The Equity Premium Puzzle- A Model For Its Behavior

ECONOMICS

The Equity Premium Puzzle: A Model for Its Behavior Timothy P. Lavelle

I

n the United States, there exist independent markets for risk free government bond securities

and risky equity securities. A rational/sophisticated investor will build a portfolio out of these two components in such a way as to optimize his portfolio in terms of the ratio of riskiness (or, more specifically, idiosyncratic variability) and expected return (Black and Litterman 1992). Once this equilibrium condition is met (an investor-specific optimal allocation between risky and risk free securities) the rules of arbitrage pricing should keep the return spread between these two securities at an equilibrium level (Brennan, Schwartz and Lagnado 1997). From this notion of an arbitrage pricing model, one can look at historical returns in the United States and appreciate that the aggregate return spread between the market index for equity securities and government debt instruments should be characterized by a function of expected returns, risk, and investor preference for risk. This premium has been roughly 6% over the long run (Kocherlakota 1996). Given the observed level of risk for each asset class, one can then derive the implied risk preference profile of the investing community in aggregate. This stream of analysis has been applied to what has come to be known as the equity premium puzzle. While the fact that a premium exists is not a puzzle to even the most naïve of market observers, it is the magnitude of this premium that is interesting and perplexing. No theory to date has been able to explain it fully, and thus, the focus of this article will be to demystify its enigmatic behavior using a dynamical systems approach. As stated previously, the premium return of equity securities over government debt securities is inevitable due to the greater systematic and idiosyncratic risk of equity returns in the market. The problem, however, is determining whether this premium is appropriate given the level of risk and investors’ risk aversion, or if it is, in fact, extreme. Siegel and Thaler use a utility preference model to describe the level of risk aversion implied by the long run 6% equity premium (Siegel and Thaler 1997). The following anecdote demonstrates this level of risk aversion. Given a fair bet with 50% probability of winning and exactly doubling one’s entire wealth and a 50% probability of losing and exactly halving one’s entire wealth, an individual with the implied level of risk aversion derived from the equity premium would be willing to pay 49% of his or her entire wealth to avoid the gamble (Siegel and Thaler 1997). Thus, if one has wealth of one million dollars and is given a 50% chance to either be worth two million dollars or 500 thousand dollars, that individual would rather pay 490 thousand dollars to avoid this bet. This person would rather sacrifice 490 thousand dollars with 100% certainty than have a 50%

chance of winning more money and a 50% chance of losing only ten thousand dollars more. While there is no way to prove mathematically that this implies the equity premium is implausible and ludicrous, one can see from this demonstration that it is self-evident. It would be difficult, if not impossible, to find an individual who would behave under these conditions in a similar manner. Thus, if it is hard to find one person, it is certainly impossible to assume that the average market participant would behave such as this. From this brief (and relatively rough) anecdote, one can innately see that the equity risk premium is too large to be described using a simple rational investor model. In fact, Mehra and Prescott resolve that the only way to explain or make sense of the equity premium is if the typical investor is implausibly, and thus unnaturally, averse to risk (Mehra and Prescott 1985). As a result, many economists have attempted to explain and describe the causes of this premium, but as of yet, none have fully succeeded (Kocherlakota 1996). Accordingly, the question of this study is not what accounts for the equity risk premium, or even why it is so large. Rather, this study will attempt to describe the dynamic nature of its level through time. Ever since the advent of Modern Portfolio Theory, it has been widely accepted that the best way to examine an investment in a financial security is not to analyze its historical returns and deviations, but rather to make estimates of the future behavior of these securities (Fabozzi, Gupta, and Markowitz 2002). That being said, often the best proxy for the future is the performance of the past. Many of the attempts at describing the equity premium puzzle have concentrated on predicting the future level of the premium based on forecasted future levels of consumption, return, and risk aversion (Kocherlakota 1996). In contrast, this study will attempt to explain the oscillations of the equity premium as an entity in and of itself rather than as a spread between two unrelated (or related) securities. The central hypothesis of this study is that the equity premium has a mean, which is non-stationary as a result of market shocks, and behaves in a constant, predictable manner around that mean after accounting for noise.

