Does High-Order Consumption Risk Matter? Evidence from the Consumer Expenditure Survey (CEX)

Transcription

1 Does High-Order Consumption Risk Matter? Evidence from the Consumer Expenditure Survey (CEX) Marco Rossi May 31, 2007 Abstract High order moments of consumption growth cannot adequately explain the equity premium puzzle with standard approximation techniques. Using consumption data from the Consumer Expenditure Survey (CEX), I construct a stochastic discount factor (SDF) for excess returns, calculated as a weighted average of individual intertemporal marginal rates of substitution (IMRS). In a Taylor series expansion of this SDF, high order terms are not able to account for the variation in the equity premium. This finding lends credence to the conclusion of Cogley (2002) that higher order moments of consumption growth are at best weak instruments for explaining return variation. Keywords: incomplete markets, household consumption data, idiosyncratic risk, Euler equations. Department of Finance, The Pennsylvania State University; marco.rossi@psu.edu. I gratefully acknowledge the assistance of Tim Simin and Olesya Grishchenko. All errors are my own. 1

2 1 Introduction Since the failure of standard representative agent models to explain the equity premium (Mehra and Prescott (1985)), there has been an ongoing debate on whether market incompleteness can explain this puzzle. I contribute to this debate by calibrating a factor model obtained from an approximate equilibrium model in which higher order consumption risk is priced. The model nests the representative agent model as a special case. I decompose the approximate stochastic discount factor (SDF) coming out of this model into a systematic component and an idiosyncratic component. I find that neither component is able to explain the equity premium. Furthermore, the approximate SDF yields the counter intuitive result that the equity premium should be even larger than it is. I argue that these results might depend on the inappropriate use of a Taylor expansion to approximate the SDF and on the implicit restriction that this method imposes on the sample selection process. In what follows, I briefly summarize the theory and the empirical evidence constituting the ongoing debate. Using aggregate consumption data, Hansen and Singleton (1982) and Mehra and Prescott (1985) are among the first to reject a model with a constant-relative-risk-aversion (CCRA) utility function. As argued by Hansen and Jagannathan (1991), for reasonable values of risk aversion, the pricing kernel in this model inherits the low volatility of aggregate consumption and thus cannot explain the observed equity premium. Two possible explanations are in order: either the specification of the utility function is not correct, or markets are incomplete and aggregate data are inappropriate. The present work focuses on the the issue of incomplete markets. Aggregate data are not appropriate if markets are incomplete because, in the presence of undiversifiable idiosyncratic risk, in equilibrium, individual marginal rates of substitutions will not be equal across states and periods. In this case, individual consumptions will not be perfectly correlated (as in an Arrow-Debreu equilibrium) and every euler equation will reflect the fact that consumers are exposed to both idiosyncratic and systematic risk. On the other hand, the Euler equation of a representative consumer (using aggregate consumption) will reflect only the systematic component of risk present in the economy. The importance of cross-sectional moments of consumption (growth) is stressed by Constantinides and Duffie (1996) who provide a theoretical framework to study the effect of incompleteness on market equilibrium. They derive a stochastic discount factor that depends on aggregate consumption as well as the cross sectional variance of per capita log 2

3 consumption growth. This model has the potential to explain the equity premium puzzle if the cross-sectional variance of income (consumption) goes up during recessions. Empirical and numerical research on incomplete markets has had mixed success. Telmer (1993) studies a general equilibrium model in which two agents face both systematic and idiosyncratic income shocks and are allowed to trade in only one risk-free bond. By simulating this economy, he finds that agents are able to self insure almost entirely by trading in this single asset. Mace (1991) tests the implications of full consumption insurance by regressing changes in individual consumption on changes in aggregate consumption and other right-hand-side variables related to idiosyncratic shocks. She finds that the latter are rarely significant. The most negative evidence is provided by Cogley (2002) who documents a scarce correlation between cross-sectional moments of household per-capita consumption growth and stock returns. He finds that including these cross-sectional factors in the SDF can only produce an equity premium of approximately 2 percent for low values of the coefficient of relative risk aversion. Recent work has been more successful in exploiting heterogeneity. Using a stochastic discount factor calculated as the weighted average of individual households marginal rates of substitution, Brav, Constantinides, and Geczy (2002) perform a calibration exercise and present evidence that the equity premium is consistent with plausible levels of the coefficient of relative risk aversion. Jacobs and Wang (2004) use the same stochastic discount factor to derive a two-factor consumption-based asset pricing model that compares well with the Fama-French three-factor model and significantly outperforms the CAPM. The two factors are the cross sectional mean and variance of consumption growth calculated from the Consumer Expenditure Survey (CEX). Lastly, Balduzzi and Yao (2005) derive an aggregate stochastic discount factor by averaging across marginal utilities rather than marginal rates of substitution. The authors reach similar conclusions to those of Brav, Constantinides, and Geczy (2002) and also claim that their pricing kernel performs better in the presence of measurement errors. From an analysis of the evidence presented, it is clear that the debate on the role of idiosyncratic risk in explaining equity premia is not settled yet. What is more striking is that the divergence of opinions is based on the same data set, i.e. the CEX. With this paper, I disentangle the contribution of idiosyncratic and systematic risk and, most importantly, I argue that the divergence of results in this literature might depend on the approximation techniques, i.e. Taylor series expansion, used to describe the SDF. The paper is organized as follows. In section 2, I describe the model and the decompo- 3

