Define the following:

(ii) An office offers an optional waiver of premium benefit on sickness of any duration in respect

of a 25-year with or without profits endowment assurance policy with weekly premiums payable

for 25 years or until earlier death. There is a waiting period of 12 months for the waiver of

premium benefit, and, during the second year of the policy, only half the premium (including

the extra premium for the waiver benefit) is waived. The sum assured under the endowment

policy is payable immediately on death, or on survival until the end of the term.

Using the basis given below, calculate the percentage by which the normal weekly premium

(i.e. the premium for a policy without the waiver benefit) for a life aged exactly 30 at entry

should be increased in order to provide the waiver benefit.

mortality: English Life Table No.12-Males;

sickness: Manchester Unity 1893-97, Occupation Group AHJ;

interest: 4% per annum;

expenses: 15% of each extra premium for the waiver benefit.

18.2 A policy issued by a life office to a male life aged exactly 35 is subject to level weekly premiums

ceasing at exact age 65. If the man has been sick for 6 months or more when a premium falls

due, the premium is waived. The policy provides the following benefits:

(a) on survival to exact age 65, an annuity of 5,000 per annum payable monthly in advance,

(b) on death before age 65, a return of all premiums paid (including those waived during

sickness) together with compound interest at 4% per annum to the date of death.

There is no waiting period and the off periods are the same as those underlying the tables in

Formulae and Tables for Actuarial Examinations. Calculate the weekly premium.

Basis: English Life Table No.12-Males, Manchester Unity Sickness Experience 1893-97, Occupation Group AHJ, interest 4% per annum, no expenses.

(a) Ten years ago, a man then aged exactly 30 effected an insurance policy providing sickness

benefits of 100 per week for the first six months of sickness, 50 per week for the remainder

of the first year and 30 per week thereafter, with benefit ceasing at age 60. Calculate the

weekly premium payable to age 50 on the following basis:

Mortality: English Life Table No. 12 - Males;

Sickness: Manchester Unity 1893-97, Occupation Group AHJ;

Interest: 4% per annum;

Expenses: 10% of each premium.

(b) The man now wishes to alter his policy so that premiums will in future be waived during

all periods of sickness. Calculate the revised premium payable assuming that the alteration

basis follows the premium basis above.

Note. Expenses are incurred even when premiums are waived.

a. What is the present-value function for this insurance? b. Set up a formula analogous to (4.4.2) for the actuarial present value, A5," for this insurance. c. Show algebraically that, under the assumption of a uniform distribution ( deaths over the insurance year of age, . Show, under the assrmiption of a constant force of lnortalit),r between integl'; ages, that where u,(k} = log bank. on 4.5 . a. Show that (4.2.6), for an aggregate mortality basis, can be rewritten as _ 1 m A = P le Uyyp Myldy x20. 10 '1 b. Differentiate the formula of {a} to establish (4.5.1), dx dx c. Use the same technique to show = [MID + 3151, - MI) 3: a- o. _l d3? = me) + ales + star + was as x 2 0. . Solve the differential equation (4.5.1) as follows: a. Use the integrating factor e:tp{Lr [5 + p.(z)] dz} 3!, = JED p.(x} exp{ -J: [5 + u.(z}] dz} dx. 5" to obtain b. Use the integrating factor 8"\" to obtain 3 = J: p.(x} III311 III) (II. . a. By an algebraic or probabilistic argument, verify the following backward recursion formula of an 31-year term insurance with a unit benet: n+1 an qx+n + \"35:: Axlt'l' b. Determine an appropriate starting value for use with this formula. c. Use your Illustrative Life Table with i = 0.06 to calculate the actuarial present value of a 10-year term insurance issued at ages I = 13, . . . , 130. Ala = mg, v . a. Use recursion relation (g) at the end of Section 4.3 and your Illustrative Life Table to calculate (124)23 at i = 0.06. b. Modify the recursion relation of part {a} to obtain one for (DI),, and deter- mine a starting value for it. c. Modify the recursion relation of part (b) to obtain one for (Li), and deter- mine a starting value for it. d. Make the recursion relations of parts (b) and (c) specic to the assumption of a uniform distribution of deaths over each year of age. . Use your Illustrative Life Table to verify the numerical solutions to parts (a) and (b) of Example 4.2.4. [Hint Set 3 = 0.00 and A = 0.04 in your Makeham law parameters, 1' = 6\"\" 1, and use recursion formula (d) at the end of Section 4.3. Remember that the insurance in Example 4.2.4 is payable at the moment of death.] . a. By an algebraic or probabilistic argument, verify the following backward recursion formula for the actuarial present value of a unit benet endow- ment insurance to age y with the death benet payable at the moment of death: Any-T} 2 Aer! + \"pr Awnm x = D: 1! ' * ' r If _ 1* b. Determine an appropriate starting value for use with this formula. c. Use you: Illustrative Life Table with the assumption of uniform distribu- tion of deaths over each year of age and i = 0.06 to calculate the actuarial present value of a unit benet endowment inSurance to age 65 with the death benet payable at the moment of death for issue ages x = 13, . . . , 64. d. By an algebraic or probabilistic argument, verify the following backward recursion formula for the actuarial present value of a unit benet n-year endowment insurance with the death benet payable at the moment of death: Ana = Aim + v" \"p; (1 Upg+n 3111;} + \"Pi: Ax+lil x=0,1,...,w-*1. . Let 2 be the present-value random variable for a 100,000 unit 20-year endow- ment insurance with the death benet payable at the moment of death. Use your Illustrative Life Table to calculate the mean and the variance of Z on the basis of a Makeham law with A = 0.001, B = 0.00001, c = 1.10, and E = 0.05