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I am calculating GMV and TAN mu and sigma as well as weights using the straightforward derivations, such as:

\begin{equation} \mu_{gmv}=\frac{\mathbf{1}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}{\boldsymbol{\mu}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}, \end{equation}

\begin{equation} \sigma_{gmv}=\frac{1}{\sqrt{\boldsymbol{\mu}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}}, \end{equation}

\begin{equation} \mathbf{w}_{gmv}=\frac{\boldsymbol{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}'\boldsymbol{\Sigma}^{-1}\mathbf{1}}, \end{equation}

\begin{equation} \mu_{tan}=\frac{\boldsymbol{\mu}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}{\mathbf{1}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}, \end{equation}

\begin{equation} \sigma_{tan}=\frac{\sqrt{\boldsymbol{\mu}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}}{|\mathbf{1}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}|}, \end{equation}

\begin{equation} \mathbf{w}_{tan}=\frac{\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}{\mathbf{1}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}. \end{equation}

If we let \begin{equation} \begin{split} a &= \mathbf{1}'\boldsymbol{\Sigma}^{-1}\mathbf{1}\\ % ones(j) * UTU(j, k) * ones(k) b &= \mathbf{1}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}\\ % ones(j) * UTU(j, k) * fbar(k) c &= \boldsymbol{\mu}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}\\ % fbar(j) * UTU(j, k) * fbar(k) d &= ac - b ^ 2,\\ \end{split} \end{equation}

then the $\sigma_{eff}$ for each mean, $\mu_{eff}$, in the orthogonal portfolios on the efficient frontier line is \begin{equation} \sigma_{eff} =\sqrt{\frac{a \mu_{eff}^2 - 2 b \mu_{eff} + c}{d}}. \end{equation}

However, when I plot the efficient frontier of the daily log-returns for the DOW30 for the last two years (plotted log-scale on returns just to show the GMV mean), I noticed the mean and sigma for the tangency portfolio are much greater than the mean and sigma for the stocks themselves. The tangency weights are also quite large as well.

By the way, the sum of the stock-specific weights for the GMV and tangency portfolios are both one, respectively.

Should I not use log-returns, but rather use the straightforward price returns to generate the covariance matrix $\boldsymbol{\Sigma}$ and the stock-specific mean return $\mu$?

mean and sigma weights

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    $\begingroup$ To me "tangency portfolio" requires knowing the interest rate $r_f$. But in your calculations above I don't see $r_f$ anywhere. What do you mean by tangency portfolio? Or are you using 0 for the interest rate? $\endgroup$ – noob2 Mar 8 '17 at 17:31
  • $\begingroup$ Yes, using zero for the interest rate, but the mean and sigma for the tan portfolio seem almost an order of magnitude too large. There's something going on with weights in spite of summing to unity? $\endgroup$ – 4k3x9d7r Mar 8 '17 at 17:49
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    $\begingroup$ OK, formulas look alright. What kind of data did you use to estimate your covariance matrix? 24 monthly returns? And you have how many securities 30? $\endgroup$ – noob2 Mar 9 '17 at 3:58
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    $\begingroup$ Another thing to check, as you said, is the size of the weights. Is there one weight that is a big positive number and another that is a big negative? $\endgroup$ – noob2 Mar 9 '17 at 4:16
  • $\begingroup$ The covariance matrix was based on daily log-returns for the 30 DOW stocks from the last two years, so the number of days is in the high 400s. I also compared results of calculations with other online return and covariance data for calculating GMV and TAN weights, and the resulting weights, means, and sigmas were the same as those published. Maybe the CAPM line truly intersects the efficient frontier line way out at high mean and sigma values? Maybe I'll cut the #stocks down to 5-10 and see what happens. $\endgroup$ – 4k3x9d7r Mar 9 '17 at 14:15
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Here is a solution which I discovered via comparison of using log-returns vs. returns. I also trimmed down the number of stocks. When using log-returns vs returns we get the following efficient frontiers. The log-returns results in the most textbook looking plot, but note that the TAN point is way out to the right -- and that the slope of the CAPM line is low. Whereas, the returns results in the TAN point intersecting the CAPM line near the GMV point, and the CAPM line has a very steep slope. Overall, I think using log-returns is the way to go.

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