Valuation of Reverse Mortgages with Default Risk Models

Abstract

Reverse mortgages are designed to offer additional sources of financing incomes to senior homeowners. In the United States, home equity conversion mortgages (HECMs) are nonrecourse reverse mortgage loans insured by the Federal Housing Administration (FHA). Based on a fairly recent stream of the reverse mortgage literature, the relatively high loan-to-value ratio has jeopardized the financial soundness of such contracts. In the wake of the 2008 financial crisis, rising property taxes and homeowner insurance defaults impaired HECM solvency; hence, policy changes were implemented to help prevent borrower default. In this paper, we propose a pricing solution which, as we demonstrate in the paper, effectively improves program solvency by fairly matching the benefits and liabilities of HECM participants. The methodology allows for customization of fair mortgage loan payments and premiums and improves program accessibility based on borrowers’ individual credit and default risk. Our proposed pricing solution and the corresponding newly designed rating system provide HECM policymakers with a better payment arrangement and offer important policy implications for the current HECM program. Rather than borrower property taxes and insurance delinquency, we demonstrate that the mispricing of HECM mortgage insurance premiums and the corresponding loan payments could be the primary reasons for program insolvency.

Appendices

Appendix A: Revisiting the Parity Condition (6)

In “Fair Pricing Conditions for RM Guarantees”, we proposed the parity condition (6) for the MIP and annuity loan payment determination, which helps mitigate the HECM insolvency problem. In the following, we explain this parity condition from the perspectives of benefit and liability matching of all HECM participants. In presence of the event of death or default, the EPVs of the benefits and liabilities of the HECM program, its borrower, and the lender, are respectively given by

$$ \begin{array}{@{}rcl@{}} \text{EPV of HECM benefits } \!&=&\!\text{EPV of MIPs}, \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} \text{EPV of HECM liabilities} \!&=&\! \text{EPV of loss} + \text{EPV of rental incomes}, \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} \text{EPV of borrower's benefits } \!&=&\! \text{I{\kern-.2pt}n{\kern-.2pt}i{\kern-.2pt}t{\kern-.2pt}i{\kern-.2pt}a{\kern-.2pt}l{\kern-.2pt} w{\kern-.2pt}i{\kern-.2pt}t{\kern-.2pt}h{\kern-.2pt}d{\kern-.2pt}r{\kern-.2pt}a{\kern-.2pt}w{\kern-.2pt}a{\kern-.2pt}l{\kern-.2pt}s{\kern-.2pt} + {\kern-.2pt} E{\kern-.2pt}P{\kern-.2pt}V{\kern-.2pt} o{\kern-.2pt}f{\kern-.2pt} a{\kern-.2pt}n{\kern-.2pt}n{\kern-.2pt}u{\kern-.2pt}i{\kern-.2pt}t{\kern-.2pt}y{\kern-.2pt} l{\kern-.2pt}o{\kern-.2pt}a{\kern-.2pt}n{\kern-.2pt} p{\kern-.2pt}a{\kern-.2pt}y{\kern-.2pt}m{\kern-.2pt}e{\kern-.2pt}n{\kern-.2pt}ts},\\ &&\!+ \text{EPV of rental incomes} \end{array} $$

(32)

$$ \begin{array}{@{}rcl@{}} \text{EPV of borrower's liabilities} \!&=&\! \text{EPV of house value - EPV of nonnegative balance} \\ \!&=&\! \text{$H(0)$} - \text{EPV of rental incomes} \\&&\!- \text{EPV of nonnegative balance}, \end{array} $$

(33)

$$ \begin{array}{@{}rcl@{}} \text{EPV of lender's benefits} \!&=&\! \text{EPV of loan accumulation}, \end{array} $$

(34)

$$ \begin{array}{@{}rcl@{}} \text{EPV of lender's liabilities} \!&=&\! \text{I{\kern-.2pt}n{\kern-.2pt}i{\kern-.2pt}t{\kern-.2pt}i{\kern-.2pt}a{\kern-.2pt}l{\kern-.2pt} wi{\kern-.2pt}t{\kern-.2pt}h{\kern-.2pt}d{\kern-.2pt}r{\kern-.2pt}a{\kern-.2pt}w{\kern-.2pt}a{\kern-.2pt}l{\kern-.2pt}s + EPV of annuity loan payments}\\ &&\!+ \text{EPV of MIPs} , \end{array} $$

