Reading 18: Currency Exchange Rates-LOS h习题精选 2011, face, interest, forward, percent

土豆妮

Session 4: Economics for Valuation
Reading 18: Currency Exchange Rates

LOS h: Explain interest rate parity, and illustrate covered interest arbitrage.

 

 

Assume the 1 year USD:EUR forward rate is 0.89348, the German interest rate is 3.38 percent, and the U.S. interest rate is 1.90 percent. If interest rate parity (IRP) holds, the USD:EUR spot rate is approximately:

A) 0.91204.

B) 1.56670.

C) 0.88069.

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Interest rate parity is given by:

Forward FCC = Spot FCC × [(1 + r_domestic) / (1 + r_foreign)], or alternatively
Spot FCC = Forward FCC × [(1 + r_foreign) / (1 + r_domestic)] = 0.89348 × (1.0190 / 1.0338) = 0.88069

Note that in this question, the dollar is the foreign currency and the Euro is the domestic currency.

土豆妮

Bob Bowman, CFA, is an analyst who has been recently assigned to the currency trading desk at Ridgeway Securities, a hedge fund management firm based in New York. Ridgeway’s stellar reputation as a top tier hedge fund manager has been built upon many years of its portfolio outperforming both the market and its peer group. Ridgeway’s portfolio is globally diversified, with less than 35% of its assets currently invested in U.S. securities. Ridgeway seeks to enhance its portfolio returns through the active use of currency futures that correspond to its investments. From time to time, Ridgeway will also take advantage of arbitrage opportunities that arise in the currency markets.

In his new position, Bowman will be reporting to the head currency trader, Jane Anthony. Among Bowman’s new responsibilities, he will be performing an ongoing analysis of global currency rates. His analysis is expected to include projections of future exchange rates and a sensitivity analysis of exchange rates in a variety of interest rate scenarios. Using his projections as a starting point, he will then be expected to suggest possible trading strategies for Ridgeway. Bowman knows that his analysis will begin with the underlying principles of the five basic international parity relationships. However, he does realize that certain principles will be more useful than others when applied to a “real-world” situation. To test his knowledge of the subject, Anthony has asked Bowman to prepare a presentation on the interrelationships between exchange rates, interest rates, and inflation rates. For the presentation, Bowman will need to prepare a brief analysis of current market conditions and formulate some basic trading strategies based upon his projections. He also will need to demonstrate his ability to calculate predicted spot rates for currencies, given some basic inflation rate and interest rate assumptions.

Bowman begins his task by gathering the following current market statistics:

1 year U.S. Interest Rates = 8%

1 year U.K. Interest Rates = 10%

1 year $/? forward rate = 1.70

Current $/? spot rate = 1.85

Bowman knows that if the forward rate is lower than what interest rate parity indicates, the appropriate strategy would be to borrow:

A) pounds, convert to dollars at the forward rate, and lend the dollars.

B) pounds, convert to dollars at the spot rate, and lend the dollars.

C) dollars, convert to pounds at the spot rate, and lend the pounds.

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If the forward rate is lower than what the interest rate parity indicates, the appropriate strategy would be: borrow pounds, convert to dollars at the spot rate, and lend dollars. (Study Session 4, LOS 18.h)

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Bowman also knows that if the forward rate is higher than what interest rate parity indicates, the appropriate strategy would be to borrow:

A) dollars, convert to pounds at the forward rate, and lend the pounds.

B) dollars, convert to pounds at the spot rate, and lend the pounds.

C) pounds, convert to dollars at the spot rate, and lend the dollars.

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If the forward rate is higher than what interest rate parity indicates, the appropriate strategy would be: borrow dollars, convert to pounds at the spot rate, and lend the pounds. (Study Session 4, LOS 18.h)

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Based on the information above, Bowman would like to calculate the forward rate implied by interest rate parity. The answer is:

A) 1.88 $/?.

B) 1.67 $/?.

C) 1.82 $/?.

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Given the above relationship, interest rate parity does not hold.

(If interest parity held, 1.70 = 1.85 × (1.08 / 1.10), but 1.85 × (1.08 / 1.10) = 1.82).

Therefore, an arbitrage opportunity exists.

