Equity Premium
Yun Xu
11/14/2013
The Magnitude and Concept of the Equity Premium
in 150 Textbooks
Pablo Fernandez
Pricewaterhouse Coopers Professor of Corporate Finance
IESE Business School
July 16, 2011
Predicting the Equity Premium with Dividend Ratio
Amit Goyal • Ivo Welch
Goizueta Business School, Emory University • Yale School of
Management, Yale University
Management Science, Vol. 49, No. 5 (May, 2003), pp. 639-654
May 2003
Contents
1
INTRODUCTION OF EQUITY PREMIUM
2
2
OVERVIEW OF THE EQUITY PREMIUM PREDICTION
11
3
IN-SAMPLE FIT
14
4
OUT-OF-SAMPLE FORECAST & ALTERNATIVE SPECIFICATIONS 17
5
FURTHER RESEARCH ON DIVIDEND RATIO
24
6
CONCLUSIONS
29
7
SUMMARY
31
1
Part I
INTRODUCTION OF EQUITY
PREMIUM
2
Four Concepts of Equity Premium
• The equity premium is one of the most important parameters in
finance. The term equity premium is used to designate four different
concepts:
• Historical equity premium (HEP): historical differential return of the
stock market over treasuries.
• Expected equity premium (EEP): expected differential return of the
stock market over treasuries.
• Required equity premium (REP): incremental return of a diversified
portfolio (the market) over the risk-free rate required by an investor. It is
used for calculating the required return to equity.
• Implied equity premium (IEP): the required equity premium that arises
from assuming that the market price is correct.
3
Assumptions and recommendations of the
150 textbooks
4
Assumptions and recommendations of the
129 textbooks that assume REP=EEP
5
Review of the recommendations
• Figure 1 contains the evolution of the Required Equity Premium (REP) used or recommended by
150 books with average is 6.5%.
• Figure 2 shows that the 5-year moving average has declined from 8.4% in 1990 to 5.7% in 2008
and 2009.
6
Comments on the four concepts of the equity
premium
• The HEP: easy to calculate and is equal for all investors, provided they
use the same time frame, the same market index, the same risk-free
instrument and the same average (arithmetic or geometric).
• Different stock index chosen will make HEP different:
7
Comments on the four concepts of the equity
premium(cont’d)
• The EEP: Some authors try to find the Expected Equity Premium by
conducting surveys because investors and professors do not share
“homogeneous expectations,” do not hold the same portfolio of risky
assets and may have different assessments of the expected equity
premium.
8
Comments on the four concepts of the equity
premium(cont’d)
• The REP: is the answer to such a question: what incremental return do
investors require for investing in a diversified portfolio of shares over the
risk-free rate. Different investors and companies use different REPs.
• The IEP: is the implicit REP used in the valuation of a stock that
matches the current market value. The most widely used model to
calculate the IEP is the dividend discount model.
• P0 = d1 / (Ke - g), which implies: IEP = d1/P0 + g - RF
Where:
P0: the current price per share
Ke: the required rate of return
D1: the dividend (equity cash flow) per share expected to be received at time 1
d1: the expected long term growth rate in dividends per share
• The estimates of the IEP depend on the particular assumption made for
the expected growth. Even if market prices are correct for all investors,
there is not an IEP common for all investors: there are many pairs (IEP,
g) that accomplish equation.
9
Author’s conclusion & discussion
• The recommendations regarding the equity premium of 150 finance and
valuation textbooks published between 1979 and 2009 range from 3% to
10%. Several books use different equity premia in different pages and
most books do not distinguish among the four different concepts that
the phrase equity premium designates: Historical equity premium,
Expected equity premium, Required equity premium and Implied
equity premium.
• There is not a generally accepted equity premium point estimate and
that there is not either a common method to estimate it, even for the
HEP.
• Which equity premium is used to value companies and investment
projects by the author?
• for Europe, REPs between 3.8% and 4.3%
• for the U.S., given the yields of the T-Bonds, an additional 4%
compensates the additional risk of a diversified portfolio.
10
Part II
OVERVIEW OF THE EQUITY
PREMIUM PREDICTION
11
Abstract
• The use of aggregate dividend ratios to predict equity premium has a
long tradition in finance.
• Dividend ratios:
• Dividend yield: D(t)/P(t-1) & Dividend-price ratio: D(t)/P(t)
• A typical regression specification:
Where: D(t) = dividends paid by all stocks at the end of year t
P(t) = total stock market capitalization at the end of year t
Rm(t) = return on the stock market
Rf(t) = return on a short-term risk –free treasury bill
• Do dividend ratio have the predictive ability of equity premium?
