The assessment of insurance risk premium with a simulation study using Archimedean copulas

The risk premium and the probability premium are used to determine appropriate coefficients of absolute risk aversion under CARA utility. A defensible range of risk aversion coefficients is defined by the coefficients that correspond to risk premiums falling between 1 and 99% of the amount at risk or to probability premiums falling between .005 and .49 for a lottery that pays or loses a given sum. The consequences of ignoring risk premiums when selecting risk-aversion coefficients for representative decision makers are illustrated by calculation of the implied risk premium associated with the levels of absolute risk aversion assumed in six selected studies.

It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of d-dimensional l 1 ; -norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189-207] in an analogous manner to the well-known Bernstein-Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the d-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.

The notion of risk aversion originates from Bernoulli’s proposal of introducing the certainty equivalent of a lottery through its expected utility. This proposal implies, if the utility function is concave, a positive risk premium, where the risk premium of a lottery is the difference between its expected value and its certainty equivalent. Pratt (1964) based his notion of risk aversion on the risk premium and he introduced a measure of risk aversion (the well known Arrow-Pratt measure of risk aversion), which strictly depends on the concavity of the utility function. This result occurs since Pratt (as well as {ptde} Finetti, 1952, and Arrow, 1965) adopted the expected utility theory. However, we can also use the risk premium when a non-expected utility theory is adopted. With regard to these theories, which represent the preference preordering on the set of lotteries by means of a utility function which is non-linear in the probabilities of consequences, we find that the risk premium depends on the non-linearity in the probabilities much more than on the non-linearity in the consequences (i.e., on the concavity of the utility function of the consequences). This result leads to the distinction between the risk aversion of the first order and the risk aversion of the second order (Montesano, 1988 and 1991, Hilton, 1988, and Segal and Spivak, 1990).

Theories of rational decision making hold that decision makers should select the best alternative from the available choices, but it is now well known that decision makers employ heuristics and are subject to a set of psychological biases. Risk aversion or risk seeking attitude has a framing effect and can bias the decision maker towards inaction or action. Understanding decision-makers’ attitudes to risk is thus integral to understanding how they make decisions and psychological biases that might be at play. This paper presents the Engineering-Domain-Specific Risk-Taking (E-DOSPERT) test to measure the risk aversion and risk seeking attitude that engineers have in four domains of engineering risk management: identification, analysis, evaluation and treatment. The creation of the instrument, an analysis of its reliability based on surveying undergraduate engineering students in Australia and the United States, and the validity of the four domains are discussed. The instrument is found to be statistically reliable to measure engineering risk aversion and risk seeking, and to measure engineering risk aversion and risk seeking to risk identification and risk treatment. However, factor analysis of the results suggest that four other domains may better describe the factors in engineers’ attitude to risk.

Defining propensity to self-protect as the maximum amount an individual is willing to pay for a one-unit reduction in the probability of loss, this article studies its basic behavior and its relationship to the individual's degree of risk aversion and the initial loss probability. It is shown that if the initial loss probability is below a threshold, a more risk-averse individual has a higher propensity to self-protect, and the threshold is controlled by individuals' aversion to general downside risk increases and aversion to overall riskiness measured in variance. INTRODUCTION Self-protection is defined as the expenditure on reducing the probability of suffering a loss. Despite its relevance to a wide range of economic issues, [1] self-protection--especially its relationship with an individual's attitude towards risk--has not been adequately understood. In their pioneering work, Ehrlich and Becker (1972) noted that self-protection may be attractive to both risk-averse people and risk lovers and that unlike self-insurance (the expenditure on reducing the severity of loss), self-protection and market insurance can be complements. More recently, Dionne and Eeckhoudt (1985) showed that a more risk-averse individual does not always purchase more self-protection. And Briys and Schlesinger (1990) explained this phenomenon by showing that self-protection in general does not reduce the riskiness of individuals' final wealth. Sweeney and Beard (1992a) went one step further in showing that for a general loss probability function of self-protection spending, it is impossible to characterize th e preferences of an individual who always chooses a higher level of self-protection. In their attempt to verify an interesting intuition that insurance is reducing small chances of bad outcomes and gambling is increasing small chances of good outcomes, McGuire, Pratt, and Zeckhauser (1991) came closest to identifying a relationship between an individual's degree of risk aversion and his or her choice of self-protection. They show that if a less risk-averse individual's optimal choice of self-protection is such that the resulting loss probability is less than a critical switching level, then a more risk-averse individual's optimal choice will be higher than the less risk-averse. [2] From these previous contributions, it is clear that the relationship of an individual's attitude towards risk with his or her propensity to purchase self-protection is not as straightforward as that with market insurance or self-insurance. What they do not imply, however, is that a more primitive and more precise characterization of the relationship between risk aversion and individuals' propensity to purchase self-protection is impossible insofar as one accepts the Expected Utility paradigm. Staying within the Expected Utility framework, one can look at the problem from a slightly different angle. Specifically, instead of investigating the optimal choice of self-protection given an assumed relationship between self-protection spending and the loss probability, one can explicitly consider an individual's willingness (or propensity) to purchase self-protection-the maximum an individual is willing to pay for a given reduction in the probability of loss. [3] So far only scant effort has been made in understanding the problem from this perspective. Under various restrictions, Eeckhoudt, Godfroid, and Gollier (1997) compared the effects of risk-aversion on the risk premium [defined in Pratt (1964)] and on the willingness to pay and conclude that some properties of the risk premium are not shared by the willingness to pay. Eeckhoudt and Godfroid (1998) showed in a quite different context with exponential utility functions that the lower the initial probability of accident, the greater the market value o f a reduction in the probability. The aim of this article is to provide a comprehensive study of individuals' propensities to purchase self-protection. …

