Printer Friendly

PRICING AN INSURANCE PRODUCT THAT INTEGRATES REVERSE MORTGAGE WITH LONG-TERM CARE INSURANCE.

1. INTRODUCTION

Reverse Mortgage is an inviting financial lending product offered to any senior citizen who owns a house. In the case of standard home mortgage loan, the borrower repays the loan periodically to the lender (say, bank). As the name suggests, the 'reverse mortgage' derives this nomenclature due to the periodic cash flow from the lender (i.e., the financial institution issuing the reverse mortgage) to the borrower (namely, the senior homeowner). The homeowner pledges his property as collateral to the particular lending institution in exchange for cash, rather than selling the property for cash. Some special financial institutions, such as the state and local housing agencies, credit unions, insurance companies, and the banks, may offer the reverse mortgage. Any amount that a homeowner can receive depends primarily on the value of the property, the homeowner's age, and the current interest rate. Borrowers can choose one of several types of payment options, such as a lump sum, fixed monthly payment (until death), line of credit, or a combination of payment options. Reverse mortgage has some advantage, including: the reverse mortgage recipient can continue to live in the house; the utilization of the obtained cash is flexible and diverse; the homeowner can defer handing over the house until the homeowner dies; there are no tax requirements for the reverse mortgage borrower. Also, the cash obtained by releasing the borrower's home equity can be used for daily living costs, house maintenance, and long-term care insurance, and other relevant expenses.

Reverse mortgage is normally categorized by law into two categories, viz., collateral reverse mortgage and ownership conversion reserve mortgage. The collateral reverse mortgage can be redeemed and ownership conversion reverse mortgage not. In the case of the collateral reverse mortgage, the borrower is able to redeem the reverse mortgage by repaying the loan principal and accumulated interests through property sale at any time from the mortgage's effective date to due date. Of course, when the reserve mortgage contract is due, the borrower can choose a financial institution to auction off the property to repay loan and due interest. Home Equity Conversion Mortgage System is a typical collateral reverse mortgage in USA. In the case of the ownership conversion reverse mortgage, the borrower enters into a contract with a lending institution to obtain an annuity until his death, and at death the property ownership is transferred to the lender. Rente Viager is a typical ownership conversion reverse mortgage in France.

It is for some time now that the funds from the reverse mortgage are being used to pay for long-term care costs. The relevant research mainly discuss qualitatively its feasibility, effectiveness, opportunity, and challenges. Firman (1983) suggests that one should analyze the potential of utilizing home equity conversion mortgage system as a means of supplementing cash income and long-term care insurance. Jacobs and Weissert (1987) estimates that when home equity conversion plan is available, two-thirds of all elderly homeowners could afford an adequate long term care insurance policy. Benejam (1987), Benjamin (1992) and Gibbs (1992) further discuss the utility of home equity conversions to fund long term care for the elderly. Rasmussen et al. (1997) presents a more detailed view of reverse mortgage as a financial tool for tapping housing equity for various purposes and at various stages in the life cycle; one of main uses is to fund long-term care. Ahlstrom et al. (2004) explains the basic features of reverse mortgage and the long term care insurance. They outline the opportunities and challenges of linking reverse mortgage with the long term care insurance. Stucki (2005) discusses the consumer's attitude toward using reverse mortgages to fund the long term care insurance, product design barriers, and the role of government. Stucki and Group (2006) supports reverse mortgage as a financial instrument to fund the long-term care and shows that it would be beneficial for improving the senior citizen's life after retirement. In addition to the quantitative results, Murtaugh et al. (2001) designs a product combining reverse mortgage with long term care insurance. The idea there is to uitilize reverse mortgage to finance the long-term care insurance. Using the data from the 1986 National Mortality Followback Survey in USA, they investigate the premium, annuity, and risk according to the sex, age, and disability. Their research shows that this particular combination of products will give a 21% increases in the potential customers, and hence it is beneficial for the insurer to develop the bundled reverse mortgage and the long-term care insurance.

The risk of integrating the reverse mortgage with long term care insurance involves particularly the housing price risk, interest rate risk, disability risk, and the longevity risk. In order to rationally price this combined product, one must build an appropriate model that takes into account the above risks. In general, the risk of housing price is modeled in two ways. The first one is to fit the time series model based on the historical data of the housing price, as discussed by Nothaft et al. (1995), Chinloy et al. (1997), Chen et al. (2010b), and Li et al. (2010). The second one is to assume directly that the dynamics of housing price is driven by a forward stochastic differential equation, as in Bardhan et al. (2006), Wang et al. (2008), Mizrach (2008), Huang et al. (2011), Chen et al. (2010a), Lee et al. (2012), and Tsay et al. (2014).

The literature on classical interest rate model includes: the Dothan (1978) model, Vasicek (1977) model, Cox, Ingersoll and Ross (1985) model, Exponential Vasicek model, Hull and White (1990) model, Black and Karasinski (1991) model, Mercurio and Moraleda (2000) model, the CIR++ model, and the Extended Exponential Vasicek model (Brigo and Mercurio, 2006). Du Pasquier (1912, 1913) introduce a three-state (active, disabled and dead) Markov model to describe the invalid or sickness process, and derive the full differential equations for the transition probabilities. His work lays the foundations for the application of the multi-state Markov model to the long-term care insurance, disability insurance, and critical illness. Fong et al. (2015) estimates the transition intensities of the above three-state Markovian model with the generalized linear model based on a large sample of elderly in the USA.

This article is organized as follows. Section 2 presents the models of risk factors. In Section 3, we first design the product that integrates reverse mortgage with long term care insurance, and then derive the pricing model for the bundled product by the principle of expected balance between gain and payment. We also analyze the monotonicity of the lump sum, annuity, and annuity payment factors for the parameters of housing price and interest rate model. Section 4 provides numerical results to examine how the housing price risk, interest rate risk, and longevity risk impact the annuity payment, the expectation of total annuity present value, and the annuity payment factors. Finally, in Section 5 we draw conclusions about our findings.

2. MODELS OF RISK FACTORS

The main risks involved with reverse mortgage, as pointed out by Szymanoski (1994), include longevity risk, interest rate risk, and property value risk. In addition to these risks, the product integrating reverse mortgage with long-term care still faces other risks such as the disability risk. In order to obtain a suitable model to price the combined product, we must first explore how to describe these risk factors that the combined product enforces. In this section we employ the Black-Scholes model to simulate the dynamics of home price, the Ornstein-Uhlenbeck process to drive the instantaneous interest rate, and a three-state Markov chain to model the disability risk and longevity risk.

