Term Structure of Interest Rates

Definition

The term structure of interest rates (equivalently, the yield curve) is the mapping from bond maturity $n$ to the continuously-compounded zero-coupon yield $y_t^n$ prevailing at time $t$ :

$y_t^n = -\frac{1}{n}\log P_t^n$

where $P_t^n$ is the time- $t$ price of a zero-coupon bond that pays one unit at maturity $n$ . The yield curve summarises the market's collective pricing of riskless cash flows at all horizons, and is therefore the foundational object of fixed income, monetary economics, and term-structure modelling.

Key Ideas

Yield decomposition: any zero-coupon yield splits into (i) the expected average future short rate under the physical measure and (ii) a term premium — the risk compensation demanded for holding a long-duration bond. No-arbitrage models make this decomposition exact through the pricing kernel.
Three-factor structure: empirically, the first three principal components of yield changes explain $>99\%$ of variation, traditionally labelled level (parallel shift of the entire curve), slope (differential between long and short yields), and curvature (medium-maturity yields relative to the short and long ends). These correspond to the Nelson-Siegel (1987) basis functions.
Standard shapes: upward-sloping (normal) reflects positive term premia and/or expectations of rising rates; inverted curve signals expected rate cuts, often associated with recessions; flat or humped curves represent transitional states.
Forward rates: the instantaneous forward rate $f_t^n = -\partial \log P_t^n / \partial n$ is the marginal cost of borrowing at horizon $n$ . The yield equals the average forward rate up to maturity: $y_t^n = \frac{1}{n}\int_0^n f_t^s\,ds$ .
Predictive content: the yield spread $(y_t^{\text{long}} - y_t^{\text{short}})$ is one of the most robust leading indicators of real activity; an inverted curve has preceded every U.S. recession since 1960 (Estrella-Hardouvelis 1991).

How It Works

Bond Pricing Fundamentals

Under no-arbitrage, there exists a stochastic discount factor (SDF) $M_{t+1}$ such that for any asset:

$P_t^n = E_t\!\left[M_{t+1}\,P_{t+1}^{n-1}\right]$

For zero-coupon bonds this recursion, initialised at $P_t^0 = 1$ , prices the entire yield curve once $M_{t+1}$ is specified. The one-period short rate is $r_t = -\log E_t[M_{t+1}]$ (ignoring Jensen's-inequality correction under log-normality).

Classical Theories

Theory	Mechanism	Implication
Pure Expectations Hypothesis (EH)	Term premium $= 0$ ; $y_t^n = \frac{1}{n}\sum_{k=0}^{n-1}E_t[r_{t+k}]$	Slope predicts future short-rate changes one-for-one; no excess return from duration
Liquidity Preference	Investors prefer short bonds; must be paid for duration risk	Positive term premium at all maturities; explains the typical upward slope
Market Segmentation	Distinct clienteles at each maturity	Local supply/demand at each tenor drives yields independently

Empirical evidence decisively rejects the pure EH: Fama-Bliss (1987) and Campbell-Shiller (1991) show the slope predicts excess returns rather than future short-rate changes, implying a time-varying term premium.

Affine No-Arbitrage Models

The dominant approach specifies the SDF as log-normal with affine market prices of risk $\lambda_t = \lambda_0 + \lambda_1 X_t$ , giving yields that are affine in a state vector $X_t$ :

$y_t^n = A_n + B_n'\, X_t$

The coefficients $A_n, B_n$ are not free — they satisfy recursive no-arbitrage equations derived from the pricing kernel (see Affine Term Structure Model). This class enforces internal consistency: every maturity is priced by the same state process and the same risk prices.

Why It Matters

Monetary policy transmission: the central bank directly controls only the overnight rate; the yield curve shows how that policy propagates to borrowing costs at all maturities through expectations and time-varying risk premia.
Real activity forecasting: yield-curve slope is among the most reliable recession indicators and is embedded in most composite leading-indicator frameworks.
Fixed-income pricing: coupon bonds, mortgages, swaps, and interest-rate derivatives are all priced by discounting cash flows along the appropriate yield curve.
Inflation expectations: the breakeven inflation rate (nominal yield minus Treasury Inflation-Protected Securities (TIPS) yield) extracts market-implied long-run inflation expectations directly from term-structure data.

Open Questions

Term premium identification: decomposing yields into expected short rates vs. term premia requires an assumed model; different affine term structure models (ATSMs) give substantially different decompositions for identical yield data.
Macro-finance channels: Ang-Piazzesi (2001) establish a statistical link between macro factors and yields, but the structural mechanism — why inflation uncertainty or recession risk raises the term premium at specific maturities — remains debated.
Convenience yields and liquidity: government yields embed a liquidity/safety premium absent from corporate curves; disentangling the riskless rate from convenience distorts ATSM estimates.
Non-linearity and regime changes: linear Gaussian ATSMs fit poorly in environments with binding zero lower bounds or sudden monetary-regime shifts; extending the affine framework while retaining tractability is an active area.