Risk Aversion and the State Preference - Assignment Example

RISK AVERSION Introduction According to Gravelle and Rees (2004), risk aversion refers to the investor preferring less risk to more risk, since the possibility of loss is considered to be more serious than the possibility of gain. The most rational form of risk aversion is through insurance. Certainty equivalent in relation to an uncertain economic gain is the particular sum of money received by an individual, and is considered equal to the risk or gamble. An individual may implement ways to maximise net benefit, based on the certainty equivalents (Tisdell 1968). The factor of optimally increasing expected net gains does not take into consideration risk attitudes or an individual’s attitude to carrying the risk of uncertainty, it is risk neutral. On the other hand, the individual’s attitude to risk bearing is taken into account if expected utility rather than expected gain is to be maximised. Risk premium, equity premium or market premium refers to the additional allowance for risk which results in high rates of interest in the private sector. “The market risk premium is the expected rate of return in the aggregate stock market in excess of the risk-free interest rate” (Fernandez 2002, p.201). Thesis Statement: The purpose of this paper is to compare the State Preference and Machina triangle diagrams, explain an Edgeworth Box diagram, and discuss the factors that determine the efficient allocation of risk. Discussion An individual is risk averse if for any probability distribution the expected value of the distribution is preferred to the distribution itself. An individual who prefers a certain income rather than an uncertain one is said to be risk averse. “In contrast, a risk-neutral person is one who is indifferent to all alternatives with the same expected value” (Katz and Rosen 1998, p.168). For the consumer, uncertainty in the economic market could relate to a combination of or any one of the following factors: income, product price, product quality, and product availability, besides future income, interest rates and inflation rates (McKenna 1986). According to Eeckhoudt and Gollier (1995), the inverse relationship between marginal utility and wealth in the context of expected utility, explains why the largest loss should be covered first through insurance. The State Preference and Machina Triangle Diagrams The State Preference and the Machina Triangle diagrams can be compared and contrasted, as indifference maps for risk averse expected utility maximisers. The expected utility model as an approach to the theory of individual behaviour towards risk is distinctive due to the simplicity of its axioms, employing utility function and mathematical expectation, its utility functions such as risk aversion by concavity characterizing its various types of behaviour (Machina 1982). The expected utility hypothesis of behaviour towards risk is essentially that the individual decision maker possesses a von Neumann-Morgenstern utility function U(.) or von Neumann-Morgenstern utility index U1 “defined over some set of outcomes, and when faced with alternative risky prospects over these outcomes will choose that prospect that maximizes the expected value of U(.) or U1 (Machina 1988) The State Preference approach can be applied naturally to Insurance, since it is a clear “state-contingent” contract, that is, it pays an indemnity if a loss occurs. With conventional expected utility, the same problem of optimal insurance can be formulated (Arrow 1965; Eisner and Strotz 1961; Borch 1968). Figure 1. below depicts Optimal Insurance, by the State Preference diagram. Figure 1. Optimal Insurance (Fonseca 2009) “The simplest model is a two-state model with a fixed premium per dollar of coverage, g. As shown in Figure 1. state-dependent endowment w lies below the 45� certainty line. The existence of a state-independent utility function defined over payoffs, is assumed as follows: U(w) = p su(wA) + (1-p s)u(wN)” (Fonseca 2009). This represents the expected utility of the agent at the endowment point, shown in Figure 1. as U(w). The insurance agent aims to find the optimal amount of insurance coverage C, with some premium per dollar of coverage, y. It is found that in the above case, the marginal utility of a bad state is equal to that of a good state. The agent takes full coverage, hence there is recovery of the entire amount in the case of an accident. The optimal coverage is shown in Figure 1. by the “allocation c = (wN-a , wA + b ) where the highest indifference curve U(c) is tangent to the fair insurance line F on the 45 certainty line” (Fonseca 2009). Full coverage occurs only on the assumption of “fair insurance”, or Y = pi. The marginal utility of a good state is less than the marginal utility of a bad state, and with quasi-concave utility, the insurance agent’s utility in a good state exceeds his utility in a bad state. Hence the agent would still make some loss in the case of an accident, and cannot provide full coverage. In Figure 1. “unfair” insurance