An equilibrium model for U.S. Treasury yields explains how macroeconomic trends and related expectations for future short-term interest rates shape the yield curve. Long-term yield trends arise from learning about stable components in GDP growth and inflation. They explain the steady rise of Treasury yields in the 1960s-1980s and their decline in the 1990s-2010s. Cyclical movements in yields curves result from learning about transitory deviations of GDP growth and inflation. They explain why curves have been steep out of recessions and inverted in mature economic expansions. Finally, since the 2000s pro-cyclical inflation expectations and fears for secular stagnation have accentuated the steepness of the Treasury curve; positive correlation between inflation and growth expectations means that the Fed can cut rates more drastically to support the economy.

based on Zhao, Guihai, “Learning, Equilibrium Trend, Cycle, and Spread in Bond Yields”, Bank of Canada Staff Working Paper 2020-14

The below are quotes from the paper. Cursive text and text in brackets have been added for clarity.

The post ties up with this site’s summary on systematic value and macroeconomic trends.

### What treasury yield equilibrium theory can explain

“Key features in the historical dynamics of U.S. Treasury bond yields [include] a __trend in long-term yields, business cycle movements in short-term yields, and a level shift in yield spreads__…This paper presents a new equilibrium model to jointly explain these key features.”

“[The graphs] below shows some salient features in the data: (1) a hump-shaped trend in the yields, (2) business cycle (cyclical) movements in short-term yields and in the spreads between long- and short-term yields, (3) more-frequent and deeper inverted curves (accompanied by more-frequent recessions) pre-1990s than post-1990s, and (4) a positive yield spread on average.”

“We show that the __historical dynamics of the yield curve can largely be explained by movements in the short-rate expectations__ and provide a joint-equilibrium understanding of these salient features in the data.”

“The __trend is generated by learning from the stable components in GDP growth and inflation__, which share similar patterns to the neutral rate of interest and trend inflation…__Cyclical movements in yields and spreads are mainly driven by learning from the transitory components in GDP growth and inflation__. The less-frequent inverted yield curves observed after the 1990s are due to the recent secular stagnation and procyclical inflation expectation.”

### Long-term economic trends

“It has long been recognized that nominal interest rates contain a slow-moving trend component. Recent empirical studies propose macro trends as the driving force behind this low-frequency variation…[Academic papers] document the __empirical importance of trend inflation for explaining the secular decline in Treasury yields__ since the early 1980s [and also] show that it is crucial to also __include the neutral interest rate of interest which has driven the downward trend in long term yields__ over the last 20 years.”

“Unlike standard Bayesian learning, where the variance of the posterior [subjective view of probability distribution after seeing data] converges to zero…__learning is perpetual__…due to the agent’s fading memory. The posteriors for the mean inflation and mean growth rates, as state variables, capture the trends in inflation and growth.”

“The agent in this paper takes into consideration the __risk of belief updating__. Hence, the posteriors for the long-run means are state variables, which move closely with the macro trend estimates.”

“Using the U.S. real GDP growth and the rate of inflation from the GDP deflator, __we can calculate the posterior beliefs for the output growth and inflation__. We then show that they __closely match the estimated natural rate of interest and long-term inflation expectations__…We can calculate the model-implied 10-year nominal and real bond yields, which match the historical movements in the data well.”

“Not only does the model capture the low-frequency variations, but…also exhibits a moderate business cycle component. Furthermore, the posterior of the mean inflation matches the survey-based trend inflation. As a result, __the hump-shaped trend in the 10-year Treasury yield (from the late 1960s to the late 1990s) reflects an increase in inflation expectations before the mid-1980s __and a secular decline afterwards. More recently, as inflation expectations stabilized, the decline in the posterior for mean output growth and, hence, the 10-year real yield, has been the main driver of the downtrend in nominal yields.”

### Business cycle fluctuations

“To address…cyclical movements…__GDP growth and the inflation rates are decomposed into…one stable and one transitory/volatile [component].__..The representative agent learns about the unconditional mean output growth and inflation rates from the stable component, and…about the stationary deviations from the mean by using the transitory/volatile component…The __posteriors for the transitory deviations from the long-run mean…have larger impacts on the short-term yield than on the long-term yield__, which implies a positive (negative) spread when the short-run beliefs are negative (positive) [giving rise to] cyclical movements in the short-term yields and, hence, in the spreads.”