4 sition of the equity premium into systematic and idiosyncratic risk. In section 3, I present the data and the sample selection criteria. In section 4, I present the findings. In section 5, I draw conclusions and lay out the path for further research. 2 The Model 2.1 The Stochastic Discount Factor Consider an economy populated by a set of households, i = 1,..., I, that participate in capital markets. Household members have identical, strictly concave, utility functions and trade securities with gross returns R j, j = 0, 1,..., J, where the subscript j = 0 indicates the risk-free asset. The optimality condition for asset holders that are expected-utility maximizers is given by the usual Euler equation: u (c i,t ) = E t βu (c i,t+1 )R j,t+1 ], i, j (1) where β captures consumers impatience, and the expectation is conditional on the shared information set. The economic intuition of expression (1) is that, in equilibrium, today s marginal cost of giving up one unit of consumption must be equal to tomorrow s discounted expected marginal benefit. Dividing both sides of (1) by u (c i,t ), the optimality condition can be rewritten as ] E t β u (c i,t+1 ) u (c i,t ) R j,t+1 = 1, (2) which shows that every investor s marginal rate of substitution is a valid pricing kernel. The pricing implication of individual optimality conditions could be tested directly using individual consumption data. However, the data needed for this approach is plagued by measurement errors. One way to mitigate this problem is to exploit the linearity of the expectation operator and construct an aggregate stochastic discount factor using a linear combination of the marginal rates of substitution of individual households. Brav, Constantinides, and Geczy (2002) average (2) across i and obtain an aggregate stochastic discount factor given by m 1 t = β I i=1 u (c i,t+1 ) u /I. (3) (c i,t ) Balduzzi and Yao (2005) average (1) across i and then divide the resulting expression by current aggregate marginal utility: m 2 t = β I i=1 u (c i,t+1 ) I i=1 u (c i,t ). (4) 4

5 Following Brav, Constantinides, and Geczy (2002), I use a constant-relative-risk- aversion (CRRA) specification for utility, i.e. u(c) = c1 γ 1 γ, and apply (2) to a risky asset, R j, and to the risk-free asset, R 0. This yields the following optimality condition: ] E t βg γ i,t Re j,t+1 = 0, (5) where Rj,t+1 e R j,t+1 R 0,t+1, g i,t c i,t+1 c i,t, and γ is the RRA coefficient. 2.2 Approximation and Risk Decomposition To study the effect of higher order moments of consumption growth on agents optimal decisions, I take a Taylor expansion of the marginal rate of substitution of every investor around the cross-sectional sample mean, g t 1 I I i=1 g i,t. Setting β = 1, for every i = 1,..., I, we have g γ i,t = g γ t 1 + N ( 1) n n=1 n! n ( ) ] gi,t g n t (γ + l 1). (6) l=1 Averaging over individuals, we obtain the aggregate stochastic discount factor proposed by Brav, Constantinides, and Geczy (2002): m 1 t = g γ t 1 + N ( 1) n n=1 n! g t ] n (γ + l 1)Mt n, (7) where Mt n ( ) I gi,t g n t i=1 g t is the sample n th cross-sectional moment. Substituting (7) into (5) we can explicitly represent expected returns, for any asset j = 1,..., J, as l=1 E t R e j,t+1 ] = λt β gr,t + ε t, (8) where λ t V AR t g γ t+1 E t g γ t+1 COV t g γ β gr,t ] ] (9) t+1, Re j,t+1 V AR t g γ t+1 ] ] (10) E t g γ N ( 1) n n t+1 n=1 n! l=1 (γ + l 1)M t+1 j,t+1] n Re ε t ]. (11) E t g γ t+1 5

