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.

Move the loading $\theta$ , the expected loss and the certain cost $c$ : $\Pi(X)=(1+\theta)\mathbb E[X]$ $θ) E [X]$ , and because the certain cost is loaded too, $\Pi(X+c)$ exceeds $\Pi(X)+c$ by $\theta c$ .

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