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Risk Management in Retirement
– What is the Optimal Home Equity Release Product?
Katja Hanewald1, Thomas Post2 and Michael Sherris3 15 July 2012
Draft prepared for the ARIA 2012 Annual Meeting
Please do not cite or circulate without prior consent by the authors
Abstract: This paper studies the optimal choice of home equity release products. The decision problem of a retiring couple is modeled that holds the major fraction of their wealth as home equity and faces longevity, long-term care, house price, and interest rate risk. The couple can choose to buy annuities, long-term care insurance, and to borrow against the home using different equity release products. These decisions involve the timing problem of when to optimally release home equity. The framework is used to compare the welfare effects of different home equity release products and to study the role of government-provided long-term care insurance.
JEL Classification: D14, D91, G11, R20
Keywords: Retirement, inter-temporal optimization and decision making, home equity release, reverse mortgage, annuities, long-term care insurance
1 [Contact author] School of Risk and Actuarial and ARC Centre of Excellence in Population Ageing Research (CEPAR), Australian School of Business, University of New South Wales, Sydney, Australia. Email:
2 Department of Finance, School of Business and Economics, Maastricht University and Netspar, Email:
3 School of Risk and Actuarial Studies and ARC Centre of Excellence in Population Ageing Research (CEPAR), Australian School of Business, University of New South Wales. Email: [email protected].
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1 Introduction
This paper studies the optimal product choice of home equity release products from the homeowner’s perspective in the presence of longevity, long-term care, house price, and interest rate risk. A retiring couple chooses among different home equity release contracts, and long-term care and longevity insurance products. Home equity release contracts differ substantially in the way house price risks are shared and transferred between the homeowner and the lender. Due to this the optimal choice is strongly dependent on the homeowners’ individual characteristics (including risk aversion and a bequest motive) and on the interaction with longevity and long- term care risks, which vary strongly between different institutional settings.
Home equity is a very special asset class. The home is an investment and a residence, providing non-pecuniary services. For example, people value the ability to “age in place” (Davidoff, 2010c) and, even with substantial mortgage balances outstanding, people are happy about being a homeowner (Whitehead and Yates, 2010). Homeownership rates are high and are between 50%
and 80% for most OECD countries (Andrews and Caldera Sánchez, 2011). Home equity represents the major share of the elderly’s total assets. For example, for households aged 65+, in the U.S., the value of the primary residence comprises on average (median) 49% (52%) of households’ total assets, with 82% of households aged 65+ actually owning a house (2009 wave of the Survey of Consumer Finances). A home is not an ideal asset to meet the financial needs of elderly households especially when having no other substantial sources of income. It concentrates a large amount of household savings in a single asset, exposing the household to substantial idiosyncratic risk (Case, Cotter, and Gabriel, 2011). It is illiquid, and in case of urgent
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cash needs, for example, due to health shocks it cannot be sold in parts to pay for out-of-pocket costs.
Home equity release products convert home equity into liquid wealth but allow homeowners to continue residing in their home. Markets for equity release products are growing in the U.S.
(Shan, 2011), in the UK (KRS, 2011) and elsewhere (Reifner et al., 2009; Deloitte and SEQUAL, 2012). A range of different contract designs exist that share and transfer house price risks in various ways between the homeowner and the lender. The most common product in most markets is a reverse mortgage with rolled-up interest (Oliver Wyman, 2008; Davidoff 2010c).
This product allows homeowners to participate in house price appreciations, while giving protection to adverse house price developments through a no-negative equity guarantee (NNEG) typically embedded in the product. Home reversion schemes available, for example, in the UK and in Australia allow homeowners to transfer a proportion of house price risks to the lender in exchange for a lump-sum payment and a lease-for-life agreement (Oliver Wyman, 2008).
Contract designs also differ in other dimensions such as having a fixed or variable interest rate (Oliver Wyman, 2008). Markets are very dynamic and new products are being constantly developed (Australian Securities and Investment Commission, 2005). Comparing and choosing among various products with different risk and return features has become an increasingly important, but also increasingly difficult task for today’s homeowners.
Several papers examine home equity release products in optimal household portfolios. Artle 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
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utility. Fratantoni (1999) models the product choice between two reverse mortgage designs—
annuity payout plan and line-of-credit plan—for an elderly homeowners facing non-insurable expenditure shocks. He finds that line-of-credit plans are generally preferable 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 the reverse mortgages itself is utility-enhancing and finds interaction effects with annuities and long-term care insurance. For example, home equity may substitute long term care insurance. Davidoff (2010c) introduces house price risk into a reverse mortgage choice model. He shows that amortization of interest (rolled-up interest), a feature inherent in many currently sold contracts types, is not always Pareto optimal. Likewise, Yogo (2009) considers stochastic housing prices (and stochastic health depreciation), confirming that reverse mortgages are utility enhancing. The decision between fixed-rate and adjustable-rate products so far has been studied for “normal” mortgages, with adjustable-rate products being found more attractive to homeowners (Campbell and Cocco, 2003).
