Kindly!
1.
2.Model A is a stationary AR(1) model, which follows the equation:y = ? +? +? ?
where ?t is a standard white noise process.
(i) State two approaches for estimating the parameters in Model A. [2]
Mary, an actuarial student, wishes to revise Model A such that the error terms ?t
no
longer follow a Normal distribution.
(ii) Explain which of the approaches in part (i) she should now use for parameter
estimation. [2]
(iii) Propose a method by which Mary will be able to calculate estimates of the
parameters ? and ?2, with reference to any relevant equations. [3]
Mary, has now constructed Model B. She has done this by multiplying both sides of
the equation above by (1 ? cB), where B is the backshift operator, so that Model B
follows the equation:y= ? +? ? cB y cB ? .
(iv) Explain why Model A and Model B are identical. [2]
(v) Explain for which values of c Model B is stationary.
3.
(i) Outline the benefits that can be provided to members of a defined contribution pension scheme. [4] (ii) Compare the financial risks of a defined contribution pension scheme with those of a defined benefit pension scheme for: (a) a member and (b ) an employer. [7] A company sponsoring a non-contributory defined benefit pension scheme has decided to amend the scheme to provide benefits on a defined contribution basis for future service. The benefits payable for service prior to the date of amendment are set out below: Normal Pension Age (NPA) 60 Accrual rate 1/60ths of Final Pensionable Salary for each year of pensionable service Commutation at NPA Maximum of 20% of member's pension with each fl p.a. of pension providing f12 of cash Final Pensionable Salary Average of basic pay at 1 April for the three year period prior to leaving Pension increases in payment 3% p.a. Spouse's pension on death before NPA 1/3"d of basic pay on 1 April prior to death Lump sum on death before NPA 5 times basic pay on 1 April prior to death Spouse's pension on death after NPA 50% of member's pension at date of death An extract of the minutes of the company board meeting where the decision was taken is as follows: "The company's objective is to reduce the financial risks associated with the current pension scheme, and to offer a replacement scheme having the same expected long term cost for future service." (iii) Calculate the rate (as a percentage of basic pay) at which the company will need to contribute to the defined contribution scheme in order to try to achieve this objective. State the assumptions you have made and show your workings. [8] (iv) Discuss why the cost of the new pension benefits might differ from the cost of the current scheme, assuming that the company pays contributions at the rate calculated in part (iii). [6]Consider a non-dividend-paying security with price S, at time . The security price follows the stochastic differential equation: dS, = S,(udt + odZ) where: . Z, is a standard Brownian motion H = 16% per annum 6 = 25% per annum * is the time from now measured in years So = 1 (i) Derive the distribution of S,- [4] An investor has taken out a house loan, with a repayment of $100,000 due in six years' time. (ii) Determine the amount that the investor would need to invest in the security to give a 75% probability of having an investment value of at least f100,000 in six years' time. [4] The investor only has $50,000 available, which he invests in this security at time 1= 0. (iii) Calculate the following risk measures applied to the difference between the value of the security and f100,000 at time ( = 6: (a) 90% Value at Risk relative to f100,000 ( b ) expected shortfall or surplus relative to f100,000 [5] (iv) Comment on the implications for the investor of your answers to part (iii). [2] (v) Suggest two changes that the investor might therefore make to his portfolio
