The Hidden Assumption

Transcription

1 The Hidden Assumption Min (Berg) Cui September 14, 2018 Abstract I provide a new explanation towards the Equity Premium Puzzle. The benchmark Consumption CAPM equation can be derived from three different model setups; each model setup requires a different set of assumptions. Except for the most restrictive representative agent setup, two other heterogeneous agents model setups require the assumption of Aggregating with Consumption Weights. I argue, both theoretically and empirically, that consumption weights contribute largely to the quantitative discrepancy suggested by the Puzzle. By simply replacing consumption weights with wealth weights, we are able to lower the required RRA below 20 to match the historical stock market risk premium data. Keywords: Weights Classification: G12 Consumption-based Asset Pricing, Heterogeneous Agents, Aggregation 1 Introduction Since Shiller (1982) and Mehra & Prescott (1985), the Equity Premium has been puzzling the academia for more than three decades. The puzzle asserts that the consumption-based asset pricing model cannot generate a risk premium large enough to match the stock market data with the aggregate consumption and a reasonably small Relative Risk Averse (RRA) The author thanks Jessie He, Siming Liu, Yoosoon Chang, Noah Stoffman and Todd B. Walk for valuable comments. Department of Economics, Indiana University. cuim@indiana.edu. 1

2 parameter for the agent s preference. Or equivalently, the model needs an RRA that is far above the micro supported upper bound of 10 for the model to match the data. Numerous studies have attempted to solve the puzzle by challenging one or multiple assumptions associated with the model. In this paper, I provide an alternative explanation to the puzzle by challenging one assumption, which was missing from the previous literature Aggregating with Consumption Weights. I also provide both theoretical arguments and some empirical evidences to support the new explanation. The original empirical test of the Consumption CAPM using the aggregate consumption is valid under three different sets of assumptions. Except the most restrictive set of assumptions that delivers the existence of a representative agent, both other two sets of assumptions include one assumption that we aggregate agents using consumption weights. Theoretically, I argue that consumption weights are problematic because they underweight the rich while overweight the poor, and underweight the saver while overweight the consumer (agents who have the same income as the saver but save less). Because agents who are more constrained by their current resources are overweighted by consumption weights, their opinions of low returns are overweighted which leads to a low model implied risk premium. Empirically, I show that a large part of the low risk premium generated by the model with the aggregate consumption can be explained by this consumption weights. I show, using the Panel Study of Income Dynamics (PSID) data, with a power utility and an RRA at 10, the model with consumption weights generates a risk premium at 0.98% level while the one with wealth weights generates a risk premium at 4.20% level. Equivalently, an RRA at together with wealth weights can deliver a risk premium that aligns with the 7.8% risk premium we see in the financial market historical data. The rest of the paper is structured as follows. In section 2, I lay out the benchmark model and list all sets of assumptions needed to test the model in its original form. In section 3, I review literature and categorize previous studies into the corresponding assumption(s) they challenge. In section 4, I challenge the assumption of Aggregating with Consumption Weights with both theoretical arguments and empirical evidences. Finally in section 5, I conclude and lay out some possible paths for future research. 2

3 2 Model and Sets of Assumptions The benchmark model of the Consumption CAPM assumes a simple power utility u(c) = c 1 γj 1 γ j for agent j. The agent s optimization decision delivers the following asset pricing equation, E t r i t+1 r f t+1 = γ j Cov t [g j t+1, r i t+1], (1) where γ j is agent j s Relative Risk Averse (RRA) parameter (or the inverse of the agent s Elasticity of Inter-temporal Substitution EIS), g j t+1 is the consumption growth rate for agent j and r i t+1 is asset i s return. r f t+1 is the risk free rate which is determined in time t and E t r i t+1 is the prediction of asset i s t + 1 period return. The original empirical test of the model utilizes the following pricing equation, E t r m t+1 r f t+1 = γ Cov t [g t+1, r m t+1], (2) where rt+1 m is the market return and E t rt+1 m r f t+1 is the next period s risk premium. g t+1 is the aggregate consumption growth rate and γ is the RRA. Based on market returns and risk-free rates data from Kenneth French s website, the risk premium is about 7.8% between 1929 and With the annual aggregate consumption data from the FRED (NIPA data), a simple calculation shows the covariance between consumption growth rates and market returns is about Hence, an RRA at 10 generates only 1.4% risk premium, while the actual premium 7.8% is 5.6 times of that. Or equivalently, we need an RRA at = 56 to match the data. This quantitative discrepancy between the model and the data creates the Equity Premium Puzzle. However, there is a gap between Equation (1) in which an individual agent s own consumption growth rate serves as a risk factor and Equation (2) in which only the aggregate consumption growth rate serves as the risk factor. The gap requires extra assumptions to bridge. To achieve that, we either need assumptions that deliver the existence of a representative agent or assumptions that deliver an aggregation result for heterogeneous agents. 3

