Forecasting Returns

Forecasting Returns#

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Risk Premia Across Assets#

The risk premium of asset $i$ is its expected excess return,

\[ \E\left[\rx[i]\right] \]

Linear factor models (LFMs) describe how risk premia vary across assets.
Most theories attribute the variation to differences in risk.

CAPM#

The CAPM says risk premia across different assets $i$ are:

\[ \E\left[\rx[i]\right] = \left(\fbeta[i,\mkt]\right)\lambda_\mkt \]

All risk premia are proportional (by beta) to the market risk premium.

But the above form does not condition on time.
The beta and both risk premia are estimated as stationary time series averages.

Risk Premia Over Time#

So how do risk premia change over time?

\[ \E_t\left[\rx[i]_{t+1}\right] \]

Is the expected excess return of asset $i$ always the same, no matter what period the investor considers?

\[ \E_t\left[\rx[i]_{t+1}\right] = \E\left[\rx[i]\right] \]

Or is the risk premium of an asset a function of time-varying factors, $x_t$?

\[ \E_t\left[\rx[i]_{t+1}\right] = f\left(x_{t}\right) \]

Linear Methods#

If we believe risk premia vary over time, we must specify a functional form for $f(x)$ in

\[ \E_t\left[\rx[i]_{t+1}\right] = f\left(x_{t}\right) \]

as well as specifying the factor(s), $x_t$.

A linear specification makes estimation straightforward.
Linear regression gives the best linear approximation to $f(x)$.

Conditional Expectations#

Suppose

\[ y = \alpha + \beta x + \epsilon \]

Then the expectation of $y$ conditional on $x$ is

\[ \E\left[y|\ x\right] = \alpha + \beta x \]

Thus, if $\beta\ne 0$, the conditional expectation varies as $x$ varies.

Forecasting Regressions#

A forecasting regression for returns is of the form:

\[ \rx[i]_{t+1} = \alpha + \beta x_t +\epsilon_{t+1} \]

If $\beta\ne 0$, then the conditional expectation of $\rx[i]_{t+1}$ varies over time as $x_t$ varies.

\[ \E\left[\rx[i]_{t+1}|\ x_t\right] = \alpha + \beta x_t \]

LFMs use regressions to measure variation in risk premia across assets.

Constant Risk Premia#

The classic view says risk premia are constant over time.

Thus, in any forecasting regression of returns, $\beta=0$.

\[ \rx[i]_{t+1} = \alpha + \beta x_t +\epsilon_{t+1} \]

The classic view also says price growth is a random walk with drift.

\[ \log P_{t+1} - \log P_t = \text{constant} + \epsilon_{t+1} \]

So $\E_t\left[\frac{P_{t+1}}{P_t}\right] = \text{constant}$.

Testing Return Predictability#

Test the classic view on the risk premium of the market index, $\lambda_\mkt$.

Consider using this period’s return to forecast that of next period:

\[ \rx[\mkt]_{t+1}= a +\beta \rx[\mkt]_t +\epsilon_{t+1} \]

This test of the classic view of market return predictability uses the lagged return as the predictor variable.

Market Return Autocorrelation#

\[ \r_{t+1}= a +\beta \r_t +\epsilon_{t+1} \]

	Monthly $r^m$	Monthly $\tilde{r}^m$	Annual $r^m$	Annual $\tilde{r}^m$
$b$	0.09	0.09	-0.01	-0.01
$t(b)$	1.72	1.77	-0.11	-0.05
$R^2$	0.008	0.009	0.000	0.000
Out-of-sample $R^2$	-0.005	-0.004	-0.025	-0.028

Table: Autoregression estimates for market returns and excess market returns.

CRSP value-weighted market and one-month Treasury, monthly from January 1927 to June 2026 (annual: calendar years 1927-2025), heteroskedasticity-robust standard errors.
Out-of-sample $R^2$: forecasts from 1947 on, each from the regression on all earlier data, against the mean of that earlier data.

