In our master’s thesis we studied if forecasting of future dividends can improve the predictability of stock returns with price-dividend ratio. We analyzed this question from different perspectives including in-sample and out-of-sample analysis, as well as economic significance of results. Another dimension of our analysis was dedicated to comparison of two competing methodologies: latent variables approach in the spirit of Binsbergen and Koijen (2010) and adjusted price-dividend ratio proposed by Lacerda and Santa-Clara (2010). Since the question of stock returns predictability is very general and complex, we did not expect to get an exact and simple answer to our research questions. Nevertheless, we would like to focus on the key findings, which were discovered during our analysis.
First of all, we have shown that the theory behind adjusted price-dividend ratio methodology finds a convenient empirical support. By forecasting dividend growth with either historical mean or AR(1)-process, we were able to filter out some variation in price-dividend ratio due to variation in expected dividends.
Thus, we obtained a substantial improvement in predictability of stock returns relative to regular price-dividend ratio. Moreover, we documented that AR adjustment always delivered slightly better results than historical mean adjustment proposed by Lacerda and Santa-Clara(2010). This difference in results became especially important when we had constructed simple trading strategies, where AR adjustment delivered more substantial benefits to investors. Moreover, out-of-sample results were robust and relative performance of different predictors was not sensitive to the choice of sample period.
The comparison latent-variables approach with adjusted price-dividend ratio methodology was two-fold. On the one hand, the model of Koijen and van Binsbergen did not show any improvements in predictability of stock returns above price-dividend ratio, while all adjusted ratios explained larger part of variation in stock market returns. On the other hand, the benchmark model was definitely the best from out-of-sample perspective, though AR adjustment was consistently close to it. Therefore, we conclude that there is a strong predictive potential behind the latent variables approach, which can be exploited in real-time delivering significant value gains for investors.
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Quite surprisingly, ARMA adjustment, which is structurally equivalent to the model of Koijen and van Binsbergen have shown relatively poor performance, especially out-of-sample. For some sample periods it was not even able to outperform regular price-dividend ratio. We conclude that its mediocre performance is mostly driven by extra noise generated by inclusion of moving-average term into the realized dividend growth rate model. Moreover, we believe that the assumption that the market uses ARMA(1,1) model to forecast dividend growth rates could be overcomplicated, which was supported by our empirical results.
Finally, our results show that the aggregate stock market returns are predictable to reasonable extent both in- and out-of-sample. This predictability is significant, especially when we analyze actual value gains for trading strategies based on our predictors. We believe that there is still huge uncovered potential behind both adjusted price-dividend ratio and latent variables approach methodologies. In particular, one can estimate future dividends from the current market conditions instead of historical data. For example, Golez (2012) extracts dividend forecasts from the prices of futures and options traded in the market, which also leads to the adjusted price-dividend ratio, which captures significant part of variation in realized returns. If one can propose better and more precise way to forecast dividends, he would be able to improve returns predictability even further.
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Appendix A
We first reformulate the model in the standard state-space form. We define an expanded state vector as:
[ ̂
] which satisfies:
with
[
]
[
]
[ ]
which are assumed to be jointly normally distributed.
The measurement equation with an observable vector , is:
with
[ ]
[ ]
[
] The Kalman Filter is then constructed as follows:
|
|
| |
| |
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| |
|
| |
| |
The likelihood function is further computed based on prediction errors and their covariance matrix:
∑ ∑
Finally, the covariance matrix of the shocks is:
([
]) [
]
We furthe maximize the likelihood function with simmulated annealing procedure.
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Appendix
BSimulated annealing is a probabilistic algorithm designed to solve global optimization problem. A common problem of conventional algorithms based on numerical derivatives is inability to leave the local maximum point when they reach it. In other words, with conventional algorithms one has to make a very good initial guess of parameters to be sure that the algorithm will reach global rather than local maximum point. Simulated annealing overcomes this problem.
