Solve the following questions clearly explaining your answers.

A Markov chain X _has a discrete state space S. The initial probability distribution is given by P X, =1 =q;. The one-step transition probabilities are denoted by P[Xm+1 = im+| |X'm =im| = Pimm+1 (m,m+1) (i) State the Markov property for such a process. [1] (ii) Write down expressions for the following in terms of p's and q's. (a) PXO = 1, X1=,...,X, = in (b) P[X4 = i ] [2] [Total 3]A moving average (stochastic) process, X,, has a discrete time domain and a continuous state space and is defined as: where {Z,, neZ) are independent and identically distributed N (0,o?) random variables and o , 2, ; are constants. (i) Prove that X,, is weakly stationary. [5] (ii) Explain whether the Markov property holds. [2] (iii) Deduce whether the process has independent increments. [1] [Total 8]13 An unquoted company wishes to raise a large loan in order to invest in a major new project. The finance director has proposed issuing bonds that have a significant number of warrants attached. Describe the advantages and disadvantages to the company's existing shareholders of attaching warrants to the bond issue. [5] 14 Describe the advantages of raising additional equity finance by means of a rights issue. [5] 15 A small company engaged a firm of consultants to evaluate a very complicated investment opportunity. The consultancy devised a Monte Carlo simulation and ran a very large number of iterations. They discovered that the project generated a positive net present value for 85% of the simulations. The directors of the company believe that this result is sufficient for them to justify investing in the project. Explain how the directors should go about interpreting the results of this simulation before making a final decision on the project. [5] 16 Describe the purpose of a cash flow statement. [5]