Long-run regression models using the trailing earnings over price ratio to predict future returns suggested by Campbell and Shiller (1988, 2001) work quite well. However, in this note we show that this variable might result in a downward biased proxy for expected future returns. Instead we suggest using a moving average of the log of 1 plus the earnings price ratio when forecasting long-run returns. The empirical results for the S&P 500 show the superiority of our approach to existing ones.Earnings yield, Stock Return, Forecasting

Bjorn E. Tuypens

Charlotte S. Hansen

English

Research Papers in Economics

  1
Proxying for Expected Returns with Price Earnings 
Ratios
∗ 
 
 
Charlotte S. Hansen
† 
 
Bjorn E. Tuypens
‡ 
Zicklin School of Business    Oak Hill Platinum Partners, L.L.C. 
Baruch College, New York, NY    Rye Brook, NY 
 
 
 
 
 
October 24, 2004 
 
 
 
 
Abstract 
Long-run regression models using the trailing earnings over price ratio to predict future 
returns suggested by Campbell and Shiller (1988, 2001) work quite well. However, in 
this note we show that this variable might result in a downward biased proxy for expected 
future returns. Instead we suggest using a moving average of the log of 1 plus the 
earnings price ratio when forecasting long-run returns. The empirical results for the S&P 
500 show the superiority of our approach to existing ones.  
 
Keywords: Earnings yield, Stock Return, Forecasting 
JEL classification: C52, G12, G14 
 
 
 
 
1. Introduction 
 
There is a vast body of literature on variables that predict expected stock returns; we will 
refer to Ferson, Sarkissian, and Simin (2003) for a recent overview of this literature. In 
this note we discuss the difference between using a moving average of earnings yields 
                                                 
∗ Charlotte Hansen gratefully acknowledges the financial support provided by the Eugene Lang Junior 
Faculty Research Fellowship and from the PSC-CUNY Research Foundation of the City University of New 
York.  
† Corresponding author: Charlotte S. Hansen, Mailing address: Department of Economics and Finance, 
Zicklin School of Business, Baruch College, One Baruch Way, Box B10-225, New York, NY 10010, Tel: 
(646) 312-3519, Fax: (646) 312-3451, Email: charlotte_hansen@baruch.cuny.edu.  
‡ The views, thoughts and opinions expressed in this paper are those of the authors in their individual 
capacity and should not in any way be attributed to Oak Hill Platinum Partners L.L.C. or to Bjorn Tuypens 
as a representative, officer, or employee of Oak Hill Platinum Partners L.L.C.    2
and a ratio of a moving average of earnings relative to the current price to predict long-
run stock returns. The latter ratio involves averaging earnings over a period from 4 to 30 
years to smooth business cycle fluctuations. This smoothed earnings series is then 
divided by the most recent price level, which is assumed to contain all relevant 
information to predict future returns. 
 
We provide two arguments for why one should use a moving average of the earnings 
yield. First, we present a simple model to interpret the earnings yield as a proxy for 
expected returns, and argue that a moving average of earnings over the current price 
results in a biased estimation of expected returns. Second, the q-period moving average 
of earnings yields can be interpreted as an expectation of future returns formed over q 
periods, where q goes to infinity as the sample size increases. Under the usual ergodicity 
assumption this quantity converges to the required return. Finally, we present empirical 
results for the S&P 500 index returns over the period 1920-2003. Using the Fair and 
Shiller (1990) approach we show that the moving average of earnings yields subsumes all 
information contained in the smoothed earnings over the current price ratio. 
 
 
 
2. The model 
 
We consider a Gordon (1959) constant growth firm, with its stock price  t P  given by  
 
 
(1 ) t
t
Dg
P
R g
+
=
−
, 
 
where  t D  is the dividend per share, g is the growth rate of the dividends, and R  is the 
required rate of return. 
 
Let  t E  denote earnings per share. From Modigliani and Miller (1958), we can show that, 
if the internal rate of return equals the required rate of return, the earnings price ratio 
equals the required return, i.e.  
 
                                                                   ,
t
t
E
R
P
=                   (1) 
 
with  (1 ) , gR π =− where π  is the payout ratio defined by  . 1 tt DE π = +  This is the basis for 
using the earnings yield to estimate  or proxy for future returns. Accordingly, the 
logarithm of 1 plus the earning price ratio, ln(1+E/P), is a proxy for the log return 
ln(1 ). rR =+
1 The question in the literature on predicting stock returns is how to form an 
estimator for future returns. In the next section we consider two methods, currently used 
in practice, to proxy for expected returns using earnings price ratios. 
                                                 
1 We will use the term log earnings yield for the term ln(1+E/P) throughout this note.    3
 
 
3. Two proxies for expected returns 
 
3.1 Trailing earnings over price 
 
The first proxy, 
Th
t F = () ( )
1
1 ln 1 /
h
thj t h j EP −+ = + ∑ , involves an average of the past h 
earnings divided by the most recent price. This corresponds to forming the expectation 
based on only one observation, namely the smoothed earnings over the current price. The 
argument for this proxy is that earnings fluctuate over the business cycle and therefore 
need to be smoothed, but only information in the current price is relevant for future 
returns.  
 
