Answer the questions below

Contributions to a pension scheme by employees are made at a rate of 5% of salary when aged under 35, 6% between ages 35 and 45, and 75% when aged 45 or over. Calculate the present value of the future contributions payable by a member aged exactly 30 who in the past year has received a total salary of f12,718. A company pension scheme provides the following benefits for all members: 1) a pension on retirement (on grounds of ill-health or of age) of one-eightieth of final pen- sionable salary for each year of service (including fractions), (2) a lump sum on retirement of 3 times the annual pension, (3) on death in service, a lump sum of f30,000, (4) on withdrawal from service, a return of the employee's contributions, accumulated at 3% per annum compound. Final pensionable salary is defined 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 benefit (1) above in respect of a new entrant aged a with annual salary rate SAL. (You need not define the service table functions.) (b) In respect of a new entrant aged a with annual salary rate SAL, give formulae for valuing benefits (2), (3) and (4) above, using suitable commutation functions. (You need not derive the formulae.) (c) Hence find a formula for the employer's contribution rate for a new member aged r and a starting salary rate of f10,000 p.a. (ii)(a) Using the basis given in the pension fund section of the Formulae and Tables (and the supplement), find the value of each of the benefits (1), (2), (3) and (4) for a new entrant aged 45 with salary rate f10,000 per annum. (b) Hence or otherwise find the employer's contribution rate for this new member. 3 The pension scheme of a certain company provides an annual pension on retirement (for '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 member dying in service there is payable at the time of death a lump sum of 30,000. There is no benefit 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 benefits for a new entrant to the fund at age 30 with an initial salary rate of f10,000 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 recordedExercise 9.1 In order to value the benefits in a final salary pension scheme as at 1 January 2008, a salary scale, $r, has been defined so that $ 41/$, is the ratio of a member's total earnings between ages x + t and x + 1 + 1 to the member's total earnings between ages x and x + 1. Salary increases take place on 1 July every year. One member, whose date of birth is 1 April 1961, has an annual salary rate of $75 000 on the valuation date. Using the salary scale in Table 9.1, estimate the member's expected earnings during 2008. Exercise 9.2 Assume the salary scale given in Table 9.1 and a valuation date of 1 January. (a) A plan memberaged 35 at valuation received $75 000 in salary in the year to the valuation date. Given that final average salary is defined as the average salary in the four years before retirement, calculate the member's expected final average salary assuming retirement at age 60. (b) A plan member aged 55 at valuation was paid salary at a rate of $100 000 per year at the valuation date. Salaries are increased on average half-way through each year. Calculate the expected average salary earned in the two years before retirement at age 65.Exercise 9.3 A pension plan member is aged 55. One of the plan benefits is a death in service benefit payable on death before age 60. (a) Calculate the probability that the employee dies in service before age 60. (b) Assuming that the death in service benefit is $200 000, and assuming that the death benefit is paid immediately on death, calculate the EPV at age 55 of the death in service benefit. (c) Now assume that the death in service benefit is twice the annual salary rate at death. At age 55 the member's salary rate is $85 000 per year. Assuming that deaths occur evenly throughout the year, estimate the EPV of the death in service benefit. Basis: Service table from Table 9.2. Interest rate 6% per year effective. Salary scale follows Table 9.1; all salary increases occur half-way through the year of age, on average. Exercise 9.4 A new member aged 35 exact, expecting to earn $40 000 in the next 12 months, has just joined a pension plan. The plan provides a pension on age retirement of 1/60th of final pensionable salary for each year of service, with fractions counting proportionately, payable monthly in advance for life. There are no spousal benefits. Final pensionable salary is defined as the average salary over the three years prior to retirement. Members contribute a percentage of salary, the rate depend- ing on age. Those under age 50 contribute 4% and those aged 50 and over contribute 5%. The employer contributes a constant multiple of members' contributions to meet exactly the expected cost of pension benefits. Calculate the multiple needed to meet this new member's age retirement benefits. Assume all contri- butions are paid exactly half-way through the year of age in which they are paid. Basis: Service Table: from Table 9.2 Survival after retirement: Standard Ultimate Survival Model Interest: 4% per year effective Exercise 9.5 (a) Anew employee aged 25 joins a DC pension plan. Her starting salary is $40 000 per year. Her salary is assumed to increase continuously at a rate of 7% per year for the first 20 years of her career and 4% per year for the following 15 years. At retirement she is to receive a pension payable monthly in advance, guaranteed for 10 years. She plans to retire at age 60, and she wishes to achieve a replacement ratio of 70% through the pension plan. Using the