4 The following aggregation result is commonly used for heterogeneous preferences models, E t r m t+1 r f t+1 = [ J j=1 ] 1 c j t 1 Cov C t γ j t [g t+1, rt+1], m (3) where rt+1 m is the market return and E t rt+1 m r f t+1 is next period s risk premium. g t+1 is the aggregate consumption growth rate and γ j is agent j s RRA. c j t is agent j s consumption and C t is the aggregate consumption. With this aggregation result, we can still utilize Equation (2) in the empirical test. The only difference is that the estimated γ is no longer the RRA of the Representative Agent, it is the inverse of the consumption-weighted average of all agents EIS 1/γ j. Given the existence of this aggregation result and the availability of the aggregate consumption data, most studies focus on the explanatory power of the aggregate consumption on asset prices by simply assuming the existence of a representative agent. Among them, the vast majority of empirical tests of Consumption CAPM are based on Equation (2), thus I focus my following discussion based on Equation (2) as well. 2.1 Three Equivalent Setups The Equation (2) can be derived from three equivalent model setups, the first one is the Representative Agent model setup, the other two are Heterogeneous Agents model setups, in which the heterogeneity arise from either the Market Incompleteness or the Preference Heterogeneity or both. γ Cov t [g t+1, r m t+1] = = = γ Cov t [g t+1, rt+1] m }{{} Rep Agent (Homegeneous P reference with Complete Market) J j=1 c j [ t γ Covt [g C t+1, j rt+1] ] m t }{{} Homogeneous P reference with Incomplete Market J j=1 c j [ t γ C j Cov t [gt+1, j rt+1] ] m t }{{} Heterogeneous P references (with Complete/Incomplete Market), 4

5 where γ is the estimated RRA, γ is the RRA under the homogeneous preference and γ j is the RRA for agent j under heterogeneous preferences. g t+1 is the aggregate consumption growth rate and g j t+1 is agent j s consumption growth rate. rt+1 m is the market return. c j t/c t is agent j s consumption weight and c j t/ C t = [ c j t/γ j] [ J ] / j=1 cj t/γ j is agent j s risk-aversenessadjusted consumption weight. An agent with larger γ is more risk averse, hence less likely to invest in the stock market. Therefore his weight should be downward adjusted by the factor of γ accordingly. Naturally, the simple consumption weight c j t/c t is a special case of this risk-averseness-adjusted consumption weight c j t/ C t with γ j = γ, j J. For all three setups that utilize the aggregate consumption, we need to assume the Full Market Participation otherwise market non-participants consumption should be excluded from the equation. Further, to deliver the pricing kernel in the form of Equation (2), we need to assume that all agents utility function is the Simple Power Utility. Besides those two, we need different extra assumptions for different model setups. The first model setup assumes a Representative Agent, which is equivalent to assume both the Preference Homogeneity and the Market Completeness. estimated γ is the representative agent s RRA γ. Under this setup, the The second model setup assumes the Homogeneous Preference with an Incomplete Market, which is equivalent to assume only the Preference Homogeneity but not the Market Completeness. However, to get the aggregation in the form of Equation (2), we need to additionally assume Aggregating using Consumption Weights. The third model setup assumes the Heterogeneous Preferences with either a Complete or an Incomplete Market, which is equivalent to assume only the Market Completeness but not the Preference Homogeneity (Heterogeneous Preference with a Complete Market), or assume neither the Preference Homogeneity nor the Market Completeness (Heterogeneous Preference with an Incomplete Market). In addition, to get the aggregation result in the form of Equation (2), we need to assume Aggregating using Consumption Weights. From models restrictiveness perspective, both the Homogeneous Preference with an Incomplete Market setup and the Heterogeneous Preferences with either a Complete or an Incomplete Market setup are less restrictive than the Representative Agent model. They relax the Representative Agent model s assumptions requiring both the Preference Homogeneity 5

