 months? a. Long in € b. Short in € c. Long in $ (Correct) d. Short in $ What will the $ value of the investment in be after 6 months? a. €102,960,000 / 0.99 = $104,000,000.00 b. €105,040,000 x 0.99 =$103,989.00 c. €104,535,000 / 0.99 = $105,590,909.09 (Correct) d. €102,465,000 x 0.99 = $101,440,350.00 e. €102,960,000 / 1.01 = $101,940,594.06 f. €105,040,000 x 1.01 = $106,090,400.00 g. €104,535,000 / 1.01 = $103,500,000.00 h. €102,465,000 x 1.01 = $103,489,650.00
2. While you were visiting London, you purchased a Jaguar for £35,000, payable in three months. You have enough cash at your bank in New York City, which pays 0.35% interest per month, compounding monthly, to pay for the car. Currently, the spot exchange rate is $1.45/£ and the three-month forward exchange rate is $1.40/£. In London, the money market interest rate is 2.0% for a three-month investment. There are two alternative ways of paying for your Jaguar. (a) Keep the funds at your bank in the U.S. and buy £35,000 forward. (b) Buy a certain pound amount spot today and invest the amount in the U.K. for three months so that the maturity value becomes equal to £35,000. Evaluate each payment method. Which method would you prefer? Why?
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Solution: The problem situation is summarized as follows: A/P = £35,000 payable in three months iNY = 0.35%/month, compounding monthly iLD = 2.0% for three months S = $1.45/£;
F = $1.40/£.
Option a: When you buy £35,000 forward, you will need $49,000 (= £35,000 × $1.40/£) in three months to fulfill the forward contract. The present value of $49,000 is computed as follows: FV/(1+i)n = PV
$49,000/(1.0035)3 = $48,489.
Thus, the cost of Jaguar as of today is $48,489. Option b: The present value of £35,000 is £34,314 = £35,000/(1.02). To buy £34,314 today, it will cost $49,755 = 34,314x1.45. Thus the cost of Jaguar as of today is $49,755. You should definitely choose to use “option a”, and save $1,266, which is the difference between $49,755 and $48489. Online problem format original question: While you were visiting London, you purchased a Jaguar for £35,000, payable in three months. You have enough cash at your bank in New York City, which pays 0.35% interest per month, compounding monthly, to pay for the car. Currently, the spot exchange rate is $1.45/£ and the three-month forward exchange rate is $1.40/£. In London, the money market interest rate is 2.0% for a three-month investment. There are two alternative ways of paying for your Jaguar. (a) Keep the funds at your bank in the U.S. and buy £35,000 forward. (b) Buy a certain pound amount spot today and invest the amount in the U.K. for three months so that the maturity value becomes equal to £35,000. Evaluate each payment method. Which method would you prefer? Why? In what currency and for what amount should the forward contract be entered into if you keep the funds in your US ? a. Long in pound = £35,000 (Correct) b. Long In dollar = £35,000 x $1.40/£ = $49,000 c Short in pound = £35,000
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d. Short In dollar = £35,000 x $1.40/£ = $49,000 What will be the cost of the Jaguar as of today in $ if you enter into the forward contract and keep the funds in your US ? a. PV =(£35,000/(1.02) x 1.40 = $48,039.22 b. PV = $49,000/(1.0035)3) = $48,489.08 (Correct) What amount in pound will you purchase today to invest in the UK if you apply option b? a. PV = $49,000/(1.0035)3) / 1.40 = £34,635.06 b. PV = £35000/(1.02) = £34,313.73 (Correct) Which method is the best and why? a. Alternative A because cheaper (Correct) b. Alternative B because cheaper 3. Currently, the spot exchange rate is $1.50/£ and the three-month forward exchange rate is $1.52/£. The three-month interest rate is 8.0% per annum in the U.S. and 5.8% per annum in the U.K. Assume that you can borrow as much as $1,500,000 or £1,000,000. a. Determine whether the interest rate parity is currently holding. b. If the IRP is not holding, how would you carry out covered interest arbitrage? Show all the steps and determine the arbitrage profit. c. Explain how the IRP will be restored as a result of covered arbitrage activities. Solution: Let’s summarize the given data first: S = $1.5/£; F = $1.52/£; i$ = 2.0%; i£ = 1.45% Credit = $1,500,000 or £1,000,000. a. (1+i$) = 1.02 (1+i£)(F/S) = (1.0145)(1.52/1.50) = 1.0280 Thus, IRP is not holding exactly. b. (1) Borrow $1,500,000; repayment will be $1,530,000. (2) Buy £1,000,000 spot using $1,500,000. (3) Invest £1,000,000 at the pound interest rate of 1.45%; maturity value will be £1,014,500. (4) Sell £1,014,500 forward for $1,542,040
