Provide answers to all the questions in the attachments below.

On 1 October 2003 a pension scheme member was aged exactly 45, and had earned (40,000 over the previous year. The salary scale s, is defined such that for a life aged exactly x and for any integer 130: expected earnings between ages x + / and x + ( +1 expected earnings between ages x and x + 1 Final salary is defined as the earnings received in the year immediately prior to retirement. Salaries are increased on 1 January each year. Write down an expression for the expected final salary for this member, given that he intends to retire on 31 December following his 60th birthday.A stochastic interest rate model assumes that the annual growth factors for each future year are independently distributed lognormal random variables with parameters # =0.05 and of =0.01. Calculate the mean and standard deviation of the annual rate of return. [3] Calculate the median and the upper and lower quartiles of the distribution of the annual rates of return. [4] (iii) Comment on the relative sizes of the mean, median and mode of the annual rate of return. [2] (iv) A single payment of $10,000 must be made at the end of 5 years. Calculate the probability that a single initial investment of $7,250 will be sufficient to meet the liability for this payment. [3] (v) State, with reasons, whether the probability calculated in (iv) would be greater or smaller if the interest rate model assumed an unknown interest rate that was constant in all future years (with the same basic distribution), rather than assuming that the rates for each year are independent. [3] [Total 15] You are given that the mode of a lognormal distribution is at x = #-"Write brief notes on the following: (a) Scarcity and choice. (4 marks) (b) Diminishing marginal utility. (4 marks) (c) Price elasticity of demand. (4 marks) (d) Income elasticity of demand. (4 marks) (e) Substitution and income effects of a price change. (4 marks) (Total: 20 marks)