6 and the Market Completeness to either or neither. The latter two models are valid without the assumption of Aggregating with Consumption Weights, this extra assumption solely brings us the benefit of using the aggregate consumption as the risk factor. 2.2 Sets of Assumptions We have in total Five Assumptions about the Model. M1 - Full Market Participation (FMP) M2 - Simple Power Utility (SPU) M3 - Homogeneous Preference (HMP) M4 - Complete Financial Market (CFM) M5 - Aggregating with Consumption Weights (ACW) Each of those three model setups discussed above requires a different subset of those Five Assumptions The Representative Agent model requires the Full Market Participation (FMP), the Simple Power Utility (SPU), the Homogeneous Preference (HMP) and the Complete Financial Market (CFM). 2. The Homogeneous Preference with an Incomplete Market model requires the Full Market Participation (FMP), the Simple Power Utility (SPU), the Homogeneous Preference (HMP) and the Aggregating with Consumption Weights (ACW). 3. The Heterogeneous Preferences with A Complete Market model requires the Full Market Participation (FMP), the Simple Power Utility (SPU), the Complete Financial Market (CFM) and the Aggregating with Consumption Weights (ACW). 1 The assumption sets are different if we only require the aggregate consumption to be a risk factor. No need for SPU for Representative Agent and Homogeneous Preference with Incomplete Market. 6

7 An Incomplete Market model requires the Full Market Participation (FMP), the Simple Power Utility (SPU) and the Aggregating with Consumption Weights (ACW). By assuming any one of those three subsets of assumptions, we can have the model in the form of Equation (2). To correctly test the model in this form, we further need to assume no measurement error on the data. Hence, we need following Three Assumptions about the Data. D1 - Correct Measure of Returns (CMR) D2 - Correct Measure of Consumptions (CMC) D3 - Correct Measure of Covariances (CMV) If we believe the economic mechanism of consumption-based asset pricing models is correct, then any attempt to test the Consumption CAPM in the form of Equation (2) is equivalent to a test of the joint validity of any one of the following sets of assumptions. S1 - FMP, SPU, HMP, CFM, CMR, CMC and CMV. S2 - FMP, SPU, HMP, ACW, CMR, CMC and CMV. S3 - FMP, SPU, (CFM), ACW, CMR, CMC and CMV. All three sets of assumptions share five common assumptions, they are the Full Market Participation (FMP), the Simple Power Utility (SPU), the Correct Measure of Returns (CMR), the Correct Measure of Consumptions (CMC) and the Correct Measure of Covariances (CMV). They also differ on three assumptions, we either need both the Homogeneous Preference (HMP) and the Complete Financial Market (CFM), or need the Aggregating with Consumption Weights (ACW) together with either or neither of HMP and CFM. 7

8 3 Literature and Its Merits Almost all past studies can be summarized as attempts to challenge one or multiple assumptions listed above. In this section, I review literature and categorize past studies into the corresponding assumption(s) they challenge. I begin with those five common assumptions, since they are needed regardless which model setup we assume. 3.1 Full Market Participation (FMP) Mankiw & Zelds (1991) start to crack the Puzzle by challenging the assumption of the Full Market Participation. They argue that since not all agents participate in the financial market, some agents Euler equations do not hold. Therefore we should not simply assume the existence of a representative agent or an aggregation across all agents, both of which imply that the aggregate consumption is the risk factor. Vissing-Jørgensen (2002) provides an empirical investigation into implications of the violation of the FMP assumption. She finds that by using market participants consumption only, we are able to get a much more realistic estimation on EIS (RRA). 3.2 Simple Power Utility (SPU) Another branch of literature focus on the assumption that agents have a simple power utility. One extension of this simple power utility with one consumption good is to allow multiple consumption goods, especially those with more volatile growth rates in the utility function. However, with a Cobb-Douglas utility setup, other goods disappear in the pricing kernel because of the separability of the Cobb-Douglas function. Dunn & Singleton (1986), Eichenbaum & Hansen (1990) and Yogo (2006) bring non-separable utility functions into the model to keep those goods in the pricing kernel. A closely related study is Aït-Sahalia et al. (2004), they achieve low level estimates on the RRA with a non-homothetic utility defined on both basic consumption goods and luxury goods. Another extension is to use the recursive utility. Epstein & Zin (1989), Weil (1989) and Epstein & Zin (1991) show that with the recursive utility, we are able to bring the RRA lower to a more realistic level. The recursive utility function separates the RRA and the 8

