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Portfolio choice in retirement--what is the optimal home equity release product?

ABSTRACT

We study the decision problem of the optimal choice between home equity release products from a retired homeowner's perspective in the presence of longevity, long-term care, house price, and interest rate risk. The individual can choose to release home equity using reverse mortgages or home reversion plans, to buy annuities, and long-term care insurance. The individual enjoys utility gains from having access to either one of the two equity release products. Higher utility gains are found for the reverse mortgage, as its product features allow for higher lump-sum payouts and provide downside protection for house prices.

INTRODUCTION

We study the decision problem of the optimal choice between different home equity release products from the perspective of a retired homeowner in the presence of longevity, long-term care, house price, and interest rate risk. For elderly homeowners, the home's equity is often the most significant asset. For example, the value of the primary residence for U.S. households aged 65+ comprises on average (median) 49 percent (52 percent) of total assets, with 82 percent of households owning a house (2009 Survey of Consumer Finance). To use the home's equity for consumption purposes generally would require selling the home and renting a place instead (a decision problem covered, for example, in Yao and Zhang, 2005). However, many homeowners are reluctant to sell the home. They prefer to "age in place" (Davidoff, 2010c). For these homeowners, home equity release products allow elderly homeowners to convert the equity in their home into liquid wealth without having to move. Home equity release contracts differ substantially in the way house price risks, interest rate risk, and longevity risk are shared between the homeowner and the provider of the product. To make the right product choice is an important question for an elderly homeowner--we address this normative research question in this article.

Markets for equity release products for retirees exist in numerous countries including the United States, the United Kingdom, Australia, Canada, New Zealand, and several countries in the European Union. The two main forms of equity release are reverse mortgage schemes ("loan model") and home reversion schemes ("sale model") (see, e.g., Hosty et al., 2008; Reifner et al., 2009a). Reflecting those market conditions, we model a retiree's choice between a reverse mortgage and a home reversion plan.

Reverse mortgages are the most common products internationally and dominate the U.S. market (Consumer Financial Protection Bureau, 2012). When taking out a reverse mortgage, the homeowner receives a lump-sum payment (or annuity or line of credit) through borrowing against the home's value. There are no regular interest payments on the mortgage; instead, interest is added (rolled up) to the loan balance over time. The loan is paid back when the homeowner moves out or dies. Even if the loan balance becomes larger than the home's value, the homeowner has the right to continue residing in the home and the loan amount that has to be paid back is typically capped by the home's value (no negative equity guarantee [NNEG]).

Home reversion has existed for a long time in the form of private arrangements, for example, in France, Portugal, and Poland (Reifner et al., 2009b). Commercial home reversion is available, for example, in Australia, France, Finland, New Zealand, and the United Kingdom. With a home reversion plan, the homeowner sells (a part of) his home in exchange for a lump sum. The homeowner keeps the right to live in the home as long as he lives. When the homeowner moves out or dies there is no payment to the provider of the home reversion plan. However, as compensation for the life-long right to live in the home, the provider of the plan reduces the upfront lump-sum payment by the present value of future rent payments.

Alai et al. (2014) compare the cash flows and risk profile of stylized reverse mortgage and home reversion plans from the perspective of the product provider. The comparison shows that for loan-to-value ratios (FTVs) of less than 50 percent reverse mortgages are more profitable and less risky for the provider than home reversion plans. The opposite is true for higher LTVs (which are rare outside of the U.S. market). This finding may explain why more reverse mortgages than home reversion providers exist internationally. At the same time it raises the question: Is a home reversion plan more beneficial for homeowners?

In addressing this question, we add to a growing literature examining the role of equity release products in optimal household portfolios. Artie and Varaiya (1978) show that the possibility of borrowing against home equity in retirement and thereby relaxing liquidity constraints and smoothing consumption over the life cycle enhances utility. Fratantoni (1999) models the product choice between two reverse mortgage designs--annuity payout plan and line-of-credit plan--for a homeowner facing noninsurable expenditure shocks. He finds that line-of-credit plans are generally preferred since they are more flexible and can provide large sums of money in case of the expenditure shock. Davidoff (2009, 2010a, 2010b) extends this research by allowing for health and longevity risks. He confirms that the availability of reverse mortgages is utility enhancing and finds interaction effects with annuities and long-term care insurance (LTCI). For example, home equity may substitute for LTCI. Yogo (2009) and Nakajima and Telyukova (2013) consider stochastic house prices (and stochastic health depreciation), confirming that reverse mortgages are utility enhancing.

We provide the following contributions to the literature. (1) While the previous literature focused on reverse mortgages, we compare the two main forms of equity release products, reverse mortgages and home reversion plans, in a model that allows for longevity risk, uncertain long-term care costs, house price risk, and interest rate risk. That is, the decision problem we address is a retired homeowner's optimal choice of home equity release products. (2) Both equity release products are offered at different points in time and we study the timing decision of when to optimally release home equity. (3) We analyze the optimal choice in different institutional settings for LTCI and examine the resulting interactions. We distinguish between a currently relevant setting, in which costs have to be paid out of pocket with private insurance available, and a setting potentially relevant in the future, in which most long-term care costs are partly born by a government-sponsored system. Suggestions to introduce government-provided LTCI have been made in the United Kingdom and Australia (Commission on Funding of Care and Support, 2011; Productivity Commission, 2011). Through introducing government-provided LTCI, the choice of reverse mortgages and home reversion plans may be impacted as retirees are relieved from a major risk, the risk of high out of pocket LTC costs.

We find that the individual enjoys utility gains from having access to either one of the two equity release products. Higher utility gains are found for the reverse mortgage; thus, the homeowners' optimal choice is to release home equity with a reverse mortgage. This product gives larger upfront lump-sum payments and provides downside protection against house price risk. Both features are valuable for risk-averse and impatient individuals. The individual chooses to unlock home equity early in retirement. These key results emerge consistently across a range of cases with different parameter values. The availability of a government-provided LTCI does not significantly change the optimal choice between the equity release products.

THE MODEL

General Structure of the Model and Timing

The decision problem of a single individual is modeled who holds the major fraction of her wealth in her home. The individual faces longevity risk, long-term care risk, house price risk, and interest rate risk. The individual can always choose to purchase annuities and LTCI. In addition, there is either a reverse mortgage or a home reversion plan available.

The individual's decisions are studied in an augmented life cycle model that extends previous work by Davidoff (2009, 2010b, 2010c) by allowing for interest rate risk, by including home reversion plans in addition to reverse mortgages, and by modeling the timing decision of when to release home equity. The model has two periods (three dates) to capture the individual's decisions at retirement and at an advanced age. The model's input parameters are calibrated such that each period reflects a multiyear horizon. Figure 1 illustrates the decision and timing structure of the model.

At time t = 0, the individual is in good health. The initial endowment consists of a mortgage-free home and liquid wealth. The individual decides on consumption, on saving over the first period of her retirement, on purchasing annuities, on LTCI, and on taking out the equity release product available (either the reverse mortgage or the home reversion plan). Equity release products increase liquid wealth available for consumption, saving, and purchasing insurance products.

