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Solving it algebraically: As seen in the above provided reference (just above " 1) "), the general formulation for the unconstrained Markowitz portfolio optimization scheme, is given by: \begin{align} &\text{arg}\max_{w} \; w^T\mu-\frac{\delta}{2} w^T\Sigma w.\\ \end{align} In absence of any constraints, the above optimization scheme have the ...


If your Sharpe ratio is positive when you are long your high portfolio and short your low portfolio, your high portfolio had larger returns than the low portfolio. Your mean return of the long-short portfolio is positive. This is only possible if, on average, the sum of the long return and the short return is positive.

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