9 EIS. A consequence of this is the relative size between the RRA and the EIS creates a new element in the agent s attitude towards risks, the ambiguity aversion. Jeong et al. (2015) obtain promising results. Some argue that the deviation from the consumption habit, instead of the consumption level itself, matters to the consumption smoothing incentive, hence to asset prices. Campbell & Cochrane (1999) twist the simple power utility functional to incorporate this idea, and explained a variety of asset pricing phenomena. 3.3 Correct Measure of Returns (CMR) Another possible reason for the empirical failure of the Consumption CAPM is that the data is problematic. Dimson et al. (2002) argue that the high risk premium in the late 20th century is simply due to good luck. Looking at a longer time horizon, the risk premium is lower and thus there is no puzzle about it. Fama & French (2002) and McGrattan & Prescott (2005) notice the decline in real stock returns. Naturally, if the trend continues, the Puzzle will be gone. McGrattan & Prescott (2003) suggest the returns we use are pre-tax returns, the capital gain tax reduces returns and hence narrows the risk premium gap between the one in the market and the one predicted by the model. 3.4 Correct Measure of Consumptions (CMC) Another possible error in data is from the measurement of consumptions. Savov (2011) argues the level of garbage is a better proxy to household s actual consumption than the filtered consumption data in NIPA. He achieves an estimate of the RRA under 20, which is far below the level suggested by the Puzzle. Kroencke (2017) uses the unfiltered NIPA consumption data and achieves similar result. 3.5 Correct Measure of Covariance (CMV) However, even with correctly measured returns and consumptions data, the calculation on covariances between returns and consumptions might be wrong. One question that needs to 9

10 be addressed is which frequency we should use to measure rates and hence to calculate covariances. Daniel & Marshall (1997) and Parker & Julliard (2005) argue long-run consumption growth rates should be used instead of short-run rates. Even under the same frequency, timing in measuring consumption growth rates to calculate covariances matters too. Jagannathan & Wang (2007) find consumptions in the fourth quarter provides more volatile growth paths, therefore calculated covariances are larger. Different assumptions on the evolution of outputs (returns) also result different covariances. Barro (2006), based on the same intuition of Rietz (1988), adds a term of rare disasters into the evolution of outputs. Hence, covariances between consumption growth rates and assets returns become larger because of extra terms in the equation. 3.6 Complete Financial Market (CFM) Both the Complete Financial Market (CFM) and the Homogeneous Preference (HMP) are needed for the existence of a representative agent. An attempt to challenge any one of those two is hence a challenge of the representative agent setup. Beyond this point, we are in the territory of heterogeneous agents. Constantinides & Duffie (1996) point out an important issue the financial market is likely incomplete. Without a complete market, agents cannot fully insure their consumptions. Therefore, agents differ in terms of uninsured idiosyncratic risks and the heterogeneity rises in idiosyncratic consumption growth rates. Brav et al. (2002) and Cogley (2002) provide empirical investigations into the impact of heterogeneous agents on asset prices 2, although they end up with opposite conclusions. 3.7 Homogeneous Preference (HMP) Because of the existence of the aggregation result as in Equation (3), no study has been done in challenging the Homogeneous Preference assumption alone. However, there are attempts to challenge this assumption, along with other assumptions, as discussed in the next section. 2 Brav et al. (2002) assume the heterogeneity comes from both the market incompleteness and limited market participation. Cogley (2002) does not emphasize the source of the heterogeneity, instead he focuses on the fact that the heterogeneity in consumption growth rates exists. 10

