Lecture 7: Forecasting Returns

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Title

Outline

• Forecasting Regressions

• Dividend-Yield Forecasting

Risk premia across assets

The \textcolor{structure}{risk premium} of an asset, \(i\), is defined as the \textcolor{structure}{expected} excess return,

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

• \textcolor{structure}{Linear Factor Models} (LFM’s) describe how risk premia vary across assets.

• Most theories attribute the variation to difference in risks.

Example: 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), \(x_t\).

• If we specify a linear function, the statistics/numerics are much easier.

• Recall that a linear regression gives the best linear estimator of such a function \(f(x)\).

Regressions to measure 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 \textcolor{structure}{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.

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

• We similarly used regressions in LFM’s to discover variation in risk premia across assets.

Classic view

The \textcolor{structure}{classic view} says risk premia are constant over time.

• Thus, in any forecasting regression of returns, \textcolor{structure}{\(\beta=0\)}.

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

• The classic view also says price growth is a \textcolor{structure}{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 the classic view

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.

Evidence: Is the market return autocorrelated?

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

\begin{table}[h] \label{tab:uni} \caption{Auto-regression estimates for market returns, excess market returns.} \begin{center} \begin{tabular*}{.75\textwidth}{@{\extracolsep{\fill}}lrrclrr} \hline \noalign{} & \multicolumn{2}{c}{Monthly} & & & \multicolumn{2}{c}{Annual} \ \cline{2-3} \cline{6-7} \noalign{} & \(\r[\mkt]\) & \(\rx[\mkt]\) & & & \(\r[\mkt]\) & \(\rx[\mkt]\)\ \noalign{} \(b\) & 0.11 & 0.12 & & & 0.01 & 0.02\ \(t(b)\) & 2.02 & 2.05 & & & 0.09 & 0.15\ \(R^2\) & 0.01 & 0.01 & & & 0.00 & 0.00 \ \hline\hline \end{tabular*} \end{center} \end{table}

• CRSP value-weighted equity markets, 1927-2010.

• CRSP 3-month U.S. treasury bill.

• GMM standard errors.

Evidence: Is the risk-free return autocorrelated?

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

\begin{table}[h] \label{tab:uni} \caption{Auto-regression estimates for the risk-free return.} \begin{center} \begin{tabular*}{.5\textwidth}{@{\extracolsep{\fill}}lrclr} \hline \noalign{} & Monthly & & & Annual \ \noalign{} \(b\) & 0.89 & & & 0.92 \ \(t(b)\) & 30.38 & & & 13.31 \ \(R^2\) & 0.80 & & & 0.83 \ \hline\hline \end{tabular*} \end{center} \end{table}

• CRSP value-weighted equity markets, 1927-2010.

• CRSP 3-month U.S. treasury bill.

• GMM standard errors.

Conclusions from the return auto-regressions

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 data estimates suggest stock returns, particularly excess returns, are i.i.d.

• The monthly data shows some autocorrelation, but not much explanatory power.

• Furthermore, trading costs would seem to make this small predictability a novelty of no economic importance.

Other Ways to See Predictability?

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

• This reinforced classic view that returns are unpredictable—that 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?

Outline

• Forecasting Regressions

• Dividend-Yield Forecasting

Signals

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 famous examples.

Dividend-yield

The \textcolor{structure}{dividend-yield} \(\DP_t\) refers to the \textcolor{structure}{dividend-price} ratio, \(\frac{D_t}{P_t}\).

• Other common cash-flow-to-value measures include earnings-price and book-price (book-market) ratios.

• Obviously, using value-to-cashflow ratios such as dividend-price works the same.

• 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:

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

Classic view of dividend yield

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.

Evidence: Stock-market Predictability

\label{slide:retpredict}

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

\begin{table}[h] \caption{Stock Return Predictability Regressions.}

\label{tab:retpredict} \begin{center} \begin{tabular*}{.8\textwidth}{@{\extracolsep{\fill}}lrrrr} \hline\noalign{} & \multicolumn{4}{c}{Horizon} \ \cline{2-5}\noalign{} & 1 month & 1 year & 5 years\ \noalign{} \(b\) & 0.25 & 4.08 & 21.27\ \(t(b)\) & 1.01 & 2.45 & 4.43 \ \(R^2\) & .01 & .09 & .31 & \ \hline\hline \end{tabular*} \end{center} \end{table}

Regression of cumulative excess returns on dividend-price ratio.

• NYSE/AMEX/NASDAQ value-weighted equity markets.

• Monthly data, 1927-2010.

• GMM standard errors.

Interpreting the regression estimates

\underline{At a one-month horizon,}

• Slope coefficient is insignificant—statistically and economically.

• Agrees with the implications of the auto-regression.

• More evidence seemingly supportive of the classic view.

\underline{At longer horizons,}

• Coefficient is economically significant.

• At one-year, a one-point increase in dividend-price forecasts a four-point increase in returns!

Illustration of return predictability

Modern view of dividend yield

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.

• Instead, estimates show prices move the wrong way—increase expected returns even more.

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

When prices are low, we (used to / now) expect…

Long Horizons as a Way to See Predictability

Predictability of market returns by dividend-yield only seen in long horizons regressions.

• Due to persistent nature of the forecasting variable, \(\DP_t\).

• Autoregressive coefficient at a monthly frequency is about .98!

\[ \DP_tt = a + b~ \DP_t + \epsilon_{t+1} \]

Other Forecasting Variables?

Other variables seem to have similar ability to forecast returns.

• Cyclically-adjusted price-earnings ratios

• Macro-economic indicators. (Investment, consumption, etc.)

• Inflation and rates. (The ``Fed Model’’)

Statistical concerns

The dividend-price predictability is controversial.

• DP is a persistent variable, (high autocorrelation.)

• Regressions where \(x\) has high autocorrelation can be biased or mis-specified.

This is a very active area of research to find the best predicting variables and models.

References

• Campbell, John. Financial Decisions. 2016. Chapter 5.

• Cochrane, John. Asset Pricing. 2005. Chapters 12, 20.1.

• Cochrane, John. Discount Rates. 2011.

• Tsay, Ruey. Analysis of Financial Time Series. 2010. Chapters 1, 7.