We now show that the proxy,
Th
t F , is a downward biased estimate of expected returns. 
From Gordon's growth model it follows that prices grow at rate g, 
 
1
1
(1 )
(1 ) ,
t
tt
gD
Pg P
Rg
+
+
+
== +
−
 
and by (1), earnings also grow at a rate g since 
 
11 (1 ) (1 ) tt t t EP R g P R g E ++ == + = +. 
 
Consequently the ratio of the h-period moving average of earnings and the current price  
 
1
1 1
h
tj j
t
hE
P
−
+ − = ∑
   =  
1
1 1
1
(1 )
(1 )
h j
th j
h
th
Eh g
Pg
−
+− =
+−
+
+
∑
 
 
                        =     
1
1
1
(1 )
h
hj
j
Rh
g
−
−
= + ∑  
 
                                  =     
1 (1 ) 1
1
(1 )
h
g
Rh
gg
−  +
−  + 
 
 
                                                              →   0  as  h  →  ∞.                                            (2) 
 
Slutsky's Theorem implies that 
1
1 1 ln 1
h
tj j
t
hE Th
P F
−
+− =  ∑ =+ 

→ 0 as h→ ∞.  Hence this is a 
downward biased estimator of the required return.  
To assess the importance of this result, assume that companies pay out 50% of their 
earnings as dividends. The historical real rate of return on the S&P 500 is 7%. Then the 
growth rate, g, equals    4
(1 ) 0.5 0.07 0.35 gR π =− = ⋅ =  
    
or a growth rate of 3.5%. Taking a moving average of earnings over 30 years, but not of 
prices, leads to an earnings yield (or expected return) of  
 
1
1 1
h
tj j
t
hE
P
−
+ − = ∑
   =  
1 (1 ) 1
1
(1 )
h
g
Rh
gg
−  +
−  + 
 
                                =   ()
()
1
30
1 0.035 1
0.07 30 1 0.04417
0.035 1 0.035
−  +
⋅⋅ − = 
 + 
 
 
or only 4.417% as opposed to the historical 7%.  
 
 
3.2  Moving average of the log earnings yield 
 
The second proxy, 
MA
t F , is constructed as the average of the past q log earnings price 
ratios,  ()
1
1ln 1
tqj
tqj
q E
qP j
−+
−+ = + ∑ . In effect, this is a time-series average of the log earnings yield 
over the past q periods, which under ergodicity converges to the expected value of the log 
earnings yield as q →∞. The more observations we use to form the expectation the 
closer it comes to its true value, so this should give an advantage to the proxy 
MA F . Note 
that averaging the log earnings yield does not give rise to a downward biased estimate. 
 
 
3.3 Comparison 
 
The realized log gross return on a portfolio held for one period is  ()
1 ln
tt
t
PD
t P r
−
+ =  and the 
realized gross return over a period of q years is given by 
1 ()
q
tq tj j rq r + + = =∑ . Figure 1 
depicts the average realized real return measured over 10 periods,  1 () tq q rq + , and the three 
proxies: ,
MA F  
10,
T F  and 
30 T F  for the S&P 500 index from 1920-2003.
2 All variables are 
adjusted for inflation. It turns out that 
30 T F  is smaller than 
10 T F  for almost every period 
from 1920 through 2003 with the exception of the periods 1935-1939 and 1948-1951, 
where there is hardly any difference between the two forecasts. This is what we would 
expect given result (2) from the previous section saying that 
Th F  decreases as h 
increases. However, it is not clear-cut from the graphs which variable works better when 
proxying for expected returns. Further analysis is needed. 
 
 
 
                                                 
2 Data is from Prof. Shiller's web page: www.robertshiller.com.   5
[Insert figure 1 about here] 
 
 
 
4. Forecasting long-run returns 
 
We use annual data for the S&P 500 index covering the period 1920-2003. Table 1 
reports the descriptive statistics of the realized return and the different proxies. 
 