Method Subjects. The subjects used in this analysis are the New York Stock Exchange (NYSE) aggregate index and the 90-day U.S. Treasury bill. The NYSE aggregate index was selected as a broad market index to track the general movement of the U.S. security prices for an extended period of time. This index has been available for over 100 years and has always represented a wide range of diversified industries present and operating within the United States. Unlike some other market indices, it has exhibited somewhat more damped market cyclicality as a result of its broad industry base. The constituents of the index include all equity securities actively traded on the New York Stock Exchange as of the date of each data measurement. Therefore, a survivorship bias is not introduced and thus the index is constantly evolving. The value-weighted version of this market index was chosen so that securities that are larger on the market are thus larger within the market index and the opposite for small securities. The 90-day U.S. Treasury bill was selected as the representative risk-free rate for several reasons. Since it is a U.S. Treasury debt instrument, it is backed by the full faith and trust of the United States Government. In the financial community, this is seen as the equivalent of default risk-free. Given that this is a fixed maturity, zero coupon 90-day bill, inflation risk is virtually nonexistent as well. For the purposes of this study, it is assumed that U.S. inflation was relatively constant for any 90-day period over the last 100 years. For the few time periods that may have exhibited more turbulent inflation, it is assumed that these effects are damped in the analysis by a

corresponding effect on the equity index levels observed. While using a shorter maturity bill would reduce the inflation risk just slightly more, the more volatile nature of these securities outweighs the risk-reduction benefit and may in fact add unwanted noise to the analysis. Apparatus. The apparatus used to collect the price information for the study’s time period (1934 to 2004) for each security is the Center for Research in Security Prices (CRSP). The CRSP is operated and maintained by the University of Chicago Graduate School of Business and is widely considered the industry standard in historical security prices. The data was downloaded on October 2, 2005 and should be considered current and accurate as of the most recent CRSP updates to that point. The data will be analyzed using R statistical software in an effort to isolate the non-stationary mean and fit the resulting trends to an appropriate model. Procedure. For each security, the monthly nominal price levels from 1934 to 2004 were used to calculate monthly nominal returns. Monthly data points were used as they provide many data points while also not introducing an undue level of noise to the analysis from weekly or daily returns. The period 1934 to 2004 is used as it is the longest period available for which both securities have a complete data set on record in the CRSP database. Once the monthly returns are calculated, the risk free return is subtracted from the equity return to calculate the equity risk premium. A plot of this time series is available in appendix A.

Results Descriptive statistics. Given that this data set represents a time series of returns, the most appealing measure of central tendency is the geometric average return. This return (GAR) is calculated as follows (1):

Where n is equivalent to the number of discrete periods (852 months) and r is equivalent to the period return. Thus, for this time series, the resultant geometric average return is 0.57% per month. In order to get an annual estimation of the equity premium, the rate can be compounded for 12 months (2):

Consequently, the annual geometric average equity risk premium from 1934 to 2004 is 7.10% (SD = 4.49%). This is consistent with Mehra and Prescott’s calculation of the market risk premium as 6.90% (notice the discrepancy as a result of calculations based on different years). Given that this is a time series that incorporates regular compounding, the geometric average return is the most accurate and appropriate measure of central tendency. Though, for the sake of comparison, the arithmetic monthly return annualized is 8.41% and the median monthly return annualized is 11.90%. Thus, the geometric return is a considerably more conservative description of the central tendency of the time series than either the arithmetic average or median. In order to determine whether this time series phenomenon can be described as a series of overlapping sinusoidal waves, a Fourier analysis was completed. This Fourier analysis should be able to indicate whether there is some sort of a recurring cycle underneath the string of seemingly white noise. In order to perform the analysis, the Fast Fourier Transform function in

