- Email: [email protected]
A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles
Finance Research Letters xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles Paul E. Godek ⇑ MiCRA – Microeconomic Consulting and Research Associates, Washington, DC, United States
a r t i c l e
i n f o
Article history: Received 27 January 2015 Accepted 28 March 2015 Available online xxxx JEL classification: G10 General financial markets G11 Investment decisions G14 Information and market efficiency G17 Financial forecasting and simulation Keywords: Trend reversion Ornstein–Uhlenbeck process Bubbles and inverse bubbles Efficient markets
a b s t r a c t Appraising the current valuation of equity markets is a popular pastime for academics, investors, and pundits alike. Here I consider a measure of valuation based on a century-long trend of U.S. equity returns and the tendency of returns to revert (eventually) to that trend. The approach here is to incorporate a simple trend into an Ornstein–Uhlenbeck process. The empirical results offer some support for the theoretical description, though not to an extent that would cause harm to the concept of efficient markets. The reconciliation of trend reversion with market efficiency lies in the weakness of the trend’s ‘‘gravitational pull.’’ The results do, however, provide an operational measure for describing markets as over- or under-valued, which indicates that ‘‘bubbles’’ and ‘‘inverse bubbles’’ are both common. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Observers of equity markets – academics, investors, and pundits alike – often appraise the current ‘‘valuation’’ of those markets. Are equities under- or over-valued, cheap or expensive, relative to
⇑ Tel.: +1 202 467 2505. E-mail address: [email protected] http://dx.doi.org/10.1016/j.frl.2015.03.006 1544-6123/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
2
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
various metrics?1 Without contradicting any of that analysis, I consider an alternative measure of valuation, based on the century-long trend of U.S. equity returns and the tendency of returns to revert (eventually) to that trend. The procedure described here involves the incorporation of a simple trend into an Ornstein–Uhlenbeck process. The empirical results are generally consistent with the theoretical description, though not to an extent that would cause harm to the concept of efficient markets. The reconciliation of trend reversion with market efficiency lies in the weakness of the trend’s ‘‘gravitational pull.’’ In any case, the approach not only provides an operational measure of whether equity markets can, at any point in time, be characterized as over- or under-valued, it also indicates that ‘‘bubbles’’ and ‘‘inverse bubbles’’ are both common, with inverse bubbles occurring somewhat more frequently.2 2. Expected returns, actual returns Consider an index that reflects the value of a defined set of equities, which takes the value It at time t and which returns an average rate c. Based on the average return, the expected value of It is:
EðIt Þ ¼ ðI0 Þect
ð1Þ
The starting point (I0 ) can be set to 1 and the index is conveniently expressed in terms of its natural logarithm (log). The value of the log of the index at time t is:
V t ¼ lnðIt Þ
ð2Þ
And the expected value (again, based on the average return) of the log index is a linear function of time:
EðV t Þ ¼ EðlnðIt ÞÞ ¼ ct
ð3Þ
This approach will be used later to derive a prospective trend for equity returns. The change over time in the value of the index ðV t Þ:
V tþy V t
ð4Þ
equals the percentage return over the period t to t + y. If the index follows a path known as a random walk with drift, then the first differences of the index are described by the process:
V tþ1 V t ¼ a þ et
ð5Þ
where a is the average one-period return and e is a white-noise residual. An equivalent characterization results from moving V t to the right-hand side of the equation:
V tþ1 ¼ a þ V t þ et
ð6Þ 3
which is the standard form for a unit-root process with drift. As is well known, equation 6 fits the data with precision. For 100 years of inflation-adjusted, totalreturn, logarithmic, U.S. equity index values at quarterly intervals, an ordinary least-squares
1 Measures of valuation typically use price-earnings ratios, dividend yields, or other metrics. Some of that research has been recognized by the 2013 Nobel Prize in Economics shared by Eugene F. Fama, Lars Peter Hansen, and Robert J. Shiller. Shiller (2014) provides a comprehensive survey of the relevant literature. The approach here is preceded by many related studies. See, in particular, Lo and MacKinlay (2002): [chapters 2 and 6], Campbell et al. (1997): [chapter 9], and Poterba and Summers (1988). On the application of mean-reversion models to interest rates and option values, see Lo and Wang (1995) and Vasicek (1977). Additional discussions of mean-reversion models can be found in Hull (2005): [chapter 28], Barndorff-Nielsen and Shephard (2001), and Dixit and Pindyck (1994): [chapter 3]. 2 In the physical sciences, a bubble is described as a gas enclosed within a liquid sphere; an anti- or inverse-bubble is a liquid enclosed within a gas sphere. Here, the term inverse bubble refers to periods when cumulative equity returns are substantially below their long-run trend. Fama (2014) and Shiller (2014) both discuss the concept of bubbles and provide surveys of the relevant literature. General overviews of the long-run tendencies of equity returns are provided in Malkiel (2011), Dimson et al. (2002), Shiller (2000), Shleifer (2000), Cornell (1999), and Siegel (1998). 3 In continuous time the model would be described as geometric-Brownian-motion-with-drift, a type of Wiener process. See Dixit and Pindyck (1994): [chapter 3].