 
Table 1: Descriptive Statistics 
 
   () rq 
MA F  
10 T F  
30 T F  
Mean          0.074  0.077  0.070  0.062 
Median        0.076  0.077  0.064  0.060 
Std. Dev.     0.058  0.019  0.031  0.028 
ρ   0.847 0.944 0.720 0.747 
Descriptive statistics of annual realized real  1 () () q rq rq =  for q=10 and the proxies 
MA F , 
10 T F , and 
30 T F . All variables are adjusted for inflation. We let ρ  denote the first order autocorrelation coefficient. 
The sample period for the S&P 500 index covers 1920 to 2003.  
 
 
The proxies have a smaller standard deviation than the realized return, which is consistent 
with the notion that the proxies are expectations. Now consider the means of the proxies. 
The average value for 
MA F  is 7.7%, which is slightly larger than the average realized 
return of 7.4%. However, the median of 
MA F  and the realized return come quite close 
with values of 7.7% and 7.6%, respectively. Both of the proxies 
10 T F  and 
30 T F  are 
smaller on average than the realized return, and even  more so when earnings are 
smoothed over a longer period, confirming the result in equation (2).   
 
 
Table 2: Correlation Matrix 
 
   () rq 
MA F  
10 T F  
30 T F  
() rq  1  0.761 0.609 0.599 
MA F    1  0.493  0.457 
10 T F     1  0.798 
30 T F      1  
Correlation matrix between the annual realized real  1 () () q rq rq =  for q=10 and the proxies 
MA F , 
10 T F , 
and 
30 T F . All variables are adjusted for inflation. The sample period for the S&P 500 index covers 1920 
to 2003.  
 
   6
The correlation matrix between the realized return and the proxies is reported in Table 2. 
The proxy 
MA F  has the highest correlation with the future realized return with a 
correlation of 0.76. Each of the proxies 
10 T F  and 
30 T F  has a correlation of about 0.6 with 
the future realized return. So, from this preliminary analysis 
MA F  seems to be a better 
forecast of future realized return. 
 
We can evaluate the forecasts for expected return along the lines of Fair and Shiller  
(1990) by regressing the actual real return on the three forecasts: 
 
10 30
10 30 () .
MA T T
tq M A t T t T t tq rq F F F u αβ β β ++ =+ + + +           (3) 
 
If neither of the forecasts contain any information relevant for the q period realized return 
then the estimates of  MA β ,  10 T β , and  30 T β  should be zero. If each of the forecasts contain 
independent information then  MA β ,  10 T β , and  30 T β  should all be non-zero. If all forecasts 
contain information but the information in one of the forecasts, say 
10 T F , is completely 
contained in the other forecasts, F
MA and F
T30, and these other forecasts contain additional 
information, then  MA β  and  30 T β  are non-zero, and  10 T β  is zero.  
We apply the Fair and Shiller (1990) approach using partial regressions; from the Frisch-
Waugh theorem both approaches are equivalent. The reason for using partial regressions 
is twofold. Firstly, the forecasts are highly correlated with correlations ranging between 
0.5 and 0.8, so the multivariate regression (3) is likely to be contaminated by 
multicollinearity. Secondly, we want to use the inference methodology for long-run 
regressions developed in Hansen and Tuypens (2004). While the inference methodology 
works for multivariate regression models with long-run or short-run explanatory 
variables, it does not work for a mixture of short-run and long-run explanatory variables. 
Therefore, we run univariate regressions instead, where we regress the actual return on 
each of the three forecasts separately: 
 
10 10
10 10
30 30
30 30
() ,
() ,
() .
MA MA
t q MA MA t t q
TT
tq T T t tq
TT
tq T T t tq
rq F u
rq F u
rq F u
αβ
αβ
αβ
++
++
++
=+ +
=+ +
=+ +
 
 
We can test the exact same hypotheses as above using univariate regressions. The 
hypotheses can be formulated as follows: If neither of the forecasts contain any 
information relevant for the q period realized return then the estimates of  MA β ,  10 T β , and 
30 T β  should be zero. If each of the forecasts contain independent information then MA β , 
10 T β , and  30 T β  should all be non-zero. In addition, if we regress the residual from each of 
the univariate regressions on the remaining forecasts, they should enter with non-zero 
coefficients as well. For example, if all forecasts contain information, but the information   7
in one of the forecasts, say F
T10, is completely contained in the forecast F
MA, then  10 T β  
should be non-zero in the regression:
3 
 
 10
10 .
MA TM A
tq Tt t q uF γ βε + + =+ +  
 
We report the regression results for the regressions of demeaned realized return on the 
demeaned proxies for expected future return in Table 3. Inference is carried out using the 
methods developed in Hansen and Tuypens (2004) for long-run regressions.  
 
From Table 3 we see that each of the forecasts contain information about expected future 
returns; each of them enters with a significantly positive coefficient with t-statistics larger 
than 1.96.  In line two and three we examine whether the forecasts 
10 T F or 
30 T F  contain 
information beyond that in 
MA F . It turns out that neither 
10 T F nor 
30 T F enters with 
coefficients significantly different from zero, indicating that neither of these forecasts 
contains information that is not already contained in 
MA F .  
 
