Recent empirical analyses on aggregated datasets have revealed a common exponential behavior in the shape of the probability density of the corporate growth rates. In this paper we present clearcut evidence on this topic using more disaggregated data. We propose a very simple model that, under rather general assumptions, provides a robust explanation for the observed regularities. It is based on the idea that the firms' ability of taking up new business opportunities increases with the number of opportunities already exploited. A theoretical result is presented for the limiting case in which the number of firms and opportunities go to infinity. Moreover, using simulations, we show that even in a moderately small industry the agreement with asymptotic results is almost complete.