Expected Value Premium Principle¶

Definition¶

For integrable loss $X\ge0$ $0$ , the expected value principle sets $\Pi(X)=(1+\theta)\mathbb{E}[X]$ $θ) E [X]$ , where $\theta\ge0$ is a fixed loading coefficient: the loading $\Pi(X)-\mathbb{E}[X]=\theta\mathbb{E}[X]$ is a fixed proportion of the mean, so the principle has nonnegative loading. At $\theta=0$ $0$ , it is the net premium principle, $\Pi(X)=\mathbb{E}[X]$ $E [X]$ .

Interpretation¶

The loading is a constant proportion of expected loss. Different risks with the same mean receive the same price, regardless of their dispersion.

Premium plotted against expected loss: the red line (1+θ)E[X] lies above the dashed net-premium line E[X], with the loading 20 marked at E[X]=100 and the premium 132 at E[X]=110.

The figure draws the Examples case $\theta=0.2$ $0.2$ . The premium line $(1+\theta)\mathbb{E}[X]$ is a ray through the origin with slope $1.2$ , so the loading, the vertical gap to the net-premium line $\mathbb{E}[X]$ , grows in proportion to the mean: at $\mathbb{E}[X]=100$ the premium is 120 and the gap is 20. Only the mean enters, so any two risks with that mean sit at the same point. Adding a certain 10 moves the mean to 110 and the premium to 132, so the certain amount is loaded as well, which is why translation consistency fails.

Key Results¶

The principle is proportional and additive for integrable losses, even without independence.
It has nonnegative loading and is monotone under stochastic or stop-loss order.
For $\theta>0$ $0$ , translation consistency fails: adding certain cost $c$ adds $(1+\theta)c$ to the premium.
Iterativity follows from total expectation only in the net case; repeating a positive loading generally applies it twice.

Examples¶

For expected loss 100 and $\theta=0.2$ $0.2$ , premium is 120. Adding a certain 10 raises the premium to 132, rather than 130.

Prerequisites¶

Premium Principles and Their Properties

Sources¶

Risk Insurance · Lecture 11

Aliases: mean loading principle, expected value principle, expected-value premium principle Keywords: net risk premium, net premium principle, Π(X) = (1 + θ) E[X], loading coefficient θ, proportional loading Symbols: pi, theta

Last updated: 14 October 2026 08:21