is shown at Point c}, “ the tangency of the unfair insurance line G with the highest indifference curve U (c} )” (Fonseca 2009). Since the allocation is out of the 45} certainty line, full coverage is not given for the loss. Figure 2. Indifference Curves in the State Preference Diagram (Powerpoint Slide No.12) Figure 2. above represents the indifference curves in the State Preference diagram. The red curved line depicts the expected utility indifference curve. Figure 3. below represents the Expected Utility Indifference Curves in a Machina Triangle diagram. Figure 3. Machina Triangle Diagram: Expected Utility Indifference Curves (Machina 1988, p.233) In modern consumer theory, the expected utility model specifies a set of objects of choice, represented by a real-valued maximand or preference function V(.), one object being preferred over another if it is assigned a higher value by this preference function. However, the expected utility model is different from “the theory of choice over nonstochastic commodity bundles” (Fonseca 2009) in two main ways: being a theory of choice under uncertainty the objects of choice are not deterministic outcomes but rather probability distributions over these outcomes. Secondly, the expected utility model applies a very specific restriction on the functional form of the preference function V(.). To demonstrate the risk averse property of preferences the lines made up of dashes in Figure 3. above represent the loci of solutions to specific equations which are equivalent to a constant. An individual is stated to be risk averse if increases in risk always lead to less preferred indifference curves represented in graphs as steeper indifference curves than the isoexpected value loci, and their utility indices would satisfy specific equations. The indifference map in Figure 3 indicates that the Lottery P is indifferent to the source. The lottery yielding amount x2 is believed to be the certainty equivalent of the lottery P. The origin lies on a lower isoexpected value locus than P shows a general characteristic of risk averse preferences, which is that the certainty equivalent of any lottery will always be less than its mean. When the outcomes are parts of the real line the aspects of preferences can be represented in the shape of the von Neuman-Morgenstern utility function U (.). The certainty equivalent of this lottery is given by that sure outcome c which also yields a utility level of u. For the fixed utiltiy indices U1, U2, U3, “the indifference curves of an expected utility maximizer consist of parallel straight lines in the triangle of slope [U2 – U1]/ [U3 – U2], as illustrated by the solid lines in Figure 1 (Fonseca 2009). Based on the evidence researchers are in the process of developing alternatives to the expected utility model, usually generalizations of the same. These are required to be capable of exhibiting nonlinearity as well as other standard properties of risk preferences such as first order stochastic dominance preference and risk aversion. “A general framework for the analysis of differential nonexpected utility preference functions in terms of their local linear approximations or expected utility approximations is developed by Machina (1982). The Edgeworth Box Diagram: The Optimal Sharing of Risks In some transactions, both parties may be risk averse, hence they would “generally contract to share the total risks and returns” (La Manna 1997). Figure 4. below, depicting the Edgeworth Box illustrates this concept. The indifference curves for each agent have the same absolute slope along their respective 45 degree certainty lines. It is clear that both traders A and B cannot achieve certainty positions. Geometrically, it is evident that the contract curve lies between the two 45 degree certainty lines, hence some of the risk will be borne by both A and B. Fiigure 4. The Edgeworth Box Diagram X (Wolfram 2010) The Edgeworth Box is a “traditional visualization of the benefits potentially available from trade” (Wolfram 2010). The purpose is to investigate the set of reallocations of goods that would benefit the two individuals A and B, by means of taking some starting allocation of goods. In the above figure as a result of the blue area, this set is known as “the core”. The green region shows the locus of allocations that are “Pareto optimal”, known as the “contract curve”. Pareto optimal allocations denote the condition when no changes would benefit both individuals. In the bottom graph are seen “the utility of A and B along the contract curve (the Pareto frontier), the utility they possess at the starting allocation, and a mapping of the core into the utility space” (Wolfram 2010). Factors that Determine the Efficient Allocation of Risk According to Shavell (2007), the efficient allocation of risk is as efficient a determinant of social welfare as the production of goods and services or the reduction of accident losses. In order to pursue the best allocation of risks, it is essential to ensure that estimation risk is accounted for: “first, determine