In summary, a number of studies using different models find that reverse mortgages are utility- enhancing. The utility gains are shown to depend on the interaction with health and longevity risks, and on the availability of products to insure against these risks.
This study provides the following contributions to the literature. First, we extend previous models by considering longevity, long-term care, house price, and interest rate risk and by modeling the household as a couple as reverse mortgage decisions are often triggered by the
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death of one spouse.1 Second, a general model is developed covering a range of different equity release products and the timing problem of when to release home equity. Third, we analyze the optimal choice in different institutional settings for long term care insurance (LTCI) and examine the resulting interactions. We distinguish between a setting in which most costs have to be paid out-of-pocket with private insurance available and a setting in which most long-term care costs are born by a government-sponsored system.
The results of this study show that the couple enjoys large utility gains from having access to either one of the two equity release products. Higher utility gains are found for the reverse mortgage. The household chooses to unlock home equity early on 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 change the use of equity release products significantly, but does change the demand for annuities.
The paper is structured as follows. Section 2 introduces the life cycle model. Section 3 presents the results. Section 4 concludes and discusses policy recommendations.
1 Shan (2011) reports that 45% in the U.S. equity release program are single females, 34% are couples, and 11% are single males (based on 2007 data). In Australia, the majority of equity release customers are couples between 70-75 years old (Deloitte and SEQUAL, 2012).
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2 The Model
2.1 General Structure of the Model and Timing
The decision problem of a retiring couple is modeled that holds the major fraction of wealth in their home. The index ∈ , is used to denote probabilities, payouts or utility values of the husband (m) and the wife (f), respectively. The couple faces longevity risk, long-term care risk, house price risk, and interest rate risk. Different insurance and home equity release products are available to the couple.
The decisions of the couple are modeled in an augmented life cycle model. The model extends previous work by Davidoff (2009, 2010b, 2010c) by considering a couple, by allowing for interest rate risk, by including different types of equity release products and modeling the timing decision of when to release home equity. A two-period model (three points in time) is developed that captures the couple’s decisions at retirement and at an advanced age. The model’s input parameters are calibrated such that each of the two periods reflects a multi-year horizon. Figure 1 illustrates the decision and timing structure of the model.
-- Figure 1 here --
At time t = 0, both spouses are in good health. The initial endowment consists of the home and liquid wealth. The couple decides on consumption, on saving over the first period of their retirement, on purchasing annuities, long-term care insurance (LTCI) and on taking out an equity release product. Equity release products increase liquid wealth available for consumption, saving, and for purchasing insurance products.
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At time t = 1, as in Davidoff (2009), each husband and wife independently will be in one of four health states, implying different health care expenses. The random 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. There are the following main cases at t = 1.
1) Both spouses are dead: Their remaining liquid wealth and housing wealth (net of mortgage repayments) are left as a bequest.
2) At least one partner is still alive: The household receives payments from insurance contracts and from equity release products bought at t = 0. Health state-dependent care expenses not covered by insurance are paid out-of-pocket. The couple decides on consumption and saving over the second period.
2a) Both spouses are in a nursing home or one partner is in a nursing home and the other one is dead: The house is sold and all outstanding loans are paid back. Additional sale proceeds are added to liquid wealth.
2b) At least one partner is still living at home: The couple decides whether to take out another equity release product.
At t = 2, both partners are dead with certainty. Remaining liquid wealth and housing wealth (net of mortgage repayments) are bequeathed.
2.2 Interest Rates, Mortgage Rates, House Price Growth and Savings Growth
The risk-free interest rate r0 over the first period is known at t = 0. The interest rate r1 over the second period is a random variable realized at t = 1. Mortgage rates are derived from interest
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rates by adding a margin ∆RM to r0 and r1 (see Sections 2.6 and 2.8 for more details). Savings at t = 0, S0, and at t = 1, S1, accrue the respective one-year interest rates r0 and r1. At t = 0, the couple owns a mortgage-free house worth H0. At t = 1 the house value is H1 = H0 · (1 + g1) and at t = 2 it is H2 = H1 · (1 + g2), where the growth rates g1 and g2 are i.i.d. random variables, uncorrelated with the interest rate.
2.3 Health States and Care Costs
At t = 0, both husband and wife are in good health and no care expenses have to be paid. At time t = 1, husband and wife are each independently in one of four health states, requiring different levels of health care costs. The four states are: staying in good health and having no long-term care costs (state h) with probability ph, needing some care at home at cost LTCc (state c) with probability pc, needing to move to a nursing home at care costs LTCn (state n) with probability pn, and being death (state d) with probability pd (ph + pc + pn + pd = 1).