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
3
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx Table 1 Random walk model of index value. Intervals Dependent variable Observations
Quarterly, 1st Quarter 1914 – 4th Quarter 2013 Vt+1 399
Independent variable
Coefficient
Standard error
t-Stat
Intercept Vt R-square Durbin Watson
0.019 0.999 0.998 2.054
0.009 0.002
2.12 411.12
Data source: www.econ.yale.edu/~shiller/data.htm.
regression generates the results in Table 1.4 The Durbin-Watson statistic indicates an absence of significant autocorrelation in the residual and the coefficient on V t is not significantly different from one.5 In this model, the best estimate of next period’s value of the index is this period’s value plus the average periodic return. Of course, the high R-square does not indicate a high degree of predictability of the one-period return. It indicates only that the level of the index at any given point in time is highly informative about the level at the next point in time. 3. Trend reversion The same U.S. equity index along with a linear trend, over the last 100 years, is shown in Fig. 1. The trend line is constructed by adding the average periodic return (.016) each quarter, beginning at the same starting value as the index. Because the trend line is not constructed by regressing the value of the index on time, the deviations from the trend do not sum to zero. Given this construction, there is no forced symmetry around the trend and, therefore, no built-in tendency toward trend reversion. Determining whether a random walk with drift is consistent with a stationary trend is inherently difficult. The difficulty follows, in part, from the dominance of the drift over the variance of the series in the long run.6 Nonetheless, there is a specific form of trend reversion that generates straight-forward and testable implications. In this context, trend reversion implies that returns from a point in time forward will tend to be above average to the extent that returns to that point in time have been below average, and mutatis mutandis. A tractable way to incorporate trend reversion into an empirical model of equity returns is with a discrete-time version of an Ornstein–Uhlenbeck process, as follows:
V tþ1 V t ¼ d þ bðV t V t Þ þ et
ð7Þ
V t
where is the expected value of the index (which in this case is the trend line described above) at time t. The trend-reversion parameter b measures the effect on the return of the index being above or below the expected value.7 This parameter (which should be negative in this form of the model) is said to measure the ‘‘speed of adjustment’’ or the ‘‘gravitational pull’’ relative to the trend. 4 The empirical results in this paper are derived from the publicly-available data base on U.S. equity returns generated by Robert J. Shiller, available at www.econ.yale.edu/~shiller/data.htm. Those data are at monthly intervals; quarterly intervals are generated by using the values for March, June, September, and December. The choice of any particular interval is, of course, arbitrary. 5 In this model – with an intercept term and a white-noise residual – it is not necessary to use a revised test criterion, such as Dickey-Fuller, to evaluate the value of the slope coefficient. See Hamilton (1994): [chapter 17]. A standard statistical test for whether a coefficient is equal to a particular value is an F test with (q, n–k) degrees of freedom, where q is the number of restrictions being tested, n is the number of observations, and k is the number of parameters in the model including the intercept. In this case, the F statistic is .16 with 1 and 397 degrees of freedom. The hypothesis that the coefficient on V t is equal to one is not rejected at standard confidence levels, even if the Dickey-Fuller criterion is applied. See Greene (2003): [chapter 6]. 6 See Hamilton (1994): [chapters 15–17] for an extensive discussion of that issue. Dixit and Pindyck (1994): [chapter 3] and Shiller (2014) also discuss the difficulty of the exercise here: distinguishing a ‘‘Wiener process from an Ornstein–Uhlenbeck process.’’ Shiller (2014): [page 1490.] 7 See Steele (2001): [chapter 9], Campbell et al. (1997): [chapter 9], and Dixit and Pindyck (1994): [chapter 3] for general discussions of the process. In this form the process is sometimes referred to as a trending Ornstein–Uhlenbeck process.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
4
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0
Data source: www.econ.yale.edu/~shiller/data.htm Fig. 1. Log equity index and average-return trend: 1914–2013 (100 years at quarterly intervals).
