Help o!!! Correct solns only

1. (i) Define the term Standard Contribution Rate. [1]

A company sponsors a defined benefit pension scheme that provides a pension at

age 65 of 1/60th of pensionable salary for each year of service. Pensions in payment

increase in line with price inflation, and have an attaching 50% spouse's pension.

(ii) Estimate the Standard Contribution Rate using the Projected Unit Method for

a 40 year old male joining the scheme, stating any assumptions you make. [5]

The company wishes to reduce the ongoing cost of providing pension benefits through

the defined benefit scheme.

(iii) Suggest six distinct ways of changing the scheme benefits such that the

Standard Contribution Rate calculated in part (ii) reduces. [3]

(iv) Without performing any calculations, comment on the effectiveness of each of

the changes suggested in part (iii). [3]

(v) Suggest other ways the company could reduce its ongoing pension costs

2

Let X, =a+bt + Y,, where Y, is a stationary time series, and a and b are fixed non- zero constants. Show that X, is not stationary. [2] Let AX, = X, - X,. (ii) Show that AX, is stationary. [1] (iii) Determine the autocovariance values of AX, in terms of those of Y- [4] Now assume that Y, is an MA(1 ) process, i.e. Y,= &, + PE,_ (iv) Set out an equation for AX, in terms of b, B , &, and L, the lag operator. [1] (v) Show that AX, has a variance larger than that of Y. [4]Consider a one-period binomial tree model for the stock price process S, Let So = $100 and assume that in three months' time the stock price is either $125 or $105. No dividends are payable on this stock. Assume also that the continuously compounded risk-free rate is 5% per annum. (i) Verify that this market is not arbitrage-free by considering the relationship between the risk-free rate and the stock price movements. [2] (ii) (a) Identify a portfolio which would generate an arbitrage profit. (b) Calculate this profit. [4] Now assume that the continuously compounded risk-free rate is 20% per annum. Consider a European put option on this stock, expiring in three months' time and with strike price K = $120. (iii) Calculate the current price of this put option. 131(i) State the Cameron-Martin-Girsanov theorem. [3] (ii) State an important property of the discounted value of a security price process under the risk-neutral measure. [1] The price process S, of a traded security satisfies the following stochastic differential equation: dS, =US,di + GS,dw,, where W, is a standard Brownian motion under the real-world probability measure, and u and G are constants, with o > 0. Let r > 0 be the continuously compounded risk-free rate of interest. (iii) Show, using parts (i) and (ii), that W, + 2.r is a Brownian motion under the risk-neutral probability measure, ifa = (H-) [3] (iv) Calculate the value of 2. in the case in which u = 0.04 + r and o = 0.4. [1] Another traded asset has a price process satisfying the stochastic differential equation dA, = (0.06 +r)A,di + YA,dW,. (v) Determine the value of the volatility coefficient y, using your result from part (iv). [2]