Equity Release to supplement pension: Risk analyses of reverse mortgages

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1 Master s Thesis Executive Master Actuarial Science Equity Release to supplement pension: Risk analyses of reverse mortgages C.C. Werner-Huibers 14 th November 2013

2 1 Contents 2 Introduction Equity release products The equity release market abroad The Dutch Mortgage market Behavioural aspects or other barriers of equity release products Behavioural and psychological obstacles Regulatory issues Research Questions Equity Release Product Design A Lump-Sum Reverse Mortgage Cash Flow Pattern Income Stream Reverse Mortgage Cash Flow Pattern Pricing Framework: Simulation models for reverse mortgage risks Modelling the Probability of Termination Deriving arbitrage free scenarios for reverse mortgage products State variables of the VAR Model Mortgage rates VAR(1) model Stochastic discount factors Simulation results The Base Case Scenario Reverse mortgage products; Base case results Loan balance and house price Transforming the constant term of the VAR(1)model Transform to an average growth rate of Sensitivity to the mortgage margin Sensitivity to the Loan Value Ratio Sensitivity to mortality, costs and average delay of sales Transform to the long-term historical average growth rate Backtest using historic data starting in

3 5.4.1 Main risk drivers Current barriers for development of Dutch equity release market Conclusion Suggestions for further research Appendix A: List of variables Bibliography

4 2 Introduction Increased life expectancy and healthcare costs, a shrinking working population and a financial crises puts pressure on our pensions system and poses the question how to finance adequate pension incomes in the (near) future. With pensions and accrual rates decreasing and risks increasingly shifted toward households, there will be a need for households to make individual life cycle saving. There is a school of thoughts that advocates to look at life income in a broader sense 1. A strong retirement income system requires attention to other components of wealth, like housing, health and human capital 2. In this thesis I will focus on the component house as a possible funding source for retirement. Dutch elderly hold a large proportion of their savings in the form of home equity, which can be used to provide post-retirement income and health care costs. The majority of the Dutch population are homeowners and the widest gap between house value and mortgage loan is found among older house owners. In 2012 the average housing value of over-60s exceeds 300 thousand euros 3. So for the majority of elderly people their homes represent a key source of personal wealth. This locked equity could become an important component of financing retirement needs of an ageing population. Figure 1: mortgage loans and house values 2012 The process of converting home equity into cash while the homeowner is still alive is called home equity conversion (Phillips and Gwin 1992). Home equity conversion could facilitate consumption smoothing over the life cycle. Because release home equity might also be used to pay for health care expenditure, home equity conversion could also be of interest for public policy experts who are looking for ways to reduce healthcare costs. In order to help people releasing house equity it will be 1 In 2010 at the Pensioen Expeditie 2010, a congress of the Achmea Academy (www.pensioenexpeditie.nl). Prof. Gerry Dietvorst presented his vision on pension thinking: from three to five pillars or layers. 2 Innovation in retirement planning, pension research council CBS; Web magazine, 06 June

