Chapter 7

Pricing and Valuation of Interest Rates and Other Swaps

Understanding swaps as portfolios of forwards and bond replication strategies

7.1

Understanding Swaps and Their Relation to Forwards

A swap is a derivative contract where two parties agree to exchange a series of future cash flows at specified times according to predetermined terms. Interest rate swaps, a common type, involve the exchange of interest rate payments.

A swap can be thought of as a portfolio of forward contracts. For example, a three-year swap with semi-annual payments is conceptually equivalent to a series of six forward contracts, each with a different maturity but all using the same fixed price (the swap rate).

Key Characteristics of Swaps

  • They are typically customized, over-the-counter (OTC) agreements
  • By design, swaps have a value of zero at initiation
  • The fixed rate, known as the swap rate, remains constant throughout the life of the swap
7.2

Swap Values and Prices

Pricing and Valuation Principles

Pricing a swap involves determining the fixed rate of the swap at the contract's initiation date. This swap rate is set at a level that makes the present value (PV) of the expected fixed-rate payments equal to the present value (PV) of the expected floating-rate payments. This equalization ensures the swap's initial value is zero.

Valuing a swap is the process of determining its market value at any point after initiation. As interest rates fluctuate, the PVs of the fixed and floating legs will diverge, giving the swap a positive or negative value.

An important operational aspect is that the floating rate is reset at the beginning of each settlement period based on the current market reference rate (MRR).

Periodic Settlement Calculation

At the end of each period, a net cash flow is exchanged. The settlement amount from the perspective of the fixed-rate payer is calculated as follows:

Periodic Settlement = (MRR – Swap Rate) × Notional Amount × Period

Where:

  • MRR: The Market Reference Rate (the floating rate) for that period
  • Swap Rate (sN): The fixed rate agreed upon in the contract
  • Notional Amount: The principal amount upon which interest is calculated
  • Period: The fraction of a year corresponding to the settlement period (e.g., 0.5 for semi-annual, 0.25 for quarterly)

Numerical Example

Scenario: Consider an interest rate swap with a notional amount of $20 million and quarterly payments (Period = 0.25). The fixed-rate payer agrees to a swap rate of 4%. At the beginning of a new quarter, the 3-month MRR is reset to 5%.

(0.05 – 0.04) × $20,000,000 × 0.25 = 0.01 × $20,000,000 × 0.25 = $50,000

Result: Since the floating rate (5%) is higher than the fixed rate (4%), the fixed-rate payer will receive a net payment of $50,000 for this period.

Replicating a Swap with Bonds

A powerful way to understand and value a swap is to view it as a combination of a fixed-rate bond and a floating-rate note (FRN). The position of a fixed-rate payer (who is also a floating-rate receiver) is economically equivalent to holding the following two positions:

  1. A long position in a floating-rate note (FRN): This provides the cash inflows from the floating rate (MRR)
  2. A short position in a fixed-rate bond: This creates the cash outflows of the fixed coupon, which is set equal to the swap rate

Therefore, the value of the swap to the fixed-rate payer at any point in time is simply the value of the FRN minus the value of the fixed-rate bond.