As described above in Eqs. (1-3), the trend is constructed using only the average periodic return c:
V t ¼ ct With that value for
ð8Þ V t ,
returns would follow the path:
V tþ1 V t ¼ d þ bðV t ctÞ þ et
ð9Þ
Moving V t to the right side and rearranging terms generates the following specification:
V tþ1 ¼ d þ ðb þ 1ÞV t ðbcÞt þ et
ð10Þ
The empirical implications of this model differ from the random-walk-with-drift model in several ways. First, the coefficient on V t differs from one by the trend-reversion parameter b; thus the coefficient on V t should be significantly different from (and less than) one. Second, given that b should be negative, the estimated coefficient on time should be positive. Finally, the model implies an inherent and testable constraint: one minus the regression coefficient on V t multiplied by the average periodic return will equal the negative of the coefficient on time (t).8 Applied to the same data as before, an ordinary-least-squares procedure can again be employed to estimate the model. The results are shown in Table 2. In this model, the Durbin-Watson statistic again indicates an absence of significant autocorrelation in the residual. How do the implications of the model fare? The implied trend reversion parameter (b = .962 – 1.0 = –.038) is negative and the coefficient on time is positive. However, the standard statistical test cannot reject the joint hypothesis that the coefficient on V t is equal to one and the coefficient on time
8 To my knowledge, these characteristics of a simple trend incorporated into a discrete-time, trending Ornstein–Uhlenbeck process have not been described before. The literature on this subject is vast and encompasses several fields of study, however, so that contention may well be incorrect.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
5
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx Table 2 Trend reversion model of index value. Intervals Dependent variable Observations
Quarterly, 1st Quarter 1914 – 4th Quarter 2013 Vt+1 399
Independent variable
Coefficient
Standard error
t-Stat
Intercept Vt t R-square Durbin Watson
0.006 0.962 0.001 0.998 2.018
0.010 0.014 0.000
0.60 70.76 2.74
Data source: www.econ.yale.edu/~shiller/data.htm.
is equal to zero; although that is not particularly surprising given the values of the coefficients.9 Finally, recall the constraint implied by the model: one minus the coefficient on V t multiplied by the average periodic return will equal the negative of the coefficient on t. The average periodic return is .016, so the constraint implies the following calculation:
ð:038Þð:016Þ :001
ð11Þ
The standard statistical test finds that the constraint is not rejected.10 In sum, the empirically results are broadly consistent with implications of the model. One should appreciate, however, the lack of meaningful predictive power in the model. While the trend-reversion parameter does have a substantial effect on the predicted return (by .038 times the deviation from the trend), running the regression with return ðV tþ1 V t Þ as the dependent variable results in an R-square of only .02. The model explains almost none of the variance of the return. Predictions based on this model would still be imprecise, as they are with the simpler randomwalk-with-drift model.
4. Bubbles A modified version of Fig. 1 more clearly reveals both the variability of the index and the weakness of the gravitational pull of the trend line. Fig. 2 plots the difference between the index and the trend line, so the vertical axis measures the percentage difference between the index and the trend. The horizontal lines at plus and minus 30 percent are highlighted. It is clear that the index can stay well above or below the trend line for long periods of time: see 1917 through 1925 on the low side and 1958 through 1969 on the high-side. In addition, substantial deviations, say greater than 30 percent, are somewhat more likely to occur below the trend than above it. Of the 400 quarters in the sample, the index value is 30 percent or more below the trend for 115 quarters; the index value is 30 percent or more above the trend for 86 quarters. Indeed, bubbles and inverse bubbles are not at all unusual. The market is about as likely to be outside of the range plus or minus 30 percent (201 out of 400 quarters) as to be within that range (199 out of 400 quarters). One might say that the index is prone to substantial deviations above and below the trend line, which sooner or later reverse – like a wayward dog on a long leash.