 
Table 3: Fair-Shiller test results 
 
Dependent 
variable 
l
MA β   t-stat 
l
10 T β   t-stat 
l
30 T β   t-stat  R
2 
() rq  2.787*  (2.207)      58.0% 
     resid
MA    0.432  (1.473)    12.9% 
     resid
MA      0.501  (1.512)  13.4% 
         
() rq    1.130*  (2.097)    37.0% 
     resid
T10  1.688*  (2.504)      33.8% 
     resid
T10      0.149  (0.325)      0.8% 
         
() rq      1.385*  (2.300)  43.1% 
     resid
T30  1.463*  (2.191)      28.1% 
     resid
T30    -0.045  (-0.121)        0.1% 
Regression estimates of 10-year real S&P 500 returns on various proxies, and regression estimates of 
residuals on each of the alternative proxies for the expected future return. The t-statistics (numbers in 
brackets) are calculated using the methods from Hansen and Tuypens (2004); * indicates significance at the 
2.5% levels. The sample period for the S&P 500 index covers 1920 to 2003.  
 
 
As a check of robustness we also regress the residuals from each of the other univariate 
regressions, and we cannot reject that F
MA contains information beyond that in 
10 T F and 
30 T F .    
                                                 
3 Note that since we use estimated residuals, the t-statistics are likely to be biased upwards, implying that 
the test is more likely to reject the null hypothesis of β =0 in the residual regressions. Since we are not 
able to reject the null in any of the relevant cases, we can conclude that these results are robust.     8
 
[Insert figure 2 about here] 
 
In addition, we can compare the R-square from each of the univariate regressions. The R-
squared is greater for 
MA F  than for either 
10 T F  or 
30 T F , indicating that using a moving 
average of the log earnings yield explains a greater portion of the total variation in future 
realized returns. Figure 2 exhibits the realized return and the returns predicted using each 
of the three proxies. The figure reinforces the conclusion that a moving average of the log 
earnings yield is a better proxy for the expected future return.  
 
 
 
5. Conclusion 
 
This note shows, using the Gordon growth model, that the trailing earnings over current 
price ratio is a downward biased estimate for expected stock returns. Instead we suggest 
using a moving average of 1 plus the earnings price ratio. The empirical results for the 
S&P 500 index support the theoretical finding that trailing earnings over price ratio 
results in a biased estimator and shows that a moving average of the log of 1 plus the 
earnings price ratio subsumes all information contained in the trailing earnings over price 
ratio. 
 
 
 
References 
 
Campbell, J. and Shiller, R. (1988) Stock Prices, Earnings and Expected Dividends, 
Journal of Finance 43, 661–676. 
 
Campbell, J. and Shiller, R. (2001) Valuation Ratios and the Long-Run Stock Market 
Outlook: An Update, Cowles Foundation Discussion Paper No. 1295, Yale University. 
 
Fair, R. and Shiller, R. (1990) Comparing Information in Forecasts from Econometric 
Models, American Economic Review 80, 375–389. 
 
Ferson, W. E., Sarkissian, S. and Simin, T. (2003) Spurious Regressions in Financial 
Economics?, Journal of Finance 58, 1393–1413. 
 
Gordon, M.  J. (1959) Dividends, Earnings, and Stock Prices, The Review of Economics 
and Statistics 41, 99–105. 
 
Hansen, C. S. and Tuypens, B. (2004) Long-Run Regressions: Theory and Application to 
US Asset Markets, Working Paper, Baruch College. 
 
Modigliani, F. and Miller, M. H. (1958) The Cost of Capital, Corporation Finance and the 
Theory of Investment, The American Economic Review 48, 261–297.    9
 
 
 
 
 
 
 
 
 
 
 
Figure 1: 
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
Realized return F_MA F_T10 F_T30
   10
Figure 2: 
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
Realized return F_MA predicted return F_T10 predicted return F_30 predicted return
 
 

Comparing Information in Forecasts from Econometric Models,

Dividends, Earnings, and Stock Prices,

Long-Run Regressions: Theory and Application to US Asset Markets, Working Paper,

Spurious Regressions in Financial Economics?,

Stock Prices, Earnings and Expected Dividends,

The Cost of Capital, Corporation Finance and the Theory of Investment,

Valuation Ratios and the Long-Run Stock Market Outlook: An Update, Cowles Foundation Discussion Paper No.

Proxying for Expected Returns with Price Earnings Ratios

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Proxying for Expected Returns with Price Earnings Ratios

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