In a two-period exchange economy where consumers have external additive habits, face an uninsurable background risk, and differ in initial endowments, Giannikos and Koimisis (2021) showed analytically that wealth inequality may either raise or lower the equity risk premium — an ambiguity result extending the analysis of Gollier (2001a), in which the same small uninsurable background risk makes inequality unambiguously raise the premium and lower the risk-free rate — and numerically that inequality's quantitative effect on the premium remains small. The authors explicitly leave two questions open: empirical estimation of habit strength, background-risk magnitude, and wealth-distribution parameters with a comparison of predicted premia and risk-free rates to observed returns; and extension to higher-moment consumption-based capital asset pricing model (C-CAPM) specifications, such as skewness-preference stochastic discount factors, that incorporate background risk and inequality. Their central ambiguity proposition leaves a third, derived question: a systematic characterization of when inequality raises versus lowers the premium and risk-free rate.

Background

The equity premium puzzle is the finding of Mehra and Prescott (1985) that a standard consumption-based capital asset pricing model (C-CAPM — the model that prices assets by the covariance of their payoffs with aggregate consumption growth through a representative consumer's marginal utility) requires an implausibly high coefficient of relative risk aversion to reproduce the historically observed gap between average equity returns and the risk-free rate; Weil (1989) showed the same framework simultaneously mismatches the low observed risk-free rate. Among the proposed remedies, habit formation lets utility depend on consumption relative to a habit level rather than on its absolute level: Constantinides (1990) and Campbell and Cochrane (1999) used internal and external habit specifications to sharpen risk aversion in bad states. In an external ("catching up with the Joneses") additive habit, utility in each period is defined over consumption minus a fraction γ\gamma of per-capita consumption, so an agent cares about consumption relative to the aggregate (Abel-type externality). A separate literature studies background risk — an uninsurable, independent income risk such as labor income or non-marketable asset holdings that creates market incompleteness; under risk vulnerability (the property that a zero-mean background risk raises effective aversion to all other independent risks), background risk can raise the equilibrium premium, though its magnitude is debated. Distributional effects enter through a device of Gollier (2001a): a mean-preserving transfer of initial endowments introduces wealth inequality without changing the aggregate endowment. Gollier showed in a complete-markets economy without background risk that the effect of such inequality on asset prices hinges on the curvature of absolute risk tolerance (the reciprocal of absolute risk aversion, i.e., how much risk a consumer willingly bears as wealth grows): with hyperbolic absolute risk aversion (HARA) utility — the class whose absolute risk tolerance is linear in consumption — the distribution of wealth has no effect on the equity premium or the risk-free rate, while convex (concave) tolerance makes inequality decrease (increase) the premium. Gollier then introduced an uninsurable background risk of the same kind and showed that, for all HARA utilities, a small background risk combined with wealth inequality raises the equity premium and lowers the risk-free rate (a result he showed to be reversed for larger background risks), while the quantitative effect of inequality remains marginal. Giannikos and Koimisis (2021, Finance Research Letters) embedded external additive habits in this economy and showed that habits bend the effective absolute risk tolerance, so that with convex (concave) tolerance inequality decreases (increases) the premium. Their companion 2021 paper in the Journal of Risk and Financial Management (JRFM) then added a non-hedgeable background risk y~\tilde y, independent of the market payoff with non-positive mean, to agents' second-period wealth in the same two-period economy. Their central proposition shows that with HARA preferences, habit formation, relative risk aversion above one, and a background risk small relative to wealth, the absolute risk tolerance of habit-forming agents may be concave or convex — so wealth inequality may either increase or decrease the equity risk premium, even when income uncertainty is low — in contrast to Gollier's no-habit HARA benchmark, in which a small background risk makes the sign unambiguous — and their Mehra–Prescott-style calibration (two states matching US 1889–1978 consumption moments) shows both signs occur numerically, while inequality continues to have only a small quantitative effect on the premium. The paper's discussion also notes that stochastic discount factors (SDFs — the random variables giving state-contingent marginal utilities with which payoffs are discounted) built from higher-order Taylor expansions of smooth utility, such as the conditional-skewness (coskewness-preference) C-CAPM of Harvey and Siddique (2000), can display essentially arbitrary curvature, but that follow-up work (Post, van Vliet and Levy 2008; Potì and Wang 2010) found such higher-moment models empirically more limited than first expected — and none of these models account for background risk and wealth inequality. It is in this state that the authors leave their questions open.

Problem Statement

The common objective is to determine whether, when, and by how much the distribution of wealth affects the equity risk premium and the risk-free rate in an exchange economy whose consumers have external additive habits over consumption (utility defined over consumption minus a fraction of per-capita consumption), HARA preferences, and an uninsurable background risk that is independent of the market payoff and has non-positive mean, with wealth inequality introduced by a mean-preserving transfer of initial endowments — the economy of Giannikos and Koimisis (2021), which extends Gollier (2001a). Within this objective, the source leaves three linked questions open: (1) Empirical test — estimate the model's parameters (habit strength, background-risk magnitude, and wealth-distribution parameters) and compare the model-implied equity premium and risk-free rate to historically observed asset returns, establishing whether the habit-plus-inequality-plus-background-risk mechanism can match observed pricing moments with realistic parameters. (2) Higher-moment extension — evaluate the effect of background risk and wealth inequality in specifications where the stochastic discount factor is based on higher-order Taylor expansions of smooth utility, such as skewness/coskewness-preference C-CAPM models in the lineage of Harvey and Siddique (2000), determining whether the qualitative and quantitative conclusions of the additive-habit model survive when the SDF can display arbitrary curvature. (3) Characterization of the ambiguity — replace the paper's possibility result (that wealth inequality may increase or decrease the equity premium when a background risk is present) by an analytical or systematic numerical characterization of the parameter regions — habit strength, background-risk size relative to wealth, degree of wealth inequality, risk aversion, and endowment-process moments — in which wealth inequality raises versus lowers the equity premium and the risk-free rate, including the paper's robustness claim beyond its two-state calibration.

References

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