Correlated defaults and risk retention: Can prices be increasing in risk

• Closed-form aggregation of correlated insured Bernoulli defaults. • Endogenous subordination can make senior prices increasing in correlation. • Mortgage insurance reduces pool variance by the square of uninsured LGD. • Default correlation θ inflates variance by 1/(1 − 2θ). • Sensitivity analysis confirms robustness to risk aversion and default rates. Pooling is the core diversification technology in securitization, but default correlation limits diversification and shifts the optimal mix of credit enhancement. This note combines a correlated Bernoulli default process with a simple securitization design problem in which the sponsor chooses subordination and (optionally) private mortgage insurance. The technical contribution is an aggregation result for insured correlated Bernoulli cash flows: insuring each loan induces a linear "floor" that scales down its variance by the square of the uninsured loss given default. This tractability makes it immediate to characterize how correlation alters equilibrium credit enhancement. Higher correlation raises the value of loss-given-default insurance relative to pure subordination, providing a clean mechanism through which senior-tranche prices need not fall monotonically with correlation once credit enhancement is chosen endogenously.