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22b_Sharpe_ratios

22b_Sharpe_ratios - 1 Sharpe Ratio of a Portfolio Recall...

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1 Sharpe Ratio of a Portfolio Recall that a security (or portfolio) with payo ff s given by the random variable x per dollar of investment has Sharpe ratio SR ( x ) = E γ x (1 + r ) σ x where r is the riskless rate of interest. It is very useful to fi nd a formula for the highest possible Sharpe ratio that can be obtained from a collection of securities, given the Sharpe ratio of each security in the portfolio. Suppose that there are two securities with returns x and y per dollar invested. What combination ax + (1 a ) y produces the portfolio with the maximum Sharpe ratio? If the securities have inde- pendent payo ff s, the Markowitz formula tells us the optimal combination. From that we can determine the Sharpe ratio of the optimal combination. Theorem: Suppose a security paying x per dollar has a Sharpe ratio SR ( x ) , and a security paying y per dollar has a Sharpe ratio SR ( y ) . Suppose x and y are inde- pendent. Then the optimal combination z = ax + (1 a ) y of x and y has Sharpe ratio SR ( z ) = p SR 2 ( x ) + SR 2 ( y ) Proof: Since the risky assets are independent, we know that the optimal Markowitz portfolio mixes risky assets in the proportion SR ( x ) σ x to SR ( y ) σ y . Thus the Sharpe maxi- mizing z gives Sharpe ratio SR ( z ) = E [ ax + (1 a ) y ] (1 + r ) q a 2 σ 2 x + (1 a ) 2 σ 2 y = E [ SR ( x ) σ x x + SR ( y ) σ y y ] SR ( x ) σx + SR ( y ) σy (1 + r ) r SR 2 ( x ) σ 2 x σ 2 x + SR 2 ( y ) σ 2 y σ 2 y / ( SR ( x ) σ x + SR ( y ) σ y ) = E [ SR ( x ) σ x ( x (1 + r )) + SR ( y ) σ y ( y (1 + r ))] p SR 2 ( x ) + SR 2 ( y ) = SR ( x ) σ x ( Ex (1 + r )) + SR ( y ) σ y ( Ey (1 + r ))] p SR 2 ( x ) + SR 2 ( y ) = SR ( x ) σ x SR ( x ) σ x + SR ( y ) σ y SR ( y ) σ y p SR 2 ( x ) + SR 2 ( y ) = SR 2 ( x ) + SR 2 ( y ) p SR 2 ( x ) + SR 2 ( y ) = p SR 2 ( x ) + SR 2 ( y ) . 1
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2 Why managers overestimate their chances This formula sheds light on why hedge fund managers ususally overestimate how well they are going to do. Recall that the function f ( SR 1 , SR 2 ) = p SR 1 2 + SR 2 2 is convex.
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