5 essential that the financial industry develops useful sustainable products. In this respect several initiatives have suggested 4 the need for new equity release or reverse mortgage products for the Dutch market. With the future limited possibilities to acquire sufficient pension income and the current financial situation, households need to take more financial decisions, and releasing home equity may be a useful instrument for optimal financial planning. 2.1 Equity release products Selling the home, and renting a smaller one is the most direct form of home equity conversion. This form requires moving to another home, which is usually unacceptable for older people. The willingness to move decreases with age). Another option is a combination of an immediate life annuity and a regular interest only mortgage. Stringent underwriting criteria of current interest only mortgages and the considerable difference between mortgages rates and the interest rates used for valuing annuities makes this form either impossible or the resulting additional income too small to be of interest. While home equity conversion can be achieved by many forms of loan, lease or sale (second mortgages, overdraft credit, leases, or sale and- move arrangements) there are products that are exclusively designed for the purpose of equity release. Equity release products can take two forms. A sale model, also known as home reversions, which involves an immediate sale of the property but provide for the right to remain in occupation. Or a loan model, also known as reverse mortgages. Reverse mortgage payments can be dispensed in several options, an up-front lump sum payment; an annuity of monthly payments as long as the borrower stays in the house (the tenure plan or lifetime mortgage); an annuity of monthly cash payment for a fixed period (the term plan); a line of credit; or a combination of the above (Ma and Deng, 2006). In a home reversion scheme, the provider purchases the ownerships right over the customer s property (or a share of the property) and the property is leased back to the consumer for life. The homeowners receive a discounted payment in exchange for a fixed proportion of their home. The discount represents the value of the lease for life agreement. The provider of this contracts faces property risk and in respect to the lease for life agreement faces rental yield appreciation and termination risk. In a home reversion scheme the customer is always better off prolonging the duration of the contract (Alai et al 2013). A reverse mortgage operates as the reverse of a regular mortgage or a forward sale of the house. For the borrower it s a capitalised interest loan or roll-up mortgage. The mortgage loan is paid out as regular income or as a lump sum and during the lifetime there is no obligation to pay interest. At the time the borrower sells the property, moves out or dies the loan gets repaid with accumulated interest through the sale of the property. So reverse mortgages involve accumulation of debt while home reversions are debt free. Reverse mortgages are issued to couples or single borrowers and carry fixed or variable interest rates. Reverse mortgage contracts are usually non-recourse. Which means that in order to protect borrowers from negative equity, reverse mortgages provide a no negative equity guarantee (NNEG). A NNEG limits the loan repayment to the sale proceeds of the property (net liquidation value of the property). Contracts often allow for refinancing or early repayment. 4 Taskforce Verzilveren 5

6 Reverse mortgage products will pose other service challenges to the provider. The provider is not required to process payments, but instead needs to determine a property s occupancy status and condition, and ensuring adequate insurance (Zhai 2000). The risks involved As there is no obligation to make interest payment, the lender does not face credit risks during the lifespan of the loan. Instead lenders of reverse mortgages are exposed to cross-over risk. The Crossover point is a time in the future when the loan balance will be equal to the net liquidation value (Zhai, 2000). Due to the No Negative Equity Guarantee, the lender incurs a loss if the value of the property does drop below the loan balance. This risk is called the cross-over risk. Cross-over risk Figure 2: Cross-over point Cross-over risk is induced by three risk factors : Property depreciation risk, Interest rate risk and the risk of delayed termination. For example, a delayed termination caused by longevity increases the probability that the loan balance will exceed (`crosses over' ) the property value due to a longer loan duration. So an increase of the lifespan of the loan resulted from longevity or reduced mobility rates will impose a higher cross-over risk. Since the loan balance beyond the Crossover point is capped by the property value, a high interest rate aggravates the Crossover risk. The Crossover point will also be reached sooner than expected If property value does not appreciate as much as was expected. 6

7 Other risks Figure 3: main drivers of cross-over risk Other risk inherent to reverse mortgage a lender might face are early redemption, adverse selection and moral hazard risk. If the reverse mortgages loans gives the borrower an option to repay the loan at any point of time, the lender is faced with the risk of early redemption, as the borrower will pay back the loan when it is most beneficial to him (high property values) which might not be in the interests of the lender. Adverse selection arises if consumers expecting an unusually long life or low mobility enter into reverse mortgages at a rate disproportionate to their share of the population (Davidoff and Welke 2004). Extended staying beyond the optimal length for an otherwise identical non-mortgager might also by related to the additional funds released through the mortgage which can make life in the home more attractive. A change of behaviour after entering is one form of moral hazard. Another one is the risk that a borrower when facing default has no incentive to maintain property values, which could let to improper or negligent maintenance and not taking home insurance. Moral hazard could either negatively affect the property value or delay termination of the contract. Moral hazard problems extend well beyond the maintenance phase. It is predictable that in cases in which the crossover point is passed before the homeowner dies, the sale will wind up being handled by relatives who have no stake in the sale price. Research from Davidoff and Welke (2004) showed that selection in the US reverse mortgage market to that date had been advantageous rather than adverse. They concluded that although reverse mortgages enabled longer stays at home, the kind of people who did cash out their housing equity by a reverse mortgage, relatively soon disposed the entire asset by moving out. So in their opinion high reserves or risk capital did not justify the minor 7