2.4 Long-Term Care Insurance and Annuity Products
Long-term care insurance (LTCI) covering the care costs in state c and the care costs in state n is available at t = 0. Husband and wife buy separate LTCI contracts. The couple chooses the proportion of insurance coverage % by choosing the amount of wealth spent on LTCI for each partner i = m, f. LTCI is priced according to the actuarial principle of equivalence plus a proportional loading LTCI. The premium for partial coverage of an individual’s care costs is given by:
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= (1 + ) ∙ % ∙ ∙(! " #)∙ , = , . (2-1)
Furthermore, single life annuities are available at t = 0. Annuities are priced based on the actuarial principle of equivalence plus a proportional loading A. The premium for an annuity paying $ at t = 1 conditional on survival is given by:
% = (1 + %) ∙&!' (! "( )∙%#) , = , . (2-2)
The annuity payment $ is determined by the amount of wealth %,* the couple decides to invest in individual i’s annuity according to formula (2-2).
2.5 Government-Provided Long-Term Care Insurance
Scenarios are considered in which both public and private long-term care insurance (LTCI) are available. Social insurance arrangements for long-term care services exist in a number of OECD countries—German, Japan, Korea, the Netherlands and Luxembourg (for an overview, see Productivity Commission, 2012).
In this study, government-provided LTCI is modeled as a compulsory coinsurance arrangement with a stop loss limit. The insurance scheme covers a percentage %+,-.. of all care costs up to an out-of-pocket spending limit. This arrangement abstracts from the details of the different 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, and agrees with
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suggestions by the Productivity Commission in Australia (Commission on Funding of Care and Support, 2011; Productivity Commission, 2012). The retired household faces no costs for this insurance, but the cost is levied on the working generation. The couple can decide to buy private LTCI coverage for the proportion of care costs not covered by the public LTCI. Because the expected care costs are lower, a lower premium for private LTCI results.
2.6 Equity Release Products
2.6.1 Overview
The model developed above can accommodate a range of different home equity release products.
In this study we focus on lump-sum reverse mortgages and home revision plans (also called sale- and-lease-back plan or shared equity mortgage). Both products are offered to the household at t = 0 and t = 1. With these types, the analysis covers the main types of equity release schemes currently available in Australia, Canada, UK, and the US (Oliver Wyman, 2008, Davidoff, 2010c).2 We focus on reverse mortgages with variable interest rates and a NNEG because this is the dominant product design in most markets.
Often, reverse mortgages are offered only to households that own a debt-free home. We model this situation by considering scenarios in which equity release products are only offered once (at t = 0). The comparison allows us to determine optimal equity release choices from the households’ perspective.
2 Because the reverse mortgage is available at t = 0 and 1 and private annuities are available for purchase, the line- of-credit and annuity payout plan types of reverse mortgage additionally studied in Fratatoni (1999) are covered (implicitly) in our analysis.
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2.6.2 Lump-sum reverse mortgage with variable interest rates and NNEG
A lump-sum reverse mortgage (RM) with variable interest rates and a no-negative-equity guarantee is available at t = 0 and t = 1. LSRM,τ denotes the loan value of a contract taken out at time τ, paid out as a lump sum at time τ.
RMτ_balancet is the time t value of the outstanding loan balance of a reverse mortgage taken out at time τ. This balance is given by compounding LSRM,τ at the respective mortgage rate (rolled-up interest). Mortgage rates are calculated by adding a margin ∆RM to the random interest rate. The margin reflects the price of the no-negative-equity guarantee. The value of this guarantee is different for reverse mortgages taken out at t = 0 and at t = 1, resulting in different margins. For a reverse mortgage taken out at t = 0, the following mortgage rates apply: r0 + ∆RM,,0 over the first period and r1 + ∆RM,0 over the second period (or r0 + ∆RM,0 over both periods in case of fixed interest rates). The margin r1 + ∆RM,1 applies over the second period for a reverse mortgage taken out at t = 1,.
The loan amounts LSRM, 0 and LSRM, 1 are decision variables. Loan amounts are restricted by a maximum loan-to-value ratio, which is defined in terms of the house value Hτ. Different (age- specific) maximum loan-to-value ratios LTV0max
and LTV1max
apply to reverse mortgages taken out at t = 0 and t = 1. The maximum loan-to-value ratio at t = 1, LTV1max, is defined as a combined loan-to-value ratio.
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A reverse mortgage taken out at t = 0 is repaid at t = 1 if both husband and wife are in a nursing home, one partner is in a nursing home and the other one is dead, or both partners are dead.
These cases correspond to the cases 1) and 2a) described in Section 2.1. In the remaining cases, the couple can decide to take out another reverse mortgage at t = 1 and the outstanding loan balances of both contracts are paid back at t = 2.
In case of repayment, the house is sold and the proceeds of the sale are used to pay back the total outstanding reverse mortgage balance RM_balancet = RM0_balancet + RM1_balancet. The total repayment of both reverse mortgage loans is capped by the house value Ht at time t due to the no-negative-equity guarantee. To simplify the pricing, repayment of LSRM,1 has priority over repayment of LSRM,1 if at time t = 2 LSRM_balance2 < H2.
