Multivariate Risk Aversion Research Articles

Multivariate risk attitude: a comparison of alternative approaches in sustainability policies

AbstractRisk aversion has an unambiguous meaning in the univariate context: But, what does it mean to be risk averse in the multivariate case? Concave Risk Aversion (CRA) and Multivariate Risk Aversion (MRA) are relevant extensions of the risk aversion concept used in the univariate case to the multivariate case, corresponding to concave and ultramodular utility classes, respectively. Although CRA and MRA can coexist, they are dramatically different in some ways, leading to opposite preferences under some circumstances, as in the face of irreversible risks. We introduce the notions of purely concave and purely multivariate risk aversion, related to disjoint utility classes. We apply the purely risk aversion notions to the field of sustainability, where catastrophic and irreversible outcomes can be faced, in order to highlight and compare the consequences of the two approaches on sustainability policies. In this respect, we provide three main results. First, the kind of risk aversion determines the pursued goal. Second, the principle of “rejecting any fair bet” is not always preserved. Third, sustainability policies induced by different risk aversions, if repeated, produce final states in which mean-variance criterion holds.

Multivariate risk aversion utility, application to ESG investments

This paper uses the concept of multivariate or multi-attributive utility to attach different risk aversion levels to different sources of wealth (e.g. sectors, stocks, asset classes). In this context, we address the topic of environmental, social, and corporate governance (ESG) investments from the perspective of an investor with different risk aversion levels to green and brown stocks. We obtain closed-form solutions for the optimal allocations, value function, and wealth equivalent losses (WEL) from suboptimal choices. The numerical analysis demonstrates the significant increase, of up to 33%, in green investments when accounting for a differential in risk aversions levels, with up to 65% in WEL when using same risk-aversion levels.11We deeply thank anonymous reviewers and journal’s leadership for very interesting suggestions on relevant papers, keywords, and ideas that have enriched this work.

Multivariate uncertain risk aversion with application to accounts receivables pricing

Multivariate uncertain risk aversion with application to accounts receivables pricing

Allocation of tasks for reliability growth using multi-attribute utility

In reliability growth models in particular, and project risk management more generally, improving the reliability of a system or product is limited by constraints on cost and time. There are many possible tasks which can be carried out to identify and design out weaknesses in the system under development. This paper considers the allocation problem: which subset of tasks to undertake. While the method is applicable to project risk management generally, the work has been motivated by reliability growth programmes. We utilise a model for reliability growth, based on an efficacy matrix, developed with engineering experts in the aerospace industry. We develop a general multi-attribute utility function based on targets for cost, time on test and system reliability. The optimal subset is identified by maximising the prior expected utility. We derive conditions on the model parameters for risk aversion and loss aversion based on observed properties of preference. We give conditions for multivariate risk aversion under the general form of the utility function. The method is illustrated using an example informed by work with aerospace organisations.

Local Utility and Multivariate Risk Aversion

We revisit Machina’s local utility as a tool to analyze attitudes to multivariate risks. We show that for nonexpected utility maximizers choosing between multivariate prospects, aversion to multivariate mean preserving increases in risk is equivalent to the concavity of the local utility functions, thereby generalizing Machina’s result [Machina M (1982) “Expected utility” analysis without the independence axiom. Econometrica 50:277–323]. To analyze comparative risk attitudes within the multivariate extension of rank dependent expected utility of Galichon and Henry [Galichon A, Henry M (2012) Dual theory of choice with multivariate risks. J. Econom. Theory 147:1501–1516], we extend Quiggin’s monotone mean and utility preserving increases in risk and show that the useful characterization given in Landsberger and Meilijson [Landsberger M, Meilijson I (1994) Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion. Ann. Oper. Res. 52:97–106] still holds in the multivariate case.

Local Utility and Multivariate Risk Aversion

We revisit Machina's local utility as a tool to analyze attitudes to multivariate risks. Using martingale embedding techniques, we show that for non-expected utility maximizers choosing between multivariate prospects, aversion to multivariate mean preserving increases in risk is equivalent to the concavity of the local utility functions, thereby generalizing Machina's result in Machina (1982). To analyze comparative risk attitudes within the multivariate extension of rank dependent expected utility of Galichon and Henry (2011), we extend Quiggin's monotone mean and utility preserving increases in risk and show that the useful characterization given in Landsberger and Meilijson (1994) still holds in the multivariate case.

Local Utility and Multivariate Risk Aversion

We revisit Machina's local utility as a tool to analyze attitudes to multivariate risks. Using martingale embedding techniques, we show that for nonexpected utility maximizers choosing between multivariate prospects, aversion to multivariate mean preserving increases in risk is equivalent to the concavity of the local utility functions, thereby generalizing Machina's result in [18]. To analyze comparative risk attitudes within the multivariate extension of rank dependen t expected utility of [10], we extend Quiggin's monotone mean and utility preserving increases in risk and show that the useful characterization given in [17] still holds in the multivariate case.

