The fixed income term premium is the difference between the yield of a longer-maturity bond and the average expected risk-free short-term rate for that maturity. Abstractly, it is a price for commitment. The term premium is not directly observable but needs to be estimated based on the assumptions of a term structure model that separates expected short-term rates and risk premia. Model assumptions become a lot more realistic if one includes macroeconomic variables. In particular, long-term inflation expectations plausibly shape the long-term trend in yield levels. Also cyclical fluctuations in inflation and unemployment explain slope and curvature to some extent. A recent IMF paper proposes a methodology for integrating macroeconomic variables in a conventional affine term structure model.

*The post ties in with SRSV’s summary lecture on fundamental value estimates, particularly the section on fixed income.*

*The below are quotes from the paper. Emphasis and cursive text have been added. *

### What is a term premium?

“The term premium for a bond of maturity is the __difference between the bond’s yield and the expectation of the risk-free rate over the life of the bond__…The term premium is the compensation investors require for holding a long-term bond compared to rolling over a series of short-term bonds with lower maturity. In that sense, interest rates are driven by investors’ expected average level of the risk-free rate and a compensation for the longer holding period.”

“Under this definition, the term premium is a residual between observed yields and expectations of risk-free rates. As under normal conditions expectations of future short-term rates evolve rather slowly, and __term premia are the primary drivers of yield movements__, particularly for longer maturity bonds.”

“The analysis of term premia is not straightforward as both the expected future short-term interest rate and the term premium are not directly observable or measurable. __Deriving the term premium from the term structure requires making certain assumptions__.”

### Conventional term premium models

“In the finance literature, term structure models are mainly used to describe the joint evolution of different yields, with a focus on cross-sectional relationships…Securities with the same risk characteristics need to have the same price…These so-called __affine Gaussian term structure models assume that yields depend linearly on the risk factors__ (hence, affine), and Gaussian refers to the distributional assumption for the risk factors. “

“[*Thus*] modern term premium models do share common structural features:

- They are
__almost always Gaussian [__This greatly simplifies the computation…*normally distributed*]. - The
__yields of the underlying bonds and the risk-free rate are affine (linear) in some state variables__. These state variables are sometimes entirely endogenous to the yield curve but they may also include macroeconomic and financial variables, and survey data. __The dimension of the yield curve is reduced__. This can be done through unstructured cross-sectional models such as principal components analysis…or a parsimonious structured model like that of Nelson and Siegel. These factors are then given some intertemporal structure in a vector-autoregressive type model.”

“[*The figure below*] gives 10-year term premium estimates resulting from the Kim and Wright (2006), a semi-structural model, and Adrian and others (2013) models, a purely statistical model. Both term premium time series show a long-term downward trend, suggesting that the term premium has become smaller over the past 25 years.”

“Typically, the __risk factors which drive these models are assumed to be stationary__…However, if the underlying bond yields are trending and not stationary this assumption could bias the dynamics and intra-factor relationships of the model.”

### The term premium and macro factors

“__Term premia are very sensitive to the expected future path of growth, inflation, and monetary policy__… Expectations about structural factors like inflation or potential growth are revised only slowly, even after large shocks, pointing to high and persistent inertia in their reassessment…The body of literature on estimating future short term interest rates using expectations data is growing [and includes] the use of surveys to anchor the model’s dynamics.”

“Adding macroeconomic information enriches medium-term forecasts. Given the cyclical patterns in the yield curve, __forecasts based on most of the existing models are, effectively, factors slowly reverting to their means__. While this may give an accurate representation in the very short run (which is what finance models typically focus on), it does not capture well observed macro-financial dynamics.”

“The term structure model presented in this paper exploits both the cross-sectional and temporal information embedded in the term structure of interest rates, and __adds macroeconomic factors to a vector autoregressive model that is constrained to introduce cyclical dynamics with a focus on medium- to long-run forecasts__. Expectations about structural factors like inflation or potential growth are revised only slowly, even after large shocks, pointing to high and persistent inertia in their reassessment. For instance, the Survey of Professional Forecasters suggests that it has taken economists several years following the crisis to adapt their views on what the average level of interest rates should be. A similar pattern is apparent in the Federal Reserve’s Summary of Economic Projections and the CBO’s long-run forecasts of potential growth.”

“[*Empirical analysis from 1962 to 2018*] shows that the __slope of the yield curve is particularly affected by the unemployment gap__, and the __level of interest rates is strongly correlated with the level of inflation__, but not the unemployment gap…Both inflation and the level of the yield curve are subject to unit roots [*not mean reverting*], whereas the yield curve’s slope and curvature, as well as the unemployment rate, appear to be mean reverting series. This is plausible since the __rate of inflation and nominal interest rates are widely thought of as being determined by monetary policy choices in the long-run__.”

“The model’s transition dynamics are composed of two blocks.

- The
**cyclical component**comprises stationary series that include the cyclical component of the level of yields (our primary modelling innovation),__slope, curvature, and the cyclical components of inflation and unemployment__. - The
**trend component**of the__level of interest rates__: the long-run level of expected inflation and the long-run level of real yield curve.”

“__Over the last few years, the term premium has been relatively small or even negative__…At the same time, inflation expectations in the United States remained very well anchored. An inflation surprise could require monetary policy to tighten faster than anticipated, inducing to a sudden decompression of term and other risk premia, thus tightening financial conditions.”