Loimaranta Efficiency¶

Definition¶

For a positive differentiable stationary premium $b(\lambda)$ at claim rate $\lambda>0$ $0$ , Loimaranta efficiency is $e(\lambda)=\lambda b'(\lambda)/b(\lambda)=\mathrm{d}\log b(\lambda)/\mathrm{d}\log\lambda$ $d lo g b (λ) / d lo g λ$ .

Interpretation¶

The elasticity measures the percentage change in long-run premium produced by a one-percent change in claim rate. A value near one means a proportional response locally, but does not establish accurate individual pricing.

Loimaranta efficiency e(λ)=λ/(2e^λ−1) of the two-level one-year scheme plotted against λ, staying below 0.24, far under the dashed level 1 of a proportional premium; e(0.5)≈0.2176 is marked.

The figure plots the efficiency of the two-level one-year scheme from the Examples, $b(\lambda)=c(2-\mathrm{e}^{-\lambda})$ $e^{- λ})$ . At $\lambda=0.5$ it is about 0.2176: a one-percent rise in claim rate raises the long-run premium by only about 0.22 percent. The curve peaks at about 0.23 near $\lambda\approx0.77$ and then falls, so at every claim rate this premium responds far less than the proportional premium $b(\lambda)=k\lambda$ with efficiency 1.

Move $\lambda$ and switch schemes: $e(\lambda)=\lambda b'(\lambda)/b(\lambda)$ is the slope of $\log b$ against $\log\lambda$ , and it stays far below the efficiency 1 of a proportional premium. The default two-level scheme gives $e(0.5)\approx0.2176$ $0.2176$ .

Derivative and differentiability gives the local premium response $b'(\lambda)$ . Scaling it by $\lambda/b(\lambda)$ converts an absolute slope into the relative response of premium to claim rate.

Key Results¶

A proportional premium $b(\lambda)=k\lambda$ has efficiency one.
A flat premium has efficiency zero.
For the two-year discount scheme, $e(\lambda)=2\lambda(c-a)\mathrm{e}^{-2\lambda}/[c(1-\mathrm{e}^{-2\lambda})+a\mathrm{e}^{-2\lambda}]$ $a e^{- 2 λ}]$ .
Numerical evaluation can use nearby log-rate and log-premium differences; check sensitivity to the step size.

Examples¶

For a two-level one-year scheme, $b(\lambda)=c(2-\mathrm{e}^{-\lambda})$ $e^{- λ})$ . At $\lambda=0.5$ $0.5$ , efficiency is $0.5/(2\mathrm{e}^{0.5}-1)\approx0.2176$ $0.2176$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 14

Aliases: bonus-malus premium elasticity, BMS efficiency, Loimaranta elasticity, efficiency of a bonus-malus system Keywords: e(λ) = λ b'(λ)/b(λ), d log b / d log λ, elasticity of the stationary premium, proportional premium has efficiency one Symbols: lambda

Last updated: 17 October 2026 21:53