2.1. Home Pricing Model. We assume that the home price at time t follows the Black-Scholes model

(1) dH(t) = H(t) [[[mu].sub.h]dt + [[sigma].sub.h]d[W.sub.h](t)], 0 [less than or equal to] t [less than or equal to] T, H(0) = [H.sub.0],

where H(t) is defined on a complete filtered probability space ([OMEGA], F, P, [{[F.sub.t]}.sup.T.sub.t=0]) with the natural filtration F = [sigma]{[F.sub.t], 0 [less than or equal to] t [less than or equal to] T}. Here, {[W.sub.h](t), 0 [less than or equal to] t [less than or equal to] T} is a P- standard Brownian motion, [[mu].sub.h] is the annual average return rate of housing price, and ah is the annual volatility of housing price, assuming [[sigma].sub.h] > 0.

Applying Ito's formula to ln(H(t)), we have

(2) d[ln(H(t))] = ([[mu].sub.h] - 1/2 [sigma]2/h) dt + [[sigma].sub.h]d[W.sub.h](t).

Integrating both sides of the Equation (2) from s to t, we obtain the explicit solution

H(t) = H(s) exp {([[mu].sub.h] - 1/2[[sigma].sup.2]/h) (t - s) + [[sigma].sub.h]([W.sub.h](t) - [W.sub.h](s))}, 0 [less than or equal to] s [less than or equal to] t [less than or equal to] T.

Thus, the conditional mean of H(t) given [F.sub.s] is

(3) [mathematical expression not reproducible].

In particular, the mean of H(t) is

(4) [mathematical expression not reproducible].

2.2. Interest Rate Model. We next assume that the instantaneous short-rate dynamics evolves as an Ornstein-Uhlenbeck process with constant coefficients (i.e., the Vasicek model (Vasicek, 1977)). Specifically, in the (complete) filtered probability space ([OMEGA], F, P, [{[F.sub.t]}.sup.T.sub.t=0]), the interest rate process is governed by the following stochastic differential equation

(5) dr(t) = [[alpha].sub.r]([[mu].sub.r] - r(t))dt + [[sigma].sub.r]d[W.sub.r](t), 0 [less than or equal to] t [less than or equal to] T, r(0) = [r.sub.0],

where [r.sub.0], [[alpha].sub.r], [[mu].sub.r], [[sigma].sub.r] are positive constants, and {[W.sub.r](t), 0 [less than or equal to] t [less than or equal to] T} is a P-standard

Brownian motion, independent of {[W.sub.h](t), 0 [less than or equal to] t [less than or equal to] T}.

Applying Ito's formula to [mathematical expression not reproducible], we obtain

(6) [mathematical expression not reproducible].

Integrating both sides of Equation (6) over (s, t], we arrive at

(7) [mathematical expression not reproducible].

With some trivial computations, we have

(8) [mathematical expression not reproducible].

For the derivation of Eq. (8) we refer to Norberg (2004).

2.3. Temporally Inhomogeneous Markov Chain Model. We designate t = 0 to be the time at which a policy combining reverse mortgage with long-term care is signed. We assume that the process {[[xi].sub.t], t [greater than or equal to] 0} is a temporally inhomogeneous Markov chain taking values in the state space S = {1, 2, 3}, where {[[xi].sub.t], t [greater than or equal to] 0} is defined on the filtered probability space (Q, F, P, [{[F.sub.t]}.sup.T.sub.t=0]), independent of {[W.sub.h](t), 0 [less than or equal to] t [less than or equal to] T} and {[W.sub.r](t), 0 [less than or equal to] t [less than or equal to] T}. Here, the states 1, 2, and 3 respectively represent that the insured is not-disabled, disabled, and dead. Clearly, the states 1 and 2 are transient and the state 3 is absorbing. Herein and after, [[xi].sub.t] denotes the status of the insured at time t [member of] [0, T). Then, the events {[[xi].sub.t] = 1}, {[[xi].sub.t] = 2} and {[[xi].sub.t] = 3} respectively represent that the insured is not-disabled, disabled, and dead at time t. We disregard the possibility of recovery from disability to health. Hence, the directed line from 2 to 1 does not exist in Figure 1. The transition probabilities and intensities are denoted respectively by

[mathematical expression not reproducible].

With the Chapman-Kolmogorov equation, the transition probabilities and intensities satisfy the following system of differential equations:

(9) d/dt[p.sub.11](s, t) = -p11(s, t)[[[lambda].sub.12](t) + [[lambda].sub.13](t)],

(10) d/dt[p.sub.12](s, t) = [p.sub.11](s, t)[[lambda].sub.12](t) - [p.sub.12](s, t)[[lambda].sub.23](t),

d/pt[p.sub.13](s, t) = [p.sub.11](s, t)[[lambda].sub.13](t) + [p.sub.12](s, t) [[lambda].sub.23](t),

(11) d/dt [p.sub.22](s, t) = [p.sub.22] (s,t) [[lambda].sub.23](t),

d/dt [p.sub.23](s, t) = [p.sub.22](s, t)[[lambda].sub.23](t).

Using the boundary conditions [p.sub.ii](s, s) = 1 (i = 1, 2) and [p.sub.ij](s, s) = 0 (i = j) to Equations (9) and (11), we obtain

(12) [p.sub.11](s, t) = exp [-[[integral].sup.t.sub.s]([[lambda].sub.12](u) + [[lambda].sub.13](u)) du],

(13) [p.sub.22](s, t) = exp [-[[integral].sup.t.sub.s] [[lambda].sub.23] (u) du].

We solve explicitly for [p.sub.12](s, t) in two steps using Equation (10). First solve

d/dt [p.sub.12](s, t) = [p.sub.12](s, t)[[lambda].sub.23](t),

and then utilize the method of variation of constants to solve Equation (10). We then obtain

[p.sub.12](s, t) = [[integral].sup.t.sub.s] [p.sup.11](s, u)[[lambda].sub.12](u)[p.sub.22](u, t) du.

It is clear that

[p.sub.13](s, t) = 1 - [p.sub.11](s, t) - [p.sub.12](s, t),

[p.sub.23] (s, t) = 1 - [p.sub.22](s, t).

Let L be the limit of age in the Life Table, [x.sub.0] be the age of the insured at the start of the contract (t = 0), and T [??] L - [x.sub.0] be the insured's maximum future lifetime. Let [[tau].sub.i], (i = 1, 2), denote the sojourn time in state i. Since the states 1 and 2 are strictly transient, [[tau].sub.1] will be the time of first exit from state 1, and [[tau].sub.1] + [[tau].sub.2] is the time that the insured dies. The following Proposition 1 presents the density function for [[tau].sub.1] and the joint density function for ([[tau].sub.1], [[tau].sub.2]).

Proposition 1 Assume that the time-inhomogeneous Markov chain {[[xi].sub.t], t [greater than or equal to] 0} is a separable process. Then, (1) the density function for [[tau].sub.1] is

(14) [mathematical expression not reproducible],

(2) the conditional density for [[tau].sub.2] given {[[tau].sub.1] = x}, is

(15) [mathematical expression not reproducible]

and (3) the joint probability density function for ([[tau].sub.1], [[tau].sub.2]) is given by

(16) [mathematical expression not reproducible].

where

[mathematical expression not reproducible],

are as in Equation (12) and Equation (13), and the set D is given by

D := {(x, y) |0 < x < T, 0 < y < T - x}.

