Measure Of Absolute Risk Aversion Research Articles

Utility function for monetary gains (and losses) satisfying Friedman-Savage hypothesis


 
 
 In this note we provide a closed form utility function for gains that is S-shaped around the origin and satisfies the Friedman-Savage hypothesis. We obtain the corresponding Arrow–Pratt measure of absolute risk aversion (ARA) as well as Arrow–Pratt measure of relative risk aversion (RRA) for it.
 
 

Greater Arrow–Pratt (Absolute) risk aversion of higher orders

Higher-order risk attitudes are related to higher-order moments of risk, and are unequivocally characterized by the signs and levels of higher-order derivatives of utility functions. In contrast to the direction of higher-degree risk aversion, the intensity of higher-degree risk aversion beyond the Arrow–Pratt measure of absolute risk aversion is far from conclusive. The purpose of this paper is to develop a unified framework of greater (m,n)th-degree mixed risk aversion in the Arrow–Pratt tradition, which includes many competing notions of greater higher-degree (absolute) risk aversion proposed in the extant literature as special cases. Properties of greater (m,n)th-degree mixed risk aversion are studied, a choice-based characterization is established, and several applications are presented.

The dependence of the incremental risk rate of interest on absolute risk aversion - Applying the Laplace transform to risk preference evaluation

In this paper we base our study on the application of the Laplace transform to risk preference theory. With a constant measure of absolute risk aversion (Pratt, 1964; Arrow, 1965), the Certainty Monetary Equivalent (CME) of a risky project previously has been developed into an expression involving the logarithm of the bilateral Laplace transform of the probability density of its stochastic economic outcome. The internal risk aversion (IRA) is the break-even level of the absolute risk aversion, between making the project favourable or unfavourable.Below, we apply this methodology to determining an expression for the incremental risk rate of interest in the case of standard investments with payments having probability distributions. A simple approximate formula is derived explaining how the incremental risk rate of interest depends on the absolute risk aversion and the discount rate, i.e. on measures of risk preference and of time preference.

Risk preference evaluation – a fourth dimension of the application of the Laplace transform

In this paper, we apply the Laplace transform to risk preference theory (decision theory, decision analysis). We show that with a constant measure of absolute risk aversion, the certainty monetary equivalent (CME) can be developed into an expression involving the logarithm of the bilateral Laplace transform of the probability density of the outcome of risky projects. We also introduce a measure named internal risk aversion (IRA). This is in analogy with the earlier application of the transform to net present value problems. Properties of the CME function are examined. Basic examples of application are introduced followed by applications to portfolio theory and option pricing.

Changes in Polish Farmers’ Attitudes Toward Risk After Poland’s Accession to the EU

ABSTRACTThis article presents the econometric estimation of the risk attitudes of Polish farmers. For that purpose, a model of production under risk has been employed. The research is based on data from the Polish Farm Accountancy Data Network (FADN), from the years 2004–2010. For the Polish farming sector, risk-related considerations became increasingly important after Poland’s EU accession in 2004. After the accession, Polish agriculture was affected by market liberalization, which was somewhat mitigated by CAP support. To examine the risk aversion of Polish farmers, an estimation procedure based on Antle’s (1987) approach has been applied. This study suggests that, as was the case in other countries, risk aversion decreased following EU accession, that is, the Arrow-Pratt measure of absolute risk aversion based on producers specializing in field crops systematically decreased over the analyzed period.

On convergence of the utility indifference pricing in the model preserving the CGMY minimal entropy martingale measure

Motivated by financial instrument for pricing and hedging and also by the field of risk management, we consider measure changes for CGMY Levy processes which stay in the CGMY class. Within this two-parametric Esscher class of measure changes, we focus on the martingale measure with minimal relative entropy, called 'model preserving minimal entropy martingale measure' (MPMEMM). We link this measure to the utility-based indifference pricing. We precisely show that for bounded payoffs, the (exponential) utility indifference price goes to the option's price evaluated under the MPMEMM as the Arrow-Pratt measure of absolute risk aversion goes to zero. We also show that the class of Esscher martingale measures preserving the CGMY character is reduced to a single measure, described in full detail. Our results are new and contribute to the theory of option pricing under Levy models.