9 This model does require the Dickey-Fuller criterion for statistical tests. See Hamilton (1994): [chapter 17]. In the joint test, the F statistic is 3.84 with 2 and 396 degrees of freedom. The joint hypothesis that the coefficient on V t is equal to one and the coefficient on time is equal to zero is not rejected at standard levels of significance. 10 In this case, the F statistic is .69 with 1 and 396 degrees of freedom. The hypothesis regarding the linear combination implied by the model is not rejected at standard levels under the Dickey-Fuller criterion.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
6
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.101910 -0.20 -0.30 -0.40 -0.50 -0.60 -0.70 -0.80 -0.90 -1.00
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
Data source: www.econ.yale.edu/~shiller/data.htm Fig. 2. Difference between index and trend: 1914–2013 (100 years at quarterly intervals).
5. Conclusion The basis for trend reversion in equity returns is straightforward. The return on equity is determined by two components: dividends and appreciation. The appreciation component and, therefore, the return on equity can only be observed over intervals of time. If the long-run return on equity has a reasonably stable equilibrium value, then deviations from the average should eventually (over time) be resolved. Indeed, the empirical results here provide some support for characterizing the market as over- or under-valued depending on its level relative to the trend. The tension between trend reversion and market efficiency is also straightforward. Trend reversion might imply a degree of predictability that is inconsistent with an informationally-efficient market. The reconciliation of trend reversion with market efficiency lies in the weakness of the trend’s gravitational pull. Acknowledgements The author thanks without implicating Hal Baseman, Frank Fabozzi, Brian Lucey, Craig McCann, Todd Hasson, Sameer Mukherjee, and an anonymous referee. References Barndorff-Nielsen, Ole E., Shephard, Neil, 2001. Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. Roy. Stat. Soc., Ser. B (Stat. Methodol.) 63 (2), 167–241. Campbell, John Y., Lo, Andrew W., Craig MacKinlay, A., 1997. The Econometrics of Financial Markets. Princeton University Press. Cornell, Bradford, 1999. The Equity Risk Premium: The Long Run Future of the Stock Market. John Wiley & Sons. Dimson, Elroy, Marsh, Paul, Staunton, Mike. 2002. Triumph of the Optimists: 101 Years of Global Investment Returns. Princeton University Press. Dixit, Avinash K., Pindyck, Robert S., 1994. Investment Under Uncertainty. Princeton University Press. Fama, Eugene F., 2014. Two pillars of asset pricing. Am. Econ. Rev. 104 (6), 1467–1485. Greene, William H., 2003. Econometric Analysis, fifth ed. Prentice Hall. Hamilton, James D., 1994. Time Series Analysis. Princeton University Press. Hull, John C., 2005. Options, Futures, and other Derivatives, sixth ed. Prentice Hall. Lo, Andrew W., Craig MacKinlay, A., 2002. A Non-Random Walk Down Wall Street. Princeton University Press. Lo, Andrew W., Wang, Jiang, 1995. Implementing option pricing models when asset returns are predictable. J. Finance 50 (1), 87–129. Malkiel, Burton G., 2011. A Random Walk Down Wall Street, tenth ed. Norton & Company.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
7
Poterba, James M., Summers, Lawrence H., 1988. Mean reversion in stock returns: evidence and implications. J. Financ. Econ. 22, 27–60. Shiller, Robert J., 2000. Irrational Exuberance. Princeton University Press. Shiller, Robert J., 2014. Speculative asset prices. Am. Econ. Rev. 104 (6), 1486–1517. Shleifer, Andrei, 2000. Inefficient Markets, An Introduction to Behavioral Finance. Oxford University Press. Siegel, Jeremy J., 1998. Stocks for the long run, second ed. McGraw-Hill. Steele, J. Michael, 2001. Stochastic Calculus and Financial Applications. Springer. Vasicek, Oldrich, 1977. An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles Paul E. Godek ⇑ MiCRA – Microeconomic Consulting and Research Associates, Washington, DC, United States
a r t i c l e
i n f o
Article history: Received 27 January 2015 Accepted 28 March 2015 Available online xxxx JEL classification: G10 General financial markets G11 Investment decisions G14 Information and market efficiency G17 Financial forecasting and simulation Keywords: Trend reversion Ornstein–Uhlenbeck process Bubbles and inverse bubbles Efficient markets
