An Analysis of Pricing and Risks. of Reverse Mortgage Loans and. Long-Term Care Insurance

Transcription

1 An Analysis of Pricing and Risks of Reverse Mortgage Loans and Long-Term Care Insurance Wenqiang Shao A thesis submitted for the degree of Doctor of Philosophy School of Risk and Actuarial Studies UNSW Business School University of New South Wales Sydney, Australia June 2014

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6 Acknowledgements I would like to express my deepest appreciation to my supervisor, Professor Michael Sherris, for his expert and sincere guidance. This thesis would not have been possible without his invaluable and constant support. I would also like to place my sincere gratitude to my co-supervisor, Dr. Katja Hanewald, for her invaluable suggestions and always supportive encouragement. I also wish to thank Dr. Joelle Fong, with whom I have been collaborating on two research projects during my PhD studies. Her comments and suggestions have a large influence on Chapter 5 and Chapter 6. Many thanks to all faculty members and research students at the School of Risk and Actuarial Studies and to all staff at the Australian Research Council Centre of Excellence in Population Ageing Research for their help and encouragement. In particular, I thank Dr. Ralph Stevens, Dr. Yang Shen, Shang Wu, and Mengyi Xu for their valuable comments. I acknowledge financial support from the China Scholarship Council, the Australian School of Business, and the Australian Research Council Centre of Excellence in Population Ageing Research. Special thanks to the Australian Research Council Centre of Excellence in Population Ageing Research for providing me with a wonderful research environment. Last but not least, I am grateful to my parents for their encouragement, constant support and unconditional love. v

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8 List of Included Publications The following is a list of publications that are a direct result of my research towards the writing of this thesis: A version of Chapter 3: Individual House Price Models is under revision for submission to a peer review journal. An earlier version of this chapter is a working paper of Australian School of Business, University of New South Wales: Shao, A. W., Sherris, M., and Hanewald, K. (2013). Disaggregated house price indices. UNSW Australian School of Business Research Paper No. 2013ACTL09. A version of Chapter 4: Pricing and Risk Analysis of Reverse Mortgage Loans is under review with Insurance: Mathematics and Economics. An earlier version of this chapter is a working paper of Australian School of Business, University of New South Wales: Shao, A. W., Hanewald, K., and Sherris, M. (2014). Reverse mortgage pricing and risk analysis allowing for idiosyncratic house price risk and longevity risk. UNSW Australian School of Business Research Paper No. 2014ACTL01. This chapter is an extension of the work presented in the six-page conference paper: Shao, A. W., Hanewald, K., and Sherris, M. (2012). Equity release products allowing for individual house price risk. In Proceedings of the 11th Emerging Researchers in Ageing Conference, Brisbane, Australia. vii

9 A version of Chapter 5: Health Dynamics is under review with North American Actuarial Journal. An earlier version of this chapter is a working paper of Australian School of Business, University of New South Wales: Fong, J. F., Shao, A. W., and Sherris, M. (2013). Multi-state actuarial models of functional disability. UNSW Australian School of Business Research Paper No. 2013ACTL14. viii

10 Abstract The thesis develops new pricing and risk analysis frameworks for reverse mortgage loans and long-term care insurance. These are two important products that are increasingly used by individuals to finance their health costs and retirement needs. The results provide insights into the design of affordable products and risk management for product providers. The developed pricing framework for reverse mortgage loans takes into account a wide range of risks, including idiosyncratic house price risk, longevity risk, occupancy risk, and interest rate risk. The pricing model is based on projections of future state variables, including house price, rental yields, GDP, and zero-coupon yield rates. Idiosyncratic house price risk has not been widely studied in the literature due to limited public access to detailed individual house price transactions data. The thesis employs a new data set on detailed property transactions in Sydney that allows for the quantification of the idiosyncratic component of house price risk. Individual house prices are found to have very different trends and volatility compared to the aggregate house price index. Failing to take into account idiosyncratic house price risk considerably underestimates the risks of reverse mortgage providers. Health transition rates are graduated using flexible and comprehensive Generalised Linear Models (GLMs) under a multi-state Markov model framework. GLMs are shown to be more accurate in capturing the sex- and age-specific patterns in the transition rates, compared to the widely used Robinson s method. Thiele s differential equation is used in the base case analysis for pricing and reserving of generic ix