Proof: From the separability assumption and Markov property we obtain, for any t > 0, s > 0, i = 1, 2, and the separability set R = {k/2n k, n = 0, 1 ...}, that

(17) [mathematical expression not reproducible].

Setting i = 1, s = 0, t = x, and 0 < x < T in Equation (17), we obtain from Equation (12) that

(18) [mathematical expression not reproducible]

From Relations (12), (18), and

[mathematical expression not reproducible],

we obtain the density function (14) for [[tau].sub.1]. This proves Part 1.

Taking i = 2, s = [[tau].sub.1] = x, t = y, and 0 < y < T - [[tau].sub.1] in Equation (17) and using Equation (13), we have

(19) [mathematical expression not reproducible].

Now, Relation (15) results from Relations (13), (19), and

[mathematical expression not reproducible].

Also, the joint probability density function for ([t.sub.1], [t.sub.2]) is

[mathematical expression not reproducible].

Remarks 1 (1.) From Relations (14) and (15), we notice that the sojourn times [[tau].sub.1] and [[tau].sub.2] do not follow the exponential distribution. This is different from that of the time-homogeneous Markov chain case.

(2.) From Relations (18) and (19), we get

[mathematical expression not reproducible].

This implies that the occupancy probabilities for states 1 and 2 coincide with their respective transition probabilities.

3. PRICING THE INSURANCE PRODUCT INTEGRATING REVERSE MORTGAGE WITH LONG-TERM CARE

In this section, we first design an insurance product integrating reverse mortgage with long-term care. We then build the actuarial pricing model for the combined product based on the principle of balance between expected gain and expected payment. We also present the closed-form solutions of pricing models under a certain assumptions. Finally, we analyze the monotonicity of the lump sum, annuity, and annuity payment factors with respect to the parameters of home price and interest rate models.

3.1. Integrating Reverse Mortgage with Long Term Care. The bundled 'reverse mortgage--long term care' contract that we design has the following basic features:

(I) The insurer starts the payments of annuity to the insured at the beginning of the year following the signing of the contract. The annuity payment is terminated upon the death of the insured. More precisely, had the insured survived through the k-th year, (k [greater than or equal to] 1), the insurer would have paid the annuity sums [A.sub.1], [A.sub.2], ..., [A.sub.k] to the insured at the beginning of the second, third, ..., (k + 1)-st years, respectively.

(II) At time [[tau].sub.1], the insurer will take over the insured's mortgaged property, sell it in the market, and keep all the proceeds from the sale of the mortgaged property. If the insured became disabled, the insured would be requested to move into the nursing home to benefit from the long term care service. If the insured died, the integrated 'reverse mortgage--long term care' contract would automatically terminate.

The Feature (I) above implies that the insurer will pay annuity to the insured from the start of the year following the signing of the contract, and this payment continues until the insured dies. Moreover, the cash flow going to the insured need not be a fixed amount. For instance, let [[tau].sub.1] = 2.3 years and [[tau].sub.2] = 1.6 years. This means that the insured first lives in his/her home and claims two cash payments [A.sub.1], [A.sub.2], one at the beginning of the second year and the other at the beginning of the third year of the contract, respectively. At time [[tau].sub.1] = 2.3, the insured leaves home (i.e. leaves State 1) and moves into the nursing home (i.e. State 2). The insured lives in the nursing home for 1.6 years and claims the one-time cash payment of A3 at the beginning of fourth year.

The Feature (II) implies the following. When the insured leaves home (State 1) to enter into the nursing home (State 2) or dies (State 3), the insurer will take over the insured's mortgaged property and sell it. The cash that is acquired from the sale of the insured's house is used to repay loan (including annuity and accumulated interests) that the insured owes to the insurer. Thus, it provides capital reserve for the insurer in future. Compared with the taking-over of the mortgaged property after the insured dies, the terms of the above contract integrating reverse mortgage with long term care stipulates that the insured's mortgaged home equity is assumed by the insurer when the insured either moves into the nursing home or dies (whichever happens first).

3.2. Actuarial Pricing Model. We assume that we are in the perfectly competitive market. We price the 'reverse mortgage--long term care' contract by the principle of balance between expected gain and expected payment (i.e. the expected discounted present value of future sale of the pledged property is the same as the expected discounted present value of cash flow that the insurer pays).

At time [[tau].sub.1], ([[tau].sub.1] [greater than or equal to] 0), the insurer takes over the insured's mortgaged property, and sells it at time [[tau].sub.1] + [t.sub.0] where [t.sub.0] [greater than or equal to] 0 is the delay time between the insurer taking over the mortgaged property and the sale of the mortgaged property. We assume that [t.sub.0] is fixed and not a random variable. Then the expectation of discounted present value of the sale price of the morgaged property (i.e., the insurer's expected gain) is

(20) [mathematical expression not reproducible],

where H(t) is the value of the mortgaged property at time t given by the stochastic differential equation (1), and [v.sub.t] is the discount factor at time t given by the Equation

(21) [v.sub.t] := exp (- [[integral].sup.t.sub.0] r(s)ds),

and r(s) is the interest rate given by the stochastic differential Equation (5).

The expectation of discounted present value of the insured's annuities (i.e., the insurer's expected payment) is

(22) [mathematical expression not reproducible],

where [mathematical expression not reproducible] is the indicator function of the event [S.sub.0] [intersection] [[bar.S].sub.1] in which

[mathematical expression not reproducible],

[[bar.S].sub.1] being the complementary set of [S.sub.1], and the function [x] returns the largest integer not greater than x.

Then, the principle of balance between expected gain and expected payment yields

(23)

[mathematical expression not reproducible]

We shall now proceed to derive the explicit solution for annuities. Though the explicit solution of annuity payment is difficult to obtain from the Equation (23), we can obtain the explicit solution under suitable conditions.

Proposition 2 Assume that the dynamics of home price follows the Black-Scholes model given by the Equation (1), and the instantaneous short interest rate is governed by the Equation (5). Then the annuity payments [A.sub.k] (k = 1, 2, ..., T - 1) satisfy the following pricing equation

(24) [mathematical expression not reproducible],

where [mathematical expression not reproducible] is given by (14), and

(25) [mathematical expression not reproducible]

(26) [mathematical expression not reproducible],

(27) [mathematical expression not reproducible]

(28) [P.sub.1](k) = exp {- [[integral].sup.k.sub.0] [[[lambda].sub.12](u) + [[lambda].sub.13](u)] du} - exp {- [[integral].sup.T.sub.0][[[lambda].sub.12](u) + [[lambda].sub.13](u)] du},

(29) [mathematical expression not reproducible],

(30) [p.sub.12] (0, x) = [[integral].sup.x.sub.0] [p.sub.11](0, u)[[lambda].sub.12](u)[p.sub.22](u, x) du.

