Pricing an Interest Rate Swap

Pricing an interest rate swap means determining the fixed rate or market value that makes the contract economically fair given the current term structure of interest rates. At inception, a plain vanilla swap is usually priced so that the present value of the fixed leg equals the present value of the floating leg. After inception, the market value moves as rates, curves, and spreads change.

Students do not need to become curve-building specialists to understand the logic. The key is to understand what is being valued, why discounting matters, and why the swap’s mark-to-market can move materially after the trade is executed.

The Pricing Objective at Inception

For a new swap, the market typically looks for the par swap rate. This is the fixed rate that makes the contract’s initial value approximately zero to both counterparties, ignoring transaction costs and credit adjustments.

Conceptually:

If the fixed rate is set too high, the fixed-rate receiver begins with an advantage. If it is set too low, the fixed-rate payer benefits. The par rate is therefore the market-clearing fixed rate implied by the relevant curve inputs.

Valuing the Fixed Leg

The fixed leg is the easier side to value because the coupon is known once the swap is agreed.

In a standard framework:

Where:

• N is the notional
• K is the fixed rate
• \alpha_i is the accrual fraction for payment period i
• P(0,t_i) is the discount factor to payment date t_i

The fixed leg is therefore the discounted value of known periodic coupons.

Valuing the Floating Leg

The floating leg depends on projected future benchmark rates. In practice, market participants use forward curves and relevant overnight discounting conventions to estimate those future floating cash flows.

Conceptually, the floating leg is valued by:

• projecting the benchmark for each reset period
• calculating each expected floating cash flow
• discounting each projected cash flow back to present value

For a newly reset floating leg under standard assumptions, the floating side is often conceptually close to par. But once time passes and the curve changes, the mark-to-market can shift significantly.

Discounting and the Current Market Framework

Discount factors matter because a dollar paid in the future is worth less than a dollar paid today. Modern swap pricing also requires the student to think in a current benchmark framework.

In current Canadian overnight-rate markets:

• CORRA is the key overnight benchmark
• overnight-style products are priced with overnight-rate conventions in mind
• discounting and projection must be consistent with the current market benchmark structure

That means old CDOR-style intuition should not automatically be treated as the default framework for pricing a current Canadian overnight-rate swap.

    flowchart LR
	    A["Projected Floating Rates"] --> B["Floating Cash Flows"]
	    C["Fixed Rate"] --> D["Fixed Cash Flows"]
	    B --> E["Discount Factors"]
	    D --> E
	    E --> F["Present Value Comparison"]

Why the Swap Value Changes After Inception

Once the swap is live, the fixed rate is locked but market rates are not. If interest rates rise, a swap where the holder pays fixed may become more valuable because the fixed rate they locked in is now lower than prevailing swap rates. If rates fall, that same position may lose value.

This is why swaps are marked to market over time. The value depends on:

• movements in the curve
• remaining maturity
• accrued interest
• benchmark and basis changes
• credit and collateral context where relevant

Students should therefore separate two ideas:

• pricing at inception, which solves for the fair fixed rate
• valuation after inception, which measures the gain or loss on the existing contract

A Simple Intuition Example

Suppose a firm entered a five-year swap last year and agreed to pay fixed at 3.50%. If the market now requires 4.20% for a comparable new swap, the firm’s pay-fixed position has improved relative to the market because it is paying a lower fixed rate than a new payer would pay today.

That does not mean the swap generated a realized profit immediately. It means the existing contract now has positive mark-to-market value to that payer, subject to the usual valuation framework.

Common Pitfalls

• confusing the par swap rate with a guaranteed future profit
• ignoring day count fractions and payment dates
• mixing benchmark conventions from old and new market regimes
• assuming the floating leg never needs careful projection
• forgetting that mark-to-market moves can trigger collateral demands

Key Takeaways

• A new swap is usually priced so that the present values of the fixed and floating legs are equal.
• The fixed leg is valued by discounting known coupon cash flows.
• The floating leg is valued by projecting future benchmark cash flows and discounting them.
• Swap value changes after inception because market curves change while the contract’s fixed rate remains locked.
• Current Canadian overnight-rate swap pricing should be read in a CORRA-aware framework.

Sample Exam Question

A new plain vanilla swap is being priced in the market. What does the par swap rate represent?

• A. The floating rate that will apply on every reset date for the life of the swap
• B. The fixed rate that makes the present value of the fixed leg equal to the present value of the floating leg at inception
• C. The collateral rate posted under the credit support annex
• D. The penalty rate applied if the swap is terminated early

Correct Answer: B. The fixed rate that makes the present value of the fixed leg equal to the present value of the floating leg at inception

Explanation: The par swap rate is the fair fixed rate that makes the swap start with approximately zero market value.