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6 Mutual and proprietary insurers A mutual insurance company is one that has no shareholders. The insurer is owned by the with-profit policyholders. All profits are distributed to the with-profit policyholders through dividends or bonuses. A proprietary insurance company has shareholders, and usually has withprofit policyholders as well. The participating policyholders are not owners, but have a specified right to some of the profits. Thus, in a proprietary insurer, the profits must be shared in some predetermined proportion, between the shareholders and the with-profit policyholders.

So, if we count time in years from t = 0 at the start of the contract, the first premium is paid at t = 0, the second is paid at t = 1, and so on, to the tenth premium paid at t = 9. Similarly, if the premiums are monthly, then the first monthly instalment will be paid at t = 0, and the final premium will be paid at the start 11 of the final month at t = 9 12 years. 7 Life annuities Annuity contracts offer a regular series of payments. When an annuity depends on the survival of the recipient, it is called a ‘life annuity’.

24) k=1 Note that the lower limit of summation is k = 1. Similarly, ∞ E[Kx2 ] = k 2 ( k px − k+1 px ) k=0 = (1 px − 2 px ) + 4(2 px − 3 px ) + 9(3 px − 4 px ) + 16(4 px − 5 px ) + · · · ∞ =2 ∞ k k px − k=1 k px k=1 ∞ =2 k k px − e x . k=1 34 Survival models As with the complete expectation of life, there are a few lifetime distributions that allow E[Kx ] and E[Kx2 ] to be calculated analytically. For more realistic models, such as Gompertz’, we can calculate the values easily using Excel or other suitable software.