# Forward Rate Agreement Convexity Adjustment

Forward Rate Agreement Convexity Adjustment: An Overview

When financial institutions enter into forward rate agreements (FRAs), they are essentially creating contracts that establish interest rates for future periods. These agreements are often used as a way to manage risk and uncertainty in an ever-changing market. However, when calculating the value of a FRA, it is necessary to include a convexity adjustment. This adjustment is crucial in ensuring that the value of the FRA is accurately represented, as it takes into consideration the potential fluctuation in interest rates over time.

What is Convexity?

Convexity is a measure of the curvature of a bond`s price-yield relationship. In the context of FRAs, convexity can be defined as the sensitivity of the FRA`s value to changes in interest rates. Essentially, it measures how much the value of the FRA will change in response to a change in interest rates.

Why is a Convexity Adjustment Necessary for FRAs?

When calculating the value of a FRA, it is important to take into consideration the possibility of interest rates changing over time. This is particularly relevant when dealing with long-term FRAs. The longer the time period of the FRA, the greater the chance of interest rate fluctuations.

If interest rates were to increase over the life of the FRA, the holder of the FRA would have to pay more than the prevailing market rate. Similarly, if interest rates were to decrease, the holder of the FRA would receive a payment that is less than the prevailing market rate.

Therefore, to account for this potential interest rate fluctuation, a convexity adjustment is applied to the FRA`s value. The adjustment reflects the change in value that would result from a change in interest rates, taking into account the curvature of the FRA`s price-yield relationship.

How is the Convexity Adjustment Calculated?

The formula for calculating the convexity adjustment of an FRA is:

Convexity Adjustment = (FRA Rate – Forward Rate) x (Time to Maturity)^2 x Convexity Factor

The convexity factor is a multiplier that takes into account the curvature of the FRA`s price-yield relationship. It is calculated using the following formula:

Convexity Factor = (1 + r)^(-2) x ((1 + r)^(Time to Maturity) – 1)/r^2

Where r is the prevailing market interest rate.

Conclusion

In conclusion, a convexity adjustment is an important factor to consider when determining the value of a FRA. By taking into account the potential for interest rate fluctuations over time, a convexity adjustment ensures that the value of the FRA is accurately reflected. As such, financial institutions should be aware of the importance of calculating the convexity adjustment when entering into FRAs.