At time t = 1, the individual can be dead or in one of three health states, facing different health care expenses (as in Davidoff, 2009). The stochastic house value, as well as the interest rates and mortgage rates for the second period are realized. Annuities and LTCI are not available for purchase at t = 1. At t = 1 there are the following main states:

1. The individual is alive: She receives payments from insurance contracts and from equity release products contracted at t = 0. Health state-dependent care expenses not covered by insurance are paid out of pocket. The individual decides on consumption and saving over the second period.

(a) The individual is still living at home: She decides whether to take out another equity release product of the product type available (reverse mortgage or home reversion plan).

(b) The individual is in a nursing home: The house is sold and all outstanding loans are repaid from the sale proceeds of the property. Additional sale proceeds are added to her liquid wealth.

2. The individual is dead: Her remaining liquid wealth and housing wealth (net of mortgage repayments) are left as a bequest.

At t = 2, the individual is dead with certainty. Her remaining liquid wealth and housing wealth (net of reverse mortgage repayments) are bequeathed.

Interest Rates, Mortgage Rates, House Price Growth, and Savings Growth

We model all economic variables in real (inflation-adjusted) terms. The risk-free interest rate [r.sub.0] over the first period is known at t = 0. The interest rate [r.sub.1] over the second period is a random variable, realized at t = 1. Mortgage rates are derived from interest rates by adding a margin [[pi].sub.RM] to [r.sub.0] and [r.sub.1] (see the "Equity Release Products" and "Numerical Calibration of Baseline Parameters" sections). Savings, [S.sub.t], accumulate interest [r.sub.t] between time t and t + 1.

The house value is [H.sub.0] at t = 0, [H.sub.1] = [H.sub.0] x (1 + [g.sub.1]) at t = 1, and [H.sub.2] = [H.sub.1] x (1 + [g.sub.2]) at t = 2, where the growth rates [g.sub.1] and [g.sub.2] are i.i.d. random variables, uncorrelated with the interest rate.

Health States and Care Costs

At time t = 1, the individual is in one of four states. With probability [p.sub.h] she is still in good health and does not need long-term care (state h), with probability [p.sub.c] she needs some care at home at costs [LTC.sub.c] (state c), with probability [p.sub.n] she needs to move to a nursing home at costs [LTC.sub.n] (state n), and with probability [p.sub.d] = 1 - [p.sub.h] + [p.sub.c] + [p.sub.n] + [p.sub.d] she is dead (state d).

LTCI and Annuity Products

LTCI covering the care costs [LTC.sub.c] in state c and [LTC.sub.n] in state n is available at t = 0. The individual chooses the proportion of insurance coverage [%.sub.LTCI] by choosing the amount of wealth [[PI].sub.LTCI] spent on LTCI. The insurance is priced according to the actuarial principle of equivalence. The premium for partial coverage of an individual's care costs is given by:

[[PI].sub.LTCI] = [%.sub.LTCI] x ([p.sub.c] x [LTC.sub.c] + [p.sub.n] x [LTC.sub.n])/(1 + [r.sub.0]). (1)

Life annuities are available at t = 0. Annuities are also priced based on the actuarial principle of equivalence. The premium for an annuity paying the amount A at t = 1 conditional on survival is given by:

[[PI].sub.A] = A x (1 - [p.sub.d])/(1 + [r.sub.0]). (2)

The annuity payment A is determined by the amount of wealth [[PI].sub.LTCI] the individual decides to invest in the annuity according to Equation (2).

Government-Provided LTCI

Scenarios are considered in which both public and private LTCI plans are available. Social insurance arrangements for long-term care services exist in a number of OECD countries, including German, Japan, Korea, the Netherlands, and Luxembourg (for an overview, see Productivity Commission, 2011).

Government-provided LTCI is modeled as a compulsory coinsurance arrangement with a stop-loss limit. The insurance scheme covers a percentage [%.sub.govt.LTCI] of all care costs up to an out-of-pocket spending limit. This arrangement abstracts from the details of different national systems and focuses on the impact of possible structures of sharing care costs. The arrangement is in line with suggestions by the UK Commission on Funding of Care and Support, which suggests introducing a social insurance scheme with coinsurance and a cap. The arrangement also agrees with the suggestions by the Productivity Commission in Australia (Commission on Funding of Care and Support, 2011; Productivity Commission, 2011). The retired individual faces no costs for this insurance: the cost is levied on the working-age population. The individual can decide to buy private LTCI coverage for remaining care costs not covered by the public LTCI. Because the remaining care costs are lower, a lower premium for private LTCI results.

Equity Release Products

We model a lump-sum reverse mortgage and a home reversion plan (also called sale-and-lease-back plan). These two contract designs are the main types of equity release schemes currently available in Australia, Canada, the United Kingdom, and the United States (Oliver Wyman, 2008; Davidoff, 2010c). Reverse mortgages and home reversion plans are offered to the individual at t = 0 and t = 1. In several markets today, equity release products are only offered to individuals that own a debt-free home. To model this situation, we also consider scenarios in which equity release products are only offered at t = 0 or t = 1. The comparison allows us to determine the optimal timing of equity release.

The Reverse Mortgage. We focus on reverse mortgages with a lump-sum payout, variable interest rates, and an NNEG, which is currently the most common equity release product internationally. In the United States almost 70 percent of products newly originated in 2011 are lump-sum products. Consumers of alternative products (primarily line-of-credit plans) typically borrow amounts close to the maximum lump sum available (Consumer Financial Protection Bureau, 2012). We also note that because the reverse mortgage is available at t = 0 and f = 1 and private annuities are available for purchase, the line-of-credit and annuity payout plan types of reverse mortgage studied by Fratantoni (1999) are covered (implicitly) in our analysis.

Let [LS.sub.RM,t] denote the loan value of a reverse mortgage taken out at time t = 0,1, which is paid out in full at time t. Let [RM.sub.0_][balance.sub.t] and [RM.sub.1_][balance.sub.t] be the time t values of the outstanding loan balances of reverse mortgage loans taken out at time t = 0 and t = l. The outstanding loan balances are calculated by compounding [LS.sub.RM,t] at the respective mortgage rate.

The NNEG ensures that the individual's loan repayment does not exceed the value of the home. The costs for the NNEG are charged to the individual in the form of a mortgage insurance premium [[pi].sub.RM], which is added to the interest rate (see Cho, Hanewald, and Sherris, 2013; Alai et al., 2014). The value of the NNEG is different for reverse mortgages taken out at t = 0 and at t = 1, resulting in different insurance premiums. The following mortgage rates apply for a reverse mortgage taken out at t = 0: [r.sub.0] + [[pi].sub.RM,0] over the first period and [r.sub.1] + [[pi].sub.RM,0] over the second period. For a reverse mortgage taken out at t = 1, the mortgage rate [r.sub.1] + [[pi].sub.RM,1] applies over the second period. There are no other charges or lending margins.