11 long-term care insurance policies. A simulation-based model is then developed to take into account typical product features, such as the elimination period and the maximum benefit period. The elimination period is shown to be effective in making long-term care insurance more affordable whereas the maximum benefit period is more effective in extreme loss control. Solvency capital requirements taking into account longevity risk and disability risk are derived for a wide range of long-term care insurance policies under Solvency II. It is found that rider benefit policies and life care annuities have considerable capital reductions compared to stand-alone policies. x

12 Contents List of Figures xv List of Tables xix 1 Introduction 1 2 Literature Review Equity Release Products Reverse Mortgage Loans Risk Analysis of Reverse Mortgage Loans Pricing Models of Reverse Mortgages Individual House Price Models Hedonic Models Repeat-Sales Models Hybrid Models Comparison of House Price Models Long-Term Care Insurance Background Types of Long-Term Care Insurance Policies Quantifying Health Dynamics Pricing and Reserving Methods Solvency Capital Requirements xi

13 2.4 Research Questions of the Thesis Individual House Price Models Introduction House Price Models Hedonic Models Repeat-Sales Models Index Construction Based on Nine House Price Models Three Non-Regression-Based Models Restricted Hedonic and Repeat-Sales Models Unrestricted Hedonic and Hybrid Models Data Original Data Merged Data Results Estimation Results for the Regression-Based Models Model Validation Aggregate House Price Indices Property Portfolio Indices: Illustration Disaggregated House Price Indices: Identifying Factors Conclusions Pricing and Risk Analysis of Reverse Mortgage Loans Introduction Data Reverse Mortgage Pricing Framework Reverse Mortgage Loans The Hybrid House Price Model Projection of Future House Prices and Discount Factors xii

14 4.3.4 Termination of Reverse Mortgages Results Base Case Results Sensitivity Analysis: Deterministic Mortality Sensitivity Analysis: The Two-Factor Cairns-Blake-Dowd (CBD) Model Sensitivity Analysis: LTC Incidence, Prepayment and Refinancing Conclusions Health Dynamics Introduction Model Framework Data Estimation of Transition Intensities Crude Transition Rates Graduated Transition Rates Results Comparison with the Robinson s Method Conclusions Pricing and Risk Analysis of Long-Term Care Insurance Introduction Methodology Thiele s Differential Equation Simulation-Based Approach Best-Estimate Reserves and Solvency Capital Requirements Long-Term Care Insurance Premiums Base Case Results: Generic Policies xiii

15 6.3.2 Demographic Characteristics of the Simulated Cohorts Policies with Typical Product Features Reserves and Capital Requirements Best-Estimate Reserves Solvency Capital Requirements under Solvency II Conclusions Conclusions 185 References 189 Appendix A: Parameter Estimates of House Price Models 211 A.1 The Restricted Hedonic Model A.2 The Repeat-Sales Models A.3 The Unrestricted Hedonic Model A.4 The Hybrid Model Appendix B: Recursive Bond Prices Adapted to the VAR(2) Model 221 Appendix C: Best-Estimate Reserves of Stand-Alone Policies Paid with Regular Premiums 223 xiv

16 List of Figures 3.1 Number of houses against sales frequency and average time between consecutive sales Age effect for log house prices Estimated aggregate index growth rates Comparison of growth rates of the aggregate index and the portfolio ρ s index Estimated time-varying coefficients of property characteristics Average log mortality rates of the Australian population Projection of price indices for houses in different regions Average changes in log mortality rates along the cohort direction Binary black-white residuals from the Wills-Sherris model The generalised Cholesky decomposition of the age dependence matrix Simulated survival probabilities Average changes in log mortality rates over time and a comparison of the survival probabilities: Deterministic v.s. Wills-Sherris Binary black-white residuals from the CBD model Comparison of residuals from the Wills-Sherris model and the twofactor CBD model Comparison of survival probabilities: Wills-Sherris v.s. CBD Four-state Markov transition diagram xv

17 5.2 Log crude transition rates between transient states Log transformation of crude mortality rates Graduated disability and recovery intensities for males Graduated mortality rates for males Graduated disability and recovery intensities for females Graduated mortality rates for females Deviance residuals Competing risks for males and females Comparison with published sources Comparison of estimated disability and recovery probabilities from GLM and the Robinson s method Comparison of estimated death probabilities from GLM and the Robinson s method Number of disabled among the simulated cohorts Best-estimate reserves for individuals in each alive state in generic stand-alone policies with lump sum premiums Best-estimate reserves for generic stand-alone policies paid with lump sum premiums VaR of liabilities of generic stand-alone policies paid with lump sum premiums VaR v.s best-estimate reserves for lump sum premium stand-alone policies with typical product features The ratio of the total capital requirement to the best-estimate reserve for generic stand-alone policies sold to the healthy The ratio of the total capital requirement to the best-estimate reserve for generic stand-alone policies The ratio of risk-specific SCRs to the best-estimate reserve for generic stand-alone policies xvi