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Arbitrage profit will be $12,040 (=$1,542,040 - $1,530,000). c. Following the arbitrage transactions described above, The dollar interest rate will rise; The pound interest rate will fall; The spot exchange rate will rise; The forward exchange rate will fall. These adjustments will continue until IRP is restored. Online problem format – original Question: Currently, the spot exchange rate is $1.50/£ and the three-month forward exchange rate is $1.52/£. The three-month interest rate is 8.0% per annum in the U.S. and 5.8% per annum in the U.K. Assume that you can borrow as much as $1,500,000 or £1,000,000. a. Determine whether the interest rate parity is currently holding. b. If the IRP is not holding, how would you carry out covered interest arbitrage? Show all the steps and determine the arbitrage profit. c. Explain how the IRP will be restored as a result of covered arbitrage activities. Calculate the forward rate, based on IRP ($(1+i $)/£(1+i£)), to determine whether the interest rate parity is currently holding – What rate do you find? a. $1.53/£ b. $1.52/£ c. $1.51/£(Correct) Does interest rate parity hold or not? a. Yes b. No (Correct) What currency will you borrow? a. $ (Correct) b. £ c. Will not borrow What is the interest rate at which you will borrow?
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a. 2% for three months (Correct) b. 1.45% for three months c. Not applicable What is the total end-of-period borrowed amount? a. $1,521,750 b. £1,014,500 c. $1,530,000 (Correct) d. £1,020,000 e. Not applicable In what currency will you invest? a. $ b. £ (Correct) c. Not applicable Will you go short or long in a forward contract? a. Short $1,521,750 b. Long $1,521,750 c. Short £1,014,500 (Correct) d. Long £1,014,500 e. Short $1,530,000 f. Long $1,530,000 g. Short £1,020,000 h. Long £1,020,000 i. Not applicable How much arbitrage profit will you make in $? a. $22,185 b. €22,185 c. $12,040 (Correct) d. £12,040 e. No profit due to no arbitrage opportunity
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Indicate which of the following situations may occur to restore the IRP as a result of covered arbitrage activities: a. The dollar interest rate will rise b. The pound interest rate will fall c. The spot exchange rate will rise d.The forward exchange rate will fall e. a and b f. c and d g. a, b, c and d (Correct)
7. (CFA question) Omni Advisors, an international pension fund manager, uses the concepts of purchasing power parity (PPP) and the International Fisher Effect (IFE) to forecast spot exchange rates. Omni gathers the financial information as follows: Base price level Current U.S. $ price level Current South African price level Base rand spot exchange rate Current rand spot exchange rate Expected annual U.S. inflation Expected annual South African inflation Expected U.S. one-year interest rate Expected South African one-year interest rate
100 105 111 $0.175 $0.158 7% 5% 10% 8%
Calculate the following exchange rates (ZAR and USD refer to the South African rand and U.S. dollar, respectively). a. The current ZAR spot rate in USD that would have been forecast by PPP. b. Using the IFE, the expected ZAR spot rate in USD one year from now. c. Using PPP, the expected ZAR spot rate in USD four years from now. Solution: a. ZAR spot rate under PPP = [1.05/1.11](0.175) = $0.1655/rand. This answer is retrieved from the following: The base price level for both currencies were 100 and the base rand spot exchange rate of 0.175 was determined by these 100 base price levels. The S = P$/PZAR was 0.175 = 100$/100ZAR. Now it changed and determine the new spot exchange rate based on these changes in price levels.
Price levels were 1:1, now 0.945945 :1, thus 0.175 changed to
0.945945 of original 0.175 to .1655.
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b. Expected ZAR spot rate using IFE 1 year from now = [1.10/1.08] (0.158) = $0.1609/rand. Same principle applies as in a. Only difference we use interest rates to conduct estimate c. Expected ZAR under PPP four years from now= [(1.07)4/(1.05)4] (0.158) = $0.1704/rand. Same as b, but use inflation rates instead of interest rates. IMPORTANT NOTE: IFE suggests that the nominal interest rate difference reflects the expected change in exchange rates p.158 in textbook.
