Insurance Indifference Premiums¶

Definition¶

For initial policyholder wealth $w$ , strictly increasing utility $u$ and full insurance of loss $X$ , the maximum acceptable premium $P^+$ solves $u(w-P^+)=\mathbb{E}[u(w-X)]$ $X)]$ . For insurer wealth $W$ and strictly increasing utility $U$ , its minimum acceptable premium $P^-$ solves $U(W)=\mathbb{E}[U(W+P^--X)]$ $X)]$ , when finite solutions exist.

Interpretation¶

One equation compares insured and uninsured wealth for the customer; the other compares taking the contract with taking no risk for the insurer. A trade is acceptable to both when the premium lies between their thresholds.

$Concave exponential utility u(x)=−e^{−0.1x} with the chord between wealth w−10 and w; the chord midpoint E[u(w−X)] meets the curve at w−P^+, and a green segment marks premiums 5 to 6.20.$

The figure draws the Examples case with $\alpha=0.1$ $0.1$ . The chord midpoint is the expected utility of the uninsured position, and the wealth on the curve with the same utility is $w-P^+$ $P^{+}$ , with $P^+\approx6.20$ $6.20$ . Because $u$ is concave, this point lies to the left of $w-\mathbb{E}[X]=w-5$ $5$ . For a risk-neutral insurer $P^-=\mathbb{E}[X]=5$ $5$ , so every premium in the green range from 5 to 6.20 is acceptable to both parties.

Absolute risk aversion measures the local curvature of the utility used in each indifference equation. For exponential utility it is the constant $\alpha$ , so the example's coefficient specifies how strongly large losses affect the acceptable premium.

Key Results¶

Under concavity and well-defined expectations, $P^+\ge\mathbb{E}[X]$ and $P^-\ge\mathbb{E}[X]$ $E [X]$ .
A mutually acceptable full-insurance premium exists if $P^-\le P^+$ $P^{+}$ .
A risk-neutral insurer has $P^-=\mathbb{E}[X]$ in this expense-free model.
Exponential utilities with different coefficients produce different thresholds; finiteness requires the respective exponential loss moments.

Examples¶

For exponential utility parameter $\alpha$ and loss 0 or 10 with equal probabilities, the indifference premium is $\alpha^{-1}\log[(1+\mathrm{e}^{10\alpha})/2]$ $e^{10 α}) /2]$ , where $\log$ is the natural logarithm. At $\alpha=0.1$ it is about 6.20, above mean loss 5.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 8

Aliases: maximum acceptable premium, minimum insurer premium, certainty equivalent loss, indifference premium, utility indifference premium, minimum acceptable premium Keywords: u(w - P+) = E[u(w - X)], U(W) = E[U(W + P- - X)], range of acceptable premiums, exponential utility Symbols: alpha

Last updated: 17 October 2026 21:53