a b s t r a c t Appraising the current valuation of equity markets is a popular pastime for academics, investors, and pundits alike. Here I consider a measure of valuation based on a century-long trend of U.S. equity returns and the tendency of returns to revert (eventually) to that trend. The approach here is to incorporate a simple trend into an Ornstein–Uhlenbeck process. The empirical results offer some support for the theoretical description, though not to an extent that would cause harm to the concept of efficient markets. The reconciliation of trend reversion with market efficiency lies in the weakness of the trend’s ‘‘gravitational pull.’’ The results do, however, provide an operational measure for describing markets as over- or under-valued, which indicates that ‘‘bubbles’’ and ‘‘inverse bubbles’’ are both common. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Observers of equity markets – academics, investors, and pundits alike – often appraise the current ‘‘valuation’’ of those markets. Are equities under- or over-valued, cheap or expensive, relative to
⇑ Tel.: +1 202 467 2505. E-mail address: [email protected] http://dx.doi.org/10.1016/j.frl.2015.03.006 1544-6123/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
2
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
various metrics?1 Without contradicting any of that analysis, I consider an alternative measure of valuation, based on the century-long trend of U.S. equity returns and the tendency of returns to revert (eventually) to that trend. The procedure described here involves the incorporation of a simple trend into an Ornstein–Uhlenbeck process. The empirical results are generally consistent with the theoretical description, though not to an extent that would cause harm to the concept of efficient markets. The reconciliation of trend reversion with market efficiency lies in the weakness of the trend’s ‘‘gravitational pull.’’ In any case, the approach not only provides an operational measure of whether equity markets can, at any point in time, be characterized as over- or under-valued, it also indicates that ‘‘bubbles’’ and ‘‘inverse bubbles’’ are both common, with inverse bubbles occurring somewhat more frequently.2 2. Expected returns, actual returns Consider an index that reflects the value of a defined set of equities, which takes the value It at time t and which returns an average rate c. Based on the average return, the expected value of It is:
EðIt Þ ¼ ðI0 Þect
ð1Þ
The starting point (I0 ) can be set to 1 and the index is conveniently expressed in terms of its natural logarithm (log). The value of the log of the index at time t is:
V t ¼ lnðIt Þ
ð2Þ
And the expected value (again, based on the average return) of the log index is a linear function of time:
EðV t Þ ¼ EðlnðIt ÞÞ ¼ ct
ð3Þ
This approach will be used later to derive a prospective trend for equity returns. The change over time in the value of the index ðV t Þ:
V tþy V t
ð4Þ
equals the percentage return over the period t to t + y. If the index follows a path known as a random walk with drift, then the first differences of the index are described by the process:
V tþ1 V t ¼ a þ et
ð5Þ
where a is the average one-period return and e is a white-noise residual. An equivalent characterization results from moving V t to the right-hand side of the equation:
V tþ1 ¼ a þ V t þ et
ð6Þ 3
which is the standard form for a unit-root process with drift. As is well known, equation 6 fits the data with precision. For 100 years of inflation-adjusted, totalreturn, logarithmic, U.S. equity index values at quarterly intervals, an ordinary least-squares
1 Measures of valuation typically use price-earnings ratios, dividend yields, or other metrics. Some of that research has been recognized by the 2013 Nobel Prize in Economics shared by Eugene F. Fama, Lars Peter Hansen, and Robert J. Shiller. Shiller (2014) provides a comprehensive survey of the relevant literature. The approach here is preceded by many related studies. See, in particular, Lo and MacKinlay (2002): [chapters 2 and 6], Campbell et al. (1997): [chapter 9], and Poterba and Summers (1988). On the application of mean-reversion models to interest rates and option values, see Lo and Wang (1995) and Vasicek (1977). Additional discussions of mean-reversion models can be found in Hull (2005): [chapter 28], Barndorff-Nielsen and Shephard (2001), and Dixit and Pindyck (1994): [chapter 3]. 2 In the physical sciences, a bubble is described as a gas enclosed within a liquid sphere; an anti- or inverse-bubble is a liquid enclosed within a gas sphere. Here, the term inverse bubble refers to periods when cumulative equity returns are substantially below their long-run trend. Fama (2014) and Shiller (2014) both discuss the concept of bubbles and provide surveys of the relevant literature. General overviews of the long-run tendencies of equity returns are provided in Malkiel (2011), Dimson et al. (2002), Shiller (2000), Shleifer (2000), Cornell (1999), and Siegel (1998). 3 In continuous time the model would be described as geometric-Brownian-motion-with-drift, a type of Wiener process. See Dixit and Pindyck (1994): [chapter 3].
