Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

Discount Rates: II

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The Background

The empirical question that John Cochrane is interested is how do returns vary over time?

  • Do returns persist over time: i.e. is there “momentum”?
  • Do returns over-react: i.e. is there “mean-reversion”?
  • Is there something like a “bubble”?

Of course, at a more fundamental level these questions are inseparable from the subject matter of asset pricing – why do prices vary?

To think about these questions in a systematic manner it is useful to have an analytical framework. For the most part we can address the questions using a simple forecasting regression equation:

R_{t + 1} = a + bx_t + \epsilon_{t + 1}

or equivalently,

E_t[R_{t + 1}] = a + bx_t

where x_t is a signal that we believe contains information about future returns.

The Classical Efficient Markets View

The efficient markets hypothesis tells us that such signals, even if they temporarily exist, cannot last in a competitive market. As everybody seek to take advantage of the signal, it would lose all its predictability.

Of course, efficient markets is not just an a priori hypothesis, but has been well supported by empirical evidence in the last three decades. As an illustration consider for signal the lagged return r_t. A momentum (mean-reversion) trader we would expect the coefficient b > 0 (< 0). So, what does the data tell us? Here are the results.

So, not only is the coefficient b numerically close to 0, but also statistically. That is, stock returns do follow a “random walk”. In contrast, for T-bill (just because a series is ‘labelled’ return doesn’t mean it has to be unpredictable), returns are clearly persistent, with a highly significant t-statistic. While in stock markets a signal can be exploited by borrowing/lending in the bond market – and thus providing a ‘pure’ exposure to risk – one can’t do so in the bond market. Borrowing at high interest rates when the yields are high to take advantage of such high yields doesn’t take one very far. It only makes people change their inter-temporal consumption patterns (which isn’t very easy.) So, for studying prices / discount rates it make sense to work with excess returns or risk premium.

Long-Horizon Regressions

While it is true that the “random walk” view of the world holds pretty well for stock (excess) returns at short horizons (most relevant for traders), stock returns do seem to be predictable at long horizons. The attached chart here present results for excess returns forecasting regression on dividend yield. It’s clear that high dividend yields forecast subsequent excess returns. Although the R^2 for the 1-year horizon at 0.09 doesn’t seem very high, because dividend yield is a highly persistent signal and forecast returns in the same direction for many years into the future, forecasts ‘build up’ over time and at long horizons we see a higher R^2. So the two R^2 are mathematically equivalent. Since this an important point let’s consider the following simple AR(1) model for dividend yields (we still call it x_t):

r_{t + 1} = b x_t + \epsilon_{t + 1}

x_{t + 1} = \rho x_t + \delta_{t + 1}

Just iterating the returns forward and adding them gives us the following expression for long horizon returns:

r_{t + 1} + r_{t + 2} + r_{t + 3} + ... = b (1 + \rho + \rho^2...) x_t + b \rho \delta_{t+ 1} + ...

That is, as the horizon increases, because \rho is positive and numerically close to 1, the coefficient in long-horizon regressions systematically increases. With a little bit of more messy algebra the same can be shown for the R^2 as well as the t-statistic (Campbell, Lo and Mackinlay’s book has indicative proofs). For a pictorial representation, take a look at this picture.

There is considerable empirical evidence supporting this predictability at long-horizons across asset classes.

Cochrane’s Predictability Story: Future Dividend Growth vs. Future Excess Returns

So does it mean that efficient markets view is “wrong”? Well, not quite, says Cochrane. Nothing in efficient markets view requires that expected returns be constant. They can very well be time varying. So if we look at the ex-ante version of our forecasting regression again:

E_t[R_{t + 1}] = a + bx_t

it just says that if returns are at all predictable, expected returns will be time varying. So, the important question is whether these forecasts correspond to variation in expected risk premium or expected future cash flows or “bubbles”.

So far we have seen that excess returns do seem predictable at long horizons. But we also have to know about expected future cash flows – i.e. the expected dividend growth rate. What does the data say about that? The results from one of other Cochrane’s paper are here. Clearly, ex-post dividend growth is not fore-castable given today’s dividend yield. Not only are the coefficients numerically close to 0 but also statistically so.

This goes against the grain of what an efficient markets guy would believe. So, if dividend yields are high today (because prices are low relative to dividends), then efficient markets would make one believe that future dividends would be low – but that that doesn’t seem to be the case. It is the expected risk premium that is expected to be high in the future, as prices rebound. Here is a graphical representation of what Cochrane describes as  changing view of the world.

One Step Deeper

It seems to be the case that predictability in asset returns at long horizons is there because dividend growth is not forecast-able, and that gets reflected in today’s dividend yield. So what is happening here? To see this let’s take a look at the linearized present value relationship (proof of which is outlined in the Appendix of Cochrane’s paper):

r_{t + 1} = \kappa - \rho dp_{t + 1} + \rho dp_{t} + \Delta d_{t + 1}

where \kappa and \rho are constants and function of the ‘average’ dividend yield (around which the linearization is done). dp_t and r_t are both the log version of returns and the dividend price ratio.

