Where are the sources of stock market mispricing and excess volatility?

Abstract

Using a simple dividend model, we illustrate and synthesize the sources of stock market mispricing and excess volatility based upon two hypotheses—inflation illusion and heterogeneous beliefs. Our theoretical framework posits that equity mispricing arises when investors have subjective expectations about discount rates or dividend growth rates. We then analyze the sources of equity mispricing and market excess volatility under a VAR framework. Empirically, we find that both inflation illusion and heterogeneous beliefs explain equity mispricing. However, heterogeneous beliefs play a more important role in explaining stock mispricing in the long run. We also find that heterogeneous beliefs cause excess volatility, but inflation illusion does not. Therefore, dispersion in investors’ beliefs is a better explanation of stock market mispricing than the investors’ inability to properly discount future cash flows.

Appendix

We define $x_{t}$ as a 4 × 1 vector for the four variables at time t, i.e.,⁹

$$ x_{t} = (\tilde{p}_{t} - \tilde{d}_{t} ,\Updelta d_{t}^{e} ,r_{t}^{e} ,\pi_{t} )^{'} . $$

(15)

The VAR system with one lag is specified as

$$ x_{t} = Bx_{t - 1} + \xi_{t} , $$

(16)

where B is a 4 × 4 matrix of VAR coefficients and $\xi_{t}$ is a 4 × 1 vector representing the shocks to the VAR system. Given (16), the multiperiod forecast is determined as

$$ E_{t} (x_{t + \tau } ) = B^{\tau } x_{t} . $$

(17)

Define further two vectors, $e2 = (0,1,0,0)^{'}$ and $e3 = (0,0,1,0)^{'}$, the discounted expected future excess log dividend growth rates, $\sum_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} (\Updelta d_{t + \tau }^{e} )$, are, therefore, given by

$$ \sum\limits_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} (\Updelta d_{t + \tau }^{e} ) = \sum\limits_{\tau = 1}^{\infty } {\rho^{\tau - 1} } e2^{'} B^{\tau } x_{t} = e2^{'} B(I - \rho B)^{ - 1} x_{t} , $$

(18)

where $\hat{E}_{t}$ is the conditional expectations calculated using the estimated VAR parameters. Likewise, the discounted expected future excess log returns, $\sum\nolimits_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} (r_{t + \tau }^{e} ),$ are given by

$$ \sum\limits_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} (r_{t + \tau }^{e} ) = \sum\limits_{\tau = 1}^{\infty } {\rho^{\tau - 1} } e3^{'} B^{\tau } x_{t} = e3^{'} B(I - \rho B)^{ - 1} x_{t}. $$

(19)

With the estimated VAR parameters, we can decompose the realized, demeaned log price-dividend ratio, $\tilde{p}_{t} - \tilde{d}_{t}$, into three components: (a) the discounted expected excess log dividend growth rates, $\sum_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} \left( {\Updelta d_{t + \tau }^{e} } \right);$ (b) the discounted expected excess log stock returns, $\sum_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} \left( {r_{t + \tau }^{e} } \right);$ and (c) the mispricing term, $\tilde{\varepsilon }_{t}$, as follows,

$$ \tilde{p}_{t} - \tilde{d}_{t} = \sum\limits_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} (\Updelta d_{t + \tau }^{e} ) - \sum\limits_{\tau = 1}^{\infty } {\rho^{\tau - 1} } \hat{E}_{t} (r_{t + \tau }^{e} ) + \tilde{\varepsilon }_{t}. $$

(20)

The demeaned fundamental component, $p_{t} - d_{t}$, consists of the first two terms in the right-hand side of (A6), whereas the mispricing term, $\tilde{\varepsilon }_{t}$, is the difference between the realized price-dividend ratio, $\tilde{p}_{t} - \tilde{d}_{t}$, and the fundamental component, $p_{t} - d_{t}$.