Here, [p.sub.11](0, x) and [p.sub.22](x, T) are given by Equation (12) and Equation (13), respectively.

Proof: Let [mathematical expression not reproducible] denote the probability density function for [[tau].sub.1]. Noting that H(t), r(t) and [[tau].sub.1] are independent, we get

(31) [mathematical expression not reproducible],

The Equations (4) and (8), characterize G(x + [t.sub.0]), and F(x + [t.sub.0]), respectively, as corresponding Equations (25) and (26). From the independence of r(t) and [[tau].sub.1], we have

(32) [mathematical expression not reproducible]

where F(k) is characterized as (27) by Equation (8). Recalling that the probability density function for [[tau].sub.1] is given by the Relation (14), we get the Equation (28). Noting that r(t) and ([[tau].sub.1], [[tau].sub.1], [[xi].sub.t]) are independent, we have

(33) [mathematical expression not reproducible].

Thus, Proposition 2 is proved.

The following Propositions 3, 4 and 5 are special cases of the Proposition 2. They present the pricing formulas for the growing (decreasing) perpetuity annuity, the state annuity, and the level annuity, respectively.

Proposition 3 The payments for the growing (decreasing) perpetuity annuity are characterized as follows. At the beginning of period (k + 1), the annuity payment is [A.sub.k] := [A.sub.0] + d x k, k = 1, 2, ..., n with [A.sub.0] and d positive constants (as the insured is alive). Here, [A.sub.0] and d are determined by the simultaneous equations

(34) [mathematical expression not reproducible],

(35) [mathematical expression not reproducible],

where

(36) [mathematical expression not reproducible],

(37) [mathematical expression not reproducible],

(38) [mathematical expression not reproducible]

and G(x + [t.sub.0]), F(x + [t.sub.0]), F(k), [P.sub.1](k) and [P.sub.2](i, k) are given by Relations (25)-(29), respectively.

Proposition 4 In the case of state annuity, let B stand for the annuity payment when the insured continues to live at home and let [lambda]B ([lambda] > 0) as the payment after the insured enters the nursing home, i.e.

(39) [mathematical expression not reproducible].

Then

(40) [mathematical expression not reproducible],

where [mathematical expression not reproducible] are the same as those in Proposition 3.

Proposition 5 For the level annuity, the fixed amount A of annuity is paid during the whole insurance period and is given by

(41) [mathematical expression not reproducible],

where [mathematical expression not reproducible] are the same as those in Proposition 3.

The expected discounted present value of housing price [??] is the average lump sum that the insured can claim at time t = 0; we shall call it the lump sum herein and hereafter. It is easy to see that [mathematical expression not reproducible], can affect the amount of each annuity payment, and therefore we shall call them the annuity payment factors. The following Propositions 6-8 analyze how the annuity payments, lump sum, and annuity payment factors vary with the parameters ([[mu].sub.h], [[sigma].sub.h], [t.sub.0], [r.sub.0], [[mu].sub.r], [[sigma].sub.r]) involved in the home pricing model and the interest rate model.

Proposition 6 The annuity payment A, the lump sum [??], and the annuity payment factors [[??].sub.i] (i = 1, 2, 3, 4) have the following properties:

1. PARAMETER [[mu].sub.h]: (a) The lump sum [??] is an increasing function of the mean return of house price [[mu].sub.h]. (b) The annuity payment factors [mathematical expression not reproducible] are totally independent of [[mu].sub.h]. (c) The [A.sub.0] and d in Proposition 3, the annuity payment B in Proposition 4, and the level annuity payment A in Proposition 5 are increasing functions of [[mu].sub.h].

2. PARAMETER [[sigma].sub.h]: (a) The lump sum G and the annuity payment factors [mathematical expression not reproducible] are completely independent of the volatility [[sigma].sub.h] of the house price. (b) The [A.sub.0] and d in Proposition 3, the annuity payment B in Proposition 4, the level annuity payment A in Proposition 5 also do not dependent on [[sigma].sub.h].

Proof: Noting that [mathematical expression not reproducible] do not depend on ph, the partial derivative of the integrand in the definition of [??], (see Relation (36)), is

[mathematical expression not reproducible].

This implies that the lump sum [??] is the increasing function of ph.

From the definitions of the annuity payment factors [mathematical expression not reproducible] (see Relations (37) and (38)), we note that these annuity payment factors are independent of [[mu].sub.h].

Furthermore, from the Equations (34), (35), (40) and (41), we have the [A.sub.0] and d in Proposition 3, the annuity payment B in Proposition 4, and the level annuity payment A in Proposition 5 are increasing functions of [[mu].sub.h]. Thus, Part 1 of the proposition is proved.

From the definitions of [mathematical expression not reproducible] it is easy to see that they have nothing to do with [[sigma].sub.h]. Part 2 of the proposition is now obtained from the Equations (34), (35), (40) and (41).

Proposition 7 DELAY TIME [t.sub.0] BETWEEN TAKING OVER OF THE MORTGAGED PROPERTY AND THE SALE OF THAT PROPERTY:

(a) The annuity payment factors [mathematical expression not reproducible] do not depend on [t.sub.0].

(b) Now set

(42) [mathematical expression not reproducible],

and

(43) [mathematical expression not reproducible].

(b-1) In the case of any one of the following conditions

[mathematical expression not reproducible],

the lump sum [??] is an increasing function of to. Also, the [A.sub.0] and d in Proposition 3, the annuity payment B in Proposition 4, and the level annuity payment A in Proposition 5 are increasing functions of to.

(b-2) If

(44) [mathematical expression not reproducible],

holds then the lump sum [??] is a decreasing function of [t.sub.0]. The [A.sub.0] and d in Proposition 3, the annuity payment B in Proposition 4, and the level annuity payment A in proposition 5 are decreasing functions of to.

Proof: Putting

[mathematical expression not reproducible],

the minmium of [q.sub.1](z) is

[2[[sigma].sup.2.sub.r]/[[alpha].sup.2.sub.r] ([[mu].sub.h] - [r.sub.0]) - [([[mu].sub.r] - [r.sub.0]).sup.2]][[alpha].sup.2.sub.r]/2[[sigma].sup.2.sub.r].

Noting that

[mathematical expression not reproducible],

we have

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

If the condition 2 [[sigma].sup.2.sub.r]/[[alpha].sup.2.sub.r]([[mu].sub.h] - [r.sub.0]) - [([[mu].sub.r] - [r.sub.0]).sup.2] [greater than or equal to] 0 holds, we then have [g.sub.1](z) [greater than or equal to] 0, and thus [??] is the increasing function of [t.sub.0].

