5.2 One-Year Covers 77 If we consider just the average claim frequency as a function of the age, we have respectively: x = y nx (1 + i) 1 2 (5.2.18) and: x = b d nx (1 + i) 1 2 , (5.2.19)

For practical reasons, it can be useful to factorize the quantities nx , yx and dx , according to the logic of a multiplicative model. Then: nx = n tx , yx = y ux , dx = d vx , (5.2.20a) (5.2.20b) (5.2.20c) where the quantities n, y and d do not depend on age, whereas the ageing coefcients tx , ux , and vx express the impact of the age as a risk factor. Assuming that the specic age effect does not change throughout time, the claim monitoring can be restricted to the quantities n, y, d observed over the whole portfolio, so that more reliable estimates can be obtained. Example 5.2.2 Consider a policy which provides a daily benet b = 100. We assume i = 0.02 and the following statistical basis, derived from the graduation of ISTAT data: nx = n tx = 0.1048 (cid:3) dx = d vx = 10.91 (cid:3) 0.272859 e0.029841 x (cid:4) 0.655419 e0.008796 x (cid:4) . ,

The average claim frequency, the average claim duration and the premium are shown (cid:4) in Table 5.2 for various ages. 5.3 Multi-year Covers In the following sections we focus on multi-year non-cancelable policies, whose conditions are stated at policy issue and cannot be changed throughout the whole policy duration. 78 5 Actuarial Models for Sickness Insurance Table 5.2 Average claim frequency, average claim duration and premium x 30 35 40 45 50 55 60 65 70 nx 0.07000 0.08126 0.09434 0.10952 0.12714 0.14760 0.17135 0.19892 0.23093 dx 9.30991 9.72849 10.16590 10.62298 11.10060 11.59970 12.12124 12.66623 13.23572 x 64.53 78.28 94.96 115.20 139.74 169.53 205.65 249.48 302.64

5.3.1 Some Preliminary Ideas A multi-year cover can be nanced, in particular, via: 1. a single premium; 2. natural premiums; 3. level premiums (throughout the whole policy duration). Other premium arrangements can be conceived; for example: 4. shortened level premiums (i.e. level premiums payable throughout a period shorter than the policy duration); 5. stepwise level premiums. Reasons for premium arrangements like 4 and 5 have been discussed in Sect. 1.2. For brevity, we now focus on arrangements 1 to 3 only.