Insurance demand in the presence of loss-dependent background risk

We analyze insurance demand when insurable losses come with an uninsurable zero-mean background risk that increases in the loss size. If the individual is risk vulnerable, loss-dependent background risk triggers a precautionary insurance motive and increases optimal insurance demand. Prudence alone is sufficient for insurance demand to increase in two cases: the case of fair insurance and the case where the smallest possible loss exceeds a certain threshold value (referred to as the large loss case). We derive conditions under which insurance demand increases or decreases in initial wealth. In the large loss case, prudence determines whether changes in the background risk lead to more insurance demand. We generalize this result to arbitrary loss distributions and find conditions based on decreasing third-degree Ross risk aversion, Arrow–Pratt risk aversion, and Arrow–Pratt temperance.     

Decision making with Expected Shortfall and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.     

An Insurance Market Simulation With Both Adverse and Advantageous Selection

The theory of adverse selection predicts that high‐risk individuals are more likely to buy insurance than low‐risk individuals if asymmetric information regarding individuals’ risk type is present in the market. The theory of advantageous selection predicts the opposite—a negative relationship between insurance coverage and risk type can be obtained when hidden knowledge in other dimensions (e.g., the degree of risk aversion) is present in addition to the risk type. Using the heterogeneity of insurance buyers in either risk type or risk aversion, we first introduce a classroom‐based insurance market simulation game to show that adverse selection and advantageous selection can coexist. We then explain the underlying concepts using two methods: a mathematical framework based on expected utility theory and an empirical framework based on the results of the game itself. The game is easy to implement, reinforces textbook concepts by providing students a hands‐on experience, and supplements current textbooks by bringing their content up to date with current research.     

Two Classroom Simulations in Financial Risk Management and Insurance

ABSTRACT: Few classroom experiences provide as much educational value as a simulation exercise. A properly structured simulation provides students with the motivation to learn, the opportunity to explore strategies in an environment conducive to experimentation, and the immediate instructional benefit of watching their decisions affect the outcome of the collective simulation experience. This article describes the procedure and relative success of two classroom simulations for students in introductory and intermediate risk management and insurance courses. The first simulation replicates the risk management function of futures contracts through the use of hypothetical traders in the corn market with different risk management needs. The corn futures trading simulation achieves several goals for an introductory course in risk management and insurance: (1) students learn the importance of capital market risk management mechanisms; (2) students understand the transfer of risk among hedgers and speculators; and (3) students receive exposure to the concept that risk management is both possible and necessary for both speculative and pure risks. The second simulation mimics the operation of the market for homeowners insurance. By dividing students into consumer groups and insurer groups, participants experience the effect of chance events and insurance purchase decisions on their wealth. Small groups of upper‐level students act as insurers, and must price, package, and sell their product with a limited amount of surplus. Introductory students serve as consumers with limited resources who must survey the market and decide what product to buy and from whom. The competitive element and relatively unregulated market provide students with the incentive to innovate in a market for a common type of insurance and also demonstrates the need for some amount of insurance regulation. These simulations supply a simple way to enhance students' understanding of important basic concepts in a format that provides a welcome break from the traditional lecture format.     

A Gentle Introduction to Risk Aversion and Utility Theory

While the topics of risk aversion and utility theory have been discussed extensively in the academic literature on risk and insurance, this literature does not include a pedagogical discussion that is widely accessible for classroom use. This article provides a practical introduction to risk aversion that is designed for readers with little prerequisite course work in economics or statistics. We describe a simple model of insurance demand that can be applied to the property, liability, life, and health insurance markets. We also demonstrate how risk aversion affects a variety of real-life insurance decisions made under conditions of uncertainty, including how much the market will bear to pay for insurance administrative expenses and how demand varies for different types of auto insurance coverage. Exercises and practice problems are provided so that readers can test their mastery of the concepts presented in the article. An instructional note on using this article to teach risk aversion in the classroom is also provided.     

Portfolio Choice and Life Insurance: The CRRA Case

We solve a portfolio choice problem that includes life insurance and labor income under constant relative risk aversion (CRRA) preferences. We focus on the correlation between the dynamics of human capital and financial capital and model the utility of the family as opposed to separating consumption and bequest. We simplify the underlying Hamilton–Jacobi–Bellman equation using a similarity reduction technique that leads to an efficient numerical solution. Households for whom shocks to human capital are negatively correlated with shocks to financial capital should own more life insurance with greater equity/stock exposure. Life insurance hedges human capital and is insensitive to the family's risk aversion, consistent with practitioner guidance.     

Risk Management Case Project

Case projects are valuable tools for teaching risk management and insurance (RMI). This article describes a flexible case study approach that can be used as a comprehensive capstone project for the RMI major or in modules for graduate and undergraduate RMI courses. The project incorporates both fundamental RMI concepts and emerging trends in the field. The use of the case project provides benefits to both students and the RMI program, especially given the increasing pressure for assessment in college curricula.     

