Solution for the weights of the optimal risky portfolio can be found by solving the following maximization problem:
where E(r p ) = expected rates of return for portfolio P
r f = risk free rates of return
S p = sharpe performance measure, and
σ p as defined in equation (E1) of Appendix E
We can solve the maximization problem by differentiating the S p with respect to w D , and setting the derivative equal to 0 i.e., we want to solve
In the case of two securities, we know that
From above equations (F2), (F3), and (F4), we can rewrite E(r p ) - r f and σ p as:
Equation (F1) becomes
From equation (F7),
Now, plugging f(w D ), g(w D ), f’(w D ), and g’(w D ) [equations (F5), (F6), (F8), and (F9) ] into equation (F10), we have
Multiplying by 
on both sides of equation (F11), we have
Rearrange all terms on both hand sides of equation (F12), i.e.,
Left hand side of equation (F12)
Right hand side of equation (F12)
Subtracting
and 
...
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© 2006 Springer Science+Business Media, Inc.
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Lee, CF., Lee, A.C. (2006). Derivation of an optimal weight portfolio using the sharpe performance measure. In: Lee, CF., Lee, A.C. (eds) Encyclopedia of Finance. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-26336-6_82
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DOI: https://doi.org/10.1007/978-0-387-26336-6_82
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