Online problem format original question: Formulas to be applied: The absolute version of purchasing power parity (PPP): S = P$/P£. The relative version is: e = $ - £. The international Fisher effect: E(e) = i$ - i£. You are provided with the following solutions to the questions in this problem: i. [1.10/1.08] (0.158) = $0.1609/rand. ii. [(1.07)4/(1.05)4] (0.158) = $0.1704/rand. iii. [1.05/1.11](0.175) = $0.1655/rand. Which one of the solutions calculates the ZAR spot rate under PPP? a. i. b. ii. c. iii. (Correct) Which one of the solutions calculates the expected ZAR spot rate one year from now with IFE? a. i. (Correct) b. ii. c. iii. Which one of the solutions calculates the expected ZAR spot rate in USD four years from now under PPP?
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a. i. b. ii. (Correct) c. iii. 11. James Clark is a foreign exchange trader with Citibank. He notices the following quotes. Spot exchange rate
SFr1.2051/$
Six-month forward exchange rate
SFr1.1922/$
Six-month $ interest rate
2.5% per year
Six-month SFr interest rate
2.0% per year
a. Is the interest rate parity holding? You may ignore transaction costs. b. Is there an arbitrage opportunity? If yes, show what steps need to be taken to make arbitrage profit. Assuming that James Clark is authorized to work with $1,000,000, compute the arbitrage profit in dollars. Solution: a. When using the SFr/$ exchange rate as provided in the question we can consider interest rate parity as follows: For six months , iSFr = 1.0% and i$ = 1.25%. the spot exchange rate is SFr1.2051/$ and the forward rate is SFr1.1922/$. Thus, (1 + iSFr) = 1.0100 and (F/s)(1+ i$ ) = = (1.1922/1.2051)(1.0125) = 1.0017 Because the left and right sides of IRP are not equal, IRP is not holding. When applied to $/SFr we can consider interest rate parity as follows: (We show this way as well, since in of quoting dealers use $ as currency (direct quote way) for testing interest rate parity.) For six months, iSFr = 1.0% and i$ = 1.25%. the spot exchange rate is $0.8298/SFr and the forward rate is $0.8388/SFr. Thus, (1+ i$ ) = 1.0125 and (F/s) (1 + iSFr) = (0.8388/0.8298) (1.01) = 1.02095 Because the left and right sides of IRP are not equal, IRP is not holding.
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b. Because IRP is not holding, there is an arbitrage possibility: Because 1.0125 < 1.02095, we can say that the SFr interest rate quote is more than what it should be as per the quotes for the other three variables. Equivalently, we can also say that the $ interest rate quote is less than what it should be as per the quotes for the other three variables. Therefore, the arbitrage strategy should be based on borrowing in the $ market and lending in the SFr market. The steps would be as follows: Borrow $1,000,000 for six months at 1.25%. Need to pay back $1,000,000 × (1 + 0.0125) = $1,012,500 six months later. Convert $1,000,000 to SFr at the spot rate to get SFr 1,205,100. Lend SFr 1,205,100 for six months at 1.0%. Will get back SFr 1,205,100 × (1 + 0.01) = SFr 1,217,151 six months later. Sell SFr 1,217,151 six months forward. The transaction will be contracted as of the current date but delivery and settlement will only take place six months later. So, six months later, exchange SFr 1,217,151 for SFr 1,217,151/SFr 1.1922/$ = $1,020,929. The arbitrage profit six months later is $1,020,929 – $1,012,500 = $8,429. ONLINE QUESTION FORMAT ORIGINAL QUESTION Calculate the forward rate, based on IRP (SFr(1+i SFr)/$(1+i$)), to determine whether the interest rate parity is currently holding – What rate do you find? a. SFr1.2021$ (Correct) b. SFr1.0210/$ c. SFr1.2102/$ Does interest rate parity hold or not? a. Yes b. No (Correct) What currency will you borrow? a. $ (Correct) b. SFr c. Will not borrow What is the interest rate at which you will borrow?
