Equity prices and fundamentals: a DDM–APT mixed approach

For an overview of the literature on stock price indices and fundamentals’ relationships, see Jawadi and Prat (2012).

We leave aside portfolio choice theories based on the international CAPM or international APT where national stock market returns are considered from the perspective of building an efficient world portfolio. Studies generally focus on modeling the time-varying degree of integration of national stock markets and gains from international diversification. The empirical results show that international variables play a part in determining domestic market risk premium (in particular the return on the world stock index).

However, by introducing a time-varying risk premium in the Gordon dividends growth formula, Prat (1992, 2013) showed that it is possible to generate a fundamental value of the S&P index, which is more volatile than that obtained under REH. However, the author did not analyze the stock price adjustment towards the fundamental value.

Lansing (2016) explained changes in price variance through the effects of news about dividends.

Kocherlakota (1996) and more recently Prat (2007) provided comprehensive on the equity premium puzzle debate.

Adam et al. (2016) confirm for the US stock market that the consumption-based asset pricing model might solve the “volatility puzzle” if one allows for deviations from rational expectations. However, it is worth noting that, as for Cechetti et al. (2000), this study does not examine the correlation over time between theoretical and observed returns.

See Appendix of the present paper.

In particular, the VAR system used by Campbell and Shiller (1988b) (that includes the log dividend-price ratio, stock returns, growth in dividends and other variables representing the risk-free rate and the risk premium) puts into evidence that these variables enable to improve the forecast of the US stock market, which suggests that expectations are in fact not rational. This result was confirmed by Campbell (1991) and Campbell and Shiller (2001) who showed for twelve countries that, contrary to the efficient market hypothesis, price-earnings and dividend-price ratios are relevant variables to predict future stock price changes. Among others, Zhi et al. (2014) confirmed recently this result for G7 countries.

For each of the 5 years covering the period 1961–1965, the authors showed that between 50 and 75 % of variance in the observed price/earnings ratios of 175 American firms can be explained by the DDM wherein beta coefficients (deduced from the CAPM) are considered as a risk indicator. For a given firm, this means that the risk premium is given by the product of beta by its regression coefficient. It can then be seen that this approach is a mixture of the DDM and a simple version of the CAPM.

In particular, see Chapter 5 by Barucci (2003) that gives an overview of the empirical results arising from the two asset-pricing models.

Recall that the APT risk premia for n equities are generally estimated in two steps. The first one consists of regressing (time series) each of the n observed one-period returns on a set of the zero-mean unexpected components of k common variables and possibly changes in their expected values (typically, an ARIMA model is used to estimate the one-period ahead expected value). Accordingly, the slopes of these regressions give the estimated values of the sensitivities of returns to common factors while the intercepts represent the expected returns (a preliminary factor analysis is sometimes used in this first stage to choose the k relevant common factors, which of course does not avoid the arbitrary choice in the initial set of common variables). The second step consists of regressing (cross-section) the n expected returns on the estimated values of sensitivities, the k factorial premia being given by the values of the slopes while the riskless rate is captured by the intercept of this regression. Accordingly, the significant common factors are those for which the slopes are significant, the risk premium of any equity being measured by cumulating the effects of the k-independent factors.

In the Appendix of this paper, we compare this traditional formula with its so-called dynamic version proposed by Campbell and Shiller (1988b) and explain why we used the first one rather than the second one.

Unlike aggregate dividends associated with stock price indices, the monthly time series of dividends follows stepped patterns for any given firm.

With respect to the usual regressive expectation process, the observed rate of growth \( g_{it} \) was here replaced by the long term rate \( \bar{g}_{it} \).

This value can be viewed as characterizing the dynamic equilibrium condition regarding dividends growth.

This condition means that the purchase of securities is offset by short sales. Interestingly, the absence of the arbitrage hypothesis is in accordance with Welch (2000) who showed that investors have a strong view in favor of this condition.

If the only common factor is the domestic stock market, the risk premia of equities given by Eq. (3) correspond to the CAPM; this case is characterized by \( {}_{1}\varphi_{t} = \tilde{R}_{mt} - r_{ot} \), where \( \tilde{R}_{mt} \) stands for the expected market return and \( {}_{j}\varphi_{t} = 0\,\;\forall j,t \) for \( j \ne 1 \).

Of course, one should expect different sensitivities to common factors for small firms than for large firms.

Unlike to empirical tests of the APT in the literature [see footnote (14) above].

