Modern Actuarial Risk Theory

The insurance industry exists because people are willing to pay a price for being insured. There is an economic theory that explains why insureds are willing to pay a premium larger than the net premium, that is, the mathematical expectation of the insured loss. This theory postulates that a decision maker, generally without being aware of it, attaches a value u(w) to his wealth w instead of just w, where u(·) is called his utility function. To decide between random losses X and Y, he compares E[u(w − X)] with E[u(w − Y)] and chooses the loss with the highest expected utility. With this model, the insured with wealth w is able to determine the maximum premium P⁺ he is prepared to pay for a random loss X. This is done by solving the equilibrium equation E[u(w − X)] = u(w − P). At the equilibrium, he does not care, in terms of utility, if he is insured or not. The model applies to the other party involved as well. The insurer, with his own utility function and perhaps supplementary expenses, will determine a minimum premium P⁻. If the insured’s maximum premium P⁺ is larger than the insurer’s minimum premium P⁻, both parties involved increase their utility if the premium is between P⁻ and P⁺.

In this chapter we focus on the distribution function of the total claim amount S for the portfolio of an insurer. We are not merely interested in the expected value and the variance of the insurer’s random capital, but we also want to know the probability that the amounts paid exceed a fixed threshold. The distribution of the total claim amount S is also necessary to be able to apply the utility theory of the previous chapter. To determine the value-at-risk at, say, the 99.5% level, we need also good approximations for the inverse of the cdf, especially in the far tail. In this chapter we deal with models that still recognize the individual, usually different, policies. As is done often in non-life insurance mathematics, the time aspect will be ignored. This aspect is nevertheless important in disability and long term care insurance. For this reason, these types of insurance are sometimes considered life insurances.

In the insurance practice, risks usually cannot be modeled by purely discrete random variables, nor by purely continuous random variables. For example, in liability insurance a whole range of positive amounts can be paid out, each of them with a very small probability. There are two exceptions: the probability of having no claim, that is, claim size 0, is quite large, and the probability of a claim size that equals the maximum sum insured, implying a loss exceeding that threshold, is also not negligible. For expectations of such mixed random variables, we use the Riemann-Stieltjes integral as a notation, without going too deeply into its mathematical aspects. A simple and flexible model that produces random variables of this type is a mixture model, also called an ‘urn-of-urns’ model. Depending on the outcome of one drawing, resulting in one of the events ‘no claim or maximum claim’ or ‘other claim’, a second drawing is done from either a discrete distribution, producing zero or the maximal claim amount, or a continuous distribution. In the sequel, we present some examples of mixed models for the claim amount per policy.

In this chapter, we introduce collective risk models. Just as in Chapter 2, we calculate the distribution of the total claim amount, but now we regard the portfolio as a collective that produces a random number N of claims in a certain time period. We write where X _i is the ith claim. Obviously, the total claims S = 0 if N = 0. The terms of S in (3.1) correspond to actual claims; in (2.26), there are many terms equal to zero, corresponding to the policies that do not produce a claim. We assume that the individual claims X _i are independent and identically distributed, and also that N and all X _i are independent. In the special case that N is Poisson distributed, S has a compound Poisson distribution. If N has a (negative) binomial distribution, then S has a compound (negative) binomial distribution.

In this chapter we focus again on collective risk models, but now in the long term. We consider the development in time of the capital U(t) of an insurer. This is a stochastic process that increases continuously because of earned premiums, and decreases stepwise at times that claims occur. When the capital becomes negative, we say that ruin occurs. Let ψ(u) denote the probability that this ever happens, under the assumption that the annual premium and the claims process remain unchanged. This probability is a useful management tool since it serves as an indication of the soundness of the insurer’s combination of premiums and claims process, in relation to the available initial capital u = U(0). A high probability of ultimate ruin indicates instability: measures such as reinsurance or raising premiums should be considered, or the insurer should attract extra working capital.

The activities of an insurer can be described as a system in which the acquired capital increases because of (earned) premiums and interest, and decreases because of claims and costs. See also the previous chapter. In this chapter we discuss some mathematical methods to determine the premium from the distribution of the claims. The actuarial aspect of a premium calculation is to calculate a minimum premium, sufficient to cover the claims and, moreover, to increase the expected surplus sufficiently for the portfolio to be considered stable.

This chapter presents the theory behind bonus-malus methods for automobile insurance. This is an important branch of non-life insurance, in many countries even the largest in total premium income. A special feature of automobile insurance is that quite often and to everyone’s satisfaction, a premium is charged that for a large part depends on the claims filed on the policy in the past. In experience rating systems such as these, bonuses can be earned by not filing claims, and a malus is incurred when many claims have been filed. Experience rating systems are common practice in reinsurance, but in this case, it affects the consumer directly. Actually, in case of a randomly fluctuating premium, the ultimate goal of insurance, that is, being in a completely secure financial position as regards this particular risk, is not reached. But in this type of insurance, the uncertainty still is greatly reduced. This same phenomenon occurs in other types of insurance, for example when part of the claims is not reimbursed by the insurer because there is a deductible.

