I need help with the following questions

1 Explain what is meant by the following terms, in the context of mean-variance portfolio theory: efficient frontier (ii) indifference curves (iii) optimal portfolio. .2 An investor can invest in only two risky assets A and B. Asset A has an expected rate of return of 10% and a standard deviation of return of 20%. Asset B has an expected rate of return of 15% and a standard deviation of return of 30%. The correlation coefficient between the returns of Asset A and the returns of Asset B is 0.6. Calculate the expected rate of return if 20% of an investor's wealth is invested in Asset A and the remainder is invested in Asset B. (ii) Calculate the standard deviation of return on the portfolio if 20% of an investor's wealth is invested in Asset A and the remainder is invested in Asset B. (iii) Explain why an investor who invests 20% of his wealth in Asset A and the remainder in Asset B is risk-averse. 3 Using mean-variance portfolio theory, prove that the efficient frontier becomes a straight line in the presence of a risk-free asset. 4 Consider a portfolio, P, which consists of / assets held in equal proportions. Let Ap represent the return on the portfolio, and let R; represent the return on asset / . The covariance of the return on asset / with that on asset j is Cf. State the total number of data items needed to calculate [Rp] and Var(RP). (ii) Write down an expression for Var(Rp). (iii) Using your expression from part (ii), show that the specific risk of the portfolio (ie the risk associated with the individual assets) tends to zero in a well-diversified portfolio..5 Consider two independent assets, Asset A and Asset B, with expected returns of 6% po and style 11% po and standard deviations of returns of 5% po and 10% po, respectively. Let x; denote the proportion of the portfolio invested in Asset / . (i) If only Assets A and B are available, determine the equation of the efficient frontier in expected return-standard deviation space. [3] A third Asset, Asset C, is risk-free and has an expected return of 4% pa. A Lagrangian function is to be used to calculate the equation of the new efficient frontier. (ii) Write down, but do not solve, the five simultaneous equations that result from this procedure. [3] (iii) Use your simultaneous equations to derive the relationship between x, and x on the new efficient frontier. [2] (iv) Hence derive the equation of the new efficient frontier in expected return-standard deviation space. [4] [Total 12] .6 Consider a world in which there are only 2 securities, 1 and 2, such that: style 61 = 5%, V1 = (10%)2 E2 = 10%, V2 = (20%)2 Let p denote the correlation coefficient between the returns yielded by the two securities. (i) Derive the equation of the opportunity set in E-V space. [5] (ii) Derive expressions for the portfolio expected return E and the portfolio proportion *1 invested in Security 1 at the point of global minimum variance and hence comment briefly on how E and x1 vary with p. [5] [Total 10] 7 (i) Describe in detail the assumptions underlying the use of mean-variance portfolio theory. [3] style Consider a two-security world in which the returns yielded by Assets 1 and 2 are perfectly positively correlated, though they have different expected returns. (ii) Using the method of Lagrangian multipliers or otherwise, derive the equation of the efficient frontier in expected return-standard deviation space. [6] (iii) Use your answer to part (ii) to: (a) determine the gradient of the efficient frontier (b) show that the efficient frontier is a straight line in expected return-standard deviation space that passes through the points representing Assets 1 and 2. [4] [Total 13]