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
3
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx Table 1 Random walk model of index value. Intervals Dependent variable Observations
Quarterly, 1st Quarter 1914 – 4th Quarter 2013 Vt+1 399
Independent variable
Coefficient
Standard error
t-Stat
Intercept Vt R-square Durbin Watson
0.019 0.999 0.998 2.054
0.009 0.002
2.12 411.12
Data source: www.econ.yale.edu/~shiller/data.htm.
regression generates the results in Table 1.4 The Durbin-Watson statistic indicates an absence of significant autocorrelation in the residual and the coefficient on V t is not significantly different from one.5 In this model, the best estimate of next period’s value of the index is this period’s value plus the average periodic return. Of course, the high R-square does not indicate a high degree of predictability of the one-period return. It indicates only that the level of the index at any given point in time is highly informative about the level at the next point in time. 3. Trend reversion The same U.S. equity index along with a linear trend, over the last 100 years, is shown in Fig. 1. The trend line is constructed by adding the average periodic return (.016) each quarter, beginning at the same starting value as the index. Because the trend line is not constructed by regressing the value of the index on time, the deviations from the trend do not sum to zero. Given this construction, there is no forced symmetry around the trend and, therefore, no built-in tendency toward trend reversion. Determining whether a random walk with drift is consistent with a stationary trend is inherently difficult. The difficulty follows, in part, from the dominance of the drift over the variance of the series in the long run.6 Nonetheless, there is a specific form of trend reversion that generates straight-forward and testable implications. In this context, trend reversion implies that returns from a point in time forward will tend to be above average to the extent that returns to that point in time have been below average, and mutatis mutandis. A tractable way to incorporate trend reversion into an empirical model of equity returns is with a discrete-time version of an Ornstein–Uhlenbeck process, as follows:
V tþ1 V t ¼ d þ bðV t V t Þ þ et
ð7Þ
V t
where is the expected value of the index (which in this case is the trend line described above) at time t. The trend-reversion parameter b measures the effect on the return of the index being above or below the expected value.7 This parameter (which should be negative in this form of the model) is said to measure the ‘‘speed of adjustment’’ or the ‘‘gravitational pull’’ relative to the trend. 4 The empirical results in this paper are derived from the publicly-available data base on U.S. equity returns generated by Robert J. Shiller, available at www.econ.yale.edu/~shiller/data.htm. Those data are at monthly intervals; quarterly intervals are generated by using the values for March, June, September, and December. The choice of any particular interval is, of course, arbitrary. 5 In this model – with an intercept term and a white-noise residual – it is not necessary to use a revised test criterion, such as Dickey-Fuller, to evaluate the value of the slope coefficient. See Hamilton (1994): [chapter 17]. A standard statistical test for whether a coefficient is equal to a particular value is an F test with (q, n–k) degrees of freedom, where q is the number of restrictions being tested, n is the number of observations, and k is the number of parameters in the model including the intercept. In this case, the F statistic is .16 with 1 and 397 degrees of freedom. The hypothesis that the coefficient on V t is equal to one is not rejected at standard confidence levels, even if the Dickey-Fuller criterion is applied. See Greene (2003): [chapter 6]. 6 See Hamilton (1994): [chapters 15–17] for an extensive discussion of that issue. Dixit and Pindyck (1994): [chapter 3] and Shiller (2014) also discuss the difficulty of the exercise here: distinguishing a ‘‘Wiener process from an Ornstein–Uhlenbeck process.’’ Shiller (2014): [page 1490.] 7 See Steele (2001): [chapter 9], Campbell et al. (1997): [chapter 9], and Dixit and Pindyck (1994): [chapter 3] for general discussions of the process. In this form the process is sometimes referred to as a trending Ornstein–Uhlenbeck process.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
4
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0
Data source: www.econ.yale.edu/~shiller/data.htm Fig. 1. Log equity index and average-return trend: 1914–2013 (100 years at quarterly intervals).