At each step it generates a random point in the function domain. If the value of function in this point is greater than currently reached maximum, it moves to this point. However, if it is less than the current maximum, it still can move to this point with the probability that is decreasing with number of iterations. It allows looking for global maximum points beyond the neighborhood of current maximum point. More formally, assume that and and currently reached maximum point and value of the function at this point respectively. We also have , which is called a temperature parameter. At each step we generate a new random point with multivariate normal distribution. We further compute – value of the function at point . Then, we search process moves from to with the following probabilities:
{
Hence, the algorithm moves to the point with lower function value with probability, which declines with difference between the old and new value and also declines with . Initially is very high, which makes the algorithm chaotically jumping between the local maximum areas, however it gradually declines after fixed number of iterations (200 in our case) according to the pre-specified cooling schedulre:
Algorithm also tracks currently reached global maximum, so it returns to this point each time temperature is reduced. Thus, as temperature decreases, the algorithm focuses on the most promising areas. The algorithm is terminated when temperature becomes below some pre-specified level. The drawback of the algorithm is its slow speed of convergence, which is fully compensated by its benefits when computational efficiency is not a major issue.
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Appendix C
This appendix contains all programming codes in MATLAB that were used for estimation of Binsbergen and Koijen model.
Likelihood function:
function [LL] = Likelihood (parameters,ro_gD, Y, F, G, ro, cappa)
M2=[1 1 0 0; B2*(parameters(3)-parameters(4)) 0 B2 -B1];
COV=[parameters(7)^2 parameters(5)*parameters(7)*ro_gD
%Bounds for parameters are introduced to increase estimation speed
Bounds=[.001 .151; .001 .151; .001 .999; .001 .999; .001 .601; .001 .601; .001 .601; -.999 .999; -.999 -.999]
%Scaling vector for generation of new points C=[1 1 7 7 4 4 4 13 13]
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Page 46 X=X_opt;
f=f_opt;
T=T*rT;
end
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Bibliography
Alizamir, S., Rebennack, S., Pardalos, P., Improving the Neighborhood Selection Strategy in Simulated Annealing Using the Optimal Stopping Problem, Simulated Annealing, Cher Ming Tan (Ed.), ISBN: 978-953-7619-07-7, InTech, DOI:
10.5772/5571. Available from:
http://www.intechopen.com/books/simulated_annealing/improving_the_neighbor hood_selection_strategy_in_simulated_annealing_using_the_optimal_stopping_pr ob
Ang, A., and J. Liu, 2004. How to discount cashflows with time-varying expected returns. Journal of Finance, 59: 2745–2783.
Banzal, R., Yaron, A., 2004. Risks for the long-run: a potential resolution of asset pricing puzzles. Journal of Finance, 59(4), 1481-1509.
Ball, Ray, 1978. Anomalies in relationships between securities' yields and yield-surrogates. Journal of Financial Economics, 6: 103-126.
Bekaert, G., E. Engstrom, and S. R. Grenadier, 2001. Stock and bond pricing in an affine economy. NBER Working Paper No. 7346.
Brennan, M. J., and Y. Xia, 2005. Persistence, predictability, and portfolio planning. Working Paper, UCLA.
Burnside, C., 1998. Solving asset pricing models with gaussian shocks. Journal of Economic Dynamics and Control, 22: 329–340.
Campbell, John Y., 1991. A variance decomposition for stock returns. Economic Journal, 101, 157-179.
Campbell, John Y. and Robert Shiller, 1988. The dividend-price ratio and expectations of future dividends and discount factors. Review of Financial Studies, 1:195–227.
Campbell, John Y. and Thompson, S., 2007. Predicting excess stock returns out of sample: can anything beat the historical average? Review of Financial Studies 21(4): 1509-1531.
Cochrane, J. H., 1991. Explaining the variance of price-dividend ratios. Review of Financial Studies, 5(2): 243-280.
Cochrane, J. H., 2007. The dog that did not bark: A defense of return predictability. Review of Financial Studies, 21 (4): 1533–1575.
Cochrane, J. H.,2008. State-space vs. VAR models for stock returns. Working paper. University of Chicago.
Dow, Charles H., 1920. Scientific stock speculation. The Magazine of Wall Street (New York).
Fama, E. and Kenneth French, 1988. Dividend yields and expected stock returns.
