ASSET PRICING WITH RETURN EXTRAPOLATION
2.4 Comparative Statics
In this section, we examine the sensitivity of the model’s implications to changes in parameter values. We focus on parameters that either have dispersed estimates in the literature or cannot be directly observed from the data. Specifically, for the utility parameters, we study how changes in γ, the coefficient of relative risk aversion, andψ, the reciprocal of the elasticity of intertemporal substitution, affect the equity premium and the volatility of stock market returns. For the belief parameters, we look at how changes inθ affect the equity premium, the volatility of stock market returns, the price-dividend ratio, and the average interest rate. We also examine how changes in θ, χ, and λ affect return predictability and the persistence of the price-dividend ratio.
Utility parameters
Figure B.5 plots the long-run average of the equity premium and the volatility of stock market returns, each as a function of γ or ψ. The coefficient of relative
21See Xiong (2013) for more discussion of this amplification mechanism.
risk aversion is positively related to the equity premium but negatively related to the volatility of returns. Lower risk aversion naturally leads the agent to require a lower equity premium for risk compensation; reducingγ from 10 to 5, the model still explains 75% of the observed equity premium. At the same time, lower risk aversion strengthens the feedback loop described earlier since it increases the agent’s demand for risky assets. Therefore, it leads to higher return volatility.
[Place Figure B.5 about here]
Within the examined range, changes in the elasticity of intertemporal substitution do not significantly affect the average equity premium and the average volatility of returns. As a result, our model implications are quantitatively robust to changes in ψ. We provide a detailed explanation of this observation in the next section when we compare our model with the model of Bansal and Yaron (2004).
Belief parameters
Belief Parameters.—Figure B.6a plots the long-run average of the equity premium, the volatility of stock market returns, the price-dividend ratio, and the interest rate, each as a function of θ, the parameter that measures the extent to which the agent is behavioral. Setting θ to zero gives the agent fully rational beliefs. In this case, the equity premium is 0.23%; the return volatility equals the fundamental volatility of 11%; the price-dividend ratio stays constant at 135; and the interest rate stays at 2.35%. With θ = 0, the model fails to match the long-run properties of the stock market. Increasingθ from zero allows the feedback loop described above to emerge, which generates excess volatility, pushes up the perceived equity premium, and significantly reduces the price-dividend ratio. At the same time, a higherθ—
and the agent’s more extrapolative beliefs about stock market returns—leads the agent to perceive a higher persistence in dividend growth, which, under Epstein-Zin preferences, is significantly priced, causing the perceived equity premium to further rise. Finally, a higherθ leads to a true equity premium that is significantly higher than the perceived one.
[Place Figure B.6a about here]
Overall, a 1% increase in θ leads to a 0.09% increase in the equity premium and a 0.29% increase in the volatility of returns. On the other hand, the effect of θ on the interest rate is small. A higher θ increases the extrapolation bias in the agent’s
beliefs about stock market returns, but does not significantly affect her beliefs about consumption growth, which determine the equilibrium interest rate.
Figure B.6b examines how changes inθ, χ, andλaffect the predictability of stock market returns and the persistence of the price-dividend ratio. Here the predictability of returns is measured by the slope coefficient in a regression of the next year’s log excess return on the current log price-dividend ratio; the persistence of the price- dividend ratio is measured by the one-year autocorrelation of log price-dividend ratios.
[Place Figure B.6b about here]
Figure B.6b shows that higher values ofθ, χ, andλlead to stronger predictability of returns and a lower persistence of the price-dividend ratio. Higher values of χand λ—that is, higher perceived transition intensities between the high- and low-mean price growth regimes—suggest that the agent focuses on a shorter history of past return realizations when forming beliefs about future returns. A higher value ofθ has a similar implication: it suggests that the agent exhibits a stronger extrapolation bias, which means that the agent deviates more from a rational agent whose beliefs depend on both recent and distant past returns. In other words, with a higherθ, the agent relies more heavily on recent past returns when forming beliefs about future returns. Therefore, higher values ofθ, χ, andλall lead to a stronger degree of mean reversion in sentiment, which in turn gives rise to stronger predictability of returns and a lower persistence of the price-dividend ratio.
The comparative statics results in Figure B.6b are consistent with recent empirical findings. At the aggregate level, Cassella and Gulen (2015) find that, during periods when investors’ expectations about future returns depend on both recent and distant past returns, the price-dividend ratio does not strongly predict the next year’s return.
Conversely, during periods when investors’ expectations depend primarily on recent past returns, the price-dividend ratio strongly predicts the next year’s return. In our model, higher values of θ, χ, and λ lead to a higher φ, and therefore the agent’s expectations about future returns depend more heavily on recent past returns. In the meantime, they also lead to stronger return predictability. Overall, the model implies that, when the agent forms beliefs based on a short history of past returns, the predictability of returns is strong. To give an example, increasingθ from 0.05 to 0.5 changes φ from 0.37 to 0.43. At the same time, it changes β1, the slope coefficient for a regression of the next year’s log excess return on the current log
price-dividend ratio, from−0.32 to−0.72; the corresponding R-squared increases from 0.001 to 0.13. In the cross-section, Da et al. (2017) show that stocks associated with a larger extrapolation bias—beliefs of forecasters on these stocks depend more strongly on recent past returns—exhibit stronger return reversals. Applying our model to individual stocks gives the same prediction.