The loan amounts [LS.sub.RM,0] and [LS.sub.RM,1] are decision variables. The loan amounts are restricted by a maximum loan-to-value ratio, which is defined in terms of the house value [H.sub.t]. Different (age-specific) maximum loan-to-value ratios [LTV.sup.max.sub.0] and [LTV.sup.max.sub.1] apply for reverse mortgages taken out at t = 0 and t = l. [LTV.sup.max.sub.1] is defined as a combined loan-to-value ratio:

([RM.sub.0-][balance.sub.1] + [RM.sub.1_][balance.sub.1])/[H.sub.1] [less than or equal to] [LTV.sup.max.sub.1]. (3)

A reverse mortgage taken out at t = 0 is repaid at t = 1 if the individual is in a nursing home or dead (states 1(b) and 2) described in the "General Structure of the Model and Timing" section). In case the individual is still living at home, she can decide to take out another reverse mortgage at t = 1 and the outstanding loan balances of both contracts are repaid at t = 2. In case of repayment, the house is sold and the sale proceeds are used to pay back the total outstanding loan balance [RM.sub.0_][balance.sub.t] + [RM.sub.1_][balance.sub.t]. To simplify the pricing, the repayment of [LS.sub.RM,1] has priority over repayment of [LS.sub.RM,1] if at the total loan balance is less than the house value time at t = 2.

The Home Reversion Plan. Home reversion is offered at t = 0, 1. Under this arrangement, the individual sells a share [%.sub.HR,t] of the home equity [H.sub.t] at time t to the product provider and receives a lump-sum [LS.sub.HR,t] in return. The lump-sum is less than the market value of the equity share sold, reflecting the value of a lease-for-life agreement and house price risk (Alai et al., 2014). The individual does not have to pay a regular rent on the equity share sold to the bank, but the equivalent present value of rental payments is deducted from the lump sum payout.

A home reversion plan taken out at t = 0 ends at t = 1 if the individual is in a nursing home or dead. If still at home, the individual can decide to take out another home reversion plan at t = 1 and both contracts end at t = 2. When the contract ends, the house is sold and the sale proceeds are divided according to equity shares. The individual's share is added to the liquid wealth that is bequeathed.

The Individual's Maximization Problem

The individual's lifetime utility function V includes a bequest motive, as, for example, in Inkmann, Lopes, and Michaelides (2011):

V(C, W) = [2.summation over (t=0)][[delta].sup.t][[I.sub.t] x U([C.sub.t]) + (1 - [I.sub.t]) x [beta] x B([W.sub.t])], (4)

where [delta] denotes the subjective discount factor of the individual, [beta] is the utility weight of the bequest motive, [I.sub.t] is an indicator variable taking the value one if the individual is alive and zero otherwise, and [C.sub.t] is the consumption in real terms. The wealth bequeathed, [W.sub.t], is composed of liquid wealth and the individual's share of the proceeds from the sale of the house (net of loan repayments). As in the Campbell and Coceo (2003) article on (conventional) mortgage choice, the utility is defined over consumption only and not also over housing. Similar to those authors, we are interested in the (reverse) mortgage choice, but not in the choice of the housing stock over time (as this topic is covered elsewhere, e.g., in Yao and Zhang, 2005). The choice of the utility function is further motivated by the stylized fact that most elderly have strong emotional ties to their home and thus the decision to live there is treated to be always preferred over selling the home and moving when the individual is still in relatively good health (Whitehead and Yates, 2010; Consumer Financial Protection Bureau, 2012). (1) In particular, the modeling choice of the individual's utility bases the consumption value of the home on the whole house (and its characteristics) and not just on the home equity share the individual owns. This modeling choice is realistic, given that people with very small home equity shares of 10 percent or even less perceive themselves as homeowners (Whitehead and Yates, 2010). Therefore, we propose that the consumption value of the home originates from living in their own home--a place of strong emotional ties--and not from the details of the financial arrangement allowing individuals to reside in the home. In consequence, the consumption value of the home does not change with taking out a reverse mortgage or home reversion plan.

The one-period utility functions of the individual, U, is given by:

U([C.sub.t]) = [C.sub.t.sup.1-[gamma]]/1 - [gamma], (5)

where [gamma] is the relative risk aversion parameter. The bequest utility function, B, exhibits the same relative risk aversion as U and is given by:

B([W.sub.t]) = [[W.sub.t.sup.1-[gamma]] / [1 - [gamma]]]. (6)

The individual's objective is to maximize the expected value of Equation (4) subject to a set of constraints. Her optimization problem is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where the index j refers to cash flows from the equity release schemes (j = RM, HR), which are alternatively available. The optimization problem is subject to

(i) Consumption and bequest constraints:

[C.sub.0] = [W.sub.0] - [S.sub.0] - [[PI].sub.A] - [[PI].sub.LTCI] + [LS.sub.j,0,] j = RM, HR, (8)

[C.sub.1] = [S.sub.0] x (1 + [r.sub.0]) - [S.sub.1] + A - (1 - [%.sub.govt.LTCI] - [%.sub.LTCI]) x LTC + [LS.sub.j,1,] j = RM, HR. (8)

* Bequest constraints with the reverse mortgage:

[W.sub.1] = So x (1 + [r.sub.0]) + max[[H.sub.1] - [RM.sub.0]-[balance.sub.1],0],

[W.sub.2] = [S.sub.1] x (1 + [r.sub.1]) + max[[H.sub.2] - [RM.sub.0]-[balance.sub.2] + [RM.sub.1]_[balance.sub.2,]0].

* Bequest constraints with the home reversion plan:

[W.sub.1] = [S.sub.0] x (1 + [r.sub.0]) + (1 - [%.sub.HR,0]) x [H.sub.1],

[W.sub.2] = [S.sub.1] x (1 + [r.sub.1]) + (1 - [%.sub.HR,0] - [%.sub.HR,1]) x [H.sub.2].

(ii) Borrowing constraints:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

(iii) No short sale constraints for equity release and insurance products:

0 [less than or equal to] [LS.sub.j,0], [LS.sub.j,1], [[PI].sub.A], [[PI].sub.LTCI], j = RM, HR. (10)

and

(iv) further product constraints:

* Maximum loan-to-value ratios for the reverse mortgage:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

* Maximum home reversion rate:

[%.sub.HR,0] - [%.sub.HR.1] [less than or equal to] 1. (12)

* LTCI benefits capped by actual care expenses:

[%.sub.LTCI] [less than or equal to] 1. (13)

Numerical Calibration of Baseline Parameters

This section describes the numerical calibration of the model's baseline parameters. The parameter values are chosen to reflect the U.S. market. Alternative parameter values are introduced in the "Results" section. Table 1 summarizes the numerical calibration. To focus on product design effects (rather than pricing effects), all products are priced such that the product provider makes a zero expected profit. The pricing of the insurance and equity release products reflects the risks inherent in these products.

The Individual's Preferences and Endowment. The parameters defining the individual's preferences are set within the range typically used in life cycle models. The relative risk aversion y is set to 2, the subjective discount factor [delta] is set to 0.98 per year, and the strength of the bequest motive [beta] is set to 0.5 (see, e.g., Laibson, Repetto, and Tobacman, 1998; Coceo, Gomes, and Maenhout, 2005; Inkmann, Lopes, and Michaelides, 2011).