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a. 1% for six months b. 1.25% for six months (Correct) c. Not applicable What is the total end-of-period borrowed amount? a. $1,012,500 (Correct) b. SFr 1,217,151 c. $1,010,000 d. SFr1, 220,164 e. Not applicable In what currency will you invest? a. $ b. SFr (Correct) c. Not applicable Will you go short or long in a forward contract? a. Short $1,012,500 b. Long $1,012,500 c. Short SFr 1,217,151(Correct) d. Long SFr 1,217,151 e. Short $1,010,000 f. Long $1,010,000 g. Short SFr1, 220,164 h. Long SFr1, 220,164 i. Not applicable How much arbitrage profit will you make in $? a. $8,428.54 (Correct) b. SFr 8,932.88 c. $ 8,932.88 d. SFr8,428.54 e. No profit due to no arbitrage opportunity
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than is warranted by PPP, that will hurt (strengthen) the country’s competitive position in the world market. 6. Explain and derive the international Fisher effect. Answer: The international Fisher effect can be obtained by combining the Fisher effect and the relative version of PPP in its expectational form. Specifically, the Fisher effect holds that E($) = i$ - $, E(£) = i£ - £. Assuming that the real interest rate is the same between the two countries, i.e., $ = £, and substituting the above results into the PPP, i.e., E(e) = E($)- E(£), we obtain the international Fisher effect: E(e) = i$ - i£. The international Fisher effect holds that the expected rate of exchange rate change is equal to the interest rate differential between a pair of countries. 7.
Researchers found that it is very difficult to forecast the future exchange rates more
accurately than the forward exchange rate or the current spot exchange rate. How would you interpret this finding? Answer: This implies that exchange markets are informationally efficient. Thus, unless one has private information that is not yet reflected in the current market rates, it would be difficult to beat the market.
10. Explain the following three concepts of purchasing power parity (PPP): a. The law of one price. b
b. Absolute PPP. c. Relative PPP.
Answer: a. The law of one price (LOP) refers to the international arbitrage condition for the standard consumption basket. LOP requires that the consumption basket should be selling for the same price in a given currency across countries. b. Absolute PPP holds that the price level in a country is equal to the price level in another country times the exchange rate between the two countries.
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c. Relative PPP holds that the rate of exchange rate change between a pair of countries is about equal to the difference in inflation rates of the two countries.
PROBLEMS 1. Suppose that the treasurer of IBM has an extra cash reserve of $100,000,000 to invest for six months. The six-month interest rate is 8 percent per annum in the United States and 7 percent per annum in . Currently, the spot exchange rate is €1.01 per dollar and the six-month forward exchange rate is €0.99 per dollar. The treasurer of IBM does not wish to bear any exchange risk. Where should he/she invest to maximize the return? $105,590,909The market conditions are summarized as follows: i$ = 4%; i€ = 3.5%; S = €1.01/$; F = €0.99/$. If $100,000,000 is invested in the U.S., the maturity value in six months will be Equation: FV = PV(1+i/m): $104,000,000 = $100,000,000 (1 + .08/2 interest). Alternatively, $100,000,000 can be converted into euros and invested at the German interest rate, with the euro maturity value sold forward. In this case the dollar maturity value will be = ($100,000,000 x €1.01) invested in = €101,000,000 that will increase to FV = PV(1+i/m): €104,535,000 = €101,000,000 (1 + .07/2 interest) and which will be exchanged back to $ at €0.99: €104,535,000/.99 = =$105,590,909 Clearly, it is better to invest $100,000,000 in with exchange risk hedging. Online problem format – original question: What will the value of the investment be in six months if invested in US$? a. $100,000,000 x 1.04 = $104,000,000 b. $100,000,000 x 1.035 = $103,500,000 c. $100,000,000 x (1.04 -1.035) = $100,500,000 d. None of the above
What will the value of the investment in Euros be if invested in for 6 months?
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a. $100,000,000 x 1.04 = $104,000,000 x 0.99 = €102,960,000 b. $100,000,000 x 1.01 x 1.04 = €105,040,000 c. $100,000,000 x 1.01 x 1.035 = €104,535,000 (Correct) d. $100,000,000 x 0.99 x 1.035 = €102,465,000 If he considers to do an investment in Euros – then what forward position will he take with preference to the currency that he will exchange his investment to after 6 months? a. Long in € b. Short in € c. Long in $ (Correct) d. Short in $ What will the $ value of the investment in be after 6 months? a. €102,960,000 / 0.99 = $104,000,000.00 b. €105,040,000 x 0.99 =$103,989.00 c. €104,535,000 / 0.99 = $105,590,909.09 (Correct) d. €102,465,000 x 0.99 = $101,440,350.00 e. €102,960,000 / 1.01 = $101,940,594.06 f. €105,040,000 x 1.01 = $106,090,400.00 g. €104,535,000 / 1.01 = $103,500,000.00 h. €102,465,000 x 1.01 = $103,489,650.00
2. While you were visiting London, you purchased a Jaguar for £35,000, payable in three months. You have enough cash at your bank in New York City, which pays 0.35% interest per month, compounding monthly, to pay for the car. Currently, the spot exchange rate is $1.45/£ and the three-month forward exchange rate is $1.40/£. In London, the money market interest rate is 2.0% for a three-month investment. There are two alternative ways of paying for your Jaguar. (a) Keep the funds at your bank in the U.S. and buy £35,000 forward. (b) Buy a certain pound amount spot today and invest the amount in the U.K. for three months so that the maturity value becomes equal to £35,000. Evaluate each payment method. Which method would you prefer? Why?
