Bank Diversification: Laws and Fallacies of Large Numbers

Conventional wisdom states that large banks are safer than small banks because they can diversify more. This conventional wisdom, however, confuses risk with probability of failure. While the law of large numbers does imply that a large bank is less likely to fail than a small bank, equating this tendency with lower risk falls into what Samuelson [1963] termed the fallacy of large numbers. A $10 billion bank may be less likely to fail than a $10 million bank, but it may also saddle the investor with a $10 billion loss. In this article, I hope to clarify what this distinction means for banks. Banks diversify by growing—by adding risks—something distinctly different from the subdivision of risk behind standard portfolio theory. A simple meanvariance example will make the point that a risk-averse bank owner need not value diversification by addition. After that, I take a regulator’s perspective and consider how a bank guarantee fund, such as the deposit insurance agency, views bank growth and diversification. After a short review of why diversification by adding risks decreases the probability of bank failure, I look at how such diversification alters the expected value of deposit insurance agency payments, then turn to diversification’s impact on the deposit insurance agency’s expected utility, using recent results from the theory of standard risk aversion. To concentrate on the cleanest example, this article stays with the case of independent and identically distributed risks. This admittedly ignores the alleged ability of large banks to diversify regionally1 or the possibly adverse incentives of deposit insurance (Boyd and Runkle [1993], Todd and Thomson [1991]).