By D C M Dickson; Mary Hardy; H R Waters
Balancing rigour and instinct, and emphasizing functions, this contemporary textual content is perfect for collage classes and actuarial examination preparation.
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No annuity beneﬁt is paid while the insured life survives. On the death of the insured life, if the annuitant is still alive, the annuitant receives an annuity for the remainder of his or her life. 4 Other insurance contracts The insurance and annuity contracts described above are all contingent on death or survival. There are other life contingent risks, in particular involving shortterm or long-term disability. These are known as morbidity risks. Income protection insurance When a person becomes sick and cannot work, their income will, eventually, be affected.
If the annuity is paid for some maximum period, provided the annuitant survives that period, it is called a term life annuity. Annuities are often purchased by older lives to provide income in retirement. Buying a whole life annuity guarantees that the income will not run out before the annuitant dies. Single Premium Deferred Annuity (SPDA) Under an SPDA contract, the policyholder pays a single premium in return for an annuity which commences payment at some future, speciﬁed date. The annuity is ‘life contingent’, by which we mean the annuity is paid only if the policyholder survives to the payment dates.
Pr[Kx = k] = Pr[k ≤ Tx < k + 1] = k |qx = k px − k+1 px = k px − k px px+k = k px qx+k . The expected value of Kx is denoted by ex , so that ex = E[Kx ], and is referred to as the curtate expectation of life (even though it represents the expected curtate lifetime). So E[Kx ] = ex ∞ = k Pr[Kx = k] k=0 ∞ = k (k px − k+1 px ) k=0 = (1 px − 2 px ) + 2(2 px − 3 px ) + 3(3 px − 4 px ) + · · · ∞ = k px . 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 .