Interest Rate Swaps

An interest rate swap is a private deal between two parties to exchange interest payments. One side pays a fixed rate, agreed up front and unchanging for the life of the deal. The other pays a floating rate that resets each period to whatever short-term rates happen to be (these days a benchmark like SOFR). They swap these payment streams on an agreed notional amount, and here's the thing to hold onto: the notional itself is never actually exchanged. It's just the number the interest rates are applied to. Only the difference between the two interest amounts changes hands each period.

Why would anyone do this? Picture a company with a floating-rate loan. Its borrowing cost jumps around every quarter, and the treasurer hates the uncertainty. By entering a swap where it receives floating and pays fixed, the floating leg cancels the loan's floating payments and leaves the company with a clean, predictable fixed cost. The swap converted floating debt into fixed debt without refinancing anything. That's the bread and butter of swaps: reshaping interest-rate exposure cheaply.

The two legs

The clean way to value a swap is to see it as two bonds. To the party receiving fixed, the swap is like owning a fixed-coupon bond (the fixed leg) and shorting a floating-rate bond (the floating leg). So the swap's value is the value of the fixed bond minus the value of the floating bond. A floating-rate bond, right after a reset, is worth its face value, which makes the algebra tidy.

The par swap rate

At the start, a swap is set up to be worth zero to both sides, nobody pays a premium. The fixed rate that makes that happen is called the par swap rate $S$. Using discount factors from the yield curve, it works out to

$$S = \frac{1 - Z_n}{\sum_{i=1}^{n} \tau_i\, Z_i}.$$

Reading the symbols: $Z_i$ is the discount factor for payment date $i$ (today's value of $1 delivered then), $Z_n$ is the discount factor for the final date, $\tau_i$ is the length of period $i$ in years, and the sum in the denominator is called the annuity, the present value of receiving $1 of fixed rate each period. The numerator $1 - Z_n$ is exactly the value of the floating leg per dollar of notional. Set the fixed leg's value equal to it and you get the fair fixed rate.

Worked example

Consider a 2-year swap paying annually on a notional of $10 million. Suppose the discount factors from the curve are $Z_1 = 0.97$ (one year out) and $Z_2 = 0.93$ (two years out), with $\tau_1 = \tau_2 = 1$ year. The par swap rate is

$$S = \frac{1 - 0.93}{1 \times 0.97 + 1 \times 0.93} = \frac{0.07}{1.90} = 0.0368 = 3.68\%.$$

So a fair fixed rate is about 3.68%. A company that pays fixed at 3.68% and receives floating starts the swap at zero cost.

Now suppose rates jump after you've entered as the fixed payer. Your fixed payments stay at 3.68% while the floating payments you receive climb, so the swap now has positive value to you. Concretely, if the fair fixed rate for a fresh swap has risen to 4.68%, you're paying 1% below market on $10 million for two years. The gain is roughly that 1% edge times the annuity: $0.01 \times 10\text{M} \times 1.90 = 0.19\text{M}$, about $190,000. Pay-fixed positions gain when rates rise; receive-fixed positions gain when rates fall, because receive-fixed behaves like owning a bond.

Where it gets subtle

• The notional is a phantom. No $10 million ever changes hands, only the netted interest difference each period. But the rate risk is very real and scales with that full notional, a small swap notional can carry large duration.
• Floating is only fixed for one period at a time. Each floating payment is known only at the start of its own period, when the rate resets. Before that, future floating payments are estimated from forward rates, which is why a swap is really a bundle of forward rate agreements stitched together.
• The discount curve matters. The valuation above is only as good as the discount factors you feed it. Post-2008, desks discount collateralised swaps off a separate curve from the one that projects the floating payments, a distinction that trips up newcomers.
• Counterparty and clearing. Like forwards, swaps were once purely private and carried counterparty risk. Most standard swaps now clear through a central counterparty, much like futures, to contain that credit exposure.

Swaps are how the interest-rate world manages risk at scale: chain enough forwards on interest payments together, agree to net them, and you've built the single most-traded derivative on the planet.