Answer all the questions below.the questions are complete

Let 1, 2 be the modes of decrement in a double-decrement table. Suppose that 1 is

uniformly distributed over the year of age from x to x + 1 in its associated single- decrement

table, and

2

x+t = c for 0 t 1.

Find formulae for (aq)

1

x and (aq)

2

x

in terms of q

1

x and q

2

x

.

For a certain group of married women, for whom remarriage is not permitted, the dependent qtype rate of widowhood at each integer age x from 70 to 72 inclusive is twice the corresponding

dependent q-type rate of mortality. The independent rates of mortality of wives follow a(55)

ultimate (females).

(i) Using a radix of 100,000 and assuming a uniform distribution of each mode of decrement in

its associated single-decrement table, construct a double-decrement table for married women

from age 70 to age 72 inclusive, giving also the value of (al)73, the number of wives at age 73.

(ii) Find the probabilities that a wife now aged 70 will

(a) be alive and married at age 73; and

(b) be widowed within 3 years.

In a certain country, widowed and divorced men are subject to the following independent

q-type rates of decrement:

mortality: English Life Table No. 12 - Males

remarriage: rates depend on the age at, and the duration since, the end of former marriage;

the following table is an extract from these rates:

exact

age at end duration duration

of former 0 1

marriage year year

50 0.050 0.025

51 0.045 0.023

52 0.042 0.020

Calculate the probability that a man aged exactly 50 whose marriage has just ended will

remarry within 2 years.

A large industrial company recruits a constant number of school leavers aged exactly 18 years

on 1 July each year.

Upon joining, workers undergo training for one year. Of those who complete this period of

training, ten per cent fail a final test of competence and are dismissed. Employees may also

leave service voluntarily at any time. The central rate of voluntary withdrawal from service

is 0.15 for trainees and 0.10 at each age for fully trained employees.

The occupation is hazardous and all workers, including trainees, are exposed to the risk of

injury. The independent q-type rate of injury is 0.051219 at age 18 and 0.050030 at ages 19

and above. An employee who is injured is transferred to alternative work with a subsidiary

company, at a relocation cost of 1,000.

The mortality of all employees follows English Life Tables No.12 - Males. The number of

employees attaining age 21 each year is 500.

(i) Construct a service table covering the first 3 years of employment with the original company, distinguishing between those about to take the final test of competence.

a. Integrate by parts to show that A=1sfe-e1+x dt. 1+:r+t 1:). Use the expression in (a) to show that dxfdx c: t] for all x :2: '1}. Show that dxfdi = u{f]r. Show that the expressions for the variance of the present value of an nyear endowment insurance paying a unit benet, as given by [4.2.10] and (4.2.13), are identical. Let Z1 and 22 be as dened for equation (4.2.11). a. Show that limH'1 Cov(Z,, 22} = lirnM CovIIZl, 22} = D. 13. Develop an implicit equation for the term of the endowment for which Coth1r 22) is minimized. c. Develop a formula for the minimum in (b). d. Simplify the formulas in {b} and {c} for the case when the force of mortality is a constant a. Assume mortality is described by tr = 1m x for t} E x E 100 and that the force of interest is E = {1.05. a. Calculate Aia' b. Determine the actuarial present value for a 25year term insurance with benet amount for death at time L bi = 90.05:? for a person age 40 at policj.r 1ssue. Assuming De Moivre's survival function with m =. l and 1' = (1.10, calculate a. AER-m b. The variance of the present value, at policy issue, of the benet of the insurance in (a). If 3, = 0.2,.\"(1 + 0115f) and Ex = lt] x for ID 5 x 5 1m, calculate a. For a whole life insurance issued at age I, the actuarial present value and the variance of the present value of the benets 1:. (Dig. a. Show that fix is the moment generating function of T, the future lifetime of (1:), evaluated at S. b. Show that if T has a gamma distribution with parameters or and {3, then = (1 + sxsra. . Given [1, = t, l-int} = u, and at = E for all t 2:- , derive expressions for a. {IA}: = EIbTuT] b. VaeroT)