8 moral hazard effects 5. The Taskforce Verzilveren also concluded in their paper that, based on a small sample of 200 home-owners, on average people with reverse mortgages moved out much earlier than expected. 2.2 The equity release market abroad While in the Netherlands the products for equity release are very limited these products are available in numerous countries including Australia, Canada, the US, the UK, India, Japan and Singapore. While the financial crisis has initially slowed down market growth, several markets including Australia and the UK have recovered after the financial crises and show strong growth rates (Deloitte and SEQUAL, 2012; Key Retirement Solutions, 2013). Most markets are dominated by reverse mortgages with lump-sum payments instead of income stream reverse mortgages. In most countries there are industry or self-regulating bodies which oversee the issuance of No- Negative Equity Guarantees or other regulation to protect consumers. The US has the National Reverse Mortgage Lenders Association (NRMLA) where Australia knows the Senior Australians Equity Release Association (SEQUAL). These associations commit their members to appropriate product design, high standards of practice and responsible borrowing and often serve as an educational source. In the UK many providers signed up to SHIP ('Safe Home Income Plans') a voluntary code of conduct which provides several guarantees. SHIP was formed in an attempt to improve the equity release market and its previous poor reputation. Ship has been rebranded as Equity Release Council (ERC) in There are several reverse mortgage products in the United States but the predominant reverse mortgage is the Home Equity Conversion Mortgage (HECM). HECM are standardised governmentbacked reverse mortgages, via the Federal Housing Administration. To encourage the development of the equity release market, the US government insures mortgages with a No-Negative Equity Guarantee. The FHA provides for the No-Negative Equity Guarantee, financed by the borrowers. The HECM is considered the safest reverse mortgage and accounts for 95% of the market share (Ma and Deng, 2006). The Australian market has shown a considerable growth over the last years and has nearly tripled in terms of the total loan book size over the last decade and is expected to continue growing (Ai et al, 2013). The reverse mortgage is predominant, home reversion schemes also exist in Australia but are relatively new. In Australia, SEQUAL members have to issue the no negative equity guarantee to ensure the borrower can never owe more than the value of his house. The most popular product in Australia are variable rate loans, who are on average 1% above the standard variable home loan (Ai et al, 2013) In the UK equity release products have been available for 10 to 30 years and is basically made up of two types of equity release plan. The most popular plan is a lifetime reverse mortgage. The other type is a reversion plan - where the homeowners sells all or part of the property to the equity release provider in return for the right to remain there rent free. Both lifetime mortgages and home 5 They did suggest that it would be interesting to observe whether the favorable selection and relatively minor moral hazard effects observed to that date would continue in any period during which interest rates would far exceed price appreciation. 8

