Case study 1: comparing premium setting principles

We examine the use of different premium setting principles under different models for an aggregate insurance loss S. We want to compare the different approaches – in particular we want to obtain measures of the uncertainty associated with the premiums set so that we can ascertain how “precise” and reliable the premiums set by the different principles are.

We will use theory and simulated data as far as we can, and also turn to the bootstrap resampling technique for additional enlightenment.

We will compare premiums set using some or all of the following principles (as described in §4.1):

1. (1) EVP (expected value principle)

2. (2) SDP (standard deviation principle)

3. (3) VP (variance principle)

4. (4) QP (quantile principle)

5. (5) EPP (exponential premium principle)

under various distributional assumptions for the risk/aggregate loss S. We study first the case in which we have an assumed model for the distribution of S, and then the case in which we base our premiums solely on the information in an observed sample of values of S (with no assumed model).

Case 1 – in the presence of an assumed model

Let us assume that S has a compound Poisson distribution S ∼ CP(λ, FX), where λ is the claim rate and X is the individual loss variable. Further, let us assume, for illustrative purposes, that X ∼ Exp(1/μ) (with mean μ).