Arbitraging and Creating Forward Rate Agreements (FRA) in Excel

Arbitraging is the practice of exploiting price differences or inefficiencies in the market to make risk-free profits. For example, if the same asset is trading at different prices in two markets, an arbitrageur can buy the asset in the lower-priced market and sell it in the higher-priced market, earning the difference as profit.

A forward rate agreement (FRA) is a contract between two parties to exchange interest payments on a notional principal amount at a future date, based on an agreed-upon interest rate. The FRA rate is the fixed rate that the buyer of the FRA pays to the seller, while the reference rate is the floating rate that the seller pays to the buyer. The reference rate is usually based on LIBOR or another interbank rate.

An FRA can create arbitrage opportunities when the FRA rate is different from the forward rate implied by the spot interest rates. The forward rate is the interest rate that would make an investor indifferent between investing in a short-term bond and a long-term bond. If the FRA rate is higher than the forward rate, the buyer of the FRA can borrow money at the spot rate for the longer period, lend money at the spot rate for the shorter period, and enter into the FRA to receive the reference rate and pay the FRA rate. The net cash flow at the end of the contract will be positive, as the reference rate will be higher than the FRA rate, which is higher than the forward rate, which is higher than the spot rate. This is an arbitrage opportunity, as the investor can lock in a risk-free profit at the start of the contract.

Conversely, if the FRA rate is lower than the forward rate, the seller of the FRA can borrow money at the spot rate for the shorter period, lend money at the spot rate for the longer period, and enter into the FRA to receive the FRA rate and pay the reference rate. The net cash flow at the end of the contract will be positive, as the FRA rate will be higher than the reference rate, which is lower than the forward rate, which is lower than the spot rate. This is also an arbitrage opportunity, as the investor can lock in a risk-free profit at the start of the contract.

Basic Theory

Arbitrage

Arbitrage is the practice of exploiting price differences in different markets to gain a profit with no risk. In the context of interest rates, it often involves taking advantage of discrepancies between the spot and forward rates.

Forward Rate Agreement (FRA)

A Forward Rate Agreement is a financial contract that allows two parties to lock in an interest rate for a future period. It’s essentially an agreement to borrow or lend at a specified interest rate, known as the forward rate, for a predetermined time.

Procedures

Arbitrage in Interest Rates

Identify a discrepancy between the current spot rate and the expected future rate.
Calculate the arbitrage profit by exploiting the difference.

Creating Forward Rate Agreements (FRA)

Determine the current spot rate and the desired future rate.
Calculate the forward rate using the formula:
$FRA = \left(1 + r_{\text{desired}} \times \frac{t_{\text{days}}}{360}\right) / \left(1 + r_{\text{spot}} \times \frac{t_{\text{days}}}{360}\right) - 1$

where $r_{\text{desired}}$ is the desired future rate, $r_{\text{spot}}$ is the current spot rate, and $t_{\text{days}}$ is the number of days until the future period.

Scenario: Arbitrage Opportunity and FRA Creation

Assume the current spot rate is 5%, and you anticipate a future rate of 6% in 90 days. The arbitrage opportunity arises because the spot rate doesn’t reflect the expected increase.

Excel Table

Spot Rate	Future Rate	Days	Arbitrage Profit	FRA Rate
5%	6%	90	(to be calculated)	(to be calculated)

Excel Formulas

Arbitrage Profit:
$\text{Arbitrage Profit} = \text{Principal} \times \left(\frac{r_{\text{future}} - r_{\text{spot}}}{1 + r_{\text{spot}} \times \frac{t_{\text{days}}}{360}}\right)$

Assuming a principal of $1,000:

$\text{Arbitrage Profit} = 1000 \times \left(\frac{0.06 - 0.05}{1 + 0.05 \times \frac{90}{360}}\right)$
FRA Rate:
$FRA = \left(1 + 0.06 \times \frac{90}{360}\right) / \left(1 + 0.05 \times \frac{90}{360}\right) - 1$

Results

After plugging the values into Excel, the calculated Arbitrage Profit is $2.47, and the FRA Rate is approximately 0.0144 or 1.44%.

Other Approaches

Using Goal Seek in Excel:
Excel’s Goal Seek tool can be employed to find the spot rate that makes the arbitrage profit zero. Set the arbitrage profit cell as the target, and adjust the spot rate until the profit is minimized.
Monte Carlo Simulation:
For a more sophisticated analysis, consider using Excel to perform a Monte Carlo simulation. This can help model various scenarios and assess the risk associated with arbitrage strategies.