Zero-Utility Premium¶

Definition¶

For strictly increasing concave utility $u$ , the zero-utility premium $\Pi(X)$ solves $\mathbb{E}[u(\Pi(X)-X)]=u(0)$ $u (0)$ , when a finite solution exists. It is the insurer's indifference price at reference wealth zero. Utility must be defined on every relevant net outcome.

Interpretation¶

Premium income offsets the insured loss until expected utility equals the no-contract reference. The name refers to zero change in utility, not necessarily a utility function whose value is zero.

$Concave utility u(w) = -e^{-w} with the insurer's net outcomes Pi - 2 and Pi for a loss of 0 or 2; the chord midpoint has height u(0) and lies to the right of w = 0.$

The figure takes $u(w)=-\mathrm{e}^{-w}$ (so $\alpha=1$ $1$ ) and a loss $X$ equal to 0 or 2 with probability $1/2$ each, numbers chosen for illustration. The zero-utility premium is then $\Pi=\log\frac{1+\mathrm{e}^2}{2}\approx1.43$ $1.43$ , with net outcomes $\Pi$ and $\Pi-2$ $2$ . Their expected utility, the midpoint of the dashed chord, equals $u(0)=-1$ and sits above $w=\Pi-\mathbb{E}[X]\approx0.43$ $0.43$ . Because the concave curve lies above its chords, it reaches the level $u(0)$ already at $w=0$ $0$ , so $\Pi-\mathbb{E}[X]\ge0$ $0$ : the premium carries a nonnegative loading.

Change $\alpha$ , the loss $b$ and its probability $q$ : $\Pi(X)$ is the premium that leaves expected utility equal to $u(0)$ , and it never falls below $\mathbb E[X]$ . Switch to linear utility and the loading vanishes.

Insurance indifference premiums gives the insurer's equation at general reference wealth; setting that wealth to zero gives this equation. Among premium principles and their properties, nonnegative loading follows from concavity: Jensen's inequality gives $u(\Pi(X)-\mathbb{E}[X])\ge u(0)$ $u (0)$ , hence $\Pi(X)\ge\mathbb{E}[X]$ under the stated conditions.

Key Results¶

If $u$ is strictly increasing, any finite solution is unique.
Concavity implies $\Pi(X)\ge\mathbb{E}[X]$ under the needed integrability conditions.
Exponential utility gives the exponential premium principle.
For bounded losses, a solution is between the smallest and largest possible loss under suitable continuity and domain conditions.
For general utilities, reference wealth affects the indifference price.

Examples¶

With risk-neutral $u(w)=w$ $w$ , the equation is $\Pi(X)-\mathbb{E}[X]=0$ $0$ , giving the net premium. With $u(w)=-\mathrm{e}^{-\alpha w}$ it gives the exponential premium.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 12

Aliases: utility-equivalence insurer premium, principle of zero utility, zero-utility principle Keywords: E[u(Π - X)] = u(0), insurer indifference price, exponential utility gives the exponential premium, nonnegative loading by Jensen Symbols: alpha, pi

Last updated: 17 October 2026 21:53