Risk-Free Return Autocorrelation#

\[ \r[f]_{t,t+1}= a +\beta \r[f]_{t-1,t} +\epsilon_{t+1} \]

	Monthly	Annual
$b$	0.99	0.91
$t(b)$	90.74	14.63
$R^2$	0.97	0.83

Table: Autoregression estimates for the risk-free return.

One-month Treasury yield divided by 12, monthly from January 1927 to June 2026 (annual: calendar years 1927-2025), heteroskedasticity-robust standard errors.

Regression Results#

The excess market return has a regression coefficient near zero, which fits the classic view of risk premia.

High returns do not indicate particularly high or low returns going forward.
The annual estimates show little linear autocorrelation, particularly for excess returns.
The monthly estimates show some autocorrelation but little explanatory power.
Out of sample, the monthly autoregression forecasts worse than the historical mean.

Other Predictors and Horizons#

For many years, academics and practitioners have found these same results.

This reinforced the classic view that returns are unpredictable and prices are essentially a random walk.
But even if past returns do not predict future returns, how about some other predictor, $x_t$?
How about forecasting at longer horizons?

Dividend-Yield Forecasts#

Notwithstanding the “classic” view, asset managers use many signals to try to forecast returns with linear regression.

Macroeconomic signals
Asset return signals
Short-term signals / forecast horizons
Long-term signals / forecast horizons

The dividend-price ratio, also known as the dividend yield, is one of the most common examples.

Dividend Yield#

The dividend yield $\DP_t$ is the dividend-price ratio $\frac{D_t}{P_t}$.

Other common cash-flow-to-value measures include earnings-price and book-price (book-market) ratios.
The inverse value-to-cash-flow ratios contain the same information.
For an individual stock, dividends are not paid continuously, but for the market index, there is a steady stream for analysis.

Returns and the Dividend Yield#

By definition, stock returns are

\[\begin{split} \begin{aligned} R_{t+1}\equiv & \frac{P_{t+1}+D_{t+1}}{P_t}\\ R_{t+1}\equiv & \left(\frac{D_t}{P_t}\right)\frac{D_{t+1}}{D_t} + \frac{P_{t+1}}{P_t} \end{aligned} \end{split}\]

This identity holds for horizon, $t+k$, and in expectation:

\[ \begin{aligned} \E_t\left[R_{t,t+k}\right] =&~ \DP_t~ \E_t\left[\frac{D_{t+k}}{D_t}\right] + \E_t\left[\frac{P_{t+k}}{P_t}\right] \end{aligned} \]

Constant Expected Returns#

In the classic view of risk premia,

Expected returns are constant: $\E_t\left[\r_{t,t+k}\right] = \theta_r$.
Price appreciation is a random walk, $\E_t\left[\frac{P_{t+k}}{P_t}\right] = \theta_p$.

\[ \begin{aligned} \theta_r =&~ \DP_t~ \E_t\left[\frac{D_{t+k}}{D_t}\right] + \theta_p \end{aligned} \]

So under the classic view,

An increase in the dividend-yield is offset by a decrease in expected dividend growth.

Stock-Market Predictability#

\[ \rx[\mkt]_{t,t+k}= a +\beta \DP_t +\epsilon_{t+k} \]

	1 month	1 year	5 years
$b$	0.23	2.99	14.70
$t(b)$	1.78	2.59	2.97
$R^2$	.00	.04	.14

Table: Stock-return predictability regressions.

Regression of cumulative excess returns on dividend-price ratio.

CRSP value-weighted market, monthly from January 1927 to June 2026.
Newey-West standard errors with lags matching the overlap.

Regression Estimates#

At a one-month horizon,

The slope coefficient is statistically insignificant.
Agrees with the implications of the auto-regression.
More evidence seemingly supportive of the classic view.

At longer horizons,

Coefficient is economically significant.
At one year, a one-point increase in dividend-price forecasts an increase in returns several times as large.

Time-Varying Expected Returns#

The empirical evidence above shows:

Expected returns increase one-for-one with the dividend-yield.
This is not offset by dividend growth or price appreciation.
Estimates of expected price appreciation reinforce rather than offset the return forecast.