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Solution: The problem situation is summarized as follows: A/P = £35,000 payable in three months iNY = 0.35%/month, compounding monthly iLD = 2.0% for three months S = $1.45/£;
F = $1.40/£.
Option a: When you buy £35,000 forward, you will need $49,000 (= £35,000 × $1.40/£) in three months to fulfill the forward contract. The present value of $49,000 is computed as follows: FV/(1+i)n = PV
$49,000/(1.0035)3 = $48,489.
Thus, the cost of Jaguar as of today is $48,489. Option b: The present value of £35,000 is £34,314 = £35,000/(1.02). To buy £34,314 today, it will cost $49,755 = 34,314x1.45. Thus the cost of Jaguar as of today is $49,755. You should definitely choose to use “option a”, and save $1,266, which is the difference between $49,755 and $48489. Online problem format original question: While you were visiting London, you purchased a Jaguar for £35,000, payable in three months. You have enough cash at your bank in New York City, which pays 0.35% interest per month, compounding monthly, to pay for the car. Currently, the spot exchange rate is $1.45/£ and the three-month forward exchange rate is $1.40/£. In London, the money market interest rate is 2.0% for a three-month investment. There are two alternative ways of paying for your Jaguar. (a) Keep the funds at your bank in the U.S. and buy £35,000 forward. (b) Buy a certain pound amount spot today and invest the amount in the U.K. for three months so that the maturity value becomes equal to £35,000. Evaluate each payment method. Which method would you prefer? Why? In what currency and for what amount should the forward contract be entered into if you keep the funds in your US ? a. Long in pound = £35,000 (Correct) b. Long In dollar = £35,000 x $1.40/£ = $49,000 c Short in pound = £35,000
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d. Short In dollar = £35,000 x $1.40/£ = $49,000 What will be the cost of the Jaguar as of today in $ if you enter into the forward contract and keep the funds in your US ? a. PV =(£35,000/(1.02) x 1.40 = $48,039.22 b. PV = $49,000/(1.0035)3) = $48,489.08 (Correct) What amount in pound will you purchase today to invest in the UK if you apply option b? a. PV = $49,000/(1.0035)3) / 1.40 = £34,635.06 b. PV = £35000/(1.02) = £34,313.73 (Correct) Which method is the best and why? a. Alternative A because cheaper (Correct) b. Alternative B because cheaper 3. Currently, the spot exchange rate is $1.50/£ and the three-month forward exchange rate is $1.52/£. The three-month interest rate is 8.0% per annum in the U.S. and 5.8% per annum in the U.K. Assume that you can borrow as much as $1,500,000 or £1,000,000. a. Determine whether the interest rate parity is currently holding. b. If the IRP is not holding, how would you carry out covered interest arbitrage? Show all the steps and determine the arbitrage profit. c. Explain how the IRP will be restored as a result of covered arbitrage activities. Solution: Let’s summarize the given data first: S = $1.5/£; F = $1.52/£; i$ = 2.0%; i£ = 1.45% Credit = $1,500,000 or £1,000,000. a. (1+i$) = 1.02 (1+i£)(F/S) = (1.0145)(1.52/1.50) = 1.0280 Thus, IRP is not holding exactly. b. (1) Borrow $1,500,000; repayment will be $1,530,000. (2) Buy £1,000,000 spot using $1,500,000. (3) Invest £1,000,000 at the pound interest rate of 1.45%; maturity value will be £1,014,500. (4) Sell £1,014,500 forward for $1,542,040
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Arbitrage profit will be $12,040 (=$1,542,040 - $1,530,000). c. Following the arbitrage transactions described above, The dollar interest rate will rise; The pound interest rate will fall; The spot exchange rate will rise; The forward exchange rate will fall. These adjustments will continue until IRP is restored. Online problem format – original Question: Currently, the spot exchange rate is $1.50/£ and the three-month forward exchange rate is $1.52/£. The three-month interest rate is 8.0% per annum in the U.S. and 5.8% per annum in the U.K. Assume that you can borrow as much as $1,500,000 or £1,000,000. a. Determine whether the interest rate parity is currently holding. b. If the IRP is not holding, how would you carry out covered interest arbitrage? Show all the steps and determine the arbitrage profit. c. Explain how the IRP will be restored as a result of covered arbitrage activities. Calculate the forward rate, based on IRP ($(1+i $)/£(1+i£)), to determine whether the interest rate parity is currently holding – What rate do you find? a. $1.53/£ b. $1.52/£ c. $1.51/£(Correct) Does interest rate parity hold or not? a. Yes b. No (Correct) What currency will you borrow? a. $ (Correct) b. £ c. Will not borrow What is the interest rate at which you will borrow?
