SCHOOL OF FINANCE AND ECONOMICS UTS:BUSINESS WORKING PAPER NO. 116 APRIL, 2002 Solving the Price-Earnings Puzzle Carl Chiarella Shenhuai Gao ISSN: 1036-7373 http://www.business.uts.edu.au/finance/
Working Paper 116 Solving the Price-Earnings Puzzle * Carl Chiarella and Shenhuai Gao School of Finance and Economics University of Technology, Sydney April 2002 Abstract Accounting and finance professionals have empirically known that in the long run stock prices are roughly proportional to earnings. However, econometric testing could not been able to verify this expected contribution of earnings to stock prices, thus formed the price-earnings (PE) puzzle in the accounting literature. This paper seeks to solve this puzzle by allowing the earnings response coefficient to be a variable instead of a constant, and shows that the PE puzzle turns out to be a phenomenon of type I spurious regression in econometrics. JEL classification: C22; G12; M41; Key words: Price (return)-earnings relation; earnings response coefficient; type I spurious regression; 1. Introduction Accounting and finance professionals have long recognised that in the long run stock prices are roughly proportional to earnings. Accounting professionals calculate and report firms earnings in order to evaluate the operating performance of firms, and financial professionals predict firms earnings in order to predict stock prices. However, three decades of effort in econometric testing has not been able to verify the anticipated contribution of earnings to stock prices, thus has formed the price (return)-earnings (PE) puzzle in the accounting literature. By definition the basic PE relation between the stock price P and the earnings per share E is expressed by P := E PER = E / EY, where PER is called the price to earnings ratio in finance and the earnings response coefficient (ERC) in accounting, and EY is the earnings yield. The basic PE relation clearly expresses the contribution of earnings to stock prices, ie the price is proportional to the earnings. When PER is called a ratio or coefficient, it is taken for granted that PER should be a constant, and its reciprocal EY is expected to be the rate of return of firms. The constant expected rate of return is the foundation of many theoretical equity valuation models, such as the dividends discount model (DDM). Researchers try to estimate the * The authors are indebted to their UTS colleague Dr. Maxwell Stevenson who gave many valuable comments on earlier drafts of this paper. The usual caveat applies. 1
ERC by linear regression. In the accounting literature, regression of stock prices on explanatory variables is called the price model, and regression of the change in prices P k P k-1 or P k / P k-1 on explanatory variables is called the return model. Many different specifications have been used in both classes of models, however one common feature shared by all the model types of previous studies is the assumption of a constant ERC. Lev (1989) summarised the twenty years of empirical research of econometric testing since Ball and Brown (1968), and reported that the return models showed very low R 2 and very unstable ERC, suggesting that earnings contained little information in explaining stock returns. Lev suggested that the relation between returns and earnings seemed nonlinear, and the ERC was obviously not constant. These problems are not unique to the accounting literature. At the same time Roll (1988) also raised the very low R 2 problem in the financial literature. Researchers have responded to the very low R 2 problem in a variety of ways. Easton, Harris, and Ohlson (1992) found that increasing the time interval of returns could increase R 2. Collins, Kothari, Shanken, and Sloan (1994) considered that the delay of earnings data release might be the reason for the low R 2 problem. Ryan and Zarowin (1995) considered the fact that earnings data may contain value-irrelevant noise, and explained the low R 2 problem by the errors-in-variables approach. Kothari and Zimmerman (1995) compared price models and return models, and reported that price models had more economic meaning and higher R 2, while return models had lower R 2 but less econometric problems. Beaver, McAnally, and Stinson (1997) considered that both the change in prices and the change in earnings were endogenous. They specified a return model as a system of simultaneous equations, and obtained slightly higher R 2 than the single equation model. In all of the just-cited studies, the very unstable coefficient problem has been accorded too little attention. This paper will investigate the unstable coefficient problem, as well as the low R 2 problem by using an approach based on linear regression of time series data. Tippett (1990) pointed out that a financial ratio (eg. PE ratio) is not necessarily a constant. As the ratio between two stochastic processes, such as a random walk or a meaning-reverting process, a financial ratio would be another stochastic process instead of a constant. It is well known that the interest rate R (ie. bond yield) is a key financial and economic variable. Time series data show that the earnings yield EY (ie. stock yield) is highly correlated with the interest rate R, suggesting that the stock yield should be treated in the same way as is the bond yield. That is, the ERC should be considered as a variable instead of constant. Since