2.6.3 Home reversion plan
Home reversion plans (HR) are offered at t = 0 and t = 1. Under this arrangement, the household sells a share %HR,τ · Hτ , τ = 0, 1 of the home equity to the product provider and receives a lump sum LSHR,τ in return. The lump sum, LSHR,τ, is less than the market value of the equity share, reflecting the value of a lease-for-life agreement and house price risk. The household 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. The equity share of the product provider appreciates with the house price growth rates. That is, for example, for a HR contracted at t = 0, the product provider owns %HR,0 · H1 of the house value at t = 1 and %HR,0 · H2 at t = 2.
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A home-reversion plan taken out at t = 0 ends at t = 1 if both husband and wife are in a nursing home, one partner is in a nursing home and the other one is dead, or both partners are dead. In the remaining cases, the couple 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 couple’s share is added to the liquid wealth that is bequeathed.
2.7 The Couple’s Maximization Problem
The couple’s lifetime utility function V is based on Brown and Poterba (2000), but with a bequest motive, as, for example, in Inkmann, Lopes, and Michaelides (2011)3:
0( , 1) = ∑ 3?.@A .45. ∙ 67 .8, .9: + (1 − 5. ) ∙ < ∙ =(1.)>. (2-3)
δ denotes the subjective discount factor of the couple, β the utility weight of the bequest motive, 5. is an indicator variable taking the value one if at least one member of the couple is alive and zero otherwise, Ct is the consumption in real terms of the husband (m) and wife (f). The wealth bequeathed by the couple Wt is comprised of liquid wealth and the house value net of payments to repay equity release products.
3 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. Our model does not incorporate the decision to move: based on stylized facts (Whitehead and Yates, 2010), the decision to live in the own home is assumed to be always optimal when in good health and housing is not needed as an argument in the utility function (as in Campbell and Cocco, 2003).
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The one-period utility functions of the couple, U, is given by the equally weighted sum of the husband and the wife’s subutility functions, Um and Uf (Brown and Poterba, 2000):
67 .8, .9: = 5.8∙ 687 .8, .9: + 5.9 ∙ 697 .9, .8:, (2-4)
687 .8, .9: = &BC DBE)
FGH
(!'I) , (2-5)
697 .9, .8: =&BE DBC)
FGH
(!'I) ,
where 5.8 (5.9) is the indicator variable taking the value one if the husband (wife) is alive and 0 otherwise. The parameter θ controls the degree of jointness (sharing of resources) in consumption between the husband and the wife. Both spouses have their subutility function defined over consumption with an identical relative risk aversion parameter γ. The bequest utility function, B, exhibits the same relative risk aversion as U and is given by:
=(1.) = J(!'I)BFGH . (2-6)
The couple’s objective is to maximize the expectation over (2-3) subject to a set of constraints.
The couple’s optimization problem is given by:
maxB,NO,B,P
QC,PQ,EPRSTUC ,PRSTUE EW0( , 1)X, Y = 0, 1, (2-7) where the index j refers to cash flows from equity release schemes alternatively available (j = RM, HR). The optimization problem is subject to
(i) consumption and bequest constraints:
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A8+ A9 = 1A− [A− %8− %9− 8 − 9 + [\,A
!8+ !9 = [A∙ (1 + ]A) − [!+$8+ $9 − 71 − %+,-..− %8 : ∙ 8− 71 − %+,-..− %9 : ∙ 9 + [\,!
Bequest constraint in case of the reverse mortgage:
1! = [A∙ (1 + ]A) + maxW^!− _`_bcdcefg!, 0X 1? = [!∙ (1 + ]!) + maxW^?− _`_bcdcefg?, 0X Bequest constraint in case of the home reversion plan:
1! = [A∙ (1 + ]A) + 71 − %hi,A: ∙ ^!
1? = [!∙ (1 + ]!) + 71 − %hi,A−%hi,!: ∙ ^?
(2-8)
(ii) borrowing constraints:
0 ≤ [A ≤ 1A− %8− %9− 8 − 9 + [\,A (2-9) 0 ≤ [! ≤ [A∙ (1 + ]A)+$8+ $9 − 71 − %+,-..− %8 : ∙ ( 8+ 8)
− 71 − %+,-..− %9 : ∙ ( 9+ 9) + [\,!
(2-10)
(iii) no-short sale constraints for equity release and insurance products:
0 ≤ [\,A, [\,!, %8, %9, 8 , 9 (2-11)
and (iv) further product constraints
• Maximum loan-to-value ratios for the reverse mortgage:
[ik,A≤ 0ik,A8lm·^A (2-12)
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_`_bcdcefg! + [ik,!≤ 0ik,!8lm·^!