On multivariate prudence

In this paper we extend the theory of precautionary saving to the case in which uncertainty is multidimensional and we develop a matrix-measure of multivariate prudence. Furthermore, we characterize comparative prudence, decreasing and increasing prudence, the effect of uncertainty on the marginal propensity to consume out of wealth, and the Dreze-Modigliani substitution effect in this multivariate setting. We also characterize the concept of multivariate downside risk aversion as a multivariate preference for harm disaggregation. We show that our definition is equivalent to a positive precautionary saving motive. We propose an alternative measure of the intensity of downside risk aversion and show that this measure is useful in understanding several economic problems that involve multivariate preferences.

On Multivariate Prudence

In this paper we extend the theory of precautionary saving to the case in which uncertainty is multidimensional and we develop a matrix-measure of multivariate prudence. Furthermore, we characterize comparative prudence, decreasing and increasing prudence, the effect of uncertainty on the marginal propensity to consume out of wealth, and the Dreze-Modigliani substitution effect in this multivariate setting. We also characterize the concept of multivariate downside risk aversion as a multivariate preference for harm disaggregation. We show that our definition is equivalent to a positive precautionary saving motive. We propose an alternative measure of the intensity of downside risk aversion and show that this measure is useful in understanding several economic problems that involve multivariate preferences.

Decomposing the Cross Derivatives of a Multiattribute Utility Function into Risk Attitude and Value

The cross derivatives of a multiattribute utility function play an important role in the choice between multivariate lotteries and in multiattribute Taylor expansions of the utility function. This paper decomposes the cross derivatives into two components: the derivatives of a single-attribute utility function over value and the cross derivatives of the value function. This approach provides a simple method for reasoning about the signs of the cross derivatives of a multiattribute utility function using derivatives of a univariate utility function and the properties of the value function. To illustrate the approach, we relate the multivariate risk aversion concept, which involves the mixed partial derivative of the utility function, to the Arrow–Pratt risk aversion function. We show that for additive value functions, a decision maker is multivariate risk averse if and only if he is risk averse over value in the Arrow–Pratt sense. For other value functions, however, a decision maker can be risk averse or risk seeking over value and still exhibit multivariate risk aversion. The approach also derives the conditions on the value function that relate two important classes of utility functions: single attribute utility functions whose derivatives alternate in sign and multiattribute utility functions whose cross derivatives alternate in sign. These two classes are widely used in practice and form the basis of univariate and multivariate stochastic dominance. Several examples illustrate the approach.

The Welfare Impacts of Commodity Price Volatility: Evidence from Rural Ethiopia

Many governments have tried to stabilize commodity prices based on the widespread belief that households in developing countries – especially poorer ones – value price stability, defined here as the lack of fluctuations around a mean price level. We derive a measure of multivariate price risk aversion as well as an associated measure of willingness to pay for price stabilization across multiple commodities. Using data from a panel of Ethiopian households, our estimates suggest that the average household would be willing to pay 6-32 percent of its income to eliminate volatility in the prices of the seven primary food commodities. Not everyone benefits from price stabilization, however. Contrary to conventional wisdom, the welfare gains from eliminating price volatility would be concentrated in the upper 40 percent of the income distribution, making food price stabilization a distributionally regressive policy in this context.

Comparative Risk Aversion of Different Preferences

An argument of Kihlstrom and Mirman concerning comparative risk aversion with many goods is formalized. Taking more risk averse to mean that an individual is less willing to accept a risky lottery or the riskiest of two lotteries, it is shown that risk aversion cannot be compared across individuals with different preferences so concave transforms of utility functions may indeed be taken as a basis for comparative risk aversion. A characterization of multivariate comparative risk aversion is thereby obtained.

On expected utility for financial insurance portfolios with stochastic dependencies

Abstract The effect of background risks as human capital, market risks and catastrophic events has been considered in the literature in different contexts. In this note, we consider financial insurance portfolios with insurable risks and one background risk (uninsurable financial asset), such that the random losses and the background risk depend on environmental parameters. We study how dependencies between the risks influence the expected utility of the portfolio’s wealth distribution under risk aversion, when the environmental parameters are random. Stochastic bounds for the expected wealth are given from modeling the dependence between the parameters by different notions. Similar results are given for multivariate portfolios with n groups and multivariate risk aversion, besides an expected utility comparison result for the minimum and the total portfolio’s wealth.

Understanding and measuring bivariate risk attitude: what can we learn from concordance?

In past years the study of the impact of risk attitude among risks has become a major topic, in particular in Decision Sciences. Subsequently the attention was devoted to the more general case of bivariate random variables. The first approach to multivariate risk aversion was proposed by de Finetti ([2]), and Richard ([15]) and it is related to the bivariate case. More recently, multivariate risk aversion has been studied by Scarsini ([20], [21], [22]). Nevertheless even if decision problems with consequences described by more than two attributes have become increasingly important, some questions appear not completely solved. This paper concerns with a definition of bivariate risk aversion which is related to a particular type of concordance: a bivariate risk averse Decision Maker is a Decision Maker who always prefers the independent version of a bivariate random variable to the random variable itself.

