The Size Distribution of US Banks and Credit Unions

Abstract

This study examines the firm size distribution of US banks and credit unions. A truncated lognormal distribution describes the size distribution, measured using assets data, of a large population of small, community-based commercial banks. The size distribution of a smaller but increasingly dominant cohort of large banks, which operate a high-volume low-cost retail banking model, exhibits power-law behaviour. There is a progressive increase in skewness over time, and Zipf's Law is rejected as a descriptor of the size distribution in the upper tail. By contrast, the asset size distribution of the population of credit unions conforms closely to the lognormal distribution. JEL Classification: G21; L10; L16Keywords: Firm Size distribution, Zipf's Law, Gibrat's Law, Banks, Credit Unions __________________________________________________________________________ + The authors gratefully acknowledge the helpful comments of two anonymous referees on an earlier draft of this paper. The usual disclaimer applies. IntroductionThis study examines the empirical size distribution of US banks and credit unions. It is well known that most empirical firm-size distributions are highly skewed. If firm sizes are subject to proportional random growth, consistent with Gibrat's Law so that log sizes follow random walks, a lognormal cross-sectional firm-size distribution emerges over time (Gibrat, 1931;Sutton, 1997). There is, however, extensive evidence that the lognormal provides a poor approximation to empirical firm-size distributions in the upper tail, which typically exhibit greater skewness than is consistent with lognormality. Certain modifications toGibrat's Law, however, are capable of producing a cross-sectional size distribution that exhibits power-law behaviour. i A strand in the empirical literature examines the application of lognormal and power-law distributions to cross-sectional firm-size data (Simon and Bonnini, 1958;Quandt, 1966;Lucas, 1978;Cabral and Mata, 2003).Pareto (1897) describes the distribution of a collection of N subjects ranked by size, where the density function, denoted f( ), obeys a power law in the upper tail:where X is size,  is the size threshold above which the Pareto distribution applies, and  is constant. Zipf's Law describes the special case =1 (Zipf, 1949 Estimation MethodLet X i denote the assets of firm i in a particular year, and let x i =ln(X i ). Let X [i] denote the value of the i'th observation when the firms are ranked in descending order of asset size,, and let x [i] =ln(X [i] ). Let  k = X [k] denote some threshold value of k that is, initially, assumed to be pre-selected, and let  k = x [k] .We examine two candidate distributions for X i : , where  is the standard normal density function.For (ii), the likelihood function that would be formed over the two segments of the distribution is discontinuous at the truncation point  k . Accordingly, we estimate the parameters  k , 2 k  , k,  k and  k in two stages. Stage 1and sample variance of allwhere  is the sta...

Beyond the power law: Uncovering stylized facts in interbank networks

“…Our results compare well with some of the existing studies of interbank networks, but differ from others. Specifically, our findings are in line with Goddard et al [8] who study U.S. banks: the log-normal distribution fits the bulk part of the asset-size data well, while the power law does the same for the tail part. Our findings are not too different from those of Cont et al [4] for the Brazilian interbank network: while our TPL fits to the tails of various degree distributions (see Table III) deliver values of α between 1.7 and 2.2, Cont et al [4] report power-law exponents between 2.2 and 2.8 (α = 2.54 for total degree, α = 2.46 for in-degree, α = 2.83 for out-degree, and α = 2.27 for out-exposure size).…”

“…In addition, many authors propose a power law as the best-fit candidate to (at least the tail of) the empirical distribution. In particular, Goddard et al [8] find that the asset size distribution of U.S. commercial banks is well described by a truncated lognormal, while the tail part is well fitted with a power law. Cont et al [4] obtain fair fits to the tails of the in-degree, out-degree, total degree, and exposure size distributions of the Brazilian interbank network with power laws.…”

Measuring firm size distribution with semi-nonparametric densities

“…In this case, the lognormal distribution overestimates the size of the large firms. On the other hand, Goddard et al [15] examine firm size among banks and credit unions based on Zipf's law. Their study rejects Zipf's law as a descriptor of the firm size distribution in the upper tail.…”

“…A strand of the empirical literature has thus sought to examine the application of lognormal and Pareto or power-law distributions using firm size data as cross-sectional data [13] [14] [15]. However, there is evidence that, on some occasions, a poor approximation of the empirical distributions of the firm size in the upper tail, which typically exhibit greater asymmetry as a small number of large firms exist alongside a large number of smaller firms [1] [12] [16], is obtained.…”

Firm size and economic concentration: An analysis from a lognormal expansion

“…Although studies have found evidence that the lognormal distribution accurately fits to FS, favoring Gibrat's law [13][14][15], other studies feature a poor performance of this distribution, especially in the higher quantiles [16][17][18][19]. In this line, some empirical studies have shown that FS distribution can be adjusted using a Pareto or Power-law distribution [7,[20][21][22], although this latter distribution presents the shortcoming of requiring the selection of a minimum threshold to assume that FS distribution is well defined [3,[23][24][25][26][27].…”