You are given below questions about applied microeconomics,answered them

.1 Contributions to a pension scheme by employees are made at a rate of 5% of salary when ages under 35, 6% between ages 35 and 45, and 7%% when aged 45 or over. Calculate the present value of the future contributions payable by a member aged exactly 30 who in the past yea: has received a total salary of 12,718. .2 A company pension scheme provides the following benets for all members: {1} a pension on retirement {on mounds of illhealth or of age] of oneeightieth of nal pen- sionable salary for each year of service {including fractions}, {2} alump sum on retirement of 3 times the annual pension, {3} on death in service, a lump sum of 3,, {4} on withdrawal from service, a return of the employee's contributions, accumulated at 35% per annum compound. Final pensionable salary is dened as the average annual salary in the three years immediately before retirement. All members who reach age '65 retire immediately. Employees contribute to the scheme at the rate of 3% of salary, payable continuously. Salaries are revised continuously. The employer's contribution rate is assessed for each member sep arately, and is such that the prospective reserve for each new entrant is zero. Expenses are ignored. {i} {a} Derive a formula, in terms of suitable commutation functions, for valuing benet {1] above in respect of a new entrant aged 1 with annual salary rate SAL. {You need not dene the service table functions.) {b} In respect of a new entrant aged 3 with annual salary rate SAL, give formulae for valuing benets {2], {3} and {4] above, using suitable commutation functions. {You need not derive the formulae.) {c} Hence nd a formula for the employer's contribution rate for a new member aged 3 and 2 starting salary rate of 113,000 p.a. {ii}{a]l Using the basis given in the pension fund section of the Formulae and Tables {and the supplement), nd the value of each of the benets {1), {2], {3} and {4) for a new entrant aged 45 with salary rate 1,{) per annum. {b} Hence or otherwise nd the employer's contribution rate for this new member. .3 The pension scheme of a certain company provides an annual pension on retirement {fox 'age' or 'ill health" reasons} of amount equal to one per cent of the member's total earnings throughout his service. The pension is payable weekly. In addition, in the event of a membei dying in service there is payable at the time of death a lump sum of 30,000. There is ne benet on withdrawal. The company pays a constant percentage of all the members" salaries into the pension fund. The percentage is that which will exactly cover the cost of benets for a new entrant to the fund at age 30 with an initial salary rate of 10,013] per annum. Contributions are payable continuously, and the employees do not contribute to the scheme. Expenses are negligible. {a} Calculate the contribution rate paid by the company, assuming the last retirement age is 65. {b} A valuation of the fund is to be conducted. For each active member of the scheme there is recorded (i) the age nearest birthday (which is regarded as the member's exact age) at the valuation date, (ii) the annual salary rate at the valuation date, and (iii) the total past earnings in service (prior to the valuation date.) For each age, the totals of (ii) and (iii) are recorded and the following is an extract from the data. Age r No. of members Total past earnings Total of annual salary aged r for members aged r rates for members aged a 25 11 302,100 70,100 Assuming that the basis of the Tables provided is appropriate, find the liability at the valuation date for the benefits payable to the members aged 25, and determine whether the future contributions payable in respect of these members are more or less than sufficient to cover the benefits. A pension scheme provides each member who retires (for any reason) with annual pension equal to 60 x final salary per year of service. Final salary is the average income over the las 3 years of service, and fractions of a year of service are not included when calculating the pension. Assuming that equal contributions are payable by the member and his employer, that in the event of death in service a benefit is payable equal to the return without interest of both the member's and the employer's contributions, and that in the event of withdrawal from service a return without interest is made of the member's contributions, calculate the appropriate contribution rate payable by both the member and his employer in respect of a new entrant aged 40. You are consulting actuary to a small pension scheme, which has just been established. You have decided to use the pension fund tables in Formulae and Tables for Actuarial Examinations as the basis for all calculations. The scheme provides the following benefits to employees: (i) on retirement (for ill-health reasons or otherwise), a pension of th of annual pension- able salary, averaged over the previous three years, for each year of future service including fractions; (ii) on withdrawal or death in service, a return of the member's contributions, accumulated at 3% per annum compound interest. Employees contribute 2% of salary to the scheme. Pensionable salary is defined as salary less f4000. Salaries are revised continuously, and contributions are made continuously. Details of the current membership are as follows. member age current salary rate (f) 45 30,000 45 20,000 35 10,000 35 10,000 (a) The employer has decided to contribute the proportion of total salaries which, together with the employees' contributions, will exactly pay for the benefits. Calculate the employer' contribution rate. (b) A new employee, aged 35 and with current salary rate f8,000, is about to be hired. Calculate the surplus or deficiency in the pension fund after this new member joins