Standard Deviation Premium Principle¶

Definition¶

For a loss with finite second moment, the standard deviation principle sets $\Pi(X)=\mathbb{E}[X]+\alpha\sqrt{\operatorname{Var}(X)}$ $α Var (X)$ , with dimensionless fixed loading $\alpha>0$ $0$ .

Interpretation¶

The loading is measured in the same monetary units as loss. Independent pooling generally gives a smaller loading than adding the separate standard deviation loadings.

$Right triangle with legs sigma_X = 2 and sigma_Y = 2 and hypotenuse sigma_{X+Y} = sqrt 8, beside the separate premium total 24 and the pooled premium 22.83 at alpha = 1.$

For independent risks the variances add, so the standard deviations combine like the two legs of a right triangle: in the Examples case the pooled $\sigma_{X+Y}=\sqrt{2^2+2^2}=\sqrt{8}\approx2.83$ is the hypotenuse, shorter than the path $\sigma_X+\sigma_Y=4$ along the legs. At $\alpha=1$ the total loading therefore falls from 4 to about 2.83, and the premium from 24 to about 22.83; this diversification effect is why the principle is not additive.

Change $\sigma_X$ , $\sigma_Y$ and $\alpha$ : for independent risks the pooled loading $\alpha\sqrt{\sigma_X^2+\sigma_Y^2}$ is the hypotenuse of the triangle, never longer than the separate loadings $\alpha(\sigma_X+\sigma_Y)$ $σ_{Y})$ .

Key Results¶

It has nonnegative loading, translation consistency and proportionality for positive scaling.
For independent losses, $\sigma_{X+Y}=\sqrt{\sigma_X^2+\sigma_Y^2}\le\sigma_X+\sigma_Y$ $σ_{Y}$ .
Therefore it is generally not additive, and it rewards diversification.
Equal-mean convex order raises the premium when second moments are finite; stochastic order alone need not do so.

Examples¶

Two independent risks each have mean 10 and standard deviation 2. At $\alpha=1$ $1$ , separate premiums sum to 24, while the pooled premium is $20+\sqrt{8}\approx22.83$ $22.83$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 12

Aliases: standard deviation loading, SD premium principle, standard deviation principle Keywords: Π(X) = E[X] + α √Var(X), diversification, not additive, σ_{X+Y} ≤ σ_X + σ_Y Symbols: alpha, pi, sigma, square root

Last updated: 14 October 2026 08:21