Consumers In The Financial World Finance Essay - Free Essay Example

Where decisions taking place in world of certainty, consumers know for sure the utility they will receive given a choice of goods. Firms know for sure the profit they will receive from a chosen set of inputs, this does not describe the real world, technological, uncertainty, market uncertainty, many issues cannot be addressed without considering uncertainty e.g. stock market, insurance, futures markets (investment and savings decisions). In this essay I will look at attitudes towards risk and uncertainty in the insurance market. When people have to make decisions in the presence of uncertainty rational decision making does not go out the window. The standard tools for analyzing rational choice can be modified to accommodate uncertainty. A person in an uncertain environment is choosing among contingent commodities, whose value depends on the eventual outcome or state of the world. As with ordinary commodities, people have preferences for contingent commodities that can be represented by an indifference map. The slope of the budget constraint between two contingent commodities depends on the payoff associated with each state of the world. The curvature of the indifference curve depends on whether the individual is risk averse, risk loving or risk neutral. A risk-averse person will not accept an actuarially fair bet. Risk-averse people purchase insurance in order to spread consumption more evenly across states of the world. When risk-averse people are allowed to purchase fair insurance, they will insure themselves fully in the sense that their consumption is the same in every state of the world. The amount of insurance demanded depends on the premium and on the probability that the insurable event will occur. People wi th von Neumann-Morgenstern utility functions, in which the probability of each state of the world is multiplied by the utility associates with that state of the world, seek to maximise the expected value of their utility. The assumption of expected utility maximisation, together with decision trees, can be used to break up complicates decisions into simple components that can be readily solved. By comparing the expected utility of each option, the individual can determine their optimal strategy. An individuals attitude towards risk in which there is a single good (income), assuming that there are only two states of the world (state 1 and state 2), let y1 denote income in state 1 and y2, income in state 2. Let denote the probability of state 1 so that (1 ) is the probability of state 2. Given this, the expected value of income is E(y) = y1 + (1- )y2 the weighted average level of income received by the individual over the two states of the world. Utility of the expected value is u[E(y)] = u((y1) + (1- )y2). The utility received by the individual from the weighted average level of income y. Expected utility is given by E(u)= u(y1) + (1- )u(y2), which is the utility received by the individual from fluctuating income levels across the two states of the world Risk Aversion, an individual is risk averse if u[E(y)] gt; E(u). The individual prefers to have a constant amount of income rather than fluctuating amounts of a low income in state 1 of the world and a high income in state 2 of the world averages are preferred to extremes. An alternative definition uses the certainty equivalent level of income denoted by yc. This level of income must satisfy the following condition u(yc) = E(u). yc must be such that the utility received from getting yc equals the utility from facing the gamble (expected utility). In other words, yc is the level of income at which the individual is indifferent between getting that level of income for certain and facing fluctuating incomes levels. This certainty equivalent level of income therefore gives us a notion of the value of the gamble to the individual. If the individual is risk averse E(y) gt; yc, that is, the individual values the gamble at less than the expected value. Risk Neutral An individual is risk neutral if the individual values the prospect at its expected value E(y) = yc or u(y1+ + (1 )y2) = u(y1 + (1 )u(y2). Risk Loving An individual is loving if he/she values the prospect at more than its expected value yc gt; E(y) or u(y1) + (1-)u(y2) gt; u((y1+(1-)y2). The Risk Premium, consider figure 1 and recall that yc satisfies u(yc) = E(u). In figure 1, the individual is willing to give up income cd rather than face the gamble with expected income E(y). r = y yc gt; 0. If the individual is risk loving, the individual would pay to be able to face the gamble. r = E(y) yc gt; 0 Using the idea of insurance under uncertainty, it is possible to consider a risk averse individual and suppose that there are two states of the world, firstly, state 