Disparities in lending to minority applicants persist even as algorithmic lending practices proliferate. Further, disparities in interest rates charged can remain large even when loan applicants from different groups are equally creditworthy. We study the role of risk-management constraints, specifically Value-at-Risk (VaR) constraints, in the persistence of segregation in loan approval decisions. We develop a formal model in which a mainstream (low-interest) bank is more sensitive to variance risk than a subprime (high-interest) bank. If the mainstream bank has an inflated prior belief about the variance of the minority group, it may deny that group credit indefinitely, thus never learning the true risk of lending to that group, while the subprime lender serves this population at higher rates. We formalize this as a "subprime trap" equilibrium. Finally, we show that a small, finite subsidy (or partial guarantee) can help minority groups escape the trap by covering enough of the mainstream bank's downside so that it can afford to lend and learn the minority group's true risk. Once it has sufficiently many data points, it meets its VaR requirement with no further assistance, minority groups are approved for loans by the mainstream bank, and competition drives down the interest rates of subprime lenders.