# Utility theory

6 June 2017

This means the second derivative is O i. e. no penalty for risk A risk seeking individual will take the gamble I. . The increase in utility with a rise in wealth exceeds the utility experienced from avoiding a decrease in wealth and thus the curve is convex in wealth and the second derivative is positive i. e. a premium is paid for risk -To summarize this: -We assume that: – Are risk-averse – Maximize their expected utility of wealth – Prefer more wealth to less – Have diminishing marginal utility of wealth Absolute and Relative Risk Aversion -Absolute risk aversion refers to the change in amount invested in risky assets as earth changes.

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If the investor increases the amount in risky assets as wealth increases then they exhibit decreasing absolute risk aversion and etc. -It is found by: . The derivative indicates how absolute risk aversion changes as wealth U'(W) changes and generally we assume decreasing -Relative risk aversion refers to the change in the percentage investment in risky assets as wealth changes and is found by: =W A(W). The derivative indicates how relative risk aversion changes as wealth U'(W) changes and there is no consensus on this point

Utility functions used to describe investor behavior -The most commonly used functional forms for risk aversion are: Logarithmic: -The quadratic function leads to mean variance analysis being optimum. It is consistent with investors who reduce the dollar amount invested in risky assets as wealth increases and increasing relative risk aversion i. e. increasing ARA and RRA -The log function can be used for risk-averse investors who prefer more to less and whose percentage invested in risky assets remains constant i. e. creasing ARA and RRA is constant -The purpose of these is to quantify the level of utility associated with different wealth. The advantage of the quadratic form is that the c can measure the investor’s risk aversion Utility and the Investment Horizon -The certainty equivalent is the certain value that makes you indifferent between taking and not taking a gamble -For log utility investors for all classes of utility functions that display constant RRA for multiplicative investments the horizon will not affect choice. An exception to this is when returns are correlated e. g. go down after a rise.

In this case risky investments are more attractive in the long-run -Scholars have pointed out that time-indifference with uncorrelated returns depends on extreme aversion to large losses Prospect Theory (Tversky and Kahneman theory) -Prospect theory models choice in terms of gains or losses where loss-averse individuals make their decisions with respect to a particular reference points with individuals found to weight losses more heavily and do not weight probability in a in ear faction -Prospect theory uses the product of a subjective value function and non-linear probability weighting in modeling individual choice and the prospects associated with different choices

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