Copula is one of the widely used techniques to describe the dependency structure between components of a system. Among all existing copulas, the family of Archimedean copulas is the popular one due to its wide range of capturing the dependency structures. In this paper, we consider the systems that are formed by dependent and identically distributed components, where the dependency structures are described by Archimedean copulas. We study some stochastic comparisons results for series, parallel, and general $r$-out-of-$n$ systems. Furthermore, we investigate whether a system of used components performs better than a used system with respect to different stochastic orders. Furthermore, some aging properties of these systems have been studied. Finally, some numerical examples are given to illustrate the proposed results.

This dissertation explores issues regarding the effect of investor risk aversion and sentiment on financial markets. It is widely considered that a risk averse investor requires risk premium to hold risky assets, and the required risk premium is proportional to the riskiness of the underlying assets. From the myopic loss aversion perspective, investors, who get utility by frequently evaluating their portfolios and are more sensitive to losses, will require a higher risk premium as compensation for the fear of a major drop in financial wealth. Recent literature has shown that large tail jumps are often contributed to such major drop in financial markets. Rare events, which often accompanied with tail jumps, have a more drastic impact on risk averse investor, and the compensation for rare events accounts for a large fraction of the equity risk premium. However, there is no theoretical framework that has been developed to separate the risk aversion component of rare events from daily volatility.;In this work, I continue to argue the importance of the rare events have different impact on investors decision regarding to equity risk premium. Rational investor risk aversion should spike and require higher risk premium to compensate for higher risk when rare events happen. I develop a theoretical framework to decompose the risk aversion component with rare events from the part associated with daily volatility and prove that they have varying impact on risk premium. Specifically, I first extend the jump-diffusion model incorporated with disaster models and develop a theoretical framework to decompose the risk aversion component of rare events from frequent events. Then I attempt to use monetary surprises as empirical application of the theoretical framework. Bernanke and Kuttner (2005) show that the effect of monetary surprises on expected excess returns can be related to the impact of monetary policy on investor risk aversion or the riskiness of stocks. Finally I test how market response to macroeconomic news surprises conditional on investor sentiment. The results are arranged in the following order.;Chapter 1 of the dissertation makes the argument that investors should treat rare events differently from frequent events. Intuitively, a rare event (disaster) reduces the fundamental value of a stock by a time-varying amount. A rise in disaster probability lowers the expected rate of return on equity, and it also motivates investors to shift toward the risk-free asset or buy deep out-of-the-money puts. Comparing to previous models, I extend the general jump-diffusion

Risk and ambiguity aversion behavior in index-based insurance uptake decisions: Experimental evidence from Ethiopia

In this paper, Archimedean copula-based method was used to investigate the multi-state reliability analysis of a parallel system. First, the fundamental theory associated with Archimedean copula for both bi-variate and multivariate distributions as well as the tail dependence of Gumbel copula (an Archimedean copula family) were briefly introduced. Then, a general parallel system reliability problem was formulated for three identical components parallel system. Thereafter, the system reliability bounds of the parallel systems considered was derived using the copula approach. Graphical method was used to show failure space for the eight possible failure states associated with the system considered. Finally, an illustrative example is presented to demonstrate the proposed method. The results indicate that as the number of component failure increases, for parallel system with more than two components in parallel, the system probability of failure moves from the upper bound to the lower bound. Furthermore, Archimedean copula (Gumbel copula), which has been successfully used to model probability of failure for bivariate distribution, cannot successfully model the probability of failure for all possible failure state associated with more than two components parallel.

