Answer the questions below with explanations.thanks

Question 5. 10 Consider the single index model of investment returns in which for any security i: R!- =aE+Rm+ei where E(El-)={l, Enter-39:0 for re}, E(RmEI-)={l and Rm is the return on the market. '3) (ii) (iii) (ix) Assuming that this model applies, derive expressions for the mean investment return on security i, and the mean investment return on a portfolio P, containing rr securities, with a proportion x1- invested in security i . [3] H Show that Cip- = ijCH , where Cg: and (3,; are the covariance of investment .=1 returns benveen security i' and portfolio P and securities 1' and j respectively. [2] State a general expression for the variance or}, of portfolio P in terms of the covariances Cg . [1] Use your results from (ii) and (iii) to show that: do 1 a- = 3P x!- 5P Cfp . . where p = and comment briey on this result. [7'] 0% Question 5.8 A binomial lattice is used to model the price of a non-dividend-paying share up to time 7. The interval (0, 7) is subdivided into a large number of intervals of length St = T . It is assumed that, at each node in the lattice, the share price is equally likely to increase by a factor u or decrease by a factor d, where u = lotto for and d = Hot-o of The movements at each step are assumed to be independent. (i) Show that, if the share price makes a total of X", "up jumps", the share price at time 7 will be: ST = SOexp MT +OVT 2X , -2 ) where So denotes the initial share price. [4] (1i) Write down the distribution of X, and state how this distribution can be approximated when n is large. [2] (iii) Hence determine the asymptotic distribution of - for large n . [4] So [Total 10]