6 Equation (8) is a factor pricing model where factors are derived from theory. The first part of the equation captures the systematic component of the equity premium, while the new term, ε t, reflects the remuneration for idiosyncratic risk. If agents were able to hedge their idiosyncratic risk, i.e. markets were complete, equilibrium consumptions would be proportional to each other, and therefore g i,t = g i,t for all i, j. In this case we would have Mt n = 0 for all n and, as a result, ε t = 0. To sum up, the testable implication of the model is that, under the null hypothesis of complete markets, ε t = 0. 3 Data 3.1 Asset Returns The market return is given by both the equally-weighted and value-weighted nominal monthly returns on the pool of stocks traded on NYSE, AMEX, and NASDAQ. The riskfree rate is the monthly returns on 30-day Treasury Bill (CRSP variable T30RET). Real returns are calculated dividing nominal returns by (one plus) the seasonably unadjusted inflation rate (variable CPIRET from crsp.mcti file). Both returns and inflation data come from the Center for Research in Security Prices (CRSP) via Wharton Research Data Base. Descriptive statistics of the data are presented in Table 1. As can be seen, market returns are negatively skewed, which is a departure from normality and is typical of risky financial returns, while the nominal risk free rate has a positive skewness. 3.2 Consumption Data Consumption data come from the Consumer Expenditure Survey (CEX), produced by the Bureau of Labor Statistics (BLS). The survey is designed to be representative of the US civilian population. This data set is not a proper panel, but rather a series of cross sections with a limited time dimension. The CEX data are organized in quarterly files and participants interviews are conducted 3 months apart for 5 consecutive quarter, including an initial trial interview. These interviews are staggered evenly throughout the year on a monthly basis and during each interview, respondents provide the reference month in which the consumption of a particular item took place. This enables me to have up to twelve months of consumption information for each household. 6

7 I calculate per capita monthly nondurable and services (NDS) consumption by aggregating data in the following CEX categories: FOOD ALCOHOLIC BEVERAGES TOBACCO GAS UTILITIES APPAREL PUBLIC TRANSPORTATION HOUSEHOLD OPERATIONS PERSONAL CARE This definition of NDS consumption is identical to the definition adopted by Jacobs and Wang (2004). Gift items are excluded from the aggregation. I obtain per capita consumption by dividing each household consumption by the number of members in the household. I use consumption level data for the period January 1984 to November 1995 yielding 142 monthly consumption growths. Nominal consumption is divided by the seasonably unadjusted CPI level obtained from CRSP (variable CPIIND from crsp.mcti file). The base year/month is 1995: Household Selection Criteria The sample selection criteria are similar to those of Brav, Constantinides, and Geczy (2002) and Balduzzi and Yao (2005). For any period, I drop households that report as zero their NDS consumption or their food consumption. I also delete households that are not defined by the BLS as full income reporters. To mitigate the effect of measurement error, which is typical of survey data, I impose the following three-stage filter: 1. keep individual consumption growth g i,t c i,t /c i,t 1 if there is an adjacent consumption growth, i.e. if there is either g i,t 1 or g i,t+1 ; 2. drop consumption growth, g i,t 1, if g i,t 1 < (1/2) 1/3 and g i,t > (2) 1/3 (note that this filter is implemented only if g i,t 1 is followed by g i,t ); 7