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a. 2% for three months (Correct) b. 1.45% for three months c. Not applicable What is the total end-of-period borrowed amount? a. $1,521,750 b. £1,014,500 c. $1,530,000 (Correct) d. £1,020,000 e. Not applicable In what currency will you invest? a. $ b. £ (Correct) c. Not applicable Will you go short or long in a forward contract? a. Short $1,521,750 b. Long $1,521,750 c. Short £1,014,500 (Correct) d. Long £1,014,500 e. Short $1,530,000 f. Long $1,530,000 g. Short £1,020,000 h. Long £1,020,000 i. Not applicable How much arbitrage profit will you make in $? a. $22,185 b. €22,185 c. $12,040 (Correct) d. £12,040 e. No profit due to no arbitrage opportunity
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Indicate which of the following situations may occur to restore the IRP as a result of covered arbitrage activities: a. The dollar interest rate will rise b. The pound interest rate will fall c. The spot exchange rate will rise d.The forward exchange rate will fall e. a and b f. c and d g. a, b, c and d (Correct)
7. (CFA question) Omni Advisors, an international pension fund manager, uses the concepts of purchasing power parity (PPP) and the International Fisher Effect (IFE) to forecast spot exchange rates. Omni gathers the financial information as follows: Base price level Current U.S. $ price level Current South African price level Base rand spot exchange rate Current rand spot exchange rate Expected annual U.S. inflation Expected annual South African inflation Expected U.S. one-year interest rate Expected South African one-year interest rate
100 105 111 $0.175 $0.158 7% 5% 10% 8%
Calculate the following exchange rates (ZAR and USD refer to the South African rand and U.S. dollar, respectively). a. The current ZAR spot rate in USD that would have been forecast by PPP. b. Using the IFE, the expected ZAR spot rate in USD one year from now. c. Using PPP, the expected ZAR spot rate in USD four years from now. Solution: a. ZAR spot rate under PPP = [1.05/1.11](0.175) = $0.1655/rand. This answer is retrieved from the following: The base price level for both currencies were 100 and the base rand spot exchange rate of 0.175 was determined by these 100 base price levels. The S = P$/PZAR was 0.175 = 100$/100ZAR. Now it changed and determine the new spot exchange rate based on these changes in price levels.
Price levels were 1:1, now 0.945945 :1, thus 0.175 changed to
0.945945 of original 0.175 to .1655.
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b. Expected ZAR spot rate using IFE 1 year from now = [1.10/1.08] (0.158) = $0.1609/rand. Same principle applies as in a. Only difference we use interest rates to conduct estimate c. Expected ZAR under PPP four years from now= [(1.07)4/(1.05)4] (0.158) = $0.1704/rand. Same as b, but use inflation rates instead of interest rates. IMPORTANT NOTE: IFE suggests that the nominal interest rate difference reflects the expected change in exchange rates p.158 in textbook.
Online problem format original question: Formulas to be applied: The absolute version of purchasing power parity (PPP): S = P$/P£. The relative version is: e = $ - £. The international Fisher effect: E(e) = i$ - i£. You are provided with the following solutions to the questions in this problem: i. [1.10/1.08] (0.158) = $0.1609/rand. ii. [(1.07)4/(1.05)4] (0.158) = $0.1704/rand. iii. [1.05/1.11](0.175) = $0.1655/rand. Which one of the solutions calculates the ZAR spot rate under PPP? a. i. b. ii. c. iii. (Correct) Which one of the solutions calculates the expected ZAR spot rate one year from now with IFE? a. i. (Correct) b. ii. c. iii. Which one of the solutions calculates the expected ZAR spot rate in USD four years from now under PPP?
