# Are returns predictable, Campbell and Shiller (1988)

Following from the thread,

Drivers of equity returns: dividend yield, change in P/E and dividend (or earnings) growth

1) Why are returns predictable from this, is there a reason?

2) Can we expect predictability in financial markets?

## 1 Answer

Let me start with a simple example. Suppose you have a dividend strip that pays an unknown dividend $D_T$. The gross return (something like 1.05 and NOT 5%!) on this security is, by definition, $$R_{t\to T} = \frac{D_T}{P_t}$$ where $P_t$ is the current price of this security. If we use lowercase letters to denote logs (i.e., $\log D_T = d_T$ etc..) we can rewrite the previous relationship as $$p_t = d_T - r_{t \to T} \tag{1}\label{eq:1}$$ If we have a time series of log prices, returns and dividends we can take variances of both sides to obtain $$Var(p_T) = Cov(p_T, d_T - r_{t \to T}) = Cov(p_T, d_T) - Cov(p_t,r_{t \to T})$$ Finally we can divide both sides by $Var(p_T)$ to obtain $$1 = \frac{Cov(p_T, d_T)}{Var(p_T)} - \frac{Cov(p_t,r_{t \to T})}{Var(p_T)} = \beta_D - \beta_R \tag{2}\label{eq:2}$$ where $\beta_D$ $\beta_R$ are the slope coefficients you can obtain by running the following predictive time-series regressions $$d_{T} = \alpha + \beta_D p_t + \epsilon_T$$ $$r_{t\to T} = \alpha + \beta_R p_t + \epsilon_T$$

Notice that \eqref{eq:2} tells you something very important: the price of the dividend strip predicts either dividends ($\beta_D \neq 0$) or returns ($\beta_R \neq 0$) or a combination of both! You can see that, in the case of a dividend strip, you should expect to be able to predict either dividends or returns.

A stock is nothing more than a collection of dividend strips and following the steps in Drivers of equity returns: dividend yield, change in P/E and dividend (or earnings) growth you can have an approximation that is analogous to \eqref{eq:1}, i.e. :

$$pd_t = \sum \rho^j \Delta d_{t+j+1} - \sum \rho^j r_{t+j+1} \tag{3}\label{eq:3}$$

where $pd_t = \log \frac{P_t}{D_t}$, $\Delta d_{t+j+1} = \log \frac{D_{t+j+1}}{D_{t+j}}$ and $r_{t+j+1} = \log R_{t+j \to t+j+1}$. Similarly to before, you can take the variance of the log price-dividend ratio in \eqref{eq:3} to obtain $$1 = \frac{Cov(pd_t,\sum \rho^j \Delta d_{t+j+1})}{Var(pd_t)} - \frac{Cov(pd_t,\sum \rho^j r_{t+j+1})}{Var(pd_t)}$$

As before, we can say that the price-dividen ratio $pd_t$ must either predict future realized dividend growth ($\sum \rho^j \Delta d_{t+j+1}$) or future log returns ($\sum \rho^j r_{t+j+1}$) or a combination of the two.

Coming back to your question 2) if the price dividend ratio is not constant we should expect mechanically to have predictability or return, dividends or both.

Empirically, price-dividend ratios are not constant and they seem to predict mostly future returns, even though the evidence is not as straightforward as it seemed at first.