# Calculating tangency portfolio weights with the given information? (2risky +riskfree asset)

We have 2 risky and 1 risk-free asset.

E1 = 4%, STD1=10%
E2 = 5.5%, STD2 = 20%

rf=1.5%


The covariance matrix and it's inverse are given:

|0.01 0.006|
|0.006 0.04|


inverse:

|109.9  -16.5|
|-16.5   27.5|


ue (vector of excess returns)

(2.5  4)


Now the formula for the weights of the tangency portfolio should be:

$$\frac{\Sigma^{-1} U_e}{1^T \Sigma^{-1} U_e}$$

But using this formula I don't get the solution for the weights which should be (0.752,0.248)

Also what does $$1^T$$ or $$1'$$ even do? It's just the transposed vector of 1s. I looked at 500 books, videos, notes but none of them had this clearly explained so hopefully someone here could help.

The denominator $$1^T \Sigma^{-1} U_e$$ is a scalar number. First you multiply the inverse by $$U_e$$ giving a column vector which I will call $$X$$, then premultiplication of this vector by $$1^T$$ basically amounts to adding the entries in this vector.
The numerator $$\Sigma^{-1} U_e$$ is a vector, the $$X$$ vector we just talked about, it is just a matrix times column_vector product.
This whole procedure of dividing a vector $$X$$ by $$1^ TX$$ is just a trick to normalize the elements of $$X$$ so that they add up to 1.