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Replication of the Equity Premium & Risk-Free Rate Puzzles Under a CRRA-Utility Framework

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Replication of the Equity Premium & Risk-Free Rate Puzzles Under a CRRA-Utility Framework Replication of the Equity Premium & Risk-Free Rate Puzzles under a CRRA-Utility Framework Austin Bender [email protected] Presented to the Department of Economics for fulfillment of the honors requirement. __________________________________ Professor David Evans University of Oregon Eugene, Oregon March 2018 Abstract In this paper I seek to replicate aspects of the paper, “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles” by Bansal and Yaron (2004). To do this, I pull the core growth processes from Bansal and Yaron (2004) and modify them to fit a power, constant relative risk aversion (CRRA), utility model that is then utilized to showcase the equity risk premium and risk-free rate puzzles apparent in using such a method. Table of Contents Introduction..............................................3 Data Generation.......................................4 Persistence Without Recursion Issues.......5 Approach 1................................................7 Approach 2............................................... 9 Methodology...........................................10 Asset Pricing Intuition.............................10 Generating Simulations for Returns........15 Results.....................................................17 Conclusion..............................................20 Works Cited............................................21 2 Introduction Throughout history, the U.S. capital markets have produced on average an annual equity risk premium of roughly 6% over treasury bills. This risk premium is considered high in regard to the level of real interest rates and degree of risk aversion present in representative agents. In fact, the risk premium is considered to be so high that it has been a widely researched phenomenon in academic finance since the 1980s when Mehra and Prescott (1985) published a paper called “The Equity Premium: A Puzzle.” Since then the topic has been researched to try and solve the apparent puzzle, and Bansal and Yaron (2004) seek that in their paper, “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles.” The paper, Bansal and Yaron (2004), utilizes an approach to the equity premium puzzle based on Epstein and Zin recursive preferences (1989), persistent expected growth rates, and economic uncertainty (time-varying consumption volatility). It was originally intended for myself to build up to this model of preferences after working with a model containing Constant Relative Risk Aversion (CRRA), but due to the time constraints, it was soon realized that doing recursive preferences under Epstein and Zin (1989) would require another term of research replication. As a result, I focus my methodology on modeling the equity risk premium in a power utility framework while utilizing the processes in Bansal and Yaron (2004) with modified parameters to generate data to fit this framework and effectively show the puzzle. Additionally, Bansal and Yaron (2004) split their paper up into two cases. Case 1 consists of a persistent expected growth rate component, while case 2 incorporates all of case 1 plus stochastic volatility. For my replication I will focus on case 1. 3 Data Generation The goal of the data generation phase was to generate data across 840-time periods, which are interpreted as months, that will be annualized to 70 years of consumption and dividend growth rate data. The reasoning behind monthly data is due to the fact that the decision interval of the agent (assuming representative agents) is monthly according to Campbell and Cochrane (1999), and that aggregating the months up into years leads to desired features found in the real world economic data. Consumption data is assumed to be real per-capita nondurables and services data. To generate the data for the consumption and dividend growth rates contained in the paper, I utilized three different approaches to the AR(1) process contained within both rates: 1. Creation of the AR(1) process via a proprietary code of the equation for an AR(1) process, given below in equation (1). Then create the two versions of the dividend and consumption growth rates: a. Compounded b. Additive 2. Simulate the rouwenhorst procedure in Julia to generate equation (1). Then use the results in a compounding framework for the growth rate processes. All of these approaches may seem a bit redundant on the surface as they should do the same thing, but that is essentially the purpose—confirm my program for an AR(1) process is accurate in my growth rate data equations. Analysis of the three approaches will examine the difference, if any, of the rates for consumption and dividends generated via the different persistent component approaches and how each compare with one another. The equations for consumption 4 and dividend growth rate processes that incorporate the persistent component are below as (2) and (3), respectively. = + (1) +1 +1 = + + (2) +1 +1 , = + + (3) +1 +1 , , ~ . (0,1) +1 +1 +1 Persistence