Presently, one US $ is worth 140 Japanese Yen in the spot market. The interest rate in Japan on 90 days government securities is 4% per annum. What is the implied interest rate in the USA if the 3-month forward rate is 1

Interest Rate Parity Theorem and Arbitrage Step 1: Computation of implied interest rate in USA Spot Exchange Rate (S) = 140 JPY/USD 3-month Forward Rate (F) = 138 JPY/USD Interest rate in Japan (home currency, r h ) = 4% p.a. Interest rate in Japan for 3 months = 4% / 4 = 1% or 0.01 The Interest Rate Parity (IRP) theorem states: (1 + r f ) = (F/S) × (1 + r h ) Where: r f = interest rate in foreign currency (USA) r h = interest rate in home currency (Japan) F = Forward Rate S = Spot Rate Substituting the values: (1 + r f ) = (138 / 140) × (1 + 0.01) (1 + r f ) = (0.985714) × (1.01) (1 + r f ) = 0.995571 r f = 0.995571 - 1 = -0.004429 This result indicates a negative implied interest rate, which is unusual. Let's re-evaluate the IRP formula application. The standard IRP formula is often expressed as: F/S = (1 + r foreign ) / (1 + r domestic ) Or, to find the foreign interest rate (USA in this case, assuming JPY is domestic): (1 + r USA ) = (S/F) × (1 + r Japan ) Let's use the provided solution's approach which seems to be: (1 + r home ) / (1 + r foreign ) = Forward / Spot Here, Japan is home (r h ) and USA is foreign (r f ). (1 + r Japan, 3-month ) / (1 + r USA, 3-month ) = Forward Rate / Spot Rate 1.01 / (1 + r USA, 3-month ) = 138 / 140 1.01 / (1 + r USA, 3-month ) = 0.985714 1 + r USA, 3-month = 1.01 / 0.985714 1 + r USA, 3-month = 1.02462 r USA, 3-month = 0.02462 or 2.462% per quarter Implied interest rate in USA per annum = 2.462% × 4 = 9.848% p.a. Step 2: Action if actual interest rate is 7% per annum in USA Implied interest rate in USA = 9.848% p.a. Actual interest rate in USA = 7% p.a. Since the actual interest rate in USA (7%) is lower than the implied interest rate (9.848%) required for interest rate parity to hold, there is an arbitrage opportunity. Specifically, the US dollar is undervalued in the forward market relative to its interest rate. This means that money will flow out of USA to take advantage of the higher implied return elsewhere or to exploit the discrepancy. An arbitrageur would borrow in the USA (at 7%), convert USD to JPY, invest in Japan (at 4%), and simultaneously sell JPY forward to convert back to USD at the forward rate of 138 JPY/USD. This would yield a higher return than simply investing in the USA at 7%.