Financial Mathematics question

Extra: Risk-adjusted return. Define Derive the following formulas (see the class slides for the definitions of the notation). 1. Prove that ar2-28r + > 0 for any r. 2. For the efficient frontier with all risky assets: 3. For minimum variance portfolio: f/o-/Va. 4. When the riskless asset is available, the risk-adjusted excess return (the Sharpe ratio) is Efficient frontier Markowitz: Given mean return F, find the least volatile portfolio. otivation: Concave utility function more concerned about losing than winning the same amountshun volatility. Standard Deviation Standard dertamon) Constrained optimization 2 Solution by Lagrangian multipliers Lagrangian L-W-SW-x(R-W-?)-(l-W-1) where and are two Lagrangian multipliers. The optimal portfolio satisfies The two constraints (R.W-F,IW-1) become where (since any invertible covariance matrix is positive definite) Both and are linear functions of F. The portfolio variance is 2 w . w- 7A2 + 2 + 2-quadratic polynomial off with positive coefficient for 2 a hyperbola. Extra: Risk-adjusted return. Define Derive the following formulas (see the class slides for the definitions of the notation). 1. Prove that ar2-28r + > 0 for any r. 2. For the efficient frontier with all risky assets: 3. For minimum variance portfolio: f/o-/Va. 4. When the riskless asset is available, the risk-adjusted excess return (the Sharpe ratio) is Efficient frontier Markowitz: Given mean return F, find the least volatile portfolio. otivation: Concave utility function more concerned about losing than winning the same amountshun volatility. Standard Deviation Standard dertamon) Constrained optimization 2 Solution by Lagrangian multipliers Lagrangian L-W-SW-x(R-W-?)-(l-W-1) where and are two Lagrangian multipliers. The optimal portfolio satisfies The two constraints (R.W-F,IW-1) become where (since any invertible covariance matrix is positive definite) Both and are linear functions of F. The portfolio variance is 2 w . w- 7A2 + 2 + 2-quadratic polynomial off with positive coefficient for 2 a hyperbola