As described above in Eqs. (1-3), the trend is constructed using only the average periodic return c:
V t ¼ ct With that value for
ð8Þ V t ,
returns would follow the path:
V tþ1 V t ¼ d þ bðV t ctÞ þ et
ð9Þ
Moving V t to the right side and rearranging terms generates the following specification:
V tþ1 ¼ d þ ðb þ 1ÞV t ðbcÞt þ et
ð10Þ
The empirical implications of this model differ from the random-walk-with-drift model in several ways. First, the coefficient on V t differs from one by the trend-reversion parameter b; thus the coefficient on V t should be significantly different from (and less than) one. Second, given that b should be negative, the estimated coefficient on time should be positive. Finally, the model implies an inherent and testable constraint: one minus the regression coefficient on V t multiplied by the average periodic return will equal the negative of the coefficient on time (t).8 Applied to the same data as before, an ordinary-least-squares procedure can again be employed to estimate the model. The results are shown in Table 2. In this model, the Durbin-Watson statistic again indicates an absence of significant autocorrelation in the residual. How do the implications of the model fare? The implied trend reversion parameter (b = .962 – 1.0 = –.038) is negative and the coefficient on time is positive. However, the standard statistical test cannot reject the joint hypothesis that the coefficient on V t is equal to one and the coefficient on time
8 To my knowledge, these characteristics of a simple trend incorporated into a discrete-time, trending Ornstein–Uhlenbeck process have not been described before. The literature on this subject is vast and encompasses several fields of study, however, so that contention may well be incorrect.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
5
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx Table 2 Trend reversion model of index value. Intervals Dependent variable Observations
Quarterly, 1st Quarter 1914 – 4th Quarter 2013 Vt+1 399
Independent variable
Coefficient
Standard error
t-Stat
Intercept Vt t R-square Durbin Watson
0.006 0.962 0.001 0.998 2.018
0.010 0.014 0.000
0.60 70.76 2.74
Data source: www.econ.yale.edu/~shiller/data.htm.
is equal to zero; although that is not particularly surprising given the values of the coefficients.9 Finally, recall the constraint implied by the model: one minus the coefficient on V t multiplied by the average periodic return will equal the negative of the coefficient on t. The average periodic return is .016, so the constraint implies the following calculation:
ð:038Þð:016Þ :001
ð11Þ
The standard statistical test finds that the constraint is not rejected.10 In sum, the empirically results are broadly consistent with implications of the model. One should appreciate, however, the lack of meaningful predictive power in the model. While the trend-reversion parameter does have a substantial effect on the predicted return (by .038 times the deviation from the trend), running the regression with return ðV tþ1 V t Þ as the dependent variable results in an R-square of only .02. The model explains almost none of the variance of the return. Predictions based on this model would still be imprecise, as they are with the simpler randomwalk-with-drift model.
4. Bubbles A modified version of Fig. 1 more clearly reveals both the variability of the index and the weakness of the gravitational pull of the trend line. Fig. 2 plots the difference between the index and the trend line, so the vertical axis measures the percentage difference between the index and the trend. The horizontal lines at plus and minus 30 percent are highlighted. It is clear that the index can stay well above or below the trend line for long periods of time: see 1917 through 1925 on the low side and 1958 through 1969 on the high-side. In addition, substantial deviations, say greater than 30 percent, are somewhat more likely to occur below the trend than above it. Of the 400 quarters in the sample, the index value is 30 percent or more below the trend for 115 quarters; the index value is 30 percent or more above the trend for 86 quarters. Indeed, bubbles and inverse bubbles are not at all unusual. The market is about as likely to be outside of the range plus or minus 30 percent (201 out of 400 quarters) as to be within that range (199 out of 400 quarters). One might say that the index is prone to substantial deviations above and below the trend line, which sooner or later reverse – like a wayward dog on a long leash.
