Yield Curve Construction and Bootstrapping

Bootstrapping solves a zero coupon curve one maturity at a time, turning par bond and swap quotes into consistent discount factors.

Bootstrapping builds a zero coupon curve one maturity at a time, using the market prices of par instruments and the zero rates already solved for shorter maturities to isolate each new rate in turn. The method converts a handful of quoted par yields, deposit rates, or swap rates into a full set of discount factors that price every other instrument on the curve consistently.

A par rate is the coupon that would price a bond at exactly 100, given the market's current view of future cash flows. A zero rate is the single rate that discounts one cash flow, paid at one date, back to its present value today. A discount factor is the present value of $1 received at that date, and it is the quantity every other rate is derived from. These three objects describe the same market from three angles, and a rates desk moves between them constantly depending on whether it is quoting a bond, valuing a swap, or marking a curve.

A single published number such as the ten year yield is a blend of everything the market expects to happen to short rates between now and ten years out, plus a term premium for holding that duration. Pricing an actual instrument, whether a bond, a swap, or a swaption, requires the full curve rather than that one blended number, because every cash flow in the instrument needs its own discount factor.

Take a one year par bond priced at 100 with an annual coupon of 4 percent. Because the bond pays a single cash flow of 104 at the end of year one, the par condition, 100 equals 104 divided by one plus the one year zero rate, solves directly. The one year zero rate comes out at exactly 4 percent, since with one cash flow the par rate and the zero rate are the same number.

Now take a two year par bond priced at 100 with an annual coupon of 4.5 percent. Its price splits into a first year coupon of 4.50 and a final payment of 104.50 in year two. Discounting the first coupon at the one year zero rate already solved, 4 percent, leaves 95.67 to be explained by the year two cash flow. Solving one plus the two year zero rate, squared, equal to 104.50 divided by 95.67 gives a two year zero rate of approximately 4.51 percent.