Why Wealth-Independent Risk Preferences Imply CARA Utility
Summary
This note connects constant absolute risk aversion (CARA) to preferences over risky gambles that do not depend on initial wealth. It considers a small, fair gamble with equal chances of a gain or loss and defines the risk premium as the amount an investor would pay to avoid it. A Taylor expansion of utility around current wealth approximates that premium in terms of the gamble’s size and the ratio of the second to first derivatives of utility.
For small gambles, the premium is independent of wealth when absolute risk aversion, defined by the negative ratio of those derivatives, is constant. Solving the resulting differential equation yields the exponential CARA utility form, up to transformations that preserve preferences. The argument is an approximation for small gambles; the note does not establish the result for arbitrary large gambles, despite the broad framing of its prompt.
Key ideas
- For small fair gambles, the risk premium depends on absolute risk aversion at current wealth.
- Absolute risk aversion is the negative ratio of the second utility derivative to the first.
- A constant value of absolute risk aversion makes the approximate premium independent of initial wealth.
- The utility function associated with constant absolute risk aversion has an exponential form.
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Full text
# Independence of initial wealth for Constant Absolute Risk Aversion
# Independence of initial wealth for Constant Absolute Risk Aversion
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).
## Answer by RRL (score 2)
https://quant.stackexchange.com/a/39618
Suppose a fair gamble pays $G = \pm \epsilon$ where $\displaystyle P(G=\epsilon ) = P(G = -\epsilon) = \frac{1}{2}$.
From the classic work described in
> Pratt, J.W. (1964) "Risk-Aversion in the Small and in the Large'" Econometrica 55,143-54
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.
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
$$U(W_0 - \delta) = E[U(W_0+G)]=\frac{1}{2}U(W_0 +\epsilon) + \frac{1}{2}U(W_0- \epsilon).$$
Using the Taylor expansion for U around $W_0$ we have
$$U(W_0) -U'(W_0)\delta + \frac{1}{2} U''(W_0)\delta^2 + \ldots \\ = \frac{1}{2} [U(W_0) +U'(W_0)\epsilon + \frac{1}{2} U''(W_0)\epsilon^2 + \ldots ] \\ +\frac{1}{2} [U(W_0) -U'(W_0)\epsilon + \frac{1}{2} U''(W_0)\epsilon^2 + \ldots ] \\ = U(W_0) + \frac{1}{2}U''(W_0) \epsilon^2 + \ldots $$
Solving for $\delta$ for small $\epsilon$ we get
$$\delta \approx \frac{\epsilon^2}{2}\left[- \frac{U''(W_0)}{U'(W_0)} \right].$$
The premium to avoid a gamble is independent of initial wealth if and only if there is a constant $\gamma$ such that
$$- \frac{U''(W)}{U'(W)} = \gamma .$$
Solving this differential equation we get
$$U(W) = C_1 - e^{C_2} \frac{e^{-\gamma W}}{\gamma}.$$
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
$$U(W) = - \frac{e^{-\gamma W}}{\gamma}.$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.