Affine Term Structure Model

Definition

An Affine Term Structure Model (ATSM) is a no-arbitrage model of the yield curve in which bond yields are affine (constant plus linear) functions of an underlying vector of state variables $X_t$ . The state vector may contain observable macroeconomic variables, latent yield factors, or both. The affine structure follows from a log-normal stochastic discount factor (pricing kernel) with time-varying market prices of risk that are also affine in $X_t$ , together with Gaussian state dynamics. ATSMs belong to the general affine class introduced by Duffie and Kan (1996).

Key Ideas

Bond prices are exponential-affine in state: $p_t^n = \exp(\bar{A}_n + \bar{B}_n' X_t)$ , so log prices (and yields) are linear in $X_t$ .
The coefficients $\bar{A}_n, \bar{B}_n$ are not free parameters — they must satisfy recursive no-arbitrage equations derived from the pricing kernel, imposing cross-equation restrictions across all maturities simultaneously.
Risk premia are time-varying: the market prices of risk $\lambda_t = \lambda_0 + \lambda_1 X_t$ govern the difference between data-generating and risk-neutral dynamics.
When $\lambda_1 = 0$ (Vasicek 1977): constant risk premia; yields are level-shifted but risk premia don't vary with the state. When $\lambda_1 \neq 0$ : time-varying risk premia that can explain why the yield curve is on average upward sloping and why standard expectations-hypothesis tests fail.
Macro-augmented ATSMs (Ang-Piazzesi 2001): state vector partitioned into observable macro factors $X_t^o$ and latent yield factors $X_t^u$ . Both blocks enter the short rate and the pricing kernel, so the model prices all maturities using both macro and yield information.

How It Works

State Dynamics

The state vector $X_t \in \mathbb{R}^K$ follows a first-order Gaussian vector autoregression (VAR):

$X_t = \mu + \Phi X_{t-1} + \Sigma\epsilon_t, \qquad \epsilon_t \sim \mathrm{iid}\; N(0, I_K) \tag{1}$

In macro-augmented models, $X_t = ((X_t^o)', (X_t^u)')'$ where $X_t^o$ are observable macro factors (possibly including lags) and $X_t^u$ are latent yield factors.

Short Rate Equation

The one-period short rate is an affine function of all state variables:

$r_t = \delta_0 + \delta_1' X_t \tag{2}$

This generalizes the Taylor rule: if $\delta_1$ places weight only on observable macro variables, equation (2) is a strict Taylor rule; if it also weights latent factors, the short rate responds to unexplained yield-curve movements.

Stochastic Discount Factor (Pricing Kernel)

No-arbitrage guarantees the existence of a pricing kernel $m_{t+1}$ . Under log-normality:

$m_{t+1} = \exp\!\left(-r_t - \tfrac{1}{2}\lambda_t'\lambda_t - \lambda_t'\epsilon_{t+1}\right) \tag{3}$

with time-varying market prices of risk:

$\lambda_t = \lambda_0 + \lambda_1 X_t \tag{4}$

The vector $\lambda_0$ controls the unconditional slope of the yield curve; the matrix $\lambda_1$ governs how risk premia vary with the state (needed to match the failure of the Expectations Hypothesis).

Bond Price Recursion

The price of an $n$ -period zero-coupon bond $p_t^n = E_t[m_{t+1} p_{t+1}^{n-1}]$ has the closed-form solution:

$p_t^n = \exp(\bar{A}_n + \bar{B}_n' X_t) \tag{5}$

with $\bar{A}_1 = -\delta_0$ , $\bar{B}_1 = -\delta_1$ , and no-arbitrage recursions:

$\bar{A}_{n+1} = \bar{A}_n + \bar{B}_n'(\mu - \Sigma'\lambda_0) + \tfrac{1}{2}\bar{B}_n'\Sigma\Sigma'\bar{B}_n - \delta_0 \tag{6a}$

$\bar{B}_{n+1}' = \bar{B}_n'(\Phi - \Sigma\lambda_1) - \delta_1' \tag{6b}$

Yield Equation

Continuously-compounded yields are linear in state:

$y_t^n = A_n + B_n' X_t, \qquad A_n = -\bar{A}_n/n,\quad B_n = -\bar{B}_n/n \tag{7}$

The loading vectors $B_n$ describe how yield $n$ loads on each factor — they are the initial impulse responses of the yield to a factor shock. The shape of $B_n$ as a function of maturity $n$ determines whether a factor is a "level" (flat $B_n$ ), "slope" (upward-sloping), or "curvature" (hump-shaped) factor.

Risk-Neutral Dynamics

Under the equivalent martingale measure $Q$ (risk-neutral measure), the state follows:

$X_t = (\mu - \Sigma'\lambda_0) + (\Phi - \Sigma\lambda_1)X_{t-1} + \Sigma\epsilon_t^Q$

The shift from $(\mu, \Phi)$ to $(\mu - \Sigma'\lambda_0, \Phi - \Sigma\lambda_1)$ is entirely governed by the risk-price parameters, connecting the data-generating dynamics to bond prices. The recursions (6) can be seen as iterating this risk-neutral VAR.

