Oligopolistic competition in the banking sector and risk in the real economy are important characteristics of developed economies, but have so far mostly been abstracted from in monetary economics. We build a dynamic general equilibrium model of monetary policy transmission that incorporates both of these features and document that including them leads to important insights in our understanding of the transmission mechanism. Various equilibrium cases can occur, and policies have differing effects in these cases. We calibrate the model to the U.S. economy in 2016-2019 in order to study how changes in the degree of banking competition or the policy rate would have affected equilibrium outcomes. We find that doubling banking competition would have increased welfare by 1.02%, but at the cost of increasing the probability of bank default from 0.02% to 0.44%. We further find that the policy rate was set optimally to minimize the probability of bank default, but that a decrease in the policy rate by 1pp would have increased welfare by 0.40%. We also show that bank profits are increasing in the policy rate, in particular when interest rates are low. Thus, a 1pp reduction in the policy rate would have reduced profits per bank by 35.5% in our calibrated economy. Finally, we document that monetary policy pass-through is incomplete under imperfect competition in the banking sector, as a change in the policy rate by 1pp leads to a change of only 0.92pp in the loan rate, while pass-through to the deposit rate is nearly complete for rate increases, but almost zero for rate reductions due to the zero-lower bound.