Diagnosing Negative Tangency Portfolio Variance from an Invalid Covariance Matrix
Summary
The document examines why a two-stock tangency portfolio calculation produces negative variance while deriving a capital market line. The proposed procedure computes expected returns and variances, forms a covariance matrix, applies its inverse to excess returns, and normalizes the resulting values into portfolio weights. The response identifies a likely input error: the stated cross-asset value may be a correlation coefficient rather than a covariance. If so, covariance should be calculated by multiplying correlation by the two assets’ standard deviations; the response reports that this interpretation yields positive portfolio variance.
The example illustrates why covariance inputs must be checked for consistency before using a matrix to calculate portfolio risk. A covariance matrix must be positive semidefinite, and an invalid matrix can generate impossible negative variance. The suggested interpretation is tentative because the original data do not establish whether the reported value is correlation or covariance. The document does not provide the complete recalculation or independently confirm the inputs.
Key ideas
- A tangency portfolio’s weights can be found by applying the inverse covariance matrix to excess expected returns and normalizing.
- Negative portfolio variance signals that the covariance inputs or calculations need review.
- A correlation coefficient must be converted to covariance using both assets’ standard deviations.
- The proposed correlation interpretation is plausible but cannot be confirmed from the supplied information alone.
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# Problem with finding the efficient capital market line formula, getting negative variance
# Problem with finding the efficient capital market line formula, getting negative variance
So my goal is to write the Capital Market line formula considering this data: $\[ \begin{array}{|c|c|c|} \hline \text{Stock 1} & \text{Stock 2} & \text{Probability} \\ \hline -15\% & -20\% & 20\% \\ 15\% & 30\% & 30\% \\ 5\% & 15\% & 50\% \\ \hline \end{array} \] $ with Covariance(1,2)=1,83% and a risk free asset rf=0,035
I want to use the method using Tangency portfolio to deduce the CML formula, but my problem is that I find negative variances for the tangency portfolio...
Here's what I did so far: To define the CML, I want to use this formula: µp=((µt-rf)/σt)*σp+rf Now I'm trying to get µt and σt. To do so, first I'm calulating means and variances for each stock: $\[ \begin{array}{|c|c|c|} \hline & \text{Stock 1} & \text{Stock 2} \\ \hline \mu & 0.045 & 0.125 \\ \sigma^2 & 0.0108 & 0.0306 \\ \hline \end{array} \]$
And then I write the covariance variance matrix such as: $\[ \Sigma = \begin{bmatrix} 0.0108 & 0.0183 \\ 0.0183 & 0.0306 \end{bmatrix} \]$
And solve for Z=Σ^-1*(µ-rf) and get $\[ Z = \begin{bmatrix} 304.08 \\ -178.91 \end{bmatrix} \]$ from where I get those weights using w=zi/Sum(zi) $\[ W = \begin{bmatrix} 2,42 \\ -1,42 \end{bmatrix} \]$
But when I try to get σ^2 of the tangency portfolio using WTΣW, I get σ^2=-8,2128e-4.... I really don't get what I did wrong.
## Answer by mountshoutcap (score 1)
https://quant.stackexchange.com/a/77771
with the inputs you provided the result is negative. could it be possible that what you call Covariance(1,2) $=1.83\%$ is actually the correlation coefficient of Stock $1$ and Stock $2$ ?
Pretty much the only extra step you'd have to do if the above is true, is to compute the covariance via $Cov(1,2) = \rho\times\sigma_1\times\sigma_2$
doing so, the final result is positive. let me know if you want to see all the calculations.
hope this helpsShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.