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Background risk in consumption and the equity risk premium

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Abstract

We consider the economy in which an agent faces, in addition to market risk, an additive independent background risk in consumption. In contrast to the Lucas (Econometrica 46:1429–1445, 1978) complete consumption insurance model, under plausible assumptions about the unconditional mean and variance of the agent’s subjective distribution of background risk the model with the additive independent background risk fits the historical average excess return on the US stock market with the coefficient of relative risk aversion (RRA) below five for the subsets of households designated as assetholders. The greater the size and/or the lower the expected value of background risk, the lower (compared to the Lucas (Econometrica 46:1429–1445, 1978) model) the value of the RRA coefficient needed for the model with background risk to match the historical average equity premium. Allowing for an extremely unlike large decrease in the agent’s consumption considerably decreases the required coefficient of RRA. It is concluded that the presence of the additive independent background risk in the consumption of assetholders can account for nearly 60 % of the historical average equity premium, hence rationalizing the equity premium puzzle of Mehra and Prescott (J Monet Econ 15:145–162, 1985). With RRA below five, the model with background risk is consistent with the historical average real interest rate if the agent has the subjective time discount factor lower than, but close to, 1. The findings are robust to the assumed type of background risk, the proxy for the market portfolio, and the threshold value in the definition of assetholders.

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Notes

  1. Mehra and Prescott (1985) argue that an implausibly large coefficient of RRA is needed for the Lucas (1978) model to fit the historical average excess return on the US stock market. Weil (1989) shows that the Lucas (1978) model with this large risk aversion coefficient can match the real interest rate only if the representative agent has a time discount factor greater than 1 (i.e., has a negative rate of time preference).

  2. See, e.g., Campbell (2006) for the discussion of how labor income and housing can either increase or decrease the agent’s willingness to take investment risk depending on the considered characteristics of these proxies for background risk.

  3. Much of background risk is idiosyncratic and, therefore, unhedgeable (see Campbell 2006, e.g.). See also Heaton and Lucas (1996), Sinai and Souleles (2005), Campbell (2006), and Palia et al. (2014), among others, who argue that labor income and owner-occupied housing behave more like a risk-free asset and, hence, are not related to stock returns.

  4. The documented evidence that background risk has a negative impact on households’ stock market participation and stock holdings suggests that the presence of background risk typically makes an agent more averse to market risk compared to the no background risk case. Campbell (2006) argues that private business risk increases effective risk aversion of self-employed households and households with significant private business assets even if this risk is uncorrelated with public equity risk. Illiquidity of houses and the possibility for households to increase their labor supply when the returns on their investments are poor also, along with the risk in labor income, have a discouraging effect on financial risk taking (Campbell 2006). Guiso and Paiella (2008) construct a direct measure of the Arrow-Pratt index of absolute risk aversion and, using data from the Bank of Italy Survey of Household Income and Wealth, present empirical evidence that the agent’s aversion to investment risk is positively affected by additional background risk.

  5. See Franke et al. (1998) and Poon and Stapleton (2005).

  6. The indirect utility g is unique to the original utility function u and the distribution of \(\Phi _{it}\). As pointed out by Gollier (2001), although g inherits some properties of u, not all the characteristics of the original utility are transferred to the indirect utility function in the presence of the additive independent background risk. The indirect utility function g inherits decreasing absolute risk aversion (Pratt 1964) and decreasing absolute prudence (Kimball 1993) from utility u. In other words, if u is standard risk averse, then g is also so. Because mixed risk aversion implies standardness, g also inherits mixed risk aversion from u (Dachraoui et al. 2004). Gollier (2001) argues that if at least one of the utility functions for two agents i and \(i^{\prime }\) (\(i\ne i^{\prime }\)), \(u_{i}\) and \(u_{i^{\prime }}\), is characterized by non-increasing absolute risk aversion, then comparative risk aversion is also preserved in the presence of background risk.

