Lecture%205%20Annotated

Lecture%205%20Annotated - Life Insurance and Superannuation...

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Life Insurance and Superannuation Models Week 5: Further Analysis of Reserves March 28, 2011 1 / 17

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Week 5: Further Analysis of Reserves Summary of Lecture Recursive calculation of reserves Net premium reserves at successive durations Equations of equilibrium Interpretations Thiele’s differential equation Continuous version With-profit contracts References Chapter 8 (Bowers, et al.), Chapter 6 (Gerber), and CT5 Chapter 6 ACTL3002: Week 5 2
Benefit Reserves for General Insurances Consider a general fully discrete insurance issued to ( x ) for which: Death benefit is payable at the end of the policy year of death; Premiums are payable each year at the beginning of the year; Death benefit in the j -th policy year is b j , j = 1 , 2 , ... Benefit (net) premium payment in the j -th policy year is P j - 1 Net premiums satisfy the equation [PVFP 0 = PVFB 0 ]: s k = 0 P k v k k p x = s k = 0 b k + 1 v k + 1 k | q x . ACTL3002: Week 5 3

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Prospective loss and Reserves The insurer’s prospective loss at time h from issue is given by: h L = b K ( x )+ 1 v K ( x )+ 1 - h - K ( x ) s j = h P j v j - h , for K ( x ) = h , h + 1 , ... The net premium reserve at h is h V = E [ h L | K ( x ) h ] = s j = 0 b h + j + 1 v j + 1 j | q x + h - s j = 0 P h + j v j j p x + h = APVFB h - APVFP h . ACTL3002: Week 5 4

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The reserve in the next period h + 1 can be shown to be ( h V + P h )( 1 + i ) l x + h - b h + 1 d x + h = h + 1 V l x + h + 1 ( h V + P h )( 1 + i ) - b h + 1 q x + h = h + 1 V p x + h or h V + P h = v q x + h b h + 1 + v p x + h h + 1 V h + 1 V = ( h V + P h )( 1 + i ) - b h + 1 q x + h p x + h . Intuitively, we have: Accumulate previous reserves plus premium with interest; Deduct death benefits to be paid at the end of the year; and Divide the reserves by the proportion of survivors. ACTL3002: Week 5

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This note was uploaded on 06/12/2011 for the course ASB 1001,2522, taught by Professor Nicole during the One '09 term at University of New South Wales.

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Lecture%205%20Annotated - Life Insurance and Superannuation...

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