Journal of Financial Economics, 22:3–27.
Page 48
Fama, E. and Kenneth French, 2000. Disappearing dividends: changing firm characteristics or lower propensity to pay. Journal of Financial Economics, 60: 3-43.
Fernandez-Villaverde, J., Rubio-Ramirez, J., and Santos, M., 2006. Convergence properties of the likelihood of computed dynamic models. Economitrics, 74: 93-119.
Ferson, W. E., Sarkassian, S., Simin, T. T., 2003. Spurious regressions in financial economics? Journal of Finance, 58 (4), 1393-1413
Flood, Robert P., Robert J. Hodrick, and Paul Kaplan, 1986. An evaluation of recent evidence on stock market bubbles. Unpublished manuscript (National Bureau of Economic Research, Cambridge, MA).
Gabaix, X., 2007. Linearity-generating processes: a modeling tool yielding closed forms for asset prices. Working paper MIT.
Goetzmann W. N., and P. Jorion, 1995. A longer look at dividend yields. Journal of Business, 68: 483-508.
Goffe, W., Ferrier, G., Rogers, J., 1992. Global optimization of statistical functions with simulated annealing. Journal of Econometrics, 60: 65-99.
Golez, B, 2012. Expected returns and dividend growth rates implied in derivative markets. Available at http://ssrn.com/abstract=1271607
Gordon, Myron J., 1959. Dividends, earnings and stock prices. Review of Economics and Statistics (The MIT Press), 41 (2): 99-105.
Goyal, A., and I. Welch, 2008. A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies, 21:
1455-1508.
Hamilton, J. D., 1994. Time series analysis. Princeton University Press, Princeton, NY.
Hansen, L., Heaton, J., Li, N., 2008. Consumption strikes back? Measuring long-run risk. Journal of Political Economy, 116 (2): 260-302
Harvey, A., 1989. Forecasting, structural time series models and the Kalman Filter. Cambridge University Press, Cambridge, UK.
Julier, S., Uhlmann, J., 1997. A new extension of the Kalman Filter to nonlinear systems. Proceedings SPIE Signal Processing, Sensor Fusion, and Target Recognition VI, 3068: 182-193.
Koijen R., and J. van Binsbergen, 2010. Predictive regressions: A present-value approach. Journal of Finance, 65 (4).
Lacerda, F., and P. Santa-Clara, 2010. Forecasting dividend growth to better predict returns. Working Paper. Available at http://docentes.fe.unl.pt/~psc
Lettau, M., and Ludvigson, S., 2005. Expected returns and expected dividend growth. Journal of Financial Economics, 76: 585-626.
Page 49
Lettau, M., and S. van Nieuwerburgh, 2008. Reconciling the return predictability evidence. Review of Financial Studies, 21 (4):1607–1652.
Lewellen, J. W., 2004. Predicting returns with financial ratios. Journal of Financial Economics, 74: 209-235.
Menzly, L., T. Santos, and P. Veronesi, 2004. Understanding predictability.
Journal of Political Economy, 112(1):1–47.
Pastor, L., and P. Veronesi, 2003. Stock valuation and learning about profitability.
Journal of Finance, 58: 1749–1789.
Pastor, L., and P. Veronesi, 2006. Was There a Nasdaq Bubble in the Late 1990s?
Journal of Financial Economics, 81: 61–100.
Pastor, L., and R. F. Stambaugh, 2006. Predictive systems: living with imperfect predictors. Working Paper, University of Chicago and CRSP.
Rozeff, Michael, 1984. Stock prices and social dynamics. Brookings Papers on Economic Activity, 2: 457-498.
Rytchkov, O., 2007. Filtering out expected dividends and expected returns.
Unpublished paper, Texas A&M.
Stambaugh, R., 1999. Predictive regressions. Journal of Financial Economics, 54:
375-421.
Trojani, F., and Piatti I., 2012. Dividend growth predictability and the price-dividend ratio. Swiss finance institute research paper, 12-42. Available at SSRN:
http://ssrn.com/abstract=2077381
Yun, J., 2011. A present-value approach to variable selection. Unpublished Manuscript.
