1. Datsenka Dog Insurance Company has developed the following mortality table for dogs:

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1 1 Datsenka Dog Insurance Company has developed the following mortality table for dogs: Age l Age l Datsenka sells an whole life annuity based on the life of a dog who is age 4 The annuity pays 1000 at the beginning of each year as long as the dog is alive The interest rate used to calculate all values is 8% Calculate the Actuarial Present Value of this annuity 2 The Ismail Insurance Company sells annuities Ismail sells a 20 year certain and life annuity to Hayqal who is (40) The annuity provides guaranteed payments of 30,000 at the beginning of each year for 20 years Thereafter payments continue as long as Hayqal is alive a Hayqal s mortality follows the Illustrative Life Table b i 6% Calculate the Actuarial Present Value of Hayqal s annuity

Actuarial Present Value of this annuity 2 The Ismail Insurance Company sells annuities Ismail sells a 20 year certain and life annuity to Hayqal who is (40) The annuity provides guaranteed payments

2 3 You are given the following mortality table: l You are also given that i 5% Calculate 100( Ia ) 90:3 4 a Mortality follows the Illustrative Life Table b Deaths are uniformly distributed between integral ages c d 10% Calculate a 90:4 5 a t p 005t e b 003 Calculate a 6 a Mortality follows the Illustrative Life Table ecept at age 80 where q b i 6% Calculate a 80 7 a a b v 095 c q d q Calculate A 62

between integral ages c d 10% Calculate a 90:4 5 a t p 005t e b 003 Calculate a 6 a Mortality follows the

3 8 Deepa, (60), is receiving a whole life annuity due with non-level monthly payments The annuity pays 1000 per month for the first 10 years Thereafter, the monthly payments are 2500 per month a Mortality follows the Illustrative Life Table b Deaths are uniformly distributed between integral ages c i 6% Calculate the Actuarial Present Value of Deepa s annuity 9 Y is the present value random variable for a whole life annuity due to (70) which pays annual payments of 100 You are given that mortality follows the Illustrative Life Table and that i 8% Calculate Pr( Y 600) 10 Y is the present value random variable for a whole life annuity with continuous payments at a rate of 500 per year to (65) c Mortality follows the Illustrative Life Table d 10% e Deaths are uniformly distributed between integral ages Calculate Pr( Y 2000) 11 Y is the present value random variable for a continuous whole life annuity to (90) which is payable at an annual rate of 1000 per year a i 6% b t 2 t p90 1 for 0 t Calculate the Pr( Y 4500)

variable for a whole life annuity due to (70) which pays annual payments of 100 You are given that mortality follows the Illustrative Life Table and that i 8% Calculate Pr( Y 600) 10 Y is the present

4 12 a 1000 A A 300 b 20 c E 400 d v 096 Calculate a :20 13 a 1000 A 500 b a c d 010 Calculate p (Note that it is an annuity immediate) 14 (8 points) Y is the present value random variable for a temporary 10 year life annuity due to (80) which pays 1000 annually a Mortality follows the Illustrative Life Table b i 6% Calculate the Var( Y) 15 Y is the present value random variable for a whole life annuity payable to () with continuous payments at an annual rate of 1 a 1000 A b A c Var[ Y] 5 d Deaths are uniformly distributed between integer ages Calculate a

follows the Illustrative Life Table b i 6% Calculate the Var( Y) 15 Y is the present value random variable for a whole life annuity payable to () with

5 16 (Written Answer Question) a Mortality follows Gompertz law such that b i 6% (11 50 )(11 t p 1) ln(11) t 50 e The Wright Life Insurance Company uses this information and calculates that: a A b 2 A Wright collects a single net benefit premium of 500,000 from Erin, (50), for a whole life annuity due with level annual payments of C The single net benefit premium is the same as the actuarial present value of the annuity A Calculate C B Calculate the variance of the present value random variable for this annuity C Use the three term Woolhouse formula to determine the monthly benefit that Erin would have received if she had purchased a whole life annuity due with monthly payments 17 (Written Answer Question) A pension plan uses continuous annuities to estimate their pension liability The plan is paying benefits at a rate of 100 per year to annuitants age 65 Let Y be the present value random variable for the continuous life annuity to (65) for this plan a Mortality follows the Illustrative Life Table b i 6% c Deaths are uniformly distributed between integer ages You are given that the EY [ ] 950 to the nearest 50 Calculate EY [ ] to the nearest 1 Calculate Var[ Y ] The pension plan has 2500 annuitants who are age 65 Using the normal distribution, calculate the range of the present value of the liability for these 2500 annuitants with a 95 th percent confidence interval