“Learning about the long-run mean drives the low-frequency variations in the yields and…__learning about the short-run deviation from the mean drives the business cycle movements in the short-term yield and, hence, in the spreads__.”

“The model-implied 1-year nominal yield and the spread between 10- and 1-year nominal yields closely track their historical movements. __Out of recessions, the short-term nominal yield starts to rise when the agent begins to revise [his/her] beliefs for short-run growth and inflation upwards__ towards their long-run means…Short-run deviations are still negative and the [10-year minus 1-month yield] spread is positive…The spread starts to shrink as these short-run expectations move towards becoming positive. This pattern continues __until the late expansion stage, when the short-run growth and inflation expectations are above their long-run means__…The short-run deviations are positive now, __implying an inverted yield curve__. The agent then begins to revise short-run beliefs sharply downwards, entering a recession, and the spread switches from negative to positive.”

“Model-implied inflation expectations (posteriors) closely match…survey-based short- and long-run inflation expectations.”

### Risk and ambiguity

“The agent is assumed to have limited information about the stochastic environment and, hence, __faces both risk and ambiguity__. Here, the risk refers to the situation where there is a probability law that guides the choice. At the same time, the ambiguity-averse agent lacks the confidence to assign probabilities to all of the relevant events. Instead, they act as if they are __evaluating future prospects using a worst-case probability drawn from a set of multiple distributions__.”

“The ambiguity-averse representative agent has in mind a benchmark or reference measure of the economy’s dynamics that represents the best estimate of the stochastic process…The reference measure is the full stochastic environment…But the agent is concerned that the reference measure is misspecified and believes that the true measure is actually within a set of alternative measures that are statistically close to the reference distribution. __Using forecast dispersion to quantify the size of the ambiguity the model-implied short-rate expectations are upward-sloping under investors’ worst-case equilibrium beliefs, which generates upward-sloping nominal and real yield curves__.”

“The __ex-post predictability of excess bond returns is due to the difference between investors’ worst-case expectations and the reference measure__ – providing a rational interpretation for expectational errors.”

“The intuition of the model follows directly from the fact that interest rates reflect investors’ worst-case expectations.

- First, the
__agent chooses the lowest growth rate as their worst-case belief in equilibrium__; thus, the ambiguity about growth pushes down real yields. Given that the size of the ambiguity for growth is higher for short horizons, short-term real rates are pushed down by more than long-term real rates. Therefore, the real yield curve slopes upward. - Second, the ambiguity about the inflation rate contributes to an upward-sloping nominal yield curve but for different reasons in the two regimes.
__Pre-2000, positive inflation shocks were bad news for future growth__– the worst-case inflation was the highest rate; thus, the__ambiguity about inflation pushed up nominal yields__. Since there was more ambiguity about long-run inflation, the long-term nominal yields were pushed up by more than just the short-term nominal yields.__Post-2000, positive inflation shocks were good news for future growth__– the worst-case inflation was the lowest rate; thus,__the ambiguity about inflation pushed down the nominal yields__.”

“At each point in time, the amount of Knightian uncertainty [lack of quantifiable knowledge] the ambiguity-averse agent faces is different for the long run versus the short run, which gives rise to upward-sloping short-rate expectations under the agent’s worst-case beliefs and, hence, upward-sloping nominal and real yield curves.”

### The shift in monetary policy trade-off

“[The figure below] shows that (1) the posteriors for short-run inflation and growth deviations moved in opposite directions before the late 1990s and in the same direction afterwards, and (2) the posteriors for both short-run inflation and growth deviations were persistently negative for most of the post-2000 period…These observations imply that __short-run inflation and growth expectations drove the yields in opposite directions pre-2000, and in the same direction afterwards__.”

“Moreover, __both short-run inflation and growth expectations imply positive spreads for the past two decades__…Therefore, [the equilibrium model] generates more-frequent and deeper inverted curves for the period before the late 1990s than for the most-recent period.”

“From a monetary policy point of view, __the Federal Reserve faced a trade-off between short-run inflation and growth pre-2000 and no such trade-off afterwards__…The Fed had to raise the short-rate to either lower inflation while the output gap was negative (1970-1985) or to lower the output gap while the inflation gap was negative (1985-2000). However, starting in 2000, the U.S. economy was mostly hit by demand shocks, and the Fed could stay low for a longer period of time without facing a trade-off between the inflation and output gaps…__Hence, we observe more-frequent recessions for the period before the late 1990s and less-frequent recessions (or longer business cycles) for the period afterwards__.”