3.1. Optimal surrender region and reference probabilities

For a T-year contract, the issuer of an RM identifies the loan payments and MIPs based on the mortality table. Assume that during the underwriting, the mortality decrements about their ages are also recognized by RM borrowers with their life expectancy following what the mortality table suggests. Our goal is to determine the optimal surrender strategy, which a rational borrower should follow. In our model, the borrower’s surrender is assumed to be driven by a home appreciation. To investigate the impact of all other market factors, such as interest rate and house price volatility, on the optimal strategy, sensitivity analyses of the surrender boundary can be done further with respect to these factors.

At a random death time $\tau_x \in [0,T]$ , a borrower receives an accumulation of contract payouts over $[0,\tau_x]$ , which includes initial withdrawal, annuity payments, and rental income by depreciating the value of the home used as collateral up to time $\tau_x$ . Then, the borrower’s heirs receive a nonnegative balance $Bal(\tau_x)$ . Therefore, the accumulated payout received by the borrower and her/his heirs from 0 to $\tau_x$ can be expressed as

(3.1) \begin{eqnarray} \Psi(H_{\tau_x}, 0,\tau_x;\;\tau_x) &\;:\!=\;& \psi(H(0,\tau_x),0,\tau_x;\;\tau_x) + Bal(\tau_x), \end{eqnarray}

where

(3.2) \begin{equation} \psi(H(0,\tau_x),0,\tau_x;\;\tau_x) \;:\!=\; \omega e^{r \tau_x} + \int_{0}^{\tau_x} e^{r(\tau_x-u)} \times \delta H_u du + c \times\int_{0}^{\tau_x} e^{r(\tau_x-u)} du, \end{equation}

which represents the accumulated value at time $\tau_x$ of the contract payouts the borrower received between time 0 and $\tau_x$ . Here, $H(0,\tau_x)$ is defined as a stochastic path of the home value over the time period $[0,\tau_x]$ . As described in (3.2), the borrower receives by time $\tau_x$ an accumulated payout $\psi(H(0,\tau_x),0,\tau_x;\;\tau_x) $ , which includes the accumulation of initial withdrawals, rental income financed from the house appraisal, and annuity payments. Both the charges of MIP and interest reduce the amount of the borrower’s nonnegative bequest $Bal(\tau_x)$ to their heirs.

At each observation time t, $t \in [0,\tau_x)$ , the accumulated payout between time 0 to $\tau_x$ can be broken into two components:

(3.3) \begin{equation} \psi (H(0,\tau_x),0,\tau_x;\;\tau_x) = \psi (H(0,t),0,t;\;\tau_x)+ \psi (H(t,\tau_x),t,\tau_x;\;\tau_x), \end{equation}

where H(0, t) and $H(t,\tau_x)$ are defined as the paths of the house value in time intervals [0, t] and $[t,\tau_x]$ , respectively. By $\psi(H(0,\tau_x),0,t;\;\tau_x)$ , we denote the accumulated contract payouts represented at time $\tau_{x}$ , which is the time- $\tau_x$ value of the borrower’s loan over [0, t], given by

\begin{equation*} \psi(H(0,t),0,t;\;\tau_x)\;:\!=\; \omega e^{r\tau_x} + \int_{0}^{t} e^{r(\tau_x-u)} \times \delta H_u du + c \times \int_{0}^{t} e^{r(\tau_x-u)}du. \end{equation*}

Similarly, the value at time $\tau_x$ of the borrower’s loan receipts over $[t,\tau_x]$ is defined as

\begin{equation*} \psi(H(t,\tau_x),t,\tau_x;\;\tau_x)\;:\!=\; \int_{t}^{\tau_x} e^{r(\tau_x-u)} \times \delta H_u du + c \times \int_{t}^{\tau_x} e^{r(\tau_x-u)}du. \end{equation*}

Note that we need the above decomposition of $\psi(H(0,\tau_x),0,\tau_x;\;\tau_x)$ in order to represent the contract value of an RAM at arbitrary t, prior to the death time $\tau_x$ .

In what follows, we define the borrower surrender strategy based on the driving factor of home appreciation. For a situation when an immediate termination can lead to a substantial cash payout of the nonnegative balance, borrower’s surrender of an RM could be worth more than her/his alternative strategy of staying in the program and continuing to pay ongoing charges of MIP and interest. In practice, a prudent RM borrower will consider surrender of the contract when the current home value is high. This observation motivates us to define an optimal surrender time as

(3.4) \begin{equation} \tau_B=\inf \{t \geqslant 0 \;:\; H_t \geqslant B_t\}=\inf \{t \geqslant 0 \;:\; H_t = B_t\}, \end{equation}

where the latter representation follows from the $\mathbb{Q}$ –almost surely continuity of Brownian paths. In (3.4), the barrier $B_t$ , $t \geqslant 0$ , is a given function of time that separates the regions of surrender and no-surrender.

In the Appendix, we prove that the surrender option is a threshold strategy such that it is always worth for the borrower to lapse the contract when the value of the home reaches a prespecified deterministic barrier $B_t$ . To be specific, we show that for any fixed time t less than or equal to $\tau_x$ , there is a value $H_t^*=B_t$ of the home above which the value of the contract for the borrower is less than the benefit available immediately by terminating the contract.

For a policymaker who wants to assess the surrender decision, it is tempting to find the probability $\mathbb{Q}\left(\tau_B \leqslant t\right)$ , for borrower’s surrender time $\tau_B$ occurring at or prior to the observation time t. Due to the fact that evaluating the surrender probability $\mathbb{Q}\left(\tau_B \leqslant t\right)$ is computationally difficult, for given times $t_i$ , $i=0,1,\cdots, n$ , the probability $\mathbb{Q}\left(\tau_B \gt t_i\right)$ can be approximated by the joint probability that $H_{t_j}<B_{t_j}$ for all $j \leqslant i$ . Hence, the surrender probability for the contract in force can be approximated by

(3.5) \begin{equation} \mathbb{Q}\left(\tau_B \leqslant t_i\right) = 1-\mathbb{Q}\left(\tau_B \gt t_i\right) \approx 1 - \mathbb{Q}\left( \bigcap_{j=0}^{i} \{H_{t_j}<B_{t_j}\}\right), \end{equation}

with $0=t_0<t_1<\cdots <t_n=T$ for a sufficiently large n.