the R program was used. This function has as input the time series in question (appendix A) and outputs the full frequency power spectrum. This spectrum can be observed on the periodogram in appendix B. Notice that very little power of the overall function can be described by a sinusoidal wave at any one frequency. In fact, what little power that is described may be merely a function of random number sequences within the data. Based upon this evidence, it appears that there is no underlying recurring pattern that is being obscured by noise. In order to further describe the time series at hand, it may be appropriate to determine the autocorrelation of the discrete returns over time. While it may be customary to calculate the autocorrelation by calculating the correlation between one unit of return with the next sequential unit of return repeatedly until each period has been factored into the analysis, this may not represent the most appropriate method. Given a time lag (tau), the previously described method would calculate the autocorrelation with tau of 1. This tau may not be the appropriate tau as it may pick up too much noise in the calculation, thus obscuring the true autocorrelation. On the other hand, there is no concrete algorithm for determining the optimal tau. As such, the autocorrelation vector calculated for this analysis displays a continuum of autocorrelations for the time series as a whole, but for each tau from 1 to 851. Observe that with a tau of 1, the autocorrelation can be calculated based on 851 values; however, with a tau of 851, there is only one correlation that can be made as this time series includes 852 measurements. Thus, the lower the tau, the more measurements can be factored into the equation, but the greater the chance for picking up noise – and vice versa for a large tau. This autocorrelation data vector is displayed as appendix C. The graph shows the correlations for both positive and negative taus as the correlation calculation is commutative. Consequently, the correlations from 0 to 851 are symmetrical but flipped on the vertical axis (at tau=0) to the correlations from 0 to -851. This is done purely for illustrative purposes. At tau equal to zero, the auto correlation is equal to 1.00 by definition. As soon as tau begins to grow, the correlations fall to roughly 0.00 and then oscillate around this value until they increase to approaching 1.00 and -1.00 as tau approaches 851. This correlation expansion is a function of the decreasing number of observations when tau is large. At first glance, appendix C may appear to indicate that there is essentially no autocorrelation and that the values are categorized by pure white noise. In fact, appendix D is the same autocorrelation representation, but it was created using uniformly distributed random values between zero and one. Notice that appendix C and appendix D look very similar. Though in order to be confident that the market premium data in this analysis are in fact not autocorrelated to any power greater than mere random values, a surrogate data test is necessary. Since autocorrelation is based on the temporal nature of the data, truly unautocorrelated data should not depend on the order in which they are aligned. Thus, perhaps the small fluctuations away from zero in appendix C are simply a function of the numbers in the time series as opposed to some true autocorrelation (though minimal) at certain levels of tau. Thus, the surrogate data test will remove the time-dependency of this series and run the autocorrelation again to develop a 95% confidence interval. This is done simply by randomly assigning the values in the time series to different time positions and calculating the autocorrelation for each level of tau and then running multiple iterations of this . Then, the 95% confidence interval is determined by associating the autocorrelation for each level of tau that is 2.5% from the maximum value and 2.5% from the minimum value. This 95% confidence interval is then plotted on top of the time-dependent autocorrelation. This new merged graph is illustrated in appendix E. The dotted, red lines represent the bounds of the confidence interval 1

and the solid, black line is the actual autocorrelation. Notice that at tau equal to 348, we can reject the null hypothesis that the autocorrelation is equal to zero. For tau equal to 348, an embedded state space plot was derived. This embedded state space plots the value of the return along the x-axis and the value of the return tau months later along the y-axis (appendix F). In order to improve visibility of the clusters of data, all of the data are randomly jittered a small amount. Thus, this state space plot can illustrate the most probable future outcomes in tau periods given a current measurement based on the empirical results. Notice that for this level of tau, the data appear to make a pennant (as denoted by the two red lines) with a band of outliers both before and after the pennant. The largest clusters of data are at (0.00, 0.04), (0.00, 0.00), and (0.08, 0.01). Interestingly, there appear to be few clusters considerably below 0.00 on the y-axis. Given the significance level at tau equal to 348, the geometric average equity premium can be recalculated with this in mind. A long-term calculated equity premium given historical data is only relevant for the current period in time. Thus, the 75-year long-term average equity risk premium in 1985 may not be equal to the 75-year long-term average equity risk premium in any other year. Thus, the statistical program R can be used to calculate the moving equity risk premium given a certain lag period. The autocorrelation analysis above suggests that 348 months is the only lag period that is statistically significant. The moving equity risk premium average for this level of tau is illustrated in appendix G. For the sake of comparison, the moving averages for both 10 and 20 years are described in appendixes H and I, respectively . Notice that for each time period, the current long-term average equity risk premium is roughly 6.00% to 7.00%. Inferential statistics. Based upon the central hypothesis of this study – that the U.S. equity risk premium has a non-stationary mean around which the premium fluctuates – a damped linear oscillator model seems the most appropriate method by which to attempt to fit the data. In a damped linear oscillator model, a system is assumed to be regulated by a certain set of parameters that dictate both the frequency of oscillation and the rate of damping. The greater the level of damping, the more quickly the function reaches its equilibrium level. The greater the frequency, the more times the function crosses its stable equilibrium level in a given time period. The central concept is that forces act to bring the system to its equilibrium (or perhaps, forces act to move the system away from its equilibrium). In the case of the equity risk premium, the market forces of assumed symmetrical information, rational investors, and endogenous market entry allow arbitrage pricing to regulate the system to presumably its stable equilibrium over time. The parameters for this model are zeta and eta as described by the following equation (3): 2