Understanding and Measuring Bivariate Risk Attitude: What Can We Learn from Concordance?

In past years the study of the impact of risk attitude among risks has become a major topic, in particular in Decision Sciences. Subsequently the attention was devoted to the more general case of bivariate random variables. The first approach to multivariate risk aversion was proposed by de Finetti (1952) and Richard (1975) and it is related to the bivariate case. More recently, multivariate risk aversion has been studied by Scarsini (1985, 1988, 1999). Nevertheless even if decision problems with consequences described by more than two attributes have become increasingly important, some questions appear not completely solved. This paper concerns with a definition of bivariate risk aversion which is related to a particular type of concordance: a bivariate risk averse Decision Maker is a Decision Maker who always prefers the independent version of a bivariate random variable to the random variable itself.

The Becker-DeGroot-Marschak mechanism is not necessarily incentive compatible, even for non-random goods

This paper uses results from [Karni, E., Safra, Z., 1987, Preference reversals and the observability of preferences by experimental methods, Econometrica 55, 675–685] and [Karni, E., 1989, Generalized expected utility analysis of multivariate risk aversion, International Economic Review 30, 297–305] to show that the Becker-DeGroot-Marschak mechanism is not always incentive compatible, even when the item to be valued involves no uncertainty, contrary to several recent assertions.

Wishart Quadratic Term Structure Models

This paper reveals that the class of affine term structure models introduced by Duffie and Kan (1996) is much larger than it has been usually considered in the literature. We study fundamental risk factors, which represent multivariate risk aversion of the consumer or the volatility matrix of the technological activity returns, and argue that they can be defined as symmetric positive matrices. For such matrices we introduce a dynamic affine process called the Wishart autoregressive (WAR) process; this process is used to reveal the associated term structure. In this framework: i) we derive very simple restrictions on the parameters to ensure positive yields at all maturities; ii) we observe that the usual constraint that the volatility matrix of an affine process be diagonal up to a path independent linear invertible transformation can be considerably relaxed. The Wishart Quadratic Term Structure Model is the natural extension of the one-dimensional Cox-Ingersoll-Ross model and of the quadratic models introduced in the literature.

A strong (Ross) characterization of multivariate risk aversion

Using the ‘addition of uncorrelated noise’ as a natural definition of increasing risk for multivariate lotteries, I interpret risk aversion as the willingness to pay a (possibly random) vector premium in exchange for a reduction in multivariate risk. If no restriction is placed on the sign of any co-ordinate of the vector premium then (as was the case in Kihlstrom and Mirman's (1974) analysis) only pairs of expected utility maximizers with thesame ordinal preferences for outcomes can be ranked in terms of their aversion to increasing risk. However, if we restrict the premium to be a non-negative random variable then comparisons of aversion to increasing risk may be possible between expected utility maximizers withdistinct ordinal preferences for outcomes. The relationship between their utility functions is precisely the multi-dimensional analog of Ross's (1981)global condition forstrongly more risk averse.

Many good risks: An interpretation of multivariate risk and risk aversion without the Independence axiom

What can be inferred, without assuming the Independence axiom, about an agent's preferences over many-good lotteries from knowledge that the agent is income risk averse? We show that income risk aversion corresponds to an intuitive substitution property of the many-good lottery preferences that is, itself, equivalent to a standard definition of many-good risk aversion. We apply our approach to derive some well-known results directly from our interpretation of income risk aversion and neutrality.

Multivariate Risk Aversion and Uninsurable Risks: Theory and Applications

The objective of this paper is to develop conditions for global multivariate comparative risk aversion in the presence of uninsurable, or background, risks, and thus generalize Kihlstrom and Mirman [1974] and Karni [1979, 1989]. We analyze von Neumann-Morgenstern (VNM) utility functionsas well as smooth preference functionals which are nonlinear in distribution but locally linear in probabilities. In each case we provide an economic application which illustrates how our theorems can be used. We analyze a risk sharing, a portfolio choice, and a labor supply problem for VNM utility functions, and the optimal allocation of effort to risky technologies in the presence of a random supply (or quality) of a public good for nonlinear preference functionals. We consider thecase where the random variables are mean-independent as well as the case where they are independent. In the labor supply application for VNM utility functions, we show that if the two risks are independent, the comparative statics effect of greater risk aversion on labor supply in the presence of a background non-wage income risk is determined by a monotonic relationship between labor supply and the wage rate under certainty. That is, we extend the applicability of the Diamond-Stiglitz [1974]-Kihlstrom-Mirman [1974] single-crossing property to the case where an independent background risk is present.