(35)

with

$$ \begin{array}{@{}rcl@{}} \text{EPV of loan accumulation} &=& \text{Initial withdrawals + EPV of interest charges}\\ && \text{ + EPV of MIPs} + \text{EPV of annuity payments}, \end{array} $$

and

$$ \text{EPV of house value} = H(0) - \text{EPV of rental incomes}. $$

Equations 30-35 can be explained as follows. On behalf of the borrower, a lender pays mortgage insurance premiums to the HECM program, which is committed to insuring the lender’s crossover risk. Throughout the contract, the HECM program is also liable for allowing the borrower to finance additional rental incomes by depreciation of her/his home appraisal. In addition to the HECM program’s rental financing, the borrower receives initial withdrawals and ongoing annuity loan payments from the lender until the contract terminates. At contract termination, the borrower transfers home ownership to fulfill her/his liability. The borrower only needs to repay the accumulated loan up to the amount of house value, with a rental financing discount. Then, the lender sells the collateral house to reclaim the loan. Any excess of the loan accumulation over the home sale is considered a nonnegative balance and thus is returned to the borrower. Since the presence of the HECM program immunizes the lender from the crossover risk, she/he receives full compensation from the HECM insurance fund for the loss caused by the crossover event. Therefore, the lender receives the loan repayment and profits from her/his interest charges.

By equating the sum of the EPV of benefits in Eqs. 30, 32 and 34 and the sum of the EPV of liabilities in Eqs. 31, 33 and 35, it follows that

$$ \begin{array}{@{}rcl@{}} H(0) &-& \text{EPV of rental incomes} = \text{Initial withdrawals + EPV of interest charges} \quad \\ &+& \text{EPV of annuity payments + EPV of MIPs + EPV of nonnegative balance} \\ &-& \text{EPV of loss}, \end{array} $$

which confirms the parity condition in Eqs. 5 and 6.

Appendix B: Proof of Theorem 3.1 and Derivation of Eq. 10

According to Bielecki and Rutkowski (2013), the processes, $M_{t}^{\tau _{1}}:=\chi (t)-{\int \limits }_{0}^{t\wedge \tau _{1}}h(s)\left (1-\chi (t)\right )ds$ and $M_{t}^{\tau _{2}}:=\gamma (t)-{\int \limits }_{0}^{t\wedge \tau _{2}}\mu _{x}(s)\left (1-\gamma (t)\right )ds$, are martingales with respect to their natural filtrations. We thus have $\mathbb {E}^{\mathbb {Q}}\left [dM_{t}^{\tau _{1}}|\mathcal {F}_{t}\right ]=0$ and $\mathbb {E}^{\mathbb {Q}}\left [dM_{t}^{\tau _{2}}|\mathcal {F}_{t}\right ]=0$ respectively. This implies that for any integrable adapted integrand g, we have

and

Therefore, by Fubini’s theorem and the mutual independence of τ₁ and τ₂, and the payoffs of Bal(t) and Loss(t), we can rewrite Eq. 9 as follows:

(36)

where L(0) = ω + p₀H(0), h(t) and μ_x(t) are assumed to be nonstochastic. Equation 36 represents the different payoffs depending on the values of τ₁ and τ₂.

Note that

(37)

and

(38)

and

$$ e^{-rt}\mathbb{E}^{\mathbb{Q}} \left[H(t)\right]=e^{-\delta t}H(0). $$

(39)

Substituting (37), (38) and (39) into (36), it follows that

$$ \begin{array}{@{}rcl@{}} H(0) &=& L(0) + \delta H(0) \times {{\int}_{0}^{T}} {~}_{u} p_{x,m} \times {~}_{u} p_{x,d} \times e^{-\delta u} du \\ &&+ (r+\pi_{r}) \times {{\int}_{0}^{T}} e^{-ru} \times {~}_{u} p_{x,m} \times {~}_{u} p_{x,d} \times L(u)du + c \times {{\int}_{0}^{T}} e^{-ru} \\&&\times {~}_{u} p_{x,m} \times {~}_{u} p_{x,d} du \\ &&+ {{\int}_{0}^{T}} e^{-ru} \times \left( h(u)+\mu_{x}(u)\right) \times {~}_{u} p_{x,m} \times {~}_{u} p_{x,d} \times \mathbb{E}^{\mathbb{Q}}\left[Bal(u)\right] du. \end{array} $$

Furthermore, we use Fubini’s theorem and mutual independence to derive Eqs. 10 and 11. Substituting (37) and (38) into (10) and (11), it follows that

and

Thus, we have

$$ \begin{array}{@{}rcl@{}} &&p_{0} H(0) + p_{a} \times {{\int}_{0}^{T}} e^{-ru} \times {~}_{u} p_{x,d}\times{~}_{u} p_{x,m} \times L(u)du\\ &=& {{\int}_{0}^{T}} e^{-ru} \times \left( h(u)+\mu_{x}(u)\right) \times {~}_{u} p_{x,d} \times {~}_{u} p_{x,m}\times \mathbb{E}^{\mathbb{Q}} \left[Loss(u)\right]du, \end{array} $$

which completes the proof of Theorem 3.1.