18 6.9 The ratio of the total capital requirement to the best-estimate reserve for generic policies The ratio of risk-specific SCRs to the best-estimate reserve for generic rider benefit policies and life care annuities C.1 Best-estimate reserves of generic stand-alone policies with continuous and annual premiums xvii

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20 List of Tables 3.1 Summary statistics of the merged data set after filtering Comparison of characteristics of houses with single and multiple transactions Goodness of fit of regression-based models Summary statistics of standardised residuals Summary statistics the aggregate house price growth rate Comparison of index growth rates for the aggregate market and portfolio ρ Summary statistics of time-varying implicit prices of house characteristics Correlation coefficients between growth rates of house characteristics implicit price Summary statistics of disaggregated house price growth rates Information criteria for VAR models with different lags Parameter estimates and covariance matrix in the VAR(2) model Correlation coefficients between stochastic discount factors and state variables Information criteria for VARX models with different lag lengths ( p, q) Parameter estimates for the Wills-Sherris model based on data for Australia, xix

21 4.6 Covariance matrix of estimated parameters in the Wills-Sherris model Assumptions on termination triggers adopted from Ji et al. (2012) and Cho et al. (2013) Valuation of the mortgage insurance premium rate π and the NNEG for reverse mortgages with different loan-to-value (LTV) ratios Sensitivity analysis: valuation of the mortgage insurance premium π and the NNEG for reverse mortgages for alternative assumptions about LTC incidence, prepayment and refinancing probabilities Crude transition counts and exposure years Poisson GLM: goodness-of-fit of nested models Parameter estimates of the Poisson GLM with log link Parameter estimates of the Robinson (1996) s model based on HRS data Premiums of generic stand-alone long-term care insurance policies Premiums of generic rider benefit policies and life care annuities Proportion of survivors in each health state Demographic characteristics of the simulated cohorts Premiums of generic stand-alone policies using the simulation method Premiums of stand-alone policies with typical product features Lump sum premiums of rider benefit policies and life care annuities with typical product features A.1 Parameter estimates of the restricted hedonic model, Part A.2 Parameter estimates of the restricted hedonic model, Part A.3 Parameter estimates of the three repeat-sales models A.4 Parameter estimates in the regression of residuals from the standard repeat-sales model against the time interval between sales A.5 Parameter estimates of the unrestricted hedonic model, Part xx

22 A.6 Parameter estimates of the unrestricted hedonic model, Part A.7 Parameter estimates of the unrestricted hedonic model, Part A.8 Parameter estimates of the hybrid model, Part A.9 Parameter estimates of the hybrid model, Part A.10 Parameter estimates of the hybrid model, Part xxi

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24 Chapter 1 Introduction There will be a substantial shift in the age structure of Australia s population over the next 50 years. It is projected in Productivity Commission of Australia (2013) that by % of Australia s population will be aged 65 or older and 15% will be aged 75 or older. Other developed countries face similar changes. The ageing population will have higher needs for health care, and especially long-term care, services. Health care expenditures as a proportion of GDP in developed countries were high and showed an increasing trend in the last few years (Colombo et al., 2011; Shi and Zhang, 2013). Long-term care cost is one of the major health shocks faced by the elderly. Congressional Budget Office (2004) reports that in 2004 long-term care costs in the U.S. accounted for more than 8.5% of health expenditures and 1.2% of GDP. In 2012, long-term care costs reached more than USD billion which was 1.47% of GDP (O Shaughnessy, 2014). Real long-term care costs are predicted to triple over the next 35 years onward from 1999 due to the rapidly rising medical costs and because of population ageing (Congressional Budget Office, 1999). In Australia, long-term care costs as a proportion of GDP in 2010 are around 0.8% (The Treasury of Australian Government, 2010). This proportion is projected to triple by 2050 (Colombo et al., 2011). 1