Note that the surrender probability (3.5) does not account for the effect of mortality decrement occurring by time $t_i$ . For a market pool where the surrender experience is jointly considered with mortality decrement, RM policymakers may want to look at the joint distribution

(3.6) \begin{equation} \mathbb{Q}\left(\tau_B \leqslant t_i, \tau_x \gt t_i\right) = \mathbb{Q}\left(\tau_B \leqslant t_i | \tau_x \gt t_i\right) \times {}_{t_i} p_{x}. \end{equation}

An alternative way to show the financial incentive of an RM is based on the reference probability

(3.7) \begin{equation} \mathbb{Q}\left(H_t \geqslant B_t\right) = \Phi \left(d_2 \left(H_0,B_t,0,t\right)\right)\!, \end{equation}

which, compared to the surrender probability (3.5), is computationally tractable and hence we use it for assessing the surrender decision in the numerical results of Section 4.3. For instance, given a prespecified level of MIP and loan payment, the borrower observes at loan origination and perceives the chance of the value of the home exceeding the corresponding barrier level of $B_t$ over the future times. If the reference probability (3.7) is smaller and thus this chance is lower, the borrower will be more inclined to stay with the contract. We should note that the reference probability is a bound to the surrender probability by time t, in the sense that $\mathbb{Q}\left(H_t \geqslant B_t\right) \leqslant \mathbb{Q}\left(\tau_B \leqslant t\right)$ . For larger values of t, this bound turns to be less accurate since the surrender probability for such t becomes larger. Since the borrower surrenders for the event $\{H_t \geqslant B_t\}$ , the reference probability $\mathbb{Q}\left(H_t \geqslant B_t\right)$ can be also considered as a margin added to the surrender probability $\mathbb{Q}\left(\tau_B \leqslant t\right)$ , whose steepness/rate of change of the curve then reflects the level of the reference probability at time t. For a real-world return of the underlying home equity $\mu_H \gt r$ (under $\mathbb{P}$ ), it is clear that the reference probability $\mathbb{P}\left(H_t \geqslant B_t\right)$ is greater than its risk-neutral counterpart (3.7).

A rational borrower only surrenders when the value of the home exceeds the loan amount; otherwise, she/he can stay in the home with small cost by paying home insurance and property tax, and even continue to receive annuity payments and rental incomes. In consequence, the value of the barrier $B_t$ should be at least equal to the level of loan accumulation over time, that is $H_t \geqslant B_t \geqslant L_t$ . That is to say, for a borrower who wants to terminate the contract at time $t \in [0,\tau_x)$ , she/he will receive a surrender payout

(3.8) \begin{equation} \psi(H(0,t),0,t;\;t) +Bal(t) =\psi(H(0,t),0,t;\;t) + H_t-L_t, \quad \text{for $H_t \geqslant B_t$}. \end{equation}

Note that the nonnegative balance $Bal(t)=\left(H_t-L_t\right)^+$ in (3.8) will be immediately available to the borrower via foreclosure if the contract is surrendered at time $t \in [0,\tau_x)$ . Meanwhile, the borrower stops receiving future annuity payment and rental income with the accumulated payout $\psi(H(0,t),0,t;\;t)$ by time t. Then, the borrower’s obligation to continue the MIP and interest payments accrued to $L_t$ ceases.

3.2. Derivation of the surrender boundary

In this section, we follow the approach of Carr et al. (Reference Carr, Jarrow and Myneni1992) and derive the value function for determining the surrender boundary for RMs. We define the “book value" to reflect the fact that RM providers may want to know the values of the contract in their books of account, including the sum of any realized cash values and an unpaid “residual value" as their cost of ongoing liability.

As we state in the Theorem 3.1 below, the residual value $\widetilde{V}(H_t,t)$ of an RAM contract can be decomposed into a European loan guarantee in absence of a surrender activity until the event of death and a surrender premium charged against a potential rational termination by the borrower. To be specific, for $u \gt t \geqslant 0$ , the time-t price of the European part is given by

(3.9) \begin{eqnarray} v(H_t,t) &=& \int_{t}^{T} {}_{u-t} p_{x+t} \mu_{x+u} \left[\left(1-e^{-\delta (u-t)}\right) H_t + \frac{c}{r} \times \left(1-e^{-r(u-t)}\right)\right] du \\ \nonumber &+& \int_{t}^{T} {}_{u-t} p_{x+t} \mu_{x+u} \left[H_t e^{-\delta (u-t)}\Phi\left(d_1(H_t,L_u,t,u)\right) - L_u e^{-r(u-t)}\Phi\left(d_2(H_t,L_u,t,u)\right)\right] du, \end{eqnarray}

where $f_{x+t}(u)\;:\!=\; {}_{u-t} p_{x+t} \mu_{x+u}$ is the probability density function of a continuous lifetime of a borrower aged $(x+t)$ .