That lucky policyholders pay for the damages caused by less fortunate insureds is the essence of insurance (probabilistic solidarity). But in private insurance, solidarity should not lead to inherently good risks paying for bad ones. An insurer trying to impose such subsidizing solidarity on his customers will see his good risks take their business elsewhere, leaving him with the bad risks. This may occur in the automobile insurance market when there are regionally operating insurers. Charging the same premiums nationwide will cause the regional risks, which for automobile insurance tend to be good risks because traffic is not so heavy there, to go to the regional insurer, who with mainly good risks in his portfolio can afford to charge lower premiums.

Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical concepts and tools to do this, and derives some important results of non-life actuarial science that can be derived. There are two reasons why a risk, representing a non-negative random financial loss, would be universally preferred to another. One is that the other risk is larger, see Section 7.2, the second is that it is thicker-tailed (riskier), see Section 7.3. Thicker-tailed means that the probability of extreme values is larger, making a risk with equal mean less attractive because it is more spread and therefore less predictable. We show that having thicker tails means having larger stop-loss premiums. We also show that preferring the risk with uniformly lower stop-loss premiums describes the common preferences between risks of all risk averse decision makers. From the fact that a risk is smaller or less risky than another, one may deduce that it is also preferable in the mean-variance order that is used quite generally. In this ordering, one prefers the risk with the smaller mean, and the variance serves as a tie-breaker. This ordering concept, however, is inadequate for actuarial purposes, since it leads to decisions about the attractiveness of risks about which there is no consensus in a group of decision makers all considered sensible.

In insurance practice it often occurs that one has to set a premium for a group of insurance contracts for which there is some claim experience regarding the group itself, but a lot more on a larger group of contracts that are more or less related. The problem is then to set up an experience rating system to determine next year’s premium, taking into account not only the individual experience with the group, but also the collective experience. Two extreme positions can be taken. One is to charge the same premium to everyone, estimated by the overall mean \(\bar X\) of the data. This makes sense if the portfolio is homogeneous, which means that all risk cells have identical mean claims. But if this is not the case, the ‘good’ risks will take their business elsewhere, leaving the insurer with only ‘bad’ risks. The other extreme is to charge to group j its own average claims \(\bar X_j\) as a premium. Such premiums are justified if the portfolio is heterogeneous, but they can only be applied if the claims experience with each group is large enough. As a compromise, one may ask a premium that is a weighted average of these two extremes: The factor Z _j that expresses how ‘credible’ the individual experience of cell j is, is called the credibility factor; a premium such as (8.1) is called a credibility premium. Charging a premium based on collective as well as individual experience is justified because the portfolio is in general neither completely homogeneous, nor completely heterogeneous. The risks in group j have characteristics in common with the risks in other groups, but they also possess unique group properties.

Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depending of the mean, one would expect the coefficient of variation to be constant. Furthermore, the phenomena to be modeled are rarely additive in the collateral data. A multiplicative model is much more plausible. If other policy characteristics remain the same, moving from downtown to the country would result in a reduction in the average total claims by some fixed percentage of it, not by a fixed amount independent of the original risk. The same holds if the car is replaced by a lighter one.

In the past, non-life insurance portfolios were financed through a pay-as-you-go system. All claims in a particular year were paid from the premium income of that same year, no matter in which year the claim originated. The financial balance in the portfolio was realized by ensuring that there was an equivalence between the premiums collected and the claims paid in a particular financial year. Technical gains and losses arose because of the difference between the premium income in a year and the claims paid during the year.

The claims originating in a particular year often cannot be finalized in that year. For example, long legal procedures are the rule with liability insurance claims. But there may also be other causes for delay, such as the fact that the exact size of the claim is hard to assess. Also, the claim may be filed only later, or more payments than one have to be made, such as in disability insurance. All these factors will lead to delay of the actual payment of the claims. The claims that have already occurred, but are not sufficiently known, are foreseeable in the sense that one knows that payments will have to be made, but not how much the total payment is going to be. Consider also the case that a premium is paid for the claims in a particular year, and a claim arises of which the insurer is not notified as yet. Here also, we have losses that have to be reimbursed in future years.

In this chapter, we first recall the statistical theory of ordinary linear models. Then we define Generalized Linear Models in their full generality, with as a random component any distribution in the exponential dispersion family of densities. This class has a natural parameter θ, determining the mean, as well as a dispersion parameter, and contains all the examples we introduced in Chapter 9 as special cases. Starting from the general form of the density, we derive properties of this family, including the mean, variance and mgf. Inference in GLMs is not done by a least squares criterion, but by analysis of deviance; the corresponding residuals are deviance residuals. We discuss some alternatives. We also study the canonical link function. Then we derive the algorithm of Nelder and Wedderburn to determine maximum likelihood estimates for this family, and show how to implement it in R. The algorithm can be applied with any link function. A subfamily of the exponential family of densities not studied in Chapter 9 consists of the compound Poisson-gamma distributions with fixed shape parameter α for the claim sizes. They have a variance equal to ωμ ^p , with μ the mean, ψ the dispersion parameter and \(p = {2+\alpha \over 1+\alpha} \in (1, 2)\). Such mean-variance structures were first studied by Tweedie.