1, where initial income is y, and state 2, where income is y L. The consumer can insure against the loss (cannot affect the loss or the probability of the loss implies a moral hazard issue). The insurance company sells insurance at a premium rate p (0 lt; p lt; 1). Let q denote the amount of insurance cover purchased by the consumer y1 = y pq and y2 = y L pq + q. Let 1 denote the probability of state 1 and let 2 denote the probability of state 2 E(y) = 1(y pq) + 2(y L pq + q) Supposing that the individual maximizes expected utility max q 1u(y1) + 2u(y2) = max q 1u(y pq) + 2u(y L pq + q) Rearranging the first order condition gives the following familiar condition u'(y2) 2 = p u'(y1) 1 1-p An insurance company has an actuarially fair premium (p) if it does not alter the insured individuals expected income 1(y pq) + 2(y L pq + q) =gt; 1y + 2(y L) Hence, the fair premium is p= 2. In fact, perfect competition implies a fair premium. Competition in the industry forces the expected profits of the firm to zero 1pq + 2(pq q) = 0 =gt; q(p 2) = 0 Suppose p = 2 (premium is fair). Then, from the first order condition, can write u'(y1) = u'(y2) Since u(y) lt; 0, this implies y1 = y2 y pq* = y L + (1 p)q* =gt; L = q* (the insurer completely insures against the loss) Suppose p gt; 2 (premium is unfair). This implies 1 p lt; 1 2 = 1. Then, from the first order conditions, u'(y1) lt; u'(y2) which implies y1 gt; y2 y pq* gt; y L + (1 p)q* =gt; q*lt; L if the individual faces an unfair premium then the individual will buy less than complete cover. Using comparative statics, the first order condition gives us our demand for insurance as a function of the parameters in the model q* = D(y, L, p, 2). It is now possible to analyse how this demand changes as we change one of these parameters (say income) holding the others constant. This involves taking derivates of the first order condition with respect to the relevant variable. This results in an increase in income y increases the demand for insurance if and only if the individual has increasing absolute risk aversion (less willing to take risks and more willing to buy insurance). An increase in the price of insurance p is ambiguous and depends on the usual substitution and income effects. An increase in the loss L (due to an accident) increases the demand for insurance. An increase in the accident probability 2 increases the demand for insurance. As a theory of individual behavior, the expected utility model shares many of the underlying assumptions of standard consumer theory. Yet, the expected utility theory comes under criticism by Rabin and Thaler (2001, pp. 219-232). They argue that expected utility theory is inadequate to explain risk aversion and hence should be discarded as a theory of choice under risk and uncertainty. Watt (2002) addresses this argument stating that all the exercises in the Rabin and Thaler paper demonstrate only that an unrealistically high degree of risk aversion produces preposterous results. For a person with a high level of wealth to turn down a bet for moderate stakes that has a positive expected value will require either an unreasonably high level of risk aversion, or some other unusual peculiarity in the utility function (like a utility function that is bounded from above). Otherwise, under standard models of risk aversion, their large-scale bets will not be rejected-and neither will their m oderate-scale bets. Expected utility theory certainly faces problems in explaining certain kinds of empirical evidence, as do other competing theories. But in this case, it reveals a useful truth: namely, that risk-averse, wealth-loving people should be willing to accept certain moderate bets with positive expected value, even though at first glance, the bets may not appear attractive to them. There are alternative theories of choice under uncertainty, one is the prospect theory, which was developed by [Kahneman and Tversky (1979)]. They formulate that uncertain outcomes are defined relative to a reference point, which is typically current wealth (p.274). Outcomes are interpreted as gains and losses. Risky outcomes are referred to as prospects and the decision maker is assumed to choose among alternative prospects by choosing the one with the highest value. The value of a prospect is expressed in terms of two scales, first, a decision weight function , which associates with each probability p giving (p) reflecting the impact of p. (p) is not a probability p in evaluating a prospect. The value function assigns to each outcome x a number v(x), which encodes the decision makers subjective value of outcome. Kahneman and Tverskys (1979) formulation focuses on simple prospects which have at most two non-zero outcomes. The theory can be extended to more complicated prospects, but this poses certain difficulties