# Spot and Forward rates - arbitrage?

Working on a problem to devise an arbitrage strategy. 1 year sport rate is 5% and 3 year spot rate is 5.5%, 1x3 Forward rate is 5.6%. I calculated the 2 year spot rate as 5.75%

(1+0.055)^3 = (1+0.05)(1+S)^2 implies S = 5.75%


Using these spot rates, the actual 1x3 forward rate should be 11.83%

((1+S3)^3/(1+S1)) - 1 implies 11.83%, S3 is 3 year spot and S1 is 1 year spot


I can borrow $1 today, I will owe 1.05 in 1 year, I will enter into a 2 year forward and pay 5.6% and at the end of 3 years, I will owe 1.056. At the same time, at the end of year 1, I will lend a dollar at the current prevailing market rate of 5.75% and earn 1.1742 at the end of 3 years. This makes me a profit of 1.1742 - 1.056. Am I on the right track ? Any feedback is much appreciated. Yosh ## 1 Answer Since the implied forward rate (5.75%) is higher than the actual one (5.6%), the strategy should involve borrowing at the actual forward rate and investing at the implied one. In what follows, I show the steps that you should take at the beginning/end of each year to exploit the arbitrage opportunity, along with the cash flows. t = 0: • Borrow X\$ for 1 year at 5.0%: +X\$• Invest X\$ for 3 years at 5.5%: -X\$• Enter into the forward contract to borrow X$(1+5%) 1 year from now for 2 years: 0\$Net cash flow: 0$

t = 1 (i.e. end of 1st year):

• Borrow using the forward contract: +X\$(1+5%) • Repay the loan: -X\$(1+5%)

Net cash flow: 0\$t = 3 (i.e. end of 3rd year): • Repay the forward contact: -X\$(1+5%)(1+5.6%)^2
• Gain from the investment: +X\$(1+5.5%)^3 Net cash flow: X\$(0.00334857)