Kindly assist in solving the questions given below. Thank you

(i) State Wilkie's updating equation in respect of the force of inflation and explain carefully what each of the components of the equation represents. State also which type of time series process the equation is an example of. [5] (ii) Show that the total return on an equity from time : to ? +1 is given by: YO 1 Y ( + 1 ) +1 exp[K (1+1)] where Y() is the equity dividend yield at the end of year z and K(t) is the force of dividend growth during year I. [3] (iii) If /() is the force of inflation during year ? and R() is the real yield on perpetual index-linked bonds at the end of year , show that the risk premium on equities in relation to index-linked bonds is equal to: log + E log -log +1 +K(+1)-1(+1)|U(0) R(D) R(+1) where U() is the state variable vector for the model concerned. [3] [Total 11]A company is analysing the number of accidents that occur each year on the factory floor. It believes that the number of accidents per year / has a geometric distribution with parameter 0.8, so that: P(N = n) =0.8x0.2", n=0,1,2. ... For each accident, the number of employees injured is Y', where Y = X +1, and X is believed to have a Poisson distribution with parameter 2.2. The company has taken out an insurance policy, which provides a benefit of fl,(00 to each injured employee, up to a maximum of three employees per accident. irrespective of the level of injury. There is no limit on the number of accidents that may be claimed for in a year. (1) Show that E(S) -0.634 and var(S) -2.125, where S is the total number of employees claiming benefit in a year under this policy. [7] (ii) Hence find the mean and variance of the aggregate amount paid out under this policy in a year. [Total 8]