25 This thesis addresses the important question of how to finance the increasing health care, and especially long-term care, costs of an ageing population such as in Australia. The thesis focuses on two financial products that can provide additional funding sources for retirees to finance their consumption needs and health care costs in retirement, namely reverse mortgage loans and long-term care insurance. Both products have been suggested as a source of funding for health care costs and longterm care costs of the elderly in two recent reports by Productivity Commission of Australia (2011, 2013). However, the markets for both products are not well developed in Australia. The thesis develops new pricing and risk analysis frameworks for reverse mortgage loans and long-term care insurance. It contributes the following aspects to the existing literature: (1) develop house price models to assess the idiosyncratic component of house price risk; (2) develop a new framework for the pricing and risk analysis of reverse mortgage products allowing for idiosyncratic house price risk, longevity risk and stochastic discount factors; (3) quantify health transition rates using Generalized Linear Models (GLMs) which are accurate and flexible in capturing age- and sex-specific patterns in the transition rates; and (4) derive premiums, reserves and solvency capital requirements for a broad range of long-term care insurance policies allowing for typical policy features and compare risk-based capital across policies of different types and policies issued to different cohorts. The first contribution of the thesis is the development of house price models that quantify individual house price returns and risks. The study is based on a 2

26 large data set on detailed transactions of individual houses in Sydney. A broad range of house price models in the literature are analysed and compared in this thesis. It is found that the hybrid hedonic-repeat-sales model and the unrestricted hedonic model outperform other candidate models in many ways and are therefore selected for further analyses. In particular, the two selected house price models allow for time-changing implicit prices of house characteristics by including interactions of attributes and time dummy variables. The two models provide valuable tools for banks and insurance companies that hold portfolios of mortgage contracts, as well as for investors that have direct property investments or hold financial derivatives that are linked to house prices. Using the estimation results from the two selected house price models, indices for the aggregate Sydney market and for portfolios of houses with certain characteristics are constructed and compared with indices derived from other models. The results show that the stratified median index accumulates at a relatively slow pace in early years and has higher growth rates in later years. On the contrary, the repeatsales index overestimates house price growth rates in early years and underestimates house price growth rates in later years. The hybrid index is slightly lower than the unrestricted hedonic index in later years. The three most important house characteristics affecting house price dynamics are found to be the location, the number of bathrooms and the total land size. The second contribution is to develop a new pricing and risk analysis framework for reverse mortgage loans, taking into account idiosyncratic house price risk as quantified in the first part of the thesis and other key risks including longevity risk and interest rate risk. The pricing and risk analysis framework allows for multiple termination triggers of reverse mortgage loans, including death, entrance into Long-Term Care (LTC) facilities, prepayment due to non-health related reasons, and refinancing. This thesis analyses the combined impact of house price risk and longevity risk on the pricing and risk profile of reverse mortgage loans in a stochas- 3

27 tic multi-period model. The model incorporates the previously developed hybrid hedonic-repeat-sales pricing model for individual houses with heterogeneous characteristics, as well as a stochastic mortality model for mortality improvements along the cohort direction (the Wills-Sherris model). The results show that pricing based on an aggregate house price index does not accurately assess the risks undertaken by reverse mortgage lenders, and that failing to take into account cohort trends in mortality improvements substantially underestimates the longevity risk involved in reverse mortgage loans. Specifically for reverse mortgage loans with low loan-to-value ratios, the risk is substantially underestimated if idiosyncratic house price risk or longevity risk is not taken into account. For example, a 65-year-old female homeowner in the Central Business District of Sydney taking out a reverse mortgage with a 40% loan-to-value ratio should be charged a 50% lower mortgage insurance premium if house price risk is assessed using the aggregate Sydney house price index. The third contribution focuses on developing a sophisticated and flexible model to graduate transition rates between different health states in a Markov model framework. The developed model is flexible in capturing differing age patterns in the different transition rates for males and females. The developed model is calibrated to the U.S. Health and Retirement Studies (HRS) data, which is a biennially ongoing nationally representative survey of Americans at the age of 50 or older and their spouses. The results demonstrate that many of the transition rates have more curvature in the age pattern than an exponentially linear function of age. It is also found that transition rates of females generally show more curvature in the age pattern than those of their male counterparts. In particular, disability rates of females go up at a faster increasing rate than those of males, implying that females have increasingly higher risks of getting disabled as they become older. The graduated transition rates are consistent with public sources but show very different trends from those 4

