optimization - Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz Portfolio Theory?

I have seen the following formula for the tangency portfolio in Markowitz portfolio theory but couldn't find a reference for derivation, and failed to derive myself. If expected excess returns of $N$ securities is the vector $\mu$ and the covariance of returns is $\Sigma$, then the tangent portfolio (maximum Sharpe Ratio portfolio) is:

\begin{equation} w^* = (\iota \Sigma^{-1} \mu)^{-1} \Sigma^{-1} \mu \end{equation}

Where $\iota$ is a vector of ones. Anyone know a source of the derivation?

Answer

The unconstrained mean-variance problem $$w_{mv,unc}\equiv argmax\left\{ w'\mu-\frac{1}{2}\lambda w'\Sigma w\right\} $$ can easily be found by taking the derivative $$\frac{\partial}{\partial w}\left(w'\mu-\frac{1}{2}\lambda w'\Sigma w\right)=\mu-\lambda\Sigma w $$ setting it to zero, and solving for $w$. This gives $$w_{mv,unc}\equiv\frac{1}{\lambda}\Sigma^{-1}\mu $$ To find the portfolio constraining all the weights to sum to $1$, it is as simple as dividing by the sum of the portfolio weights $$w_{mv,c}\equiv\frac{w_{mv,unc}}{1'w_{mv,unc}}=\frac{\Sigma^{-1}\mu}{1'\Sigma^{-1}\mu} $$which after canceling out the risk aversion variables gives what you have above.

For more general constraints, such that $Aw=b$, the formula is more complex. I often refer to the derivation in this paper for the formula.