Prices and Returns

Warning: this will only be interesting to academic finance people.

One of the fun things about teaching is that it forces me to look back at old ideas and refine them. Last week, I needed a problem set for my MBA class. It occurred to me, why not have them do for returns what Shiller did for dividends?

Here it is

At each date \( t \) I plot the return and final terms of the Campbell-Shiller identity

\( p_t - d_t = \sum_{\tau=t}^T \rho^{\tau-t-1} \Delta d_{\tau} - \sum_{\tau=t}^T \rho^{\tau-t-1} r_{\tau} + \rho^{T-\tau} \left( p_T-d_T \right) \)

where p = log price, d = log dividend, r = return, \(\rho = 0.96\)

In words, plot at each date the actual price-dividend ratio, the corresponding ex-post dividends, the corresponding ex-post return, and ex-post terminal price. (There is no expectation on the right hand side.) Shiller plots the price, dividend and terminal price term, (see here). I'm just adding the return term.

What does this mean? Shiller's plots contrast actual prices with what prices would be if clairvoyant investors knew what actual dividends would be and discounted at a constant rate. It's a total bust. Here, we're looking at, what would prices be if clairvoyant investors knew what actual returns were going to be, but thought dividends would never change. As you can see, actual prices almost exactly mirror these "ex-post rationally discounted" prices!

A second, deeper, meaning. It is more conventional to make this decomposition using expected values,

\( p_t - d_t = E_t \left[ \sum_{\tau=t}^T \rho^{\tau-t-1} \Delta d_{\tau} - \sum_{\tau=t}^T \rho^{\tau-t-1} r_{\tau} + \rho^{T-\tau} \left( p_T-d_T \right) \right] \)

For \(E_t\) we use regression based forecasts, for example by running long-run returns on dividend yields and a vector of other variables. If you use dividend yields as the forecasting variable then each term is just a number times dividend yield at time t. To be specific, if you run

\( \sum_{j=1}^k \rho^{j-1} r_{t+j} = a + b^r \times (p_t-d_t)+\varepsilon^r \)

Then the three terms are

\( p_t - d_t = b^d \times (p_t-d_t) - b^r \times (p_t-d_t) + b^{pd} \times (p_t-d_t) \)

Since \(b^r \approx -1 \) (i.e. the regression coefficient of long run returns on d-p has a coefficient of about +1) \( b^d \approx 0 , \ b^{pd}\approx 0 \) we see that the discount rate term accounts for all price volatility. Plotting the terms is pretty boring: 

Yes, there are separate red and blue lines. Price-dividend ratios do not forecast dividend growth so the green line is flat. Price dividend ratios do forecast returns, just enough to account for the volatility of prices.

Now, to the point:  What if we add more variables to forecast returns and dividend growth? Investors surely use lots of information.  That would surely change our understanding of the sources of price volatility, no?  In "Discount rates" I tried Lettau and Ludvigson's cay variable. It did a great job of forecasting short run returns. But it decays quickly, and doesn't change this long run picture much at all.

Ok, but surely there are other variables out there that can forecast returns and dividend growth, that could upend the whole picture, no?

My top picture answers that question. Even if you can perfectly forecast returns, you will not substantially change the decomposition of price-dividend ratio volatility. The ex-post values are a sort of upper bound for how much things can ever change, no matter how much more information we stick in the VAR. And the answer is, no matter how we change short-run return forecasts, no matter what information set we use, the decomposition of price volatility will still say the vast majority of price-dividend ratio variation comes from expected returns. (And, likewise, Shiller's plots for ex-post dividends say that no matter how many variables you try, dividend forecasts will not explain much price-dividend ratio volatility.)

You may either pity or admire my MBAs who put up with this sort of thing on a weekly basis. If you want more details or documentation, it's problem set 3 here. Now, back to writing Problem set 5.