Background risk and small-stakes risk aversion
aa r X i v : . [ ec on . T H ] O c t Background Risk and Small-Stakes Risk Aversion ∗ Xiaosheng Mu † Luciano Pomatto ‡ Philipp Strack § Omer Tamuz ¶ October 19, 2020
Abstract
We show that under plausible levels of background risk, no theory of choice underrisk—such as expected utility theory, prospect theory, or rank dependent utility—cansimultaneously satisfy the following three economic postulates: (i) Decision makers arerisk-averse over small gambles, (ii) they respect stochastic dominance, and (iii) theyaccount for background risk.
How humans evaluate the trade-off between risks and rewards is one of the core questions ineconomics. In solving this trade-off many people exhibit small-stakes risk aversion : small,actuarially favorable gambles—such as a lottery where one loses $10 or wins $11 with equalprobability—are often rejected. In this paper we study how background risk (stemming, for instance, from investments inthe stock market or health conditions) affects risk attitudes towards small gambles. We argue ∗ We want to thank Ned Augenblick, Nick Barberis, Sebastian Ebert, Armin Falk, Stefano DellaVigna,Paul Heidhues, Matthew Rabin, Todd Sarver, Charlie Sprenger and Juuso V¨alim¨aki for helpful commentsand discussions. † Princeton University. Email: [email protected]. Xiaosheng Mu acknowledges the hospitality ofColumbia University and the Cowles Foundation at Yale University, which hosted him during parts ofthis research. ‡ Caltech. Email: [email protected]. § Yale University. Email: [email protected]. ¶ Caltech. Email: [email protected]. Omer Tamuz was supported by a grant from the Simons Founda-tion ( See, e.g., Hey and Orme (1994), and the survey by Harrison and Rutstr¨om (2008) for evidence of risk-averse behavior. amble StDeviation of Background Risk: σ Gain/Loss Laplace Logistic Normal * $11 / $10 σ ≥ $156 σ ≥ $200 σ ≥ $3319$55 / $50 σ ≥ $779 σ ≥ $999 σ ≥ $7422$110 / $100 σ ≥ $1557 σ ≥ $1997 σ ≥ $10 , / $500 σ ≥ $7785 σ ≥ $9984 σ ≥ $23 , / $1000 σ ≥ $15 , σ ≥ $19 , σ ≥ $33 , Table 1:
Standard deviation of background risk sufficient for every decisionmaker with monotone preferences to accept various fifty-fifty gambles underdifferent distributional assumptions on the background risk. The numbersdisplayed are bounds derived from our Theorem 1 and Corollary 2. * Thenormal distribution has mean $100 , $
0, according to a limited liability assumption which wediscuss in § X underwhich she gains G dollars or loses L dollars with probability 1/2. If the gamble was takenin isolation, then the choice of whether or not to accept it would depend on her preferencesbetween X and a sure outcome of 0. But if the decision maker is facing an independentbackground risk W regarding her wealth, then the relevant choice is between W , if thegamble is rejected, and W + X , if the gamble is accepted. Table 1 displays different levelsof standard deviations and distributional assumptions for the background risk under which2 amble StDeviation of Background Risk: σ Gain/Loss Laplace Logistic Normal$11 / $10 σ ≥ $62 σ ≥ $46 σ ≥ $44$55 / $50 σ ≥ $306 σ ≥ $230 σ ≥ $217$110 / $100 σ ≥ $612 σ ≥ $460 σ ≥ $434$550 / $500 σ ≥ $3058 σ ≥ $2299 σ ≥ $2169$1000 / $1100 σ ≥ $6115 σ ≥ $4598 σ ≥ $4338 Table 2:
Standard deviation of background risk sufficient for a CPTdecision maker to accept various fifty-fifty gambles under different distri-butional assumptions on the background risk. The specific CPT prefer-ence considered here has gain/loss probability weighting functions w + ( p ) = p γ ( p γ +(1 − p ) γ ) /γ , w − ( p ) = p δ ( p δ +(1 − p ) δ ) /δ with γ = 0 . , δ = 0 .
69, loss aver-sion parameter λ = 2 .