Regarding the seven significant common factors \( {}_{j}X_{t}^{{}} \) considered hereafter, for \( {}_{1}X_{t}^{{}} \) (domestic stock market) and \( {}_{5}X_{t}^{{}} \) (world stock market), we have \( {}_{j}a = 1 \) (and \( {}_{j}b = 0 \))  since these factors are directly viewed as market and world risk premia respectively; for \( {}_{2}X_{t}^{{}} \) (economic sentiment), we have \( {}_{j}a < 0 \) for \( {}_{3}X_{t}^{{}} \) and \( {}_{4}X_{t}^{{}} \) (interest rate spreads) and for \( {}_{6}X_{t}^{{}} \) (oil price) we have \( {}_{j}a > 0 \); for \( {}_{7}X_{t}^{{}} \) (Euro/USD exchange rate), the sign of \( {}_{j}a \) is a priori undetermined since this variable both increases the cost of imports (negative effect) and improves competitiveness (positive effect), so that the overall effect might vary according to the companies.

However, some important companies were excluded either because (1) data on dividends was not available over the period, (2) these companies were subject to a merger or acquisition (i.e. EADS, EDF/GDF), or (3) the companies were listed on the stock market after 1989.

It can be seen that the French market factor in Fig. 2 confirms this point.

The magnitude of the deviation for each of the four groups is measured as the average of the standard errors of individual deviations (as estimated from the DDM-APT model) relative to the equities belonging to the group.

The world inflation rate was also considered, but this variable appeared significant for only 4 of the 27 equities considered, so it was excluded from the set of potential factors.

We alternatively expressed this variable in real terms (i.e. deflated by the world commodity price index) but the results remained quite similar.

Stock price indices, exchange rate and oil price are dated at the end of the month (as are equity prices), while interest rates and the Economic Sentiment Index are monthly averages.

We do not report the results to save space.

Of course, under the unrealistic conditions that the DDM is “the true” valuation formula, that the common factors of risk premia are perfectly known, and that share prices adjust instantaneously to their fundamentals, the correlations between the contemporaneous residuals \( \varepsilon_{it} = \eta_{{i{\kern 1pt} t}} + \zeta_{{i{\kern 1pt} t}} \) of the 27 firms would be null and then the OLS relevant, since the error terms \( \zeta_{{i{\kern 1pt} t}} \) would be white noises while the terms \( \eta_{{i{\kern 1pt} t}} \) that represent specific components of risk are a priori uncorrelated across firms.

The average of the \( R^{2} \) between the 351 pairs of the 27 SUR-residuals is 0.25, with a minimum of zero and a maximum of 0.64 (about 80 % of \( R^{2} \) were found to be significant). To compare, we estimated Eq. (6) for the 27 firms using OLS and found a drop up to 20 % of the number of significant common factors for all firms with respect to the SUR method, hence confirming the interest of using the latter. However, the SUR method brings no correction to the variance of estimates when errors terms are auto-correlated. To appreciate the consequence of this issue, we compared the OLS results to those obtained using the Newey-West method—according to which the variances of estimates converge towards their true values when residuals are serially correlated—and found that the two methods lead practically to the same set of significant common factors, hence suggesting that, with our data, the autocorrelation of residuals does not alter the selection of factors. In addition to being robust to correlation of residuals across firms, the SUR method has the advantage to avoid the arbitrary choice of instrumental variables, as it would be the case especially with the GMM. Finally, note that, although a panel approach was feasible with our cylindered data, it is not appropriate since all slopes related to the 27 firms are a priori firm-dependent.

Surprisingly, oil price appeared insignificant for Total. This suggests that, on average, the negative effect on risk premium arising from an expected increase in the supply side of the oil market is compensated for by the positive effect resulting from an expected decrease in oil demand. Otherwise, the fact that the WTI market relates to crude oil price while Total is concerned with production and refining for both oil and gas, adds uncertainty with regard to the sign of the global effect.

The tenuous results concerning the ADF tests are mainly due to the existence of a few extreme values in the residuals of Eq. (6). We did not eliminate these points by using dummies since the extreme deviations between equity prices and fundamental values may intervene in the adjustment process of share prices towards fundamentals analyzed hereafter.