• Maximum home reversion rate:
%hi,A−%hi,! ≤ 1. (2-13)
• LTCI benefits capped by actual care expenses:
% ≤ 1, = , . (2-14)
2.8 Numerical Calibration of Baseline Parameters
This section describes the numerical calibration of the model’s baseline parameters. The parameters are chosen to reflect the U.S. market and to allow comparison with previous studies.
Alternative parameter values are introduced in Section 3. Table 1 summarizes the numerical calibration. To distinguish product design effect from pricing effect, especially in the different equity release products, 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.
-- Table 1 here --
2.8.1 The Couple’s Preferences and Endowment
The standard parameters for the couple’s preferences (relative risk aversion, subjective discount factor, strength of the bequest motive) are set within the range typically used in life cycle models
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in the literature. Relative risk aversion γ is set to 2, the subjective discount factor δ is set to 0.98, and the strength of the bequest motive β is set to 0.2 (see, e.g., Laibson, Repetto, and Tobacman 1998; Cocco, Gomes, and Maenhout 2005; Inkmann, Lopes, and Michaelides 2011). The jointness in consumption parameter θ is set to 0.2. This value is lower than the value of 0.5 used by Brown and Poterba (2000) to reflect that jointness of consumption is less effective when one or both partners are in a nursing home.
The U.S. HECM equity release program, to which most reverse montages originated belong, requires both spouses to be at least 62 years old to access mortgages. Thus, the initial age of both spouses is set to 62 at t = 0.The maximum age in the model (at t = 2) is set to 100, and for having identical period lengths, the age at t = 1 is set to 81 making one period 19 years long. The initial endowment consists of liquid wealth of W0 = $135,000 and a house worth H0 = $250,000, which reflect the median values for financial assets and primary residences for couples aged 60 to 65 in the 2009 wave of the Survey of Consumer Finances.
2.8.2 Interest Rates and House Price Growth
Interest rates are modeled as in Campbell and Cocco (2003) in their analysis of standard mortgages. That is, future one-year interest rates are given by the mean rate plus a transitory i.i.d.
shock. Based on one-year U.S. Treasuries, Campbell and Cocco estimate the mean of real interest rates to be 2% with a standard deviation of 2.2%. The interest rate over the first period, r0, is set equal to the mean real rate.
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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 Cocco (2003) based on the Panel Study of Income Dynamics (PSID): the mean real growth rate is 1.6%
with a standard deviation of 11.7%.4
2.8.3 Health States, Care Costs, Long-Term Care Insurance and Annuity Products
For the calibration of 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) the same values are used as in 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 then priced according to formula (2-1) with the interest rate of 2%. Likewise, annuities are priced according to formula (2-2) using the respective survival probabilities. A zero loading is assumed for both the LTCI and annuities (LTCI = LTCI = 0).
2.8.4 Fair Pricing of the Reverse Mortgage
The reverse mortgage is priced such that provider of the product makes on average across all future states a zero profit. The profit is calculated as the expected present value of the loan repayment (discounted using interest rates) minus the initial loan amount. A margin ∆RM is
4 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.
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determined that compensates the product provider for the equity guarantee (NNEG) embedded in the reverse mortgage.
Figure 2 gives the margin ∆RM,0 for the variable interest rate reverse mortgage taken out at t = 0 for different actual loan-to-value ratios (LTVs). Given the calibration of interest rate, house price and health states, the value of the house will always be enough 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 this reflects in a positive margin on the interest rate. The margins vary between 0.05% and 1.86%. These values fall into the range reported by Shan (2011), who reports that for U.S.
HECM loans the lender’s margin is typically between 1-2%. For LTVs higher than 0.85, the profit of the insurer is always negative on average, independent of the margin, and this establishes a maximum LTV.
-- Figure 2 here --
The pricing of the variable interest rate reverse mortgage offered at t = 1 is similar: a margin
∆LS,1 is determined to compensate the product provider for the NNEG. The value of the NNEG depends on how much the household has already 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 ∆RM,1 for different loan amounts, represented as “additional loan-to-value ratio”. Results are presented for
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different values of initial borrowing (i.e. for different LTVRM,0 ratios), assuming low house price growth over the first period and low interest rates over the second period.
-- Figure 3 here --
2.8.5 Fair Pricing of the Sale and Lease Back Plan
The sale and lease back plan is priced such that product provider makes on average across all future states a zero profit. The provider’s profit is calculated as the expected present value of the proceeds from sale of the equity share minus the initial lump sum paid out to the household. This initial lump sum is reduced to account for the expected present value of rental yields.
To determine present values of sale proceeds and rental yields, discount factors are used 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 Section 2.8.2 and rental yields over the first period.
A discount factors for the second period is determined in the same way. Rental yields over the first and the second period are computed from an annual rental yield %rent, which is a percentage of the equity share, %HR,τ · Hτ , sold at the beginning of the period τ = 0, 1.
