Independence of initial wealth for Constant Absolute Risk Aversion - Quantitative Finance Stack Exchange most recent 30 from quant.stackexchange.com 2019-08-20T00:51:49Z https://quant.stackexchange.com/feeds/question/39584 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://quant.stackexchange.com/q/39584 2 Independence of initial wealth for Constant Absolute Risk Aversion Fasiledes Fetene https://quant.stackexchange.com/users/34080 2018-05-03T08:35:19Z 2018-05-04T22:17:43Z <p>Suppose a consumer's preference over wealth gambles (lotteries) can be represented by a twice differentiable Von Neumann Morgenstern utility function. Show that the consumer's preference over gambles are independent of his initial wealth if and only if his utility function displays Constant Absolute Risk Aversion (CARA).</p> https://quant.stackexchange.com/questions/39584/-/39618#39618 2 Answer by RRL for Independence of initial wealth for Constant Absolute Risk Aversion RRL https://quant.stackexchange.com/users/8153 2018-05-04T22:05:08Z 2018-05-04T22:17:43Z <p>Suppose a fair gamble pays $G = \pm \epsilon$ where $\displaystyle P(G=\epsilon ) = P(G = -\epsilon) = \frac{1}{2}$.</p> <p>From the classic work described in </p> <blockquote> <p>Pratt, J.W. (1964) "Risk-Aversion in the Small and in the Large'" <em>Econometrica</em> 55,143-54</p> </blockquote> <p>for small gambles, the absolute amount an agent is willing to pay to avoid a gamble of a given size is determined by the coefficient of absolute risk aversion.</p> <p>For a rough argument, we have a risk averse agent with initial wealth $W_0$ and utility function $U$ willing to pay $\delta$ to avoid the gamble such that $\delta$ is determined by</p> <p>$$U(W_0 - \delta) = E[U(W_0+G)]=\frac{1}{2}U(W_0 +\epsilon) + \frac{1}{2}U(W_0- \epsilon).$$</p> <p>Using the Taylor expansion for U around $W_0$ we have</p> <p>$$U(W_0) -U'(W_0)\delta + \frac{1}{2} U''(W_0)\delta^2 + \ldots \\ = \frac{1}{2} [U(W_0) +U'(W_0)\epsilon + \frac{1}{2} U''(W_0)\epsilon^2 + \ldots ] \\ +\frac{1}{2} [U(W_0) -U'(W_0)\epsilon + \frac{1}{2} U''(W_0)\epsilon^2 + \ldots ] \\ = U(W_0) + \frac{1}{2}U''(W_0) \epsilon^2 + \ldots$$</p> <p>Solving for $\delta$ for small $\epsilon$ we get</p> <p>$$\delta \approx \frac{\epsilon^2}{2}\left[- \frac{U''(W_0)}{U'(W_0)} \right].$$</p> <p>The premium to avoid a gamble is independent of initial wealth if and only if there is a constant $\gamma$ such that</p> <p>$$- \frac{U''(W)}{U'(W)} = \gamma .$$</p> <p>Solving this differential equation we get</p> <p>$$U(W) = C_1 - e^{C_2} \frac{e^{-\gamma W}}{\gamma}.$$</p> <p>Without loss of generality (from the invariance properties of utility functions) we can set $C_1 = C_2 = 0$ to obtain the CARA utility function</p> <p>$$U(W) = - \frac{e^{-\gamma W}}{\gamma}.$$</p>