Risk aversion, intertemporal substitution, and the aggregate investment-uncertainty relationship

We analyze the role of risk aversion and intertemporal substitution in a simple dynamic general equilibrium model of investment and savings. Our main finding is that risk aversion cannot by itself explain a negative relationship between aggregate investment and aggregate uncertainty, as the effect of increased uncertainty on investment also depends on the intertemporal elasticity of substitution. In particular, the relationship between aggregate investment and aggregate uncertainty is positive even if agents are very risk averse, as long as the elasticity of intertemporal substitution is low. A negative investment-uncertainty relationship requires that the relative risk aversion and the elasticity of intertemporal substitution are both relatively high or both relatively low. We also show that the implications of our model are consistent with the available empirical evidence.     

When Is a Coinsurance‐Type Insurance Policy Inferior or Even Giffen?

This article derives the necessary and sufficient conditions for a coinsurance‐type insurance policy covering a particular risk to be inferior and to be Giffen. Mossin's decreasing absolute risk aversion assumption for insurance to be inferior is avoided. The result generalizes Hoy and Robson and Briys, Dionne, and Eeckhoudt's results to the case with a continuum of states and relaxes their assumption of constant relative risk aversion. It is shown that knowledge about the distribution of risk can be used to relax assumptions on an utility function for a coinsurance‐type insurance policy to be inferior and to be Giffen.     

Option Pricing Bounds and the Elasticity of the Pricing Kernel

In this paper we use power functions as pricing kernels to derive option-pricing bounds. We derive option pricing bounds given the bounds of the elasticity of the true pricing kernel. The bounds of the elasticity of the true pricing kernel are closely related to the bounds of the representative investor's coefficient of relative risk aversion. This methodology produces a tighter upper call option bound than traditional approaches. As a special case we show how to use the Black–Scholes formula to obtain option pricing bounds under the assumption of lognormality.     

A Shortcut Way of Pricing Default Risk Through Zero-Utility Principle

Pricing the default risk is a hot challenge for every risk manager. The problem is tackled in the framework of the zero‐utility principle. According to Pratt (1964) , an approximation of the risk premium should be proportional to the Arrow–Pratt absolute risk aversion coefficient and the variance. Is that still true as a default risk is concerned? The answer appears to be negative, because the variance does not look to be an appropriate tool for asymmetrical risk. On the other hand, fear of ruin coefficient and probability of default are proved to be well‐tailored tools for a preliminary pricing. Bid and ask price approximations are both elicited and a necessary condition for risk exchange set out.     

Relative Importance of Risk Sources in Insurance Systems

Abstract

Actuaries, and other managers of uncertainty, identify factors in modeling insurance risks because they believe (1) that these factors affect the outcome of a risk or (2) that the factors can be managed, thus allowing analysts a degree of control over the insurance system. This paper shows how to use a statistical measure, the coefficient of determination, for quantifying the relative importance of a source of uncertainty. With a quantitative measure of relative importance, risk managers can sharpen their intuition about the relative importance of risk factors and become better custodians of financial security systems.

This paper shows that the coefficient of determination is intuitively appealing in assessing the effectiveness of basic risk management techniques including risk exchange, pooling, and financial risk management. A single source common to all risks reduces the effectiveness of a pool; the risk measure quantifies the relative importance of this common source. The coefficient of determination is shown to have roots in the economics as well as the statistics literature. This connection provides further motivation for using the coefficient of determination and also suggests alternative measures for quantifying relative importance. The risk measure is useful in multivariate situations in which several factors affect a risk simultaneously. The paper illustrates this usefulness by considering a pool of policies that is subject to mortality, a common disaster, and a common investment environment.     

The Effects of Uncertainty on the Demand for Health Insurance

This article analyzes the effects of uncertainty and increases in risk aversion on the demand for health insurance using a theoretical model that highlights the interdependence between insurance and health care demand decisions. Two types of uncertainty faced by the individuals are examined. The first one is the uncertainty in the consumer's pretreatment health and the second is the uncertainty surrounding the productivity of health care. Comparative statics results are reported indicating the impact on the demand for insurance of shifts in the distributions of pretreatment health and productivity of health care in the form of first‐order stochastic dominance, Rothschild–Stiglitz mean‐preserving spreads, and second‐order stochastic dominance. The demand for insurance increases in response to a Rothschild–Stiglitz increase in risk in the distribution of the pretreatment health provided that the health production function is in a special class and the price elasticity of health care is nondecreasing in the pretreatment health. Provided also that the demand for health care is own‐price inelastic, the same conclusion is obtained when the uncertainty is about the productivity of health care.     

