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

Question

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.

Answer 1

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.

In this case the numerator is the vector (208.75 68.75). The sum of its elements (i.e. the denominator) is 277.5. Dividing the vector by 277.5 we get the weights (0.752252252 0.247747748) which add up to 1 as required.