This paper studies two new models in which banks face a non-trivial asset allocation decision. The first model (CVH) predicts a negative relationship between banks' risk of failure and concentration, indicating a trade-off between competition and stability. The second model (BDN) predicts a positive relationship, suggesting no such trade-off exists. Both models can predict a negative relationship between concentration and bank loan-to-asset ratios, and a nonmonotonic relationship between bank concentration and profitability. We explore these predictions empirically using a cross-sectional sample of about 2,500 U.S. banks in 2003 and a panel data set of about 2,600 banks in 134 nonindustrialized countries for 1993-2004. In both these samples, we find that banks' probability of failure is positively and significantly related to concentration, loan-to-asset ratios are negatively and significantly related to concentration, and bank profits are positively and significantly related to concentration. Thus, the risk predictions of the CVH model are rejected, those of the BDN model are not, there is no trade-off between bank competition and stability, and bank competition fosters the willingness of banks to lend.