• The paper suggests a simple, recursive residuals (out-of-sample)
graphical approach to evaluate the predictive power of popular equity
premium and stock market time-series forecasting regressions.
12
Data(value-weighted CRSP index)
Rm(t)is the log of the total return on the value-weighted stock market from year t-1 to t.
EQP(t) subtracts the equivalent log return on a three-month treasury bill.
DP(t)is the dividend-price ratio, i.e., the log of aggregate dividends D(t) divided by the
aggregate stock market value P(t). DY(t),the dividend-yield ratio, divides by P(t- 1)
instead. ΔD(t)is the change in log dividends from year t - 1 to t. All variables are in
percentages. JqBr is the Jarque-Bera test for normality. The critical level to reject
normality is 5 .99 at the 95% level, 9.21 at the 99% level. ADF is the Augmented DickeyFuller test for the absence of a unit root. An A DF value of -3.5 rejects the presence of a
unit root at the 1% level( -2.9% at the 5% level;-2 .6% at the 10% level).
13
Part III
IN-SAMPLE FIT
14
In Sample Univariate Regression
Explanation. All series are described in
Table 1. This table presents the results of
the following regression:
EQP(t) =a + β x(t- 1) + ε (t).
The first row of each regression is the
coefficient, the second line it’s OLS
t-statistic, and the third line it’s Newey West adjusted t-statistic. Data frequency
is annual; s.e. is the standard error of the
regression residuals, and N is the
number of observations. The sample
period refers to the dependent variable,
EQP(t)
Interpretation: Panel A confirms the findings in Fama and French(1988, Table 3).
Prior to the 1990s, the dividend yield (DY(t)) had significant forecasting power, the
dividend-price ratio (DP(t)) had acceptable forecasting power.
However, when the sample is extended into 2002, the in-sample predictive ability
declines. The dividend-price ratio misses conventional statistical significance levels,
while the dividend-yield ratio retains good statistical significance
15
Fama and French (1988)
16
Part IV
OUT-OF-SAMPLE FORECAST &
ALTERNATIVE SPECIFICATION
17
Out-of-Sample Forecast
• Even a sophisticated trader could not use the regression in Table 2 to
predict equity premium. What can be used is only the prevailing data.
• Dividend-ratio coefficients show remarkably different patterns.
• The beta using dividend-price ratios has high s.e., but low variability. It crosses
Zero in our sample.
• The beta using dividend yields is always positive, economically larger, but also
continuously declining.
18
Other Illustrations of the changing dividend
model coefficients
By estimation subsamples
Estimated coefficients vary widely across
subperiods, casting some doubt on the
stability on the specified model
By estimation only with
prevailing data
dividend yield underperformed the prevailing mean.
dividend-price ratio sometimes outperformed the
prevailing mean out-of-sample but statistically and
economically insignificant and non-stability.
19
Main Contribution—simple graphical
diagnostic
20
Interpretation:
• Plotting the cumulative sum-squared error form the unconditional
model minus the cumulative sum-squared error from the dividend-ratio
model.
• A positive value indicates that the dividend ratio has outperformed the
unconditional model. A positive slope indicates that the dividend ratio
had lower forecasting error than the unconditional moving average
equity premium in a given year
• Relative to the simple prevailing equity premium mean, the dividend
yield shows poor predictive performance out-of-sample in the 1960s.
Both the dividend yield and the dividend-price ratio show poor
performance in the 1990s.
• Prior to the 1990s, both dividend ratios had only two very good years,
1973 and 1974.
21
Alternative Specifications
• Numerous variations:
• Reinvest the dividends, instead of summing them;
• Change in dividend ratio, because the dividend ratio is close to
stationary;
• Use simple returns and yields, instead of log returns and yields;
• Predict on different horizons(monthly, quarterly, multiyearly);
• Reconcile definitions to match exactly those of Fama and French(1988)
including using only NYSE firms, predicting stock returns rather than
premia, and 30-year estimation window;
• Use different “fixed number of years” estimation windows;
22
Alternative Specifications (Cont’d)
• Use regression prediction standard error to normalize forecast errors;
• Try convex combination of the dividend-yield model prediction and the
unconditional prediction;
• Forecast with Stambaugh(1999) correction for high serial correlation in
the dividend yield;
• Use Earnings-price, earnings-payout ratios(Lamont 1998) or more
complex measures based on analysts’ forecast(Lee et al. 1999);
• Try a similar experiment for forecasts of the equity premium using the
risk-free rate.
• In sum, none of these variations impact the conclusion that outof-sample performance has always been poor.