Estimation (Ang-Piazzesi Two-Step Procedure)

When the state contains latent factors, maximum likelihood requires inverting yields to extract the unobservable state. The Chen-Scott (1993) approach:

Designate $K_2$ yields as measured without error (as many as latent factors); these invert exactly to $X_t^u$ .
Remaining yields are measured with IID (independent and identically distributed) error, contributing to the likelihood through a Gaussian measurement-error term.
The full likelihood is the joint density of all yields and observable state variables, with a Jacobian term from the inversion.

To avoid numerical instability in single-step joint Maximum Likelihood Estimation (MLE) (which tends to produce explosive dynamics when the state is highly persistent), Ang and Piazzesi propose a two-step procedure: (1) estimate macro dynamics and short-rate macro coefficients by ordinary least squares (OLS); (2) estimate remaining parameters by MLE holding step-1 fixed.

Why It Matters

Pricing all maturities: unlike unrestricted VARs, an ATSM prices every maturity analytically, so impulse responses and variance decompositions are available for any yield without additional estimation.
Forecasting discipline: the cross-equation no-arbitrage restrictions act as regularization — Ang-Piazzesi show ~25% root-mean-square-error (RMSE) improvement over unrestricted VARs in out-of-sample yield forecasting.
Factor identification: the affine loading structure $B_n$ provides a principled definition of "level/slope/curvature" as the factor weights on the yield curve, rather than a post-hoc principal-component-analysis (PCA) label.
Macro interpretation: macro-augmented ATSMs (Ang-Piazzesi 2001, later Rudebusch-Wu 2008, Joslin-Priebsch-Singleton 2014) directly answer how monetary policy and inflation transmit through the entire yield curve.

Affine Jump-Diffusion (AJD) and the Cox-Ingersoll-Ross (CIR) Model

The Gaussian ATSM above (with $\Phi, \Sigma$ constant) is a special case of the broader affine jump-diffusion (AJD) class (Duffie-Kan 1996; Duffie-Pan-Singleton 2000). In the AJD framework the state vector follows:

$dX_t = K(\theta - X_t)\,dt + \Sigma(X_t)^{1/2}\,dW_t + dJ_t$

where $K$ is a mean-reversion matrix, $\Sigma(X_t)$ is affine in $X_t$ (its $(i,i)$ diagonal element is $\alpha_i + \beta_i' X_t$ ), and $J_t$ is a compound Poisson jump process with intensity affine in $X_t$ . Bond yields remain affine in $X_t$ under AJD.

The canonical single-factor AJD model is Cox-Ingersoll-Ross (1985b):

$dr_t = \kappa(\bar r - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t$

The square-root diffusion keeps $r_t > 0$ and has a gamma stationary distribution. Bond prices are $p(r,T) = e^{A(T) + B(T)r}$ where $A$ and $B$ solve Riccati ordinary differential equations (ODEs) — the direct precursor of the AJD coefficient recursions.

Estimation Methods for Continuous-Time Models

Sundaresan (2000) surveys six estimation strategies for continuous-time Markov models:

Method	Key Reference	Principle
Generator GMM (Generalized Method of Moments)	Hansen-Scheinkman (1995)	Infinitesimal generator eigenfunctions yield moment conditions; distribution-free
SMM (Simulated Method of Moments)	Duffie-Singleton (1993)	Match moments of simulated paths to data; works for intractable likelihoods
EMM (Efficient Method of Moments) / Indirect Inference	Gallant-Tauchen (1996)	Match SNP (semi-nonparametric, i.e., flexible auxiliary) model score to simulated paths
ML (Maximum Likelihood) via density expansion	Aït-Sahalia (1999)	Closed-form series expansion of transition density; exact under regularity conditions
CF-GMM (Characteristic Function GMM)	Singleton (2001)	Exploits known characteristic function of AJD; efficient for exponential-affine models
MCMC (Markov Chain Monte Carlo) / Bayesian	Jacquier-Polson-Rossi (1994)	Augments with latent state paths; most flexible; computationally demanding

For AJD models with known characteristic functions, CF-GMM and ML via density expansion are the most efficient; SMM and EMM are preferred when the characteristic function or transition density is unavailable in closed form.

Open Questions

The orthogonality assumption (macro and latent factors independent) rules out feedback from the yield curve to macroeconomic variables, contradicting the yield curve's well-known predictive power for output (Estrella-Hardouvelis 1991).
Non-Gaussian extensions (square-root CIR processes, jump processes, regime switches) break the Gaussian closed-form recursions (6) or complicate estimation substantially; AJD with jumps partially restores tractability via characteristic functions.
Identification of $\lambda_1$ is empirically difficult — many parameters are insignificant and estimates are sensitive to sample period.
Stochastic volatility models (Heston 1993) alone explain the cross-sectional volatility smile but fail to match its maturity term structure; adding jumps is necessary but not sufficient (Sundaresan 2000).