The Swap Curve and OIS Discounting

You already know how to price a single swap and read off its par rate (lesson 3). Now zoom out. Plot the par swap rate at every tenor — 1y, 2y, 5y, 10y, 30y — and you get the swap curve, the single most-watched interest-rate object on the planet. Dealers quote it, trading desks price off it, and central-bank watchers read the market’s whole rate forecast in its slope. But a curve of par rates is only the surface. Underneath sit the discount factors, the zero rates, and the implied forwards — and, since 2008, a quiet revolution split that one curve into two. This lesson takes you from the quoted surface all the way down to the dual-curve machinery that values every collateralised swap in the world.

The swap curve: the term structure of par swap rates

Analogy. Think of the swap curve as the menu board of a money diner: for each holding period — 1 year, 2 years, all the way to 30 — it posts the fixed rate you can lock in today by trading fixed-for-floating. One curve, every maturity, refreshed by the second.

The definition. The swap curve is the par swap rate $s_n$ plotted against tenor $n$. Recall from lesson 3 that the par rate is the fixed coupon making a brand-new swap worth exactly zero — neither side pays the other to enter. Collect $s_1, s_2, \dots, s_{30}$ and you have the curve. Because swaps are deep, liquid, and standardised, this curve is the benchmark off which corporate loans, mortgages, structured notes, and other derivatives are priced.

Reading the slope. The shape is a message from the market:

• Upward-sloping (normal): longer swaps lock in higher rates. A long rate is roughly the average of the short rates expected over its life plus a term premium (extra yield for tying money up longer). An upward slope says the market expects short rates to rise and/or demands that premium.
• Inverted: longer swaps lock in lower rates — only possible if the market expects the short rate to fall (a classic recession/rate-cut signal), enough to overpower any term premium.

Toggle the island between Normal and Inverted, then flip on the forward overlay — we’ll explain exactly why the forwards sit where they do in the next section.

Bootstrapping the curve: from par rates to discount factors to forwards

The goal. Par rates are what the market quotes; discount factors are what you actually need to value any cash flow. Bootstrapping converts one into the other, recursively, by no-arbitrage — the identical trick you used to strip a bond curve into spot rates.

The engine. From lesson 3, a par swap of tenor $n$ satisfies

where $DF_i$ is the discount factor to payment date $i$ and $\tau_i$ is its accrual fraction (the year-fraction of period $i$). The denominator $\sum_i \tau_i DF_i$ is the annuity — the PV of receiving $1 of fixed rate each period. Rearrange to isolate the new unknown $DF_n$, assuming every earlier $DF_i$ is already known:

That’s the whole recursion. Solve $DF_1$ from the 1y rate, plug it in to solve $DF_2$ from the 2y rate, and climb the ladder.

Worked example — solve the 2y discount factor. Suppose annual payments ($\tau_i = 1$), the 1y discount factor is $DF_1 = 0.9615$ (a 1y rate near 4%), and the 2y par swap rate is $s_2 = 4.50\% = 0.045$. Then

So $DF_2 = 0.9156$. From there the 2y zero (spot) rate $z_2$ follows from $DF_2 = (1 + z_2)^{-2}$:

And the implied 1y forward one year out, $f_{1,2}$, is the rate that links the two discount factors:

Notice the punchline: the 1y rate was ~4.00%, the 2y par rate is 4.50%, but the implied forward for the second year is ~5.01% — above the par curve. That’s the overlay in the island made concrete: when the par curve slopes up, forwards must sit above it, because each longer par rate is a running average of short rates, and to keep dragging that average upward the marginal (forward) rate has to lead. Invert the curve and the logic flips — forwards drop below par to pull the average down.

OIS discounting and the dual-curve world

The old world (pre-2008). One curve did both jobs. The LIBOR curve was used to project the floating cash flows (LIBOR was the floating index) and to discount them back to PV. Tidy, single-curve, and quietly wrong — it assumed LIBOR was a near-risk-free funding rate. Then 2008 revealed that LIBOR carried real bank-credit and liquidity risk, and the basis between LIBOR and truly overnight rates blew out to hundreds of basis points.

The collateral insight. Most dealer swaps trade under a CSA (Credit Support Annex): each day the out-of-the-money party posts cash collateral equal to the swap’s mark-to-market. Here’s the key fact — posted cash collateral earns interest at the overnight rate. So if you’re owed money on a swap, you hold collateral that grows at the overnight rate; that overnight rate is your true funding/reinvestment rate on the position. By no-arbitrage, the discount rate must equal the rate your collateral earns — the OIS rate (Overnight Indexed Swap rate; today SOFR-based in USD), not LIBOR.

The result: dual curves. Projection and discounting split into two separate curves:

• Projection (forward) curve — built from the floating index (e.g., 3-month LIBOR/term rate). It answers: what floating cash flows will this swap throw off?
• Discount (OIS) curve — built from OIS swap rates. It answers: what are those cash flows worth today, given collateral earns the overnight rate?

You project on one curve and discount on another. That is the dual-curve framework, and it is now the standard way every collateralised linear rate product is valued.

Why it matters: valuations, hedges, and the basis

Valuation moves. Switching the discount curve from LIBOR to OIS re-prices every swap, and the effect grows with maturity and with how off-market the fixed rate is. For long-dated or deeply in/out-of-the-money trades, the PV difference between LIBOR- and OIS-discounting can run into real money — the kind of revaluation that resized entire derivative books in 2009–2010.

Hedge ratios move. Your risk now lives on two curves, so you carry sensitivities to both. A swap’s DV01 splits into exposure to the discount (OIS) curve and exposure to the projection (forward) curve; hedging only one leaves the other naked. Dual-curve risk reporting is now standard precisely because a single-curve hedge under-hedges.

The basis becomes a traded thing. The gap between the two curves — LIBOR-OIS (and today the SOFR basis) — is itself a market with its own quotes and its own swaps (basis swaps). It widens in funding stress (a classic crisis stress gauge) and tightens in calm. What used to be assumed away is now a P&L line.

Putting it together

The swap curve is the term structure of par swap rates — the world’s benchmark rates curve — and its slope encodes expected short rates plus a term premium. Bootstrapping strips that surface into discount factors by the no-arbitrage recursion $DF_n = (1 - s_n \sum_{i=1}^{n-1} \tau_i DF_i)/(1 + s_n \tau_n)$, and from the discount factors you read off zero rates and implied forwards — which sit above the par curve when it slopes up and below when it inverts. Since 2008, the single LIBOR curve split in two: a projection curve generates the floating cash flows while the OIS curve discounts them, because cash collateral under a CSA earns the overnight rate, making OIS the true funding/discount rate. That dual-curve world changes valuations, doubles your hedge dimensions, and turns the LIBOR-OIS / SOFR basis into a traded risk — while uncollateralised trades need a separate XVA funding adjustment.

Next — SOFR and the death of LIBOR — we follow the overnight rate that took over: how a scandal-ridden survey rate was retired and a transaction-based benchmark rebuilt the curves from the ground up.