# 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.

• 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). Dec 19, 2023 at 12:11

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 $$Cov(1,2) = \rho\times\sigma_1\times\sigma_2$$