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Part-3-Notes-232-2010W

# Part-3-Notes-232-2010W - Summary of Lecture Notes ACTSC 232...

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Summary of Lecture Notes – ACTSC 232, Winter 2010 Part 3 – Life Annuities An annuity is a series of payments paid at equal intervals or continuously over a fixed term. PV of the annuity-due of 1 for n years is ¨ a n = 1 + v + · · · + v n - 1 = n - 1 X k =0 v k = 1 - v n d . PV of the annuity-immediate of 1 for n years is a n = v + v 2 + · · · + v n = n X k =1 v k = 1 - v n i . PV of the continuous annuity with annual rate of 1 for t years is ¯ a t = Z t 0 v s ds = 1 - v t δ . A life annuity on ( x ) is a series of payments paid at equal intervals or continuously over a random term when the life ( x ) is alive. 3.1 Discrete life annuities on ( x ) : (a) A discrete whole life annuity-due of 1 on ( x ) pays 1 at the beginning of each year until ( x ) dies. The present value of the payments in the annuity is ¨ a K x +1 = 1 - v K x +1 d . The APV of the annuity is denoted by ¨ a x and ¨ a x = E a K x +1 ] = X k =0 v k k p x = 1 - A x d . The variance of the present value is V ar a K x +1 ] = V ar [ v K x +1 ] d 2 = 2 A x - ( A x ) 2 d 2 . 1

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(b) A discrete whole life annuity-immediate of 1 on ( x ) pays 1 at the end of each year until ( x ) dies. The present value of the payments in the annuity is a K x . The APV of the annuity is denoted by a x and a x = E [ a K x ] = X k =1 v k k p x , where a 0 = 0. Note that ¨ a x = 1 + a x . (c) A discrete n -year temporary life annuity-due of 1 on ( x ) pays 1 at the beginning of each year as long as ( x ) survives during the n -year term. The present value of the payments in the annuity is ¨ a ( K x +1) n = ¨ a K x +1 , 0 K x n - 1 ¨ a n , K x n = 1 - v ( K x +1) n d , where a b = min { a, b } . The APV of the annuity is denoted by ¨ a x : n and ¨ a x : n = n - 1 X k =0 v k k p x = 1 - A x : n d . The variance of the present value is V ar a ( K x +1) n ] = V ar [ v ( K x +1) n ] d 2 = 2 A x : n - ( A x : n ) 2 d 2 . Recursive formulas for ¨ a x and ¨ a x : n : ¨ a x = 1 + vp x ¨ a x +1 , ¨ a x = ¨ a x : n + n E x ¨ a x + n , ¨ a x : n = 1 + vp x ¨ a x +1: n - 1 . The actuarial accumulated value at time n of the n -year temporary life annuity-due of 1 on ( x ) is denoted by ¨ s x : n and ¨ s x : n = ¨ a x : n n E x .
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Part-3-Notes-232-2010W - Summary of Lecture Notes ACTSC 232...

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