Why Tangency Portfolio Estimates Can Become Extreme

# Return $\mu$ and volatility $\sigma$ for tangency portfolio of DOW30 too large?


# Return $\mu$ and volatility $\sigma$ for tangency portfolio of DOW30 too large?












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 $\mu$?

## Answer by user6430 (score 1)

https://quant.stackexchange.com/a/32921

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.