We propose a model of a financial market with multiple assets that takes into account the impact of a large institutional investor rebalancing its positions so as to maintain a fixed allocation in each asset. We show that feedback effects can lead to significant excess realized correlation between asset returns and modify the principal component structure of the (realized) correlation matrix of returns. Our study naturally links, in a quantitative manner, the properties of the realized correlation matrix -- correlation between assets, eigenvectors and eigenvalues -- to the sizes and trading volumes of large institutional investors. In particular, we show that even starting with uncorrelated "fundamentals", fund rebalancing endogenously generates a correlation matrix of returns with a first eigenvector with positive components, which can be associated to the market, as observed empirically. Finally, we show that feedback effects flatten the differences between the expected returns of assets and tend to align them with the returns of the institutional investor's portfolio, making this benchmark fund more difficult to beat, not because of its strategy but precisely because of its size and market impact. This is joint work with Rama Cont.