– Insurance premium formula

Insurance premium is a sum of all costs of insurance company. One of the essential and the most valuable components is risk cost (money paid when insurance accident occurred). Due to the nature of the risk it is the most stochastic variable. Therefore we can build premium cost model upon risk model because other price components can be easily predicted.

– Classic model

Starting with initial reserve x > 0, an insurance risk business earns money at constant rate c > 0. Meanwhile, according to a point process with times {TN: n ≥ 1}, 0 < t1 < t2 < · · ·, and counting process {N (t)}, claims against the business occur in the amounts Bn > 0 causing the reserve to jump downward by such amounts. If at any such jump, the reserve falls ≤ 0, then the business is said to be ruined (see fig. 1.).

Risk process is defined (see fig. 2.) as:

Estimation of ruin probabilities is really diﬃcult for a large x because the probability can be really small; so long simulation is required to detect a ruin. However, there are a variety of eﬃcient methods for estimating the probability of ruin. They are: “ rare event simulation”, “ importance sampling”, “ change of measure” and “ fast simulation” methods. The basic idea of these approaches is to change the distributions of interarrival and claim size so that ruin occurs more often, and then transform the answer back to the original distribution Karl Sigman (2).

– Markovian model

Let’s take a look at another insurance risk simulation model which makes more sense when considering car insurance. Policy holders join according to a Poisson process at rate v.

Independently each such holder remains joined for an amount of time that is exponentially distributed with rate λ and then quits, and then is no longer a policy holder. Independently, each holder sends in claims according to a Poisson process at rate µ. All claim sizes are iid with distribution G. Also, each holder pays a premium to the company of $c per unit time. Some observations: At any given time t, suppose there are N policy holders. Starting from time t, let X1 denote the time until the next new holder joins, X2 the time until one of the N holders quits, and X3 the time until the next claim comes in. Then, by the Poisson/exponential assumptions and the memory less property of the exponential distribution, these three random variables are independent and exponentially distributed X1 ~expv, X1 ~expNµ, X1 ~expNλ ;

There are 3 event types in this model; e1: a new policy holder joins, e2: an existing policy holder quits and e3: a claim comes in. The time until the next event is the minimum m = min{X1; X2; X3}, is distributed exponentially with rate r = v+N λ +N µ. Thus if the current time is t and there are N policy holders, then we can schedule the “ next event” at time t+ ((1= r) ln (U). But we must determine type of the event. Noting that each of X1, X2 or X3 will be the minimum with probabilities p1 = v/r; p2 = N λ /r; p3 = N µ /r. In such case we need to independently generate a rv C with distribution P(C = 1) = p1; P(C = 2) = p2; P(C = 3) = p3; if C = i event is of type ei. Generate U. If U <= p1, then set C = 1. If p1 < U <= p1 + p2, then set C = 2. If U > p+p2, then set C = 3. Here then is the code for simulating this model up to time T. Detailed algorithm with an example can be found in Karl Sigman (2).

– The structural simulation model

## In the structural simulation, claim emergence and development is assumed to be decomposable into the following two components:

– An underlying stationary claim emergence pattern that is the culmination of underlying stochastic claim processes subject to random noise.

– One or more socio-economic factors which distort the stationary emergence pattern, “ stretching” or “ shrinking” it in some way. Each of these socio-economic factors may itself entail a stochastic process subject to random noise.

Where Cd is the expected payment generated from the underlying stationary emergence, Z is the set of socio-economic factors that modify the payment, and a, d e is the residual random error from the stochastic processes according to A Structural Simulation Model for Measuring General Insurance Risk (13) by Gault Timothy.

Typically, the socio-economic factors will be represented by a system of stochastic equations that describe the behavior of each variable over time and how each variable interacts with other variables according to Christofides Shawn. Brief algorithm schema is presented on fig. 3.

Detailed description of the algorithm and example could be found in A Structural Simulation Model for Measuring General Insurance Risk (15-20) by Gault Timothy.

## Couple more methods can be found in this source but they go beyond the scope of the paper.

– Collective risk model

## This approach goes beyond the scope of the paper so I will describe it very briefly.

The key idea of the approach is to simulate aggregated claim amount (classical models simulate individuals).

We consider a short term insurance contract covering a risk. Risk is a single policy or a group of policies. The random variable S denotes the aggregate claims paid by the insurer in the year in respect of this risk. The distribution of S is a compound distribution. We consider the most important case when N is Poisson with parameter λ. In the end of math work we will receive equations of on the fig. 4. which are used for simulation.

According to Pacáková, Viera (130)

Detailed description of the algorithm and example could be found in Modeling and Simulation in Non-life Insurance (130-132) by Pacáková Viera.

## References

Sigman, Karl. « Insurance Risk Models. ». N. p., n. d. Web. 7 July 2014.

R. E. Beard, Risk Theory, the Stochastic Basis of Insurance. London: Chapman and Hall, 1994.

Gault , Timothy , Len Llaguno, and Stephen Lowe. A Structural Simulation Model for Measuring General Insurance Risk . : Casualty Actuarial Society E-Forum, Summer 2010 Print.

Christofides , Shawn. Regression Models Based on Log-Incremental Payments,. : Institute of Actuaries Claims Reserving Manual, 1990. Print.

Pacáková, Viera . Modelling and Simulation in Non-life Insurance. ISBN: 978-1-61804-016-9 , n. d. Web. 7 July 2014.

Kaas, R., and M. Goovaerts. Modern Actuarial Risk Theory.. Boston: Kluwer Academic Publishers, 2001. Print.

Rubinstein, R. Y.. Simulation and the Monte Carlo Method. . New York: John Wiley &Sons, Inc., , 1981. Print.

Boland, P. J.. Statistical and Probabilistic Methods in Actuarial Science. . : Chapman&Hall/CRC, , 2007. Print.