28 reported in the Wolfram Life Transitions Project contributed by Chandler (2007). Chandler (2007) employs the Robinson s method (Robinson, 1996) based on a sample of individuals aged 65 and above drawn from the National Long-Term Care Survey (NLTCS) data in the 1980s. To compare the two models, the Robinson s method is replicated using the HRS data. The results show that the graduated transition rates using GLMs are more accurate in capturing the age- and sex-specific patterns than the Robinson s method. The fourth contribution is to derive premiums, reserves and solvency capital requirements for a wide range of long-term care insurance policies, including standalone policies sold to individuals in different health states, rider benefit policies (long-term care insurance combined with whole life insurance), and life care annuities (long-term care insurance combined with annuities) sold to individuals in different health states. The Thiele s differential equation approach is used in deriving premiums and best-estimate reserves for generic long-term care insurance policies where no elimination period or maximum benefit period is included. The results show that life care annuities are more affordable for disabled and older individuals. The results provide evidence for the insurability of long-term care expenses for individuals with impaired health. To take into account typical product features, a simulation-based model is employed to calculate premiums and reserves for policies with different combinations of elimination periods and maximum benefit periods. The elimination period is shown to be very effective in making stand-alone long-term care insurance more affordable while the maximum benefit period is more effective in extreme loss control. Solvency capital requirements taking into account longevity risk and disability risk for different types of policies are compared in the Solvency II standard formula. It is found that rider benefit policies and life care annuities have considerable capital reductions compared to stand-alone policies. Stand-alone policies sold to disabled individuals require lower capital per unit premium compared to those sold to healthy 5

29 individuals of the same age. The results provide insights into the design of affordable long-term care insurance and risk management for long-term care insurance providers. The remainder of this thesis is arranged as follows. The next chapter reviews literature on the pricing and risk analysis of reverse mortgage loans and long-term care insurance. Chapter 3 focuses on assessing the idiosyncratic component in house price risk faced by individual homeowners and lenders of reverse mortgage loans. Chapter 4 provides a new model framework for pricing and risk analysis of reverse mortgage loans allowing for idiosyncratic house price risk, longevity risk, interest rate risk and loan termination triggers. Chapter 5 quantifies health transition rates using the Generalised Linear Model under a Markov model framework. Chapter 6 develops a pricing and reserving framework for a wide range of long-term care insurance policies and compares solvency capital requirements for different types of long-term care insurance under the Solvency II framework. Chapter 7 concludes. 6

30 Chapter 2 Literature Review Basic financing sources of health care costs are public and personal co-contributions in many countries around the world. These financing systems are funded through copayments of public programmes, personal savings, and private insurance. A report by Productivity Commission of Australia (2011) proposes several financing sources as supplements for the basic aged are financing scheme in Australia, for example quarantined superannuation, home equity that can be unlocked using equity release products, and private long-term care insurance. This chapter reviews prior studies on equity release products and long-term care insurance that are two important retirement financial products in funding consumption and health costs in retirement (Olivieri and Pitacco, 2001; Productivity Commission of Australia, 2011). Section 2.1 discusses different types of equity release products and prior studies on pricing and risk analysis of reverse mortgage loans. Among the different risks of reverse mortgage loans, idiosyncratic house price risk is not widely recognised or studied in the reverse mortgage pricing literature due to limited public access to detailed data on individual house transactions. Section 2.2 reviews prior studies on house price models that can be used to assess the idiosyncratic component of house price risk. Section 2.3 discusses studies on methods of pricing and reserving of long-term care insurance policies. The last section of this 7

31 chapter presents the research motivation of the thesis and outlines the research questions that are investigated in the thesis. 2.1 Equity Release Products The rate of home ownership among the old population has been high in many countries around the world and retirees hold a large proportion of their savings in the form of home equity (Coile and Milligan, 2009; Ong, 2008; Poterba et al., 2010). This nest egg is recognised as an important component of financing retirement needs of an ageing population. The recent report on Caring for Older Australians by Productivity Commission of Australia (2011) recommends that home equity should be considered as a means to pay for health care costs and that home equity release products would allow individuals to unlock this wealth. Equity release products are financial products that allow homeowners to unlock home equity while living in their houses until they die or permanently leave the house. These products are especially designed for the elderly who are asset-rich but cash-poor (Wang et al., 2008). Six categories of equity release products can be identified in the market (Hosty et al., 2008; Hyde, 2008): The reverse mortgage scheme, which entitles the borrower to take out a loan against his or her house and to repay the loan only when he or she dies, enters into long-term care facilities, permanently moves out of the house, or voluntarily repays the loan due to changes in the financial situation. The reversion scheme, under which a trade of ownership and lease is involved. The product provider in essence buys a share of the house and simultaneously grants the borrower a lease to live in the house for life. The interest-only scheme, which requires borrowers to pay interests on a regular basis while living in the house and to repay the principal only when any of termination events occurs. 8