EY R, substituting the interest rate R for the earnings yield EY, time series data show that E / R is highly correlated to the stock price P, ie P = E / EY E / R. Financial practitioners use this information to predict the aggregate stock market in practice (see Wigmore, 1998). Chiarella and Gao (2002b) have shown how the process of earnings yield following interest rate may be considered as the basic adjustment process of the stock market, and have specified and estimated a dynamic model to express this adjustment process. It is the underlying adjustment mechanism that guarantees the long run equilibrium between the earnings yield and interest rates. The timing effect of the short run dynamics can be ignored for the long run equilibrium relation, and the long run equilibrium can be approximated by a static relation, which could be captured by regression. Thus, in the long run the interest rate R can be a good proxy for the earnings yield EY, and the PE relation can be well approximated by P E / R. This relation states that in the long run stock prices are directly proportional to earnings 2
and inversely proportional to interest rates. This relation fully recognises the contribution of earnings to prices, and verifies the usefulness of earnings to equity evaluation. It also shows that interest rates that represent financial market risks are equally important in determining stock prices. In log-variables the PE relation can be written as p e r, which can be understood in the econometric sense that p and e r are cointegrated. When price models capture cointegration, the return models (without an error-correction mechanism) are then misspecified (see Engle and Granger, 1987). Chiarella and Gao (2002a) showed that in regression of time series the ignored system dynamics will become systematic errors in regression equations, and the assumption of no specification errors in the statistical model underlying the regression procedure is violated. Thus, regression of differenced time series tends to reject the relation between their levels and incur type I spurious regression 1. The bad performances of the return models reported in the literature during the past three decades can be viewed as an empirical manifestation of type I spurious regression. The fact that the PE relation is still a puzzle after three decades of research is a consequence of type I spurious regression. This paper is arranged as follows. Section 2 verifies that the earnings yield is a varying quantity. Section 3 derives a price model in which the earnings yield is allowed to vary. Section 4 enters into economic and econometric considerations underlying the choice between price and return models. Finally conclusions are drawn in Section 5. 2. Earnings yield is a variable The following notations are used: Stock price index: P, p = ln(p), Earnings per share: E, e = ln(e), Earnings yield: EY, ey = ln(ey). With these notations the basic PE relation by definition is given by P := E / EY, or in log-variables by p := e ey. The stock price index instead of individual stock prices is used in this paper. The stock index is a well-diversified portfolio of stocks, in which most of nonsystemic risks of stocks have more or less cancelled each other, and the systemic risk becomes dominant. The systemic risk comes from the changing macroeconomic conditions on firms during business cycles, and is captured by interest rates. The earnings yield reflects the risk of stocks. So the correlation between the earnings yield and the interest rate can be seen more clearly from the index than from individual stocks. The stock price index is the S&P 500, and the interest rate is the discount rate of the Federal Reserve Board. We use monthly time series data for the time horizon from Jan-1979 to Aug-2001. The interest rate R, the stock price index P, and the price to earnings ratio PER are obtained from the DATASTREAM database. Then, we calculate the earnings yield by EY = 1/PER, and the earnings per share by E = P / PER. Since the earnings per share E is calculated from the price P, to avoid spurious 1 Chiarella and Gao (2002a) use the term type I spurious regression to refer to the rejection of a true relation, and type II spurious regression to refer to the acceptance of a false relation, more often by a flawed regression scheme in econometrics. Type II spurious regression is the one that is most familiar to econometricians. 3
correlation between their noise components, we smooth the time series for E by the 3- point centred moving average algorithm, and repeat the operation five times. 0.16 0.14 S&P500_monthly 0.12 0.1 0.08 0.06 0.04 0.02 R EY 0 Jan-79 Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Figure 1 The relation between the S&P 500 earnings yield EY and the FED discount rate R. 0.18 0.16 0.14 0.12 S&P500_monthly Regression EY on R R 2 = 0.85 a = 0 (0.002) b = 1.028 (0.026) 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Figure 2 The regression of the earnings yield EY (vertical axis) on the interest rate R (horizontal axis). 4