Farmers’ risk preferences play an important role in agricultural production decisions, risk takers means the farmers who are willing to take risky decisions in farming, risk aversion means an attitude of reluctance to take risky decisions in farming. Climatic change effects all regions across the globe and causes substantial agitations that can be expected to be natural systems that have foreseeable influences on the economic systems of upland regions through both direct and indirect means. Risk preferences reflect the farmers’ personal experiences and beliefs, these preferences explain how the decision-maker assesses and react to risks. This study characterizes risk behaviour among marginal and small farmers in Cauvery Delta Zone and determines how these risk preferences affects the farmers. The study was conducted in Cauvery Delta zone of Thanjavur, Thiruvarur and Nagapattinam districts with a Sample size of 366 farmers which consists of 183 marginal and 183 small farmers was selected randomly based on proportionate random sampling method. The risk behavior was measured by the measure of risk attitude and two lottery methods viz., Eckel-Grossman and Holt-Laury based lottery method. Measure of risk attitude results shows that, 27.60 per cent of farmers were moderate risk taker followed by 24.30 per cent were risk averser and 15.00 per cent of farmers were risk taker. The Eckel and Grossman lottery method result shows CRRA (Constant Relative Risk Aversion) value was 0.38 to 0.67, which shows that marginal farmers were risk aversers and small farmers were moderate risk takers. The CRRA adapted from Holt and Laury [1] range for the maximum was 1.37 and minimum -1.71 for their choices. The majority of marginal farmers were risk aversers, the socio-economic characteristics of the farmers decides the risk preference. The risk-averse farmer this may imply risk-taking behavior that is reduced by risk aversion (resulting in on-farm risk management strategies) and a reduced demand of insurance.

The behavioural components of risk aversion

A copula is a means of generating ann‐variate distribution function from an arbitrary set ofnunivariate distributions. For the class of portfolio allocators that are risk averse, we use the copula approach to identify a large set ofn‐variate asset return distributions such that the relative magnitudes of portfolio shares can be ordered according to the reversed hazard rate ordering of thenunderlying univariate distributions. We also establish conditions under which first‐ and second‐degree dominating shifts in one of thenunderlying univariate distributions increase allocation to that asset. Our findings exploit separability properties possessed by the Archimedean family of copulas.

The notion of risk aversion was originally developed with reference to the Expected Utility model. de Finetti (1952), Pratt (1964) and Arrow (1965) associated the concavity of the von Neumann-Morgenstern utility function with some relevant aspects of the decision-maker’s preferences. In particular, risk aversion can be defined in terms of risk premium (i.e., the difference between the expected value and the certainty equivalent of a lottery). With reference to the EU model the risk premium is nonnegative for all lotteries if and only if the von Neumann-Morgenstern utility function is concave. However, with reference to the EU model, other relevant aspects of the preferences also depend on the concavity of the utility function: for instance, if we compare two lotteries of which one has been obtained from the other through mean preserving spreads, the less risky lottery is (weakly) preferred for all pairs of lotteries of this kind if and only if the von Neumann-Morgenstern utility function is concave. Moreover, the EU model does not imply that a randomization of lotteries matters (for instance, according to the EU model, a lottery whose consequences are a randomization of the outcomes of two equally preferred lotteries is indifferent to them), while the possibility that a decision-maker prefers not to be involved in an additional lottery could be considered as a kind of risk aversion. Taking into consideration more general models than the EU model, it is no longer true that risk aversion only consists of positive risk premia and of aversion to riskier (in the sense of mean preserving spreads) lotteries and that these two risk aversions depend on the same characteristic of decision-maker’s preferences.

Semiparametric bivariate Archimedean copulas

The (additive) generator of an Archimedean copula is a strictly decreasing and convex function, while Morgenstern utility functions (applying to risk aversion decision makers) are nondecreasing and concave. In this presentation, relationships between generators and utility functions are established. For some well known Archimedean copula families, links between the generator and the corresponding utility function are demonstrated. Some new copula families are derived from classes of utility functions which appeared in the literature, and their properties are discussed. It is shown how dependence properties of an Archimedean copula translate into properties of the utility function from which they are constructed.