To determine the level of the surrender premium charged against potential rational termination, we first derive the corresponding instantaneous rate of charges. By terminating the contract, the borrower can cease the increasing amount of MIPs on the loan, and the ongoing interest charges on the loan, MIPs and annuity payments that may no longer match the value of the contract guarantees. In exchange, an equivalent level of surrender premium is priced and added to the corresponding European part of the residual value of an RM loan, for offering borrower’s surrender option. As shown in the Appendix, the instantaneous charge of surrender premium at time $u \in [t,\tau_x]$ is given by

(3.10) \begin{equation} \eta(u) du\;:\!=\;(\pi_r+ p_a) L_u du. \end{equation}

The instantaneous charge represented by Equation (3.10) implies that by means of an early termination, the borrower can save immediately on a surrender benefit, including the lender’s interest spread and MIP charges at the rate of $\pi_r +p_a$ on accumulation of the loan. It is noteworthy here that at the time of termination, the borrower can save interest charges at the rate of $r+\pi_r$ on the loan amount, but also she/he has to early repay the lender so that the borrower will lose the time value of the loan at the risk-free rate r. Thus, we observe from (3.10) that the borrower earns her/his instantaneous surrender benefit from the loan interest at the rate of $\pi_r L_u$ in net.

Since a rational termination can only occur before the time of death, the surrender premium at $t<\tau_x$ is then based on the expectation of the instantaneous charges over time, that is

(3.11) \begin{eqnarray} e\left(H_t,t\right) &=& \int_{t}^{T} {}_{u-t} p_{x+t} \times e^{-r(u-t)}\times \eta (u) \times \Phi \left(d_2(H_t,B_u,t,u)\right)du. \end{eqnarray}

In (3.11), the expectation of the instantaneous charge of (3.10) is discounted with borrower’s mortality decrement, while the payment can be triggered only when $H_u \geqslant B_u$ over time. Different from the MIPs financed from the loan and charged for crossover loss, the valuation of surrender premium (3.11) for a potential early exercise of nonnegative balance does not come with real cash flows. Note that rental incomes can be viewed as borrower’s implicit benefit of releasing equity value by discounting an equivalent amount from home appraisal. For a borrower’s rational surrender occurring when $H_u \geqslant B_u \geqslant L_u$ , she/he stops to receive the rental income but can save an equal amount of rental discount from the nonnegative balance Bal(u). As such, the borrower does not benefit or lose from ceasing rental income, and no rental benefit would be counted as part of the surrender premiums.

The formulas (3.9) and (3.11) lead us to the price of an RAM contract in Theorem 3.1, by which we identify the level of the boundary value $B_t$ over time. A detailed derivation of Formulas (3.9)–(3.11) and the proof of Theorem 3.1 can be found in the Appendix.

Theorem 3.1 For a filtered probability space $\left(\Omega, \mathcal{F}, \left(\mathcal{F}_t\right)_{t \geqslant 0}, P\right)$ , denote by $\mathcal{T}_{[t,\tau_x]}$ the set of all stopping times $\tau$ that are less or equal to $\tau_x$ and greater or equal to any observation time $t \in [0,\tau_x]$ . Then, it is worth for a rational borrower to terminate the contract at the observation time t when the surrender payout in (3.8) is no less than the book value $V(H_t,t)$ of a surrenderable RAM given by

(3.12) \begin{eqnarray} \nonumber V(h,t) &=& \psi(H(0,t),0,t;\;t) + \sup_{\tau \in \mathcal{T}_{[t,\tau_x]}} \mathbb{E}\left[e^{-r(\tau-t)}\left(\psi(H(t,\tau),t,\tau;\;\tau)+Bal(\tau)\right)|H_t=h\right]\\ &=& \psi(H(0,t),0,t;\;t) + \widetilde{V}(h,t), \end{eqnarray}

where the residual value of the contract by time-t admits the following decomposition

(3.13) \begin{equation} \widetilde{V}(H_t,t)=v(H_t,t)+e(H_t,t), \end{equation}

with $v(H_t,t)$ and $e(H_t,t)$ denoting, respectively, a European part and an early surrender premium.

Theorem 3.1 provides a way to calculate the value of an RAM with surrender option. However, since the surrender premium in (3.11) depends on the optimal surrender boundary $\{B_t, t \geqslant 0\}$ , one needs to compute it first. In the following, we derive the optimal surrender boundary condition in analogy to Kim and Yu (Reference Kim and Yu1996) and Bernard et al. (Reference Bernard, MacKay and Muehlbeyer2014b). For the determination of the surrender boundary, we observe that the time value of the contract guarantee fades away when it approaches T. Therefore, a rational borrower will terminate whenever the value of the home is greater than the loan accumulation at T. This implies that the barrier for the value of the home should be equal to the loan accumulation, that is, $B_{T}=L_{T}$ . For any $t \in [0,T)$ , along the surrender boundary, we have the following equation:

(3.14) \begin{equation} \Psi(H_t,0,t;\;t)=\psi(H(0,t),0,t;\;t)+H_t-L_t=\psi(H(0,t),0,t;\;t)+B_t-L_t. \end{equation}

Note that in (3.14), $\Psi(H_t,0,t;\;t)$ is a function of the value $H_t$ , while $\psi(H(0,t),0,t;\;t)$ is a function of the path H(0, t). Thus, we have

(3.15) \begin{equation} B_t-L_t = \widetilde{V}(B_t,t) = v(B_t,t)+e(B_t,t). \end{equation}

By substituting (3.9) and (3.11) into (3.15), we obtain

(3.16) \begin{eqnarray} \nonumber B_t &-& L_t = \int_{t}^{T} {}_{u-t} p_{x+t} \mu_{x+u} \left[\left(1-e^{-\delta (u-t)}\right) B_t + \frac{c}{r} \times \left(1-e^{-r(u-t)}\right)\right] du \\ \nonumber &+& \int_{t}^{T} {}_{u-t} p_{x+t} \mu_{x+u} \left[B_t e^{-\delta (u-t)}\Phi\left(d_1(B_t,L_u,t,u)\right) - L_u e^{-r(u-t)}\Phi\left(d_2(B_t,L_u,t,u)\right)\right] du, \\ &+& \int_{t}^{T} {}_{u-t} p_{x+t} \times e^{-r(u-t)}\times \eta (u) \times \Phi \left(d_2(H_t,B_u,t,u)\right)du. \end{eqnarray}

Based on Equation (3.16), in the following, we discuss methods of computing the barrier level $B_t$ for practical use.

