Introduction

Forward Rate Agreements (FRAs) are essential tools for managing interest rate risk. In this chapter, we delve into the valuation of plain vanilla interest rate swaps using a sequence of FRAs. This method allows us to determine the value of a swap based on market expectations of future interest rates. We’ll explore the step-by-step process of calculating the swap’s value using FRA rates, and provide insightful examples to illustrate the valuation technique.

Valuation Process

Understanding FRAs: A Forward Rate Agreement (FRA) is a contract between two parties to exchange a fixed interest rate for a floating interest rate on a future date. The fixed rate is agreed upon at the inception of the FRA, while the floating rate is based on a reference rate, often a benchmark such as LIBOR.

• Linking FRAs to Swap Valuation: A plain vanilla interest rate swap can be broken down into a series of FRAs, each covering a specific period. By replicating the fixed and floating cash flows of a swap using appropriate FRAs, we can derive the swap’s value.

• Calculating Fixed and Floating Cash Flows: The fixed cash flows in a swap correspond to the payments the fixed-rate payer receives, while the floating cash flows are the payments the floating-rate payer receives. The fixed cash flows can be computed directly from the FRA rates and the notional amount. The floating cash flows are determined based on the reference rate prevailing during each FRA’s term.

• Net Cash Flows: To determine the net cash flows exchanged between the swap counterparties at each settlement date, subtract the floating cash flow from the fixed cash flow.

• Discounting Cash Flows: The net cash flows are then discounted back to the present value using appropriate discount factors derived from prevailing market rates. The present value of net cash flows represents the value of the swap at that point in time.

Example: Consider a 5-year plain vanilla interest rate swap with annual settlement dates. The fixed rate in the swap is 4%, and the notional amount is USD 1 million. Let’s assume the FRA rates for the next five years are as follows:

• Year 1: 3.8%
• Year 2: 4.2%
• Year 3: 4.5%
• Year 4: 4.6%
• Year 5: 4.8%

1) Calculate the fixed cash flows for each year using the fixed rate and notional amount.
2) Calculate the floating cash flows for each year using the FRA rates and notional amount.
3) Determine the net cash flows by subtracting the floating cash flows from the fixed cash flows.
4) Calculate the discount factors for each settlement date based on market rates.
5) Discount the net cash flows for each year back to the present value.
6) Sum up the present values of net cash flows to find the value of the swap.