Bahnemann.Chapter6
Reading: Bahnemann, D., "Distributions for Actuaries", CAS Monograph #2, Chapter 6.
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In Plain English!
Premium Concepts
Most of Section 6.1 of the text should be very familiar to anyone who has worked in insurance for a while. The key points are given here but you should skim-read the source just to be safe.
The premium charged for a policy is the expected loss (including expected Allocated Loss Adjustment Expense) plus a load for general expenses, underwriting profit, and a provision for risk.
Let N be the per policy claim count random variable, m be the number of exposures, and φ be the ground-up claim frequency per exposure. Let Y be the claim size including ALAE. Then [math]E[N]=m\phi[/math] is the expected claim count and the expected loss and ALAE for a policy is given by [math]E[N]\cdot E[Y][/math]. The per policy pure premium is [math]p=\phi\cdot E[Y][/math].
Usually, the expected claim count, [math]E[N][/math], depends on the exposure base associated with the coverage. A policy may have one exposure such as in the case of a 1-year Homeowners policy, or may have multiple exposures. An example of multiple exposures on a policy would be a 6-month auto policy which covers 3 vehicles. This has an exposure of [math]0.5\cdot 3=1.5[/math] vehicle years.
Claim frequency is the expected number of claims per unit of exposure, and claim severity is the average claim size given a claim has occurred.
The risk charge (also know as the provision for risk) is extra premium collected by the insurer to cover:
- Process risk - the random fluctuation of losses about the expected values.
- Parameter risk - the uncertainty surrounding the selection of model parameters.
The rate per unit of exposure is given by [math]R=\frac{p+f}{1-v}[/math], where p is the pure premium, f is the fixed expense dollars, and v is the variable expense percentage.
The policy premium is given by [math]P=mR[/math].
If all expenses are variable then f = 0 and the quantity [math]\psi=\frac{1}{1-v}[/math] is called a loss cost multiplier (LCM). The LCM is used to load all other costs on top of the pure premium to get the final rate.