Bootstrap the Zero Curve

A rates desk quotes par swap rates at 1, 2, 5, and 10 years, then needs to discount an off-market cash flow that settles in year 7. Par rates won't do that on their own — discounting each coupon takes a discount factor at every node, which is exactly what bootstrapping produces. Feed this tool your list of annual par rates and it returns the matching zero (spot) curve, one vertex at a time. It's the step that turns quoted par/swap rates into the discount curve you use to price any stream of cash flows.

The method leans on the par-bond identity: a bond trading at par is priced at face, and its coupon equals the par rate for that maturity. Write that equality out and the coupon times the sum of the already-known discount factors, plus the factor for the new maturity, must add up to 1. Every earlier factor was solved at a previous node, so a single unknown remains — the discount factor at the current vertex — which you isolate and convert into an annual zero rate. The recursion is exact at the quoted nodes, but it assumes annual coupons and genuinely par instruments at each tenor; it does not interpolate intermediate maturities or smooth the curve, so you need one clean vertex per year with no gaps.

Enter the annual par rates in maturity order — the first for 1 year, the second for 2 years, and so on down the list. The output gives the zero rate at each maturity, meaning the single rate that discounts a cash flow paying only on that date. Quick check: the 1-year zero should match the 1-year par rate, since there's just one payment. And on an upward-sloping par curve the zero rates sit above the par rates at the long end — if they come out below, re-check the order of your inputs.