3.3. Pricing with actuarial equivalence for surrenderable RM loans

In Section 3.2, we describe how to calculate the barrier level $B_t$ as a function of a given annuity payment c. Since the level of loan payment can also impact the solvency of HECM insurance fund, in this section, we develop an actuarial equivalence for identifying a fair level of loan payment in conjunction with the borrower’s surrender option.

Due to the fact that an ongoing payment of annual MIP is ceased upon borrower’s surrender at which the crossover loss becomes due for RM insurers, we consider a proxy loss function between the accumulated annual MIPs and the crossover loss occurring at the surrender time $\tau \in [0,\tau_x]$ , which is defined by

\begin{equation*} \Psi_0^l \left(H_{\tau},\tau\right)\;:\!=\; Loss(\tau) - \int_{0}^{\tau} e^{r(\tau-u)} p_a L_u du = \left(L_{\tau}-H_{\tau}\right)^+ - \int_{0}^{\tau} e^{r(\tau-u)} p_a L_u du. \end{equation*}

For an optimal stopping time $\tau^* \in \mathcal{T}_{[0,\tau_x]}$ that has maximized the value of the contract over the period $[0,\tau_x]$ , we have recovered its associated surrender boundary $\{B_t, t \in [0,T]\}$ by Theorem 3.1. With an initial MIP, we want to identify a fair level of the loan payment by applying the assumption of zero expected loss-at-issue in the sense that

\begin{equation*} 0 = \mathbb{E} \left[e^{-r \tau^*} \Psi_0^l \left(H_{\tau^*},\tau^*\right) \right]-p_0 H_0, \end{equation*}

or equivalently

(3.17) \begin{equation} p_0 H_0 + \mathbb{E} \left[\int_{0}^{\tau^*} e^{-ru} p_a L_u du \right] = \mathbb{E} \left[e^{-r \tau^*} Loss(\tau^*) \right]\!, \end{equation}

which suggests a perfect match between the actuarial present value (APV) of MIPs and the cost of insurance accounting for the borrower’s surrender option.

In Theorem 3.2, we derive the analytical formulas for actuarial equivalence (3.17) to identify the fair level of loan payment conforming to the rational surrender. The proof of Theorem 3.2 is similar to the one in Theorem 3.1 with its detail presented in Appendix.

Theorem 3.2 For a surrenderable RM loan, a fair level of annuity payment c for tenure payment option or initial withdrawal $\omega$ (with assumed $c=0$ ) for a lump-sum loan that satisfies the actuarial equivalence (3.17) can be solved from the following analytical representation

(3.18) \begin{equation} p_0 H_0 = \widetilde{V}^l \left(H_0,0\right) = \mathbb{E} \left[e^{-r \tau^*} \Psi_0^l \left(H_{\tau^*},\tau^*\right) \right] =v^l(H_0,0) + e^l\left(H_0,0\right)\!, \end{equation}

with the European price of the loss payout $\Psi_0^l$

(3.19) \begin{eqnarray} \nonumber v^l(H_0,0) &=& \int_{0}^{T} {}_u p_x \mu_{x+u} \left[L_u e^{-ru} \Phi \left(-d_2\left(H_0,L_u,0,u\right)\right) - H_0 e^{-\delta u} \Phi \left(\!-d_1\left(H_0,L_u,0,u\right)\right)\right] du \\ &-& \int_{0}^{T} {}_u p_x \times p_a L_u e^{-ru} du, \end{eqnarray}

and an early surrender premium given by

(3.20) \begin{eqnarray} e^l(H_0,0) &=& \int_{0}^{T} {}_u p_x \times \left[p_a L_u e^{-ru} - \int_{0}^{u} r p_a L_s e^{-rs}ds\right] \times \Phi \left(d_2\left(H_0,B_u,0,u\right)\right) du, \end{eqnarray}

respectively. In (3.19) and (3.20), the surrender boundary $\{B_u,u \geqslant 0\}$ and the solution of loan payment (i.e., c or $\omega$ ) conform to both Theorems 3.1 and 3.2.

In Theorem 3.2, the value of the discounted proxy loss $\Psi_0^l$ can be written as a European part, which amounts to the cost of $\Psi_0^l$ only paid out at the time of death, together with an early exercise premium for $\Psi_0^l$ in the situation when the optimal surrender strategy in Theorem 3.1 is exercised. From the insurer’s perspective, the early repayment of the loan allows the borrower to discontinue the ongoing annual MIP that should be paid until the death, while it also causes her/him to give up the time value of holding the past premium financing at the risk-free rate r. In consequence, as described in (3.20), additional cost of insurance needs to be added over the European part of the premium to compensate the insurer for the net surrender benefit received by the borrower. The purpose of Theorem 3.2 is to characterize a fair level of loan payment, together with the resulting boundary $\{B_t, t\geqslant 0\}$ , that reconciles both Theorems 3.1 and 3.2. In the next section, we provide a recursive algorithm for the recovery of barrier level $B_t$ as a function of annuity payment c. All other things being unchanged, we then use the Matlab “fzero" function to solve the constant c that complies with the equivalence (3.18).

