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Unformatted text preview: Summary of Lecture Notes  ACTSC 232, Winter 2010 Part 2  Life Benefits 2.1 Life insurances on ( x ) A life insurance on ( x ) is a contract or policy issued by an insurer to a life currently aged x . The insurer will pay benefits to the beneficiaries of ( x ) in the future. The payment times of the benefits are contingent on the death time of ( x ). Such benefits are called death benefits or life benefits. Generally speaking, a life insurance on ( x ) is called a continuous life insurance if benefits are payable at the moment of the death of ( x ). A life insurance is called a discrete life insurance if benefits are payable at the end of the death year of ( x ). Review of the expectations of the functions of T x and K x : For a function g , E [ g ( T x )] = Z g ( t ) f x ( t ) dt = Z g ( t ) t p x x + t dt and E [ g ( K x )] = X k =0 g ( k ) Pr { K x = k } = X k =0 g ( k ) k p x q x + k . Review of present values: Let v t denote present value (PV) at time 0 of 1 (dollar or unit) to be paid at time t and v t is called discount function. If the force of interest t = is a constant, then v t = v t = e t = ( 1 1+ i ) t , where v = 1 1+ i = e and 1 + i = e . Unless stated otherwise, we assume that t = is a constant or v t = v t = e t . (a) A general continuous life insurance on ( x ) pays death benefits at the death time of ( x ). Denote the benefit by b t if T x = t or ( x ) dies at time t, t > 0. Let Z denote the present value at time 0 or at age x of the benefits to be paid by the insurance. Then Z = b T x v T x = b T x e T x and Z is a random variable, where T x is the death time of ( x ). The expectation or mean of the present value Z is E [ Z ] = E b T x e T x = Z b t e t f x ( t ) dt = Z b t e t t p x x + t dt 1 which is called the actuarial present value (APV) of the insurance, or the expected present value (EPV) of the insurance, or the pure premium of the insurance, or the net premium of the insurance, or the single benefit premium of the insurance. The second moment of the present value Z is E [ Z 2 ] = E b 2 T x e 2 T x = Z b 2 t e 2 t f x ( t ) dt = Z b 2 t e 2 t t p x x + t dt and V ar [ Z ] = E [ Z 2 ] ( E [ Z ]) 2 . The distribution function of Z is denoted by F Z ( z ) = Pr { Z z } . The distribution function may be continuous, or discrete, or mixed. (b) A general discrete life insurances on ( x ) pays death benefits at the end of the death year of ( x ). Denote the death benefit by b k +1 if K x = k or ( x ) dies in year k + 1, k = 0 , 1 , 2 ,.......
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 Summer '08
 MATTHEWTILL

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