Time Evolution of the Mutual Fund Size Distribution

We investigate the process of mutual fund growth both empirically and theoretically. The size of large mutual funds has a heavy tailed distribution that has been conjectured to be a power law; we investigate the data more carefully and show that it is better described by a log normal. To explain this we develop a stochastic growth model based on multiplicative growth, creation and annihilation. Under the simplifying assumption that these processes do not depend on fund size we obtain a time-dependent analytic solution of the model. The distribution evolves from a log normal into a power law only over long time scales, suggesting that log-normality comes about because the industry is still young and in a transient state due to its rapid growth in recent years. We make the model more realistic by taking into account size dependent effects, in particular the decay in the rates of diffusion and drift with increasing fund size. The resulting model is in good quantitative agreement with the data. Surprisingly, it appears that investor choice does not directly determine the size distribution of mutual funds.