Regime-switching affine term structure model

Definition

A no-arbitrage term structure model in which the short rate, factor dynamics under both physical and risk-neutral measures, and the prices of risk are governed by a discrete-time Markov regime variable. Conditional on a regime path, bond prices are exponential-affine in the underlying state factors with regime-dependent loadings; unconditional prices are obtained by integrating over the Markov chain.

Intuition

Standard affine term structure models assume the parameters governing factor dynamics are constant over time. Empirically this is implausible: monetary policy regimes, inflation regimes, and macro volatility regimes all shift discretely. Regime-switching ATSMs preserve the analytical tractability of the affine class — recursive, closed-form bond price coefficients — while letting the drift, volatility, and risk prices jump with the latent regime, generating much richer term structure dynamics (regime-conditional yield curves, time- varying risk premia, fat-tailed yield distributions).

Formal notation

Let $s_t\in 1,...,K$ be a discrete Markov chain with transition matrix $\Pi$. The pricing kernel evolves as

$m_{t+1}=exp(-r_t-1/2\lambda (s_{t+1}{)}^{\prime }\lambda (s_{t+1})-\lambda (s_{t+1}{)}^{\prime }{\epsilon }_{t+1})$

where the short rate $r_t={\delta }_0(s_t)+{\delta }_1(s_t{)}^{\prime }{x}_t$ is regime-dependent and the latent state follows a regime-switching VAR

$x_{t+1}=\mu (s_{t+1})+\Phi (s_{t+1})x_t+\Sigma (s_{t+1}){\epsilon }_{t+1}$.

Conditional on a regime path $s_{0:T}$, the n-period bond price is $P_n(x_t;s_t)=exp(A_n(s_{0:t})+B_n(s_{0:t}{)}^{\prime }{x}_t)$, where the coefficients $(A_n,B_n)$ satisfy regime-conditional Riccati-type recursions.

Variants

• Bansal-Zhou (2002) — two-regime model focused on bond return predictability.
• Dai-Singleton-Yang (2007) — square-root regime-switching model with explicit regime risk pricing.
• Ang-Bekaert-Wei (2008) — three-factor regime-switching model with inflation, used to decompose nominal yields into real rates + expected inflation + inflation risk premium (ang-bekaert-wei-2008-real-rates-expected-inflation).

Comparison

• vs. constant-parameter affine ATSM: regime switching delivers regime- conditional yield curves and richer term premia at the cost of additional parameters and a discrete state.
• vs. stochastic-volatility affine models (CIR-style): regime switching produces jumps in conditional moments rather than continuous diffusion; better suited to discrete monetary policy or inflation regime changes.
• vs. non-affine regime-switching models: closed-form bond prices are preserved (within each regime path) — a major computational advantage.

When to use

• The factor or short-rate dynamics are visibly non-stationary across distinct monetary or inflation regimes.
• A decomposition of nominal yields into components requires regime-conditional expectations that constant-parameter affine models cannot deliver.
• Tractability of the affine recursion is needed for likelihood-based estimation or filtering at scale.

Known limitations

• Number of regimes is a modeling choice; results can be sensitive to it.
• Latent regimes may not map cleanly onto observable monetary policy or macroeconomic regimes without further identifying restrictions.
• Identification of regime-conditional risk prices typically requires either long samples spanning multiple regimes or auxiliary data (e.g., survey forecasts).

Open problems

• Joint identification of regime-conditional risk prices and physical-measure transition probabilities under short samples.
• Extending the framework to many discrete regimes (curse of dimensionality on the regime path) or to a continuum of regimes.
• Bridging discrete regime switching with continuous-time stochastic volatility for hybrid pricing models.

Key papers

• ang-bekaert-wei-2008-real-rates-expected-inflation

My understanding

This is the immediate parent of the regime-switching architecture used in this project’s CRE asset pricing pipeline. The Ang-Bekaert-Wei specification demonstrates that regime-conditional affine bond prices identify economically distinct components (real / inflation / risk) using only nominal yields and inflation surveys — exactly the identification challenge the broader project faces when separating monetary regimes from wage-rigidity regimes in the 4-compound-state CRE setup.