Comparative Risk Aversion Research Articles

Comparative risk aversion vs. threshold choice in the Omega ratio

We study conditions under which the threshold parameter in the Omega ratio represents risk aversion in the sense of monotonicity of risk premia. To this end, we derive asymptotic expansions for risk premia associated with taking a small additional risk on top of a background risk. These risk premia have the expected monotonicity behavior if, roughly speaking, the variance of the additional risk decreases with the background risk and if the density of the background risk is log-concave. When these conditions are violated, the threshold in the Omega ratio does not represent risk aversion in general. Finally, we compare our sufficient conditions for the Omega ratio to those that are needed to guarantee monotonicity of risk premia with an expected utility criterion under background risk. We argue that the conditions that are needed for the Omega threshold to represent risk aversion are comparable to those that are needed for expected utility with exponential utility functions.

Comparative Risk Aversion vs. Threshold Choice in the Omega Ratio

Comparative Risk Aversion vs. Threshold Choice in the Omega Ratio

Risk aversion over finite domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

Comparative risk aversion in two periods: An application to self‐insurance and self‐protection

AbstractRisk management decisions provide a means to elicit individuals' risk preferences empirically. In such a context, the literature often presumes that the decision to invest in risk management and the benefit of this investment occur contemporaneously. There is, however, no consensus in the theoretical literature that one‐period results can be transferred to intertemporal settings. To address this gap, we study the effect of an increase in risk aversion on the demand for risk management in a two‐period context. Our findings reproduce the one‐period results and, thus, support the focus of previous empirical literature on the structure of the risk rather than on the timing of investments and benefits. We also contrast our results with those obtained by employing widely used but limited preferences to examine risk aversion in intertemporal settings (standard additive expected utility setting, Selden, Epstein and Zin).

Comparative risk aversion with two risks

This paper characterizes aversion to one risk in the presence of another, which is invulnerable to the size of exposure to the former risk and consistent with the common bivariate risk preferences for combining good with bad. We show that all bivariate utility functions that satisfy bivariate risk apportionment exhibit risk aversion with two risks if, and only if, the dependence structure of the two risks is characterized by the notion of expectation dependence. We then propose an intensity measure of risk aversion with two risks that is based on the utility premium normalized by the marginal utility evaluated at an arbitrarily chosen pair. We show that the intensity measure being uniformly larger is equivalent to the concept of greater generalized Ross risk aversion. An application for optimal prevention in a two-period model is presented when the dependence structure of the underlying random variables is governed by the notion of expectation dependence.

Comparative risk apportionment

A decision maker who would rather apportion an independent risk in a state with a good lottery than in a state with a bad lottery is said to have a preference for risk apportionment (Eeckhoudt and Schlesinger in Am Econ Rev 96:280–289, 2006). In this paper, we propose a measure for the strength of nth-degree risk apportionment preference based on Pratt’s probability premium (Pratt in Econometrica 32:122–136, 1964). Under expected utility theory, we analyze the relationship between a greater preference for risk apportionment and both the Ross and Arrow–Pratt versions of comparative risk aversion.

Comparative risk aversion in RDEU with applications to optimal underwriting of securities issuance

We provide a characterization of comparative weak risk aversion and comparative RDEU risk aversion for RDEU preferences and, in particular, we correct a claim made by Quiggin (1993) regarding comparative RDEU risk aversion. We then apply the analysis of comparative risk aversion to a problem of optimal design of underwriting contracts in securities issuance. Specifically, in public offerings of equity, an investment banking firm (the underwriter) plays an insurance role: through the underwriting contract, the issuing firm transfers the issue risk to the underwriter, as would an insured to an insurer. We extend a classical model proposed by Mandelker and Raviv (1977) to situations where the issuing firm and the underwriter have RDEU preferences. Assuming that the issuing company’s and the underwriter’s utility functions are concave and linear, respectively, and that either the underwriter is risk neutral or both the issuing company and underwriter are strongly risk averse, we show that a firm-commitment contract is optimal if and only if the issuing company’s probability weighting function dominates the underwriter’s.

Who should exert more effort? Risk aversion, downside risk aversion and optimal prevention

I provide new results on how risk preferences affect optimal prevention. I identify a comparative risk aversion and a comparative downside risk aversion effect and highlight those cases where both effects are aligned. Alignment depends on a probability threshold, which, in turn, only depends on the preferences of the benchmark agent. This allows to define an entire class of decision-makers who all share the same comparative static prediction relative to the reference agent. I relate my findings to different intensity measures of downside risk aversion and apply them to parametric preference changes and specific classes of utility functions.