3.4. Calculation of the surrender boundary

The integral equation in (3.16) can be used to compute the optimal surrender boundary $\{B_t, t \in [0,T]\}$ . Observe, however, that in order to obtain the value of the barrier $B_t$ at a specific time t, the optimal surrender barriers for future times must be known. Since $B_{T}=L_{T}$ at maturity T, we work backward through time to recursively recover the optimal surrender boundary. Because (3.16) does not have an analytic solution, numerical integration schemes must be used. Practically this is done by dividing the interval [0, T] into n subintervals $0=t_0<t_1\lt \cdots <t_n=T$ of equal lengths, that is, $\Delta_{t_{i}}=t_{i+1}-t_{i}=T/n$ , where times $t_i, i=0, \cdots,n-1$ , represent the only possible times for the termination due to the event of borrower’s surrender or death. For $k=1,\cdots, n$ , define

(3.21) \begin{eqnarray} && g_1(u,t_{n-k}) \;:\!=\; \left(1-e^{-\delta (u-t_{n-k})}\right) B_{t_{n-k}} + \frac{c}{r} \times \left(1-e^{-r(u-t_{n-k})}\right) \\ \nonumber &+& B_{t_{n-k}} e^{-\delta (u-{t_{n-k}})}\Phi\left(d_1(B_{t_{n-k}},L_u,t_{n-k},u)\right) - L_u e^{-r(u-t_{n-k})}\Phi\left(d_2(B_{t_{n-k}},L_u,{t_{n-k}},u)\right), \end{eqnarray}

and

(3.22) \begin{equation} g_2(u,t_{n-k}) \;:\!=\; e^{-r(u-t_{n-k})} \times \eta(u)\times \Phi \left(d_2(B_{t_{n-k}},B_u,t_{n-k},u)\right). \end{equation}

Then, the surrender premium in (3.11) can be approximated by using the method of rectangular integration, which results in the following approximations $I_1(k)$ and $I_2(k)$ , $k=1, \cdots, n$ , of the European value of the contract without surrender, and the surrender premiums over the time intervals $(t_{n-k}, t_n)$ , respectively:

(3.23) \begin{equation} I_1(k)\;:\!=\; \sum_{i=0}^{k-1} {}_{t_{n-k+i+1}-t_{n-k}} p_{x+t_{n-k}} \times {}_{{\Delta}_{t_{n-k+i+1}}} q_{x+t_{n-k+i+1}} \times g_1(t_{n-k+i+1},t_{n-k}), \end{equation}

and

(3.24) \begin{equation} I_2(k) \;:\!=\; \frac{T}{n} \sum_{i=0}^{k-1} {}_{t_{n-k+i+1}-t_{n-k}} p_{x+t_{n-k}} \times g_2(t_{n-k+i+1},t_{n-k}), \end{equation}

for $i \leqslant k-1$ and $k=1,2,\cdots,n$ , where n denotes the number of rectangles used for approximating the integral. In (3.23), we approximate the death probability ${}_{{\Delta}_{t_{n-k+i+1}}} q_{x+t_{n-k+i+1}} = 1 - {}_{{\Delta}_{t_{n-k+i+1}}} p_{x+t_{n-k+i+1}} \approx \mu_{x+t_{n-k+i+1}} \times {\Delta}_{t_{n-k+i+1}}$ , for a sufficiently large n.

Note that for approximating both the European part (3.23) and the early surrender premium (3.24) in discrete-time periods, the time of borrower’s death occurs after the observation time $t_{n-k}$ . Furthermore, borrower’s surrender can only happen prior to the death determination and at the beginning of each time interval, at which the ongoing annuity payment and MIP are paid. The numerical method of integration that we have employed in (3.23) and (3.24) has produced stable results, but it is possible that other quadrature methods may lead to more efficient algorithms. Next, we use the following iteration steps proposed by Bernard et al. (Reference Bernard, MacKay and Muehlbeyer2014b) for the recovery of the borrower’s surrender boundary, described in (3.15) and (3.16):

1. Step 1. $B_{t_n}=B_T=L_T$ .

2. Step 2. Recursively, for $k=1,\cdots, n$ , compute $I_1(k)$ and $I_2(k)$ in (3.23) and (3.24), respectively, to approximate the right part of (3.16) and solve the following equation for the only unknown $B_{t_{n-k}}$ :

   (3.25) \begin{equation} B_{t_{n-k}} - L_{t_{n-k}} = I_1(k) + I_2(k). \end{equation}

3.5. Calculation of the surrender boundary with prepayment penalty

Although rational borrowers will only terminate the contract when $H_t \geqslant L_t$ , RM insurers may still experience a loss due to a crossover event, when the contract is foreclosed due to the borrower’s mobility issues such as relocation or the financial hardship of maintaining their contractual obligations. As suggested by some Canadian RM providers, such as Equitable Bank, insurers may impose additional management fees and financing costs caused by the lapse of the contract. Also, borrowers may surrender for refinance and/or home sale, which involve a transaction cost and/or a cost of moving and renting. To account for these expenses, we propose a prepayment penalty charge at the percentage rate of $\kappa_t=e^{\kappa t}-1$ , with constant $\kappa \gt0$ at a surrender time t, representing an assessment of the additional equity release so that the borrower would get much less than $H_t-L_t$ through surrender. Due to the nonrecourse provision, the loan accumulation is alway increasing and hence the corresponding cost of a crossover event tends to increase over time. Thus, we assume that $\kappa_t$ is an exponential function in t, which is levied on the loan accumulation $L_t$ at the surrender time $t \in [0,T]$ . In what follows, we describe a method of finding the surrender boundary with the proposed penalty charges. The details of the derivation can be found in the Appendix. The numerical approximation to determine the surrender boundary with prepayment charges follows a backward procedure, which can be summarized in two main steps:

1. Step 1. $B_{t_n}=B_T=L_T$ .

2. Step 2. Recursively, for $k=1,\cdots, n$ , in analogy to the integral expressions in (3.23) and (3.24), compute $I_{1,\kappa}(k)$ and $I_{2,\kappa}(k)$ to approximate the right part of (3.26) and solve the following equation for the only unknown $B_{t_{n-k}}$ :

   (3.26) \begin{equation} B_{t_{n-k}} - e^{\kappa t_{n-k}} L_{t_{n-k}} = I_{1}(k) + I_{2,\kappa}(k), \end{equation}
   where $I_{1}(k)$ is given in (3.23). $I_{2,\kappa}(k)\;:\!=\; \frac{T}{n} \sum_{i=0}^{k-1} {}_{t_{n-k+i+1}-t_{n-k}} p_{x+t_{n-k}} \times g_{2,\kappa}(t_{n-k+i+1},t_{n-k})$ , for $i \leqslant k-1$ and $k=1,2,\cdots, n$ . $g_{2,\kappa}(u,t_{n-k}) = e^{-r(u-t_{n-k})} \times \eta_{\kappa}(u) \times \Phi \left(d_2(B_{t_{n-k}},B_u,t_{n-k},u)\right)$ . $\eta_{\kappa}(u)= \left(\pi_r + p_a + \kappa \right) e^{\kappa u} L_u +c \times \left(e^{\kappa u} -1\right)$ .

