How do I compute a forward swap rate from the discount factor curve?

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AcadiFi · Accepted Answer

The forward swap rate F for a swap starting at T_s and ending at T_e is the fixed rate that makes the present value of the forward-starting swap zero at inception. It is a weighted combination of forward rates along the swap's payment dates.

Formula (par forward swap rate):
F = [DF(T_s) - DF(T_e)] / sum_{i=1}^{n} tau_i x DF(T_i)

where T_i are payment dates, tau_i are accrual fractions, and the denominator is PVBP (annuity factor).

Example: 2x5 forward swap, starting in 2 years, 5-year tenor, semi-annual fixed payments. From the DF curve:
DF(2.0) = 0.908
DF(2.5) = 0.884, DF(3.0) = 0.861, DF(3.5) = 0.839, DF(4.0) = 0.817, DF(4.5) = 0.796, DF(5.0) = 0.775, DF(5.5) = 0.755, DF(6.0) = 0.735, DF(6.5) = 0.715, DF(7.0) = 0.696.

Sum of DFs from T=2.5 to T=7.0 (ten payments, tau=0.5):
Sum = 0.5 x (0.884 + 0.861 + 0.839 + 0.817 + 0.796 + 0.775 + 0.755 + 0.735 + 0.715 + 0.696) = 0.5 x 7.873 = 3.937

Forward swap rate = (0.908 - 0.696) / 3.937 = 0.212 / 3.937 = 0.0538 = 5.38%

This is the strike that makes a 2x5 swaption at-the-money-forward. Dealers quote swaption premiums in basis points relative to this F. A key intuition: F is higher than the 2-year spot swap rate because the forward curve is upward sloping between 2 and 7 years.

In practice, quants use SOFR OIS DFs for discounting and forecast DFs (from term SOFR or compounded in-advance rates) for floating leg projections.

How do I compute a forward swap rate from the discount factor curve?

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