Previous studies on optimal consumption and portfolio choices have considered a rental yield of 6% per year (Yao and Zhang, 2005). At this value and given the calibration of interest rate, house price and health states, only 24% of the value of the equity share is paid out to the couple
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under a home reversion plan taken out at t = 0 (and this value is independent of the percentage
%HR that the couple decides to sell). In this study, a lower (after-tax) rental yield of 2% is used, resulting in 54% of the value of the equity share paid out to the couple. Alternatively, a rental yield of 4% is considered.
2.8.6 Government-Provided Long-Term Care Insurance
With the government-provided LTCI, the individual has to cover (1 − %+,-..) =50% of the care costs up to a maximum of care costs of $6,276 p.a. (equal to $100,000 for the 19-year horizon). For care costs higher than $6,276 p.a., the individual’s out-of-pocket costs are limited to 50%·$6,276 p.a.
3 Results
The MATLAB function fmincon was used to implement the couple’s optimization problem as a constrained nonlinear optimization problem. For the numerical solution of the model, the house price process is discretized using a binomial process (as in Yao and Zhang, 2005 or Davidoff, 2010c). The interest rate process is discretized in the same way.
3.1 Optimal Equity Release at Different Points in Time
The base case is when the couple decides on consumption, saving, on purchasing annuities, private long-term care insurance (LTCI) and on taking out an equity release product. The model parameters are the baseline parameters given in Table 1. Different scenarios are compared: One
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scenario without equity release products, two scenarios in which either the reverse mortgage or the home revision plan described in Section 2.6 are offered only at t = 0 and two scenarios in which the reverse mortgage or the home revision plan are offered at t = 0 and t = 1.
Government-provided LTCI is not available in any of the five scenarios. Table 2 gives the corresponding results.
-- Table 2 here --
The results for first three scenarios show that the couple clearly has a demand for equity release products at time t = 0. When offered the reverse mortgage only at t = 0, the couple chooses to borrow up to the maximum loan-to-value-ratio (LTV). When offered the home reversion plan at t = 0, the household chooses to convert a very large proportion %HR,0 of the home. In both cases, the changes in expected discounted utility indicate substantial welfare gains for the couple. The utility gain is higher for the reverse mortgage than for the home reversion plan.
Having access to equity release products at time t = 1 in the last two scenarios further improves the couple’s utility, but the utility gain is smaller than the utility gain from having access to equity release products at t = 0. Borrowing and home reversion mainly takes place at t = 0.
Table 2 reports the couple’s consumption, annuity premiums, LTCI premiums and savings at t = 0 as percentages of initial liquid wealth W0 before equity release. The sum of these figures
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indicates that the couple significantly increases their liquid wealth with equity release products.
The reverse mortgage results in higher lump-sum payments, which explains why this product is preferred over the home reversion plan. The additional liquidity from the two equity release products is used to increase consumption C0 and to increase savings. Annuity demand is relatively stable across scenarios. It is low, because the annuity pays only at t = 1 and because of the bequest motive. Private LTCI demand is also unchanged: The couple faces uncertain but potentially high care costs and always buys full LTCI coverage as a result.
3.2 Government-Provided Long-Term Care Insurance
Next, a variant of the model with compulsory government-provided LTCI as described in Sections 2.5 and 2.8.6 is considered. Again, the couple decides on consumption, saving, on purchasing annuities, private long-term care insurance (LTCI) for the remaining out-of-pocket care costs and on 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 revision plan described in Section 2.6 are offered at t = 0 and t = 1. The numerical results for these scenarios are given in Table 3.
Scenarios with equity release products offered only at t = 0 are not compared separately.
-- Table 3 here --
The couple benefits from the fairly generous government-provided LTCI scheme, but the utility gains are much smaller than the utility gains from having access to equity release products. This
24
can be seen by comparing the scenario without equity release products in Table 3 with the base case results presented in Table 2 in the previous section.
Similar levels of equity release are found to be optimal with the government-provided LTCI. As in the base case without public LTCI, the couple chooses to borrow up to the maximum loan-to- value-ratio (LTV) of the reverse mortgage at t = 0 and similar levels of home reversion at t = 0 and t = 1 are found to be optimal.
With the government-provided LTCI, the couple spends less of their wealth on private LTCI.
They still choose to buy full coverage for each partner but because the private insurance only covers the remaining out-of-pocket care costs, the premiums are much lower. They use the additional wealth to buy more annuities.
3.3 Sensitivity Analyses: Higher Risk Aversion, No Bequest Motive, and a Higher House Value
Table 4 gives the results for three different cases used to test the sensitivity of the base case results presented in Table 2 in Section 3.1. In each case, one model parameter is varied. The first case is when the couple has no bequest motive (β = 0), in the second case a higher risk aversion is assumed (χ = 5), and in the third case a higher initial house value of H0 = $500,000 is considered. For each case, three different scenarios are compared: One scenario without equity
25
release products and two scenarios in which the reverse mortgage or the home revision plan described in Section 2.6 are offered at t = 0 and t = 1.
