Answer the questions below.the question is complete

5 It is daired to set up a pension scheme for the group of employea dmcribed below. For each member there is recorded his exact age, his exact length of service with the company, and his annual rate of salary. The scheme will provide pensions Df lath of \"pensionable salary\" for each year of service {fractions of a year being included] and on death or withdrawal from service a return will be made of the member's contributions with 3% compound interest. All members will contribute at the same rate and the employer will contribute the same amount as each member. The contribution rate will be such as to provide exactly the benets for future service. The basis of \"Formulae and Tables\" is to be used. {a} Assuming that pensionable salary is the average annual earnings over the threeyear period ending on the retirement date, calculate the contribution rate payable by each member. {b} Calculate also the total liability for past service benets. The employer wishes to meet this liability by paying additional contributions proportional to future salary payments. At what rate should these additional contributions be made? {c} Immediately after the scheme is set up as described above the 45 year old member with 5 years of service withdraws. Is the position of the fund improved or worsened by this with drawal'? T The pension scheme of a large company provides the following benets, among others: {1} on death in service of married members: a spouse's pension of onethird of the member's annual salary at the date of death; {2] on death after normal retirement of the member at age 55 or after illhealth retirement at an earlier age: a spouse's pension of 1% of annual salary at the date of retirement for each year of scheme membership (including fractions of a year]: no pension is payable if the marriage has taken place after the member's retirement date. Assume that a service table has been constructed, and that the proportion of members mar ried at exact 11,: is by. Assume further that spousa are the same age as members, and that a unisex mortality table is used for all calculations, with the age rated up by 5 years on ill health retirement. 1You are given the age nearest birthday, current salary rate and years cf past service (including fractions} of each member. Spouse s pension is payable monthly 1n advance, beginning immediately on the death of the member. {i} Using the rate of interest 1 per annum, nd formulae for the mean present value of each J the benets (1] and {2} above for a member aged :1: (I 4:: '55]. 1You are NOT required to construct commutation functions. {ii} Suppose that the schemes rules are to be changed so that benet {2] is to be payable to any surviving spouse. Show how to modify the formulae of (i) to accommodate this change. A pension scheme only allows retirement at exact age 65. An investigation of the mortality of the retired members of the scheme was carried out over the period 1 January 2001 to 31 December 2006. The following data were obtained: Date of death Member Date of retirement (if occurred during the investigation period) 1 April 1998 30 April 2005 1 August 2000 1 February 2001 1 June 2002 31 August 2004 DO-JAVAWN 1 August 2002 31 December 2006 1 March 2004 1 May 2004 30 November 2006 1 January 2005 All months should be assumed to be of equal length. (i) Explain the form or forms of censoring that are present in these data and in this observational plan. [2] (ii) Calculate the Kaplan-Meier (product-limit) estimate of the survival function Sos (t) from these data, stating clearly any additional assumptions that you make. [10] (iii) Estimate the force of transition from alive to dead for a two-state Markov model for age 67 last birthday, using the data given. The exposed to risk should be calculated exactly. State all assumptions required, and state the age to which the estimate would be assumed to apply. [7]