23
Part V
FURTHER RESEARCH ON DIVIDEND
RATIO
24
Instrumenting the changing dividend-yield
process
• If theory is correct, changes in the dividend-yield autocorrelation and in the dividend
yield’s ability to predict changes in dividend growth could themselves imply changes in the
dividend-yield ability to predict the equity premium
•
•
•
•
Rm(t) have low correlation (recently almost no correlation with DP(T-1)).
Δ𝐷(t) used to be strongly negative correlated with DP(t-1) but it is i.i.d. today.
DP(t) had only mild autocorrelation in post-WW2 period, it is a random walk today.
Reason: prices continued to be random walk with high variance, while dividends have
remain stationary and low variance.
25
Campbell-Shiller instrumented dividendratio regressions
Assume {DP(t)} follows a stationary process with
mean 𝐷𝑃(𝑡)= d − p. Then using Taylor expansion
we can get
Define 𝛋 =
1
1+𝑒 𝐷𝑃 𝑡
, k = -log 𝛋 − (1 − 𝛋)log(1/𝛋-1)
• Take covariance with DP(t) and divide by variance of DP(t) of equation (4)
26
The forecasting ability does not improve using
Campbell and Shiller (1988) identities.
The source of poor predictive ability
• Puzzle: to WHAT dividend-price ratio really predicts?
• Rearrange the terms in Equation (4) and recurse forward,
• 𝐷𝑃 𝑡 = 𝑅𝑚 𝑡 + 1 − Δ𝐷 𝑡 + 1 + 𝛋 ∗ 𝐷𝑃 𝑡 + 1 − 𝑘
𝑖
•
= ∞
𝑖=0 𝛋 [Rm(t+1-i) - Δ𝐷(t+1+i)] + constant
• It seems that dividend-price ratio must predict the next-period stock
return, the next-period dividend growth rate, the next period dividendprice ratio.
• However, Table 6 shows:
• DP(t) is primarily forecasting itself over horizons up to about 5 years. It
is a partial forecaster of itself over 5 to 10-year horizons, and does not
forecast itself over 20-year horizons. DP(t) is primarily forecasting
future market returns(and some dividend growth) over horizons greater
than 10 years. It does not forecast market returns or dividend growths
over horizons less than 5 years.
27
Decomposition of Dividend-Yield
Components over different horizons
28
Part VI
CONCLUSIONS
29
Conclusions
Conclusions & Suggestions
• A figure that graphs comparative sum-squared model residuals out-of-sample(like
Figure 3) can act as a powerful diagnostic for equity premium and stock price
prediction.
• For simple dividend-yield models prediction equity premium, the diagnostic
suggests that good in-sample performance is no guarantee of out-of-sample
performance.
• The diagnostic further suggests that any remaining explanatory predictive ability
of the dividend ratios in the post-war period prior to the 1990s was due to two
years only, 1973 and 1974.
Explanations
• The primary source of poor predictive ability is parameter instability.
• The dividend yield has failed to forecast one-year-ahead returns or
dividend-growth rates, because it primarily forecast its own change.
• Instrumenting the model to account for the time-varying properties of the
dividend yield and dividend growth process does not aid the dividend
ratio in predicting the stock market levels.
30
Part VII
SUMMARY
31
Summary
Good points
• Fernandez documents the range of advice on equity premium in the textbooks,
which provides a reference that people can consider to apply in the practical
investment decision or valuation.
• Goyal and Welch looked at statistical techniques for predicting the risk premium
on the market, which reminds us that a market trader who just assumed that the
equity premium was “like it has been” would typically have outperformed a trader
who employed dividend-ratio forecasting regressions.
• The alternative specification section in the second paper especially gives a broader
idea on researching or verifying models which could also be useful to practice.
Questions
• The first paper gives us a general range for the equity premium but does
not show the reason for why professors use this number of equity
premium.
• The second paper tells us the weak power of the dividend ratios to predict
the equity premium with out-of-sample data or in a longer horizon but
does not provide more thoughts on which ratio might be useful to predict
the equity premium.
32
Recommendations
A Comprehensive Look at The Empirical
Performance of Equity Premium Prediction
Amit Goyal • Ivo Welch
• This paper comprehensively reexamine the performance of variables
that have been suggested by the academic literature to be good
predictors of the equity premium
• Numerous variables: Dividend ratio, Earnings Price Ratio, Dividend
payout ratio, Stock variance, Cross-sectional premium, Book-to-market
ratio, Investment-capital ratio, The consumption, wealth and income
ratio, Inflation rates, aggregate net or equity issuing activity and various
interest rates and spreads.
• Conclusion: NO models predict the equity premium well out-ofsample.(except 1973 and 1974)
33
Thank you!
34