an uncertainty range of market parameters (theta) that contain the true parameter (theta to the power of t), and yet it is as small as possible” (Meucci 2005, p.447), and then find out the optimization. For insurer pricing the price of risk should be in accordance with the type of risk under consideration. However, most risk based capital approaches distinctly use a common price of risk based on the company’s expected cost of capital for pricing. Recent developments in capital allocation for risk capital for solvency and by-line pricing show that a new technique is required. It is essential to differentiate between the risk adjustment to premiums to reflect the underwriting risk of the line of business which is usually considered as a risk loading in a premium, and the capital held to ensure that the insurer is capable of meeting his obligations under adverse circumstances. The premium risk loadings denote the market pricing of insurance risks and should also show the price elasticity if profit maximising assumptions are made. The risk based capital generally provides a level of enterprise-wide capability to meet obligations, and may be considered as a target capital level. In the case of a shareholder company, much of this capital is subscribed by equity investors. The risk measure which determines the economic value of this risk based capital is the insolvency put option value. “Fair pricing of lines of business require the allocation of the default put option value by line” (Sherris 2007, p.15). Otherwise the risk loading by line shows the underlying risk of each line of business from market factors independent of the insurer’s solvency or amount of capital. In non-competitive insurance markets, these risk loadings or profit loadings should depict the market conditions including price elasticity. Insurer pricing is also impacted by frictional costs and costs of market imperfections related to capital. Usually, these are borne by the shareholder or as a negative cost to equity value, and will be recouped only to the extent that “value maximising premium loadings and demand elasticity in the insurer market allow a loading to be included in the premiums to recover these costs” (Sherris, 2007, p.15). Conclusion This paper has highlighted Risk Aversion in insurance. The State Preference and Machina triangle diagrams have been compared and contrasted as a means of representing the indifference maps for risk averse expected utility maximisers. The Edgeworth Box diagram was used to show how risks are shared optimally between two risk averse decision makers. Further, the importance of optimal allocation of risks in insurance and the factors that determine the efficient allocation of risks were investigated, taking into account both the insurer market and business forces. References Arrow, K.J. (1965). Aspects of the theory of risk-bearing. Helsinki: Hahnsson Foundation. Borch, K. (1968). The economics of uncertainty. Princeton, New Jersey: Princeton University Press. Eeckhoudt, L. and Gollier, C. (1995). Risk, evaluation, management and sharing. London: Harvester Wheatsheaf. Eisner, R. and Strotz, R.H. (1961). Flight insurance and the theory of choice. Journal of Political Economy, 69: pp.355-368. Fernandez, P. (2002). Valuation methods and shareholder value creation. Edition 6. London: Academic Press. Fonseca, G.L. (2009). The State-Preference approach. The History of Economic Thought Website. Retrieved on 6th January, 2010 from: http://homepage.newschool.edu/het//essays/uncert/statepref.htm Gravelle, H. and Rees, R. (2004). Microeconomics. 3rd Edition. The United Kingdom: Longman. Katz, M.L. and Rosen, H.S. (1998). Microeconomics. 3rd Edition. New York: McGraw-Hill Irwin. La Manna, M. (1997). Readings in microeconomic theory. London: Cengage Learning. Machina, M.J. (1988). Expected utility hypothesis. Retrieved on 6th January, 2010 from: http://dss.ucsd.edu/~vcrawfor/ExpectedUtilityHypothesis.pdf Machina, M.J. (1982). “Expected utility” analysis without the independence axiom. Econometrica, 50 (2): pp.277-323. McKenna, C.J. (1986). The economics of uncertainty. London: Oxford University Press. Meucci, A. (2005). Risk and asset allocation. The United Kingdom: Springer Publishers. Powerpoint Slide No.12. (2008). Indifference curves in the State Preference diagram. Risk, Information and Insurance, Week 4, 2008. Shavell, S. (2007). Economic analysis of accident law. Massachusetts: Harvard University Press. Sherris, M. (2007). Risk based capital and capital allocation in insurance. Paper presented at the Institute of Actuaries. Biennial Convention 23-26 September, 2007. New Zealand. Retrieved on 8th January, 2010 from: http://www.versicherung.unikoeln.de/download/versich/sherris/sherris_literatur_1.pdf Tisdell, C.A. (1968). The theory of price uncertainty, production and profit. New Jersey: Princeton University Press. Wolfram (Wolfram Demonstrations Project). (2010). The Edgeworth Box. Wolfram Web Resources. Retrieved on 8th January, 2010 from: http://demonstrations.wolfram.com/TheEdgeworthBox/ Read More