I'm reading about the mean-variance optimization of active portfolios. A bit of prior background from the book I'm reading: the author discusses the mean-variance optimal portfolios without cash, which amounts to solving the following optimization problem:

Maximize: $w^Tf - \frac{1}{2}\lambda (w^T\Sigma w)$, subject to $w^Ti = 1$,

where $w$ and $f$ are column vectors of the weights and returns (respectively) of all securities, $\Sigma$ is the covariance matrix, $\lambda$ the risk tolerance and $i = (1, ..., 1)^T$ is just a column vector of all $1$'s. The optimal weight vector turns out to be:

$$w^* = \frac{\Sigma^{-1}i}{i^T \Sigma^{-1}i} + \frac{1}{\lambda}\frac{(i^T \Sigma^{-1}i)\Sigma^{-1}f\ -\ (i^T \Sigma^{-1}f)\Sigma^{-1}i}{i^T \Sigma^{-1}i}$$

Next, while considering an active portfolio, we can decompose the portfolio into benchmark and active weights - $w = b+a$. Since $w^T i = 1$ and $b^T i = 1$, $a^Ti = 0$. So the optimization problem in this case is:

Maximize: $a^Tf - \frac{1}{2}\lambda (a^T\Sigma a)$, subject to $a^Ti = 0$. The optimal active weight vector is:

$$a^* = \frac{1}{\lambda}\frac{(i^T \Sigma^{-1}i)\Sigma^{-1}f\ -\ (i^T \Sigma^{-1}f)\Sigma^{-1}i}{i^T \Sigma^{-1}i}$$.

So far so good. Now the interpretation given in the book is as follows:

...it (the active weights) is independent of the benchmark. Consequently, the expected active return or alpha and the active risk are also independent of the benchmark.

I can't understand how this claim follows from the equations above. Secondly,

It is therefore theoretically feasible to utilize or port it on any benchmark. In other words, two active equity portfolios managed against two different equity benchmarks could have the same active weights. For instance, the active weights of an equity portfolio managed against S&P 500 index could be the same as the weights of a long-short market-neutral hedge fund. This is the idea behind the so-called portable alpha strategies, i.e., the alpha or excess return generated from a strategy can be ported onto another different benchmark.

Could someone please explain what is meant by "porting" on to a benchmark? How can you port an $n$-dimensional active weight vector meant for an index with $n$ stocks on to another benchmark with $m \neq n$ stocks?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.