Problem 3 (Optimal portfolio for two risky assets). Recall our portfolio has a weight of w on stock, and 1w on bonds. 1. MVP or minimal variance portfolio is the one with minimal variance (identify it on the efficient frontier). Show that the weight of MVP is wMVP=Var(rSrB)B(BS) 2. We want to achieve a portfolio that has a smaller variance than either stocks or bonds, without shorting any of them, i.e. w[0,1]. Show that we can only achieve this, if the correlation satisfies 1min{SB,BS} Can we reduce variance by diversification when correlation is too high? 1 3. Let r be the return of our portfolio. We want to solve the mean-variance optimization problem of minwE[r]Var(r) Show that the solution is w=wMVP+2Var(rSrB)SB Explain what kind of change in mean return and risk-aversion parameter would let the optimal solution be more towards stocks. Hint: Notice that wMVP is not affected by and at all. Problem 3 (Optimal portfolio for two risky assets). Recall our portfolio has a weight of w on stock, and 1w on bonds. 1. MVP or minimal variance portfolio is the one with minimal variance (identify it on the efficient frontier). Show that the weight of MVP is wMVP=Var(rSrB)B(BS) 2. We want to achieve a portfolio that has a smaller variance than either stocks or bonds, without shorting any of them, i.e. w[0,1]. Show that we can only achieve this, if the correlation satisfies 1min{SB,BS} Can we reduce variance by diversification when correlation is too high? 1 3. Let r be the return of our portfolio. We want to solve the mean-variance optimization problem of minwE[r]Var(r) Show that the solution is w=wMVP+2Var(rSrB)SB Explain what kind of change in mean return and risk-aversion parameter would let the optimal solution be more towards stocks. Hint: Notice that wMVP is not affected by and at all