Recall the definitions of [z.sub.1] and [z.sub.2] given above by the Relations (42) and (43), respectively. Now, if the condition 2[[sigma].sup.2.sub.r]([[mu].sub.h]- [r.sub.0]/[[alpha].sup.2.sub.r]) - [([[mu].sub.r] - [r.sub.0]).sup.2] [less than or equal to] 0 holds, then [g.sub.1]([z.sub.i]) = 0, i = 1,2. Moreover, it is obvious that [mathematical expression not reproducible] in case of [[alpha].sub.r] > 0 and x + [t.sub.0] [greater than or equal to] 0. Thus the lump sum G is an decreasing function of [t.sub.0] if the condition (44) holds. One similarly obtains the rest of the proposition.

The following Proposition 8 analyzes how the lump sum and annuity payment factors vary with the parameters involved in the interest rate model.

Proposition 8 The lump sum [??] and annuity payment factors [[??].sub.i] (i = 1, 2, 3, 4) have the following properties:

1. PARAMETER [r.sub.0]: If [[alpha].sub.r] [not equal to] 0, then [mathematical expression not reproducible] are decreasing functions of [r.sub.0].

2. PARAMETER [[mu].sub.r]: If [[alpha].sub.r] > 0, then [mathematical expression not reproducible] are decreasing functions of [[mu].sub.r]. If the opposite case [[alpha].sub.r] < 0 holds, then [mathematical expression not reproducible], are increasing functions of [[mu].sub.r].

3. PARAMETER [[sigma].sub.r]: If [[alpha].sub.r] [not equal to] 0, [[sigma].sub.r] > 0, then [mathematical expression not reproducible], are increasing functions of [[mu].sub.r].

Proof: Noting that both [mathematical expression not reproducible] are totally independent of [r.sub.0], the partial derivative w.r.t [r.sub.0] of the integrand in the definition of [??] is

[mathematical expression not reproducible].

The partial derivatives w.r.t [r.sub.0] of the annuity payment factors [[??].sub.i], i = 1, 2, 3,4, are

[mathematical expression not reproducible].

Since [mathematical expression not reproducible] whenever [[alpha].sub.r] [not equal to] 0 and z [greater than or equal to] 0, we obtain Part 1 of the Proposition.

Define

[mathematical expression not reproducible].

Since [mathematical expression not reproducible] are free from [[mu].sub.r], the partial derivative w.r.t. [[mu].sub.r] of the integrand in the definition of [??] is

[mathematical expression not reproducible].

The partial derivatives w.r.t [[mu].sub.r] of the annuity payment factors are

[mathematical expression not reproducible].

Note that [g.sub.2](z) [less than or equal to] 0 in case of [[alpha].sub.r] > 0, z [greater than or equal to] 0, and [g.sub.2](z) [greater than or equal to] 0 in case of [[alpha].sub.r] < 0, z [greater than or equal to] 0. We thus obtain Part 2.

Define

[mathematical expression not reproducible].

Because [mathematical expression not reproducible] do not depend on [[sigma].sub.r], the partial derivative w.r.t [[sigma].sub.r] of the integrand in the definition of [??] is

[mathematical expression not reproducible].

The partial derivatives w.r.t [[sigma].sub.r] of the annuity payment factors are

[mathematical expression not reproducible].

We have

[mathematical expression not reproducible].

Consequently, [g.sub.3](z) [greater than or equal to] [g.sub.3](0) = 0 in case of z [member of] [0, +[infinity]). This proves Part 3. Remarks 2: Obviously, the following properties are also true:

1. The [A.sub.0] in Proposition 3 is a decreasing function of d, and in turn, d is a decreasing function of [A.sub.0].

2. The annuity payment B in Proposition 4 is a decreasing function of [lambda].

4. NUMERICAL RESULTS

We devote this section to the numerical analysis of the annuity for an insurance product integrating reverse mortgage with long-term care. To keep the computation simple, we only consider the case of a single insured. We will illustrate that the impacts of risks involving the home price, interest rate, and the longevity on the annuity payment, the expected discounted present value of total annuity, and the annuity payment factors.

We take the parameters of the standard case with the following values:

* at the initial time, the home price is [H.sub.0] = 100;

* the annual mean return rate of home price [[mu].sub.h] = 0.04;

* the delay time of selling the mortgaged house [t.sub.0] = 0 (that is, as soon as the insured leaves the State 1, the house is sold off);

* the initial interest rate [r.sub.0] = 0.04;

* the mean reversion level of interest rate [[mu].sub.r] = 0.06;

* the volatility rate of interest rate [[sigma].sub.r] = 0.01;

* the mean reversion speed of interest rate [[alpha].sub.r] = 0.25;

* the limiting age L = 110;

* the age of the insured at time 0 is [x.sub.0] = 65;

* the incremental creep of the growing (decreasing) perpetuity annuity d = 0 (this means that the perpetuity annuity degenerates into the level annuity);

* the proportion [lambda] = 1 (it implies that the state annuity is simplified to the level annuity);

* the transition intensities of three-state Markov model are given by

(45) [mathematical expression not reproducible].

The model is employed to value the premiums of the disability policy by the Danish companies (Ramlau-Hansen, 2001). The [[lambda].sub.13](t) = [[lambda].sub.23](t) implies that the mortality for active lives and for disabled lives are not discriminated. With these parametric values,

(1) the level annuity A = 13.003,

(2) the lump sum [??] = 90.252,

(3) the annuity payment factors [[??].sub.1] = 6.033 and [[??].sub.2] = 0.908.

4.1. Sensitivity Analysis for Parameters of Home Price. We start the numerical analysis of how the average rate of return of the home price impacts the annuity, lump sum, and annuity payment factors, while we keep fixed the other parametric values given above.

Table 1 shows how the mean return of home price [[mu].sub.h] affect the annuity, lump sum and the annuity payment factors, while other parameters are kept fixed. Here we note the following.

(A) The higher the mean return of the home price, the greater the lump sum and annuity are; however, the annuity payment factors remain constant, [[??].sub.1] = 6.033 and [[??].sub.2] = 0.908 (as they are not affected by [[mu].sub.h], which coincides with the conclusion in Proposition 6).

The essence of this insurance product is to exchange the profit from selling the mortgaged house with the insured's annuity until his/her death. The higher mean return of the home price contributes to increased profit from the sale of the mortgaged house in future, and similar is the case with lump sum and annuity.

When [[mu].sub.h] increases from 0.02 to 0.16, (a) the lump sum leaps up from 77.092 to 290.761, with the the average rate of change 290.7166-761-77.092/0.16-0.02 = 1526.207; and (b) the annuity rises from 11.107 to 41.892, with the average rate of change 41.892 - 11.107/0.16-0.02 = 219.893.

Compared with others parameters of the home price and interest rate model, the mean return of house price has the dominating influence on both the lump sum and annuity.

