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Equation (13) in Treynor and Black How to Use Security Analysis to Improve Portfolio Selection (1973) gives the optimal weights using the formula:

In practice, I have seen the rightmost fraction calculated as:

But where does this come from? Equation (11) states that:

And also:

So dividing those equations leaves the lambda. The best I can do is equation (19) where:

So how do I prove that?

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    $\begingroup$ The $h_i$ have to add up to $1$. Does that help? $\endgroup$ Commented Dec 23, 2024 at 16:13
  • $\begingroup$ My question is: how does the rightmost fraction in equation (13), as it is usually calculated in practice, follow from the variable definitions given in equation (11) (or am I missing something)? $\endgroup$ Commented Dec 23, 2024 at 18:44
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    $\begingroup$ First of all eqn (13) is $h_i=\frac{\mu_i}{\sigma^2_i}\frac{\sigma^2_p}{\mu_p}$. Let's call the second fraction on the right K, a number which we don't know. We do know that $\sum_i h_i = 1$ so $\sum_i h_i = K\sum_i \frac{\mu_i}{\sigma^2_i}=1$ So $1/K=\sum_i \frac{\mu_i}{\sigma^2_i}$ and $h_i= \frac{\mu_i}{\sigma^2_i} /\sum_j \frac{\mu_j}{\sigma^2_j}$ where all terms on the right side are known. $\endgroup$ Commented Dec 23, 2024 at 22:40
  • $\begingroup$ Solved. Thank you very much. Please post an Answer so I can give you some points. Also, just wondering why the change in subscript from i to j? $\endgroup$ Commented Dec 24, 2024 at 12:25
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    $\begingroup$ That's a computer programming technicality, we are already using the variable $i$ so we need anothe variable $j$ which is "bound" to the summation sign, it is like a "do-loop" variable nested within a code block that uses $i$ for a different purpose. If you are comfortable with using "i" throughout that's fine, a compiler might not like it. $\endgroup$ Commented Dec 24, 2024 at 13:20

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