We analytically derive optimal central bank portfolios in a minimum variance framework with two assets and "transaction …

This paper studies the long-run evolution of bank risk and its links to the macroeconomy. Using data for 17 advanced … economies, we show that the riskiness of bank assets declined materially between 1870 and 2016. But even though bank assets have … become safer, the losses on these assets are associated with increasingly large output gaps. Before 1945, bank asset returns …