Problem
Between June 2004 and July 2005, the FOMC raised the federal funds rate by 225 basis points, yet long-term yields fell (the 10-year yield declined 50 bps; the 10-year forward rate dropped 150 bps). This unusual behavior during a tightening episode requires decomposing yields into expected future short rates and term premiums. The paper estimates a three-factor Gaussian affine term structure model to perform this decomposition.
Key idea
Use the Gaussian affine term structure framework (Duffie-Kan 1996, Duffee 2002) with three latent factors, augmented by survey data on interest rate expectations (Blue Chip Financial Forecasts). Survey data helps identify the persistence of factors and expected future short rates, which are notoriously difficult to pin down from yields alone. The model provides a four-way decomposition: expected future real rates, expected future inflation, real term premiums, and inflation risk premiums.
Method
- Nominal model: 3 latent factors x(t) follow an Ornstein-Uhlenbeck process (dx = Kx dt + Sigma dB). Short rate is affine in factors: r(t) = rho_0 + rho’x(t). Market prices of risk are affine: lambda(t) = phi + Phi*x(t).
- Bond prices: exponential-affine P_n,t = exp(a(n) + b(n)‘x(t)) where (a,b) solve Riccati ODEs.
- Estimation: state-space form with weekly zero-coupon yields (3m to 10yr) as observables and Kalman filter. Survey expectations (6-month, 12-month, and 6-11yr ahead T-bill forecasts from Blue Chip) enter the measurement equation as noisy observations of model-implied expectations.
- Real term structure extension: Kim (2004) adds an inflation process dQ/Q = pi(t)dt + sigma_0 dW + sigma’dB with pi(t) affine in factors. Combined with CPI data and inflation survey expectations (SPF) to decompose nominal into real + inflation components.
- Identification: K is lower triangular, Sigma is diagonal.
- Sample: July 1990 to July 2005.
Results
- Term premium decline: the model attributes most of the 150 bps decline in the 10-year forward rate since June 2004 to a fall in term premiums (estimated ~120 bps decline). If term premiums had not changed, 10-year yields would have risen modestly.
- Real vs. inflation decomposition: about two-thirds of the decline in nominal term premiums is attributed to falling real term premiums; estimated inflation risk premium declined ~30 bps.
- Historical patterns: the 10-year forward term premium has trended lower since 1990; it rose during the 1994 tightening (contrasting with 2004-05), dipped in October 1998 (flight-to-quality), and rose during recessions.
- Long-horizon convergence caveat: with stationary factors, distant-horizon forward rate expectations converge to rho_0, so sufficiently distant forward rate movements are always attributed to term premiums. This is shared by all stationary affine models including those with regime switching.
- Correlation with Cochrane-Piazzesi (2005): the estimated 10-year forward term premium correlates 0.83 with the Cochrane-Piazzesi return-forecasting factor.
Limitations
- Gaussian factors cannot capture stochastic volatility or time-varying uncertainty directly.
- No regime switching: the model assumes time-invariant parameters, potentially missing structural breaks.
- The stationarity of factors forces distant-horizon expectations to converge, which may be unrealistic.
- TIPS data not incorporated (only available since 1997 with liquidity issues); real term structure relies on CPI data.
- Short sample (1990-2005); structural stability is assumed but not tested.
Open questions
- How do regime-switching extensions (as in regime-switching-affine-term-structure) change the term premium estimates?
- Can the model be extended with time-varying volatility (stochastic volatility affine models) while retaining tractability?
- How sensitive are the term premium estimates to the choice of survey data and the assumption of Gaussian measurement errors?
My take
The Kim-Wright model became the Federal Reserve’s standard term premium decomposition tool, widely cited in policy discussions. Its clean Gaussian-affine structure with survey augmentation makes it a benchmark for term structure analysis. The connection to the CRE project is through the affine pricing framework: the CRE model’s regime-switching-affine-term-structure extends this class by adding Markov-switching regimes. Kim-Wright’s stationarity caveat (distant-horizon forward rates always attributed to term premiums) motivates regime-switching extensions where regime shifts can create persistent level shifts in expectations. The inflation decomposition showing both real and inflation-risk-premium components declining is relevant to understanding the macro backdrop for commercial real estate pricing. The Riccati ODE bond pricing (a(n), b(n)) is the continuous-time analogue of the discrete recursions in the CRE model’s term structure module.