This paper investigates the financial stability consequences of banks' interest rate risk exposure and uninsured deposit funding share. We develop a model incorporating insured and uninsured deposits, interest rate-sensitive securities, and credit-risky loans to understand how banks respond to interest rate risk and the potential for deposit runs. The model delivers the concentration of uninsured deposits in larger banks and examines how banks' portfolio- and funding choices impact financial stability. When banks anticipate volatile bond returns, they seek exposure to this interest rate risk. We study the effects of recent Federal Reserve rate hikes on banks and analyze micro-prudential policy tools to enhance the banking sector's resilience. Higher liquidity requirements that target uninsured deposits are effective at curbing run risk of large banks but cause misallocation in the lending market. Size-dependent capital requirements are equally effective at mitigating run risk, with minimal unintended consequences.