9 reversion plans fall under the remit of the Financial Services Authority (FSA). The guarantees of the SHIP include a no negative equity guarantee. A government-insured reverse mortgage program has been launched by the Korean government in July In Korean reverse mortgage, the used types of cash advances are tenure payment and tenure payment with line of credit. The Korean Housing Finance Corporation, the guarantor of the Korean reverse mortgage program, reported the number of reverse mortgage borrowers was rising rapidly (Ma and Deng, 2006). 2.3 The Dutch Mortgage market The ability to release equity in any form are highly dependent on the difference between the mortgage rates and the rates used for discounting pension liabilities (Task force Verzilveren 2013). Mortgage rates in the Netherlands are relatively higher compared to other European countries 6, while the discount rates who are based on government bonds are low. Explanations mentioned for the high mortgage rates are besides market imperfections, inter alia the Dutch funding gap and the high loan to value ratios. Reports of the DNB show a large gap between the savings deposits in banks and outstanding residential mortgages in the Netherlands 7. About two-thirds of the consolidated balance sheet of the Dutch banking system is funded in the financial markets. Due to this funding-gap, Dutch banks rely heavily on the capital market as a source of funding, mainly with short maturities. Short-term market funding increases banks vulnerability. To reduce vulnerability it is essential to either recapitalise banks or reduce the size of mortgage loans portfolios. Figure 4: Funding Gap ;Source: Special Report Rabobank Quick Scan Hypotheekrente; NMA, november DNBulletin: Mortgage lending makes banks dependent on market funding; june 6 th

10 New regulation is trying to reduce the funding gap by limiting interest-only mortgages. Strengthening the capital position of banks in line with Basel III is another instrument, but will as a result limit the supply of mortgages for Dutch households. Various parties are making efforts to attract other institutional investors like Dutch pension funds and insurers to invest in Dutch residential mortgages and thereby use the significant pension-savings to solve the funding gap. Forthcoming Capital requirements (Basel III) and the limited funding sources will have to be taken into account in product designs of mortgage related products. 2.4 Behavioural aspects or other barriers of equity release products In the Netherlands and many other European countries the market for equity release products is non-existent or very thin. There has been considerable debate in the literature on the economic potential of the reverse mortgage market in other countries. Yet even the most pessimistic assessment suggests that the reverse mortgage market is much smaller than could be expected. This could be due to suboptimal supply and/or demand. A number of behavioural aspects or other aspect on the demand side may have prevented the reverse mortgage market from developing. Shan (2011) mentioned behavioural and psychological obstacles to be of importance and affect if equity release products are seen as a possible option Behavioural and psychological obstacles First of all, elderly homeowners with strong bequest motives may not find reverse mortgages attractive because reverse mortgages reduce the amount of wealth they can leave to their heirs. According to Chaplin a strong motive for bequest could lead to avoidance of the reverse mortgage market (Chaplin 2000). Recent research by Andersson and Sandström (2013) in Sweden showed that potential reverse mortgage borrowers did not consider leaving their house as a bequest as their main priority. Instead they were positive to spend money instead of only focusing on saving to their heirs. Another behavioural aspect is the consumer uncertainty about future preferences or needs. For an elderly household planning to move in the near future, a reverse mortgage would seem to be a very bad idea. Uncertainty about future increasing medical expenses in time might also cause that many elderly homeowners like to hold on to their housing equity in case they might need it in the future. Reverse mortgages are complex financial products and can be particularly challenging for elderly homeowners. Lang (2008) concludes that the great obstacle to the acceptance of reverse mortgage products in Germany is a lack of understanding among the public. It is suggested that people that have a higher educational background will find it easier to understand the concept of a complex financial product such as reverse mortgage (Chou et al., 2006). In the research of Anderson and Sandström (2013) concern was expressed of the possibility to be tricked into a bad deal if entering a reverse mortgage by some potential borrowers in the focus group. This aspect and the issue regarding the costs associated with taking the loan are something mentioned in the focus group to be perceived drawbacks of equity release products Another aspect in complex psychology of reverse mortgages is the aspect that many elderly households are reluctant to take on debt. They value owning their homes free and clear so much that they are averse to the idea of borrowing against them. Having spent so much of their lives trying to pay off their initial mortgage. The study of Anderson Sandström (2013) also showed that some potential reverse mortgage borrowers in the focus group would feel ashamed or embarrassed if they needed to take the loan. She concluded that these feelings of shame and embarrassment might be 10