7.6 7.2 Log(S&P500)_monthly 6.8 e - r 6.4 6 p 5.6 5.2 1.9e 4.8 4.4 Jan-79 Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Figure 3 Time series in log-variables. The relation mong the stock price index p, earnings less interest rate e r, and the 1.9 times earnings 1.9e. 7.5 Log(S&P500)_monthly 7 6.5 6 5.5 5 4.5 Regression p on e R 2 = 0.86 a = 0.544 (0.131) b = 1.723 (0.042) Regression p on (e - r) R 2 = 0.93 a = -0.429 (0.107) b = 1.071 (0.018) 4 2 3 4 5 6 7 Figure 4 Regression in log-variables. The left panel is the regression of the price p (vertical axis) on the earnings e (horizontal axis), and the right panel is p on e r. Figure 1 shows the close relation between the earnings yield EY and the interest rate R over the last two decades. Such a comovement has in fact existed for four decades since the late 1950s (see the time series plot in Asness, 2000). The figure shows that like the interest rate (bond yield), earnings yield (stock yield) should be treated as a variable rather than a constant. 5
Figure 2 shows the regression of EY on R, which indicates that the relation is linear, R 2 is high, and the residual is well behaved. More importantly, as expected there is strong statistical support for the hypothesis that the slope b = 1 and the intercept a = 0. Figure 3 suggests that when taking interest rate r as the proxy for earnings yield ey, the price can be well approximated by p e r. For comparison the earnings e multiplied by 1.9 is also plotted. We see that e r is closer than e to p. Closeness can be measured by the sum of squares of the gap. Note that the original variables P and E exhibit strong exponential characteristics, and so their logarithmic transforms are linear. Thus, log-variables should be used in linear regression, in order to get rid of the hertroscedasticity in residuals. Figure 4 shows the regression, in log-variables, of stock price p on earnings e and of p on earnings less interest rate e r. The estimates from the regression are given by p k = 0.544 + 1.723 e k (1) (0.131) (0.042) R 2 =0.86, and p k = -0.429 + 1.071 (e k - r k ) (2) (0.107) (0.018) R 2 =0.93, where the numbers in parentheses are standard errors. From Figure 4 we see that when the interest rate r is included, both R 2 and the behaviour of the residual are improved a lot. More importantly, as expected it may be asserted that statistically the slope b = 1 and the intercept a 0. The rationale underlying the above observation is that here R is the Federal Reserve discount rate that represents monetary policy, and the stock market responds to this policy. The earnings yield represents the risk premium required by investors. Since investors spread their money among different financial instruments, and the stock market adjusts to maintain no-arbitrage with the general financial market, therefore the earnings yield should reflect the performance of the general financial market. This is the reason why the earnings yield should be closely related to interest rates. The basic PE relation P = E / EY implies that stock prices link to firms performances through earnings, and link to risks in financial markets through earnings yields. The time-varying earnings reflect the changing macroeconomic conditions on firms, and the time-varying interest rates and earnings yields reflect the changing financial market conditions. Therefore, both earnings and earnings yields are varying quantities. 3. Regression when earnings yield is a variable The earlier cited empirical research is based on linear regression built around the basic PE relation P = E PER = E / EY. Assuming the ERC is a constant, the basic specification of the price model is then given by = α1 + β1ek + ε1 k. (3) Where the intercept α 1 is expected to be zero, and the slope β 1, ie the ERC, is expected to be the reciprocal of the firm s expected rate of return. In order to avoid (type II) spurious regression, the regression model is often specified in term of returns, and the return model is given by = α 2 + β2 Ek + ε2 k. (4) 6
The return models actually used in the accounting literature are variants of the basic specification (4). For example E k = α3 + β3 + ε 3k, (5) 1 E k 1 was used by Beaver et al (1997), whilst Kothari and Zimmerman (1995) employed the form E k = α4 + β4 + ε 4 k, (6) 1 1 and Easton et al (1992) wrote + Dk E k = α5 + β5 + ε 5k. (7) 1 1 Of course, the different specifications of price and return models were assumed to be equivalent in order to infer the ERC (see Kothari and Zimmerman, 1995, and Christie, 1987). Otherwise, there would be no point in comparing the performances of different models using the β estimation. For example, in (7) the factor 1/P k was explained as a scaling factor, which was introduced to improve the econometric conditions. The model had been explained on the basis that the level of earnings had an effect on stock returns. In reality, since P is a variable rather than a constant, its effect in the equation cannot be understood as that of a scaling factor. After dividing by P the original variables in fact become new variables. For time series data the percentage capital gain P k / P k-1 is virtually a white noise series, and (7) actually expresses the relation between the dividend yield D k / P k-1 and the earnings yield E k / P k-1, but says nothing about the PE relation. Similarly, (6) is actually a relation between the percentage return P k / P k-1 and the earnings yield E k / P k-1, and so is not equivalent to (3). Of course, one should not be surprised that quite different ERC estimates are obtained from the different models. The empirical evidence in Kothari and Zimmerman (1995) showed that price and return models give quite different results, suggesting