Bootstrapping

The technique rests on a single idea: a coupon bond is a portfolio of zero-coupon claims, so its price must equal the sum of its cash flows discounted at the spot rate matching each cash flow date. If every spot rate up to the second-to-last payment is already known, the final unknown spot rate can be isolated algebraically from the observed price.

Take a one-year spot rate of 4% and a two-year bond paying a 5% annual coupon that trades at par. Discounting the first coupon at 4% gives roughly 4.81, leaving 95.19 to be accounted for by the final payment of 105. Dividing 105 by 95.19 gives about 1.1030; the square root, minus one, yields a two-year spot rate near 5.02%. That figure now feeds the three-year calculation, and the process advances outward until the curve is complete.

Bootstrapping is the bridge between the three representations of the term structure. The par curve states the coupon rate that makes a bond of each maturity trade at face value. The spot curve states the single discount rate applicable to a lone cash flow at each date. The forward curve states the rates implied for future periods. Each can be derived from the others, so they carry identical information expressed differently.

Because the par curve blends cash flows across dates, it cannot be used directly to discount an arbitrary stream. The spot curve can. Once bootstrapped, spot rates convert into discount factors, and discount factors price any fixed cash-flow instrument in an internally consistent, arbitrage-free way. When the par curve slopes upward, bootstrapped spot rates lie above par yields at the same maturities, and implied forwards lie above the spots.

Government bond desks, swap dealers and valuation teams all maintain bootstrapped curves. In the swap market the inputs are money-market rates at the short end, futures or forward rate agreements in the middle, and par swap rates further out; the resulting discount factors value floating-rate notes, swaps and structured cash flows. Since the shift toward collateral-based discounting, overnight indexed swap rates have become the standard input for building discount curves.

Practical execution requires judgment. Liquid, on-the-run issues are preferred because stale or thinly traded prices contaminate every subsequent step. Maturities rarely align neatly, so par yields are interpolated first, and the interpolation method visibly shapes the extracted zeros. Coupon frequency, day-count conventions and settlement dates must be handled consistently, and any embedded option or tax distortion disqualifies a bond as an input.

Level I questions typically supply two or three par yields, or a short spot rate plus a coupon bond price, and ask for the next spot rate. Expect to set up the pricing equation, isolate the unknown discount factor and take the appropriate root. You should also be able to run the chain in reverse: given spot rates, compute the par yield or the implied forward rate between two dates, and recognise that a bond priced off the spot curve may differ from one priced at a single yield to maturity.

Beyond calculation, the concept underpins valuation logic used later. Spot rates derived this way are the benchmark against which Z-spreads and option-adjusted spreads are measured, and they explain why identical cash flows from different issuers command different prices. Errors in the early steps propagate outward, so the first solved rate deserves the most care.