Imagine that we are insuring a policyholder against the total damage X that might be accrued in automobile accidents over a year. The number of accidents the policyholder might have is modelled as Pólya(0.26,0.73). The damages incurred in any one accident is \$Lognormal(300, 50). The insurer has to determine the premium to be charged.

The premium must be at least greater than the expected payout E[X] otherwise, according to the law of large numbers, in the long run the insurer will be ruined. The expected payout is the product of the expected values of the Pólya and Lognormal distributions: in this case = 0.1898 * \$300 = \$56.94. The question is then: how much more should the premium be over the expected value? Actuaries have a variety of methods to determine the premium. Four of the most common methods are listed below.

### Expected value principle

This calculates the premium in excess of E[X] as some fraction q of E[X]:

Ignoring administration costs q represents the return the insurer is getting over the expected capital required E[X] to cover the risk.

This method is implemented in ModelRisk with the VosePrincipleEV function.

### The Standard deviation principle

This calculates the premium in excess of E[X] as some multiple a of the standard deviation of X

The problem with this principle is that, at an individual level, there is no consistency in the level of risk the insurer is taking for the expected profit  since s has no consistent probabilistic interpretation.

This method is implemented in ModelRisk with the VosePrincipleStdev function.

### Esscher principle

The Esscher method calculates the ratio of the expected values of XehX to ehX

The principle gets its name from the Esscher transform which converts a density function from f(x) to a*f(x)*Exp[b*x] where a, b are constants. It was introduced by Bühlmann (1980) in an attempt to acknowledge that the premium price for an insurance policy is a function of the market conditions in addition to the level of risk being covered. Wang (2003) gives a nice review.

This method is implemented in ModelRisk with the VosePrincipleEsscher function.

### Risk adjusted or PH principle

This is a special case of the Proportional Hazards Premium Principle based on coherent risk measures (see, e.g. Wang, 1996). The survival function (1-F(x)) of the aggregate distribution which lies on [0,1] is transformed into another variable that also lies on [0,1]

where F(x) is the cumulative distribution function from the aggregate distribution.

This method is implemented in ModelRisk with the VosePrincipleRA function.

- calculation of premiums under four different principles.

The mean and variance of the aggregate distribution, shown with a grey background, provide a reference point (e.g. the premiums must exceed the mean of 56.94).

For the concrete implementation of the ModelRisk functions for these calculations, see the following topics: VosePrincipleEsscher, VosePrincipleEV, VosePrincipleRA, VosePrincipleStdev.