To determine whether to borrow dollars or pounds, express the foreign rate in hedged US$ terms (by manipulating the equation for IRP). We get:

(1.70 / 1.85) × 1.10 = 1.0108, which is less than 1.08 (U.S. rate), so we should start by borrowing British pounds and lending U.S. dollars.

Arbitrage Example:

1. Today:
   1. borrow 5,000 GBP @ 10%
   2. buy $9,250 with the proceeds of the loan (5,000 GBP × 1.85).
   3. lend $9,250 @ 8%
   4. buy 5,500 GBP one year in the future @ 1.70 $/￡. This guarantees your loan repayment of 5,000 GBP × 1.1 = 5,500 GBP.
2. One year later, close out your position:
   1. collect the proceeds of your loan: $9,990 = $9,250 × 1.08
   2. buy 5,500 GBP with your forward contract → cost = 5,500 GBP × 1.70 = $9,350
   3. pay off your loan of 5,500 GBP
   4. reap your profits: $9,990 ? $9,350 = $640
   5. Alternately, you could say that the arbitrage profit is 376.47 GBP. Bob Bowman is a US investor so we left his profits in USD. 640 USD = 1.70 × 376.47 GBP.

(Study Session 4, LOS 18.h)

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A junior colleague asks Bowman for the mathematical equation that describes interest rate parity. Which of the following equations most accurately describes interest rate parity? (S₀ is the spot exchange rate expressed in dollars per unit of foreign currency, F_0,T is the forward exchange rate, and r_US and r_FX are the risk-free rates in the U.S. and foreign country.)

A) F0,t = S0 [(1+rUS) / (1+rFX)].

B) S1 = F0,t [(1+rUS) / (1+rFX)].

C) F0,t = S0 [(1+rFX) / (1+rUS)].

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Interest Rate Parity

Interest rates between countries and their exchange rates (spot and futures) must be in equilibrium at all times or else there will be arbitrage opportunities. Interest rate parity says that:

F_0,t = S₀ [(1+r_US) / (1+r_FX)]

Where:

S0	=	the current exchange rate in the spot market
F0,t	=	the current exchange rate in the forward of futures market
rUS	=	the risk-free interest rate in the U.S.
rFX	=	the risk-free interest rate in the foreign market

Note: the above currency quotes are expressed in $/FX. (Study Session 4, LOS 18.h)

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Now, suppose Bowman has the following information available to him: the current spot exchange rate for Indian Rupees is $0.02046. Inflation over the next 5 years is expected to be 3% in the U.S. and 5% in India. Bowman must calculate the U.S. Dollar/Indian Rupee expected future spot exchange rate in 5 years implied by purchasing power parity (PPP). The answer is:

A) $0.02250.

B) $0.01858.

C) $0.02010.

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The PPP assumption is that the future spot exchange rate will change exactly as the inflation rates affect the values of each currency. For the computation, raise the U.S. inflation rate to the 5th power (because of 5 years) and divide it by the Indian inflation rate raised to the 5th power. Then multiply the result by the spot exchange rate. ((1.03)⁵ / (1.05)⁵) × 0.02046 = $0.01858. (Study Session 4, LOS 19.h)

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Bowman routinely calculates the expected spot rate for the Japanese Yen per U.S. dollar. He knows that the current spot exchange rate is 189.76 Yen/USD. He is also aware that the interest rates in Japan, Great Britain, and the U.S. are 8%, 4%, and 5% respectively. Calculate the expected spot rate for Yen/USD in a one year period.

A) 195.18 Yen/USD.

B) 187.95 Yen/USD.

C) 184.49 Yen/USD.

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The exact methodology of the covered interest rate parity (IRP) is:

expected spot rate in one period (FC/DC) = spot rate today (FC/DC) × [(1 + R_FC) / (1 + R_DC)].

Setting up this equation gives us E(S₁) = 189.76 Yen/USD × (1.08 / 1.05) = 195.18 Yen/USD. (Study Session 4, LOS 18.h)

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土豆妮

The spot rate between the Canadian dollar and the British pound is 1.265 CAD/? and the forward rate is 1.193 CAD/?. The interest rate in Canada and England are 6.13% and 6.01%, respectively. A person living in Toronto, Canada can borrow either Canadian dollars or pounds. If an arbitrage opportunity exists, which currency would they lend or borrow?