BI Norwegian Business School – Preliminary Thesis Report
Adjusted price-dividend ratio and stock returns predictability
Thesis supervisor: Bruno Gerard
Hand-in date:
15.01.2013
Campus:
BI Oslo
Programme:
Master of Science in Financial Economics
Master of Science in Business and Economics with major in Finance
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1. INTRODUCTION ... 1
2. MODEL ... 6
2.1HISTORICAL MEAN ADJUSTMENT ... 7
2.2ARMA ADJUSTMENT ... 8
2.3AR ADJUSTMENT ... 9
3. STATISTICAL METHODOLOGY ... 9
4. DATA ... 10
5. PRELIMINARY ESTIMATION RESULTS ... 11
6. BIBLIOGRAPHY ... 13
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The question of predictability of stock returns has always played an important role in financial economics. Researchers have been concerned with major economic forces, which drive a capital gain process and implications of various macroeconomic shocks for the equity returns. Moreover, this question was the one of the great importance for ordinary participants of financial markets, because any evidence of predictability would generate feasible trading strategies and lead to better understanding of portfolio management. Deep study in this field has been started quite recently, because in 60-70s the efficient market hypothesis was assumed to reflect the reality by majority of researchers, so predictability of stock returns was considered to be impossible. Nevertheless, the hypothesis that dividend price ratio (D/P) forecasts returns has always existed among researchers and practitioners (Dow, 1920 and Ball, 1978). The intuition of the hypothesis is that dividends is high relative to stock prices when expected returns are high given that future dividend growth rates remain constant. This intuition comes directly from the famous Dividend Discount Model (DDM) initially proposed by Gordon in 1959. Number of researchers found statistical evidence (mostly for annual data) that support the hypothesis. See, for example, Rozeff (1984), Flood, Hodrick and Kaplan (1986) and Campbell and Shiller (1987). However, the paper of Fama and French (1988) is often considered to be fundamental in this field. They not only confirmed statistical significance of dividend yields for prediction of future returns, but also discovered that forecasting power increases with the return horizon and provided strong economical intuition to support their findings.
Another impact on the empirical research of stock returns predictability came from the paper of Campbell and Shiller (1988). They proposed log-linearized present-value model, which produces a simple relation between log price-dividend ratio and expectations of log dividend growth rates and log returns at the price of small approximation error. Thus, they stress the fact that variation of price-dividend ratio may reflect not only changes in expected returns, but also changes in expected future dividends and we should not expect parsimonious results by using regular price-dividend ratio solely to predict future returns. This fact was also pointed out by Menzly, Santos and Veronesi (2004) and Goetzmann and Jorion (1995). Moreover, many researchers claim that dividends are highly persistent implying at least some persistency in dividend yields. Hence, returns predictability may be mistakenly supported due to this persistency. See, for
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a conclusion that stock returns and dividend growth predictability are best studied jointly.
There were many other valuation ratios, whose ability to forecast future returns was tested. However, what makes dividend yield special is strong intuitive interpretation and decent theoretical basis behind it (DDM). In turn, it is not as easy to construct convenient economical arguments, for example, for book-to-market ratio to support the hypothesis that it predicts future returns. Hence, it is not a big surprise that since the publication of Campbell and Shiller paper (1988), many other papers with different versions of present-value model appeared. See Koijen and van Binsbergen (2010), Lacerda and Santa-Clara (2010, working paper), Trojani and Piatti (2012), Cochrane (2007), Lettau and Van Niewerburgh (2008), Pasto and Veronesi (2003), Pastor and Veronesi (2006), Bekaert, Engstrom and Grenadier (2001), Burnside (1998), Ang and Liu (2004), Brennan and Xia (2005) and Rytchkov (2007). In cases of Pasto and Veronesi (2003), Pastor and Veronesi (2006), Bekaert, Engstrom and Grenadier (2001), Ang and Liu (2004) and Brennan and Xia (2005) price-dividend ratio is presented as an indefinite integral of exponentially-quadratic terms making empirical part of the work much more involved relative to other papers. They employ either generalized method of moments or two-step procedure to estimate their model. In turn, Koijen and van Binsbergen (2010), Trojani and Piatti (2012), Rytchkov (2007) and Cochrane (2007) combine the present-value model with the assumption that both expected returns and expected dividend growth are latent variables that follow an exogenously-specified time-series model. Then, they assume normality of the shocks to estimate the model with the maximum likelihood. Finally, they use filtering techniques to uncover expected returns and expected dividend growth rates. Rytchkov (2007) and Cochrane (2007) focus more on methodology construction, derivation of main properties of state-space models, applicability of Kalman filter and relaxation of different assumptions and the consequences for estimation techniques. Koijen and van Binsbergen (2010) concentrate on the empirical side of the latent-variables approach and aggregate the whole history of the price-dividend ratio and dividend growth rates to deliver predictors for future returns and dividend growth rates. Since our master's thesis is closely related to the paper of Koijen and van Binsbergen (2010), we proceed to a more detailed discussion of their work.
