"EU Hypothesis, Risk and Insurance

"EU Hypothesis, Risk and Insurance - The Expected...

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The Expected Utility Hypothesis, Risk Aversion, and Insurance: Brief Introduction Consider a bet of x dollars on the flip of an unbiased coin. If W 0 is current wealth then future wealth W 1 becomes random under this bet, equal to W 0 + x with probability 0.5 and equal to W 0 - x with probability 0.5. This is an actuarially fair bet , in that: E W 1 = Expected Future Wealth = 0.5(W 0 + x) + 0.5(W 0 - x) = W 0 Refusal of an actuarially fair bet is evidence of risk aversion. Expected future wealth equals current wealth, regardless of whether the bet is accepted or not. The only difference between acceptance versus refusal of the bet lies in randomness of future wealth, in risk. John Von Neumann and Oskar Morgenstern ( Theory of Games and Economic Behavior , 1944) argued that refusal of the bet may be explained by differences in expected utility - assuming diminishing marginal utility of wealth. I.e., we have EU(W 1 ) = 0.5U(W 0 + x) + 0.5U(W 0 - x) < U(W 0 ), and therefore the bet will be refused. The key to understanding this approach is realizing that it is utility of outcomes, NOT the dollar amounts per se, that matter in making choices under uncertainty. This point is echoed in Friedman’s Law’s Order, Chapter 6. If we weight the gain in utility from winning and loss in utility form losing equally, with diminishing marginal utility it is not surprising that such a bet will be refused: “It hurts more to lose than it feels good to win!” APPLICATION TO INSURANCE Graphically, Wealth W 0 + x W 0 - x W 0 E W 1 Utility of Wealth U(W) U(W 0 + x) U(W 0 ) U(W C )=EU( W 1 ) U(W 0 - x) W C π Here W C is the “Certainty Equivalent” level of wealth, the amount of certain wealth that gives the same utility as random wealth W 1 . The Arrow-Pratt Risk Premium,
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&amp;quot;EU Hypothesis, Risk and Insurance - The Expected...

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