A) Borrow pounds.

B) Lend pounds.

C) Borrow Canadian dollars.

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Use the following formula to determine if an arbitrage opportunity exists and which currency to borrow.

if 1 + r_D > [(1 + r_F)(Forward rate)] / Spot rate, then borrow foreign.

1.0613 > [(1.0601)(1.193)] / 1.265

1.0613 > 1.265 / 1.265

1.0613 > 1 therefore borrow foreign (British pound) and lend domestic (Canadian dollar).

土豆妮

The interest rates in the U.S. and Great Britain are 7.23% and 6.94% respectively. The forward rate is 1.70$/? and the spot rate is 1.73$/?. Which currency would an investor borrow, if any, to make an arbitrage profit?

A) Lending pounds.

B) Borrow pounds.

C) Borrow dollars.

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Use the following formula to determine if an arbitrage opportunity exists and which currency to borrow.

if 1 + r_D > [(1 + r_F)(Forward rate)] / Spot rate then borrow foreign.

1.0723 > [(1.0694)(1.70)] / 1.73

1.0723 > 1.81798 / 1.73

1.0723 > 1.0509, therefore borrow foreign (pounds).

Alternatively, the dollar is appreciating. [(1.73 ? 1.70) / 1.70] = 1.76% and the $U.S. interest rate is higher. Clearly, investing in $U.S. (and borrowing pounds) is the way to go.

土豆妮

The annual interest rate is 8.02% in Mexico and 7.45% in Canada. The spot peso-dollar exchange rate is 569.87 MXN/CAD, and the one-year forward rate is 526.78 MXN/CAD. If an arbitrage opportunity exists, how much would a person living in Mexico make borrowing 15,000,000 pesos or the equivalent in Canadian dollars?

A) 1,292,410 pesos.

B) 1,284,230 pesos.

C) 1,304,207 pesos.

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Note that peso is at a forward premium (less pesos per CAD in the future) and that peso interest rate is higher. Therefore it is clear there are arbitrage profits from lending in pesos and borrowing CAD.

First convert to Canadian dollars to determine the amount of interest due at the end of the year. (15,000,000 MXN) × (CAD/569.87 MXN) = 26,321.79 CAD.

26,321.79 CAD × 0.0745 = 1,960.97 CAD interest due at the end of the year.

Lend out pesos 15,000,000 pesos × 1.0802 = 16,203,000 pesos received at the end of the year.

Convert to Canadian dollars (16,203,000 MXN) × (CAD/526.78 MXN) = 30,758.57 CAD.

Subtract the original loan amount and interest: 30,758.57 ? 26,321.79 (original loan) ? 1,960.97 (interest) = 2,475.81 CAD profit.

Convert the remainder back to pesos: (2,475.81 CAD) × (526.78 MXN/CAD) = 1,304,207.19 peso profit.

土豆妮

The annual interest rates in England and New Zealand are 6.54% and 7.03%, respectively. The one-year forward exchange rate between the British pound and the New Zealand dollar is 0.45 GBP/NZD and the spot rate is 0.41 GBP/NZD. If a person living in London can borrow 10,000 pounds or the equivalent amount in New Zealand dollars, how much arbitrage profit, if any, can he make?

A) 1,043.61 GBP.

B) 1,093.20 GBP.

C) 1,124.88 GBP.

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Borrow 10,000 GBP at 6.54% = 654 GBP interest due at the end of the year.
Convert to NZD: (10,000 GBP) × (1 NZD/0.41 GBP) = 24,390 NZD.
Lend out NZD at 7.03% interest: (24,390 NZD) × (1.0703) = 26,104.88 NZD.
Convert back to GBP: (26,104.88 NZD) × (0.45 GBP/NZD) = 11,747.20 GBP.
11,747.20 GBP ? 10,000 GBP (original amount borrowed) ? 654 GBP interest = 1,093.20 GBP profit.

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土豆妮

The forward rate between the Mexican peso and the U.S. dollar is 556.75 MXN/USD and the spot rate is 581.23 MXN/USD. The Mexican interest rate is 5.89%, and the U.S. rate is 5.75%. If a person lives in Mexico and can borrow $10,000 or the equivalent in pesos, how much can she make if currency arbitrage opportunities exist?

