Predictability In Addition To Correlation

Today to a greater extent than or less other picayune Federal Reserve annotation that I discovered spell teaching. Warning: this volition exclusively live on of whatsoever involvement at all to time-series finance academics. I'll endeavour to come upward dorsum amongst something practical soon!

Does the predictability of stock returns from variables such every bit the dividend yield imply that stocks are safer inwards the long run? The response would seem to live on yeah -- toll drops hateful expected provide rises, bringing prices dorsum too making stocks safer inwards the long run. In fact, the response is no: it is possible to run across potent predctability of returns from dividend yields, yet stocks are completely uncorrelated on their own.

I've been through iii versions of showing how this paradox works. In  Asset Pricing the best I could come upward up amongst was a complex factorization of the spectral density matrix inwards social club to derive the univariate procedure for returns implied past times the VAR. In subsequently Ph.D. classes, I industrial plant life a agency to practise it to a greater extent than simply, past times seeing that returns accept to follow an ARMA(1,1), too and then matching coefficients. This year, I industrial plant life a agency to exhibit it fifty-fifty to a greater extent than only too intuitively. Here goes.


Background: The hateful too variance of 2 twelvemonth 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, too both hateful too variance scale linearly amongst horizon. The ratio of hateful 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 resources allotment is too then the same at whatsoever horizon. If the covariance term is negative, stocks bounce dorsum after declines, so the variance scales less than linearly amongst horizon, too stocks are safer for long-run investors. (Yes, I'm mixing logs too levels similar crazy here, but this is a dorsum of the envelope weblog post.)

So, our number is, are stock returns correlated over time? Stock returns are, inwards fact, predictable from variables such every bit dividend yields. But that does non hateful they are predictable at all from past times returns, correlated over time, too thus whatsoever safer for long-run investors. This is the  nice picayune paradox.

Sum upward the simplest version of stock predictability amongst 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 too \(\Delta d\) is log dividend growth. The provide coefficient is virtually \(b_{r}\approx 0.1\) too the dividend increment coefficient is virtually nothing \(b_{d}\approx 0\), too \(\phi \approx 0.94.\)

So, depression prices hateful high subsequent returns, too high prices (relative to dividends) hateful depression subsequent returns. It would seem that stocks are indeed much safer for long-run investors, every bit at that topographic point truly is a feel that depression prices are temporarily depression too volition revert if you lot tin hold back long enough.

More seductively, if you lot plot the impulse-response run to dividend increment \(\varepsilon _{t}^{d}\) shocks too dividend yield \(\varepsilon _{t}^{dp}\) shocks, you lot run across the quondam is a cashflow shock, giving a one-time stupor to returns too a random walk inwards prices. (Top) But the dividend yield stupor is an expected provide shock, yielding a completely temporary constituent to prices -- greenish employment bottom right. If prices learn upward too dividends learn upward too, the crusade is permanent. If prices learn upward too dividends practise non move, the toll crusade is completely transitory, too perfectly prophylactic inwards the long run. Stocks are similar long term bonds summation iid cashflow risk.

You would intend thence that just seeing prices learn up, amongst no data virtually dividends, you lot would accept something betwixt the two; a partially transitory crusade inwards stock prices that is somewhat safer inwards the long run.

You would live on wrong.

Let's figure out the correlation of returns \(cov(r_{t+1},r_{t+2})\) implied past times this picayune 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 kickoff term \(b_{r}^{2}\phi \sigma ^{2}(dp)\) induces a positive autocorrelation or momentum, making stocks truly riskier for long term investors. \(dp_{t}\) moves slow over time, so if returns \(r_{t+1}\) are higher than usual, too then returns \(r_{t+2}\) are probable to live on higher than green every bit well. This term makes stocks riskier inwards the long run.

The 2d term \(b_{r}cov(\varepsilon _{t+1}^{r},\varepsilon _{t+1}^{dp})\) is strongly negative. If at that topographic point is a positive stupor to expected returns, this sends electrical flow prices too hence electrical flow returns down. In this agency stocks are similar bonds: if yields rise, prices autumn too electrical flow returns fall. This is the safer inwards the long-run term.

The fun part: In the measure parameterization, these effects almost just offset.

To exhibit this fact, nosotros take away to respect just how negative \(cov(\varepsilon _{t+1}^{r},\varepsilon _{t+1}^{dp})\) is. Campbell too Shiller's linearized provide 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 too 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 increment shocks too dividend yield shocks are just virtually 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 accept an aspect for the covariance of provide too dp shocks, so nosotros tin croak on
\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 nosotros had \(\phi =\rho =0.96=0\), nosotros learn the result, \(cov(r_{t+1},r_{t+2})=0.\) Now, \(\phi \approx 0.94\) is the green estimate. But \(\phi \) is an OLS\ justice of a real persistent series, so biased down. \(\phi =0.96\) is non that far off.

So, pretty much, dividend yield predictability coexists amongst consummate lack of provide autocorrelation. The momentum upshot of a slow-moving forecast variable just offsets the bond yield upshot that high prices hateful depression subsequent returns.

That does non hateful that predictability is unimportant for long term investors. Influenza A virus subtype H5N1 properly done (Merton) portfolio theory isn't every bit unproblematic every bit
\begin{equation} w= \frac{1}{\gamma }\frac{E(R^{e})}{\sigma ^{2}(R^{e})} \end{equation}.
It turns out at that topographic point is a marketplace position timing need too a hedging demand. The hedging need is positive inwards our case. So, it's soundless possible that long-run investors should pose to a greater extent than into stocks, fifty-fifty though unproblematic Sharpe ratios are non improve at long horizons. I haven't yet industrial plant life a unproblematic agency to calculate hedging demands however.

Update: Influenza A virus subtype H5N1 correspondent tells me this illustration is inwards a railroad train of John Campbell lecture notes he "inherited distantly."