Actuarial mathematics for life contingent risks by D C M Dickson; Mary Hardy; H R Waters

By D C M Dickson; Mary Hardy; H R Waters

Balancing rigour and instinct, and emphasizing purposes, this contemporary textual content is perfect for college classes and actuarial examination preparation.

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3 The force of mortality 23 This relationship gives a way of finding µx+t given Sx (t). 9) to develop a formula for Sx (t) in terms of the force of mortality function. 9) we have µx = − d log S0 (x), dx and integrating this identity over (0, y) yields y µx dx = − (log S0 (y) − log S0 (0)) . 0 As log S0 (0) = log Pr[T0 > 0] = log 1 = 0, we obtain y S0 (y) = exp − µx dx , 0 from which it follows that Sx (t) = S0 (x + t) = exp − S0 (x) x+t t µr dr = exp − x µx+s ds . 11) 0 This means that if we know µx for all x ≥ 0, then we can calculate all the survival probabilities Sx (t), for any x and t.

Joint life annuity A joint life annuity is issued on two lives, typically a married couple. The annuity (which may be single premium or regular 12 Introduction to life insurance premium, immediate or deferred) continues while both lives survive, and ceases on the first death of the couple. Last survivor annuity A last survivor annuity is similar to the joint life annuity, except that payment continues while at least one of the lives survives, and ceases on the second death of the couple. Reversionary annuity A reversionary annuity is contingent on two lives, usually a couple.

Thus, in a proprietary insurer, the profits must be shared in some predetermined proportion, between the shareholders and the with-profit policyholders. Many early life insurance companies were formed as mutual companies. More recently, in the UK, Canada and the USA, there has been a trend towards demutualization, which means the transition of a mutual company to a proprietary company, through issuing shares (or cash) to the with-profit policyholders. Although it would appear that a mutual insurer would have marketing advantages, as participating policyholders receive all the profits and other benefits of ownership, the advantages cited by companies who have demutualized include increased ability to raise capital, clearer corporate structure and improved efficiency.

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