that these models are not equivalent at all. Despite the variety of specifications, one common feature shared by the previous studies so far is the assumption of the constant ERC. The constant coefficient is required by the linear regression, which would otherwise be invalid. What seems to have been overlooked is that if the estimated coefficients are very unstable, the correct conclusion is that the coefficient is not constant at all, and the model should be rejected. If the earnings yield EY (and therefore the ERC) is a variable, then the basic PE relation P = E / EY is nonlinear. It is then invalid to estimate 1/EY for given P and E with linear regression. Doing so would result in very unstable estimate. The differenced form of the basic PE relation is then given by P E EY =, (8) P E EY rather than P = E / EY that results from the constant EY assumption. When the relation is viewed in this way then (3) and (4) are no longer equivalent. In logvariables the PE relations p = e ey and p = e ey are linear. But, when regressing price on earnings only, an important variable will be missing. Even though the logarithmic relations p = e ey and p = e ey are equivalent, when estimating the relation from the proxy variables, for example letting r be the proxy for ey, then the relations p e r and p e r are no long equivalent in regression. Such confusion arises when one seeks to analyse dynamic 7
problems with static intuitions. The next section will elaborate upon this point, though for a more detailed explanation we refer the reader to Chiarella and Gao (2002a). 4. Understanding cointegration in accounting applications Kothari and Zimmerman (1995) compared price and return models empirically, and reported that the results from price models were closer to theoretical inferences, while the return models encountered less econometric problems. How should researchers trade-off between economic and econometric considerations? Since the purpose of empirical accounting research is to test economic relations by econometric models, rather than to verify econometric methods by economic facts, so economic inference is important for applied work, and econometric considerations can only be a reference point. Indeed econometric textbooks always warn that model selection cannot be based on purely econometric reasoning. 4.1 The price model Unlike the ordinary regression in econometrics, the relations among time series are dynamic relations. When estimating the relation among levels of variables by static regression, such as pk = α 6 + β 6 ( ek rk ) + ε 6k, (9) the researcher is actually concerned with the long-term equilibrium between the levels of variables. For the long-term equilibrium relations the dynamic (ie. temporal) effect can often be ignored. This implies that one can use a static model to approximately capture the long-term equilibrium relation. By doing so the ignored dynamic effects become systematic errors included into the residuals. This can be understood in the framework of cointegration in econometrics. To guarantee the long-term equilibrium relation to be significantly identified by a static regression, the variables (regressand and the linear combination of regressors) are required to be integrated of order 1, ie be an I(1) processes (which exhibit trends), and the residual be an I(0) process (which exhibits no trend). 2 If the residual is I(1) then the regression will be spurious; if the variables are I(0) then the estimation will be downward biased and insignificant, see Chiarella and Gao (2002a) for a more detail explanation. The static model (9) is purposely used to approximate the underlying dynamic relation, and the residual from cointegration contains both random and systematic components and is an I(0) process, so in this situation the non-white residual should not be considered as an econometric problem for price models. Also, in applied research one usually does not have sufficient prior knowledge about the data generating process in order to specify a probability model. In such cases making statistical inferences from the estimated coefficients can often be misleading. However, one can still compare the competing models at the whole model level, according to the parameter estimates, the pattern of the fitted curves and residuals, as well as statistical tests. Econometric tests can be a reference point, but more important is the economic inference. This is the viewpoint adopted in macroeconomic research (see Hodrick and Prescott, 1997). For example, we compare the two competing models (1) and (2) according to R 2, the behaviour of residuals (see Figure 4), as well as the parameter estimates. These criteria for (2) are better than those for (1), and we 2 If a nonstationary time series becomes a stationary one after differencing once, then it is said to be integrated of order 1, and is denoted by I(1). Thus, the stationary time series after differencing is said to be integrated of order 0, and is denoted by I(0). 8