After some algebra, we can rewrite above as:

p_t = \kappa + \rho p_{t + 1} + (1 - \rho) d_{t + 1} - r_{t + 1}

If the above equation holds at time t, so it would at all times. We can then iterate it forward till infinity and we end with an expression for price dividend yield as:

\displaystyle p_t - d_t = \frac{\kappa}{1 - \rho} +\sum_{j = 1}\rho^{j - 1} (\Delta d_{t + j} - r_{t + j}) + \lim_{j\to\infty}\rho^j (p_{t + j} - d_{t + j})

Since the above equation holds ex-post (in realized terms), it must also hold ex-ante. So if we take expectations on both sides conditional on information at time t (and ignoring the constant term), we have:

\displaystyle p_t - d_t = \sum_{j = 1}\rho^{j - 1} (E_{t}[\Delta d_{t + j}] - E_{t}[r_{t + j}]) + E_{t}[\lim_{j\to\infty}\rho^j (p_{t + j} - d_{t + j})]

i.e. price-dividend ratio can only move if either there is news of future expected dividend growth (E_{t}[\Delta d_{t + j}]), or future expected discount rates (E_{t}[r_{t + j}]) or a rational “bubble” (E_{t}[\lim_{j\to\infty}\rho^j (p_{t + j} - d_{t + j})]). So, if dividend growth rates are unpredictable (as we saw earlier), then price-dividend ratio can only move if discount rates are predictable (which seems to be the case as we saw earlier). Note that the summation to infinity only reflects that we are looking at long-horizon returns and cash-flows (\rho in the above formula is very close to 1).

As it turns out this above price-dividend relationship (also known as Campbell-Shiller approximation) is very useful to quantitatively address issues related to time-series predictability of asset returns.

On the Nature of the Forecasting Regression

One thing to note is that while our economic explanation of price dividend ratio is how it reflects the expectation of future dividend growth and / or discount rates, our regression equation runs the other way round. That is, our ex-ante description of price dividend yield ratio is:

\displaystyle p_t - d_t = \sum_{j = 1}\rho^{j - 1} (E_{t}[\Delta d_{t + j}] - E_{t}[r_{t + j}]) + E_{t}[\lim_{j\to\infty}\rho^j (p_{t + j} - d_{t + j})]

While our regression equation for (returns) is:

r_{t + 1} = a + b(p_t - d_t) + \epsilon_{t + 1}

So when we run regression of returns of dividend growth/returns on dividend-yield ratio, what we learn is that dividend-yield ratio is moving around, and not the other way round. The statistical justification behind this regression is that the RHS variable is uncorrelated with the error term. This is the reason regressions are run this way.

Economically also it makes sense. Think of it this way. If traders believe that future dividend growth rates would be high, their actions impound that information in today’s price. And even though an individual trader may be wrong, the average trader must be right. So, if price variation comes news about dividend growth (discount rates), then on average dividend growth (discount rates) should be higher (lower) after the price rise. This is what our forecasting regression equation does ex-post (i.e. in realized terms).

Volatility and Predictability

One of Robert Shiller’s famous arguments against efficient market guys is that dividend growth just doesn’t covary enough (with price-dividend yields) to justify the volatility of asset prices. Our Campbell-Shiller approximation allows us to address this question. With a little bit of rearrangement of terms it can be shown that the Campbell-Shiller approximation implies:

\displaystyle var_{t}[p_t - d_{t}] = cov_{t}[p_t - d_t, \sum_{j = 1}\rho^{j - 1} \Delta d_{t + j}] -

cov_{t}[p_t - d_t, \sum_{j = 1}\rho^{j - 1} r_{t + j}] + \rho^j cov_{t}[p_t - d_t, p_{t + j} - d_{t + j}]

That is, variance of price dividend yield should be able to be totally explained by how (ex-post) future dividend growth, future discount rates and possibility of bubbles covary with the price-dividend yield. It also a very testable ‘model’ to study predictability of price-dividend yields. Just divide both sides by var_t[p_t - d_{t}], and we end up with an equation that neatly links the coefficients on price dividend yields in dividend growth / discount rates / “bubbles” regression as follows:

1 \approx \beta^k_r - \beta^k_{\Delta d} + \beta^k_{dp}

where dp_t represents the dividend-yield in log terms, and \beta the relevant regression coefficients . (The reason that the identity above is approximate and not exact is that in regression we are forcing ourselves to a finite long-horizon of length k.) The coefficient on returns is near 1 and all rest statistically and numerically 0. Here are the regression results, and here a pictorial representation.

To quote from Cochrane, the bottom line is:

…that all price-dividend ratio volatility corresponds to variation in expected returns. None corresponds to variation in expected dividend growth, and none to “rational bubbles”.

This pretty much covers the main ideas in time series predictability portion of Cochrane’s paper. He ends by pointing out that phenomena is pervasive across asset classes, and that multivariate regressions only accentuate the contribution of long-run expected returns to price-dividend variation. This, of course, is Cochrane’s story and one can point to issues. For example, all his results seem to depend on the Campbell-Shiller approximation of the present value formula. Also, are long-horizon econometric results really reliable?


Written by Vineet

July 9, 2011 at 11:56 am

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