Six months US T bills has nominal rate of 4 percent, while default free Japanese bonds that mature in 6 months have a nominal rate of 2.5 percent. In the spot exchange market, JPY 1 equals to USD…

Solution to the Forward Exchange Rate Problem

Given:

• US 6-month T-bill nominal rate (r₁): 4% (0.04)
• Japanese 6-month default-free bond nominal rate (r₂): 2.5% (0.025)
• Spot exchange rate (S): JPY 1 = USD 0.013 (i.e., USD/JPY = 1/0.013 ≈ 76.92)
• Interest Rate Parity (IRP) holds.

Objective:

Find the 6-month forward exchange rate (F) under IRP and interpret the result.

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Step 1: Understanding Interest Rate Parity (IRP)

IRP states that the forward exchange rate (F) should eliminate arbitrage opportunities between two currencies by equating the returns from investing in risk-free assets in both countries.

The formula for covered interest rate parity (CIRP) is:

$$F=S\times \frac{(1+r_1)}{(1+r_2)}$$

Where:

• $F$ = Forward exchange rate (USD/JPY)
• $S$ = Spot exchange rate (USD/JPY)
• $r_1$ = US nominal interest rate
• $r_2$ = Japanese nominal interest rate

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Step 2: Plugging in the Values

Given:

• $S=\frac{1}{0.013}\approx 76.92$ (USD/JPY)
• $r_1=0.04$
• $r_2=0.025$

$$F=76.92\times \frac{(1+0.04)}{(1+0.025)}$$

$$F=76.92\times \frac{1.04}{1.025}$$

$$F=76.92\times 1.0145\approx 78.02$$

Thus, the 6-month forward exchange rate (F) is USD 1 = JPY 78.02 (or JPY 1 = USD 0.0128).

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Step 3: Interpretation of the Result

1. Forward Premium/Discount:

   • The forward rate (78.02) is higher than the spot rate (76.92).
   • This means the USD is expected to depreciate against the JPY in 6 months (since it takes more JPY to buy 1 USD forward).
   • Alternatively, the JPY is expected to appreciate against the USD.

2. Why?

   • The US interest rate (4%) is higher than Japan’s (2.5%).
   • Under IRP, higher interest rates in the US attract foreign capital, increasing demand for USD in the spot market.
   • To prevent arbitrage, the forward market adjusts by discounting the USD (i.e., requiring more JPY in the future to buy USD).

3. Economic Implication:

   • Investors expect the USD to weaken relative to the JPY due to higher US rates (which may signal inflation or monetary tightening).
   • This aligns with the forward premium observed.

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Final Answer:

The 6-month forward exchange rate (USD/JPY) is approximately 78.02, indicating that the USD is at a forward discount against the JPY. This reflects the higher US interest rate compared to Japan, consistent with Interest Rate Parity.