A General Pricing Model for a Mortgage Insurance Contract Considering the Effects of Multivariate Random Variables on Termination Probabilities and Loss Rate

Abstract

This article describes the derivation of a general closed-form formula for determining a fair premium for both Federal Housing Administration (FHA) and private mortgage insurance (MI). Our model incorporates the regulations appearing in MI contracts, the changes in economic situations, the termination hazard rates (i.e., prepayment and default), and the loss rate given default. We then give an example to show how one uses our model to calculate an MI premium with FHA regulations by using real mortgage data. Our pricing formula can also be used to calculate the implied default hazard rates given the FHA's current MI. The comparison of this implied rate with the actual rate should help mortgage insurers decide whether the current MI premium should be adjusted. Further analysis shows how sensitive MI premiums are to changes in the model parameters, such as the volatility of the interest rate and the house price appreciation rate. Our pricing formula should make it easier for mortgage insurers to determine fair MI premiums and employ sophisticated risk-management procedures.