Lecture%206%20Annotated - Life Insurance and Superannuation...

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Life Insurance and Superannuation Models Week 6: Gross Premiums and Reserves April 4, 2011 1 / 25
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Week 6: Gross Premiums and Reserves Summary of Lecture Expense-loaded premiums Gross premium calculations based on the equivalence principles Types of expenses Acquisition, maintenance, settlement, etc. First-year vs renewal expenses Reserving based on gross premiums benefit and expense reserves gross premium vs net premium reserves Cash surrender values/options References Chapter 15 (Bowers, et al.), Chapter 10 (Gerber), and CT5 Chapter 7 ACTL3002: Week 6 2
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Types of Life Insurance Contract Expenses Investment-related expenses (e.g. analysis, cost of buying, selling, servicing) Insurance-related expenses: Acquisition (agents’ commission, underwriting, preparing new records) Maintenance (premium collection, policyholder correspondence) General (research, actuarial, accounting, taxes) Settlement (claim investigation, legal defence, disbursement) ACTL3002: Week 6 3
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First year vs. Renewal Expenses Most life insurance contracts incur large losses in the first year because of large first year expenses: Agents’ commission Preparing new policies and records administration These large losses are hopefully recovered in later years. How then do these first year expenses spread over the policy life? Anything not first year expense is called renewal expense (used for maintaining and continuing the policy). ACTL3002: Week 6 4
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Gross Premium Calculations Gross premium calculations can be done using the principle of equivalence: APV (Future Gross Premiums) = APV (Future Benefits + Expenses). To illustrate, consider a 5 - year endowment policy issued to ( 60 ) with the following characteristics: Sum insured is $ 100 payable at the end of the year of death or on maturity; Renewal expense, including the first, of $ 1 . 50 per policy; Valuation basis: A1967-70 Ultimate, 4 % interest, where A 60 : 5 | = 0 . 82733 and ¨ a 60 : 5 | = 4 . 489 ACTL3002: Week 6 5
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Gross Premium Calculations cont... Denote by G the gross premium payable annually in advance. Thus, we have G ¨ a 60 : 5 | = 100 A 60 : 5 | + 1 . 5 ¨ a 60 : 5 | so that G = 100 A 60 : 5 | + 1 . 5 ¨ a 60 : 5 | ¨ a 60 : 5 | = 100 A 60 : 5 | ¨ a 60 : 5 | + 1 . 5 = 19 . 93 Note that the first term is indeed the net annual premium (ignoring expenses): P = 100 A 60 : 5 | ¨ a 60 : 5 | = 18 . 43 Therefore G = P + e = net premium + expense loadings = 18 . 43 + 1 . 5 = P + 1 . 5 ACTL3002: Week 6 6
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Gross Premium Reserve Calculations Now let us obtain the gross premium (GP) reserve after t < 5 years allowing for renewal expenses. Use the same mortality and interest basis to calculate the premium. We have GP reserve = t V = ( 100 A 60 + t : 5 t | + 1 . 5 ¨ a 60 + t : 5 t | ) - G ¨ a 60 + t : 5 t | = 100 A 60 + t : 5 t | + 1 . 5 ¨ a 60 + t : 5 t | - ( P + 1 . 5 a 60 + t : 5 t | = 100 A 60 + t : 5 t | - P ¨ a 60 + t : 5 t | = Net premium reserve This is true only when there is a level renewal expense incurred
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