A) $459.39.

B) $479.59.

C) Arbitrage opportunities do not exist.

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First determine if arbitrage opportunities exist by using the following equation:

if 1 + r_D > [(1 + r_F)(Forward rate)] / Spot rate, then borrow foreign (dollars).

1.0589 > [(1.0575)(556.75)] / 581.23

1.0589 > 588.763 / 581.23

1.0589 > 1.01296, therefore, borrow foreign (dollars).

Borrow $10,000 at 5.75%, interest = $575 due at the end of the year. Convert to pesos using the spot rate: ($10,000) × (581.23 MXN/USD) = 5,812,300 pesos.

Lend out at 5.89%: (5,812,300 pesos) × (1.0589) = 6,154,644.47 pesos. Convert to dollars: (6,154,644.47 MXN) × (USD/556.75 MXN) = $11,054.59. $11,054.59 ? $10,000 (original amount borrowed) ? $575 (interest) = $479.59 profit.

土豆妮

Given currency quotes in FCC, if:  1 + rDC <	(1 +rFC)(forward rate)	funds will:
	spot rate

A) flow in and out of the domestic country.

B) flow into the domestic country.

C) flow out of the domestic country.

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This equation is Interest Rate Parity rearranged! If the term on the left (1 + r_DC), is less than the term on the right, it means that the domestic rate is low relative to the hedged foreign rate. Therefore, there is a profitable arbitrage from borrowing the domestic currency and lending at the foreign interest rate.

Because we lend in the foreign market, we say that the funds flow out of the domestic economy.

土豆妮

The spot rate for the dollar is 0.1432 $/ADF. Andorran and U.S. interest rates are 6.6% and 7.2%, respectively. If the 1-year forward rate is 0.1430 $/ADF, a U.S. investor could earn an arbitrage dollar profit per ADF of:

A) $0.0010.

B) $0.0011.

C) $0.0075.

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Let us first check if an arbitrage opportunity exists. Applying the interest rate parity theorem, we have:

Forward rate = 0.1432 × 1.072/1.066 = 0.1440 $/ADF > 0.1430 $/ADF (quoted forward rate)

This implies that an arbitrage opportunity exists. The inequality implies that ADF is mispriced (weak) in the forward market or is underpriced relative to the dollar. We should buy ADF in the forward market and sell the dollar in the spot market. This requires that we borrow in Andorra and convert the francs into dollars at the spot rate. Invest the proceeds in U.S. securities @ 7.2%, and simultaneously enter into a forward transaction where we sell the dollars for ADF @ 0.1430 $/ADF. Assuming that we borrow 1 ADF today and convert it into dollars, we will have 0.1432 dollars to invest at 7.2% for one year. After one year we will have 0.1432 × 1.072 = 0.1535 dollars. At that point, we will owe an Andorran bank 1 × 1.066 or 1.066 ADF, including interest. We will need to convert enough dollars at the forward rate to pay off this loan. At the forward contract rate, we will need to convert 1.066 × 0.1430 = 0.1524 dollars into ADF to pay off our obligation. This will leave us with an arbitrage profit of 0.1535 ? 0.1524 = 0.0011 dollars.

土豆妮

The forward rate between Swiss francs and U.S. dollars is 1.8 SF/$ and the current spot rate is 1.90 SF/$. The Swiss interest rate is 8.02% and the U.S. rate is 11.02%. Assume you can borrow francs or dollars and you live in Switzerland. If an arbitrage opportunity exists, how can you take advantage of it?

A) Borrow domestic currency.

B) Lend foreign currency.

C) Borrow foreign currency.

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Borrow foreign if 1 + r_D> [(1 + r_F)(forward rate)] / spot rate

1 + 0.0802 > [(1 + 0.1102)(1.8)] / 1.9

1.0802 > 1.99836 / 1.9

1.0802 > 1.0518 therefore borrow foreign (dollars) and lend domestic (francs).

Alternatively, U.S. rate is 11.02 ? 8.02 = 3% higher and USD is at (1.8 ? 1.9) / 1.9 = 5.3% discount since USD will fall more than the extra 3% interest, better to lend francs.