accept (2). Even so, we still remain on the lookout for better models. In reality, the performance of the dynamic model in Chiarella and Gao (2002b) is better than regression models. 4.2 The return model The return model is used to capture the short-term informational effect of earnings on stock returns. The desire to avoid econometric problems is also an important consideration in preferring return models over price models (see Christie, 1987). Using differenced time series instead of their levels, the return model corresponding to (9) is given by pk = α 7 + β7 ( ek rk ) + ε 7k. (10) As Chiarella and Gao (2002a) pointed out, the mechanism underlying the long-term equilibrium in (9) is the economic adjustment process. The adjustment processes of long-term economic relations are conditioned on the observation of the levels of economic variables. For example, the financial market evaluating stock prices conditioned on firms earnings and interest rates. The adjustment process towards long-term equilibrium is a level matching process, and it does not manifest itself in differenced variables (without an error correction mechanism). Indeed, econometric textbooks always warn that the long-term information contained in the levels of variables will be lost in differenced time series. The reader should notice the difference between = and. Many mistakes in applied econometric work actually come from the confusion between an exact relation and an approximation to it. Since (9) is based on the approximate relation p e r rather than an exact relation p = e ey, the static regression models (9) and (10) are not equivalent for dynamic processes. When the long-term equilibrium (9) holds, then the corresponding short-term relation (10) does not hold. Comparing the pattern of the relation between the trends of the differenced data in Figure 5 and the pattern between the corresponding variables in Figure 3, one may detect a similarity. This pattern shows a dynamic relation that can be captured by dynamic models, such as differential or difference equations. Regression is used to capture the correlation among variables, and the correlation depends on the integrated order of the variables. As mentioned earlier, regression may capture the long-term equilibrium relation if the variables are I(1) as in price models, but it is very difficult to do so if they are I(0) as in return models. This is the reason that the very low R 2 problem is an inherent problem of return models. If the purpose is to infer the long run PE relation, then return models are ill-defined and will convey spurious information. Figure 5 suggests that the differenced time series p and (e r) are roughly white noises. From the econometric perspective, since white noise is a well-behaved random variable, it is not difficult for these series to pass econometric tests. Thus return models have less econometric problems, but also contain less information. The noise in (10) is roughly the difference of that in (9). Since differencing a stochastic process will reduce its signal to noise ratio, the signal to noise ratio of return models will be much smaller than that of price models. The effect of the low signal to noise ratio is to reduce statistical significance. Intuitively, the noise terms in price models and return models contain different elements. For example, speculation has a heavy influence on short-term stock returns, but not on long-term stock values. The speculative effect will manifest itself more significantly in return models than in price models, and this will dilute the effect of earnings. If the purpose of return 9
models is to detect the short-term informational effect of earnings on stock returns, then the informational effect, if any, will be buried in the effect of speculations, and will be difficult to be detected by regression. p Log(S&P500)_monthly (e-r) (e-r) p Jan-79 Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Figure 5 Time series of differenced log-variables p and (e r), as well as their trends. 1.8 1.4 1 0.6 0.2-1.8-1.4-1 -0.6-0.2-0.2 0.2 0.6 1 1.4 1.8 2.2 Log(S&P500)_monthly Regression p on (e -r) R 2 = 0.003 a = 0.103 (0.023) b = 0.084 (0.061) -0.6-1 -1.4-1.8-2.2 Figure 6 Regression of the time series in Figure 5. The vertical axis is p and the horizontal axis is (e r). 10
Figure 6 shows the regression of the differenced time series that are plotted in Figure 5. Model (10) is fitted as p k = 0.103 + 0.084 (e k - r k ) (11) (0.023) (0.061) R 2 =0.003. Comparing to (2) one can see that the long-term information contained in the levels of variables has disappeared from the differenced variables. 5. Conclusions The previous empirical studies in the accounting literature dealing with the PE relation have been based on the assumption that the ERC is a constant. By allowing the ERC to be a variable instead of a constant, this paper sheds new light on the PE puzzle. This paper finds that in the long run the PE relation can be well approximated by P E / R. This relation states that in the long run stock prices P are directly proportional to earnings E, and inversely proportional to the interest rate R. This relation fully recognises the contribution of earnings to stock prices, as well as the usefulness of earnings to equity valuation. This paper also argues that since the PE relation P E / R is a nonlinear relation between levels of variable, rather than a relation between differenced variables, therefore return models in the previous literature are not equivalent to price models when seeking to infer the ERC. It has also been show why the very low R 2 problem is an inherent problem of return models, and indeed the empirical analysis of Section 4 indicates that return models are ill-defined. 11
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