Consider the following two-period (today and tomorrow) economy. A representative investor has an initial wealth of...

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Consider the following two-period (today and tomorrow) economy. A representative investor has an initial wealth of Wo to be allocated across two risky assets, A and B. Today's prices for one unit of these risky assets are, respectively, PAO and PBO. Asser process tomorrow (PAI and PB₁) are random variables, possibly correlated. Assuming all wealth is invested in these risky assets, we have that Wo = QAPAO + QBPB0, Where QA and Qв are the number of units of each risky asset purchased. (a) Assuming that no dividends or coupons are paid, express the expected rate of return, E(y) and the associated risk, S(y) (as measured by the standard deviation of returns), of any portfolio of the two risky assets formed today, as a functions of o, the portfolio share of asset A, and the expected returns for each asset, their variance and covariance. (b) Define, in words, the minimum variance portfolio. Derive the expression for the minimum variance portfolio share of asset A and denote it by ov. (c) Assuming that short selling is not allowed, give conditions on return variances and return correlations such that a minimum variance portfolio (with strictly positive ov) exists. What is the efficient frontier? Is a positive minimum variance portfolio always part of the efficient frontier? Explain. (d) Now introduce an additional, risk-free, asset in this economy. For each unit purchased today, this asset yields a return gf with certainty tomorrow. Further, assume the representative investor has utility function U, which is increasing in expected returns and decreasing in the variance of returns. Explain how the optimal portfolio consisting of the two risky assets and the risk-free asset is determined. Illustrate your argument graphically, using risk, S(y), on the x-axis and expected return, E(y), on the y-axis. Consider the following two-period (today and tomorrow) economy. A representative investor has an initial wealth of Wo to be allocated across two risky assets, A and B. Today's prices for one unit of these risky assets are, respectively, PAO and PBO. Asser process tomorrow (PAI and PB₁) are random variables, possibly correlated. Assuming all wealth is invested in these risky assets, we have that Wo = QAPAO + QBPB0, Where QA and Qв are the number of units of each risky asset purchased. (a) Assuming that no dividends or coupons are paid, express the expected rate of return, E(y) and the associated risk, S(y) (as measured by the standard deviation of returns), of any portfolio of the two risky assets formed today, as a functions of o, the portfolio share of asset A, and the expected returns for each asset, their variance and covariance. (b) Define, in words, the minimum variance portfolio. Derive the expression for the minimum variance portfolio share of asset A and denote it by ov. (c) Assuming that short selling is not allowed, give conditions on return variances and return correlations such that a minimum variance portfolio (with strictly positive ov) exists. What is the efficient frontier? Is a positive minimum variance portfolio always part of the efficient frontier? Explain. (d) Now introduce an additional, risk-free, asset in this economy. For each unit purchased today, this asset yields a return gf with certainty tomorrow. Further, assume the representative investor has utility function U, which is increasing in expected returns and decreasing in the variance of returns. Explain how the optimal portfolio consisting of the two risky assets and the risk-free asset is determined. Illustrate your argument graphically, using risk, S(y), on the x-axis and expected return, E(y), on the y-axis.

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a To express the expected rate of return Ey and the associated risk Sy of a portfolio of two risky assets A and B formed today as a function of the po... View the full answer