# Predictability and correlation

Today another little note that I discovered while teaching. Warning: this will only be of any interest at all to time-series finance academics. I'll try to come back with something practical soon!

Does the predictability of stock returns from variables such as the dividend yield imply that stocks are safer in the long run? The answer would seem to be yes -- price drops mean expected return rises, bringing prices back and making stocks safer in the long run. In fact, the answer is no: it is possible to see strong predctability of returns from dividend yields, yet stocks are completely uncorrelated on their own.

I've been through three versions of showing how this paradox works. In  Asset Pricing the best I could come up with was a complex factorization of the spectral density matrix in order to derive the univariate process for returns implied by the VAR. In later Ph.D. classes, I found a way to do it more simply, by seeing that returns have to follow an ARMA(1,1), and then matching coefficients. This year, I found a way to show it even more simply and intuitively. Here goes.

Background: The mean and variance of two year log returns is
\begin{eqnarray} E\left( r_{t+1}+r_{t+2}\right) &=& 2E(r) \\ \sigma ^{2}\left( r_{t+1}+r_{t+2}\right) &=& 2\sigma ^{2}(r)+2cov(r_{t+1},r_{t+2}). \end{eqnarray}
If returns are independent over time, the covariance term is zero, and both mean and variance scale linearly with horizon. The ratio of mean to variance $$E\left( r_{t+1}+r_{t+2}\right) /\sigma ^{2}\left( r_{t+1}+r_{t+2}\right)$$ which (roughly) controls portfolio allocation is then the same at any horizon. If the covariance term is negative, stocks bounce back after declines, so the variance scales less than linearly with horizon, and stocks are safer for long-run investors. (Yes, I'm mixing logs and levels like crazy here, but this is a back of the envelope blog post.)

So, our issue is, are stock returns correlated over time? Stock returns are, in fact, predictable from variables such as dividend yields. But that does not mean they are predictable at all from past returns, correlated over time, and thus any safer for long-run investors. This is the  nice little paradox.

Sum up the simplest version of stock predictability with a vector autoregression
\begin{eqnarray} r_{t+1} &=& b_{r}dp_{t}+\varepsilon _{t+1}^{r} \\ \Delta d_{t+1} &=& b_{d}dp_{t}+\varepsilon _{t+1}^{d} \\ dp_{t+1} &=& \phi dp_{t}+\varepsilon _{t+1}^{dp} \end{eqnarray}
Here $$dp$$ is the log dividend yield and $$\Delta d$$ is log dividend growth. The return coefficient is about $$b_{r}\approx 0.1$$ and the dividend growth coefficient is about zero $$b_{d}\approx 0$$, and $$\phi \approx 0.94.$$

So, low prices mean high subsequent returns, and high prices (relative to dividends) mean low subsequent returns. It would seem that stocks are indeed much safer for long-run investors, as there really is a sense that low prices are temporarily low and will revert if you can wait long enough.

More seductively, if you plot the impulse-response function to dividend growth $$\varepsilon _{t}^{d}$$ shocks and dividend yield $$\varepsilon _{t}^{dp}$$ shocks, you see the former is a cashflow shock, giving a one-time shock to returns and a random walk in prices. (Top) But the dividend yield shock is an expected return shock, yielding a completely temporary component to prices -- green line bottom right. If prices go up and dividends go up too, the movement is permanent. If prices go up and dividends do not move, the price movement is completely transitory, and perfectly safe in the long run. Stocks are like long term bonds plus iid cashflow risk.

You would think therefore that just seeing prices go up, with no information about dividends, you would have something between the two; a partially transitory movement in stock prices that is somewhat safer in the long run.

You would be wrong.