Next we vary the time delay [t.sub.0] (while keeping other parameters fixed as above) and analyze how it affects the annuity, lump sum, and annuity payment factors.

Table 2 contains the corresponding annuity, lump sums and annuity payment factors while we vary the delay time [t.sub.0], (with other parameters kept fixed). We now note:

(B) the larger the delay time in selling the house (the larger [t.sub.0]), smaller the lump sum and annuity, while the annuity payment factors remain constant, (not influenced by [t.sub.0]). This is in accord with Proposition 7(b-2) in the case of [([[mu].sub.r] - [r.sub.0]).sup.2] - 2[[sigma].sup.2.sub.r]/[[alpha].sup.2.sub.r]([[mu].sub.h] - [r.sub.0]) = 4 x [10.sup.-4] [greater than or equal to] 0, [[alpha].sub.r] = 0.25 > 0, [z.sub.1] = -24 [less than or equal to] 0, and [z.sub.2] = 1 [greater than or equal to] 1. However, it should be noted that if we changed some parameters, the lump sum might also increase with the increase of [t.sub.0] (refer to Proposition 7(b-1)).

The average rate of change of the lump sum and annuity are 90.252-85.301/1.75-0 = 2.829 and 13.003-12.290/1.75-0 = 0.407. This indicates that the lump sum and annuity remain stable as the delay time [t.sub.0] < 1.75 years.

4.2. Sensitivity Analysis for Parameters of Interest Rate. This subsection provides the numerical analysis of how the parameters of interest rate model impacts the annuity A, the lump sum [??], and the annuity factors [[??].sub.i] = 1,2; again, we keep the other parametric values fixed.

Now we note from Table 3:

(C) The lump sum [??] and annuity payment factors [[??].sub.i] = 1,2, are decreasing as the initial interest rate [r.sub.0] increases (while other parameters are kept fixed). This agrees with our Proposition 8, and the annuity A is also decreasing.

With the explicit solution of interest rate in Equation (7), we know that the higher initial interest rate means that higher level of average interest rate, which contributes to the lower level of average discounted factor of interest rate, and that in turn results in the lower lump sum and annuity payment factors. This implies that the higher risk of interest rate produces lower annuity.

The average change rate of the lump sum and annuity is respectively 95.882- 63.322/0.16-0.02 = 232.571 and 13.092-12.486/0.16-0.02 = 4.329, which illustrates that the slight fluctuations of initial interest rate leads to the large fluctuations of the lump sum, while the slight fluctuations of the annuity ensues. It implies that, while the annuity is stable for the fluctuations of initial interest rate [r.sub.0], the lump sum is sensitive for the fluctuations.

The Table 4 provides the numerical values caused by the impact of the average reversion level [[mu].sub.r] of the interest rate. Here we note:

(D) the lump sum [??] and annuity payment factors [[??].sub.i], i = 1, 2, decrease with the increase of average reversion level [[mu].sub.r] of interest rate. This conclusion is theoretically supported by the Proposition 8.

From the mean of discounted factor given by Equation (7), it is obvious that the greater [[mu].sub.r] is smaller the discounted factor. Thus, the lump sum decreases from 112.386 to 58.573 as hr increases from 0.02 to 0.16, where the average change rate of the lump sum and annuity is respectively 112.386-58.573/0.16-0.02 = 384.379 and 13.672-11.482/0.16-0.02 15.643, exerting a significant influence on both the lump sum and annuity, second in importance only to the average housing price returns.

Table 5 given below provides the impact of the volatility of interest rate on [mathematical expression not reproducible].

(E) Table 5 reveals that the lump sum [??] and annuity payment factors [[??].sub.i], i =1, 2, increase as the volatility of interest rate [[sigma].sub.r] increases, and this is consistent with our Proposition 8.

The higher volatility rate contributes to the higher average level of the discounted factor. Hence, the lump sum and annuity payment factors become greater. It appears to be unreasonable that the lump sum increases with the increase of volatility rate, which is probably caused by the flaws of pricing model itself. On the other hand, the annuity decreases with the increase of [[sigma].sub.r], and this is reasonable.

(F) The higher volatility rate generates higher market risk. Thus the insurer will certainly pay the smaller annuity to the insured in order to avoid the higher risk. Therefore, we do not advise that the model be used to pricing the lump sum of the product of reverse mortgage integrated with the long-term care; however, it is suitable for pricing the annuity. The average change rate of the lump sum and annuity is respectively 111.6283-90.125/0.08-0.01 = 305.3 and 13.680- 13.003/0.08-0.01 = 9.671, indicating that the volatility of interest rate has an important effect on both the lump sum and annuity.

We next consider the impact of the reversion speed [[alpha].sub.r] of the interest rate on [mathematical expression not reproducible].

(G) From Table 6, it is clear that the annuity, the lump sum and the annuity payment factors decrease with the increasing of the reversion speed [[alpha].sub.r] of interest rate as other parameters take fixed values. The average change rate of the lump sum and annuity are 97.375-85.678/1.75-0.05 = 6.881 and 13.245-12.904/1.75-0.05 = 0.201, respectively. This implies that the reversion speed slightly affects the lump sum and annuity as [[alpha].sub.r] takes value between 0 and 1.75.

4.3. Sensitivity Analysis for the Age of the Insured. In this subsection we discuss the impact on [mathematical expression not reproducible]. by the limiting age and initial age, impact on [A.sub.0] by the incremental creep d, and the impact on B by the proportion [lambda].

(H) Table 7 reveals the following: When the age limit (survival) is over 95, the lump sum [??] and the first annuity payment factor [[??].sub.1] almost stay unchanged. If the age limit is over 110, then the annuity A and the second annuity payment factor [[??].sub.2] almost stay unchanged. Thus, it seems reasonable to assume that the longevity is 110.

(I) Table 8 illustrates that as the initial age [x.sub.0] of the insured increases, the lump sum G and annuity A are increasing, while annuity payment factors [[??].sub.i], i = 1, 2, show a decreasing trend. Meanwhile, the decreasing speed of annuity payment factor [[??].sub.2] is significantly lower than that of annuity payment factor [[??].sub.1]. This shows that the annuity payment factor [[??].sub.2] plays more and more important role in calculating the annuity with the increase of the initial age of the insured.

(J) For the growing (decreasing) perpetuity annuity in Proposition 3, Table 9 shows that [A.sub.0] decreases with the increase of the incremental creep d, and this is in agreement with Remarks 2. The lump sum and the annuity payment factors remain constant, [mathematical expression not reproducible].

(K) For the state annuity in Proposition 4, Table 10 shows that the annuity payment B decreases with the increase of the proportion [lambda], and this is in accord with Remarks 2. Moreover, the lump sum and annuity payment factors remain constant, [mathematical expression not reproducible].