Rent Seeking Contest and Indirect Risk Preference

Rent seeking contest shapes the risk preference of the contestants. It instills in the weaker contestant who has little to lose and much to gain a preference for risk taking, and the weaker the contestant, the stronger the instilled preference for risk taking. On the other hand, it causes the stronger contestant who has much to lose and little to gain to have a preference for risk aversion and, the stronger the contestant, the more intense the instilled preference for risk aversion. Increases in the economies of scale in rent seeking amplify the effects of rent seeking contests on the risk preference of the contestants. The paper also derives the Arrow-Pratt measure of absolute risk aversion for the contestants.

Lattices, Lotteries and Choice

This paper addresses the classic Arrow-Pratt model of choice under uncertainty, relaxing one important condition, and using different tools. The condition relaxed is the single variable in preferences, by allowing for imperfect substitutability in preferences. The tools used are not topological but lattice theoretic. It is argued that when goods are imperfectly substitutable in preferences, not only attitudes to risk but also underlying preferences determine choice and therefore, a complex interaction between preferences and attititude to risk must be addressed in order to establish monotonicity of choice. Comparative static analysis is carried out with respect to changes in income, changes in uncertainty and changes in preferences and it is shown that the appropriate conditions for comparative statics are superextremal and single crossing property conditions with respect to appropriate value lattices (Antoniadou 1996). These are extensions of revealed preference conditions and in the case of a take-it-or-leave-it lottery choice, they are both necessary and sufficient. Furthermore, these conditions are implied by the standard conditions based on the Arrow-Pratt measure of absolute risk aversion in the univariate case. The case of Leontief preferences is studies extensively to demonstrate the work.

The uniqueness of pure-strategy Nash equilibrium in rent-seeking games with risk-averse players

This article proves that if each player’s measure of absolute risk aversion is nonincreasing in his/her wealth, then there exists a unique pure-strategy Nash equilibrium in the general rent-seeking game, where risk-averse players can coexist with risk-neutral players, attitude toward risk can differ among players, and concave production functions for lotteries are not necessarily the same among players.

Risk Aversion, Wealth, and Background Risk

We use household survey data to construct a direct measure of absolute risk aversion based on the maximum price a consumer is willing to pay for a risky security. We relate this measure to consumer's endowments and attributes and to measures of background risk and liquidity constraints. We find that risk aversion is a decreasing function of the endowment—thus rejecting CARA preferences. We estimate the elasticity of risk aversion to consumption at about 0.7, below the unitary value predicted by CRRA utility. We also find that households' attributes are of little help in predicting their degree of risk aversion, which is characterized by massive unexplained heterogeneity. We show that the consumer's environment affects risk aversion. Individuals who are more likely to face income uncertainty or to become liquidity constrained exhibit a higher degree of absolute risk aversion, consistent with recent theories of attitudes toward risk in the presence of uninsurable risks.

Risk-Averse Firms and Two-Part Tariffs under Uncertainty

This paper studies two-part tariffs with explicit consideration of cost uncertainty and risk aversion. It finds that firms charge a risk premium over expected marginal cost for each unit they sell. This pricing rule is socially optimal if and only if the modeled market is fully covered in equilibrium. A risk-averse monopoly tends to generate less aggregate net consumer surplus than a risk-neutral monopoly in partial-cover equilibrium but consumer welfare is indifferent when the market is fully covered. In oligopoly, consumer welfare increases (decreases) in the Arrow-Pratt measure of absolute risk aversion when the number of firms is relatively large (small).

On the intensity of downside risk aversion

The degree of downside risk aversion (or equivalently prudence) is so far usually measured by $\frac{-U^{\prime \prime \prime }}{U^{\prime \prime }}$ . We propose here another measure, $\frac{U^{\prime \prime \prime }}{U^{\prime }}$ , which has specific and interesting local and global properties. Some of these properties are to a wide extent similar to those of the classical measure of absolute risk aversion, which is not always the case for $\frac{ -U^{\prime \prime \prime }}{U^{\prime \prime }}$ . It also appears that the two measures are not mutually exclusive. Instead, they seem to be rather complementary as shown through an economic application dealing with a simple general equilibrium model of savings.

