19.4 Final Remarks

Let us first concentrate on the franchise and fixed amount deductibles. Figures 19.6 and 19.9 depict the comparison of the two corresponding premiums and the effect of increasing the parameters $ a$ and $ b$. Evidently $ P\geqslant P_{FD}\geqslant P_{FAD}$. Moreover, we can see that the deducible of about DKK $ 2$ million in the log-normal case and DKK $ 40$ million in the Burr case reduces $ P_{FAD}$ by half. Figures corresponding to the two loss distributions are similar, however we note that the differences do not lie in shifting or scaling. The same is true for the rest of considered deductibles. We also note that the premiums under no deductible for log-normal and Burr loss distributions do not tally because the parameters were estimated via the Anderson-Darling statistic minimization procedure which in general does not yield the same moments, cf. Chapter 13. For the considered distributions the mean, and consequently the pure risk premium, is even 3 times bigger in the Burr case.

The proportional deductible influences the premium in an obvious manner, that is pro rata (e.g. $ c=0.25$ results in cutting the premium by a quarter). Figures 19.7 and 19.10 show the effect of parameters $ c$, $ m_1$ and $ m_2$ of the limited proportional deductible. It is easy to see that $ P_{LPD(c,m_1,m_2)}
$ is a decreasing function of these parameters. Figures 19.8 and 19.11 depict the influence of parameters $ d_1 $ and $ d_2$ of the disappearing deductible. Clearly, $ P_{DD(d_1,d_2)} $ is a decreasing function of the parameters and we can observe that the effect of increasing $ d_2$ is rather minor.

It is clear that the choice of a distribution and a deductible has a great impact on the pure risk premium. For an insurer the choice can be crucial in reasonable quoting of a given risk. A potential insured should take into account insurance options arising from appropriate types and levels of self-insurance (deductibles). Insurance premiums decrease with increasing levels of deductibles. With adequate loss protection, a property owner can take some risk and accept a large deductible which might reduce the total cost of insurance.

We presented here a general approach to calculating pure risk premiums under deductibles. In Section 19.2 we presented a link between the pure risk premium under several deductibles and a limited expected value function. We used this link in Section 19.3 to calculate the pure risk premium in the case of the deductibles for different claim amount distributions. The results can be applied to derive annual premiums in the individual and collective risk model on a per occurrence deductible basis.

The approach can be easily extended to other distributions. One has only to calculate levf for a particular distribution. This also includes the case of right-truncated distributions which would reflect the maximum limit of liability set in a contract. Moreover, the idea can be extended to other deductibles. Once we express the pure risk premium in terms of the limited expected value function, it is enough to apply a form of levf for a specific distribution. Finally, one can also use the formulae to obtain the premium with safety loading which is discussed in Chapter 18.