11 3.8 Multiple Assumptions Gârleanu & Panageas (2015) is the first study to challenge the Homogeneous Preference (HMP) assumption, however they use the Epstein-Zin-Weil utility instead of the Simple Power Utility (SPU). Cui (2018) provides an empirical investigation with the Panel Study of Income Dynamics data, and finds that heterogeneous preferences generate larger risk premium and capture more risk premium dynamics than the homogeneous preference model under the heterogeneous Epstein-Zin-Weil utility model setup. Another study that challenges multiple assumptions is Malloy et al. (2009). They investigate market participants consumption with the Epstein-Zin-Weil utility, which is an attempt to challenge both the Full Market Participation (FMP) assumption and the Simple Power Utility (SPU) assumption. Bansal & Yaron (2004) also utilize the Epstein-Zin-Weil utility, but they focus on the evolution of consumptions. They argue that there is a long-run predictable component in the evolution of consumptions, and agents care this component more in considering the consumption smoothing. With this extra component, covariances between consumption growth rates and asset returns are different. Hence, this is an attempt to challenge both the Simple Power Utility (SPU) assumption and the Correct Measure of Covariance (CMV) assumption. 4 On Aggregating with Consumption Weights By far, there is no study has been done to explicitly challenge the assumption of Aggregating using Consumption Weights (ACW) as I am aware of. A possible reason for this is that this assumption is hidden behind the Representative Agent model. S1,S2 and S3, all three different sets of assumptions generate the same Equation (2). Therefore, an empirical test on Equation (2) can be a test on the joint validity of any one of those three sets. Because of the simplicity of the Representative Agent model and the availability of the aggregate consumption data, most literature argue the test is against S1, putting the ACW assumption in the shadow since it is the only assumption in S2 and S3 that is not in S1. 11

12 In this section, I discuss this hidden assumption. I begin with the reason why we need this assumption, then I argue why it is a problematic assumption. 4.1 Why is it needed? As discussed in Section 2, the Representative Agent model setup does not require the assumption of Aggregating with Consumption Weights (ACW). It is only required under heterogeneous agents model setups. However, comparing to a world with a representative agent, we are more likely living in a world with heterogeneous agents. Firstly, the Representative Agent model requires both the Preference Homogeneity and the Market Completeness, while heterogeneous agents models require just one or none of those two assumptions. The Representative Agent model is far more restrictive, hence far less realistic. Secondly, the Representative Agent model has strong implications on data. By assuming the existence of a representative agent, the model implies the absolute homogeneity in households consumption growth rates, which contradicts the data. With the Consumer Expenditure Survey data from 1982:Q1 to 1996:Q1, Brav et al. (2002) find substantial heterogeneity in quarterly households consumption growth rates. After excluding outliers with too large consumption increases or decreases, the cross-sectional standard deviation of consumption growth rates for households with positive assets is about 6%. The standard deviation increases as the sample selection gets narrower to richer households, it rises to about 10% for households with at least $20,000 assets. The PSID data from 1999 to 2009 supports this finding; the cross-sectional standard deviation of households biennial consumption growth rates is about 52%, which is equivalent to 6.5% quarterly. Both the theoretical argument and the data are against the Representative Agent model setup, hence Heterogeneous Agents are better alternatives for model setups. But Heterogeneous Agents model setups have data problems when we try to empirically test them. Without the assumption of Aggregating with Consumption Weights (ACW), the aggregate consumption cannot be used in the pricing kernel, and we need households level consumption data. Further, since the model requires covariances between each household s consumption growth rates and assets returns, we need a panel structure for households level consump- 12

13 tions. Unfortunately, such data is in scarcity. The CEX data provides a limited panel structure with only 5 quarterly consumption data points for each household. Although the PSID data provides a longer panel structure, its consumption measure is less accurate before 1999 and the measure frequency is lower (biennial) after On the contrary, the aggregate consumption data is readily available. The NIPA data has three long time series of aggregate consumptions at relatively high frequencies. It provides monthly data back to 1959, quarterly data back to 1947, and annual data back to In order to utilize the NIPA aggregate consumption data, we need to assume the Aggregating with Consumption Weights (ACW). Consumption Weights can be valid weights. In fact, if we assume a frictionless economy, then Equation (1) holds for all agents j. Therefore, any weights are valid. Brav et al. (2002), for example, give a positive answer to whether Equal Weights are valid weights. Moreover, under some frictionless assumptions, Consumption Weights are identical to some other weights, like Saving Weights, Wealth Weights and Income Weights. Let s consider a two periods non-random endowment economy, with heterogeneous agents j, j = 1,, J. Each individual solves the following utility maximization problem, max u(c j 0) + βu(c j 1) c j 0 s.t. c j 0 + s j = e j 0 c j 1 = Rs j + e j 1 where u(c) = c1 γ. R is the risk-less rate of return, which is determined by aggregating all 1 γ agents, hence it is given to any agent j. Agents have the same RRA coefficient γ, but they differ in endowments (wealth). The first order condition gives the following optimization condition, c j 0 = c j 1 = 1 R + (βr) 1/γ (Rej 0 + e j 1) (βr)1/γ R + (βr) 1/γ (Rej 0 + e j 1) 13

