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1 A portfolio P consists of n assets, with a proportion x, invested in asset i , i =1,2,....n (so style that Ex, = 1). i1 The annual returns Ap 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) = Pp var(RM) + var(=p) where cp denotes the component of the portfolio return that is independent of movements in the market. [3] (ii) Explain why the specific risk, var(ep), is sometimes referred to as the 'diversifiable risk', giving an algebraic justification for your answer. [4] (iii) Discuss the following statement: 'A portfolio with a beta of zero is equivalent to a risk-free asset.' [2] [Total 9] 2 Show that in the single-index model of asset returns: style E, = a + BEM V, = B/ VM + Vei and Ci = B;B;VM where V is the variance of e, - [8] 3 Consider the data in the table below, which relates to Securities 1, 2 and 3. Security 1 2 3 0.0 2.0 -2.2 1.1 D.6 2.0 Vai 2.2 1.3 1.2 You are given that: the expected return and standard deviation of the market return are 10 and 5 respectively the returns of each security can be modelled using an appropriate single-index model.(i) Calculate: (a) the expected return and standard deviation of return for each security (b) the covariance of returns between each pair of securities. (ii) Consider a portfolio which consists of Securities 1, 2 and 3 in equal proportions. Calculate: (a) the variance of the portfolio (b) the systematic risk of the portfolio (c) the specific risk of the portfolio. .4 Distinguish between the three main classes of multifactor model. 5 Consider the single-index model of investment returns in which for any security / : style R; = a; + /;Rim + ; where E()=0, E(6 6;)=0 for i # j, E(Rms;)=0 and Rm is the return on the market. (1) Assuming that this model applies, derive expressions for the mean investment return on security / , and the mean investment return on a portfolio P, containing n securities, with a proportion x, invested in security i . [3] (ii) Show that Cip = _x/Cy , where Ge and Cy are the covariance of investment returns /-1 between security / and portfolio P and securities / and / respectively. [2] (iii) State a general expression for the variance op of portfolio P in terms of the covariances Cij [1] (iv) Use your results from (ii) and (iii) to show that: Oop 1 Bip =- Ox; Up where Ap = - "if and comment briefly on this result. [7] [Total 13]