32 The shared appreciation mortgage scheme, under which the provider is entitled to share a predetermined proportion of future increases in the growth of the house price and in return borrowers are granted an interest-free loan. The home income plan, which offers borrowers regular incomes against the property while borrowers have to pay back interests on a regular basis, which is in essence a hybrid of an interest-only product and an annuity providing incomes. The property option for pensioners and investors (POPI), which entitles the property investor the right to purchase the pensioner s home at a pre-determined price upon certain events that are pre-specified in the contract. With a POPI, the homeowner essentially sells an option on the future capital growth of the house to the investor, in exchange for fixed incomes. Equity release products were first introduced in 1965 in the U.K. The products were launched in the form of reversion schemes but had many different features from modern reversion schemes. The home income product came into existence in 1972 when Allied Dunbar introduced the Home Income Plan in the U.K. In 1987 the Home Equity Conversion Mortgage (HECM) programme was authorised by the U.S. Department of Housing and Urban Development and was initiated in the U.S. market in Since then reverse mortgage loans have gained popularity in the U.S. market and many other countries. Reverse mortgage loans under the HECM programme are insured by the Federal Housing Administration (FHA) and are therefore considered the safest equity release plans in the U.S. among other programmes, including Fanie Mae s Home Keeper programme and Financial Freedom s Cash Account Advantage Plan (Chen et al., 2010). Reverse mortgage loans were first introduced to Australia in the 1990 s by St. George Bank (Reed and Gibler, 2003; Thosar, 2002). But due to lack of demand in the market, St. George withdrew the products. During recent years, the market for 9

33 equity release products has been expanding rapidly in Australia and is mainly in the form of two types: reverse mortgage schemes and reversion schemes (Australian Securities & Investments Commission, 2005; Deloitte and SEQUAL, 2012). As reported in Deloitte and SEQUAL (2012), the market size of reverse mortgage loans had nearly quadupled since 2005 and reached AUD 3.32 billion by the end of The released home equity is mainly used by Australian borrowers to improve their home conditions, to repay outstanding debts and to fund their retirement consumption (Deloitte and SEQUAL, 2012). The potention market size for reverse mortgage loans in Australia is estimated in Mitchell et al. (2006) to be 1.3 million households. Due to the overwhelming popularity of reverse mortgage loans, the thesis focuses on reverse mortgage loans Reverse Mortgage Loans Reverse mortgage loans allow homeowners to take out loans against their house up to a certain ratio and the borrowers are not required to repay the loans until they die or permanently move out. The ratio of the loan amount to the house value is called the loan-to-value ratio. The maximum loan-to-value ratio is set by the lender taking into account the home value, the borrower s age, the interest rate at loan issue and other variables. Maximum loan-to-value ratios in Australia are around 45% (National Information Centre on Retirement Investments Inc., 2014), which is substantially lower than the maximum loan-to-value ratios offered to homeowners in the U.S. The eligibility age for reverse mortgage loans in Australia is usually 65 but varies across different providers (National Information Centre on Retirement Investments Inc., 2014). Under the HECM programme in the U.S., reverse mortgage loans can only be issued to borrowers aged 62 and above. The payout of reverse mortgage loans can be a lump sum, an annuity income, a line of credit, or a combination of these three draw down options. Deloitte and SEQUAL (2012) report that a majority of reverse mortgage loans are taken out as 10