Here, zeta is the damping parameter and eta is the frequency. The length of time necessary for the system to reach equilibrium is sensitive to both the parameters described above as well as the initial value of x, though since the system will ultimately reach equilibrium no matter what the initial value of x, the system is said to have insensitive dependence on initial conditions. In order to estimate the parameters eta and zeta, two methods will be used: local linear approximation and latent differential equation modeling. In the local linear approximation method, a three dimensional state space is created for each level of tau between 1 and 850. At

each level of tau, estimates of the first derivative of the function (4) and the second derivative (5) are calculated given the following formulae:

Here, x1, x2, x3 are equal to the x value of each dimension of the state space embedding. These estimates of the derivative can then be used to develop eta and zeta estimates for each level of tau. The level of tau that provided the estimates with the greatest goodness of fit parameter (R2 = .700) is eight. At tau equal to eight, zeta is approximated as -0.004 and eta is approximated as -0.033. Given these parameters, one can create a model of the behavior of the system. a.

b.

Figure 1: (a) Damped linear oscillator model conforming to equation 3 given LLA parameters and (b) vector field representation of this model for the change in x and . This representation has initial condition of x at time one equal to seven as a proxy for the equity market premium. The model damps very slowly and begins to approach zero as time approaches 2000 periods (months). Relatively shorter periods of time, such as 120 months, result in an oscillating premium that appears much more sinusoidal and regular. a.

Figure 2: (a) Damped linear oscillator model conforming to equation 3 given LLA parameters.

However, the null hypothesis of local linear approximation is that ητ2 = -2. Thus, for this tau and all other meaningful taus observed in this analysis, we must fail to reject the null hypothesis. The data may also be fit to a latent differential equation model using the Mx software package. The structure of the model used to fit the data can be described by figure 3: a.

Figure 3: (a) Structure of latent differential equation model used in eta and zeta parameter estimation in Mx. This analysis provides somewhat different estimates for the parameters eta and zeta. This model predicts that the best estimation of eta is 0.211 and zeta is -0.098 (measure of dynamic error = 4.71x10-5). These parameters are once again used to simulate the behavior of the function. a.

b.

Figure 4: (a) Damped linear oscillator model conforming to equation 3 given LDE parameters and (b) vector field representation of this model for the change in x and . Notice here that in just five months, the level of the premium will quadruple according to the model. In Figure 3b, one can clearly see the point repellent at the origin. This point repellent is what causes the system to move violently from its equilibrium value. According to the latent differential equation model, it appears that there is no steady state equilibrium for this system.

Discussion The one definitive statement that can be made about the level of the equity risk premium is that its behavior is characterized and masked by a high amount of noise. This is evidenced by

the low descriptive power of the Fourier analysis as well as by the autoregressive plot that exhibits only one statistically significant level of tau given a 95% confidence interval. This is not to say that the level of the equity premium is entirely a random walk, however. One must use the two inferential models to analyze the random or deterministic nature of the system. According to the local linear approximation model, one cannot reject the null hypothesis, and thus, one cannot claim that the model parameters are statistically significant. According to the latent differential model, it appears that the system does not actually behave as the hypothesis predicts. In fact, the latent differential model predicts that the system is easily thrown out of equilibrium and continues towards infinity. Thus, the implication is that market shocks must bring the system back in order, as opposed to the hypothesis that market shocks cause the system to depart from a steady state equilibrium. Given the very slow nature of the damping in the linear approximation method, an examination of the premium’s fluctuations for a more practical time period (10 to 20 years) may appear to behave, to the casual observer, as a sinusoidal wave. This is consistent with the popular security valuation method of comparable multiples analysis (Reilly and Brown 2003). This method assumes that market security prices systematically fall out of sync with the market reference but will eventually return and surpass their intrinsic value. The primary concept is that the market value oscillates around its intrinsic value. The data and analyses also suggest that the popular method of estimating the current long-term equity risk premium as simply the geometric average of the historical equity risk premium is flawed and virtually useless. There are two commonly accepted practices for estimating the equity risk premium necessary for asset pricing: forward looking models and historical averages (Arnott and Bernstein 2002). The estimation of the equity risk premium is vitally important for asset pricing. Given the current paradigm for asset pricing, the Capital Asset Pricing Method (CAPM), one can observe that the expected return for any capital asset is extremely sensitive to the equity risk premium. The CAPM can be described as the following (6):

