We compile a rich dataset that links institutional investors' position level holdings with corporate bond characteristics and estimate demand elasticities with respect to critical sources of risk. Persistence in institutions' holdings provide us with an instrument to isolate exogenous movements in prices. We find significant heterogeneity in demand elasticities across the main players in the corporate bond market, namely insurers, pension funds, and mutual funds. Long-term investors are sensitive to interest rate movements and supply liquidity, whereas mutual funds, with shorter investment horizon and benchmark constraints, demand liquidity. Price impact increased post-crisis for all institutions and has remained higher than the pre-crisis levels, signaling a general decline in bond market liquidity due perhaps to regulatory changes in the corporate bond market. Price impact jumped up significantly during COVID-19, perhaps suggesting a reluctance of dealers to intermediate in the market place, and illustrating that firms' funding opportunities are highly sensitive to investors' latent demand shocks. Our results have wide ranging implications for corporate bond pricing due to heterogeneity in investors and investment mandates, and are hard to reconcile with standard, representative agent based models.