34 lump sums in Australia. The choice of the draw dawn option is important because it affects the borrower s entitlement to the Australian age pension. According to the Department of Human Services of Australia 1, annuity incomes from reverse mortgage loans are not deemed to the income test for the age pension while this is the case for lump sum payment from reverse mortgage loans. In particular, the following two situations are tested against the income test for the age pension: (1) the lump sum payment from the reverse mortgage loan is above AUD 40,000; (2) the lump sum payment is less than or equal to AUD 40,000 but is not spent within 90 days starting from the loan issue date. The interest rate on reverse mortgage loans can be fixed or variable. Compared to other loans, reverse mortgage loans usually charge higher interest rates (Reed and Gibler, 2003). Costs associated with applications, stamp duty, registration and on-going services are rolled into the loan balance and compounded with the principal and interest Risk Analysis of Reverse Mortgage Loans The Australian government has introduced a statutory regulation that requires all reverse mortgage providers to offer No-Negative Equity Guarantees (NNEGs) in new reverse mortgage loans 2. This guarantee protects reverse mortgage loan borrowers from negative home equity. In particular, the loan repayment at the termination of the contract is capped to the proceeds from the sale of the house. The NNEG is also referred to as the non-recourse provision in the U.S. (Chen et al., 2010). With NNEGs embedded in reverse mortgage loans, the risk of loss when the net house value is less than the outstanding loan amount is fully transferred to the reverse 1 Detailed information can be found at enablers/income-test-pensions 2 Detailed information can be found at superannuation-and-retirement/income-sources-in-retirement/home-equity-release/ reverse-mortgages 11

35 mortgage provider. This risk is usually referred to as the cross-over risk in the literature (e.g., Chen et al., 2010; Huang et al., 2011; Wang et al., 2008). Mortgage insurance premiums are charged to borrowers to fund the NNEGs included in reverse mortgage loans. In practice, the mortgage insurance premium is credited on a regular basis to the outstanding loan balance (Chen et al., 2010). The cross-over risk is the core risk in the analysis of reverse mortgage loans. It is caused by the combination of risks including house price risk, occupancy risk, interest rate risk, and other risks (Wang et al., 2008). House Price Risk A key risk factor - house price risk - has received relatively little research attention in the literature. Previous studies on reverse mortgage loans and other equity release products typically assess house price risk based on nationwide or city-level house price indices (see, e.g., Alai et al., 2014; Chen et al., 2010; Hosty et al., 2008; Li et al., 2010; Sherris and Sun, 2010). For example, Chen et al. (2010), Yang (2011) and Lee et al. (2012) model house price risk using a nationwide house price index for the U.S., whereas Hosty et al. (2008) and Li et al. (2010) use a nationwide index for the U.K. Wang et al. (2008) employ average house prices in eight capital cities in Australia. Sherris and Sun (2010), Alai et al. (2014) and Cho et al. (2013) use city-level data for Sydney, Australia. However, few reverse mortgage providers hold portfolios that are representative of the aggregate market. Therefore pricing and hedging methods based on aggregate house price indices only cover a limited part of the actual house price risk that reverse mortgage providers face. In addition, reverse mortgage loans typically include nonegative equity guarantees that are basically a portfolio of options on individual properties, instead of an option on a portfolio of properties. The heterogeneity of individual houses adds a substantial risk to reverse mortgage loans. Recent real estate research has shown that the trends and risks in houses prices vary substantially 12

36 across different submarkets within a city (see, e.g., Bourassa et al., 1999, 2003; Ferreira and Gyourko, 2012; Hanewald and Sherris, 2013). Hanewald and Sherris (2013) and Dröes and Hassink (2013) also show that the aggregate house price index considerably underestimates the house price risk. The risk resulting from the variability of individual house prices is called idiosyncratic house price risk. The risk arising from the difference between individual house prices and the aggregate house price index is usually referred to as the basis risk (Li et al., 2010). Employing an aggregate real estate index to assess the house price risk in equity release products therefore does not account for the idiosyncratic house price risk and is likely to underestimate the risks underwritten by product providers. One major reason that idiosyncratic house price risk is not widely accounted for in the reverse mortgage literature is the limited public access to individual house transactions data (Li et al., 2010). For this thesis, a large data set of detailed house transactions in Sydney is provided by Residex Pty Ltd that allows for an analysis of the idiosyncratic house price risk in reverse mortgage loans. This provides the research motivation for two chapters of the thesis. Chapter 3 develops individual house price models that assess the idiosyncratic component of house price risk. Chapter 4 projects price trends and uncertainty for individual houses with heterogeneous characteristics and then develops a pricing and risk analysis model for reverse mortgage loans allowing for idiosyncratic house price risk. A review of existent house price models is provided in Section 2.2. Interest Rate Risk Interest rate risk is an important risk that has a direct impact on the outstanding loan amount of a rate reverse mortgage loan and therefore affects the cross-over risk. Previous studies such as Wang et al. (2008), Chen et al. (2010) and Li et al. (2010) have used fixed interest rate to price reverse mortgage loans. In long-term contracts like reverse mortgages, it is more realistic to take into account stochastic interest 13