Note that the inclusion of prepayment charges in (3.26) increases the expected surrender costs, including the surrender premiums and penalty charges at the time of termination. However, it will not affect the value of the European part reflecting the value of the contract only with death termination.

4.1. Comparison of surrender boundary and surrender probability

In the following, we investigate the financial incentive of RMs by comparing the borrower surrender boundary and the probability of breaching it based on a variety of loan payment. Our findings are summarized as follows.

First, the level of the loan payment identified with the insurer’s pricing criterion has a considerable impact on the value of the loan and hence the level of borrower’s surrender boundary and probability. In Figure 1, we identify the respective surrender boundaries characterized in Theorem 3.1 (blue lines) and then use the Monte Carlo method with $10^6$ repetitions for approximating the corresponding surrender probabilities excluding the effect of mortality decrement that have been defined in (3.5) (red lines). Since a standard RM loan generally bears a significant amount of interest and MIP expenses, it lowers the borrower accessibility of the loan. On top of the fair values of $\omega=16.6780$ and $c=2.2343$ identified by the pricing criterion (3.17), we consider different levels of loan payment for the sensitivity test. As displayed in the left panel of Figure 1, for different $\omega$ all the initial barrier values fall below the house price, that is, $B_0 \lt H_0$ , and the corresponding surrender probability (3.5) is equal to one, at which it is financially unwise for a rational borrower to enter the loan. As explained by many authors, such as Szymanoski et al. (Reference Szymanoski, Enriquez and DiVenti2007) and Haurin et al. (Reference Haurin, Ma, Moulton, Schmeiser, Seligman and Shi2016) HECM borrowers are a highly selected sample with high liquidity needs. When the liquidity needs fade away, the borrower may optimally terminate the contract for surrender benefit. In contrast, an increase in the loan payment could reduce the borrower’s surrender incentive and lead to a higher initial barrier value than the house price. For instance, in the right panel of Figure 1, one can find that the surrender probability shifts downward for a sufficiently high level of the annuity payment that gives rise to a higher initial barrier value than the house price.

Figure 1. Surrender boundary and probability with no mortality decrement for a lump-sum withdrawal (left panel) and tenure annuity payment (right panel) respectively.

In addition, most market experience with rational surrender can happen in the early-through-mid term of the loan life. As displayed in the left panel of Figure 1, at all levels of $\omega$ the lower initial barrier level than the value of the home should cause an immediate termination for rational borrowers. In the absence of mortality decrement, when the annuity level leads to a higher initial barrier level than the price of the home, we observe from the right panel of Figure 1 that the curves of the surrender probability (3.5) (red lines) strictly increase in time, while they are concave upward in the first several years and then turn to be concave downward in the later years of the contract. As explained in Section 3.1, this concavity of the curves can be further confirmed by the reference probability (3.7), indicating that the surrender event of $\{H_t \geqslant B_t\}$ is more likely to occur in the early years than that in the later years. In the case that the occurrence of mortality decrement is jointly considered after the surrender by time t, the left panel of Figure 2 depicts the joint probabilities (3.6) given different levels of lump-sum withdrawal and annuity payment. For the respective cases $c=2.2343$ , $\omega=16.6780$ , $\omega=25$ and $\omega=35$ , compared to the probability curves (red lines) in the right panel of Figure 1, the joint distribution (3.6) illustrated in the left panel of Figure 2 is equivalent to the survival function (2.6), since the probability $\mathbb{Q}\left(\tau_B \leqslant t_i | \tau_x \gt t_i\right)$ is equal to one for all $t_i$ . For the cases when $c=5.0$ and $c=5.5$ , the probability curves slightly drop in the early years, while weighted by mortality decrements, they are significantly depressed in the later years, which leads to a right skewness of the entire distribution. In particular, it is impractical for a borrower to terminate for $t \geqslant 30$ , as the joint probability (3.6) tends to be zero. This observation confirms that for a market pool of RM policies, one can anticipate that most surrender occurrences prior to the death termination will occur in the early-through-mid term of the contract.

Figure 2. Surrender probability with mortality decrement (left panel) and EPVs of Bal(t) and Loss(t) (right panel), respectively.

Finally, the surrender probability generally decreases due to the accelerated accumulation in the later years of the loan. This reduces the borrower’s surrender incentives but could also increase the inherent riskiness of the loan to the insurer. To assess the inherent riskiness of an RM loan over time, in the right panel of Figure 2, we depict the expected present values (EPVs) of nonnegative balance and crossover loss at a varying level of generic time t. Due to a greater possibility of a crossover event occurring in the later years of the contract, for example, at the maturity $T=40$ , we examine the level of crossover risk by checking the reserve of the nonnegative balance and the size of the crossover loss with the predetermined level of loan payment. As illustrated in the right panel of Figure 2, with the base cases $\omega=16.6780$ and $c=2.2343$ , we obtain the corresponding EPVs of Bal(T) and Loss(T) in (2.4) and (2.5), respectively, as $\mathcal{C}=28.1372$ and $\mathcal{P}=2.6738$ for the lump-sum loan, and as $\mathcal{C}=4.8813$ and $\mathcal{P}=41.5151$ for the tenure annuity loan. These results show that the loan accumulation with annuity payment option exhausts the borrower’s nonnegative balance more quickly than that with the lump-sum withdrawal and hence causes a significant level of crossover loss at T. In contrast, the lump-sum option reserves more cash value in the nonnegative balance than the annuity loan, with a substantial amount of $\mathcal{C}$ that leads to a negligible $\mathcal{P}$ . One should note that the future mortality decrement of the borrower aged 70 has a significant influence on the actual risk level of the loan. As shown in the left panel of Figure 2 (black line), the survival probability is approaching zero for $t \gt 30$ , indicating that the above significant cost of crossover loss $\mathcal{P}$ illustrated as examples can rarely happen from real market experience.