5. CONCLUSION

The product integrating reverse mortgage with long-term care mainly involves several risk factors such as the home price risk, interest rate risk, disability risk, and life expectancy risk. We employ a three-state temporally inhomogeneous Markov chain model to describe the disability risk and life expectancy risk. This gives a unified and rigorous approach for the combined product reverse mortgage and long-term care insurance. We use the Black-Scholes model to describe the dynamics of the home price and the Ornstein-Uhlenbeck process for that of interest rate. This paper builds a pricing model for the lifetime annuity of the combined product, derives the closed form solutions of the growing (decreasing) perpetuity annuity, the state annuity, and the level annuity. We then discuss the impact of the parameters associated with the home price and interest rate over monotonicity of the lump sum, annuity, and annuity payment factors. We present a numerical analysis of the lump sum, the annuity, and annuity payment factors, and analyze their sensitivity to the said parameters. The result shows that the average return of home price has a major influence on the annuity and the lump sum. Next to the average return of home price, the mean reversion level and volatility of interest rate play a dominant role. Initial interest rate affects the lump sum in a significant way, while affecting the annuity only slightly. Meanwhile, when the delay time of selling the house is in the range 0 [less than or equal to] [t.sub.0] [less than or equal to] 1.75 and the reversion speed of interest rate in the range 0 < [[alpha].sub.r] [less than or equal to] 1.75, they hardly exert any effect on the lump sum and annuity.

Having noticed in this work, both theoretically and numerically, the greater sensitivity of the annuity and lump sum to the parameters of average return of home price, the mean reversion level and volatility of interest rate, we continue this problem analyzing these properties in terms of switching Markov chains.

Received July 3, 2016

REFERENCES

[1] Ahlstrom, A., Tumlinson, A., Lambrew, J., (2004): Linking reverse mortgages and long-term care insurance; Technical Report, Washington, DC: Georgetown University and Brookings Institution. http://www.urban.org/url.cfm?ID=1000628

[2] Bardhan, A., Karapandza, R., Urosevic, B., (2006): Valuing mortgage insurance contracts in emerging market economics; Journal of Real Estate Finance and Economics, 32 (1), 9-20.

[3] Benejam, A. A. (1987): Home equity conversions as alternatives to health care financing; Medicine and Law, 6(4), 329-348.

[4] Benjamin, S., (1992): Using the value of their houses to fund long term care for the elderly; Transactions of the 2fth International Congress of Actuaries, Montreal, 4, 19- 34.

[5] Black, F., Karasinski, P. (1991): Bond and Option Pricing when Short Rates are Lognormal; Financial Analysts Journal, 47, 52-59.

[6] Brigo, D., Mercurio, F. (2006): Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit; Springer Berlin.

[7] Chen, M. C., Chang, C. C., Lin, S. K., Shyu, S. D., (2010a): Estimation of housing price jump risks and their impact on the valuation of mortgage insurance contacts; Journal of Risk and Insurance, 77 (2), 399-422.

[8] Chen, H., Cox, S. H., Wang, S. S., (2010b): Is the home equity conversion mortgage in the United States sustainable? evidence from pricing mortgage insurance premiums and nonrecourse provisions using the conditional Esscher transform; Insurance: Mathematics and Economics, 46 (2), 371-384.

[9] Chinloy, P., Cho, M., Megbolugbe, I. F., (1997): Appraisals, transaction incentives, and smoothing; Journal of Real Estate Finance and Economics, 14 (1), 89-112.

[10] Cox, J. C., Ingersoll, J. E., and Ross, S. A. (1985): A Theory of the Term Structure of Interest Rates; Econometrica, 53, 385-407.

[11] Dothan, L. U. (1978): On the Term Structure of Interest Rates; Journal of Financial Economics 6, 59-69.

[12] Du Pasquier, L. G. (1912): Mathematische Theorie der Invaliditatversicherung; Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, 7, 1-7.

[13] Du Pasquier, L. G. (1913): Mathematische Theorie der Invaliditatversicherung; Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, 8, 1-153.

[14] Firman, J. (1983): Health care cooperatives: innovations for older people; Health Affairs, 4(4), 50-61.

[15] Fong, J. H., Shao, A. W., Sherris, M. (2015): Multi-State Actuarial Models of Functional Disability; North American Actuarial Journal, 19(1), 41-59.

[16] Gibbs, I. (1992): A substantial but limited asset: the role of housing wealth in paying for residential care; In, J Morten (ed.) Financing Elderly People in Independent Sector Homes: The Future, London: Age Concern Institute of Gerontology.

[17] Huang, H. C., Wang, C. W., Miao, Y. C., (2011): Securitization of crossover risk in reverse mortgages; Geneva Papers on Risk and Insurance-Issues and Practice, 36 (4), 622- 647.

[18] Hull, J., White, A. (1990): Pricing Interest Rate Derivative Securities; The Review of Financial Studies, 3, 573-592.

[19] Jacobs, B., Weissert, W. (1987): Using home equity to finance long-term care; Journal of Health Politics, Policy, and Law, 12, 77-95.

[20] Lee, Y.-T., Wang, Ch.-W., and Huang, H.-C. (2012): On the valuation of reverse mortgages with regular tenure payments; Insurance: Mathematics and Economics, 51, 430-441.

[21] Li, J. S. H., Hardy, M. R., Tan, K. S. (2010): On pricing and hedging the no-negative equity guarantee in equity release mechanisms; The Journal of Risk and Insurance, 77 (2), 499-522.

[22] Mercurio, F., Moraleda, J. M. (2000): An Analytically Tractable Interest Rate Model with Humped Volatility; European Journal of Operational Research, 120, 205-214.

[23] Mizrach, B., (2008): Jump and co-jump risk in subprime home equity derivatives; Working Paper, Department of Economics, Rutgers University.

[24] Murtaugh M. C., Spillman C. B., Warshawsky J. M. (2001): In Sickness and in Health: an Annuity Approach to Financing Long-Term Care and Retirement Income; The Journal of Risk and Insurance, 68(2), 225-254.

[25] Norberg, R. (2004): Vasicek beyond the normal; Mathematical Finance, 14(4), 585-604.

[26] Nothaft, F. E., Gao, A. H., Wang, G. H. K. (1995): The stochastic behavior of the Freddie Mac/Fannie Mae conventional mortgage home price index; American Real Estate and Urban Economics Association Annual Meeting, San Francisco.

[27] Ramlau-Hansen, H. (1991): Distribution of Surplus in Life Insurance; ASTIN Bulletin, 21(1), 57-71.

[28] Rasmussen, D. W., Megbolugbe, I. F., Morgan, B. A., (1997): The Reverse Mortgage as an Asset Management tool; Housing Policy Debate, 8(1), 173-194.

[29] Stucki, B. R., (2005): Use your home to stay at home: Expanding the use of reverse mortgages for long-term care; Washington, D.G.: National Council on Aging.

[30] Stucki, B. R., Group K., (2006): Using reverse mortgages to manage the financial risk of long-term care; North American Actuarial Journal, 10(4), 90-102.

