An example of a Monte Carlo simulation risk analysis model for general risk analysis
Minimum software requirements: ModelRisk Complete edition
Technical difficulty: 2
Techniques used: Monte Carlo simulation in Excel
ModelRisk functions used: VosePrincipleEV,VosePrincipleStdev,VosePrincipleEsscher,VosePrincipleRA
ModelRisk provides a number of functions to quickly determine the premium that should be charged for an insurance policy under several different methodologies.
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 principleThis calculates the premium in excess of E[X] as some fraction θ of E[X]:
Premium = (1+θ) E[X] θ > 0
Ignoring administration costs θ 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 principleThis calculates the premium in excess of E[X] as some multiple a of the standard deviation of X:
Premium = E[X] + α*σ[x] α > 0
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 α*σ[x] since α has no consistent probabilistic interpretation. This method is implemented in ModelRisk with the VosePrincipleStdev function.
Esscher principleThe Esscher method calculates the ratio of the expected values of Xe*Exp[hX] to e*Exp[hX] where h > 0.
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 principleThis 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. Applications of these functions are illustrated in the Premium calculations model:
The mean and variance of the aggregate distribution, provide a reference point (e.g. the premiums must exceed the mean).