8 3. drop consumption growth, g i,t 1, if g i,t 1 is either below (1/5) 1/3 or above 5 1/3. I consider a sub sample of households that are likely to hold stocks. Asset holders are defined as households that have a financial wealth greater than $1 at the beginning of their first regular interview. I measure financial wealth as the difference between the variables SECESTX and COMPSECX, which are reported in the family characteristics files compiled by BLS. Another complication is the presence of seasonalities. For instance, I observe a regular (seasonal) sharp increase in NDS consumption in December and a subsequent sharp drop of NDS consumption in January of every year in the sample. Therefore, I seasonally adjust aggregate consumption growth by regressing it on twelve monthly dummies. I use the residuals to obtain seasonally adjusted individual consumption levels at time t as c i,t 1 residual t. Descriptive statistics of consumption data are presented in Table 2. In Table 3, I also report the correlation between returns and consumption factors. From this table, it can be seen that the correlation between odd moments and returns is typically positive, while it is typically negative for even moments. The only exception to this rule is the correlation between skewness of consumption growth (for all households) and value-weighted market returns. As argued by Cogley (2002) this pattern of correlation is required to succeed in explaining the equity premium with cross-sectional moments of consumption growth. 4 Empirical Analysis 4.1 Set Up I focus on approximations up to the third order. For the second order approximation, n = 2, (12) reduces to ˆɛ j,t = 1 ( T γ(1+γ) T t=1 g γ t 2 Mt 2 1 T T t=1 g γ t+1 For the third order approximation, n = 3, we have ˆɛ j,t = 1 T T t=1 g γ t ( γ(1+γ) ) R e j,t 2 M 2 t γ(1+γ)(2+γ) 1 T T t=1 g γ t+1 6 M 3 t. (12) ) R e j,t. (13) Note that the term corresponding to the first order is always zero because M 1 t = 0, being the first centered moment. In general, even order terms have a positive sign, while 8

9 odd order terms have a negative sign. I do not directly estimate (12) or (13) with GMM to infer what value of γ is a minimizer. This exercise would be trivial, since (12) or (13) are identically zero if we set γ = 0. Instead, I compute its sample equivalent for reasonable values of γ and verify if the resulting quantity is close enough to zero. If this quantity is not zero, then it is interesting to see if one can explain any residual premium not explained by the systematic component. 4.2 Discussion of the Results In Table 4 and 5, I report the estimate of the systematic component and of the idiosyncratic component of the equity premium. Table 4 reports result for the whole sample, while Table 5 considers the sub-sample of asset holders. The predicted premium is annualized for convenience. As can be seen, the systematic component of the model predicts a very low premium for low values of the RRA coefficient. The observed real equity premium is approximately six percent, which is 15/20 times bigger than what the systematic component of the SDF can explain. This result is just another version of what is known in the literature as the equity premium puzzle (see Mehra and Prescott (1985)). The idiosyncratic component of the SDF produces apparently striking results. This second part seems to suggest that the equity premium is not big enough. We would expect, given the correlation presented in Table 3, that this component could explain the residual (unexplained) component of the observed premium. As I argue in the next subsection, this might be the result of the poor approximation provided by a Taylor expansion. After all, the correlations in Table 3 (with one exception) all go in the right direction. 4.3 Numerical Issues When using local approximations for the IMRS, it is important to identify a-priori the interval ( r, r) in which the Taylor series is guarantied to converge, where r is the radius of convergence. For a Taylor series, the radius of convergence is the distance between the approximation point and the closest singularity of the function. Consider for example a CRRA utility function. The marginal utility is given by c γ, where γ is the coefficient of relative risk aversion. Clearly, marginal utility has a singularity at zero, which implies that, if we approximate marginal utility around a point c, the radius of convergence is given by c. This observation is relevant for my study because it tells us that we cannot use data outside the interval (0, 2c). 9

10 To illustrate this point consider the function 1 c, the marginal utility of a myopic investor, and approximate it around the point c = 1. The radius of convergence in this case is equal to one, and we expect the Taylor series to converge absolutely to the true function in the interval (0, 2). In Figure 1, I plot five approximations, up to the 10 th order, and it can be seen that they do better and better. On the other hand, if we consider bigger intervals, not only will the approximation not improve, it will worsen. In Figure 2, I draw the same graph on the bigger interval (0, 2.5). It can be seen that the approximations are not reliable beyond the supremum of (0, 2). 5 Conclusion This paper provides both conceptual and empirical evidence that a simple Taylor expansion of the SDF might not capture the impact of idiosyncratic risk on the equity premium. This result is closer in spirit to the finding of Cogley (2002) than it is to the positive results of Brav, Constantinides, and Geczy (2002) and Jacobs and Wang (2004). The use of local methods forces the researcher to select a homogeneous set of investors while she is really interested in studying heterogeneity. It is important to be aware of this fact in the sample selection stage and to exploit fully the potential of Taylor approximations. One should include as many observations as the radius of convergence allows, thus preserving as much heterogeneity as possible. Unfortunately, this seems to be hardly enough. Global approximation schemes might overcome the difficulty of working with trimmed data while interested in outliers. References Balduzzi, P., Yao, T., Testing heterogeneous-agent models: An alternative aggregation approach. Journal of Monetary Economics. Brav, A., Constantinides, G. M., Geczy, C. C., Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. The Journal of Political Economy 110(4), 793. Cogley, T., Idiosyncratic risk and the equity premium: Evidence from the consumer expenditure survey. Journal of Monetary Economics (49),

