Risk aversion Assignment Example | Topics and Well Written Essays - 2000 words

Risk aversion - Assignment Example Generally, the extent of risk aversion is the degree to which the individual prefers the certain income over the uncertain income. In terms of a utility function, this translates to the distance between the utility generated by the certain income and the utility generated by the gamble which has an expected income equal to the certain income. Obviously, for a concave utility function, the utility of the certain income will lie above the utility of the uncertain income with the same expected value. For a convex utility function this will be reversed. These are explained in the diagram below (figure 1). Figure 1: Risk Aversion and the curvature of the utility function In the diagram above, a rational individual is considered whose preferences are represented by the utility function U(.) defined over money incomes X. Suppose the individual has a choice of either playing a lottery with two possible outcomes: X1 and X2, where X2 > X1. To keep things simple let us further assume that both outcomes equally likely to occur. That is, both outcomes X1 and X2 have a probability of occurrence = ?. Thus if X1 is realized the individual gets U(X1) and if X2 realizes, the individual derives U(X2). Then, the expected income from the lottery is ?[X1+X2] and the expected utility is ? [U(X1) +U(X2)]. Now, observe that whether the utility derived by the individual from a certain income of ?[X1+X2] which is equal to U?[X1+X2] lies above ? [U(X1) +U(X2)], the expected utility from the lottery with an expected earning of ?[X1+X2], depends upon the curvature of the function. When the utility function is concave, . This shows that the individual prefers a certain income over and above a lottery with an expected income that is equal to certain income. Extending this logic it is simple to show that a risk loving individual will have a convex utility function while a risk neutral person will have a utility function that has a constant slope. Also, greater the distance between U?[X1+X2] an d ? [U(X1) +U(X2)], the more risk averse is the individual, since the preference for the certain income is even greater in that case. This implies that the more concave the utility function the greater will be the risk aversion of the individual. Similarly, greater the convexity of the utility function, greater will be the individual’s love for risk. Therefore, it can be generally agreed upon that a risk-averse person will have a concave utility function while a risk lover will have a convex utility function. A risk neutral person’s preferences will be designated by a utility function with a constant slope. Now, Mr. D’s Utility function is: Then, and, Since , and thus, Mr. D’s utility function is positively sloped. A positively sloped utility function implies more income is preferred to less by Mr. D. For his attitude towards risk, the curvature (sign of the second order derivative) of the utility function has to be considered. Now, and, Therefore, the ut ility function is convex if the value of the positive parameter and it is concave if the positive parameter . If the utility function is concave, Mr. D is risk averse while if the utility function is convex, then Mr. D is in nature a risk loving person. Therefore, regarding the attitude of Mr. D towards risk, we conclude the following: Mr. D’s attitude towards risk depends on the value of the parameter . If , Mr. D loves