Risk Aversion, Wealth and Background Risk

We use household survey data to construct a direct measure of absolute risk aversion based on the maximum price a consumer is willing to pay to enter a lottery. We relate this measure to consumers' endowment and attributes and to measures of background risk. We find that risk aversion is a decreasing function of endowment - thus rejecting CARA preferences - but that the elasticity to consumption is far below the unitary value predicted by CRRA utility. We also find that households' attributes are of little help in predicting their degree of risk aversion, which is characterized by massive unexplained heterogeneity. However, consumers' environment affects risk aversion. Individuals who are more likely to face income uncertainty exhibit a higher degree of absolute risk aversion, consistent with recent theories of attitudes towards risk in the presence of uninsurable risks. We also find that risk attitudes have considerable predictive power on several household decisions, including occupation and portfolio choice, moving decisions and health status.

A Measurement of the Certainty Effect

In order to measure the certainty effect, the present paper tries to develop an axiomatic model of decision under risk which can accommodate the certainty effect and is, apart from this property, equivalent to expected utility. This leads to a new concept denoted by certainty preference which implies that a higher utility index is assigned to a consequence in the case of certainty than to the same consequence in the presence of risk. Now, a notion of comparative certainty preference can be obtained which will be characterized in terms of properties of the derived utility representation and allows for a generalization of the Arrow–Pratt measure of absolute risk aversion.

Optimal Insurance Coverage: Comment

Do you believe them? Suppose we restrict both of the above propositions to cases where the probability of a loss is and where the amount at risk is too large? Both of these propositions follow easily from a recent communication by Szpiro (1985) in this Journal. If Szpiro's results are to be believed, we only need to know the price of insurance and the measure of absolute risk aversion at the individual's level of initial wealth to determine what level of coverage is optimal -or at least as an approximation without any of great magnitudes. One of the culprits in obtaining the above results turns out to be the author's mimicking of Pratt's approximation of his risk premium formula using Taylor expansions. Szpiro argues that reasonable levels of the amount at risk, L, and the loss probability, q, do not yield serious errors in calculating his results. For his examples, Szpiro provides the magnitudes of the errors in estimating utility at post-loss wealth levels when we expand utility around the pre-loss wealth level. Even when these errors are small, what we should really care about is how close our predicted optimal insurance level is to the true optimal level. If L is small in relation to the pre-loss wealth level, W, then any predictor will have a small error as a percent of W. If W is $1 million and L is $1 thousand, then a predictor of insuring half the amount at risk has a maximal error in coverage (as a fraction of W-L) of about 0.0005. Although this may be a good predictor of final total wealth, it is doubtful that it is a good predictor of the level of insurance coverage.

Generalized Expected Utility Analysis of Risk Aversion with State-Dependent Preferences

In this paper the author develops measures of absolute risk aversion for preference relations on risky prospects that are state-dependent and are representable by Frechet differentiable, nonlinear, functionals. A comparative statics analysis of optimal air travel insurance coverage illustrates the usefulness of the proposed measures. It is also demonstrated that the analytical framework described above is consistent with the purchase of air travel insurance even in circumstances where expected utility theory is not. Copyright 1987 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Uncertain lifetime, risk aversion and life insurance

Abstract This paper develops the requisite conditions for interpersonal comparisons of attitudes toward risk in the presence of uncertain lifetime. For comparable individuals measures of absolute risk aversion are presented. Using the notion of actuarial notes the paper applies the measures of risk aversion to a comparative static analysis of the demand for term life insurance coverage.

Additional Aspects of Rational Insurance Purchasing

The purchase of insurance provides an individual with the ability to shift risk associated with either timeless or temporal uncertain prospects (see Dreze and Modigliani 1972). Previous insurance decision analysis can be further divided into cases in which the insurance contract is exogenously specified (see Mossin 1968; Smith 1968) and cases in which the optimal form of contract is to be determined (see Raviv 1979; Brennan and Solanki 1981). This note is concerned with extending some of the results for timeless uncertain prospects with the insurance contract exogenously specified. Many of the properties associated with the purchase of insurance for timeless uncertain prospects are analyzed by Mossin (1968) for different forms of contracts (see also Gould 1969). Utilizing the Arrow-Pratt measure of absolute risk aversion, the effect of wealth upon the optimal amount of insurance is determined for the different forms of contracts. Recent work by Ross (1981) demonstrates that in situations where individuals face more than one (timeless) uncertain prospect and they can purchase insurance only for a subset of these prospects, then a more risk-averse individual in the Arrow-Pratt This paper examines the use of the ArrowPratt and Ross measures of risk aversion. It is shown that in situations where the individual faces more than one uncertain prospect and can purchase insurance only for a subset of these prospects. then in many cases the Arrow-Pratt and Ross measures of risk aversion are not sufficient to describe the behavior of the individual in purchasing insurance.