11 important factors that have to be taken into consideration, in order for reverse mortgage to reach its full potential on the market Regulatory issues An important regulatory issue is whether or not countries introducing home equity release products also provide for a no negative equity guarantee and how this regulation is explained to the potential borrowers. For example Reed (2009) documents concern among Australian borrowers regarding the possibility of being forced out of their homes in case of negative equity. In addition to concerns about negative equity certain features of the equity release programs and its interaction with some welfare programs could be undesirable. For example, the additional income received from a reverse mortgage may disqualify one from public assistance or would increase excess and contributions. Ong (2003) mentioned the unfavourable tax regimes in the UK as one of the reasons behind the scarce development of reverse mortgage market, in case a reverse mortgage annuity were to be taxed hence reducing social security entitlements. The Australian industry body SEQUAL 8 identified the treatment of lump sum loans in government eligibility tests for the Age Pension as a barrier for growth of the market in Australia. Mitchell & Piggot (2003) focus on the potential for reverse mortgages. It could potentially, next to boost consumption among the elderly, mitigate the demand for long term care facilities. In this case, it would be in the best interest of the government and they should play a substantial role in improving the efficiency of capital markets and providing or facilitate a No Negative Equity Guarantee, in order to support the development of a market for reverse mortgages. 8 Senior Australian Equity Release Association of Providers 11

12 2.5 Research Questions The equity release market faces a huge potential of accumulated assets by the older populations but in order to develop has to deal with limitations and barriers of cultural environment, the current economic situations and volatility of the property market. The ability to release equity are highly dependent on the difference between the mortgage rates and the rates used for discounting pension liabilities. In order to successfully develop the potential, products should possess mechanisms for mitigating or addressing the specific risks involved and minimize the difference between mortgage and discount rates. Thereby also taking into account the effect of mortgage margins, economic and cultural barriers and the social demand for simplicity which might affect the attractiveness and availability of the products in the future. In order to find out the potentiality of home equity release products as a mean to supplement retirement income for elderly homeowners, the aim of this paper is to develop a model of economic variables to capture the interaction between macroeconomic variables which determine house prices based on Dutch data for use in quantifying the major risks of a reverse mortgage or equity release products. The following research questions are defined. 1. What are the specific risk factors related to reverse mortgage products and how could they be priced? 2. What are the main drivers of the cross-over risk? 3. What barriers for development of the equity release market can be defined and what could be done in order to develop the market for home equity conversion? The answers to the research questions will be organized as follows. In chapter 3 different products designs will be outlined, including the related risks and cash flow pattern. We set out the pricing formula for the No Negative Equity Guarantee embedded in the reverse mortgage products. We work out a pricing framework in chapter 4. In order to do so we define a termination model and use a VAR model to model the dynamics of interest rates and house prices. Stochastic discount factors are derived from the VAR model. Cash flow structures are analysed for reverse mortgage products. Numerical examples are used to compare these products in net payoff for the borrower and risk (NNEG) for the provider. To define the main contributors to cross-over risk we will report and analyse the results in chapter 5. We evaluate these results along with other characteristics of the product and answer research question 3. Chapter 6 will conclude and end with suggestion for further research. 12

