I read in several papers that a proxy for the beta of a universe of underlyings is to use the first eiggenportfolio of the PCA approach (The portfolio that corresponds to the biggest eiggenvalue of the correlation matrix)

My purpose here is to create a basket that contains the beta of the universe, eg. the non idiosyncratic risk.

1/ Why PCA? My concern is that PCA is maximizing the variance and I am not sure this is exactly what we want to have.

2/ Is there other ways to choose the weights of this basket (appart for equally weighted :) I am thinking for instance of minimizing the portfolio that is long a stock and short all others?

Thanks