Multi-Curve Framework

Before 2008, a bank used one yield curve to do two jobs at once: figure out what a floating rate would reset to in the future, and figure out how much to discount any cash flow back to today. That worked because unsecured interbank lending (LIBOR) and the safest overnight rate (what's now called OIS) moved almost in lockstep. Then the LIBOR-OIS spread blew out to over 300 basis points during the crisis, and it became obvious the two curves were pricing different things — one carried real bank credit risk, the other barely any. Using a single curve for both jobs stopped being an approximation and became a real, exploitable pricing error.

Two curves, two jobs

A floating swap leg's value at time $t$ still needs a forecast of the future rate and a discount factor, but now they come from different curves:

$$V_{\text{float}} = \sum_i \tau_i \, F_i \, D_{\text{OIS}}(t_i),$$

where $F_i$ is the forward rate for period $i$ projected off the forecast curve (built from LIBOR or SOFR instruments), and $D_{\text{OIS}}(t_i)$ is the discount factor from the OIS curve, because OIS is the rate actually earned on the cash collateral posted against the swap under a standard credit support annex. In plain English: what you expect to be paid comes from one curve, but what that future payment is worth today comes from a different, safer curve — because collateralized derivatives are funded at the safe rate, not the risky one.

Worked example 1 — the spread matters

A 1-year floating payment of a notional $10,000,000 is expected to reset at 5.30% (forecast curve), one payment in exactly one year. The OIS discount rate for one year is 5.00%; the old single-curve approach would have discounted at the forecast rate itself, 5.30%. Cash flow: 10{,}000{,}000 \times 0.053 = \530{,}000$. Under multi-curve, its present value is$530{,}000 / 1.05 = $504{,}762$. Under the old single-curve shortcut, it would have been$530{,}000 / 1.053 = $503{,}324 — a difference of \1,438 on one payment, from discounting at the wrong rate. Scaled across a bank's swap book, this kind of error is worth millions.

Worked example 2 — forecast and discount pulling apart

Suppose the forecast curve has a 3-month forward rate of 5.40% two years out, while the OIS curve's 2-year discount factor is $D_{\text{OIS}}(2) = 0.9070$. A $1,000,000 floating payment based on that forward: 1{,}000{,}000 \times 0.054 \times 0.25 = \13{,}500$(quarter-year accrual), present-valued as$13{,}500 \times 0.9070 = $12{,}244$. If a trader mistakenly used the forecast curve's own discount factor instead (say$D_{\text{fcst}}(2) = 0.9010$, since it embeds bank credit risk and is lower), they'd get$13{,}500 \times 0.9010 = $12{,}164 — understating the payment's value by \80 purely from mixing up which curve does which job.

Picture two of these curves stacked, one for OIS discounting and one for LIBOR or SOFR-plus-basis forecasting: they share the same short end when spreads are tight, but the gap between them widens under credit stress — exactly the widening this framework was built to price.

What this means in practice

Every swap desk now bootstraps two (or more, per currency and tenor) curves simultaneously: the OIS curve first, since it's needed to discount the instruments used to build the forecast curves, then each forecast curve using OIS discounting throughout. Getting the order or the curve assignment wrong doesn't just shift a valuation slightly — it can flip the sign of who owes whom on an off-market swap.