13 3 Equity Release Product Design Developing an equity release product involves creating a product that is an intersection of insurance, banking and investment. In every equity release product the essential question is who will get the possible gains and who will bear the losses (Taskforce Verzilveren 2013). To a large degree, the independence and existence of the risks depend upon the design of the product. It is possible to design products that minimize the risks by means of conservatism, diversification, hedging, assetliability matching or sharing the risks with homeowners. Conservatism would be to limit the monthly loan-payment to the home-owner to a sufficiently low amount that would nearly eliminate all the risks faced by the providers. The disadvantage of this approach would be the product s lack of marketability. Providers should accept some risks, but should charge an appropriate price for taking that risk. In contrast to regular mortgages reverse mortgages will rarely bare a loss is early years but are becoming risky in later durations. Releasing home equity by regular mortgage products is difficult due to stringent underwriting criteria of current interest only mortgages. The considerable difference between mortgages rates and the interest rates used for valuing annuities makes this form either impossible or the resulting additional income too small to be of interest. For regular mortgages a maximum loan to value ratio is desired. To provide for a significant cash income a reverse mortgage must probably use the home s full value and even expect appreciation as well. We analyse specific reverse mortgage designs. The first one is a lump-sum reverse mortgage (or rollup mortgage) with an up-front premium for the NNEG. The net lump-sum payment could be used for an annuity for life purchased at an insurer. The second product would be an income stream reverse mortgage. The reverse mortgages contracts are non-recourse, so the no negative equity guarantee (NNEG) limits the loan repayment to the sale price of the property. Products can be designed to charge for the risks using some combination of up-front or an annual premium. An front-end charge could be expressed in terms of percentage point of the home s initial value. An annual premium could be assessed as a number of basic point of the current outstanding loan balance, so in fact the loan balance will be charged by a higher loan interest rate. In this case the annual fee will grow larger during the lifespan of the loan, and could as a result lead to undesired spreading of the fee among borrowers. A fixed annual fee could provide more equal spreading. Fixed and growing annual fee s both will create prepayment risk, which will be absent in an up-front charged fee. 9 In the following,the different product designs will be described in more detail. We review the basic features and describe the risks involved in these products. All products will be based on floating interest rates and the products will all be kept to maturity. Where the termination point is the maximum attainable age of the last homeowner. 9 The current used insurance premium in the US is an upfront charge of 2% of the loan value plus a monthly premium of 0.5% of the outstanding loan. Although this charges are been criticized for being too high (Chaplin 2002) 13

14 3.1 A Lump-Sum Reverse Mortgage The first reverse mortgage product to be considered will be lump sum reverse mortgage which is the most common type of reverse mortgage. In a lump sum reverse mortgage or roll-up mortgage the borrower receives a loan in the form of a lump sum. The loan is rolled up with interest until the last borrower dies, moves into long-term care or sells the house. To convert home equity into retirement income the Lump-sum could be used for an annuity for life (two life s) at an insurer. There will be a no negative equity guarantee (NNEG) so the mortgage provider will be faced with cross-over risk. The NNEG will be fined by an upfront mortgage insurance premium, financed as part of the reverse mortgage. The insurance premium will be deducted from the initial cross lump-sum payment. Close to default the provider could be faced with moral hazard risk, but we will not address this specific risk in these thesis. Figure 7: Loan balance and property value for a lump-sum reverse mortgage Initial loan L 0 = H 0 * α (3.1) The maximum initial loan amount is determined by the loan value or borrowing ratio that is set as a proportion of the value of the property. The borrowing ratio is denoted by α. We denote the property value at time t as H The accumulated Loan balance at time t will be denoted as L t and will increase every quarter by a quarterly risk-free rate (ri), which will be based on the 3 months Euribor rate, and a quarterly lending margin π. Loan balance i (3.2) Where is the average delay in time from the point of home exit until the actual sale of the property. the delay in sales of the property after the loan termination point and T is the termination point. 14

15 At termination, the repayment amount is min. The borrower s net equity is capped by the property value due to the NNEG. Net Equity (Ct) = Ht - min. (3.3) Max. The transaction costs of selling the house, δ are defined as a percentage of the house value. The present value of providers loss is then : Loss Tx= Max - * (3.4) Where is the risk-adjusted stochastic discount factor. The value of the No Negative Equity Guarantee can be defined by: NNEG = [ ] (3.5) We assume end-of-the year terminations and so the termination probability t is the probability that a reverse mortgage is in force by time t and will be terminated by t+1. Where is the loan termination point, i.e. the maximum attainable age of the last homeowner. The net payment after deduction of the upfront premium then becomes: NP 0 = L 0 NNEG (3.6) Cash Flow Pattern The cash flows at time zero for the borrower are the lump sum payment from the lender. This lump sum payment is used for initial premium for an annuity at an insurer. During the lifetime of the loan there will be no interest payments. The borrower could use the lump-sum payment to receive periodic income by purchasing an annuity. 15

