Term Structure of Interest Rates

Each point on the spot curve is the single rate that discounts one cash flow received at that maturity back to its present value. Because zero-coupon government bonds exist at only a limited set of maturities, analysts usually extract spot rates from coupon bonds by bootstrapping. The one-year spot rate comes directly from a one-year instrument with a single payment. The two-year spot rate is then solved from a two-year bond whose price equals its first coupon discounted at the known one-year spot rate plus its final coupon-plus-principal discounted at the unknown two-year spot rate. That rate, once solved, feeds the three-year calculation, and so on outward.

The resulting sequence of spot rates converts one-for-one into discount factors, each being the present value of one unit of currency payable at that date. Any bond in the same currency and credit class should then price as the sum of its cash flows multiplied by the matching discount factors; a material deviation is a relative-value signal rather than a defect in the curve.

A spot curve contains an implied set of forward rates, the rates at which the market will effectively lend for a future period beginning at a later date. The no-arbitrage link is simple: investing for two years at the two-year spot rate must produce the same terminal wealth as investing for one year at the one-year spot rate and reinvesting for a second year at the one-year forward rate starting in one year. Compounding the one-year spot and that forward rate therefore reproduces the compounded two-year spot rate, and the forward rate is recovered by division.

This mechanic explains why curve slope and rate expectations are not the same thing. When the spot curve rises with maturity, implied forwards sit above spot rates, but those forwards embed term premia as well as expectations. Traders quote forward rate agreements, forward-starting swaps and bond forwards directly off this structure, and a carry-and-rolldown assessment of a bond position is essentially a bet that realised rates differ from the implied forwards.

Term structures are observed upward-sloping, flat, inverted and humped. The pure expectations view attributes slope entirely to anticipated future short rates; liquidity preference adds a premium that grows with maturity to compensate investors for price risk; segmented markets and preferred habitat explain shape through supply and demand within maturity buckets, with insurers and pension funds concentrated at the long end and money-market funds at the short end. In practice all three forces operate, which is why slope is a noisy signal about policy alone.

Markets quote several representations of the same structure. Government bond yields give the on-the-run yield curve; par curves state the coupon that would price a bond at par at each maturity; swap and overnight-indexed-swap curves provide a continuous, liquid benchmark used to discount derivative and corporate cash flows. Spreads such as the two-year versus ten-year gap are shorthand for the slope of whichever curve is under discussion.

At Level I a candidate should be able to price a bond using a given sequence of spot rates rather than a single yield, and to recognise that the bond's yield to maturity is a complex weighted average of those spot rates rather than any one of them. Bootstrapping a two- or three-year spot rate from par or coupon bond data is a standard item, as is deriving an implied forward rate from two spot rates and, in reverse, constructing a spot rate from a chain of forwards.

Interpretation matters as much as arithmetic. Expect to distinguish spot, par and forward curves; to state which theory of the term structure is consistent with a described slope or with the existence of a maturity premium; and to explain why the same curve produces different yields for bonds of identical maturity but different coupons. The curve is also the reference point for spread measures, since any credit or option-adjusted spread is defined relative to a benchmark term structure.