The HECM reverse mortgage program that dominates the U.S. market requires borrowers to be at least 62 years old to access mortgages. Thus, the initial age of the individual is set to 62 at t = 0. The maximum age in the model (at t = 2) is set to 100, and to have two periods of identical lengths, the age at t = 1 is set to 81, making one period 19 years long. The initial endowment consists of liquid wealth of [W.sub.0] = $135,000 and a house worth [H.sub.0] = $250,000, which reflect the median values for financial assets and primary residences for individuals aged 60-65 in the 2009 wave of the Survey of Consumer Finances.

Interest Rates and House Price Growth. Interest rates are modeled following Campbell and Coceo (2003), who analyze conventional mortgages. That is, future 1-year interest rates are modeled as a mean rate plus a transitory i.i.d. shock. Based on 1-year U.S. Treasuries, Campbell and Coceo estimate the mean of real interest rates to be 2 percent with a standard deviation of 2.2 percent. The interest rate over the first period, [r.sub.0], is set equal to the mean real rate.

Annual house price growth rates are modeled as normally distributed i.i.d. random variables. The parameters of the distribution are derived from estimates provided by Campbell and Coceo (2003) based on the Panel Study of Income Dynamics (PSID): the mean real growth rate is 1.6 percent with a standard deviation of 11.7 percent. (2)

For the numerical solution of the model, the house price process is discretized using a binomial process (as in Yao and Zhang, 2005; Davidoff, 2010c). The interest rate process is discretized in the same way.

Health States, Care Costs, Long-Term Care Insurance and Annuity Products. The probabilities of the four health states (staying in good health, needing some care at home, needing to move to a nursing home, being death) and the state-dependent care costs (0, moderate, high, 0) are the same values as those used by Davidoff (2009). That is, the probabilities for entering the different states are based on Robinson (2002) and the annual care expenses are based on Ameriks et al. (2011). Annual care costs in real terms are $10,000 in the second state, $50,000 in the third state, and zero otherwise. LTCI for a 62-year-old person is priced according to Equation (1). Likewise, annuities are priced according to Equation (2) using the survival probabilities.

Pricing of the Reverse Mortgage. The reverse mortgage is priced such that the product provider makes a zero profit on average across all future states. The profit is calculated as the expected present value of the loan repayment (discounted using interest rates) less the initial loan amount. An interest rate margin [[pi].sub.RM] is calculated such that the product provider is compensated for a possible shortfall arising from the NNEG embedded in the reverse mortgage.

Figure 2 gives the margin [[pi].sub.RM,0] for the variable interest rate reverse mortgage taken out at t = 0 for different LTVs. Given the calibration of interest rate, house price and health states, the value of the house will always be sufficient to repay the loan for small LTVs up to 0.30. For LTVs between 0.35 and 0.85, there are states where the NNEG becomes effective and the provider will charge a positive margin on the interest rate. The margins vary between 0.04 percent and 1.8 percent p.a. These values fall into the range reported by Shan (2011), who documents that for U.S. HECM loans the lender's margin is typically between 1 percent and 2 percent. For LTVs higher than 0.85, the expected profit of the lender is always negative in our model, independent of the margin, and this establishes a maximum LTV.

The pricing of the reverse mortgage offered at t = 1 is similar: a margin [[pi].sub.LS,1] is determined to compensate the product provider for the NNEG. The value of the NNEG depends on the loan amount borrowed at t = 0, on the house price growth rate over the first period and on interest rates at t = 1. Figure 3 gives the margin [[pi].sub.RM,1] for different additional LTVs, each for different [LTV.sub.0] ratios and assuming low house price growth over the first period and low interest rates over the second period.

Pricing of the Home Reversion Plan. The home reversion plan is priced such that the product provider makes a zero profit on average across all future states. The provider's profit is calculated as the expected present value of the sale proceeds of the released equity share minus the initial lump sum paid out to the individual. The lump sum is the market price of the equity share minus the expected present value of the rent on the released equity share (Alai et al., 2014). The rental yields over the first and second periods, are computed by accumulating the annual rental yield %rent on the home equity released at the beginning of the period.

The present values of the sale proceeds and rental yields are calculated using discount factors that reflect house price risk. The discount factors for the first period are determined by dividing the total value of the released equity share at t = 1 by the value of that share at t = 0. The total value includes capital growth as described in the "Interest Rates and House Price Growth" section and rental yields over the first period. The discount factors for the second period are determined in the same way. A rental yield of 2 percent (equal to the mean interest rate) is used, resulting in 58 percent of the value of the equity share paid out to the individual.

Government-Provided LTCI. With the government-provided LTCI, the individual has to cover (1-[%.sub.govt.LTCI]) = 50 percent of the care costs up to a maximum of $6,276 per year (equal to $100,000 for the 19-year horizon). For care costs higher than $6,276, the individual's out-of-pocket costs are limited to $6,276.

Implementation and Equivalent Wealth Variation. The MATLAB function fmincon is used to implement the individual's optimization problem as a constrained nonlinear optimization problem. Scenarios are compared based on maximized discounted expected utility values. We report measures of equivalent wealth variation that compare, in relative dollar terms, the maximized expected utility values in scenarios where equity release products are available against a benchmark scenario without equity release products. That is, we compute the percentage [theta] by which initial housing and liquid wealth would have to be increased in the benchmark scenario to make the individual indifferent between the optimal decisions in the benchmark scenario and in a given scenario with equity release products. The benchmark scenario varies across model variants (e.g., with different preference parameters).

RESULTS

Comparison of Reverse Mortgages and Home Reversion

The individual decides on consumption, on savings, on buying annuities and private LTCI, and on taking out one of the two equity release products. First, annuities, LTCI, and equity release products are only offered at t = 0. Government-provided LTCI is not available. The model parameters are the baseline parameters given in Table 1. We compare three scenarios: one without equity release products and two scenarios in which either the reverse mortgage or the home reversion plan described in the "Equity Release Products" section are offered.

The first three columns of Table 2 give the results. When offered the reverse mortgage at t = 0, the individual borrows up to the maximum loan-to-value ratio (LTV) of 85 percent. When offered the home reversion plan at t = 0, the individual converts a 74 percent ([%.sub.HR,0]) of the home. The individual significantly increases her liquid wealth with equity release. Her total liquid wealth is $135,000 without equity release, $347,500 with the reverse mortgage, and about $241,500 with the home reversion plan. The additional liquid wealth from equity release is used to increase consumption, savings, and the demand for annuities and private LTCI as in Davidoff (2010b). The individual spends between 22 percent and 31 percent of her t = 0 liquid wealth on annuities. Private LTCI demand is high in all three scenarios because the individual faces potentially high care costs. In both scenarios, the equivalent wealth variation factor [theta] indicates utility gains. (3) The utility gain is higher with the reverse mortgage than with the home reversion plan.

Table 2 also reports the results for a case in which the equity release products are offered only later in retirement (at t = 1) and for a full flexibility case where equity release is offered both at retirement (t = 0) and later in retirement (t = 1).

