A Semi-Markov Multiple State Model for Reverse Mortgage Terminations

Abstract

Reverse mortgages provide a mechanism for seniors to release the equity that has been built up in their home. At termination, the mortgagors are usually guaranteed to owe no more than the value of their property. The value of the reverse mortgage guarantee is heavily dependent on the maturity or termination date, which is uncertain. In this paper, we model reverse mortgage terminations using a semi-Markov multiple state model which incorporates three different modes of exit: 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 UK and Home Equity Conversion Mortgages (HECMs) in the USA. 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.

Reverse mortgage pricing and risk analysis allowing for idiosyncratic house price risk and longevity risk

“…Termination triggers of reverse mortgages include mortality, move-out due to health-related issues, voluntary prepayment and refinancing (Ji et al, 2012). To model the different termination triggers we use a variant of the multi-state Markov model developed by Ji et al (2012).…”

“…To model the different termination triggers we use a variant of the multi-state Markov model developed by Ji et al (2012). Similar models have been used by Alai et al (2014) and Cho et al (2013).…”

“…Following Ji et al (2012) and Cho et al (2013), we model the different triggers for reverse mortgage termination but allow mortality rates to be random variables. The mortality rate of borrowers is assumed to be lower than that of the population of the same age, in order to reflect the better health of retirees that still live at home compared to those that have moved to aged care facilities.…”

“…The mortality rate of borrowers is assumed to be lower than that of the population of the same age, in order to reflect the better health of retirees that still live at home compared to those that have moved to aged care facilities. At-home mortality rates are derived by applying age-specific scaling factors to the population mortality rates (Cho et al, 2013;Ji et al, 2012). The probability of moving out due to health-related reasons, mainly because of entry into a long-term care (LTC) facility, is assumed to be a proportion of the mortality rate.…”

“…Due to lack of public access to detailed data on these rates or probabilities, assumptions based on the UK experience given in Equity Release Working Party (2005), Hosty et al (2008), Ji et al (2012) and Cho et al (2013) have been employed in this paper. The parameter assumptions are summarised in Table 7.…”