Life insurance decisions under recursive utility

In this paper, we generalize recursive utility to include lifetime uncertainty and utility from bequest. The generalization applies to discrete-time as well as continuous-time recursive utility, and it is an important step forward in the development of recursive utility. We formalize the problem of optimal consumption, investment, and life insurance choice under recursive utility, and we state a verification theorem with a generalized Hamilton-Jacobi-Bellman equation. Our generalization of recursive utility allows us to study optimal consumption, investment, and life insurance choice under separation of (market) risk aversion, elasticity of inter-temporal substitution, and elasticity of substitution between bequest and future utility. The separation gives rise to hump-shaped consumption patterns as observed in realized consumption.     

Revealed Preferences of Business Students at a Major State University

ABSTRACT: Business schools are facing a rapidly changing environment as new technology and alternative delivery systems erode barriers of time and place. The authors analyze the practices for determining curricula and courses offered by a major state university's business school and conclude that the school operates as a command economy. Such a structure produces artificial demand for some courses and obfuscates the measurement of true student preferences. The authors propose that, at least for full-capacity courses, enrollment should be replaced as a demand measure by one based on attempts to enroll. The authors' demand measure provides significantly different results from the traditional enrollment measure. The authors' results also provide insight into the relative preferences of students for different courses at the business school and the factors affecting these preferences, with some positive implications for the risk management and insurance discipline.     

Precautionary Saving and Consumption Smoothing across Time and Possibilities

This paper examines how aversion to risk and aversion to intertemporal substitution determine the strength of the precautionary saving motive in a two-period model with Selden/Kreps–Porteus preferences. For small risks, we derive a measure of the strength of the precautionary saving motive that generalizes the concept of "prudence" introduced by Kimball (1990b) . For large risks, we show that decreasing absolute risk aversion guarantees that the precautionary saving motive is stronger than risk aversion, regardless of the elasticity of intertemporal substitution. Holding risk preferences fixed, the extent to which the precautionary saving motive is stronger than risk aversion increases with the elasticity of intertemporal substitution. We derive sufficient conditions for a change in risk preferences alone to increase the strength of the precautionary saving motive and for the strength of the precautionary saving motive to decline with wealth. Within the class of constant elasticity of intertemporal substitution, constant-relative risk aversion utility functions, these conditions are also necessary.     

Optimal insurance contract and coverage levels under loss aversion utility preference

This study develops an optimal insurance contract endogenously and determines the optimal coverage levels with respect to deductible insurance, upper-limit insurance, and proportional coinsurance, and, by assuming that the insured has an S-shaped loss aversion utility, the insured would retain the enormous losses entirely. The representative optimal insurance form is the truncated deductible insurance, where the insured retains all losses once losses exceed a critical level and adopts a particular deductible otherwise. Additionally, the effects of the optimal coverage levels are also examined with respect to benchmark wealth and loss aversion coefficient. Moreover, the efficiencies among various insurances are compared via numerical analysis by assuming that the loss obeys a uniform or log-normal distribution. In addition to optimal insurance, deductible insurance is the most efficient if the benchmark wealth is small and upper-limit insurance if large. In the case of a uniform distribution that has an upper bound, deductible insurance and optimal insurance coincide if benchmark wealth is small. Conversely, deductible insurance is never optimal for an unbounded loss such as a log-normal distribution.     

Strategic Market Entry Project

Successful risk management is critical to top level decision makers in any organization, involving fundamental strategic policy and planning to identify and allocate scarce resources to projects or activities that generate sustainable competitive advantage and maximize available long‐term growth opportunities, or even survival. This article describes a flexible group project wherein students of risk management and insurance (RMI) may gain additional exposure and experience with applications of fundamental strategic management theory in the context of their particular RMI major coursework. The Project may be a useful tool in helping RMI students further develop their research and presentation skills, as well as enhance critical strategic decision making; exposure to cultural, regional or globalization issues; application of fundamental strategic management concepts; and knowledge of current events. While this Project was developed primarily for RMI students, students across business disciplines also may benefit from participation.     

Liquidity Risk and the Dynamics of Arbitrage Capital

We develop a continuous‐time model of liquidity provision in which hedgers can trade multiple risky assets with arbitrageurs. Arbitrageurs have constant relative risk‐aversion (CRRA) utility, while hedgers' asset demand is independent of wealth. An increase in hedgers' risk aversion can make arbitrageurs endogenously more risk‐averse. Because arbitrageurs generate endogenous risk, an increase in their wealth or a reduction in their CRRA coefficient can raise risk premia despite Sharpe ratios declining. Arbitrageur wealth is a priced risk factor because assets held by arbitrageurs offer high expected returns but suffer the most when wealth drops. Aggregate illiquidity, which declines in wealth, captures that factor.