We find that in the full flexibility case there are virtually no additional utility gains from having access to reverse mortgages at times t = 0 and t = 1. The individual again borrows up to the maximum LTV at t = 0 and makes very similar financial decisions as in the case when the reverse mortgage is offered at t = 0 only. The utility gain of having access to the reverse mortgage is substantially lower (more than 10 percentage points) when the reverse mortgage is only available at t = 1. That is, when faced with an all-or-nothing decision between borrowing in t = 0 or t = 1, the individual prefers to borrow early.

For the home reversion plan, adding full flexibility is of some value for the individual and utility gains increase slightly. The timing of equity release changes: the individual sells a smaller proportion of home equity at t = 0 (58 percent compared to 74 percent when the product is only offered at t = 0) and releases more equity at t = 1. That is, equity release is delayed. The amount of home equity released at t = 1 depends on the realization of house prices and interest rates at t = 1. Larger shares are released when house prices and interest rates are high. Averaging across the states at t = 1 in which the individual is actually offered the home reversion plan because she is still alive and living at home, we find that she sells another 17 percent of home equity at t = 1. When access to the home reversion plan is limited to t = 1 only, utility gains are still higher than when in the case where home reversion is only available at t = 0 (but lower than under full flexibility). That is, when faced with an all-or-nothing decision between a home reversion plan in t = 0 or t = 1 the individual would prefer to contract late.

Overall, the results show that the individual generally prefers to release equity via the reverse mortgage rather than with the home reversion plan. With respect to the timing, the individual favors early equity release with the reverse mortgage. For the home reversion plan the timing matters less (in terms of utility gains), and the individual uses timing flexibility to contract a larger fraction of home reversion in the future period. (4,5)

Two product features are responsible for the greater attractiveness of the reverse mortgage and the higher preference for early equity release with this product. First, the reverse mortgage has an asymmetric payout profile that allows the homeowner to benefit from house price increases but protects him from house price decreases through the NNEG. With the home reversion plan, the homeowner is fully exposed to the house price risk for the share of the home retained. The NNEG makes the reverse mortgage more attractive for a risk-averse homeowner. Second, the reverse mortgage gives a higher payment at t = 0 than the home reversion plan but results in lower payouts at the end of the planning horizon (both products are fairly priced). The lump-sum payout from the home reversion plan is reduced because of the "sale-and-lease-back" structure of the contract in which the provider deducts the present value of future rents upfront. The reverse mortgage is better suited for shifting financial resources to early periods when the individual is more likely to be alive and utility is not heavily discounted, which explains its higher utility gains and preference for early usage.

We demonstrate these effects based on several sensitivity analyses that highlight general tendencies when changing the model's underlying assumptions. In particular, we vary the individual's preference parameters. Table 3 gives the results for different values of the parameters of the utility function: the risk aversion parameter [gamma], the subjective discount factor [delta], and the strength of the bequest motive [beta].

In Panel A of Table 3 the risk aversion parameter [gamma] is varied. Both products' utility gains increase with higher risk aversion. But because of the effects explained above, the welfare gains for the reverse mortgage (which comes with downside protection for house price risks) increase more strongly. Likewise, a more risk-averse individual sells larger fraction of the home under the home reversion plan at t = 0 to decrease exposure to house price risk.

When increasing the subjective discount factor [delta] (Table 3, Panel B), that is, when making the individual more oriented toward future consumption, the utility gains for both products decrease. Shifting consumption to earlier periods with equity release products becomes less valuable for more future-oriented individuals. But, as expected, the difference in utility gains between the reverse mortgages and the home reversion plan is largest for individuals with a higher valuation for present utility (low [delta]). For such individuals, the upfront deduction of expected rents in the home reversion plan is more undesirable. A higher bequest motive (Table 3, Panel C) leads to similar tendencies. Individuals with higher bequest motives value future utility more (they put a higher weight on bequests) and have lower utility gains from equity release products (as shown for reverse mortgages by Nakajima and Telyukova, 2013). Again, the upfront deduction of the expected rent for the home reversion plan results in the largest utility gain difference between the two products for individuals with a greater weight on present utility (i.e., no bequest motive).

As the home reversion plan is generally less attractive for the individual, the optimal strategy for using this product is more adapted toward timing its usage. Compared to the equity release with the reverse mortgage, a smaller fraction of the home is sold at t = 0 and more equity is released at t = 1, with usage based on the realization of house prices and interest rates. For example, in the base case with products available both at t = 0 and t = 1, the average fraction of the home sold at t = 1 over all house price and interest rate scenarios is 17 percent for the home reversion plan (compare Table 2). This fraction is higher when realized house prices and interest rates are high. These are scenarios where the home equity increases and high interest rates make the upfront deduction of expected rents smaller as these are more heavily discounted. Conversely, as the reverse mortgage has a payout profile that favors early usage, additional future flexibility (allowing contracting at t = 1 vs. only at t = 0) has little value to the individual (compare Table 2).

Government-Provided LTCI

Next, we consider government-provided LTCI as described in the "Government-Provided LTCI" sections. Again, the individual decides on consumption, saving, annuitization, private LTCI coverage for the remaining out-of-pocket care costs, and equity release. The model parameters are the baseline parameters given in Table 1. Three different scenarios are compared: one scenario without equity release products and two scenarios in which the reverse mortgage or the home reversion plan described in the "Equity Release Products" section are offered at t = 0 and t = 1. The numerical results for these scenarios are given in Table 4. Scenarios with equity release products offered only at t = 0 are not compared separately.

Similar levels of equity release are found to be optimal with the government-provided LTCI. As in the base case without public LTCI, the individual chooses to borrow up the maximum LTV with the reverse mortgage at t = 0 and chooses similar levels of home reversion at t = 0 and t = 1. Compared to the corresponding base case scenarios, slightly higher levels of wealth are invested into the annuity. Also, as suggested by Davidoff (2010b), the individual chooses similar levels of private LTCI coverage for the out-of-pocket care costs not covered by the government-provided LTCI. But because the premium for this is lower, less wealth is spent on private LTCI and instead is used to increase consumption and savings.

Sensitivity Analyses: The House Value and Preexisting Debt

In this section, a lower or a higher initial house value ([H.sub.0] = $200,000 or $500,000) are considered.

The last three columns of Table 5 give the results for a higher initial house value of [H.sub.0] = $500,000. In the base case, the house value was [H.sub.0] = $250,000 and made up 65 percent of the individual's total wealth at t = 0. This ratio is 60 percent (79 percent) for a house value [H.sub.0] = $200,000 ($500,000), and due to the isoelastic properties of the utility function the following results also apply to households with different levels of housing and total wealth given the same relative proportions of assets. The results show that the individual again chooses to borrow the maximum LTV at t = 0 with the reverse mortgage and increases the percentage sold with the home reversion scheme compared to the base case. In either scenario, the total amount of equity released is increased and the utility gain from having access to equity release products is higher compared to the base case. These findings show that individuals who have a higher proportion of their wealth invested in home equity benefit more from having access to equity release products. Likewise, individuals with a lower house value relative to liquid assets (first three columns of Table 5) enjoy smaller utility gains from having access to equity release products.