Without Recursion, Issues The AR(1) process of contains a very high persistence parameter at = 0.979, which +1 means the previous value of is nearly passed through to the next period completely. A paper by Barsky and DeLong (1993) have shown that a value of 1 is reasonable for , but Bansal and Yaron (2004) go with a parameter value less than one to maintain stationarity. Still, the persistence parameter is relatively large, and that is necessary to capture the volatility of price- dividend ratios and hence the risk premium on equities. This would be reasonable based on the conclusions the authors find in the paper, but only if I was using Epstein and Zin preferences (1989). Given the fact that I used a CRRA setup, the model does not fit the data very well for backing out equity risk premiums intended for Epstein and Zin preferences (1989). In fact, it produces very small if not negative results. This is largely due to the fact that agents do not care about the persistence aspect about rates, which will impact their long-term wealth and, hence, their decisions today. Instead, the agents care about the near-term shocks to their consumption and it how it impacts their consumption behavior. To address this issue, I shift from +1 dependence on to to capture an expectation of the growth rate of under a probability +1 distribution, and also, end up changing = 0.979 to = 0. This eliminates the persistence aspect of , and makes the autocorrelation coefficients generated in Bansal and Yaron (2004) very 5 different from mine, as I essentially eliminated autocorrelation moments from my processes. To maintain the ergodic standard deviation of the distribution previously dependent on a high level of persistence, I set the distribution’s standard deviation to, ( ) = = 0.216 (4) (1 2 ) � 2 ( , ) − (0, ) (1 2 ) 2 → 2 The problem still exists however, that the random shock component− on dividends and consumption is independent of one another and reduces the correlation between consumption growth rates and dividend growth rates. As I will explain in the methodology below, it is critical that the two processes move closely together to generate equity risk premiums. This leads to changing the parameters again to fit the power utility model, as opposed to the Epstein and Zin (1989) preference model. To maximize the equity risk premium, I shutdown both the independent random shock components attached to dividends and consumption, and instead feed shocks through the system via to consumption and dividends. Next, I select values for the +1 leverage parameter ( ), which is the degree of leverage on expected consumption growth according to Abel (1999) , in the dividend process so that it matches the standard deviation of dividend growth. Selecting a value of 4.12 is required for this to match, and although it is above that given in the paper, I don’t think it is unreasonable as there have been values as high as 6.2 in Kiku (2006), but that was on value stocks only and whether or not 4.12 is too high for an aggregate market dividend parameter is a topic for another analysis. Finally, the standard deviation component in , must be matched to that of the standard deviation of consumption growth. Doing these parameter changes help the processes move together, and therefore drive an 6 equity risk premium that stems from the covariance. The computation of the new parameters is as follows, = 12 = 0.353 Note that the original value of (not listed in the table) was used to compute this new value for and that is a given parameter in the paper that is multiplied with . A similar approach for was used to arrive at = 4.12. Approach 1 – Compounded Generating the AR(1) process via my own proprietary code consisted of generating a process that contained 1840 periods and 1000 simulations of that process. Then I selected the last 840 periods of each simulation so that the process would not be starting off from zero, but rather starting at a point where the benefits of large samples are realized. Using this AR(1) process, I then generated the monthly data for consumption, via a matrix that consisted of 840-time periods (every component replicated in this paper is ran through a Monte Carlo simulation with 1000 trials, hence the matrix would have 1000 rows). The next step is to compound the rates, which due to everything being in log form, everything is expressed as decimals, so a 1 needs to be added to every rate to make it compoundable. Otherwise the system will shrink to zero, and the economy and dividends would become non-existent. Finally, I take all the compounded rates th from and generate a function to pull every 12 rate and divide it by the previous 12- 1 840 month rate→ to create a vector with 70 entries of consumption growth rates, , which are now +1 considered annual. This same approach was used for generating the dividend growth process, , , just with the equation for the dividend data instead. +1 7 The results found that the average annual growth rate of consumption, about 1.8%, was able to be matched by my simulations, with the exception of some fluctuation in the hundredths of a percent from time-to-time when I would run the model.
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