In fact, while for some equities the volatility is greater for \( f_{it}^{*} \) than for \( p_{it}^{{}} \), the opposite prevails for others. Similar remarks can be made when sub-periods are distinguished for a given equity. Regarding the mean of the 27 \( f_{it}^{*} \) and the mean of the 27 \( p_{it}^{{}} \), these two averages appeared to be rather comparable (standard errors are 19.1 and 17.9, respectively); if observations corresponding to the bursting of the “dot.com bubble” are excluded, these two averaged values are not significantly different.

This visual feeling is confirmed when the common component of deviations is approximated by the average of the 27 deviations \( z_{{i{\kern 1pt} t}} \), that is \( \bar{z}_{{{\kern 1pt} t}} = \frac{1}{27}\sum\limits_{i = 1}^{27} {z_{i\,t} } \). Indeed, the mean value of the 27 coefficients \( R^{2} (z_{it} ,\bar{z}_{\,t} ) \) was 0.34, indicating that, on average, about 1/3 of the variance of an equity’s deviation is due to the common component, while 2/3 is due to a specific component (we checked that the average \( R^{2} \) between the specific components \( z_{{i{\kern 1pt} \,t}} - \,\;\bar{z}_{t} \) and the common component \( \bar{z}_{t} \), which equals 0.04, can be neglected in a first instance). Although \( \bar{z}_{t} \) was found to be significantly serially correlated, we checked that this common component is not linked with any linear combination of our seven common factors, hence confirming that these factors was captured satisfactorily in the fundamental values of equities.

The fundamental value of a given equity is supposed common knowledge for stockholders but each of them has their own appreciation of uncertainty about this value.

See Jawadi and Prat (2012) for more details.

Recall that the basic specification of the ECM can be written as \( p_{{i{\kern 1pt} t}} - p_{{i{\kern 1pt} t - 1}} = \rho_{i} (f_{{i{\kern 1pt} t - 1}}^{*} - p_{{i{\kern 1pt} t - 1}} ) + \,b_{i} (f_{{i{\kern 1pt} t}}^{*} - f_{{i{\kern 1pt} t - 1}}^{*} ) + \zeta_{it} \), \( \rho_{i} > 0 \), \( b_{i} > 0 \). Under the long-term equilibrium condition \( b_{i} = 1 \) (meaning that price and fundamental value increase at the same rate) and re-arranging the terms, we get \( z_{{i{\kern 1pt} t}} - z_{{i{\kern 1pt} t - 1}} = - \rho_{i} \,z_{{i{\kern 1pt} t - 1}} + \zeta_{it} \). By increasing this basic ECM by including past values of changes in \( p_{it} \) and \( f_{{i{\kern 1pt} t}}^{*} \), we get Eq. (9) provided that \( k_{i} = 0 \). We leave the possibility that the intercept \( k_{i} \) is different from zero since there is a priori no reason for the means of deviations \( z_{it} \) to be null over the period under consideration. In fact, we found \( k_{i} \ne 0 \) for only 3 of the 27 equities.

It should be noted that if \( z_{it} \) was viewed as an observable variable, Eq. (9) would corresponds to a classical unit root test, so that the condition found \( 0 < \rho_{i} < 1 \) in Table 3 would allow to conclude that \( z_{it} \) is stationary, meaning that \( p_{{i{\kern 1pt} t}} \) and \( f_{{i{\kern 1pt} t}}^{*} \) are cointegrated, which goes rather in the sense that \( f_{{i{\kern 1pt} t}}^{*} \) is a valid measure of the fundamental value.

As a result, the two largest firms (Total and Sanofi) are those for which prices adjust the slower to fundamentals. This suggests that, in our sample, the tendency of the market to reach efficiency is not improved with the increase of firm size.

To save space, we do not report the results of these tests. They are however available upon request.

Moreover, the short term adjustment process could also involve variables other than those considered in the right hand side of Eq. (9).

The subsequent studies based on the dynamic Gordon growth model also used a VAR system [among others, see Bekaert and Engstrom (2010)].

C&S found rather large residuals for the log dividend-price ratio Eq. (A.2). In fact, the authors reported that the Taylor approximation may in some cases produce non negligible errors with respect to the Gordon structural relation. In this regard, one can see that Eqs. (10) and (11) link the log of a percentage to percentages, while Eq. (1) leads to link a percentage to percentages, which seems a priori more natural.

In fact, this is a generally accepted assumption since one cannot probably do otherwise. For example, when expected returns are for one year ahead, it is implicitly supposed something like uniformity of expectations during this time horizon.