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    Banking / rates

    Dual-curve swap pricing

    Since collateral became standard, a swap is projected off one curve and discounted off another. This prices a five-year payer swap with a term projection curve and an OIS discount curve, then reports the collateral basis — the part of the NPV that single-curve pricing misses.

    5y projection4.40%5y OIS4.00%Basis40 bp
    Swap NPV
    $336,359
    Payer swap in the money
    Collateral basis
    $462,675
    Difference against single-curve discounting
    Par rate
    4.503%
    Fixed leg at 4.200%
    PV01
    $11,091
    Annuity $110,905,879
    Curve and cash flows
    Forwards come off the projection curve; every cash flow is discounted on OIS.
    $336,359
    Curve and cash flows
    YearProjectionForwardOISDFFixed PVFloating PV
    14.550%4.655%4.150%0.95935$1,007,317$1,116,467
    24.500%4.550%4.100%0.92127$967,336$1,048,061
    34.450%4.446%4.050%0.88559$929,871$984,334
    44.425%4.446%4.025%0.85129$893,857$946,211
    54.400%4.394%4.000%0.81873$859,667$899,333

    Annual periods, act/365 fixed. The floating leg is the sum of the discounted forward cash flows.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Projection against discount curve
    The gap between the two curves is the basis the single-curve price ignores.
    Projection against discount curve0.0001.022.043.064.085.101y1y2y2y3y3y4y4y5y5yProjection (term)OIS (discount)
    NPV against the fixed rate
    Each curve crosses zero at its own par rate; the vertical gap is the collateral basis.
    NPV against the fixed rate-$2638.4k-$1482.2k-$326.1k$830.1k$1986.2k$3142.3k2.20%2.20%2.53%2.53%2.87%2.87%3.20%3.20%3.53%3.53%3.87%3.87%4.20%4.20%4.53%4.53%4.87%4.87%5.20%5.20%5.53%5.53%5.87%5.87%6.20%6.20%Dual curveSingle curve
    Canonical equations
    The exact relationships implemented by this engine.

    f_i = (DF_{i−1}/DF_i − 1) / τ_i · From the projection curve.

    Σ f_i τ_i DF_i(OIS) · Projected on one curve, discounted on another.

    K Σ τ_i DF_i(OIS) · The same OIS discount factors.

    NPV_dual − NPV_single · What OIS discounting adds or removes.

    finance methodology

    A clear path from inputs to results.

    A collateralised swap is projected off one curve and discounted off another. The floating leg follows the term curve; the discount factors come from OIS. Pricing both legs off a single curve ignores the basis between them, and on a five-year swap that is worth hundreds of thousands of dollars on a $25m notional.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the notional and the fixed rate the swap pays.
    2. Set the 1-year, 3-year and 5-year nodes of the projection curve.
    3. Set the same nodes for the OIS discount curve.
    4. Read the NPV, the par rate and the collateral basis.
    5. Compare the dual-curve and single-curve NPV against the fixed rate.

    The formula

    NPV=N∑iτi(fi−K)DFiOIS\mathrm{NPV} = N\sum_i \tau_i \left(f_i - K\right) DF_i^{OIS}

    Reference: Post-crisis swap pricing convention (OIS discounting); see Hull & White, 'The Valuation of Collateralized Derivatives'.

    Worked example

    $25m five-year payer swap at 4.20%

    Inputs
    • Notional $25,000,000, fixed rate 4.20% paid.
    • Projection 4.55% / 4.45% / 4.40% at 1, 3 and 5 years.
    • OIS 4.15% / 4.05% / 4.00% at the same nodes.
    Calculation path
    1. 1

      Linear interpolation fills the annual zero rates; forwards run from 4.66% in year one to 4.41% in year five.

    2. 2

      Discount factors come from OIS, so the five-year factor is e^−0.04×5 = 0.8187.

    3. 3

      The floating leg is worth $4.994m against a fixed leg of $4.658m.

    4. 4

      The NPV is +$336,359. Discounting the same cash flows on the projection curve gives −$126,316.

    5. 5

      The $462,675 difference is the collateral basis, about 40 bp a year over five years.

    Result

    NPV +$336,359 · par rate 4.503% · basis $462,675.

    Interpreting the result

    The par rate is where the swap is worth zero. At 4.20% the payer is paying below the par rate of 4.503%, which is why the NPV is positive. The collateral basis is not a profit: it is the amount by which the pre-crisis pricing convention misstates the same trade, and it scales with notional, tenor and the gap between the curves.

    Assumptions & limitations

    • Annual fixed and floating payments, act/365 fixed.
    • Forward rates derived from the projection zero curve; discounting on the OIS zero curve.
    • No day-count basis, no payment lag and no credit adjustment.
    • Curves interpolated linearly between the three nodes.

    Frequently asked questions

    Why is OIS the discount curve?

    A collateralised trade is funded at the collateral rate, so cash flows are discounted at OIS. Discounting at the projection curve double-counts the credit and liquidity spread embedded in it.

    Why does the basis scale with tenor?

    It is a spread earned on the outstanding notional each year, so it accumulates with the life of the trade and with the size of the curve gap.

    What happens if both curves are equal?

    The floating leg telescopes to 1 − DF(T) and the dual-curve NPV equals the single-curve NPV exactly. The basis goes to zero.

    Is the par rate the same under both conventions?

    No. Dual-curve discounting shifts the par rate by roughly the basis, which is why legacy tickets and new trades can disagree on value.

    Methodology last reviewed 2026-10-21. Results are informational and not financial or engineering advice — see our disclaimer.