Question

Suppose that \(A(0)=100\) and \(A(1)=105\) dollars, the present price of pound sterling is \(S(0)=1.6\) dollars, and the forward price is \(F=1.50\), dollars to a pound with delivery date \(1 .\) How much should a sterling bond cost today if it promises to pay \(£ 100\) at time \(1 ?\) Hint: The forward contract is based on an asset involving negative carrying costs (the interest earned by investing in sterling bonds).

Short answer

The sterling bond should cost approximately 152.38 dollars today.

Step-by-step solution

Identify known values

Given that the amount in dollars today is A(0) = 100, and in one year it is A(1) = 105. The present price of the pound sterling is S(0) = 1.6 dollars. The forward price for delivery in one year is F = 1.50 dollars. The payment in pound sterling at the end is £100.

Calculate the interest rate

Find the interest rate earned over the year in dollars: \[ \text{Interest Rate} = \frac{A(1) - A(0)}{A(0)} = \frac{105 - 100}{100} = 0.05 \text{ or } 5\% \]

Determine equivalent amount in pounds using the forward price

Since the forward price is the price at which the pound can be exchanged in one year, use it to determine the amount in pounds: \[ \frac{A(1)}{F} = \frac{105}{1.5} = 70 \text{ pounds} \]

Calculate the cost of the sterling bond today

The given sterling bond pays £100 in one year. Therefore, the equivalent cost today in pounds must account for the yield (interest rate) over the period.Use the interest rate found in Step 2 to adjust the future payment:\[ C = \frac{100}{1 + 0.05} = \frac{100}{1.05} \approx 95.24 \text{ pounds} \]

Convert the pound cost to dollars using the spot rate

Use the current exchange rate (spot rate) to convert this price back to dollars: \[ \text{Cost in Dollars} = C \times S(0) = 95.24 \times 1.6 \approx 152.38 \text{ dollars} \]

Key concepts

Forward Price

A forward price is the agreed-upon price for a transaction that will occur at a specified future date. In the context of foreign exchange, it is the rate at which one currency will be exchanged for another in the future. This contract protects against fluctuations in currency values.
For example, in our problem, we see that the forward price of pound sterling is set at 1.50 dollars. This means that one year from now, you can exchange one pound for 1.50 dollars, irrespective of the market rate at that time.
Forward prices help businesses manage risks associated with currency exchange rate volatility. Understanding and negotiating the correct forward price can significantly impact financial planning and budgeting.

Interest Rate Calculation

Calculating the interest rate is crucial for understanding the time value of money. The interest rate reflects the cost of borrowing or the reward for saving.
In our problem, we calculated the interest rate over one year by comparing the amounts in dollars today and one year later. Using the formula:
\( \text{Interest Rate} = \frac{A(1) - A(0)}{A(0)} \)
we determined the interest rate to be 5%. This implies that for every dollar invested today, you would earn an additional 5%.
Interest rates play a key role in numerous financial calculations, such as present value, future value, and in this case, determining the cost of a bond today given its future value.

Spot Rate Conversion

The spot rate is the current exchange rate between two currencies. It represents the immediate exchange value.
In our scenario, the spot rate for pounds to dollars is 1.60. This means that currently, 1 pound can be exchanged for 1.60 dollars.
To determine the cost of a sterling bond in dollars, we initially calculated its cost in pounds accounting for the interest rate and then converted this amount to dollars using the spot rate:
\( \text{Cost in Dollars} = C \times S(0) \)
This calculation ensures that we accurately determine the dollar cost based on the current exchange rate. Spot rates are vital in foreign exchange markets as they provide a benchmark for various financial transactions and investments.