Benchmark Relative Models

Our previous mean variance model were build based on optimization on pure holdings, but sometimes we care about holdings net of the benchmark, and make decisions on those active holdings.

The alpha and beta for a portfolio can be defined as a result of the following regression model, where \(r_B\) is the benchmark portfolio return

\[ r=\alpha+\beta \cdot r_B+\epsilon \]

And we write \(u:=r-\beta \cdot r_B\) as the residual return.

Denote the portfolio holdings as \(\mathbf{x}_P\), and the benchmark holding \(\mathbf{x}_B\), the active holding is \(\mathbf{x}_P-\mathbf{x}_B\), and active return can be written as \(r_P-r_B=\left(\beta_P-1\right) r_B+u_P\). Regard the \(r_B\) as the passive returns, since \(u_p\) is the residual return, which is not correlated with the \(r_B\), therefore we could derive the variance on active return (also called the tracking error) as the following,

\[ \omega_P^2:=\operatorname{var}\left(u_P\right)=\operatorname{var}\left(r_P-\beta_P \cdot r_B\right)\\ \psi_P^2=\operatorname{var}\left(r_P-r_B\right)=\left(\beta_P-1\right)^2 \sigma_B^2+\omega_P^2 \]

If \(\mathbf{V}\) covariance matrix, then \(\psi_P^2=\left(\mathbf{x}_P-\mathbf{x}_B\right)^{\top} \mathbf{V}\left(\mathbf{x}_P-\mathbf{x}_B\right)\).

To quantify the alpha here, define the information ratio being expected value of alpha over its standard deviation, which is

\[ \frac{\mathbb{E}\left(r_P-\beta_P \cdot r_B\right)}{\sigma\left(r_P-\beta_P \cdot r_B\right)}=\frac{\alpha_P}{\omega_P} \]