Pricing a Swap as the Difference of Two Bonds

You already know a swap is a strip of forward-rate agreements — a clever way to slice it into period-by-period bets on rates. That view is great for intuition. But traders don’t reach for a stack of FRAs when they want to price an existing swap in two seconds flat. They reach for a much slicker trick: a swap is just two bonds wearing a trench coat. One leg behaves exactly like a fixed-rate bond; the other behaves exactly like a floating-rate bond. Subtract them and you have the swap’s value — no strip of forwards required. This lesson builds that decomposition from scratch, proves the one lemma that makes it magical (the floating leg is worth par), derives the par swap rate as a falling-out consequence, and then marks an off-market swap to market.

The decomposition: a receive-fixed swap = long a fixed bond, short a floating bond

Analogy. Picture two friends settling a recurring bill. Each month, one pays a flat $40 (the “fixed” friend) and the other pays whatever the variable bill happens to be (the “floating” friend). Neither hands over a deposit at the start or claws one back at the end. Now imagine they agree that on the final month each will also hand the other exactly $1,000. That extra $1,000 flows both directions on the same day — it washes out completely. Nobody is better or worse off. But suddenly each friend’s payment stream looks exactly like a loan being repaid: regular coupons, then a lump of principal at the end. That lump is the notional, and conjuring it on both sides is the entire trick.

The mechanics. A receive-fixed swap means you receive the fixed coupons and pay the floating coupons. Add a notional $N$ paid to you at maturity on the fixed side, and a notional $N$ paid by you at maturity on the floating side. The two cancel, so:

A pay-fixed swap is the exact mirror — you pay the fixed coupons and receive floating — so it is short the fixed bond and long the floating bond:

The floating leg is worth par at every reset

This is the keystone. A floating-rate bond is worth par immediately after each reset — and here’s the clean reason why.

The one-period intuition. Suppose only one period remains. The note will pay its final floating coupon plus the notional. At the reset, that coupon is set to the current market rate $r$, so the coupon is $r \cdot N$. The note’s value is everything discounted back one period at that same rate $r$:

The $(1+r)$ on top and the $(1+r)$ on the bottom annihilate each other, leaving exactly the notional $N$ — par. The coupon was built from the same rate you discount with, so they can’t help but cancel.

Why it holds for many periods, not just one. Work backwards. At the last reset the note is worth par (the one-period argument above). Step back one reset: the note pays a coupon of $r_{\text{prev}} \cdot N$ and is then worth par — so it’s again $\frac{r_{\text{prev}} N + N}{1 + r_{\text{prev}}} = N$. The “worth par next reset” fact rolls all the way back to today. The floating leg is a chameleon: it always re-prices to par the instant its coupon refreshes.

The payoff for swap pricing. If the floating bond is worth $N$, the receive-fixed value collapses to a single term:

All the work is now in pricing one ordinary fixed-rate bond and subtracting the notional. The hero island below does exactly this: it prices the fixed-rate bond at the current market rate, pins the floating bond at par, and shows their signed difference. Slide the market rate and flip the side to watch the swap drift in and out of the money.

The par swap rate: the fixed rate that makes a new swap worth zero

When two parties strike a new swap, neither should pay the other to enter — the deal must be worth zero at inception. From our boxed formula, $V_{\text{receive-fixed}} = \text{PV}(\text{fixed bond}) - N = 0$ requires the fixed bond to price to par. So we hunt for the coupon $c$ that makes a bond price to par.

Write the fixed bond’s price using discount factors $DF_i$ (the present value today of $1 paid at time $i$). With a unit notional ($N = 1$) and annual coupons $c$:

Solve for $c$ — call it the par swap rate $s$:

With realistic day-count fractions $\tau_i$ (the year-fraction of each period, e.g. 0.5 for semiannual), the general form is

The denominator $\sum \tau_i DF_i$ has its own name — the annuity factor (or PV01 building block) — and it returns over and over in swaps, so befriend it now.

Fully worked example. Suppose the discount factors stripped from today’s curve are:

Step 1 — sum the discount factors (annual, so $\tau_i = 1$):

Step 2 — read off $DF_5 = 0.8626$, so $1 - DF_5 = 0.1374$.

Step 3 — the par swap rate:

Verify it prices to par. Plug $c = 3.00\%$ back into the fixed bond:

It prices to par (the tiny miss is rounding), confirming the swap is worth zero at this rate. A swap struck at 3.00% on this curve is fair to both sides.

Marking an off-market swap to market

The par swap rate makes a new swap worth zero. But the instant the curve moves, an existing swap stops being fair — it has gained or lost value, and somebody must report that on their books. This is marking to market.

The direction of the move. You are long the fixed bond if you receive fixed. Bonds rise when rates fall. So:

• Receive-fixed gains when market rates FALL (your fixed bond rallies above par).
• Pay-fixed gains when market rates RISE (you’re effectively short that bond).

This is precisely the slider behavior in SwapValueBridge above: push the current market rate below the 4% contract and the receive-fixed bar turns positive (in the money); push it above and it goes negative.

The quick approximation. A handy shortcut values an off-market swap as the rate differential times the annuity times the notional:

where $A = \sum \tau_i DF_i$ is the annuity factor and $N$ the notional. The logic: each period you receive $s_{\text{contract}}$ but the fair rate is now $s_{\text{market}}$, so you earn an excess of $(s_{\text{contract}} - s_{\text{market}})$ per period — and the annuity discounts that recurring edge back to today.

Worked example. A $10,000,000 receive-fixed swap was struck at $s_{\text{contract}} = 4\%$ with 5 years left. Market swap rates drop to $s_{\text{market}} = 3\%$, and at the new curve the annuity factor is $A = 4.58$.

The receiver is up roughly $458,000 — they locked in a 4% stream the market will now only pay 3% for, and that 1% edge, capitalized over five discounted years, is worth nearly half a million. The pay-fixed counterparty is down the same $458,000. (The exact bond-difference method gives a very close figure; the approximation is the trader’s mental-math version.)

Putting it together

Strip away the mystique and a vanilla interest-rate swap is two bonds wearing a trench coat. Conjure a notional payment on each leg at maturity — it cancels exactly, changing nothing — and a receive-fixed swap becomes long a fixed-rate bond, short a floating-rate bond (pay-fixed is the mirror). The floating leg is worth par at every reset, because its coupon is set to the very rate you discount with, so the value collapses to $V_{\text{receive-fixed}} = \text{PV}(\text{fixed bond}) - N$. Demanding zero value at inception forces the fixed bond to price to par, which hands you the par swap rate $s = (1 - DF_n)/\sum \tau_i DF_i$ for free. And the moment the curve moves the swap stops being worth zero: a receiver gains when rates fall, a payer when they rise, with a mark of roughly $(s_{\text{contract}} - s_{\text{market}}) \times A \times N$. Master this one subtraction and you can price, quote, and risk-manage a swap on a napkin.

Next — the swap curve and OIS discounting — where those discount factors actually come from, why post-2008 desks discount swap cash flows off the overnight (OIS) curve rather than the swap curve itself, and how the whole edifice is bootstrapped from market quotes.