11 Constantinides, G., Duffie, J. D., Asset pricing with heterogeneous consumers. Journal of Political Economy 104, Hansen, L., Singleton, K., Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50, Hansen, L. P., Jagannathan, R., Implications of security market data for models of dynamic economies. Journal of Political Economy 99(2), Jacobs, K., Wang, K. Q., Lifetime portoflio selection by dynamic stochastic programming. Journal of Finance LIX(5), Mace, B., Full insurance in the presence of aggregate uncertainty. Journal of Political Economy 99(5). Mehra, R., Prescott, E. C., The equity premium: A puzzle. Journal of Monetary Economics 15, Telmer, C., Asset-pricing puzzles and incomplete markets. Journal of Finance 48,

12 Table 1: Summary Statistics for Asset Returns This table presents the means, standard deviations and skewness for the monthly equally weighted and value weighted market returns and for the risk-free rate. The market return is the nominal return on value-weighted NYSE-AMEX-NASDAQ index (CRSP variable VWINDD). The risk-free rate is the monthly nominal return on 30-day Treasury Bill Portfolio (CRSP variable T30RET). Inflation is the gross-return on the seasonally unadjusted consumer price index (CPI). Mean SD Skew Obs Value Weighted Mkt Return Equally Weighted Mkt Return day T-bill Return Inlfation (CPI) Table 2: Summary Statistics for Consumption This table presents summary statistic for household per-capita consumption growth. In the first row I report statistics for the whole sample. In the second row I report statistics for the asset holders only. Asset holders are households, who hold financial wealth at the time of their first interview. Total assets are the sum of the market value of US savings bonds, stocks, bonds, mutual funds and other such securities as defined by BLS. Households are included if they pass a three-stage filter. First, we include only households for which at least 3 consecutive consumption levels are available. Second, I drop consumption growth, c i,t/c i,t 1, if c i,t/c i,t 1 is either below (1/5) 1/3 or above 5 1/3. Third, I drop c i,t/c i,t 1 if c i,t /c i,t 1 < (1/2) 1/3 and c i,t+1 /c i,t > (2) 1/3. I seasonally adjust aggregate consumption growth by regressing it on twelve monthly dummies. I use the residuals to obtain seasonally adjusted individual consumption levels at time t as c i,t 1 residual t. No. Obs Mean SD Skew Kur Min Max NDS,ALL NDS,AH Table 3: Returns and Cross-sectional Moments of Consumption This table reports the correlation between the first three cross-sectional moments of consumption and the equity premium (both value- and equally- weighted). All Households Asset Holders Mean SD Skew Mean SD Skew Value Weighted Equally Weighted

13 Table 4: Systematic and Idiosyncratic Risk Premium: All Households This table presents the sample equivalent of the systematic component of the SDF, ˆλ ˆβ gr,t, and the idiosyncratic component, ( ) 1 T γ(1+γ) T t=1 g γ t M 2 t 2 γ(1+γ)(2+γ) M 6 t 3 Rj,t e ˆɛ j,t =. 1 T T t=1 g γ t+1 The idiosyncratic component refers to that obtained with a third order approximation. To ge the second order approximation, just eliminate the last term (the term involving the third moment). The expectation in the numerator is estimated via GMM and the p-values refer to robust t-statistics. This simplification comes from the fact that the numerator is approximately equal to one. RRA Systematic 2ND p-value 3RD p-value Panel A: Value Weighted Market Panel B: Equally Weighted Market Table 5: Systematic and Idiosyncratic Risk Premium: Asset Holders Same as Table 4, except that the the analysis is conducted on the sub-sample of asset holders. Asset holders are defined as households that have a financial wealth greater than $1 at the beginning of their first regular interview. RRA Systematic 2ND p-value 3RD p-value Panel A: Value Weighted Market Panel B: Equally Weighted Market

14 Taylor approximation vs. actual function 6 Taylor Function x Figure 1: Taylor approximation within the radius of convergence. Taylor series expansion of the function 1 x up to the 10th order. The series converges to the true function as the order of the expansion goes to infinity. 8 6 Taylor approximation vs. actual function Taylor Function x Figure 2: Taylor approximation beyond the radius of convergence. Taylor series expansion of the function 1 up to the x 10th order. The series diverges as the order of the expansion goes to infinity. 14