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a. i. b. ii. (Correct) c. iii. 11. James Clark is a foreign exchange trader with Citibank. He notices the following quotes. Spot exchange rate
SFr1.2051/$
Six-month forward exchange rate
SFr1.1922/$
Six-month $ interest rate
2.5% per year
Six-month SFr interest rate
2.0% per year
a. Is the interest rate parity holding? You may ignore transaction costs. b. Is there an arbitrage opportunity? If yes, show what steps need to be taken to make arbitrage profit. Assuming that James Clark is authorized to work with $1,000,000, compute the arbitrage profit in dollars. Solution: a. When using the SFr/$ exchange rate as provided in the question we can consider interest rate parity as follows: For six months , iSFr = 1.0% and i$ = 1.25%. the spot exchange rate is SFr1.2051/$ and the forward rate is SFr1.1922/$. Thus, (1 + iSFr) = 1.0100 and (F/s)(1+ i$ ) = = (1.1922/1.2051)(1.0125) = 1.0017 Because the left and right sides of IRP are not equal, IRP is not holding. When applied to $/SFr we can consider interest rate parity as follows: (We show this way as well, since in of quoting dealers use $ as currency (direct quote way) for testing interest rate parity.) For six months, iSFr = 1.0% and i$ = 1.25%. the spot exchange rate is $0.8298/SFr and the forward rate is $0.8388/SFr. Thus, (1+ i$ ) = 1.0125 and (F/s) (1 + iSFr) = (0.8388/0.8298) (1.01) = 1.02095 Because the left and right sides of IRP are not equal, IRP is not holding.
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b. Because IRP is not holding, there is an arbitrage possibility: Because 1.0125 < 1.02095, we can say that the SFr interest rate quote is more than what it should be as per the quotes for the other three variables. Equivalently, we can also say that the $ interest rate quote is less than what it should be as per the quotes for the other three variables. Therefore, the arbitrage strategy should be based on borrowing in the $ market and lending in the SFr market. The steps would be as follows: Borrow $1,000,000 for six months at 1.25%. Need to pay back $1,000,000 × (1 + 0.0125) = $1,012,500 six months later. Convert $1,000,000 to SFr at the spot rate to get SFr 1,205,100. Lend SFr 1,205,100 for six months at 1.0%. Will get back SFr 1,205,100 × (1 + 0.01) = SFr 1,217,151 six months later. Sell SFr 1,217,151 six months forward. The transaction will be contracted as of the current date but delivery and settlement will only take place six months later. So, six months later, exchange SFr 1,217,151 for SFr 1,217,151/SFr 1.1922/$ = $1,020,929. The arbitrage profit six months later is $1,020,929 – $1,012,500 = $8,429. ONLINE QUESTION FORMAT ORIGINAL QUESTION Calculate the forward rate, based on IRP (SFr(1+i SFr)/$(1+i$)), to determine whether the interest rate parity is currently holding – What rate do you find? a. SFr1.2021$ (Correct) b. SFr1.0210/$ c. SFr1.2102/$ Does interest rate parity hold or not? a. Yes b. No (Correct) What currency will you borrow? a. $ (Correct) b. SFr c. Will not borrow What is the interest rate at which you will borrow?
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a. 1% for six months b. 1.25% for six months (Correct) c. Not applicable What is the total end-of-period borrowed amount? a. $1,012,500 (Correct) b. SFr 1,217,151 c. $1,010,000 d. SFr1, 220,164 e. Not applicable In what currency will you invest? a. $ b. SFr (Correct) c. Not applicable Will you go short or long in a forward contract? a. Short $1,012,500 b. Long $1,012,500 c. Short SFr 1,217,151(Correct) d. Long SFr 1,217,151 e. Short $1,010,000 f. Long $1,010,000 g. Short SFr1, 220,164 h. Long SFr1, 220,164 i. Not applicable How much arbitrage profit will you make in $? a. $8,428.54 (Correct) b. SFr 8,932.88 c. $ 8,932.88 d. SFr8,428.54 e. No profit due to no arbitrage opportunity
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