4.2. General pattern of surrender boundary and reference probability

As stated in Theorem 3.1 and Equation (3.15), the value of the barrier $B_t$ , for each $t \in [0,T]$ , is a break-even point where borrower’s surrender payout is exactly equal to the book value $V(H_t,t)$ with an ongoing liability of the contract. Instead of simulating the surrender probability (3.5) or (3.6), in Sections 4.2 and 4.3 we use the reference probability (3.7) based on the barrier level $B_t$ to assess the likelihood of the borrower surrender, which, as indicated in Section 3.1, is more computationally tractable than the other two. For both types of RM such as the lump-sum withdrawal and the annuity loan, we examine the sensitivity of their surrender boundary with respect to the parameters r, $H_0$ , $\delta$ , $\sigma_H$ , $\pi_r$ , $p_0$ , $p_a$ and $\kappa$ . In this section, we delineate the general patterns of surrender boundary and reference probability based on the detailed sensitivity analyses followed in Section 4.3.

In our analyses, both curves of the surrender boundary and its corresponding reference probability are derived as a tool for identifying the timing and the likelihood of borrower surrender. In consequence, the general pattern for the curves of surrender boundary and its associated reference probability reflects the progression of the embedded financial value throughout the lifetime of the loan. With a fair level of loan payment, we observe from Figures 3–10 that the curve of the surrender boundary increases over time, while it turns to be much steeper in the later years than the early years of the contract. Correspondingly, the curve of the reference probability starts at a high level in the early years of the contract and then declines in the following years.

Figure 3. Surrender boundary and reference probability versus the interest rate r for a lump-sum option (left panel) and tenure annuity payment (right panel), respectively.

Figure 4. Surrender boundary and reference probability versus the house price $H_0$ for a lump-sum option (left panel) and tenure annuity payment (right panel), respectively.

Figure 5. Surrender boundary and reference probability versus the rental yield $\delta$ for a lump-sum option (left panel) and tenure annuity payment (right panel), respectively.

Figure 6. Surrender boundary and reference probability versus the house price volatility $\sigma_H$ for a lump-sum option (left panel) and tenure annuity payment (right panel) respectively.

Figure 7. Surrender boundary and reference probability versus the interest spread charge $\pi_r$ for a lump-sum option (left panel) and tenure annuity payment (right panel), respectively.

Figure 8. Surrender boundary and reference probability versus the initial MIP rate $p_0$ for a lump-sum option (left panel) and tenure annuity payment (right panel), respectively.

Figure 9. Surrender boundary and reference probability versus the annual MIP rate $p_a$ for a lump-sum option (left panel) and tenure annuity payment (right panel), respectively.

Figure 10. Surrender boundary and reference probability versus the prepayment charge $\kappa$ for a lump-sum option (left panel) and tenure annuity payment (right panel), respectively.

To understand the above pattern, we look more closely into both the intrinsic and time values of an option. It is well known that the intrinsic value of an option represents what it would be worth if the buyer exercised it immediately, while the time value represents the possibility that the option would increase in value before its expiration date. In the early years of the contract, the nonnegative balance $Bal(t)=\left(H_t-L_t \right)^+$ is deeply in the money, with a relatively small loan accumulation. Since at this point the liability of a crossover event, $Loss(t)=\left(L_t-H_t\right)^+$ , which can be viewed as a put option with its increasing strike $L_t$ , for $t \in [0,T]$ , is deeply out of the money, it is financially unwise for the borrower to stay in the program while adding financing charges of MIP and interest with no need for crossover loss protection, as it simply decreases the value of Bal(t) that she/he can receive by surrender. During this period, when the liquidity needs fade away, the borrower is inclined to terminate the contract as long as the value of the home is sufficiently larger than the loan amount. We thus find that the curve of the boundary stays in a lower level and the corresponding reference probability stays at a high level in the first several years.

With the passage of time, the loan accumulation grows exponentially by payment contributions from annuity payments, premiums and interest. In the later years of the contract, although the increasing level of MIP and interest added to the loan decreases both intrinsic and time values of the nonnegative balance Bal(t), the borrower tends to start receiving the nonrecourse benefit of her/his RM loan. Meanwhile, the put liability Loss(t) to the insurer is still out of the money but its time value increases. Due to her/his increasing demand for crossover loss protection, these findings indicate that the borrower is more willing to pay the MIP and interest charges for a time value of the put liability Loss(t) that potentially turns to be in-the-money soon. In the meantime, the borrower is more likely to stay in the program and take advantage of its out-of-the-money status of the nonnegative balance Bal(t), since she/he can potentially benefit from the nonrecourse clause of an RM loan by receiving a loan amount greater than the value of the home, which is equivalent to the amount of Loss(t) undertaken by the insurer. This trend lasts until the maturity of the contract, at which the time values of the nonnegative balance guarantee Bal(t) and the put liability Loss(t) both fade away. Consequently, the level of the reference probability decreases in time to a relative minimum at maturity, while the level of the corresponding surrender boundary increases more quickly and is capped at the level of $L_T$ .