-- Table 4 here --
In the case without a bequest motive, the couple decides—as in the base case—to borrow the maximum loan-to-value-ratio (LTV) allowed with the reverse mortgage at t = 0. When offered the home reversion plan at t = 0 and at t = 1, they choose to sell their home completely at t = 0 (%HR,0=100%) in exchange for a lump-sum payment and a lease-for-life agreement. Without a bequest motive, the couple buys significantly more annuities in all three scenarios than in the base case with a bequest motive, which is in line with the findings of previous studies (Brown and Poterba, 2000). Private LTCI demand is largely unaffected: the couple chooses full coverage for each partner.
The middle columns of Table 4 refer to the case when the couple has a higher risk aversion than in the base case (γ = 5 instead of γ = 2). Again, as in the base case, the couple decides to borrow the maximum loan-to-value-ratio (LTV) allowed with the reverse mortgage at t = 0. The home reversion pattern is different from the base case in Section 3.1: being more risk averse, the couple decides to release more of their home equity at t = 0 and less at t = 1 (%HR,0=80% and
%HR,1=18%, compared to 75% and 14% in the base case). The couple buys significantly more
26
annuities in all three scenarios than in the base case, but their decision to fully insurance long- term care costs with private LTCI is unchanged.
In the third case, presented in the last three columns of Table 4, a higher initial house value of H0 = $500,000 is considered. In the base case, the house value, H0 = $250,000, made up 65% of the couple’s total wealth at t = 0. This ratio is 79% for a house value H0 = $500,000. The results show that the couple chooses to borrow a similar amount with the reverse mortgage, although the loan-to-value ratios both at t = 0 and at t = 1 are reduced, More equity than in the base case is released with the home reversion scheme.
Three key findings emerge across the cases with alternative parameter values (no bequest motive, higher risk aversion, and in higher initial house value) and these findings are consistent with the base case: (i) The couple has large utility gains from having access to either one of the two equity release products, (ii) higher utility gains are found for the reverse mortgage, and (iii) equity release predominantly takes place at t = 0.
4 Summary and Conclusions
This study models the decision problem of a retiring couple that holds the major fraction of their wealth as home equity and faces longevity risk, long-term care risk, house price risk, and interest rate risk. The couple can choose to buy annuities, long-term care insurance, and to borrow against the home using different equity release products at different point in time. The numerical
27
results of this study suggest that the couple enjoys large utility gains from having access to either one of the two equity release products. Higher utility gains are found for the reverse mortgage.
The household chooses to unlock home equity early on in retirement. The key results are consistent across a range of cases with different parameter values. The availability of a government-provided LTCI does not change the use of equity release products significantly, but does change the demand for annuities.
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Table 1 Model Parameters
Parameter Baseline Value Alternative
Value
House value t = 0 H0 $250,000 $500,000
Liquid wealth t = 0 W0 $135,000
Age of both spouses t = 0 in years 62
Relative risk aversion γ 2 5
Subjective discount factor δ 0.98
Jointness of consumption θ 0.5
Strength of bequest motive β 0.2 0
Long term care expenses per year LTCi $10,000 (needing some care at home);
$50,000 (needing care in a nursing home) Mean interest rate per year (=
interest rate at t = 0)
r0 2.0%
Standard deviation of interest rate per year
Std(r0) 2.2%
Mean house price growth per year g 1.6%
Standard deviation of house price growth per year
Std(g) 11.7%
Rental yield 2% 4%
Mortgage rate margin per year mo 0 1.75%
Annuity loading factor λA 0 10%
Long term care insurance loading factor
λLTCI 0 16%
Coinsurance percentage of the
govt.-provided LTCI %+,-.. 50%
Stop loss of the govt.-provided LTCI per year,
$6,276
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 period length (19 years) in the model. All monetary values are in real terms.
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Table 2 Optimal Equity Release at Different Points in Time No Equity
Release Products
Reverse Mortgage at
t = 0
Home Reversion at
t = 0
Reverse Mortgage at t = 0 and t = 1
Home Reversion at t = 0 and t = 1 Financial decisions at t = 0
Consumption* 51% 127% 141% 148% 104%
Annuity premiums* 19% 18% 28% 18% 16%
LTCI premiums* 21% 21% 21% 21% 21%
Savings* 10% 92% 1% 70% 34%
Sum* 100% 257% 190% 257% 174%
LTCI coverage 100% 100% 100% 100% 100%
LTV0 85% 85%
%HR,0 91% 75%
Financial decisions at t = 1
Consumption** 13% 25% 77% 29% 57%
Savings* * 3% 75% 18% 71% 73%
Sum** 16% 100% 95% 100% 130%
LTV1 0%
%HR,1 14%
Expected discounted
utility -7.81E-05 -4.38E-05 -4.79E-05 -4.16E-05 -4.64E-05
Notes: RM denotes the lump-sum reverse mortgage with variable interest rates and a no-negative-equity guarantee.
