The Firm Growth Rate Riddle
(Published by JP Bouchaud | April 2024)
Year-on-year sales growth of firms show intriguing statistical regularities.
For example, the volatility sigma of growth rates depends on firm size, S. As perhaps expected, sigma decreases with size. But contrarily to the most naïve “central limit” diversification argument that would lead to a decay as 1/sqrt(S), the observed decay is much slower, like 1/S^{0.2}. In other words, large and very large firms are much more volatile than expected. (Interestingly, the volatility of stock prices follows roughly the same law).
The second striking observation is that, after rescaling by the idiosyncratic sales volatility of each firm, the growth rate distribution is still fat-tailed – actually, even more so for large firms.
In order to explain such stylized facts, Xavier Gabaix and simultaneously Mattieu Wyart & myself proposed back in 2002 that large firms are in fact much less diversified than expected.
If the distribution of sales between products or between clients follows a fat-tailed distribution, the effective size of a firm (in terms of its number of independent activities, say) would scale sub-linearly with size, say as S^{0.4}, in turn leading to the observed 1/sqrt(S^{0.4}) behaviour of volatility.
In other words, large firms rely on a handful of star products or major clients to make their sales. Correspondingly, idiosyncratic shocks do not easily diversify and become Gaussian.
This so-called “granularity hypothesis” has been the leading story in the last 20 years. However, this story leads to several other predictions that can be tested empirically.
After 4 years of heavy lifting, José Morán, Angelo Secchi and myself have finally put together our results in this paper.
The most striking result is about the distribution of sales volatility conditional to size.
The granularity hypothesis predicts such a distribution to be independent of size, provided volatility is appropriately rescaled by the size dependent average volatility. This is empirically borne out (see figure below). But for consistency, such universal rescaled distribution should have fat tails, with a decay enforced by the 1/S^{0.2} law. Unfortunately, this is not at all what is observed: the volatility distribution is much too thin-tailed to be compatible with theoretical predictions!
We conclude that the mechanisms underlying the growth of firms are not satisfactorily understood. One possibility is that shocks hitting a firm’s sub-units become more and more correlated as its size increases. We have explored this hypothesis finding encouraging evidence: both the correlation of a firm’s growth rate with that of the whole economy and the correlation of the growth rate of firms of similar sizes clearly grow with their size.
This might hold the key to solve the firm growth rate riddle, in particular explain why large firms are prone to outsized shocks.