Question

Sun Bank USA has purchased an 8 million one-year Australian dollar loan that pays 12 percent interest annually. The spot rate of U.S. dollars for Australian dollars (AUD/USD) is $0.625/A$1. It has funded this loan by accepting a British pound (BP)–denominated deposit for the equivalent amount and maturity at an annual rate of 10 percent. The current spot rate of U.S. dollars for British pounds (GBP/USD) is $1.60/£1.

a. What is the net interest income earned in dollars on this one-year transaction if the spot rate of U.S. dollars for Australian dollars and U.S. dollars for BPs at the end of the year are $0.588/A$1 and $1.848/£1, respectively? (Negative amount should be indicated by a minus sign. Do not round intermediate calculations. Round your final answer to the nearest whole number. (e.g., 32))

b. What should the spot rate of U.S. dollars for BPs be at the end of the year in order for the bank to earn a net interest income of $200,000 (disregarding any change in principal values)? (Round your answer to 5 decimal places. (e.g., 32.16161))

Answer 1

We follow these steps in order to answer this question:

a. Calculation of net interest earned.

I. First we estimate the quantum of loan taken in Great Britian Pounds - GBP.

1. Amount loaned in A$ : $8 million = 8*1000000 = $8,000,000.
2. Spot (USD/AUD) : $0.625/A$1.
3. We need to calculate the USD equivalent of the amount loaned in A$. We do this with the following formula : Equivalent of Amount loaned in USD = Amount loaned in A$ * USD/AUD spot exchange rate = A$8,000,000 * 0.625 = $5,000,000.
4. Since the A$ loan was financed with another loan borrowed in GBP, we need to find the amount borrowed in GBP using the Equivalent of Amount loaned in USD calculated in step 3 above.
5. Spot USD/GBP rate : USD1.60/GBP1
6. Amount borrowed in GBP = Equivalent of Amount loaned in USD/ (USD/GBP) spot rate = $5,000,000/1.60 = GBP 8,000,000

II. Next, we calculate the interest earned in A$ and it's equivalent in USD once year hence.

While converting the interest earned in A$ after a year into USD, we take spot rates prevailing at the year end.

1. Amount loaned in A$ : A$8,000,000
2. Interest rate per year : 12%
3. Interest earned = Amount loaned * Interest rate per year = A$8,000,000 * 0.12 = A$960,000.
4. Spot rate at year end (USD/A$) = 0.588/1
5. USD equivalent of interest earned at year end = Interest earned in A$ * (USD/A$) spot rate at year end =  A$960,000*0.588 = US$564,480.00

III. In this step, we calculate the interest to be paid in GBP and it's equivalent in USD once year hence.

In this step also, we take the spot rate prevailing at year end in order to find the interest to be paid in USD

1. Amount borrowed in GBP (From Step I above) : GBP 8,000,000
2. interest Rate : 10%
3. Interest to be paid = Amount borrowed * Interest rate = 8,000,000 *0.10 = GBP 800,000.
4. Spot USD/GBP rate at year end : $1.848/GBP 1
5. USD equivalent of interest to be paid at year end = Interest to be paid in GBP / (USD/GBP) spot rate at year end = GBP 8,000,000/1.848 = US$432,900.43

IV. We calculate Net Interest Earned in US$ in this step:

Net interest Earned = Interest Earned - Interest Paid

Net interest Earned = US$564,480.00 - US$432,900.43

Net interest Earned = US$131,579.57

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b. We need to determine the USD/GBP exchange rate so that the bank's net interest earned is $200,000.

We have:

  Net interest Earned = Interest Earned - Interest Paid

Plugging in the values we have,

US$200,000 = US$564,480.00 - Interest Paid

Transposing we get,

Interest Paid = US$564,480.00 - US$200,000

Interest Paid = US$364,480

The interest to be paid in GBP remains at GBP 800,000.

Since

Interest paid in GBP = Interest paid in USD * Spot Exchange Rate,

Spot rate of US dollars for GBPs = Interest in GBP/ Interest in USD = 800,000/364,480 = US$2.19491/GBP1