as it can violate dominance, and hence transitivity, among prospects with more than two outcomes. Potential violations may occur due to the fact that the decision weights in prospect theory are derived by applying the decision weighting function to individual probabilities rather than to the entire probability density of outcome [Quiggan, (1982) p.326] An example in the market for insurance: Let us assume the probability of the insured risk is 1%, the potential loss is $1,000 and the premium is $15. If we apply PT, we first need to set a reference point. This could be (e.g., the current wealth) or the worst case (losing $1,000). If we set the frame to the current wealth, the decision would be to either pay $15 for sure (which gives the prospect theory-utility of ) or a lottery with outcomes $0 (probability 99%) or $1,000 (probability 1%) which yields the PT-utility of . These expressions can be computed numerically. For typical value and weighting functions, the former expression could be larger due to the convexity of in losses, and hence the insurance looks unattractive. If we set the frame to $1,000, both alternatives are set in gains. The concavity of the value function in gains can then lead to a preference for buying the insurance. In this example a strong overweighting of small probabilities can also undo the effect of the c onvexity of in losses: the potential outcome of losing $1,000 is over-weighted. The interplay of overweighting of small probabilities and concavity-convexity of the value function leads to the so-called fourfold pattern of risk attitudes: risk-averse behavior in gains involving moderate probabilities and of small probability losses; risk-seeking behavior in losses involving moderate probabilities and of small probability gains. Another theory in the area of choice under uncertainty is Bell (1982, 1983, Fishburn (1981,1982) and Loomes and Sugdens (1982,1983) regret theory generalizes Savages (1951) minimax regret approach. Choice is modeled as the minimising of a function of the regret vector, defined as the difference between the outcome yielded by a given choice and the best outcome that could have been achieved in that state of nature. A decision makers preference function is defined over pairs of prospects. It is possible that prospect A is preferred to B, B preferred to C, C preferred to A. EXPAND Quiggans (1982) anticipated utility theory maintains properties of dominance and transitivity but employs a weakened version of the independence axiom. The model is consistent with a considerable ranger of choice behavior that violates von Neumann-Morgenstern expected utility theory. It also is free of the violations of dominance that can occur under prospect theory. The Allais paradox [Allais (1953)] is an example of choice behavior that can be explained under anticipate utility theory. Risk attitude under anticipated utility theory, discussed in Hilton (1988), follow Pratt (1964) and Arrows (1971) analysis of risk attitude under von Neumann-Morgenstern expected utility theory and characterize a decision makers attitude toward the risk inherent in a prospect by the decision makers risk premium for the prospect. Hilton (1988) also tests risk attitude under prospect theory, but slightly modifies the perspective on prospect theory from the theorys original statement by Kahneman and Tversky (1979), required by two features of prospect theory. First there is the problem of dominance violations, which Kahneman and Tversky (1979) state that stochastically dominated alternatives are eliminated in the editing phase of the theory, acknowledging that such a procedure raises the problem of intransitivity. Hilton (1988) concludes that after examining the concept of risk attitude under Quiggans (1982) anticipated utility theory and under a modified version of Kahneman and Tverskys (1979) prospect theory. Risk premia for prospects were characterized under booth theories in the large ad in the small. There overall risk premia were then partitioned into a Arrow-Pratt risk premium, transforming the probability distribution, and a decision weight premium, reflecting the decision makers optimism or pessimism as reflected in the implicit distortion of the probabilities. Machina (1982) has abo developed a theory of choice under risk that allows for violations of the independence axiom. Machina proves that the basic results of expected utility theory do not depend on the independence axiom, but may be derived from the much weaker assumption of smoothness of preferences over alternative probabiity distributions. Unlike anticipated utility theory, Machinas (1982) theory does not employ a uti!ity function that maps outcomes into the real line. In Machinas theory, there is no separation between outcomes and probabilities in the evaluation function.