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dividend growth rates as latent variables, which follow low-order autoregressive processes. Following Pastor and Stamabaugh (2006) and Cochrane (2008) they assume that expected returns follow AR(1)-process, however they treat expected dividend growth rates differently depending on the choice of reinvestment strategy. Since they try to avoid effects of seasonality in dividend payments, they consider an annual model, which requires taking into account how dividends received within a particular year are reinvested. Two extreme reinvestment strategies are studied in detail. First, they reinvest dividends in 30-day T-bill and call it cash-invested dividends. Second, they reinvest dividends in the aggregate stock market and refer to it as market-invested dividends. Market-invested dividends appear to be far more volatile than cash-invested dividends supporting the fact that the choice of reinvestment strategy is extremely important.
Interestingly enough, they assume that cash-invested expected growth rates are an AR(1)-process. By means of analytical argument this assumption implies that market-invested expected growth rates also exhibit moving average component and follow ARMA(1,1)-process. After specification of processes for latent variables, they employ log-linearization of realized returns in order to connect expected and realized variables through measurement equations. Then, they use Kalman filter not only to estimate unknown parameters, but also to filter out most likely values of latent variables. Later they find what fraction of realized returns and dividend growth variation can be explained by expected values, and compare these numbers to values of regular linear regressions with price-dividend ratio as a predictive variable. They discover that their model is superior to ordinary linear regression for both cash and market-invested dividends. Additionally, they emphasize that it is extremely important to study predictability of stock returns and dividend growth rates jointly because there is a tight relationship between the predictive coefficients of returns and dividend growth rates and the persistence of the dividend yield.
As long as log-linearized present-value model relates price dividend ratio to the expected returns and expected dividend growth rates, there is an alternative methodology, which does not require complex estimation techniques. The idea comes from Lacerda and Santa-Clara (2010), who argue that one can adjust price-dividend ratio for variations in expected growth rates and use the adjusted ratio to forecast future returns. However, this approach requires us to assume how market
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(2010) use moving average of historical growth rates as an estimate for future dividend growth. However, they reasonably stress the fact that the existence of better predictors is an open question and if such predictors are found, then they will presumably lead to better estimates of expected returns. Then, they transform an initial dividend-price ratio as follows:
̅̅̅
̅ where - log dividend-price ratio;
̅̅̅- historical averaged log dividend growth;
̅ (̅̅̅̅̅̅) .
Finally, they use this new adjusted ratio as a predictive variable. The intuition behind this adjustment is to distinguish between change in dividend-price ratio due to changes in expected dividend growth rate and due to changes in expected future returns. They find out that adjusted dividend yield explains more variation in future returns than ordinary dividend yield both in- and out-of-sample.
The fact that Lacerda and Santa-Clara reveal statistically significant predictive power even out-of-sample becomes even more important in the light of Goyal and Welch (2008), who strongly criticize any evidence of returns predictability. They argue that in the real world we cannot use information that is
The fact that Lacerda and Santa-Clara reveal statistically significant predictive power even out-of-sample becomes even more important in the light of Goyal and Welch (2008), who strongly criticize any evidence of returns predictability. They argue that in the real world we cannot use information that is