<p>This paper presents a generalization of principal component analysis aimed at reducing the risk of any given benchmark portfolio. Taking the benchmark investment as the starting point, we propose a method to iteratively construct an orthogonal basis of the return space. These orthogonal vectors, called pseudo-principal portfolios after suitable normalization, are combined with the benchmark through an allocation rule to achieve risk reduction. From a theoretical perspective, we connect this construction to the mean–variance framework and derive geometric properties with meaningful financial implications. From an empirical perspective, we provide in-sample and out-of-sample experiments on different datasets of real financial data to support the effectiveness of the strategy, which combines the original benchmark with the pseudo-principal portfolios to lower risk, measured in terms of return volatility.</p>