An evolutionary model of the bank size distribution is presented based on the exchange and creation of deposit money. In agreement with empirical results the derived size distribution is lognormal with a power law tail. The theory is based on the idea that the size distribution is the result of the competition between banks for permanent deposit money. The exchange of deposits causes a preferential growth of banks with a fitness that is determined by the competitive advantage to attract permanent deposits. While growth rate fluctuations are responsible for the lognormal part of the size distribution, treating the mean growth rate of banks as small, large banks benefit from economies of scale generating the Pareto tail.