Based on the results above we can also analyze the utility gains of home equity release for individuals with preexisting debt. In the United States, the share of individuals entering retirement with preexisting conventional mortgage debt is increasing, which is reflected in a larger share of individuals using reverse mortgages with preexisting mortgage debt (Consumer Financial Protection Bureau, 2012). Such debt needs to be retired by the proceeds from equity release products and decreases the amount of home equity accessible as liquid wealth. As the decisions and utility gains of an individual with CRRA preferences are constant in relative (%) terms when scaling all monetary arguments of the utility function, we can use the results in Table 5 to analyze the effect of preexisting debt.

In particular, an individual with a (lower) house value of $200,000 and $135,000 liquid wealth (house value to total wealth ratio = 65 percent) mimics in his decisions an individual who owns a house worth $250,000 as in the base case, liquid wealth of $168,750 and preexisting debt of $33,750 (the house-to-total-wealth ratio is still 65 percent and the total wealth is again 335,000). The comparison of the utility gains between the first three columns in Table 5 and the base case show that preexisting debt reduces the gains from having access to home equity release products. All other things equal, individuals with preexisting debt can access less home equity as part of the equity release proceeds are needed to retire preexisting debt and thus equity release products are of lower value for them.

SUMMARY AND CONCLUSIONS

We model the decision problem of a retired individual that holds the major fraction of her wealth as home equity and faces longevity risk, long-term care risk, house price risk, and interest rate risk. The individual wants to "age in place" and can choose to unlock home equity using a reverse mortgage or a home reversion plan at different points in time, to buy annuities, and to buy LTCI.

Consistent with previous research (Davidoff, 2009, 2010a, 2010b, 2010c; Yogo, 2009), we find that the individual enjoys utility gains from having access to (fairly priced) equity release products. The individual chooses reverse mortgage LTV ratios and home reversion rates of well over 50 percent in most scenarios according to the results of our stylized model with fairly priced products. The availability of government-provided LTCI does not change the use of equity release products significantly.

With respect to the timing of equity release, we find that the individual chooses to unlock home equity early in retirement in most scenarios studied, which agrees with the trends described by a recent study on the U.S. market reporting that reverse mortgage borrowers are taking out loans at younger ages than in the past (Consumer Financial Protection Bureau, 2012).

The utility gains from having access to reverse mortgages are generally higher because these give higher lump-sum payments than home reversion plans and provide downside protection against house price risk. In addition to the supply-side risk and profitability considerations studied in Alai et al. (2014), this finding may help to explain why reverse mortgages dominate most equity release markets. While our model's results match observed preferences between equity release products, it produces take-up rates that are higher than those currently observed in international markets. Psychological motivations of elderly borrowers and their limited product knowledge help explain this discrepancy (Davidoff, Gerhard, and Post, 2014), but are beyond the optimal choice scope of this study.

DOI: 10.1111/jori.12068

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(1) Alternatively, Davidoff (2009) considers an individual whose utility depends on both consumption and the housing stock. He introduces a utility penalty for moving out of the house when in good health and sets this parameter such that moving is never optimal, except when the individual has to go to a nursing home.

(2) The total value of a house consists of the capital value and the rental yields. The growth rate calibrated here is the capital growth rate. It excludes rental yields.

(3) The absolute values of the utility gains derived from the model are high (more than 100 percent of wealth for some later simulations). As we base our derivations on an augmented life-cycle model with two periods, these values should not be interpreted in isolation, as their magnitude may be different for different model setups (e.g., with more periods). The values for the utility gains should only be used to identify whether an equity release product increases utility (i.e., when welfare gains are larger than zero) and to perform relative comparisons between the products.

(4) The general tendencies derived hold when adding closing costs to for the products (e.g., 5 percent of the payout from the products). The individual prefers early borrowing with the reverse mortgage and a mix between a larger portion of early equity release and smaller portion later for the home reversion plan. In particular, the individual still borrows up to the maximum loan amount for the reverse mortgage, while for the home reversion plan equity release at t = 1 decreases by 2 percentage points. Utility gains, however, are lower when considering contracting costs (6 percentage points for the reverse mortgage and 3 percentage points for the home reversion plan; detailed results available from the authors on request).

(5) We have also considered scenarios were an additional risk margin (e.g., 0.1 percent or 1 percent) is added to the mortgage rate charged on the reverse mortgage loan to account for cases where the product provider is not risk neutral toward house price risk. In these scenarios, the utility gains for the reverse mortgage decrease by 1 or 2 percentage points (minus 1.1 percentage points for a risk margin of 0.1 percent and minus 2.4 percentage points for a risk margin of 1 percent), but the reverse mortgage remains the preferred equity release product. Detailed results are available from the authors on request.

Katja Hanewald is at the Australian Research Council Centre of Excellence in Population Ageing Research (CEPAR), Australian School of Business, University of New South Wales, Sydney, Australia. Hanewald can be contacted via e-mail: hanewaldk@gmail.com. Thomas Post is at the Department of Finance, School of Business and Economics, Maastricht University and Netspar. Post can be contacted via e-mail: t.post@maastrichtuniversity.nl. Michael Sherris is at the School of Risk and Actuarial Studies and Australian Research Council Centre of Excellence in Population Ageing Research (CEPAR), Australian School of Business, University of New South Wales, Sydney, Australia. Sherris can be contacted via e-mail: m.sherris@unsw. edu.au. The authors acknowledge the support of ARC Linkage Grant Project LP0883398 "Managing Risk with Insurance and Superannuation as Individuals Age" with industry partners PwC and APRA and the Australian Research Council Centre of Excellence in Population Ageing Research (Project number CE110001029). We thank the editor, Keith Crocker, and two anonymous reviewers for their suggestions to improve the paper. For their comments and suggestions we would like to thank Kevin Ahlgrim, Hua Chen, and the conference participants at the 20th Annual Colloquium of Superannuation Researchers and the 2012 Annual Meeting of the American Risk and Insurance Association. Rachel Nakhle and Yu Sun provided excellent research assistance.

