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Recall from the introduction that the risk reserve at time is given, where u is the initial reserve. The quantity is, of course, random. The ultimate hope would be to get some information on the distribution, if this is impossible, on the first few moments. As has already been pointed out, the distribution is very complicated. If we superimpose a possibly random character of the premium income, then the overall situation will become even worse.

There is one particular situation, however, where we can give a full treatment of the stochastic nature of the risk reserve. Assume that time is measured in integers. We then add all premiums collected in period in one single number, which can even be assumed to be random. Similarly we add all claims arriving in period and call this amount. The risk reserve after period is then equal. The resulting stochastic process of partial sums is called the (discrete-time) claim surplus process. In particular, is a random walk if we assume that the sequence consists of independent and identically distributed random variables?

The definition of a random walk is very similar to that of a renewal process, the only difference being that now the generic random variable is no longer concentrated on the nonnegative half-line. The theory of random walks makes up a substantial chapter in a traditional course on stochastic modeling. We will cover the most important aspects of random walk theory. If premiums are non random we can write which shows how important it, is to have workable expressions for the distribution of the aggregate claim amount.

We have used the abbreviation to denote the totality of premiums collected from the policy-holders within the portfolio up to time. Usually premiums are individually collected once a year, but the insurer can safely assume that premium income is evenly spread over the year.

The determination of a functional expression is one of the few things where the insurer can intervene in the overall process. The function should be determined in such a way that the solvability of the portfolio can be guaranteed. This requires increasing fast enough to cope with incoming claims. On the other hand, a very high value may be rather undesirable since then rival insurance companies might attract clients by offering lower premiums while covering the same risk.

The whole area of determining the specific shape of the function is called premium calculation and constitutes an essential part of the actuarial know-how. For an exhaustive treatment, see Goovaerts: De Vylder and Haezendonck (1984). We cover a few and isolated premium principles underlying the general thinking behind premium calculations. From these general principles a wide set of possible candidates emerges. Note that most often these premiums are nonrandom even if their calculation involves information on the stochastic elements within the portfolio.

The most popular form of the premium function where is the expected number of claims up to time, while EU is the mean claim size. The constant q is the safety loading which has to take care, not only of the administrative costs from handling the portfolio, but also of the necessary gain that the company wants to make ultimately. Moreover, it is clear that. Will be portfolio-dependent, as, for example, the higher the risk, the higher q has to be. Kite that in the case of a Poisson process the premium function given is of the form for some constant.

The premium principle with independent of the portfolio is called the expected value principle. Of course, the expected value principle does not keep track of the variability in the portfolio and so alternative principles include.

Another aspect of premium calculation is that not all individual policies in a portfolio must be charged with the same premium. For example: the by now classical bonw-maks premium calculation principle is widely spread in car insurance. Depending on the past history of a policy-holder, the insurer ranks the client in a certain state and charges an amount typical for that state. In the course of time the customer will move from one state to another, depending on his claim record. If we denote by X, the state of a client at the beginning of year ‘n, then the process describes a jump process adequately modeled by a discrete-time Murkow chain. No course on stochastic processes can be considered complete if it would lack a serious treatment of this important building block of stochastic modeling. We will deal with discrete-time Markov chains in general and with bonus-mauls in particular. We will then continue this discussion by changing from discrete time to continuous time.

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