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.

Move the policyholder's $\alpha$ , the insurer's risk aversion and the loss: any premium between $P^-$ and $P^+$ suits both parties. Raise the insurer's risk aversion above the policyholder's and the range closes.

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