A swap is a contract between two parties to exchange cash flows in the future, based on some underlying asset or rate. For example, an interest rate swap is a contract where one party pays a fixed rate of interest and receives a floating rate of interest, based on a notional principal amount. The fixed rate is called the swap rate, and it is determined at the inception of the swap.
The swap rate is the rate that makes the present value of the fixed cash flows equal to the present value of the expected floating cash flows. In other words, the swap rate is the rate that makes the swap have zero value at the start. This is because neither party has an advantage or disadvantage at the beginning of the swap.
To price a longer-term swap, we need to estimate the future floating cash flows, which depend on the future values of the underlying rate. One way to do this is to use the forward rates, which are the implied rates for future periods based on the current term structure of interest rates. The forward rates can be derived from the spot rates, which are the rates for borrowing or lending money for different maturities.
Using the forward rates, we can calculate the expected floating cash flows for each period of the swap. Then, we can discount these cash flows to the present using the spot rates. The sum of the present values of the floating cash flows is the present value of the floating leg of the swap.
Similarly, we can calculate the present value of the fixed leg of the swap by discounting the fixed cash flows using the spot rates. The sum of the present values of the fixed cash flows is the present value of the fixed leg of the swap.
The swap rate is the rate that equates the present value of the fixed leg and the present value of the floating leg. To find the swap rate, we can use a trial and error method, or a numerical solver, such as the goal seek function in Excel.
Basic Theory
A longer-term interest rate swap typically involves two parties exchanging interest payments over an extended period, such as 5, 10, or even 30 years. One party pays a fixed interest rate, while the other pays a floating interest rate, usually based on a reference rate such as LIBOR.
The pricing of longer-term swaps depends on the present value of future cash flows. The fixed-rate payer aims to lock in a cost-effective fixed rate, while the floating-rate payer seeks to benefit from potential decreases in interest rates.
Procedures
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- Determine Cash Flows
Identify the cash flows associated with the longer-term swap. For a fixed-rate payer, these are predetermined fixed interest payments. For a floating-rate payer, the cash flows are variable and depend on the future movements of the reference interest rate.
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- Discount Future Cash Flows
Discount each future cash flow to its present value using an appropriate discount factor. The discount factor is calculated based on the prevailing interest rates.
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- Calculate Net Present Value (NPV)
The NPV of the longer-term swap is the sum of the present values of the cash flows. The fixed-rate payer aims for a positive NPV, while the floating-rate payer seeks a negative NPV.
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- Determine the Fair Value
The fair value of the longer-term swap is the point where the NPV is zero. This is the agreed-upon market value at which the swap is considered fairly priced.
Scenario: 10-Year Interest Rate Swap
Let’s consider a scenario where Company A agrees to pay a fixed rate of 4.5% per annum to Company B, which pays a floating rate based on the 3-month LIBOR. The notional amount of the swap is $10 million, and payments are exchanged annually.
Excel Calculation
- Create a Table:
- Column A: Years (1 to 10)
- Column B: Fixed Payment (4.5% * $10 million)
- Column C: Floating Rate (Assume LIBOR starts at 3.5% and increases by 0.25% each year)
- Column D: Discount Factor (Based on prevailing market rates)
- Column E: Present Value of Fixed Payment (B / (1 + D)^A)
- Column F: Present Value of Floating Payment (C / (1 + D)^A)
- Calculate NPV:
- NPV = SUM(E2:E11) – SUM(F2:F11)
- Determine Fair Value:
- Adjust the fixed rate until NPV equals zero.
Excel Formulas
- For Discount Factor (D):
1 / (1 + Market Rate)^Year - For Present Value (PV):
Cash Flow / (1 + Market Rate)^Year - For NPV:
SUM(Present Values of Fixed Payments) - SUM(Present Values of Floating Payments)
Result
After adjusting the fixed rate in our scenario, the fair value was calculated to be approximately 4.65%.
Other Approaches
- Simulation Models: Use Excel’s Data Analysis tools to run simulations considering various interest rate scenarios.
- Monte Carlo Simulation: Incorporate randomness into interest rate movements for a more comprehensive analysis.
