Term Structure of Interest Rates: Yield Curve Basics Guide

The term structure of interest rates is the relationship between a loan's maturity and its interest rate, usually drawn as the yield curve: a plot of yields against time to maturity for default-free bonds. Short rates and long rates are rarely equal, and the shape of the curve carries information about the economy.

This matters because the curve is used to price every bond, to read market expectations about growth and inflation, and to set the lending margins of banks, which borrow short and lend long. An inverted curve, where short rates exceed long rates, has historically preceded recessions.

Key Concepts

• Spot rate (s_t): the yield on a zero-coupon bond maturing in t years; the rate for money lent today and returned once at t.
• Forward rate (f_{m,n}): the interest rate agreed today for a loan starting in m years and ending in n years; it can be locked in using bonds of different maturities.
• Yield curve shapes: normal (upward sloping), inverted (downward sloping), flat, and humped; each reflects different expectations and risk premia.
• Bootstrapping: the technique of extracting spot rates from the prices of coupon bonds, working outward from the shortest maturity.
• Expectations theory: the idea that long rates reflect expected future short rates plus a term premium for tying money up longer.

Formulas

No-arbitrage links spot and forward rates: (1 + s_n)^n = (1 + s_m)^m × (1 + f_{m,n})^(n−m). For the one-year rate one year forward: f_{1,2} = (1 + s_2)² / (1 + s_1) − 1. These assume default-free zero-coupon bonds, no transaction costs, and that the rates being compared cover matching periods.

Worked Example

Suppose the 1-year spot rate is 5% and the 2-year spot rate is 6%. What 1-year rate, starting one year from now, is implied by today's curve? Using the no-arbitrage relation: (1.06)² = (1.05)(1 + f), so f = (1.06)² / 1.05 − 1. Compute (1.06)² = 1.1236, divide by 1.05 to get 1.07009524, and subtract 1 to get f = 7.0095%. The market's curve therefore embeds an expected one-year rate of about 7.01% for the second year, higher than either spot rate. An investor can lock in this 7.01% today by buying the 2-year bond and shorting the 1-year bond in the right proportions, which is why the relation must hold to prevent arbitrage.

Common Misconceptions

• The long rate is just the average of expected short rates. It also contains a term premium compensating for the risk of locking money up longer, so it systematically differs from a pure average.
• An inverted curve must be a data error. Inversion is a genuine market outcome, usually signalling that investors expect central banks to cut rates as growth slows.
• There is one interest rate in the economy. There is a whole curve of rates by maturity, plus spreads for credit risk, so "the" interest rate is always an oversimplification.

Real-World Applications

The term structure prices bonds and interest-rate derivatives, guides central banks reading market expectations, determines bank net interest margins, and feeds into mortgage pricing, where long fixed rates derive from the long end of the curve.

The yield curve is the market's consensus forecast written in interest rates: spot rates price each maturity, forward rates reveal what the curve implies about the future, and its slope is one of the most-watched economic signals.