A Fresh Look at Prevention for Health Risks

We analyze how risk preferences affect optimal prevention for health risks using Kihlstrom and Mirman's (1974) transformation function approach. This ensures consistency of ordinal preferences. The effect of comparative risk aversion depends on a probability threshold while the effect of comparative downside risk aversion is always negative. This corroborates the role of downside risk aversion as an important determinant of optimal prevention for health risks but renders fear of sickness insignificant. We also point out several unique properties of prevention for health risks compared to financial risks.

Ex Post Uncertainty and the Consumer Problem

Wealth uncertainty may be resolved before consumption in which case consumption is optimal. But if uncertainty remains at the consumption level, after allocative decisions are made, ex post relative consumption is suboptimal. Therefore, ex ante the consumer has to balance attitudes to risk with underlying preferences. We study this problem in the context of expected utility with preferences over two goods, one of which is subject to additive ex post uncertainty. State preferences determine the utility gambles, and attitudes to risk evaluate these gambles, in order to choose an optimal consumption gamble. We study how risk aversion interacts with preferences to determine the direction and size of comparative statics effects using lattice programming, monotone comparative statics, methods. Comparative statics depend on the evaluation of comparative utility loss along the distribution of uncertainty and lattice theoretic properties allow us to determine how these relate to preferences and consumption. With the global regularity imposed by LSE (quasi-supermodularity) properties in direct value lattices we study the comparative statics of comparative risk aversion, income and underlying risk. We give conditions for a more risk averse consumer to choose a less or more risky consumption bundle and show that the direct LSE property determines comparative statics under first order stochastic dominant changes in ex ante uncertainty. Extensive examples demonstrate.

Optimal Underwriting Contracts in Securities Issuance Under Rank-Dependent Expected Utility

We provide a characterization of comparative weak risk aversion and comparative RDEU risk aversion for RDEU preferences and, in particular, we correct a claim made by Quiggin (1993) regarding comparative RDEU risk aversion. We then apply the analysis of comparative risk aversion to a problem of optimal design of underwriting contracts in securities issuance. Specifically, in public offerings of equity, an investment banking rm (the underwriter) plays an insurance role: through the underwriting contract, the issuing rm transfers the issue risk to the underwriter, as would an insured to an insurer. We extend a classical model proposed by Mandelker and Raviv (1977) to situations where the issuing rm and the underwriter have RDEU preferences. Assuming that the issuing company's and the underwriter's utility functions are concave and linear, respectively, and that either the underwriter is risk neutral or both the issuing company and underwriter are strongly risk averse, we show that a firm-commitment contract is optimal if and only if the issuing company's probability weighting function dominates the underwriter's.

Nonlinearly transformed risk measures: properties and application to optimal reinsurance

ABSTRACTWe propose the novel class of nonlinearly transformed risk measures (NTRMs) and apply them to the standard reinsurance problem in which an insurant seeks the risk-minimizing reinsurance contract. NTRMs transform both the outcomes and the probabilities of the insurant's uncertain final wealth by means of nonlinear functions. We prove relevant properties of NTRMs and show that they include popular Conditional Value-at-Risk (CVaR), Weighted Expected Shortfall (WES), Tail Nonlinearly Transformed Risk Measure (TNT), and Disutility Based Risk Measure (DBRM) as special cases. Regarding the reinsurance problem, we show that, under NTRMs, the optimal contract is of stop-loss type. We determine the optimal deductibles and provide comparative statics with respect to the insurant's risk aversion and initial wealth. Regarding comparative risk aversion, we show that the recently proposed WES, TNT, and DBRM help to overcome the restrictive all-or-nothing reinsurance decisions that prevail under CVaR. We further address the comparative statics with respect to initial wealth and show that under CVaR as well as WES and TNT, initial wealth is irrelevant: increasing initial wealth does not alter the optimal deductible. We show that NTRMs are able to overcome this shortcoming when the nonlinear transformation function of the outcomes is appropriately chosen.

Comparative Risk Aversion in Two Periods: An Application to Self-Insurance and Self-Protection

Risk management decisions provide a means to elicit individuals' risk preferences empirically. In such a context, the literature often presumes that the decision to invest in risk management and the benefit of this investment occur contemporaneously. There is, however, no consensus in the theoretical literature that one-period results can be transferred to intertemporal settings. To address this gap, we study the effect of an increase in risk aversion on the demand for risk management in a two-period context. Our findings reproduce the one-period results and, thus, support the focus of previous empirical literature on the structure of the risk rather than on the timing of investments and benefits. We also contrast our results with those obtained by employing widely used but limited preferences to examine risk aversion in intertemporal settings (standard additive expected utility setting, Selden, Epstein and Zin).

