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The Journal of Real Estate Finance and Economics

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18-08-2020

Determining an Optimal Principal Limit Factor for Reverse Mortgages under Economics-Based Models

Authors: Shu Ling Chiang, Ming Shann Tsai, Chien An Wang

Published in: The Journal of Real Estate Finance and Economics | Issue 4/2021

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Abstract

Determining an optimal principal limit factor (PLF) is important for a reverse mortgage (RM) contract because it mainly influences the development of the RM market. The goal of this study was to develop a model for calculating the optimal PLF values for both uninsured and insured RMs based on two models: breaking even and maximum profits from the lender’s standpoint. We provide numerical analyses to illustrate the application of our model and compare PLFs calculated by our models with that calculated by using the traditional model in Szymanoski (1994). The results show that our PLFs are all higher than that in Szymanoski (1994) given the same basic parameters. We also provide the sensitivity analyses of PLFs on the parameters of the longevity risk, the collateral risk and the interest rate risk. The sensitivity analyses for these risks can help policymakers and market participants modulate a reasonable PLF responsive to changes in these relevant main risks. Our model provides useful information that can help them determine a reasonable PLF and manage risks with the goal of enhancing RM market development.

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Please refer to https://www.prb.org/2018-world-population-data-sheet-with-focus-on-changing-age-structures/.

During the 1994 Q1 and the 2018 Q4, U.S. home ownership rates for homeowners age 65 or above were between 77.2% to 81.2%. For many elderly homeowners, housing wealth is the largest non-pension component of their total wealth. For example, according to the 2016 Survey of Consumer Finances (SCF), housing wealth constitutes the total wealth of 29.5% of homeowners age 65 or above. For more details, see https://www.census.gov/library/stories/2018/08/homeownership-by-age.html and https://www.federalreserve.gov/econres/scfindex.htm.

Using data from the English Longitudinal Study of Ageing 2002–2003, Sodha (2005) estimated that 10.2% of retirees in Britain had an income below the “modest but adequate” standard (£157 per week before housing costs), even though their house wealth exceeded £100,000.

The most common type of RM loan is the HECM, insured by the Federal Housing Administration. HECMs account for over 90% of all RM loans originated in the U.S. market. Data on the volume of HECM endorsements is obtainable from the U.S. Department of Housing and Urban Development (HUD).

The HUD website provides more information about the RM program. It can be found at: http://search.usa.gov/.

Please refer to “Actuarial Review of the Federal Housing Administration Mutual Mortgage Insurance Fund HECM Loans For Fiscal Year 2012.”

According to the regulations for RM contracts, homeowners have no obligation to repay the loan as long as they stay in the house. The loan should only be repaid by selling the property at the terminated date of RM. The non-recourse provision means that the loan repayments are capped at the sale of the property.

Borrowers’ options can generally be classified as follows: (1) receive a lump sum cash payment up front; (2) receive predetermined monthly cash payments from an annuity as long as the house is the primary residence (tenure plan); (3) receive predetermined monthly cash payments from an annuity for a fixed period of time determined by the borrower (term plan); (4) receive mortgage proceeds, either in unscheduled payments or installments, at times and in amounts of the borrower’s choosing until the line of credit is exhausted (line of credit plan); or (5) a combination of some or all of the above.

See the HUD website: http://search.usa.gov/.

This specification is shown on the HUD website: http://search.usa.gov/.

The calculation of NPL (0) is showed in Appendix 1.

To participate in an RM program, elders must pay the initial costs, which include the organization’s cost, third-party charges and the upfront mortgage insurance premium, and they are charged a monthly servicing fee during the RM period.

The other types of RM payment are appeared in Appendix 1.

In the RM market, a very large fraction of HECM loans are floating rate. To avoid complicated specifications that would blur our point, we let the RM contract rate be a constant value.

With the current tenure plan, the RM disbursement is paid monthly. For simplicity, we specify this payment as yearly. The formula for the present value of the annuity with a constant contract rate γ and for a constant period s is as follows: $ {\ddot{a}}_{s,\gamma }=\frac{\left({\left(1+\gamma \right)}^s-1\right)}{\gamma {\left(1+\gamma \right)}^s} $.

Many factors might lead to the termination of an RM, such as elderly homeowners moving out of their house. To avoid complicated specifications that would blur our point, we use only death as the termination event of the RM. Our model can also be extended to include other risks for terminations, such as prepayment, or moving out of or selling the house. To take account of all the termination risks, the probability of longevity risk can be replaced by the probability of all the termination risks. One can use the actual termination data for the RM to estimate this probability.

The housing market is different from the capital markets, and it is characterized by such factors as high transaction costs and low transaction frequency. However, the purpose of specifying that the house price process follows a geometric Brownian motion is to make clear that the housing return is normally distributed. This is generally an assumption in the studies about real estate issues. In addition, most of the studies using an option pricing model to valuate crossover risk are based on continuous time and let the housing price be a geometric Brownian motion process (see Szymanoski 1994; Lee et al. 2012; Pu et al. 2014).

The advantage of the extended Vasicek interest rate model is that it can be used to match an arbitrary initial forward-rate curve (see Vasicek 1977; Heath et al. 1992). Some research has shown that the Vasicek model (and hence its extension) perform well in pricing mortgage-backed securities (see Chen and Yang 1995).

See Appendix 2.

From the viewpoint of the bank’s risk management, the lender should intend to make the capital cost rate for the RM payments to be deterministic at the origination time. Even if the capital cost rate should be floated in the future, the lender can use derivative tools, such as an interest rate swap, to convert the floating capital cost rate into a fixed capital cost rate so as to hedge the interest rate risk.

All values are taken from figures published by the U.S. Department of Housing and Urban Development, available at: http://search.usa.gov/search?affiliate=housingandurbandevelopment&query=4235.

In 2015, the initial MIP was 0.5% or 2.5%, depending on the borrower’s disbursements. Over the life of the loan, the annual MIP is 1.25% of the mortgage balance. Our assumptions are only for comparative purposes.

The origination fee that HUD permits for HECMs ranges from $2500 to $6000.

Please see Appendix 3.

In some studies, the house service flow rate is determined externally and is assumed to be 2% (Chen et al. 2010; Lee et al. 2012). If we let the house service flow rate be 2%, the calculated optimal PLFs are less than that in Table 4. For uninsured RM, the optimal PLFs are 0.2433 and 0.4787 for maximum profit case and breakeven case. For insured RM, the PLFs are 0.2318, 0.2732,and 0.6584 for the traditional method, maximum profit case and breakeven case.

The estimation method is shown in Appendix 2.

Please see the definitions of the maximum claim amount in Equation (A1) in Appendix 1.

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Title: Determining an Optimal Principal Limit Factor for Reverse Mortgages under Economics-Based Models
Authors: Shu Ling Chiang
Ming Shann Tsai
Chien An Wang
Publication date: 18-08-2020
Publisher: Springer US
Published in: The Journal of Real Estate Finance and Economics / Issue 4/2021
Print ISSN: 0895-5638
Electronic ISSN: 1573-045X
DOI: https://doi.org/10.1007/s11146-020-09786-1

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