How is the fixed swap rate determined in a plain vanilla interest rate swap?

The swap rate is the fixed rate that makes the present value of fixed payments equal to the present value of expected floating payments at inception. It's essentially the par rate of the swap's fixed leg.

The Zero-Value Principle

At inception, no money changes hands — the swap has zero net value. This means:

PV(Fixed Payments) = PV(Expected Floating Payments)

The swap rate is the fixed rate that satisfies this equality.

Deriving the Swap Rate

Consider a 3-year annual-pay interest rate swap on $10M notional. We observe the following SOFR-based spot rates:

• 1-year spot (z1): 4.00%
• 2-year spot (z2): 4.30%
• 3-year spot (z3): 4.55%

Step 1: Calculate Discount Factors

• DF1 = 1 / (1.04)^1 = 0.9615
• DF2 = 1 / (1.043)^2 = 0.9194
• DF3 = 1 / (1.0455)^3 = 0.8752

Step 2: Calculate Implied Forward Rates (Expected Floating Rates)

• f(0,1) = 4.00% (the 1-year spot rate)
• f(1,2) = [(1.043)^2 / (1.04)^1] - 1 = 4.601%
• f(2,3) = [(1.0455)^3 / (1.043)^2] - 1 = 5.051%

Step 3: PV of Floating Payments

PV(Float) = f(0,1) x DF1 + f(1,2) x DF2 + f(2,3) x DF3 = 0.0400 x 0.9615 + 0.04601 x 0.9194 + 0.05051 x 0.8752 = 0.03846 + 0.04229 + 0.04420 = 0.12495 (per $1 notional)

Step 4: Solve for the Swap Rate (SFR)

PV(Fixed) = SFR x (DF1 + DF2 + DF3)

Set PV(Fixed) = PV(Float):

SFR x (0.9615 + 0.9194 + 0.8752) = 0.12495 SFR x 2.7561 = 0.12495 SFR = 4.534%

Alternative (Shortcut) Formula:

SFR = (1 - DFn) / Sum(DF1 to DFn) = (1 - 0.8752) / (0.9615 + 0.9194 + 0.8752) = 0.1248 / 2.7561 = 4.529%

(The tiny difference is rounding.)

Intuition Behind the Shortcut:

The swap can be viewed as the difference between a floating-rate bond (which prices at par at inception) and a fixed-rate bond. For the swap to have zero value, the fixed-rate bond must also price at par. The rate that makes a bond price at par is the par rate — which is the swap rate.

Why the Swap Rate Sits Between Short and Long Rates:

Notice that 4.53% is between the 1-year rate (4.00%) and the 3-year rate (4.55%). This makes sense — the fixed rate is a weighted average of the expected floating rates across the swap's life.

After Inception — How Swap Value Changes:

Once rates move, the swap gains or loses value:

• If rates rise: The floating payer benefits (fixed payer's position gains value)
• If rates fall: The fixed payer benefits (floating payer's position gains value)

Exam Tip: CFA Level II may give you spot rates or discount factors and ask you to calculate the swap fixed rate. Use the shortcut formula: SFR = (1 - DFn) / Sum(DFs). It's faster and less error-prone than the forward rate approach.

Master swap pricing in our CFA Level II Derivatives course.