[31] Szymanoski, E. J., (1994): Risk and the home equity conversion Mortgage; Journal of the American Real Estate and Urban Economics Association, 22(2), 347-366.

[32] Tsay, J.-T., Lin, C.-C., Prather, L. J., Richard J. Buttimer Jr., (2014): An approximation approach for valuing reverse mortgages; Journal of Housing Economics, 25, 39-52.

[33] Vasicek, O. (1977): An equilibrium characterization of the term structure; Journal of Financial Economics, 5, 177-188.

[34] Wang, L., Valdez, E. A., Piggott, J., (2008): Securitization of longevity risk in reverse mortgages; North American Actuarial Journal, 12(4), 345-371.

LINA MA (1,2)', JINGXIAO ZHANG (3), AND D. KANNAN (4)

(1) China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

(2) School of Science, Tianjin University of Commerce, Tianjin 300134, China

(3) Center for Applied Statistics, School of Statistics, Renmin University of China, Beijing 100087, China

(4) Department of Mathematics, University of Georgia, Athens, Georgia, USA

(1) Research of this author was supported by the National Science Foundation of China (Grant No. 71401124), Foundation of National Statistical Science Research in China (Grant No. 2015LZ03), and the Science and Technology Development Foundation of College in Tianjin (Grant No. 20131004). Corresponding Author: malinastat@163.com

Caption: FIGURE 1. The three state Markov model.
Table 1. Impacts of the average return rate of house price

[[mu].sub.h]    0.02     0.04     0.06      0.08

A              11.107   13.003   15.378    18.379
[??]           77.092   90.252   106.735   127.565

[[mu].sub.h]     0.1      0.12      0.14      0.16

A              22.206    27.130    33.522    41.892
[??]           154.127   188.301   232.662   290.761

Table 2. Impacts of the delay time of selling house

[T.sub.0]     0       0.25     0.5      0.75

A           13.003   12.907   12.808   12.707
[??]        90.252   89.584   88.898   88.198

[T.sub.0]     1       1.25     1.5      1.75

A           12.605   12.501   12.396   12.290
[??]        87.486   86.764   86.035   85.301

Table 3. Impacts of the initial interest rate

[r.sub.0]        0.02     0.04     0.06     0.08

A               13.092   13.003   12.915   12.828
[??]            95.882   90.252   84.986   80.060
[[??].sub.1]    6.345    6.033    5.738    5.459
[[??].sub.2]    0.979    0.908    0.842    0.782

[r.sub.0]        0.1      0.12     0.14     0.16

A               12.741   12.655   12.570   12.486
[??]            75.451   71.138   67.102   63.322
[[??].sub.1]    5.197    4.949    4.714    4.492
[[??].sub.2]    0.725    0.673    0.624    0.579

Table 4. Impacts of the average reversion level of interest rate

[[mu].sub.r]    0.02      0.04      0.06     0.08

A              13.672    13.342    13.003   12.669
[??]           112.386   100.334   90.252   81.760
[[??].sub.1]    6.788     6.387    6.033    5.717
[[??].sub.1]    1.432     1.133    0.908    0.736

[[mu].sub.r]    0.1      0.12     0.14     0.16

A              12.346   12.039   11.751   11.482
[??]           74.560   68.414   63.135   58.573
[[??].sub.1]   5.436    5.183    4.954    4.748
[[??].sub.1]   0.604    0.500    0.419    0.353

Table 5. Impacts of the volatility of interest rate

[[sigma].sub.r]    0.01     0.02     0.03     0.04

A                 13.003   13.036   13.092   13.169
[??]              90.252   91.094   92.531   94.616
[[??].sub.1]      6.033    6.060    6.105    6.171
[[??].sub.2]      0.908    0.928    0.962    1.014

[[sigma].sub.r]    0.05     0.06      0.07      0.08

A                 13.268   13.388    13.526    13.680
[??]              97.430   101.088   105.746   111.623
[[??].sub.1]      6.259     6.370     6.508     6.678
[[??].sub.2]      1.085     1.181     1.310     1.482

Table 6. Impacts of the reversion speed of interest rate

[[alpha].sub.r]    0.05     0.25     0.5      0.75

A                 13.245   13.003   12.934   12.915
[??]              97.375   90.252   87.838   86.873
[[??].sub.1]      6.288    6.033    5.919    5.866
[[??].sub.2]      1.064    0.908    0.872    0.860

[[alpha].sub.r]     1       1.25     1.5      1.75

A                 12.909   12.906   12.904   12.904
[??]              86.362   86.046   85.832   85.678
[[??].sub.1]      5.836    5.817    5.804    5.794
[[??].sub.2]      0.854    0.850    0.848    0.846

Table 7. Impacts of the limited age

L                95        100       105

A              13.0172   13.0044   13.0033
[??]           90.2518   90.2518   90.2518
[[??].sub.1]   6.0325    6.0325    6.0325
[[??].sub.2]   0.9007    0.9075    0.9081

L                110       115       120

A              13.0033   13.0033   13.0033
[??]           90.2518   90.2518   90.2518
[[??].sub.1]   6.0325    6.0325    6.0325
[[??].sub.2]   0.9081    0.9081    0.9081

Table 8. Impacts of the initial age

[x.sub.0]        50       55       60       65

A              6.717    8.162    10.161   13.003
[??]           75.395   80.739   85.780   90.252
[[??].sub.1]   10.776   9.310    7.702    6.033
[[??].sub.2]   0.448    0.583    0.740    0.908

[x.sub.0]        70       75       80       85

A              17.147   23.339   32.819   47.680
[??]           93.907   96.592   98.323   99.280
[[??].sub.1]   4.415    2.972    1.802    0.951
[[??].sub.2]   1.061    1.167    1.194    1.131

Table 9. Impacts of the incremental creep

d             0       0.5      1      1.5

[A.sub.0]   13.003   9.757   6.511   3.265

d             2      2.5       3       3.5

[A.sub.0]   0.019   -3.227   -6.473   -9.719

Table 10. Impacts of the proportion

[lambda]     0       0.5       1       1.5

B          14.961   13.914   13.003   12.205

[lambda]     2       2.5       3       3.5

B          11.499   10.870   10.306   9.798
COPYRIGHT 2017 Dynamic Publishers, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2017 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Ma, Lina; Zhang, Jingxiao; Kannan, D.
Publication:Dynamic Systems and Applications
Article Type:Report
Date:Mar 1, 2017
Words:9465
Previous Article:EXISTENCE AND ASYMPTOTIC STABILITY OF AN IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATION.
Next Article:IMPULSIVE CONTROL PROBLEM GOVERNED BY FRACTIONAL DIFFERENTIAL EQUATIONS AND APPLICATIONS.
Topics:

Terms of use | Privacy policy | Copyright © 2022 Farlex, Inc. | Feedback | For webmasters |