An evolutionary model of the bank size distribution is presented based on the exchange and expansion of deposit money. In agreement with empirical results the derived size distribution is lognormal with a power law tail. The key idea of the theory is to regard the creation of money as a slow process compared to exchange processes of deposit money. The exchange of deposits causes a preferential growth of banks with a fitness determined by the competitive advantage to attract permanent deposits. They generate the lognormal part of the size distribution. Sufficiently large banks, however, benefit from economies of scale leading to a Pareto tail. The model suggests that the liberalization of the banking system in the last decades is the origin of an increasing skewness of the bank size distribution.