16 Figure 8: Cash Flows mortgage borrower Figure 9: Cash Flows mortgage lender The lender provides for the lump sum payments and will not receive any interest payments. At termination the lender will receive the payoff of the loan, which is capped by the sale proceeds of the property. Figure 10: Cash Flows insurer of annuity The insurer receives a front-end premium for the joint survivor annuity and provides for periodic payments until the last survivor has died. 16

17 3.2 Income Stream Reverse Mortgage The second product design will be an income stream reverse provided by the mortgage lender. This income stream reverse mortgage has a different pay-out design. It pays out fixed amount to the borrower until terminations. The loan balance starts at zero and will increase quarterly with the accrued variable interest rate and the new payment to the borrower. There will be a No-Negative Equity Guarantee so the mortgage provider will be faced with cross over risk. Like the lump-sum reverse mortgage the lender could be faced with moral hazard risk close to default. The NNEG will be charged by a an fixed annual premium. Figure 11: Loan balance and property value of an income stream reverse mortgage. To compare the reverse mortgage products we define the periodically payment (income stream) such that the present value of all payments equals the lump-sum amount of the Lump sum reverse mortgage. Joint-Survivor annuity at T = 0 (3.7) is the in-force probability, the probability at time t that home-owners, age x:y at loan origination will still be in their homes at the end of year t. Yn,t are zero coupon yields for maturity n at time zero. For the annuity index at t=0 we used the ufr yield curve of June 2013 as published by the DNB. The initial fixed yearly payment is: Ux:y,0 = (α * H0 ) / (3.8) 17

18 The outstanding loan balance at time t is then given by: - - = ( ) (3.9) The expected present value of the lender Loss and the NNEG are derived similar to the lump-sum reverse mortgage. Loss Tx= Max * (3.10) NNEG = [t qx y oss Tx ] (3.11) The NNEG will be charged a fixed annual premium payment. We use the same annuity index for the premium payment as used to define the fixed annuity payment. x y,t t x y (3.12) The net yearly payment NYP = Ux:y,0- x y,t (3.13) 18

19 3.2.1 Cash Flow Pattern In an income stream reverse mortgage the lender pays regular payment to the borrower until the contract is terminated. The lender doesn t receive interest payments. The loan will be redeemed at termination capped to the remaining proceeds after sale of the property. Figure 12: Cash flows mortgage borrower. Figure 13: Cash flows mortgage lender. 19

20 4 Pricing Framework: Simulation models for reverse mortgage risks 4.1 Modelling the Probability of Termination The probability of termination for a borrowers couple will be derived from a Markov termination model as used by Alai et al (2013) and based on Ji (2011). A reverse mortgage may terminate for various reasons including death, entrance to a long-term care (LTC), Move out for non-health reasons and refinancing. Voluntary prepayments or refinancing occurs when the market interest rate is lower than the fixed interest rate at which the loan is accumulated. Because we will not consider fixed rates we will not include prepayment or refinancing as a possible cause of termination. Due to lacking information on termination rates we will also not include prepayment for non-health reasons and long term care as causes of termination. So we will only consider death (illustrated by the blue states in figure 17). We estimate these probabilities based on mortality data (AG ). Figure 17: A multiple state model for joint-life reverse mortgages ( Ji 2011) The probability of transition from state 0 to state I at time T is denoted by tp 0i x:y and is the probability that a reverse mortgage is in force by time t and will be terminated by t+1. This one year termination probability is of interest because the model simulations will be conducted in annual time steps. We calculate by summing the probability of transition from state 0,1 and 2 to state 5. tp 15 x:y + tp 25 x:y + tp 05 x:y (4.1) = tp y * (1- t P x ) * t q y + t P x * (1- t P y ) * t g x + t P y * t P x * t q y* t g x 20

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