Alternative Approaches to Comparative nth-Degree Risk Aversion

This paper extends the three main approaches to comparative risk aversion—the risk premium approach and the probability premium approach of Pratt (1964) [Risk aversion in the small and in the large. Econometrica 32(1-2):122–136] and the comparative statics approach of Jindapon and Neilson (2007) [Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A comparative statics approach. J. Econom. Theory 136(1):719–728]—to study comparative nth-degree risk aversion. These extensions can accommodate trading off an nth-degree risk increase and an mth-degree risk increase for any m, such that [Formula: see text]. It goes on to show that, in the expected utility framework, all of these general notions of comparative nth-degree risk aversion are equivalent and can be characterized by the concept of (n/m)th-degree Ross more risk aversion of Liu and Meyer (2013) [Substituting one risk increase for another: A method for measuring risk aversion. J. Econom. Theory 148(6):2706–2718]. This paper was accepted by Han Bleichrodt, decision analysis.

Who Should Exert More Effort? Risk Aversion, Downside Risk Aversion and Optimal Prevention

I provide new results on how risk preferences affect optimal prevention. I identify a comparative risk aversion and a comparative downside risk aversion effect and emphasize those cases where both effects are aligned. Alignment depends on a probability threshold, which, in turn, only depends on the preferences of a benchmark agent. This allows to define an entire class of decision-makers who all share the same comparative static prediction relative to the reference agent. I interpret my results in terms of Arrow-Pratt risk aversion and the Schwarzian derivative, and also study several classes of parametric preferences changes.

Risk Aversion Over Finite Domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range. By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk. This general insight, which we explain to raise several interesting questions for further research, is our main contribution.

Comparative higher-order risk aversion and higher-order prudence

In this paper, we define the nth-degree utility premium as the pain associated with facing the passage from a more favorable risk to a less favorable risk, where the risk increase is specified by the notion of more nth-degree risk. We further define the nth-degree prudence utility premium as the increase in pain when the individual suffers a sure loss. We show that the nth-degree utility premium, normalized by the (n−1)th derivative of the utility function evaluated at the initial wealth, can explain comparative risk aversion of higher orders. On the other hand, the nth-degree prudence utility premium, normalized by the nth derivative of the utility function evaluated at the initial wealth, can explain comparative prudence of higher orders.

Credibilistic risk aversion

In the probabilistic risk aversion approach, risks are presumed as random variables with known probability distributions. However, in some practical cases, for example, due to the absence of historical data, the inherent uncertain characteristic of risks or different subject judgements from the decision-makers, risks may be hard or not appropriate to be estimated with probability distributions. Therefore, the traditional probabilistic risk aversion theory is ineffective. Thus, in order to deal with these cases, we suggest measuring these kinds of risks as fuzzy variables, and accordingly to present an alternative risk aversion approach by employing credibility theory. In the present paper, first, the definition of credibilistic risk premium proposed by Georgescu and Kinnunen [Fuzzy Inf. Eng., 2013, 5, 399–416] is revised by taking the initial wealth into consideration, and then a general method to compute the credibilistic risk premium is provided. Secondly, regarding the risks represented with the commonly used LR fuzzy intervals, a simple calculation formula of the local credibilistic risk premium is put forward. Finally, in a global sense, several equivalent propositions for comparative risk aversion under the credibility measurement are provided. Illustrated examples are presented to show the applicability of the theoretical findings.

Comparative Risk Aversion in the Presence of Ambiguity

This paper suggests a characterization of increases in risk aversion within the smooth ambiguity model by Klibanoff, Marinacci, and Mukerji (2005). I show that an increase in risk aversion is qualitatively different from that under expected utility, due to the incomplete separation between risk and ambiguity attitude. The analysis clarifies how ambiguity perception and attitude depend on risk aversion. (JEL D81)

Feasible sets, comparative risk aversion, and comparative uncertainty aversion in bargaining

We study feasible sets of the bargaining problem under two different assumptions: the players are subjective expected utility maximizers or the players are Choquet expected utility maximizers. For the latter case, we consider the effects on bargaining solutions when players become more risk averse and when they become more uncertainty averse.