Variance Premium Principle¶

Definition¶

For a loss with finite second moment, the variance principle is $\Pi(X)=\mathbb{E}[X]+\alpha\operatorname{Var}(X)$ $α Var (X)$ , with fixed coefficient $\alpha>0$ $0$ . The coefficient has units inverse to those of the monetary loss.

Interpretation¶

The loading increases with squared dispersion. Changing the monetary unit requires a corresponding change in the numerical loading coefficient if prices are to describe the same contract.

Loading of the scaled loss aX under the variance principle with alpha = 0.1 and a Bernoulli(0.5) loss: the parabola 0.025a^2 against the proportional line 0.025a, marked at a = 1 and 2.

The figure plots the loading $\Pi(aX)-a\mathbb{E}[X]=\alpha a^2\operatorname{Var}(X)=0.025a^2$ for the Examples loss. At $a=1$ it meets the proportional line $a\cdot0.025$ $0.025$ , giving $\Pi(X)=0.525$ $0.525$ ; at $a=2$ the parabola reaches 0.1, twice the proportional 0.05, so $\Pi(2X)=1.1$ rather than $2\Pi(X)=1.05$ $1.05$ . This quadratic growth is why the principle is not proportional for fixed $\alpha$ .

Move the scale $a$ : the loading $\alpha a^2\operatorname{Var}(X)$ grows like $a^2$ , so $\Pi(aX)\ge a\Pi(X)$ for $a\ge1$ $1$ , with strict inequality when $a>1$ and $\operatorname{Var}(X)>0$ $0$ . Change $p$ to vary $\operatorname{Var}(X)=p(1-p)$ of the Bernoulli loss.

Variance and standard deviation supplies the squared-dispersion measure used in the loading. For independent losses these variances add, explaining additivity; multiplying all losses by $a$ multiplies this loading by $a^2$ .

Key Results¶

The principle has nonnegative loading and translation consistency.
It is additive for independent losses because their variances add.
It is not proportional for fixed $\alpha$ : $\Pi(aX)=a\mathbb{E}[X]+\alpha a^2\operatorname{Var}(X)$ $α a^{2} Var (X)$ .
It preserves equal-mean convex order, but need not preserve stochastic order when means differ.
Covariance matters when pricing a dependent portfolio.

Examples¶

For $\alpha=0.1$ $0.1$ , a Bernoulli loss with probability $0.5$ has premium $0.5+0.1(0.25)=0.525$ $0.525$ . Doubling the loss gives premium $1+0.1(1)=1.1$ $1.1$ , not $1.05$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 12

Aliases: variance loading principle, variance principle Keywords: Π(X) = E[X] + α Var(X), not proportional, additive for independent risks, variance loading Symbols: alpha, pi

Last updated: 27 October 2026 11:52