HR denotes the home reversion plan. All variables are given at the household level, summing husband’s and wife’s values. *Consumption, annuity premiums, LTCI premiums and savings at t = 0 are reported as percentages of liquid wealth W0 at t = 0 before equity release. ** Consumption and savings at t = 1 are reported as percentages of liquid wealth at time t = 1 before additional equity release.
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Table 3 The Impact of Government-Provided LTCI on Optimal Equity Release No Equity Release
Products
Reverse Mortgage at t = 0 and t = 1
Home Reversion at t = 0 and t = 1 Financial decisions at t = 0
Consumption* 74% 148% 117%
Annuity premiums* 22% 37% 27%
LTCI premiums* 4% 4% 4%
Savings* 0% 68% 21%
Sum* 100% 257% 169%
LTCI coverage 100% 97% 98%
LTV0 85%
%HR,0 70%
Financial decisions at t = 1
Consumption** 12% 40% 72%
Savings* * 7% 59% 65%
Sum** 19% 99% 137%
LTV1 0%
%HR,1 14%
Expected discounted utility -6.27E-05 -3.19E-05 -3.94E-05
Notes: All variables are given at the household level, summing husband’s and wife’s values. *Consumption, annuity premiums, LTCI premiums and savings at t = 0 are reported as percentages of liquid wealth W0 at t = 0 before equity release. ** Consumption and savings at t = 1 are reported as percentages of liquid wealth at time t = 1 before additional equity release.
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Table 4 Sensitivity Analyses: Higher Risk Aversion, No Bequest Motive, and a Higher House Value
No bequest motive (β = 0) Higher risk aversion (χ = 5) Higher house value (H0 = $500,000) No Equity
Release Products
Reverse Mortgage at t = 0 and
t = 1
Home Reversion at t = 0 and
t = 1
No Equity Release Products
Reverse Mortgage at t = 0 and
t = 1
Home Reversion at t = 0 and
t = 1
No Equity Release Products
Reverse Mortgage at t = 0 and
t = 1
Home Reversion at t = 0 and t = 1 Financial decisions at t = 0
Consumption* 46% 170% 133% 50% 119% 95% 55% 147% 165%
Annuity premiums* 33% 47% 45% 29% 34% 29% 17% 74% 11%
LTCI premiums* 21% 20% 21% 21% 21% 21% 21% 21% 21%
Savings* 0% 20% 0% 0% 84% 35% 8% 0% 62%
Sum* 100% 257% 199% 100% 257% 179% 100% 241% 258%
LTCI coverage 100% 96% 100% 100% 100% 100% 100% 100% 100%
LTV0 85% 85% 38%
%HR,0 100% 80% 83%
Financial decisions at t = 1
Consumption** 95% 55% 92% 91% 48% 66% 69% 93% 40%
Savings* * 5% 44% 1% 9% 52% 66% 32% 13% 86%
Sum** 100% 99% 94% 100% 100% 132% 101% 105% 127%
LTV1 0% 0% 3%
%HR,1 0% 18% 10%
Expected
discounted utility -7.34E-05 -3.11E-05 -3.31E-05 -2.55E-19 -9.71E-21 -2.65E-20 -7.42E-05 -2.66E-05 -3.49E-05
Notes: All variables are given at the household level, summing husband’s and wife’s values. *Consumption, annuity premiums, LTCI premiums and savings at t = 0 are reported as percentages of liquid wealth W0 at t = 0 before equity release. ** Consumption and savings at t = 1 are reported as percentages of liquid wealth at time t = 1 before additional equity release.
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Figure 1 Model Timing
Figure 2 Zero-profit margin for a reverse mortgage taken out at t = 0.
Notes: This graph shows the margin ∆LS,1for a variable interest rate reverse mortgage taken out at t = 0for different actual loan-to-value ratios.
t= 0 Period 1 t= 1 Period 2 t= 2
Realization of random
- Health status of husband and wife - House value
- Interest rate
→ Borrow against home
→ Buy annuity
→ Buy LTCI
→ Consume and save
→ If both are dead: bequeath net assets
→ Else:
→ Receive annuity and LTCI payments
→ Receive accumulated savings
→ Borrow against home
→ Cover out-of-pocket care costs
→ Consume and save
→ Bequeath net assets Husband and wife are in
good health
Husband and wife die, Realization of random house value
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Figure 3 Zero-profit margin for a reverse mortgage taken out at t = 1.
Notes: This graph shows the margin ∆LS,1for a variable interest rate reverse mortgage taken out at t = 1. The margin differs according to how much the household borrowed at t = 0. Results are given for different values of initial borrowing (i.e. for different LTVLS,0 ratios) and refer to cases with low house price growth over the first period and low interest rates over the second period.