9 This model does require the Dickey-Fuller criterion for statistical tests. See Hamilton (1994): [chapter 17]. In the joint test, the F statistic is 3.84 with 2 and 396 degrees of freedom. The joint hypothesis that the coefficient on V t is equal to one and the coefficient on time is equal to zero is not rejected at standard levels of significance. 10 In this case, the F statistic is .69 with 1 and 396 degrees of freedom. The hypothesis regarding the linear combination implied by the model is not rejected at standard levels under the Dickey-Fuller criterion.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
6
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.101910 -0.20 -0.30 -0.40 -0.50 -0.60 -0.70 -0.80 -0.90 -1.00
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
Data source: www.econ.yale.edu/~shiller/data.htm Fig. 2. Difference between index and trend: 1914–2013 (100 years at quarterly intervals).
5. Conclusion The basis for trend reversion in equity returns is straightforward. The return on equity is determined by two components: dividends and appreciation. The appreciation component and, therefore, the return on equity can only be observed over intervals of time. If the long-run return on equity has a reasonably stable equilibrium value, then deviations from the average should eventually (over time) be resolved. Indeed, the empirical results here provide some support for characterizing the market as over- or under-valued depending on its level relative to the trend. The tension between trend reversion and market efficiency is also straightforward. Trend reversion might imply a degree of predictability that is inconsistent with an informationally-efficient market. The reconciliation of trend reversion with market efficiency lies in the weakness of the trend’s gravitational pull. Acknowledgements The author thanks without implicating Hal Baseman, Frank Fabozzi, Brian Lucey, Craig McCann, Todd Hasson, Sameer Mukherjee, and an anonymous referee. References Barndorff-Nielsen, Ole E., Shephard, Neil, 2001. Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. Roy. Stat. Soc., Ser. B (Stat. Methodol.) 63 (2), 167–241. Campbell, John Y., Lo, Andrew W., Craig MacKinlay, A., 1997. The Econometrics of Financial Markets. Princeton University Press. Cornell, Bradford, 1999. The Equity Risk Premium: The Long Run Future of the Stock Market. John Wiley & Sons. Dimson, Elroy, Marsh, Paul, Staunton, Mike. 2002. Triumph of the Optimists: 101 Years of Global Investment Returns. Princeton University Press. Dixit, Avinash K., Pindyck, Robert S., 1994. Investment Under Uncertainty. Princeton University Press. Fama, Eugene F., 2014. Two pillars of asset pricing. Am. Econ. Rev. 104 (6), 1467–1485. Greene, William H., 2003. Econometric Analysis, fifth ed. Prentice Hall. Hamilton, James D., 1994. Time Series Analysis. Princeton University Press. Hull, John C., 2005. Options, Futures, and other Derivatives, sixth ed. Prentice Hall. Lo, Andrew W., Craig MacKinlay, A., 2002. A Non-Random Walk Down Wall Street. Princeton University Press. Lo, Andrew W., Wang, Jiang, 1995. Implementing option pricing models when asset returns are predictable. J. Finance 50 (1), 87–129. Malkiel, Burton G., 2011. A Random Walk Down Wall Street, tenth ed. Norton & Company.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006
P.E. Godek / Finance Research Letters xxx (2015) xxx–xxx
7
Poterba, James M., Summers, Lawrence H., 1988. Mean reversion in stock returns: evidence and implications. J. Financ. Econ. 22, 27–60. Shiller, Robert J., 2000. Irrational Exuberance. Princeton University Press. Shiller, Robert J., 2014. Speculative asset prices. Am. Econ. Rev. 104 (6), 1486–1517. Shleifer, Andrei, 2000. Inefficient Markets, An Introduction to Behavioral Finance. Oxford University Press. Siegel, Jeremy J., 1998. Stocks for the long run, second ed. McGraw-Hill. Steele, J. Michael, 2001. Stochastic Calculus and Financial Applications. Springer. Vasicek, Oldrich, 1977. An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188.
Please cite this article in press as: Godek, P.E. A simple model of market valuation and trend reversion for U.S. equities: 100 Years of bubbles, non-bubbles, and inverse-bubbles. Finance Research Letters (2015), http:// dx.doi.org/10.1016/j.frl.2015.03.006