STAT3801_2011Unit1[1]

STAT3801_2011Unit1[1] - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3801 ADVANCED LIFE CONTINGENCIES 2010-2011 2 nd semester Unit 1 Advanced life contingencies • Continuation of STAT2801 • SOA Exam M – Life contingencies segment * Life insurances and annuities * Markov chain models Life insurances and annuities • PV-of-benefit on survival-time random variables • Expected values, variances and probabilities • Considerations for life insurances and annuities • Liabilities, PV of future loss random variables • Asset shares • Recursion formulas Markov chain models • Discrete-time Markov chain models – Homogeneous and non-homogeneous • Probabilities of being in a particular state • Probabilities of transition between states S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 2 Advanced life contingencies 2010-2011, 2 nd semester 1. Analysis of Benefit Reserves 2. Multiple Life Functions 3. Transition Models 4. Multiple Decrement Models 5. Insurance Models including Expenses 1. Analysis of benefit reserves Ref: Actuarial Mathematics (AM), 8.1 - 8.5 • General formulas for benefit reserves • Recursion relations – Fully discrete benefit reserves • Benefit reserves at fractional durations • Allocation of risk to insurance years Level benefits funded by level premiums • Traditional insurance products – Purchased with level contract premiums • Single equation of equivalence principle – Yields solution for only one parameter • Incidence of expenses has not been included • Some regulatory standards – Benefit reserves defined by level premiums S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 3 In class question 1.1 Fully discrete whole life insurance of $1,000 issued on life (75) Premiums π k payable at time k k = 0 , 1 , 2 , ··· You are given: π k = π (1 . 05) k i = 0 . 05 Mortality follows de Moivre’s law with ω = 105 Premiums calculated with equivalence principle Required • Calculate π • Calculate 1 V Note: De Moivre’s law s ( x ) = 1- x ω k p x = ω- x- k ω- x = 1- k ω- x k | q x = k p x (1- p x + k ) = 1- k ω- x ¶ 1 ω- x- k ¶ = 1 ω- x S&AS: STAT3801 ADVANCED LIFE CONTINGENCIES 4 1.1 Fully discrete case For j = 1 , 2 , ··· Premiums π j- 1 payable at beginning of j th policy year Benefits b j payable at end of j th policy year time 1 2 3 4 5 ··· premiums π π 1 π 2 π 3 π 4 π 5 ··· benefits b 1 b 2 b 3 b 4 b 5 ··· Loss random variable at time h h L =        K ( x ) < h b K ( x )+1 v K ( x )+1- h- K ( x ) X j = h π j v j- h K ( x ) ≥ h Prospective formulation of benefit reserve at time h h V = E [ h L | K ( x ) ≥ h ] = E   b K ( x )+1 v K ( x )+1- h- K ( x ) X j = h π j v j- h fl fl fl fl fl fl K ( x ) ≥ h   = E   b [ K ( x )- h ]+ h +1 v [ K ( x )- h ]+1- K ( x )- h X k =0 π h + k v k fl fl fl fl fl fl K ( x ) ≥ h   Assume [ K ( x )- h | K ( x ) ≥ h ] and K ( x + h ) have same distribution h V = E   b K ( x + h...
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STAT3801_2011Unit1[1] - THE UNIVERSITY OF HONG KONG...

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