Constant, Increasing and Decreasing Risk Aversion with Many Commodities

The Arrow (1971)-Pratt (1964) theory of risk aversion has achieved important successes in the study of economic responses to risk. This theory applies to a special class of cases in which the von Neumann-Morgenstern expected utility hypothesis is used to explain these responses. Under this hypothesis, individuals who maximize the expected utility of different utility functions will in general make different choices in the face of uncertainty. For the purpose of predicting which of two different utility functions will lead to locally more risk averse economic behaviour, the essential feature of these functions is their associated Arrow-Pratt absolute risk aversion measures. In fact, the local comparisons can be extended to global comparisons if the Arrow-Pratt absolute measures are uniformly comparable. The Arrow-Pratt measure of absolute risk aversion and a closely related measure of relative risk aversion also play crucial roles in the investigation of changes, induced by wealth increases, in the risk-taking behaviour of a single expected utility maximizing individual. This study has resulted in a satisfactory theory of increasing, decreasing and constant absolute and relative risk aversion. The special cases to which the Arrow-Pratt theory is directly applicable are those in which the utility functions in question are functions of a single variable, e.g. wealth. Unfortunately, the models required to explain many economic decisions employ a version of the expected utility hypothesis in which the argument of the utility function is a multidimensional vector. One example arises in the study of the consumption-savings decision. Another occurs in the analysis of the combined consumption-savings, portfolio decision. A final example is obtained in the investigation of demand for goods of uncertain quality. The papers of Kihlstrom-Mirman (1974) and Diamond-Stiglitz (1974) have suggested a partial generalization of the Arrow-Pratt theory of absolute risk aversion. This

On Multivariate Risk Aversion

This paper develops a matrix-measure of multivariate risk aversion which is related to a notion of risk premium and states the restrictions that must be imposed upon the matrix-measures of two utility functions in order that one require a higher risk premium than another for every small multivariate risk. A necessary and sufficient condition for comparability of global attitude towards risk is that the local restrictions hold over the entire domain. The usefulness of the measures of risk aversion is discussed within the context of a multivariate risk-sharing problem. UNIVARIATE MEASURES OF ABSOLUTE and relative risk aversion were introduced in the seminal works of Pratt [8] and Arrow [1], and have since become indispensable tools for the analysis of risk bearing in situations involving unidimensional risks. Recent years witnessed numerous attempts to generalize various aspects of the Pratt-Arrow notions to the case of multivariate risk (Kihlstrom and Mirman [6], Keeney [5], Duncan [2], Paroush [7], and Stiglitz [9]). The univariate case is qualitatively different from the multivariate case in that the ordinal preferences of all decision makers are identical, whereas in the multivariate case the preference orderings may differ among decision makers. This fact has two important consequences. First, while the analysis of risk bearing in situations involving univariate risks turns out to depend solely upon the cardinal properties of the function representing the ordinal preferences of the decision makers involved, in situations involving multivariate risks, the ordinal preferences themselves play an important role in addition to that of the function representing them. Second, in the univariate case the local measure of absolute risk aversion permits a complete ordering of individuals according to the relation least as risk averse as, whereas in the multivariate case such ordering is not in general independent of the specific risk under consideration. Pratt [8] has shown that the univariate measure of absolute risk aversion satisfies a formal relation to a number which has the interpretation of risk premium. In this study I introduce a matrix measure of absolute multivariate risk aversion which satisfies a similar formal criterion. It should be noted at the outset, however, that in the multivariate case the risk premium may be regarded as a vector of commodities (see, for example, Duncan [2, 3]), and interpersonal comparison of risk aversion is not directional independent. I.e., the ranking of risk