An empirical analysis of the U.S. bond market since the 1960s emphasizes occasional abrupt regime changes, as defined by yield levels, curve slopes, and related volatility metrics. An arbitrage-free bond pricing model illustrates that bond risk premia can be decomposed into two types. One is related to continuous risk factors, traditionally summarized as the level, slope, and curvature of the yield term structure. The other type is related to regime-switching risk. Accounting for regime shift risk adds significant explanatory power to the model. Moreover, risk premia associated with regime shifts are related to the macroeconomic environment, particularly inflation and economic activity. The market price of regime shifts is strongly pro-cyclical and largely explained by these economic indicators. Investors apply a higher regime-related discount to bond values when the economy is booming.

Below are quotes from the paper. Emphasis, cursive text, and text in brackets have been added for clarity.

This post ties in with this site’s summary of the importance of macro trends, particularly the section on why macroeconomic trends matter for fixed income.

## The case for considering regime changes in bond markets

“Interest rates are subject to occasional changes over time: __periods of high interest rates and inverted yield curves are inherently different from other periods__. Aware of the possibility of sudden changes in the environment, investors will [plausibly] seek compensation for the risks associated with those changes.”

“[Based on] data on U.S. zero coupon bonds for the period Jan-1962 through Nov-2019 [we] analyze the dynamics of the yield of a 10-year government bond and the spread between the 10-year and the 3-month bonds. These variables represent the level and slope factors of the yield curve that are widely used to forecast returns and estimate risk premia…__We [document] regime changes in the level and slope of the yield curve__. The level factor is characterized by higher unconditional mean and variance during the early 1980s and during the monetary policy tightening cycle of the late 1960s.In addition, the yield curve tends to be inverted, and the volatility of the slope increases around business cycle contractions.”

[In a related model analysis] we focus on…simultaneous changes only in the long-run mean and volatility parameters…

- Specifically, the
__high mean and volatility regime__starts in the last quarter of 1979 and lasts over the entire Volcker disinflation period. It also selects observations from the mid-2003 and the 2008 recession as part of that regime, characterized by increased volatility in the level factor. Both of these events are__associated with significant changes in monetary policy uncertainty__and coincide with some form of forward guidance by the Fed… - The slope factor [reflects]
__a low unconditional mean and high volatility of the slope factor__…Changes in the regimes corresponding to the slope factor tend to be associated with__periods of rapid monetary policy__reversals, in the sense that term spreads are relatively low and highly volatile. A higher probability of this regime tends to be observed preceding all USA recessions except that in 1991.

## A model for considering regime changes in bond markets

“We develop a __statistical arbitrage-free model of bond prices that allows for priced regime shifts__. As is common in the asset pricing literature, we specify processes for the short rate and the stochastic discount factor that are both functions of a set of risk factors…At each point in time, the __economy can be in one of a finite set of possible regimes whose evolution is governed by a finite state Markov chain__. In addition, there are three other risk factors summarized by the traditional level, slope, and curvature of the yield curve, which we call the continuous risk factors.”

*N.B.: A finite state Markov chain is a model system that transitions between a finite number of states over discrete time steps. Transitions ate probabilistic, and the future state depends only on the current state, not on any previous states. Each state in a Markov chain has a specific probability of transitioning to any other state. These probabilities are summarized in a transition matrix.*

“The discrete **Markov risk factor** [in the model] serves two purposes.

- It is used to
__model nonlinearities__by assuming that the continuous risk factors evolve as first-order autoregression subject regime changes. - The parameters of the
__stochastic discount factor used to price bonds depend on the discrete Markov regime__in two ways: by directly discounting possible changes in regimes and by allowing for discrete changes in the way that fluctuations in the continuous risk factors affect bond prices.”

“We assess to what extent those discrete changes in regimes can help predict bond risk premia…The continuous risk factors and the discrete risk factor summarize the relevant information set that investors use to price bonds. Therefore, __bond prices and expected excess returns reflect two sources of risk__: the risks associated with fluctuations in the __continuous risk factors and the risk of regime shifts__…Risk premia can be decomposed into a component related to fluctuations in the continuous risk factors and a component related to regime-switching risk.”

