Using Spot and Forward Rates to Identify a Bond Arbitrage
Summary
The document examines a proposed arbitrage based on one-year and three-year spot rates and a one-year forward rate for the following two years. It first derives the intermediate spot rate from the relationship between the longer spot investment and the sequence of shorter investments. It then compares the forward rate implied by the spot curve with the quoted forward rate.
The answer outlines a cash-flow strategy: borrow for one year, invest for three years, and arrange to borrow again through a forward contract at the end of year one. The example reports a positive net amount at year three, illustrating how a discrepancy between an implied forward rate and the quoted rate can suggest an arbitrage. The proposed trade assumes the stated rates and borrowing, lending, and forward transactions are simultaneously available at those terms; practical frictions and contract details are not discussed.
Key ideas
- Spot rates can be combined to derive the forward rate implied by a term structure.
- A quoted forward rate below the implied rate may support a borrowing and investment arbitrage.
- Mapping cash flows at each date helps verify whether the strategy has a net initial cost.
- The example's conclusion depends on the stated rates and access to the specified contracts.
Tags
Full text
# Spot and Forward rates - arbitrage? # 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 $1 today, I will owe 1.05 in 1 year, I will enter into a 2 year forward and pay 5.6% and at the end of 3 years, I will owe 1.056. At the same time, at the end of year 1, I will lend a dollar at the current prevailing market rate of 5.75% and earn 1.1742 at the end of 3 years. This makes me a profit of 1.1742 - 1.056. Am I on the right track ? Any feedback is much appreciated. Yosh ## Answer by NC520 (score 0, accepted) https://quant.stackexchange.com/a/78404 Since the implied forward rate (5.75%) is higher than the actual one (5.6%), the strategy should involve borrowing at the actual forward rate and investing at the implied one. In what follows, I show the steps that you should take at the beginning/end of each year to exploit the arbitrage opportunity, along with the cash flows. t = 0: - Borrow X\$ for 1 year at 5.0%: +X\$ - Invest X\$ for 3 years at 5.5%: -X\$ - Enter into the forward contract to borrow X$(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%) 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) Additional comments: - I did not specify a monetary amount X\$, because in theory you would want to make this trade as big as possible, since it implies no investment (so you are not constrained by your capital) and no risk.
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