Pricing a swap: the par swap rate

In short

• Expected floating payments are the implied forward rates for each period; discount each one, and each fixed payment, with the zero rate for its date.
• The par swap rate $s_N$ solves $\sum IFR_{i-1,1}DF_i = s_N\sum DF_i$, so $s_N = \sum IFR_{i-1,1}DF_i \,/\, \sum DF_i$.
• It equals the coupon of a par bond priced off the same zero curve: $s_N = (1 - DF_N)/\sum DF_i$.
• The swap rate is a multi-period breakeven rate: an internal rate of return on the implied forward rates through the swap's maturity.
• The whole swap is worth zero at inception, but individual periods are not: some expected net flows are positive and some negative.