At this point, we have explained the pattern of the surrender boundary and its associated reference probability observed from the numerical examples based on the level of loan payment identified with the pricing condition (3.17). We should also mention that since both the level of loan payment and the distribution of death time can affect the initial barrier level and hence the surrender probability, some other patterns may be observed. As depicted in the right panel of Figure 1, for certain levels of loan payment that lead to a higher initial barrier value than the house price, that is, $c=5.0$ or $c=5.5$ , the concavity of the surrender probability curve (3.5) changes from up to down. As explained in Section 3.1, this indicates that the reference probability (3.7) could first climb from zero to its relative maximum with a growing amount of the loan devaluing the borrower’s nonnegative balance. Then, it decreases in time to a relative minimum at maturity as the nonrecourse feature of the loan takes into effect.

4.3. Sensitivity of surrender boundary and reference probability

In Figures 3–10, we perform the sensitivity test in regard to both types of RM loans by varying the levels of certain market parameters while keeping everything else unchanged. Unless otherwise specified, the numerical results are based on the base level of the loan payment, that is $\omega=16.6780$ for the lump-sum withdrawal or $c=2.2343$ for the tenure annuity loan. Generally, when compared to the lump-sum option, the annuitization of RM payout could save the interest expense and thus increase the loan accessibility to the borrower over time, which discourages borrower surrender with a relatively higher boundary level.

In Figure 3, we compute the surrender boundary for different levels of the interest rate r with the predetermined rates of MIP and loan payment. As indicated in the Introduction, Jiang and Miller (Reference Jiang and Miller2019) account for the refinance termination due to appreciation of the value of the home and decreases in the interest rate. Their data also suggest that borrowers are more likely to take advantage of rising housing prices and declining interest rates through refinancing in the first few years after origination. While our model is assumed to capture borrower’s surrender strategy driven by home appreciation, by performing a sensitivity test it also sheds lights on those surrender strategies driven by the interest rate. For example, we observe from Figure 3 that the surrender boundary shifts upward with the interest rate r. With the fair level of loan payment at the base rate $r=2\%$ , a decline in the rate of interest can slow down the accumulation of the loan, and then shifts the nonnegative balance value Bal(t) to be more in-the-money at the time of the termination of the contract. Meanwhile, the put liability Loss(t) becomes less valuable, which exacerbates the mismatch between the charges of MIPs and interest and the contract guarantees. Due to this value reduction, the curve of the surrender boundary (with an increasing reference probability) is pushed down, and thus, in a climate of low interest rates, a rational borrower is more inclined to terminate the contract. In general, borrower’s surrender motive can be very sensitive to interest rates. When compared to the lump-sum option, the annuitization of the loan payout leads to a higher sensitivity of surrender boundary or the associated reference probability in response to the change of interest rate.

In practice, borrowers typically have different values of their equity at the point of entering the contract. On top of the base cases that $\omega=16.6780$ and $c=2.2343$ for $H_0=100$ , we solve the fair level of loan payment for the lump-sum option as $\omega = 15.0102$ and $\omega = 18.3458$ for $H_0=90$ and $H_0=110$ , respectively, while, for the annuity loan, $c=2.0108$ and $c=2.4577$ , respectively. We observe that a borrower who has more initial equity values can receive a larger amount of loan payment, resulting that as observed from Figure 4 the surrender boundary is pushed up because of the increased value of the contract. Since the boundary improves with a larger level of loan release in proportion to the initial equity value, we observe that the reference probability does not change for both types of RM loan. With the base level of loan payment, Figure 5 displays the surrender boundaries for varying levels of the rental income $\delta$ as an equity discount. Since the rental discount reduces the appraised value of the home used as collateral over time, the higher discount rate exhausts the nonnegative balance more quickly. We thus observe that the surrender boundary shifts upward with the resulting lower curve of the reference probability over time. In Figure 6, we present the surrender boundary for varying levels of house price volatility $\sigma_H$ . We are aware that increases in house price volatility push up the surrender boundary with a reduced reference probability, since the higher level of house price volatility makes the guaranteee of nonnegative balance Bal(t) more valuable. Therefore, a rational borrower would prefer to stay in the program when the housing market is volatile.

In Figures 7–9, we present the surrender boundary for varying levels of interest spread charges $\pi_r$ and rates of initial and annual MIP, $p_0$ and $p_a$ , respectively. Generally, an increase from these parameters accelerates the loan accumulation over time. Hence, the surrender boundary is pushed upward, since the barrier level is not lower than that of the loan accumulation. All other things being unchanged, this encourages the borrower from staying in the program. Thus, we are aware that the curve of the reference probability is pushed down. As discussed in Section 3.5, for business operations RM lenders may not want their borrowers to lapse the loan and thus a penalty charge could be imposed by them. In Figure 10, we propose a penalty percentage charge in the form of $\kappa_t=e^{\kappa t}-1$ , where the force of the surrender penalty $\kappa$ is constant. Then, we show the sensitivity of the optimal surrender boundary to the proposed penalty charges for $\kappa=0\%$ , $\kappa=0.25\%$ and $\kappa=0.50\%$ . When a borrower terminates the contract, the lender sells the home used as collateral to reclaim the loan amount of $e^{\kappa t}L_t$ with penalty charges. Then, the borrower receives the balance in the amount of $H_t-e^{\kappa t}L_t$ . We find that the surrender boundary shifts upward, while the corresponding reference probability decreases with $\kappa$ . These penalty charges increase the borrower’s costs and that reduces the incentives to terminate the contract. Since it is impossible for borrowers to lapse the contract in the limiting year, the surrender boundaries drop to the level of terminal loan amount at maturity. By charging this prepayment penalty, RM programs can control the surrender boundary effectively throughout the lifetime of the contract, and then use these charges to cover program losses whenever a crossover event occurs.