  7. See Gollier (2001).

  8. Kimball (1993) shows that standard risk aversion implies risk vulnerability.

  9. The CRRA utility is in the class of “mixed risk averse” utility functions (i.e., the functions whose derivatives alternate in sign: sgn \(u^{\left( n\right) }=(-1)^{n+1}\) for all positive integers n), as defined by Caballe and Pomansky (1996). Because mixed risk aversion implies risk vulnerability (Dachraoui et al. 2004), any mixed risk averse preferences exhibit risk vulnerability.

  10. See Campbell et al. (1997).

  11. As we stressed above, in the presence of the additive independent background risk the effective RRA coefficient depends, in addition to the utility curvature parameter \(\gamma\), on the agent’s planned consumption \(C_{it}\) and the moments of the background risk distribution and, hence, may differ across the states of the world even if the agent is assumed to have state-invariant preferences (i.e., the state-invariant \(\gamma\)).

  12. See Bakshi and Chen (1996) and Gordon and St-Amour (2004), e.g.

  13. Gollier (2001) argues that in the presence of the additive independent undesirable background risk in consumption risk vulnerability of preferences implies that the indirect utility g is more concave than the original utility u (i.e., for all \(C_{it}\), \(-C_{it}g^{\prime \prime }\left( C_{it}\right) /g^{\prime }\left( C_{it}\right) >-C_{it}u^{\prime \prime }\left( C_{it}\right) /u^{\prime }\left( C_{it}\right)\)) and, hence, \(\gamma <\gamma ^{*}\).

  14. Food consumption is likely to be one of the most stable consumption components. Furthermore, as pointed out by Carroll (1994), 95 % of the measured food consumption in the PSID is noise due to the absence of interview training.

  15. See Haliassos and Bertaut (1995), Heaton and Lucas (1996, 1997, 2000), Viceira (2001), Flavin and Yamashita (2002), Haliassos and Michaelides (2003), Cocco et al. (2005), Yao and Zhang (2005), Benzoni et al. (2007), and Palia et al. (2014), e.g.

  16. Since the CE reports only some limited information about asset holdings (see Cogley 2002; Vissing-Jørgensen 2002; Malloy et al. 2009, e.g., for more details), we consider consumer units that report total assets equal to or exceeding $1000 and $5000 (in 2005 dollars) in order to reduce the likelihood of including households who do not participate in the capital markets.

  17. The CPI series are obtained from the BLS through CITIBASE.

  18. Under this assumption, the distribution of \(\Phi _{i,t+1}\) is identical across the states of the world at \(t+1\). This guaranties that in calibration background risk is independent of market risk (and, hence, the market risk premium), as required in our framework.

  19. Recall that in this model the utility curvature parameter stands for both the effective RRA coefficient and the coefficient of RRA to market risk per se.

  20. Intuitively, adding the state with a large positive (negative) uninsurable deviation of the agent’s realized consumption from his planned consumption will make the agent more optimistic (pessimistic) about his future consumption and, therefore, will make him more (less) willing to bear market risk.

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Acknowledgments

I would like to thank seminar participants at HEC (Montreal), Schulich School of Business, John Molson School of Business, and DeGroote School of Business, as well as Martin Boyer, Melanie Cao, Georges Dionne, Mark J. Kamstra, Dirk Krueger, Clarence C.Y. Kwan, Pauline Shum, Jean-Guy Simonato, and Yisong Tian for helpful comments on an earlier version of this paper. I am particularly grateful to the anonymous referee and to Cheng-Few Lee (Editor-in-Chief) for valuable suggestions that have significantly improved the paper. I gratefully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada (SSHRC).

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  1. Department of Economics, York University, Vari Hall 1028, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    Andrei Semenov

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Semenov, A. Background risk in consumption and the equity risk premium. Rev Quant Finan Acc 48, 407–439 (2017). https://doi.org/10.1007/s11156-016-0556-2

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