14 Hence, the saving s j is [ R + (βr) s j 1/γ = R + e j 1/e j 0 ] 1 c j 0 Apparently, as long as ratios of endowments e j 1/e j 0 are the same across households and there is no other friction, consumption weights equal to saving weights, as well as income weights and wealth weights. The ACW assumption becomes unimportant. This result is not surprising, because under this frictionlessness assumption, the heterogeneous agents economy is reduced to a representative agent economy. To summarize, if we believe we live in a world with heterogeneous agents, we need aggregation weights to generate a pricing kernel. Consumption Weights can be valid weights if we make assumptions about frictionlessness, and they bring us the benefit of making the aggregate consumption usable for empirical tests. 4.2 Why is it problematic? Empirical Evidence Although any weights can be valid, they do not necessarily have the same performance. In fact, the necessary condition for all weights performing the same is for every household s Euler equation to hold. Under the homogeneous preference, that is equivalent to require covariances between households consumption growth rates and asset returns being the same. However, data does not support this. The standard deviation of covariances between households consumption rates and market returns is about , which is relatively huge comparing to the average covariance which is It is also much larger than the covariance between aggregate consumption growth rates and market returns, which is I will discuss possible theoretical reasons for this huge heterogeneity in the next section, while discussing the empirical implications now. Since the necessary condition for all weights being the same does not hold, we expect to see different outcomes from different weights. Cui (2018) investigate the impact of three different weights, Equal Weights, Consumption Weights and Wealth Weights; although he uses the Epstein-Zin-Weil utility, not the simple power utility we discuss here. He finds, Consumption Weights generate the least volatile and the smallest risk premium out of 14

15 those three weights. Under different sample selection criteria, Equal Weights generate about [1.769, 2.102] times larger risk premiums than Consumption Weights, while Wealth Weights generate about [3.444, 4.205] times larger risk premiums. Recall that the Equity Premium Puzzle is a 5.6 times difference between the premium predicted by the model(a representative agent or heterogeneous agents with consumption weights) and the actual premium in data, with an RRA at 10. This 5.6 times difference will be much smaller if we incorporate Equal Weights or Wealth Weights. If the relative magnitude between risk premiums generated by different weights holds, the required RRA to match the data is reduced from 56 with Consumption Weights to [26.6, 31.7] with Equal Weights and [13.3, 16.3] with Wealth Weights. To check if this relative magnitude holds under the simple power utility, I use the same dataset and calculate the implied risk premium with an RRA at 10. Not surprisingly, Consumption Weights generate a risk premium about 0.98% while Wealth Weights generate 4.20%. That means, to get a premium at 7.8%, Consumption Weights need an RRA about 79.6 while Wealth Weights only require Although an RRA at 18.6 is still above the upper bound of 10, it is more realistic than This result is similar to Savov (2011) s [10, 26] and Kroencke (2017) s [10.32, 15.11]. Brav et al. (2002) and Cogley (2002) implicitly test the effect of Consumption Weights. Their original goal is to test the effect of the existence of heterogeneous agents. However in their configuration, they use Equal Weights instead of Consumption Weights. Therefore, the result they obtain contains both the effect from changing the representative agent to heterogeneous agents and the effect from changing Consumption Weights to Equal Weights Theoretical Arguments Ideally, a theoretical model gives us the correct weights. Sadly, such a model has not been developed yet. Although we are not able to tell what are the correct weights, we can argue that consumption weights are unlikely to be the correct ones. An agent enters the financial market with two possible roles, as a physical individual or a financial individual. Under the physical individual role, all agents are acting as the marginal trader. Hence, they all influence the market at the margin and their influences are the same, 15

