Tangency Portfolio - 2023.2 English

Vitis Libraries

Release Date
2023.2 English

Portfolios can be visualized on a graph where the x-axis is risk (portfolio variance) and the y-axis is portfolio return. The efficient frontier is the curve representing the maximum returns for given risks and it’s shape is typically:


The Tangency Portfolio is the asset weight distribution that maximises the slope (Sharpe Ratio) of a straight line (Capital Market Line) intersecting the the return axis at the risk free rate and passing through the efficient frontier. It can be calculated as: \(\boldsymbol{W_T} = \frac{\boldsymbol{\Sigma^{-1}}(\boldsymbol{\mu}-r_f.\boldsymbol{1})}{\boldsymbol{1^t}\boldsymbol{\Sigma^{-1}}(\boldsymbol{\mu}-r_f.\boldsymbol{1})}\)

Where \(\boldsymbol{W_T}\) is the tangency weights vector

\(\boldsymbol{\Sigma^{-1}}\) is the inverse of the covariance matrix \(\boldsymbol{\Sigma}\)

\(\boldsymbol{\mu}\) is the asset mean returns vector

\(\boldsymbol{1}\) is the all ones vector and \(\boldsymbol{1^t}\) its transpose

\(r_f\) is the risk free rate

This is solved using LU decomposition with back and forward substitution:

\(\boldsymbol{\Sigma^{-1}}(\boldsymbol{\mu}-r_f.\boldsymbol{1}) = Y\)

\((\boldsymbol{\mu}-r_f.\boldsymbol{1}) = Y\boldsymbol{\Sigma}\)

The Sharpe Ratio \(= \frac{Tangency\, expected\, return - r_f}{Tangency\, Standard\, Deviation}\)