Here, rfis the risk-free rate (the 90-day US Treasury rate in this study), β is the sensitivity to systemic market risk, and rm-rfrepresents the equity risk premium. Thus, for a β of 1.00, which is the market average, each 1% change in the estimation of the equity risk premium will alter the estimation of the asset return by 1% as well. This is economically significant. Currently, it is widely held that the long-term geometric average of the equity risk premium is a good proxy for its future value. This average, as described earlier, is roughly 7.10%. Given that that the autoregressive analysis in this study shows that there is almost no statistically significant lag for autocorrelation, the suggestion is that historical returns are an invalid source for determining a future proxy for the premium. Thus, one must turn to forwardlooking models. According to Arnott and Bernstein, the most appropriate estimation from forward-looking models of the equity risk premium is between 2.00% and 4.00% (Arnott and Bernstein 2002). That is a very large discrepancy from the more widely used 7.10%.

Conclusion

The results of this study are inconclusive. While the local linear approximation method to estimate parameters for the damped linear oscillator model appears to suggest that the equity risk premium in the United States is self-regulating about a certain mean level and damps itself slowly as a result of endogenous market forces, this is not statistically significant. The results also suggest that this mean reversion is periodically interrupted by exogenous market shocks. On the other hand, the latent differential model appears to suggest the opposite – that the equity premium is not stable and is easily sent spiraling to unreasonable levels. Thus, these data tend to suggest that perhaps the linear oscillator model is not an appropriate paradigm for this function. This study also finds evidence to conclude that historical levels of the equity risk premium are not a good proxy for future estimations of its value. Nevertheless, this is currently far and away the most prevalent method even among industry professionals. Welch finds that of 236 economists surveyed, the average expected future equity risk premium is 7% – clearly a function of using historical averages rather than forward-looking models (Welch 2000). Areas for further study include finding a model for the equity risk premium and a psychological study of why economists continue to use historical averages as a proxy for the future premium. Notes This analysis uses 40 iterations. 10 and 20-year historical equity risk premium averages are commonly used within the financial community for practical applications such as valuation and capital budgeting.

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References Arnott, R. D., and P.L. Bernstein (2002). “What risk premium is ‘normal’?” Financial Analysts Journal, 58, 64-85. Black, F., and R. Litterman (1992). “Global portfolio optimization.” Financial Analysts Journal, 48, 28-44. Brennan, M. J., E.S. Schwartz, and R. Lagnado (1997). “Strategic asset allocation.” Journal of Economic Dynamics and Control, 21, 1377-1403. Fabozzi, F. J., F. Gupta, and H.M. Markowitz (2002). “The legacy of modern portfolio theory.” The Journal of Investing, Fall, 7-22. Kocherlakota, N. R. (1996). “The equity premium: It’s still a puzzle.” Journal of Economic Literature, 34, 42-71. Mehra, R. and E.C. Prescott (1985). “The equity premium: A puzzle.” Journal of Monetary Economics, 15, 145-161. Reilly, F. K., and K.C. Brown (2003). Investment analysis & portfolio analysis (7e). New York, New York: Thomson. Shiller, R. J. (2003). “From efficient markets theory to behavioral finance.” Journal of Economic Perspectives, 17, 83-104. Siegel, J. J and R.H. Thaler (1997). “The equity premium puzzle.” Journal of Economic Perspectives, 11, 191-200. Welch, I. (2000). “Views of financial economists on the equity premium and on financial controversies.” Journal of Business, 73, 501-537.

Appendices