Let’s break down the problem step-by-step. --- ## **Step 1: What kind of forward contract should you use?** **Situation:** - You are a **U.S.-based importer** buying goods from the **U.K.** - You must pay in **GBP (pounds)** in 30 days. - You expect the **pound to increase** against the dollar. **Analysis:** - If the pound strengthens, it will take **more USD to buy the same amount of GBP** in 30 days. - You are exposed to the risk of a higher payment in USD. **Hedge:** - Lock in the **exchange rate** today for your future purchase. - To do this, **buy GBP forward**—i.e., **enter a long forward contract** to buy pounds at a fixed rate in 30 days. **Answer:** **You should enter a long forward contract on GBP (i.e., agree to buy GBP in 30 days at a fixed rate).** --- ## **Step 2: Calculate the 30-day no-arbitrage forward rate** We use **Interest Rate Parity** (IRP) to calculate the forward rate: \[ F = S \times \frac{(1 + r_{\text{USD}} \cdot \frac{T}{360})}{(1 + r_{\text{GBP}} \cdot \frac{T}{360})} \] Where: - \( S = \) spot rate = $1.80/GBP - \( r_{\text{USD}} = \) U.S. risk-free rate = 5.0% per year - \( r_{\text{GBP}} = \) U.K. risk-free rate = 4.0% per year - \( T = 30 \) days Plug in the numbers: \[ F = 1.80 \times \frac{1 + 0.05 \times \frac{30}{360}}{1 + 0.04 \times \frac{30}{360}} \] Calculate the numerators and denominators: - \( 0.05 \times \frac{30}{360} = 0.05 \times 0.08333 = 0.0041667 \) - \( 0.04 \times \frac{30}{360} = 0.04 \times 0.08333 = 0.0033333 \) So: - Numerator: \( 1 + 0.0041667 = 1.0041667 \) - Denominator: \( 1 + 0.0033333 = 1.0033333 \) \[ F = 1.80 \times \frac{1.0041667}{1.0033333} \] \[ \frac{1.0041667}{1.0033333} \approx 1.000831 \] \[ F = 1.80 \times 1.000831 \approx 1.80150 \] **Answer:** **The no-arbitrage 30-day forward rate is \$1.8015/GBP** --- ## **Step 3: Value of the forward contract after 10 days, when spot = $1.83** - **Today:** 10 days have passed. 20 days remain until the contract matures. - **Spot rate now:** $1.83/GBP - **Interest rates unchanged:** U.S. = 5.0%, U.K. = 4.0% - **Forward contract:** You agreed to buy GBP at $1.8015 in 20 days. **Find:** The value of the contract (marked to market). ### **Step 3.1: Find the 20-day no-arbitrage forward rate** (We need this to get the current value of the contract.) \[ F_{20} = S_{10} \times \frac{1 + r_{\text{USD}} \cdot \frac{20}{360}}{1 + r_{\text{GBP}} \cdot \frac{20}{360}} \] Where: - \( S_{10} = $1.83 \) - \( r_{\text{USD}} = 0.05 \) - \( r_{\text{GBP}} = 0.04 \) - \( T = 20 \) days - USD: \( 0.05 \times \frac{20}{360} = 0.0027778 \) - GBP: \( 0.04 \times \frac{20}{360} = 0.0022222 \) - Numerator: \( 1 + 0.0027778 = 1.0027778 \) - Denominator: \( 1 + 0.0022222 = 1.0022222 \) \[ F_{20} = 1.83 \times \frac{1.0027778}{1.0022222} \] \[ \frac{1.0027778}{1.0022222} \approx 1.000555 \] \[ F_{20} = 1.83 \times 1.000555 \approx 1.83102 \] --- ### **Step 3.2: Value of the forward contract** The value (per GBP) of a long forward contract with original price \( K = 1.8015 \), current forward price \( F_{20} = 1.83102 \), and time to maturity \( \tau = 20/360 \) years is: \[ \text{Value} = e^{-r_{\text{USD}}\tau} \times (F_{20} - K) \] - \( r_{\text{USD}} = 0.05 \) - \( \tau = \frac{20}{360} = 0.05556 \) - \( F_{20} = 1.83102 \) - \( K = 1.8015 \) \[ e^{-0.05 \times 0.05556} = e^{-0.002778} \approx 0.997225 \] \[ \text{Value per GBP} = 0.997225 \times (1.83102 - 1.8015) = 0.997225 \times 0.02952 \approx 0.02944 \] --- ## **Final Answers (Summary Table)** | Step | Calculation/Result | |------|--------------------| | 1. Hedge | **Long forward contract on GBP** | | 2. 30-day Forward Rate | **\$1.8015/GBP** | | 3. Value after 10 days | **\$0.02944 per GBP** (profit per GBP if you close out the forward contract after 10 days) | --- ### **Interpretation:** - You have a gain of about **\$0.02944 per GBP** on your forward contract after 10 days, because the spot and forward rates have moved in your favor. --- **If you need the value for a specific notional amount (say, £100,000), just multiply:** \[ \text{Total Value} = 100,000 \times 0.02944 = \$2,944 \] --- **Let me know if you need more details or calculations for a specific notional amount!**

Certainly! Here's the final answer summarized in points: 1. Since you are a U.S.-based importer expecting the British pound to increase against the dollar, you should hedge your currency risk by entering into a **long forward contract** to buy GBP in 30 days. This locks in the exchange rate now and protects you from the pound's appreciation. 2. The no-arbitrage forward rate for 30 days, based on the interest rate parity formula, is approximately **$1.8015 per GBP**. 3. After 10 days, when the spot rate has risen to **$1.83 per GBP**, the value of your forward contract (per GBP) is approximately **$0.02944**. This means if you close out the position now, you would realize a profit of about 2.944 cents per GBP. 4. To find the total profit or value of the forward position, multiply this per-GBP gain by the total amount of GBP involved. For example, for a notional amount of £100,000, the profit would be approximately **$2,944**. This approach ensures you effectively hedge against the expected appreciation of the pound, locking in a favorable exchange rate and eliminating currency risk.