same as the actuarial present value of the annuity A Calculate C B Calculate the variance of the present value random variable for this annuity C Use the three term Woolhouse formula to determine the

6 18 You are given that mortality follows the Illustrative Life Table with i 006 Calculate a 63:12 19 i Mortality follows the Illustrative Life Table ii i 006 iii Deaths are uniformly distributed between integral ages Calculate the actuarial present value of the annuity-immediate (4) a 50:15 20 Calculate a 21 i a ii 1E 095 iii 2 E 090 i Mortality follows Calculate the Var[ Y ] l ii v 090 iii Y is the present value random variable for a 2 year temporary life annuity to (75) with annual payments of 1 at the beginning of the year

50:15 20 Calculate a 21 i a 2 145 ii 1E 095 iii 2 E 090 i Mortality follows Calculate the Var[ Y ] l 75 2000 76 1600 77 960 78 480 79 120 80 0

7 22 Calculate a 2 i a 13 ii i 004 iii q 0030 iv q i a 6 when the value is calculated using an interest rate of i ii a when the value is calculated using an interest rate of i iii a 15 when the value is calculated using an interest rate of i 0% iv 1000A 400 when the value is calculated using an interest rate of i Calculate e 1 24 A temporary life annuity immediate pays 10,000 at the end of one year if (80) is alive and 20,000 at the end of two years if (80) is alive Y is the present value random variable for this annuity i q ii q iii d 10% Calculate the Var[ Y ]

calculated using an interest rate of i Calculate e 1 24 A temporary life annuity immediate pays 10,000 at the end of one year if (80) is alive and

8 25 (Written Answer Question) A temporary life annuity immediate pays 10,000 at the end of one n year if (80) is alive and 20,000 at the end of two years if (80) is alive L 0 is the present value loss at issue random variable for this annuity using the net benefit premium i q ii q iii i 10% The annuity is purchased with a single net benefit premium payable at the beginning of the annuity a Your boss estimates the single net benefit premium to be 23,500 He is accurate to the nearest 100 Calculate the single net benefit premium to the nearest 1 b Write an epression (or epressions) for the loss at issue present value random variable Include any constraints on each epression n c Calculate the Var[ L 0]

premium payable at the beginning of the annuity a Your boss estimates the single net benefit premium to be 23,500 He is accurate to the nearest 100 Calculate the single net

9 Answers: , , ,557, a 61,578 b 102,191,932,400 c 5, a b 133,349 c (2,311,464;2,383,037) ,491, a 23,48430 b

102,191,932,400 c 5,444 17 a 93898 b 133,349 c (2,311,464;2,383,037) 18

10 L n 0 =Present Value of payments less P = 0 23,48430 if (80) dies in first year which has a probability of 004 = (10,000) v 23, ,39339 if (80) dies in second year =prob of (096)(007)= (10,000) v (20,000) v 23, ,13553 if (80) survives 2 years =(096)(093)=08928 c 40,053,900

if (80) dies in second year =prob of (096)(007)=00672 2 (10,000) v

Chapter 2. 1. You are given: 1 t. Calculate: f. Pr[ T0

Chapter 2. 1. You are given: 1 t. Calculate: f. Pr[ T0 Chapter 2 1. You are given: 1 5 t F0 ( t) 1 1,0 t 125 125 Calculate: a. S () t 0 b. Pr[ T0 t] c. Pr[ T0 t] d. S () t e. Probability that a newborn will live to age 25. f. Probability that a person age