16 regardless whether the agent is trading one cent or one million. Under this role, equal weights are appropriate for the aggregation. Under the financial individual role, agents influence the market differently. A millionaire with hundreds of thousands investments surely makes larger impacts to the market than a newly graduate who just starts to save for 401k. Under this role, standing investment weights, or wealth weights as close proxies, are appropriate for the aggregation. Apparently, consumption weights are compatible with neither. In particular for the financial individual role, consumption weights underweight the Saver and the Rich while overweight the Consumer and the Poor. Consider the previous model with agents 1 and 2, Re e 1 1 = Re e 2 1 but e 1 0 < e 2 0. Two agents have the same lifetime endowment, but agent 1 has fewer endowment in period 0 while agent 2 has fewer endowment in period 1. Consumption smoothing decisions result fewer savings for agent 1. overweight the Consumer as Therefore, consumption weights underweights the Saver while c 1 0/C 0 c 2 0/C 0 = c1 1/C 1 c 2 1/C 1 > s1 /S s 2 /S. Consider the same model with agent 1 and 2, Re e 1 1 < Re e 2 1. Agent 1 is the Poor and Agent 2 is the Rich. Suppose there exists a Subsistence Consumption Constraint that c j t > c, t, j. When the constraint binds, consumption weights overweight the Poor while underweight the Rich as c 1 0/C 0 c 2 0/C 0 > s1 /S s 2 /S. The Consumer/The Poor are more likely constrained by current resources. When their subjective returns for assets are lower than objective returns, in other words they think current assets market prices are low comparing to their believed fair prices, they want to buy to push prices high and bring returns down, but cannot. As a result, real world returns keep high, while the representative agent consumption-based asset pricing model (or equivalently, heterogeneous agents model with consumption weights) overweights their opinions and suggests low risk premium (high asset prices). Thus, quantitative discrepancy emerges. The mechanism suggested by this explanation shares similar intuition as the Partial Market Participation explanation. Market non-participants are less wealthy and their Euler 16

17 equations do not hold. By including them in the pricing kernel, the model overweights the Poor and generates lower risk premium. 5 Towards the Ultimate Answer This paper does not provide the ultimate answer to the Equity Premium Puzzle, since we fail to bring the RRA below 10. It does, however, point out a new source for the quantitative discrepancy between the risk premium from the consumption-based model and the one from the data. Assuming aggregating using consumption weights cause the model to generate low risk premiums. By relaxing this assumption, we are able to match the 7.8% real world risk premium with an RRA below 20. This improvement is achieved with Wealth Weights, which are not necessarily optimal weights. In fact, we do not know what are correct weights yet. After relaxing the ACW assumption, the Consumption CAPM, in which only the aggregate consumption is the risk factor, becomes the consumption-based asset pricing model, in which all households consumptions are risk factors. The pricing kernel becomes E t r m t+1 r f t+1 = J j=1 ω j t Ω t [ γ (j) Cov t [g j t+1, r m t+1] ], (4) where ω j t /Ω t is agent j s weight at time t, and γ (j) is the homogeneous (heterogeneous) RRA. The weight hence plays an important role. Therefore, while we are removing the ACW assumption in the test of the Consumption CAPM, we introduce a new assumption in the test of the consumption-based asset pricing model, D4 - Correct Measure of Weights (CMW) Therefore, an empirical test of the model is equivalent to a test of the joint validity of either of following two sets of assumptions, S4 - FMP, SPU, HMP, CMR, CMC, CMV and CMW. S5 - FMP, SPU, (CFM), CMR, CMC, CMV and CMW. 17

18 Just like all other assumptions, the assumption of CMW is important and essential to consumption-based asset pricing models. We have neither a theoretical model nor an empirical investigation into this assumption so far. Moreover, a challenge of this assumption is compatible with challenges of other assumptions. Therefore, this assumption is worth further research efforts. Finally, the RRA under 20 in this paper is achieved by challenging the ACW alone, but with all of FMP, SPU, HMP, CMR, CMC and CMV hold. Therefore, it is compatible with, instead of contradicts to, other empirically successful explanations. Just like past literature has stated, there is no reason to believe any of those assumption holds absolutely. An failure of any assumption causes a lower risk premium, the quantitative discrepancy suggested by the Puzzle could just be the result of a combination of several such failures. The Equity Premium Puzzle is not that puzzling after all. References Aït-Sahalia, Y., Parker, J. A., & Yogo, M. (2004). Luxury goods and the equity premium. Journal of Finance, 59, Bansal, R., & Yaron, A. (2004). Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance, 59 (4), Barro, R. J. (2006). Rare disasters and asset markets in the twentieth century. Quarterly Journal of Economics, 121 (3), Brav, A., Constantinides, G. M., & Geczy, C. C. (2002). Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. Journal of Political Economy, 110, Campbell, J. Y., & Cochrane, J. H. (1999). By force of habit: a consumption-based explanation of aggregate stock market behavior. Journal of Political Economy, 107, Cogley, T. (2002). Idiosyncratic risk and the equity premium: Evidence from the consumer expenditure survey. Journal of Monetary Economics, 49,

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