Valuing Interest-Rate Swaps

A pension fund entered a $20 million, three-year interest-rate swap exactly one year ago, receiving a fixed 3.50% and paying floating. Rates have risen since then. The fund's risk officer needs a number, today, for the swap's current market value — not what it was worth at initiation, not what it will be worth at maturity, but its value right now, for the quarterly mark-to-market report. Unlike a stock, a swap has no ticker to look up. This lesson builds the number from the two pieces the swap is made of.

A swap is the difference of two bonds

The cleanest way to value a swap leans on a trick: separate the two legs and value each one as if it were a stand-alone bond. From the point of view of the party who receives fixed, pays floating, holding the swap is equivalent to

Neither bond's principal is real (no notional ever changes hands, as we established in the first lesson of this module) — but the value of the swap is still exactly the difference of what those two imaginary bonds would be worth if the principal were real, because the imaginary principal amounts are identical and cancel:

V_{\text{receive-fixed}} = B_{\text{fix}} - B_{\text{fl}}, \qquad V_{\text{pay-fixed}} = B_{\text{fl}} - B_{\text{fix}}.

B_{\text{fix}} is priced exactly like any coupon bond — discount every remaining coupon and the final "principal" back to today. The genuinely useful fact is what happens to B_{\text{fl}}.

Worked example: revaluing the pension fund's swap

The fund's swap: notional N = \$20{,}000{,}000, fixed rate 3.50\%, semiannual, three-year original term. One year (exactly two resets) has now passed, so we are valuing it right at the third reset date — two years and four semiannual coupons remain. Today's zero-coupon discount curve (continuously compounded) reads:

Maturity Zero rate Discount factor Fixed coupon PV of coupon
0.5y 4.00% 0.9802 $350,000 $343,070
1.0y 4.20% 0.9589 $350,000 $335,615
1.5y 4.40% 0.9362 $350,000 $327,670
2.0y 4.50% 0.9139 $350,000 $319,865

The fixed leg's coupons total \$1{,}326{,}220 in present value; add the "notional" repaid at 2.0 years, discounted at the same 0.9139:

B_{\text{fix}} = 1{,}326{,}220 + 20{,}000{,}000 \times 0.9139 = 1{,}326{,}220 + 18{,}278{,}000 = \$19{,}604{,}220.

Because today happens to be a reset date, the floating leg is worth par exactly: B_{\text{fl}} = \$20{,}000{,}000. The swap's value to the fund (which receives fixed) is:

V_{\text{receive-fixed}} = B_{\text{fix}} - B_{\text{fl}} = 19{,}604{,}220 - 20{,}000{,}000 = -\$395{,}780.

A negative number for the fund — sensible, since market rates have risen well above the 3.50% the fund locked in as a receiver of fixed; it is now stuck receiving a below-market rate. Its swap counterparty, who pays fixed and receives floating, holds the mirror-image position worth exactly +\$395{,}780. Swap valuation is always zero-sum this way: one side's mark-to-market gain is exactly the other side's loss.

See the value curve

Because B_{\text{fl}} always collapses to par right at a reset, the entire value of a receive-fixed swap right after a reset is just a fixed-coupon bond's price minus its own par value — and a bond priced at its own coupon rate is worth par. That's why the curve below always passes almost exactly through zero at the swap's own original fixed rate, however the current market rate wanders elsewhere.

Slide the original fixed rate and watch the whole curve — and its zero crossing — shift with it. To the right of the crossing (current rates above the swap's fixed rate) the receive-fixed position is under water; to the left, it's in the money. This is exactly the same shape as a bond's price-yield curve, because that is precisely what it is.

A $20 million swap losing $395,780 sounds dramatic; a $2 billion swap book losing the proportional amount would be a genuinely different headline. Trading desks instead track a swap's DV01 (or PV01) — the change in value for a one-basis-point (0.01%) parallel shift in the discount curve — which behaves almost exactly like a bond's duration because, as this lesson shows, a swap effectively is two bonds. A desk with many swaps on its book sums the DV01s across the whole portfolio to get one aggregate risk number instead of re-deriving a discount curve calculation for each trade individually — the same instinct that turns a warehouse full of individual positions into one manageable risk report.

It is tempting to conclude that a floating-rate leg is always worth its notional, full stop — but the theorem only holds exactly at a reset date. Between resets, the floating leg's next payment has already been locked in (reset in advance, remember) at a rate that may no longer match the market's current view, so its value can drift slightly away from par until the next reset resets it back. For most practical valuation this drift is small relative to the swap's overall size, and desks often value mid-period as if sitting exactly at the next reset for simplicity — but a risk manager marking a large book on a date that lands mid-period should account for the (usually small) known next coupon separately rather than silently assuming exact par.

There's a second, sharper subtlety worth flagging even here: which discount curve you use for B_{\text{fix}} and B_{\text{fl}} in the first place turns out to matter more than most students expect — enough that it reshaped the entire swap market after 2008, which is exactly where this module ends up.