25 and value function v ( x ) = x . for x ≥ v ( x ) = − λ ( − x ) . for x <
0. These parameter values are taken fromTversky and Kahneman (1992, pages 309–312). W + X dominates W in first-order stochastic dominance. For example, in the case of thefifty-fifty gamble with gain G = 11 and loss L = 10, it is dominant to accept whenever thedecision maker’s wealth has a standard deviation higher than $200 and is either Laplace orLogistically distributed.The standard deviation of many real-life risks plausibly exceeds this threshold by a largemargin. For example, an investor who has $100 ,
000 in an S&P 500 index fund reasonablyfaces a wealth risk with standard deviation $1 ,
000 (or 1%) for the value of her portfolio atthe end of each day , and $15 ,
000 at the end of the year. Nevertheless, small-stakes gamblesare commonly rejected. Barberis, Huang, and Thaler (2006) find that among clients of aU.S. bank with median wealth exceeding $
10 million, the rejection rate of a hypothetical$550 / $500 gamble is 71%. Table 1 suggests that this behavior is inconsistent with theseinvestors taking into account even the short-term background risk they face.Table 1 applies to all monotone preferences. For particular preference specifications, therequired background risk levels become smaller. Table 2 displays the same results for the Cu-mulative Prospect Theory preferences as proposed and calibrated by Tversky and Kahneman(1992). The levels of background risk that will lead a decision maker with such preferences to See, e.g., Bardgett, Gourier, and Leippold (2019). Our analysis shows a tension between three natural requirements for any theory of choiceunder risk: (i) risk aversion over small gambles, (ii) monotonicity with respect to first-orderstochastic dominance, and (iii) choices are made accounting for background risk. As (i)is commonly observed in real world choices, and relaxing (ii) is widely considered unap-pealing, our results suggest that theories that do not account for narrow framing—wherebyindependent sources of risk are evaluated separately by the decision maker—cannot explaincommonly observed choices among risky alternatives.
Related Literature.
Arrow (1970) and Pratt (1964) show that under expected utilityand a twice-differentiable utility function, a decision maker accepts any actuarially favorablegamble, provided that it is scaled to be small enough. Our results imply that under heavy-tailed background risk, there is a scale such that all expected utility decision makers willaccept the gamble.As argued by Rabin (2000), the degree of concavity necessary for a utility functionto explain small-stakes risk aversion leads to implausible choices when stakes are large.For example, given any risk-averse expected utility preference, if a gamble where one loses$100 or wins $110 with equal probability is rejected at all wealth levels below $300 , , ,
000 with equal probability mustalso be rejected at wealth levels below $290 , Nonetheless, Barberis, Huang, and Thaler (2006) highlight that in standarddynamic models of recursive preferences, background risk can smooth out first-order riskaversion. They show that in the presence of large background risk, Rabin’s critique extendsto preferences that admit a recursive representation and exhibit disappointment aversion.Safra and Segal (2008) also study background risk, and generalize the finding of Rabin(2000) to non-expected utility preferences that admit a Gˆateaux differentiable functional Similar considerations apply to other classes of preferencess: for example a similar effect of backgroundrisk on the willingness to accept (relatively) small gambles is shown for reference dependent utility in Table1 and Proposition 6 in K˝oszegi and Rabin (2007). In the context of stochastic choice, Khaw, Li, and Woodford (2020) provide a new model of first-orderrisk aversion based on cognitively imprecise representation of the decision environment. Later work by Gneezy and Potters (1997) andThaler et al. (1997) experimentally test the hypothesis that decision makers who account formore background risk tend to be more tolerant of independent risky gambles. To the best ofour knowledge, our paper is the first to demonstrate theoretically that whenever a plausiblelevel of background risk is taken into consideration, minimal assumptions on preferenceslead to risk-neutral behavior for small gambles. Interestingly, the connection between narrowframing and stochastically dominant choices is also explored in Rabin and Weizs¨acker (2009),who shows that any decision maker who engages in narrow framing and has non-CARApreferences will make choices that, when combined, violate stochastic dominance.Tarsney (2018) and Pomatto, Strack, and Tamuz (2020) independently show that back-ground risk can induce stochastic dominance between gambles that are not already ranked.In particular, Pomatto, Strack, and Tamuz (2020) prove that if X and Y are random vari-ables with E [ X ] > E [ Y ], then there exists an independent random variable Z such that X + Z first-order stochastically dominates Y + Z . In their result, the random variable Z is tailored to the particular X and Y involved. In contrast, in this paper we show that allsufficiently large heavy-tailed background risks ensure stochastic dominance, and explicitlycharacterize the necessary size of the background risk. The resulting background risks in As mentioned, Barberis, Huang, and Thaler (2006) also emphasize the role of narrow framing for ratio-nalizing risk attitudes toward small and large gambles. A similar theme appears in Rabin and Thaler (2001),Cox and Sadiraj (2006), and Rubinstein (2006).
A decision maker faces a choice between accepting or rejecting a gamble described by abounded random variable X that takes negative values with positive probability. We thusrule out the trivial case where X ≥ W israndom and independent of X , and accepting the gamble leads to final wealth W + X . Weinterpret W as background risk the decision maker faces when considering whether or not toaccept the gamble. We assume W is distributed according to a density g : R → R + that hasfull support, is eventually decreasing, and is piece-wise continuously differentiable. This isa weak technical assumption that holds for many common distributions, like the Normal,Logistic, or Laplace distributions.
Monotone Preferences.