Let's figure out the correlation of returns $$cov(r_{t+1},r_{t+2})$$ implied by this little VAR. Use the VAR to write
\begin{eqnarray} r_{t+1} &=& b_{r}dp_{t}+\varepsilon _{t+1}^{r} \\ r_{t+2} &=& b_{r}\phi dp_{t}+b_{r}\varepsilon _{t+1}^{dp}+\varepsilon_{t+2}^{r} \end{eqnarray}
so
\begin{eqnarray} cov(r_{t+1},r_{t+2}) &=& cov\left[ b_{r}dp_{t}+\varepsilon _{t+1}^{r},b_{r}\left( \phi dp_{t}+\varepsilon _{t+1}^{dp}\right) +\varepsilon _{t+2}^{r}\right] \\ cov(r_{t+1},r_{t+2}) &=& b_{r}^{2}\phi \sigma ^{2}(dp_{t})+b_{r}cov(\varepsilon _{t+1}^{r},\varepsilon _{t+1}^{dp}) \end{eqnarray}
The first term $$b_{r}^{2}\phi \sigma ^{2}(dp)$$ induces a positive autocorrelation or momentum, making stocks actually riskier for long term investors. $$dp_{t}$$ moves slowly over time, so if returns $$r_{t+1}$$ are higher than usual, then returns $$r_{t+2}$$ are likely to be higher than usual as well. This term makes stocks riskier in the long run.

The second term $$b_{r}cov(\varepsilon _{t+1}^{r},\varepsilon _{t+1}^{dp})$$ is strongly negative. If there is a positive shock to expected returns, this sends current prices and hence current returns down. In this way stocks are like bonds: if yields rise, prices fall and current returns fall. This is the safer in the long-run term.

The fun part: In the standard parameterization, these effects almost exactly offset.

To show this fact, we need to find just how negative $$cov(\varepsilon _{t+1}^{r},\varepsilon _{t+1}^{dp})$$ is. Campbell and Shiller's linearized return identity,
\begin{equation} r_{t+1}\approx -\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}; \rho \approx 0.96 \end{equation}
implies the VAR coefficients and errors satisfy the identities.
\begin{eqnarray} b_{r} &\approx& 1-\rho \phi +b_{d}\approx 1-\rho \phi \\ \varepsilon _{t+1}^{r} &\approx& -\rho \varepsilon _{t+1}^{dp}+\varepsilon _{t+1}^{d} \end{eqnarray}
Now, empirically, dividend growth shocks and dividend yield shocks are just about uncorrelated, $$cov(\varepsilon _{t+1}^{d},\varepsilon _{t+1}^{dp})\approx 0$$. So, the correlation we're looking for is
\begin{equation} cov(\varepsilon _{t+1}^{r},\varepsilon _{t+1}^{dp})=-\rho \sigma ^{2}\left( \varepsilon _{t+1}^{dp}\right) . \end{equation}
Now,we have an expression for the covariance of return and dp shocks, so we can continue
\begin{eqnarray} cov(r_{t+1},r_{t+2}) &=& b_{r}^{2}\phi \sigma ^{2}(dp_{t})-b_{r}\rho \sigma ^{2}\left( \varepsilon _{t+1}^{dp}\right) \\ &=& b_{r}\left( \phi \frac{1-\rho \phi }{1-\phi ^{2}}-\rho \right) \sigma ^{2}\left( \varepsilon _{t+1}^{dp}\right) \end{eqnarray}
If we had $$\phi =\rho =0.96=0$$, we get the result, $$cov(r_{t+1},r_{t+2})=0.$$ Now, $$\phi \approx 0.94$$ is the usual estimate. But $$\phi$$ is an OLS\ estimate of a very persistent series, so biased down. $$\phi =0.96$$ is not that far off.

So, pretty much, dividend yield predictability coexists with complete lack of return autocorrelation. The momentum effect of a slow-moving forecast variable exactly offsets the bond yield effect that high prices mean low subsequent returns.

That does not mean that predictability is unimportant for long term investors. A properly done (Merton) portfolio theory isn't as simple as
\begin{equation} w= \frac{1}{\gamma }\frac{E(R^{e})}{\sigma ^{2}(R^{e})} \end{equation}.
It turns out there is a market timing demand and a hedging demand. The hedging demand is positive in our case. So, it's still possible that long-run investors should put more into stocks, even though simple Sharpe ratios are not better at long horizons. I haven't yet found a simple way to calculate hedging demands however.

Update: A correspondent tells me this example is in a set of John Campbell lecture notes he "inherited distantly."