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Unformatted text preview: Life Insurance and Superannuation Models Week 2: Life Annuities (Single Life) March 5, 2011 1 / 24 Week 2: Life Annuities (Single Life) Summary Life annuities: Series of benefits paid contingent upon survival of a given life Single life considered Actuarial present values (APV) Actuarial symbols and notation Forms of annuities Discretedue or immediate Payable more frequently than once a year Continuous “Current payment techniques” APV formulas References Chapter 5 (Bowers, et al.) or Chapter 4 (Gerber) ACTL3002: Week 2 2 Whole Life AnnuityDue Pays a benefit of a unit $1 at the beginning of each year that the annuitant ( x ) survives The present value random variable can be represented as Y = ¨ a K + 1  where K is the curtate future lifetime of ( x ) The actuarial present value of the annuity ¨ a x = E [ Y ] = E [ ¨ a K + 1  ] = ∞ summationdisplay k = ¨ a k + 1  P ( K = k ) = ∞ summationdisplay k = ¨ a k + 1  · k  q x = ∞ summationdisplay k = ¨ a k + 1  · k p x · q x + k ACTL3002: Week 2 3 Current Payment Technique By using summation by parts, one can show that ¨ a x = ∞ summationdisplay k = v k k p x This is called current payment technique formula for computing a whole life annuitydue where the k p x term is the probability of a payment of size 1 being made at time k . Summation by parts (discrete analogue of integration by parts) formula can be found in the Appendix 5 of Bowers, et al. ACTL3002: Week 2 4 Relationship to Whole Life Insurance By recalling from interest theory that ¨ a K + 1  = ( 1 − v K + 1 ) / d , we have the useful relationship: ¨ a x = 1 − A x d Alternatively, we write: A x = 1 − d ¨ a x The variance formula can be calculated as: Var ( ¨ a K + 1  ) = Var parenleftbigg 1 − v k + 1 d parenrightbigg = Var ( v k + 1 ) d 2 = 2 A x − ( A x ) 2 d 2 Summation byparts can also be used to derive the recursive relationship: ¨ a x = 1 + vE [ ¨ a K ( x )+ 1   K ( x ) ≥ 1 ] P ( K ( x ) ≥ 1 ) = 1 + vp x ¨ a x + 1 ACTL3002: Week 2 5 Temporary Life AnnuityDue Pays a benefit of a unit $1 at the beginning of each year as long as the annuitant ( x ) survives for up to a total of n years The present value random variable is Y = braceleftBigg ¨ a K + 1  , K < n ¨ a n...
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