Table 1
Model Parameters

                                                            Baseline
Parameter                                                    Value

House value at t = 0                        [H.sub.0]       $250,000
Liquid wealth at t = 0                      [W.sub.0]       $135,000
Age in years at f = 0                                          62
Relative risk aversion                       [gamma]           2
Subjective discount factor                   [delta]          0.98
Strength of bequest motive                   [beta]           0.5
Long-term care expenses per year
  Needing some care at home                [LTC.sub.c]      $10,000
  Needing care in a nursing home           [LTC.sub.n]      $50,000
Mean interest rate per year                 [r.sub.0]         2.0%
  (= interest rate at f = 0)
Standard deviation of interest rate      Std([r.sub.0])       2.2%
  per year
Mean house price growth per year                G             1.6%
Standard deviation of house price            Std(g)          11.7%
  growth per year
Rental yield                              [%.sub.rent]         2%
Coinsurance percentage of the           [%.sub.govt.LTCI]     50%
  govt.-provided LTCI
Stop loss of the govt.-provided LTCI                         $6,276
  per year

                                           Alternative
Parameter                                     Values

House value at t = 0                    $200,000; $500,000
Liquid wealth at t = 0
Age in years at f = 0
Relative risk aversion                         3; 5
Subjective discount factor                  0.93; 1.00
Strength of bequest motive                     0; 2
Long-term care expenses per year
  Needing some care at home
  Needing care in a nursing home
Mean interest rate per year
  (= interest rate at f = 0)
Standard deviation of interest rate
  per year
Mean house price growth per year
Standard deviation of house price
  growth per year
Rental yield
Coinsurance percentage of the
  govt.-provided LTCI
Stop loss of the govt.-provided LTCI
  per year

Notes: This table shows baseline and alternative model parameters.
All parameters referring to multiple years (subjective discount
factor, interest rate, house price growth, mortgage rate) are scaled
by the length of one period in the model, which is 19 years. All
monetary values are in real terms.

Table 2
Optimal Equity Release at Different Points in Time

                                       No Equity   Reverse      Home
                                        Release    Mortgage   Reversion
                                       Products    at t = 0   at t = 0

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     74%
  Total liquid wealth                   135,000    347,500     241,512
  Consumption                            81,123    180,510     144,360
  Consumption %                             60%        52%         60%
  Savings                                     0     77,835      19,004
  Savings %                                  0%        22%          8%
  Annuity premium                        41,852     75,345      64,488
  Annuity premium %                         31%        22%         27%
  LTCI premium                           12,025     13,811      13,660
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             86%        99%         98%
Financial decisions at t = 1
  Additional [LTV.sub.1]
  [%.sub.HR,1]
Equivalent wealth variation [theta]                   +86%        +51%

                                         Reverse        Home
                                        Mortgage      Reversion
                                        at t = l      at t = l

Financial decisions at t = 0
  [LTV.sub.0]                                 0%
  [%.sub.HR,0]                                              0%
  Total liquid wealth                    135,000       135,000
  Consumption                            121,009       121,009
  Consumption %                              90%           90%
  Savings                                      0             0
  Savings %                                   0%            0%
  Annuity premium                              0             0
  Annuity premium %                           0%            0%
  LTCI premium                            13,991        13,991
  LTCI premium %                             10%           10%
  LTCI coverage                             100%          100%
Financial decisions at t = 1
  Additional [LTV.sub.1]                     85%
  [%.sub.HR,1]                                            100%
Equivalent wealth variation [theta]         +73%          +52%

                                         Reverse        Home
                                        Mortgage      Reversion
                                       at t = 0, 1   at t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                85%
  [%.sub.HR,0]                                             58%
  Total liquid wealth                    347,500       218,349
  Consumption                            180,510       145,054
  Consumption %                              52%           66%
  Savings                                 77,833             0
  Savings %                                  22%            0%
  Annuity premium                         75,345        59,722
  Annuity premium %                          22%           27%
  LTCI premium                            13,812        13,573
  LTCI premium %                              4%            6%
  LTCI coverage                              99%           97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                      0%
  [%.sub.HR,1]                                             17%
Equivalent wealth variation [theta]         +86%          +53%

Notes: LTV denotes the loan-to-value ratio and [%.sub.HR] is the
optimal percentage of the property sold under the home reversion
plan. Consumption %, Saving %, Annuity premium %, and LTCI premium %
are given as percentages of total liquid wealth at t = 0 (after
equity release). Additional [LTV.sub.1] and [%.sub.HR,1] are reported
as averages over those states t = 1 in which equity release products
are offered to the individual. [theta] measures the utility gain in
relative dollar terms from having access to home equity release
products. That is, [theta] measures by how much liquid wealth and the
house value would have to be increased in the "no equity release
products" scenario for the individual to have the same utility as in
the given scenario.

Table 3
Sensitivity Analyses: Preference Parameters

Panel A: Risk Aversion [gamma]

                                           Base Case: [gamma] = 2

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     58%
  Total liquid wealth                   135,000    347,500     218,349
  Consumption                            81,123    180,510     145,054
  Consumption %                             60%        52%         66%
  Savings                                     0     77,833           0
  Savings %                                  0%        22%          0%
  Annuity premium                        41,852     75,345      59,722
  Annuity premium %                         31%        22%         27%
  LTCI premium                           12,025     13,812      13,573
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             86%        99%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     17%
Equivalent wealth variation [theta]                   +86%        +53%

                                                 [gamma] = 3

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     60%
  Total liquid wealth                   135,000    347,500     221,520
  Consumption                            81,015    176,129     141,562
  Consumption %                             60%        51%         64%
  Savings                                     0     86,532       5,751
  Savings %                                  0%        25%          3%
  Annuity premium                        41,528     71,022      60,591
  Annuity premium %                         31%        20%         27%
  LTCI premium                           12,456     13,817      13,616
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             89%        99%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     22%
Equivalent wealth variation [theta]                   +94%        +57%

                                                 [gamma] = 5

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          80%
  [%.sub.HR,0]                                                     76%
  Total liquid wealth                   135,000    335,000     243,685
  Consumption                            80,900    175,600     142,810
  Consumption %                             60%        52%         59%
  Savings                                     0     83,187      30,863
  Savings %                                  0%        25%         13%
  Annuity premium                        41,136     62,640      56,480
  Annuity premium %                         30%        19%         23%
  LTCI premium                           12,964     13,573      13,532
  LTCI premium %                            10%         4%          6%
  LTCI coverage                             93%        97%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                5%
  [%.sub.HR,1]                                                     13%
Equivalent wealth variation [theta]                  +104%        +66%

Panel B: Subjective Discount Factor [delta]

                                               [delta] = 0.93

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     68%
  Total liquid wealth                   135,000    347,500     233,433
  Consumption                            93,139    229,010     182,864
  Consumption %                             69%        66%         78%
  Savings                                     0     58,543           0
  Savings %                                  0%        17%          0%
  Annuity premium                        29,423     46,135      36,899
  Annuity premium %                         22%        13%         16%
  LTCI premium                           12,438     13,812      13,670
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             89%        99%         98%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     20%
Equivalent wealth variation [theta]                  +120%        +79%

                                          Base Case: [delta] = 0.98

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     58%
  Total liquid wealth                   135,000    347,500     218,349
  Consumption                            81,123    180,510     145,054
  Consumption %                             60%        52%         66%
  Savings                                     0     77,833           0
  Savings %                                  0%        22%          0%
  Annuity premium                        41,852     75,345      59,722
  Annuity premium %                         31%        22%         27%
  LTCI premium                           12,025     13,812      13,573
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             86%        99%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     17%
Equivalent wealth variation [theta]                   +86%        +53%

                                               [delta] = 1.00

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     54%
  Total liquid wealth                   135,000    347,414     212,602
  Consumption                            75,822    157,792     128,651
  Consumption %                             56%        45%         61%
  Savings                                     0     82,849           0
  Savings %                                  0%        24%          0%
  Annuity premium                        47,210     92,781      70,406
  Annuity premium %                         35%        27%         33%
  LTCI premium                           11,967     13,991      13,545
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             86%       100%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     15%
Equivalent wealth variation [theta]                   +67%        +42%