“The stochastic discount factor __extends the usual log-linear discount factor__ with a term that adds discounting to future cash flows across different regimes given the current regime.”

“We __consider switches between two possible regimes in the level factor of the yield curve and switches between two possible regimes in the slope factor__ of the yield curve. For the level factor, we identify a regime of high unconditional mean and high volatility. As for the slope factor, we identify a period of low unconditional mean and high volatility. The combination of these two possibilities generates the Markov switching structure with four possible regimes.”

## Model findings

“Consistent with the hypothesis that occasional regime shifts are relevant risk factors, we find that __an indicator function that captures the evolution of the discrete regimes is a significant predictor of excess bond returns__ even after including the traditional level, slope, and curvature factors as additional regressors. This result implies that the regime indicator…__captures relevant information to predict expected excess returns other than that included in the usual level, slope, and curvature of the yield curve__.”

“There are nonlinear interactions between the factors that appear to be relevant correlates of bond risk premia besides those captured by the traditional level, slope, and curvature factors of the yield curve…While there are many possible definitions of risk premia, in this paper we focus on expected excess holding returns.”

“When we combine the level and slope factors to define our regime indicator, we find that this combination is a significant predictor of future excess returns, particularly so at the short end of the yield curve. In particular, __the dummy variable that simultaneously captures a high mean of the level factor and a high volatility of the slope factor predicts significantly lower excess returns__…Expected excess returns are on average lower in a regime associated with high interest rates and with an inverted and volatile slope of the yield curve.”

“[The figure below] shows, in the top panel, the evolution of the level, slope, and curvature of the yield curve and__, in the middle and bottom panels, the separation of regimes implied by the models__. The blue line in the middle panel represents the estimated smoothed probability of low slope and high volatility while the blue line in the bottom panel is the smoothed probability of high long-run mean. Likewise, the orange lines in the middle and bottom panels represent the equivalent smoothed probabilities for the restricted model without priced regime shifts.”

## How regime changes relate to macroeconomics

“We show that the component of __risk premia associated with regime shifts is related to the macroeconomic environment__. In particular, the explicit pricing of regime shifts and the nonlinearities associated with the Markov switching model generates a strong connection between bond risk premia and the macroeconomy as summarized by variables such as inflation, industrial production, and unemployment…Predictive regressions for expected excess bond returns derived from the baseline model on these macroeconomic variables yield highly significant coefficients.”

“__Bond risk premia derived from the model with priced regime shifts are highly correlated with inflation and with an indicator of economic activity__, such as the cyclical component of industrial production or unemployment. On average, expected excess returns of holding long maturity bonds are positive in normal times, in a regime with low volatility and positive slope of the yield curve, and tends to turn negative when the yield curve flattens or becomes inverted and the slope factor becomes more volatile.”

“In [the figure below] we show the estimated evolution of the estimated price of regime shifts (solid line) together with the fitted values of the regression of the expected market price of regime shifts on the cyclical component of industrial production and inflation (dashed line) and the fitted value using as correlates the cyclical component of unemployment and inflation. __The expected market price of regime shifts is strongly procyclical and a linear combination of the two macroeconomic fundamentals can explain almost a third of its variation__.”

“These results show that the expected discounting due to possible changes in regimes is highly correlated with macroeconomic fundamentals. These regressions suggest that __investors apply a higher discount due to switching risk when the economy is booming__ (as captured by the cyclical component of industrial production or unemployment) and a lower discount when inflation is higher.”

“Business cycle variations are critical to explaining excess bond returns, but they are not uncovered by [standard] factor models. In our model, the relationship between bond risk premia and macroeconomic fundamentals is revealed by the nonlinearity inherent in the price of regime switching.”

“We emphasize the importance of allowing for independent switches in the level and slope factors of the yield curve. The interaction between these independent changes in the level and slope of the yield curve imply a model with four possible regimes, which is a critical property to uncover a strong connection between bond risk premia and macroeconomic fundamentals.”