A Semi-Markov Multiple State Model for Reverse Mortgage Terminations

Reverse mortgages provide a way for seniors to release the equity that has been built up in their home to supplement their post-retirement income. The value of a reverse mortgage loan is heavily dependent on the maturity or termination date, which is uncertain. In this research, we model reverse mortgage terminations using a semi-Markov multiple state model, which incorporates three different modes of termination: death, entrance into a long-term care facility, and voluntary prepayment. We apply the proposed model specifically to develop the valuation formulas for roll-up mortgages in the U.K. and Home Equity Conversion Mortgages (HECMs) in the U.S.A. We examine the significance of each mode of termination by valuing the contracts allowing progressively for each mode. On the basis of our model and assumptions, we find that both health related terminations and voluntary (non-health related) terminations significantly impact the contract value. In addition we analyze the premium structure for US reverse mortgage insurance, and demonstrate that premiums appear to be too high for some borrowers, and substantial cross-subsidies may result. ∗A dissertation presented to the Institute and Faculty of Actuaries in partial fulfilment of the requirements for the Specialist Applications Dissertation option (SA0) for the qualification of Fellow of the Institute and Faculty of Actuaries.