Gross Premium. gross premium gross premium policy value (using dirsct method and using the recursive formula)

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1 Gross Premium In this section we learn how to calculate: gross premium gross premium policy value (using dirsct method and using the recursive formula) From the ACTEX Manual: There are four types of expenses: Initial / acquisition expenses. These are the expenses incurred when a policy is issued. Such as: cost of underwriting, clerical expenses for setting up records, medical check-up fees, commissions and advertising expenses. Maintenance / renewal expenses. After a policy is issued, there are continuing administrative expenses and commissions. For example, there is money spent on collecting premium payments and keeping track of all the accounting records. Such expenses are normally incurred each time a premium is payable. Settlement / termination expenses. Upon the termination of the policy, there are some paperwork and costs to finalize and disburse benefit payments. Occasionally, a claim investigation is required and incurs a cost. Usually, settlement expenses are insignificant when compared to acquisition and renewal expenses. Other miscellaneous expenses. There are some other expenses that may or may not be directly related to the policies but are loaded to the policies for pricing. Examples includes general expenses spent on research, legal services, taxes, licenses and other fees. 1

2 Note. The gross premium is denoted by G. Note. In calculating gross premium reserves, remember that the settlement expense s effect is at the end of period (like benefits), while the renewal expense s effect is at the beginning of period (like premiums). So, at each moment you are calculating a gross premium reserve, the settlement expense at that time does not count while the renewal expense counts (see the examples below). Example (From Dr Zhou s note). Consider a 4-year annual level premium special endowment insurance issued to a life aged 40. The death benefit is 1000 payable at the end of y.o.d., and the survival benefit is 1200 payable at the end of 4 years. The premium basis: (i) Mortality: q 40 = 0.08, q 41 = 0.09, q 42 = 0.1, q 43 = 0.11 (ii) Expenses: Initial and renewal (payable at the beginning of each policy year): Year Per policy % of premium 1st 10 25% renewal 2 5% Settlement: $5 payable at the end of year for the first 3 years and 0 afterwords. (iii) Interest rate: 6% annual effective. Calculate the gross premium G and the net benefit premium P. Solution. Step 1. ä 40:4 = 1 + v 1 p 40 + v 2 2p 40 + v 3 3p 40 = (0.92)(0.91) () 2 + (0.92)(0.91)(0.90) () 3 = E 40 = v 4 4p 40 = (0.92)(0.91)(0.90)(0.89) () 4 =

3 } { ( ) 0.06 } A 1 40:4 = A 40:4 4E 40 = {1 dä 40:4 4 E 40 = 1 ( ) = A 1 40:3 = v( 0 q 40 ) + v 2 ( 1 q 40 ) + v 3 ( 2 q 40 ) = (0.92)(0.09) () 2 + (0.92)(0.91)(0.1) () 3 = Let P be the net level premium. Then: P = 1000A1 40: E 40 ä 40:4 = Next: We now calculate the gross premium: The EPV of the initial and renewal expenses is: 3

4 G + ( G)a 40:3 = G + ( G)ä 40:4 The EPV of the settlement expense is 5A 1 40:3. The EPV of death and survival benefits is 1000A 1 40: E 40. The sum of these outgoes for the insurance company is: 1000A 1 40: E 40 +5A1 40: G + ( G)ä 40:4 We must equate this with the premium income Gä 40:4 Gä 40:4 =1000A 1 40: E A 1 40: G + ( G)ä 40:4 G = 1000A1 40: E A 1 40: G + ( G)ä 40:4 ä 40:4 =

5 Gross Premium Policy Values The gross premium reserve at time t is defined to be: tv g = E(L g t T x > t) So: tv g = EPV at time t of future benefits and expenses EPV at time t of future gross premiums Example (From Dr Zhou s note). In the previous example, find the gross premium policy values 2 V and 3 V. Solution. 2V = 3V = [ (1000) ( G) ( (0.9)(0.11) + (1000) + (1200) (0.9)(0.89) () ) ( 2 G ) = () 2 ] + (5) 0.1 [ (1000) 0.11 ] + (1200) G G = income outgo Example. Here we solved question 35.4 of practice-questions-set-5. Example. Here we solved question 35.3 of practice-questions-set-5. I have posted the details of the solution through another file. 5

6 Recursive formula for gross premium policy value Let us recall the recursive formula for net premium policy value: For the gross premium policy value, it becomes: ( t V + P t )(1 + i) = b t+1 q x+t + t+1 Vp x+t ( t V + G t e t )(1 + i) = (b t+1 + E t+1 )q x+t + t+1 Vp x+t where e t is the sum of expenses paid at the beginning of the period [x + t, x + t + 1] (such as the renewal expense), and E t+1 is the settlement expense, which is actually paid at the end of the year [x + t, x + t + 1] contingent on death. Example. Use the recursive formula to calculate the policy value 3 V g in the previous example. Solution. We have 3 V g = ( 3 V + G 3 e 3 )(1 + i) = (b 4 + E 4 )q x Vp x+3 ( 3 V + G ( G))() = ( )(0.11) + (1200)(0.89) ( 3 V G 2)() = ( )(0.11) + (1200)(0.89) substitute the value G = to get 3V = Example (Dr Zhou s note). For a fully discrete 20-year term insurance of $100,000 on (35), you are given: (i) Annual gross premiums are $200 (ii) Mortality: q 53 = nd q 54 =

7 (iii) i = 0.05 (iv) Annual expenses are 5% of premium plus 25, paid at the beginning of the year. (v) Settlement expenses are 100. Calculate the gross premium reserve at the end of year 18. Solution. Although this problem can be solved directly, however we attempt using the recursive formula. The sum of annual expenses each year is: (0.05)(200) + 25 = 35 So, we have the following picture: ( 19 V + G e)(1 + i) = (b 20 + E 20 )(q x+19 ) + (p x+19 )( 20 V) ( 19 V + G e)(1 + i) = ( )(q x+19 ) + (p x+19 )( 20 V) ( 19 V )(1.05) = (0.007) +(0.993) (0) 19 V = q 54 1 q 54 7

8 Next step: ( 18 V + G e)(1 + i) = (b 19 + E 19 )(q x+18 ) + (p x+18 )( 19 V) ( 18 V )(1.05) = (0.006) +(0.994) ( ) 18 V = q 53 1 q 53 8