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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.

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  • $\begingroup$ First off all I think that your $\mu_1$ is off. It should be $\mu_1 = 0.04$ if I did not make any mistakes in my calculations. Your given covariance seems to be a little weird. The variance with my calculations are $Var_1 = 0.0109$ and $Var_2 = 0.030625$. Using Cauchy-Schwarz you get an upper bound: $Cov_{1,2} \leq \sqrt{(Var_1 * Var_2)} \approx 1.827 \%$. This explains the results close to 0 (yet negative). $\endgroup$
    – MrLCh
    Dec 19, 2023 at 12:11

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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 helps

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