Help in solving these questions.

Question (i) A portfolio P consists of n assets, with a proportion x, invested in asset i , i = 1,2,...." (so that x, =1). Derive a formula for the portfolio beta, Sp, in i=1 terms of the individual betas for each asset. [2] (ii) The annual returns Ro on this portfolio can be assumed to conform to the single-index model of asset returns. Write down an equation defining this model and show that: var( Rp) = B; var( Ry ) + var(Ep) where Ep denotes the component of the portfolio return that is independent of movements in the market. [3] (iii) Explain why the specific risk var(ap) is sometimes referred to as the "diversifiable risk", giving an algebraic justification for your answer. [4] (iv) Discuss the following statement: "A portfolio with a beta of zero is equivalent to a risk-free asset." [2]Show that the variance of portfolio returns can be written as: Vp X + M i=1 and use this expression to show that: the contribution of the specific risk on each security to the total risk of the portfolio becomes very small as the number of securities increases and the contribution of each security to the portfolio's total risk is only the systematic risk of that security, ie Op BOOM = OM x; B; , as N - 0