Panel C: Bequest Motive [beta]

                                                 [beta] = 0

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                    100%
  Total liquid wealth                   135,000    347,500     278,896
  Consumption                            81,124    222,347     176,542
  Consumption %                             60%        64%         63%
  Savings                                     0          0           0
  Savings %                                  0%         0%          0%
  Annuity premium                        42,072    111,380      88,363
  Annuity premium %                         31%        32%         32%
  LTCI premium                           11,803     13,773      13,991
  LTCI premium %                             9%         4%          5%
  LTCI coverage                             84%        98%        100%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                      0%
Equivalent wealth variation [theta]                  +173%       +117%

                                           Base Case: [beta] = 0.5

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     58%
  Total liquid wealth                   135,000    347,500     218,349
  Consumption                            81,123    180,510     145,054
  Consumption %                             60%        52%         66%
  Savings                                     0     77,833           0
  Savings %                                  0%        22%          0%
  Annuity premium                        41,852     75,345      59,722
  Annuity premium %                         31%        22%         27%
  LTCI premium                           12,025     13,812      13,573
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             86%        99%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     17%
Equivalent wealth variation [theta]                   +86%        +53%

                                                 [beta] = 2

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     33%
  Total liquid wealth                   135,000    347,500     183,193
  Consumption                            80,870    150,930     122,514
  Consumption %                             60%        43%         67%
  Savings                                     0    135,477           0
  Savings %                                  0%        39%          0%
  Annuity premium                        40,139     47,278      47,296
  Annuity premium %                         30%        14%         26%
  LTCI premium                           13,991     13,815      13,383
  LTCI premium %                            10%         4%          7%
  LTCI coverage                            100%        99%         96%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     24%
Equivalent wealth variation [theta]                   +47%        +23%

Notes: Panel A shows the results for different levels of the risk
aversion parameter [gamma], in Panel B the subjective discount factor
[delta] is varied, and Panel C the strength of the bequest motive
[beta]. The LTV denotes the loan-to-value ratio and [%.sub.HR] is the
optimal percentage of the property sold under the home reversion
plan. Consumption %, Saving %, Annuity premium %, and LTCI premium %
are given as percentages of total liquid wealth at t = 0 (after
equity release). Additional [LTV.sub.1] and [%.sub.HR,1] are reported
as averages over those states t = 1 in which equity release products
are offered to the individual. [theta] measures the utility gain in
relative dollar terms from having access to home equity release
products. That is, [theta] measures by how much liquid wealth and the
house value need to be scaled for the individual to have the same
utility as in the scenarios without equity release products.

Table 4
The Impact of Government-Provided LTCI on Optimal Equity Release

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   f = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     56%
  Total liquid wealth                   135,000    347,500     216,208
  Consumption                            87,312    185,670     150,325
  Consumption %                             65%        53%         70%
  Savings                                     0     80,290           0
  Savings %                                  0%        23%          0%
  Annuity premium                        43,798     77,308      61,798
  Annuity premium %                         32%        22%         29%
  LTCI premium                            3,890      4,232       4,084
  LTCI premium %                             3%         1%          2%
  LTCI coverage                             89%        97%         94%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     18%
Equivalent wealth variation [theta]                   +79%        +48%

Notes: LTV denotes the loan-to-value ratio and [%.sub.HR] is the
optimal percentage of the property sold under the home reversion
plan. Consumption %, Saving %, Annuity premium %, and LTCI premium %
are given as percentages of total liquid wealth at t = 0 (after
equity release). Additional [LTV.sub.1] and [%.sub.HR,1] are reported
as averages over those states t = 1 in which equity release products
are offered to the individual. [theta] measures the utility gain in
relative dollar terms from having access to home equity release
products. That is, [theta] measures by how much liquid wealth and the
house value would have to be increased in the "no equity release
products" scenario for the individual to have the same utility as in
the given scenario.

Table 5
Sensitivity Analyses: House Value

                                             [H.sub.0] = $200,000

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     53%
  Total liquid wealth                   135,000    305,000     196,147
  Consumption                            81,108    157,385     129,295
  Consumption %                             60%        52%         66%
  Savings                                     0     68,396           0
  Savings %                                  0%        22%          0%
  Annuity premium                        41,646     65,372      53,234
  Annuity premium %                         31%        21%         27%
  LTCI premium                           12,246     13,847      13,618
  LTCI premium %                             9%         5%          7%
  LTCI coverage                             88%        99%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     19%
Equivalent wealth variation [theta]                   +70%        +41%

                                            Base Case: House Value
                                             [H.sub.0] = $250,000

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     58%
  Total liquid wealth                   135,000    347,500     218,349
  Consumption                            81,123    180,510     145,054
  Consumption %                             60%        52%         66%
  Savings                                     0     77,833           0
  Savings %                                  0%        22%          0%
  Annuity premium                        41,852     75,345      59,722
  Annuity premium %                         31%        22%         27%
  LTCI premium                           12,025     13,812      13,573
  LTCI premium %                             9%         4%          6%
  LTCI coverage                             86%        99%         97%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     17%
Equivalent wealth variation [theta]                   +86%        +53%

                                             [H.sub.0] = $500,000

                                                   Reverse      Home
                                       No Equity   Mortgage   Reversion
                                        Release       at         at
                                       Products    t = 0, 1   t = 0, 1

Financial decisions at t = 0
  [LTV.sub.0]                                          85%
  [%.sub.HR,0]                                                     68%
  Total liquid wealth                   135,000    560,000     329,356
  Consumption                            80,950    296,050     223,847
  Consumption %                             60%        53%         68%
  Savings                                16,672    125,220           0
  Savings %                                 12%        22%          0%
  Annuity premium                        25,641    125,095      92,163
  Annuity premium %                         19%        22%         28%
  LTCI premium                           11,737     13,635      13,345
  LTCI premium %                             9%         2%          4%
  LTCI coverage                             84%        97%         95%
Financial decisions at t = 1
  Additional [LTV.sub.1]                                0%
  [%.sub.HR,1]                                                     13%
Equivalent wealth variation [theta]                  +210%       +137%

Notes: LTV denotes the loan-to-value ratio and [%.sub.HR] is the
optimal percentage of the property sold under the home reversion
plan. Consumption %, Saving %, Annuity premium %, and LTCI premium %
are given as percentages of total liquid wealth at t = 0 (after
equity release). Additional [LTV.sub.1] and [%.sub.HR,1] are reported
as averages over those states t = 1 in which equity release products
are offered to the individual. [theta] measures the utility gain in
relative dollar terms from having access to home equity release
products. That is, [theta] measures by how much liquid wealth and the
house value need to be scaled for the individual to have the same
utility as in the scenarios without equity release products.
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Author:Hanewald, Katja; Post, Thomas; Sherris, Michael
Publication:Journal of Risk and Insurance
Geographic Code:8AUST
Date:Jun 1, 2016
Words:11634
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