Provide the correct answers.

Consider the single index model of investment returns in which for any security i : Riza + AR. +6 where E( )=0. E(6,E;)=0 for 1 # j, E(R., 6,)=0 and R., is the return on the market. 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 a securities, with a proportion x, invested in security i. (3] (ii) Show that Cap = ) x, Gy . where Gap and Gy are the covariance of investment returns between security / and portfolio P and securities / and / respectively. [2] (iii) State a general expression for the variance of of portfolio P in terms of the covariances Cy [1] (iv) Use your results from (ii) and (ili) to show that: Bip= ar, Op where Bo =-if and comment briefly on this result. [7] [Total 13]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 61 =T. It is assumed that, at each node in the lattice, the share price is equally likely to increase by a factor a or decrease by a factor d, where u meditodd and dmed-ofs The movements at each step are assumed to be independent. (D) Show that, if the share price makes a total of X, "up jumps", the share price at time 7 will be: Sy = Soexpo AT +OUT where So denotes the initial share price. [4] (ib) Write down the distribution of X", and state how this distribution can be approximated when a is large. [2] (iii) Hence determine the asymptotic distribution of ST for large n. So [4] [Total 10]