Valuing an interest rate swap: a three-year worked example

What we are valuing

A plain-vanilla interest rate swap exchanges a fixed rate for a floating rate on the same notional. Nothing is exchanged at the start in a par swap, so the fixed rate is set where the two legs have the same present value. Afterwards, as rates move, one side owes the other.

Here is a three-year swap on $10,000,000 notional. We pay a fixed rate of 4.00% annually and receive the floating rate, also annually. Valuation is from the payer's point of view. To keep the arithmetic visible we use annual payments and ignore day counts, credit and funding adjustments.

Step 1: discount factors from the zero curve

Take annually compounded zero rates of 3.50%, 3.80% and 4.00% for one, two and three years. The discount factor for year t is 1 / (1 + z)^t.

The sum of the discount factors, called the annuity factor, is 2.783302.

Step 2: value each leg

The fixed leg is a bond-like stream of $400,000 a year, so its present value is notional × fixed rate × annuity factor.

A floating leg that resets to the discount curve is worth par immediately after each reset. Valued at the start, it is worth the notional less the final discounted notional.

The payer pays fixed and receives floating, so the value to the payer is the floating leg minus the fixed leg.

The receiver, on the other side, is worth +$3,285. The two values always sum to zero, ignoring credit and margin.

Step 3: the par swap rate, and why the value is small

The par rate is the fixed rate that makes the swap worth zero.

We pay 4.0000%, which is 1.18 basis points above the par rate, so we are paying slightly more than the market requires. That shortfall, times the notional and the annuity factor, reproduces the value: 10,000,000 × (4.0000% − 3.9882%) × 2.7833 ≈ −$3,285.

Implied forward rates

Another way to see it: the forward rates implied by the curve are 3.50% for year 1, 4.10% for year 2 and 4.40% for year 3. The floating leg is expected to pay those, and the average (weighted by discount factors) is the par rate.

What moves the value

• Parallel rate shifts. If the whole curve rises, the payer gains, because the fixed rate they pay is now below market. A swap's rate sensitivity (its DV01) is roughly notional × annuity factor × one basis point, here about $2,783 per basis point.
• Curve shape. A swap is a package of forward rates, so a steeper or flatter curve changes the par rate even if the three-year zero rate does not move.
• Time passing. Each payment date rolls the swap closer to maturity, and the remaining annuity factor shrinks.
• Credit and funding. In practice dealers add credit valuation adjustment and funding adjustments to the discounted-cash-flow value above. Margin and collateral change how the swap is discounted.

Common exam traps

• Discounting with the wrong curve, or with one yield instead of the zero curve.
• Projecting the floating payments from forward rates but forgetting to discount them. The shortcut (notional less the final discounted notional) avoids the problem.
• Mixing up payer and receiver signs. The payer is long a floating-rate bond and short a fixed-rate bond.
• Treating the par rate as the average of the zero rates. It is a weighted average of forward rates.

More guides

• Value at Risk: the square-root-of-time ruleScale a one-day VaR to ten days with a worked example, and see where the rule breaks.
• Duration and convexity: how good is the approximation?A ten-year bond repriced at four yield shocks, exactly and with the duration and convexity estimates.
• Initial margin and the uncleared margin rules, explainedWho has to post it, how it is sized, and why the phase-in dates matter.
• Expected shortfall versus VaR: what changes in the tailWhy regulators moved from 99% VaR to 97.5% expected shortfall, with normal and fat-tailed numbers.
• Credit valuation adjustment (CVA): a worked exampleWhat counterparty credit risk costs on a derivative, step by step from exposure, default probability and recovery.