37 rates. Boehm and Ehrhardt (1994) employ the Cox-Ingersoll-Ross process to assess interest rate risk in reverse mortgage loans. The cross-over risk is exacerbated when the interest rate is higher than the appreciation rate of the house value. Therefore it is important to account for the correlation between interest rates and house price growth rates when pricing reverse mortgage loans. This thesis employs the Vector Auto-Regression (VAR) model to project future state variables, including house price growth rates, interest rates, GDP growth rates and rental yield rates. Based on projected state variables, the thesis develops a pricing model of reverse mortgage loans that takes into account correlations between these state variables. Occupancy Risk Occupancy risk is the risk that the borrower remains in the house so long that the accumulated loan amount exceeds the house value (Wang et al., 2008). It is therefore important to make accurate predictions of the borrower s life expectancy and of other termination triggers of reverse mortgage loans such as move-out due to health related reasons, voluntary prepayment and refinancing (Alai et al., 2014; Ji et al., 2012). Longevity risk is the aggregate risk that the mortality rate of the whole population decreases due to advances in medical science, technology and life-style changes (Cairns et al., 2006; Lee and Carter, 1992; Olivieri and Pitacco, 2001). A range of stochastic models are proposed in the literature to model longevity risk. Lee and Carter (1992) propose an innovative model for projecting time trends and age effects in mortality rates. The cohort effect is incorporated into the Lee- Carter model by Renshaw and Haberman (2006). The original Lee-Carter model (Lee and Carter, 1992) uses a random walk with drift to capture the time-varying improvements of mortality rates for all age groups. Time series analysis is used by many studies (e.g., Denuit et al., 2007; Lee, 2000; Renshaw and Haberman, 2006; Wang and Yang, 2013) to model the time-varying improvements of mortality rates. 14

38 Improvements for different age groups are assumed to be age-specific proportions of the aggregate improvement in the original Lee-Carter model. Therefore, mortality improvements across different age groups have perfect correlations in the original Lee-Carter model (Cairns et al., 2011, 2009; Wang and Yang, 2013). To account for non-perfect correlations between different age groups mortality improvements, Wang and Yang (2013) assume that the differenced error term in the original Lee- Carter model is correlated across different ages. An alternative approach in taking into account age-specific mortality dependence is to include more than one stochastic processes. Cairns et al. (2006) develop a model that incorporates two stochastic processes to capture the period effect on death probabilities. One time-varying parameter in the Cairns-Blake-Dowd (CBD) model represents the mortality improvements of the overall population and the other represents the age-specific improvements in mortality rates. Cairns et al. (2009) and Cairns et al. (2011) extend the two-factor CBD model proposed in Cairns et al. (2006) by including an additional stochastic process on the squared age term and a dummy variable that captures the cohort effect. Many studies (e.g., Cairns et al., 2006, 2009) model the two time-varying parameters in the two-factor CBD model (Cairns et al., 2006) as a bivariate random walk with drift. To capture serial- and cross-correlations of the two time-varying parameters, Chan et al. (2014) propose to use a Vector Autoregressive Integrated Moving-Average (VARIMA) process to model time-varying parameters. Wills and Sherris (2008) develop a multi-variate stochastic mortality model to describe the stochastic improvements of mortality rates over time. The model specifies changes in age-specific mortality rates along the cohort direction as a linear function of age plus a disturbance term driven by multiple stochastic risk factors. Observed correlations between the year-to-year changes in mortality rates of different age groups are incorporated in the multivariate distribution of the stochastic risk factors. The Wills-Sherris model allows for a more flexible and realistic age 15

39 dependence structure than, for example, the one-factor model by Lee and Carter (1992) and the two-factor model by Cairns et al. (2006). An explicit expression for the age dependence structure can be derived from the Wills-Sherris model. Li et al. (2010) employ the Lee-Carter model to assess the longevity risk in reverse mortgage loans. Chen et al. (2010) also employ the Lee-Carter model in assessing the mortality risk of reverse mortgage loans and additionally incorporate permanent and transient jumps in mortality rates (Chen and Cox, 2009). The CBD model is used by Yang (2011) to investigate the impact of longevity risk on reverse mortgage loans under the HECM programme. These studies find a pronounced impact of longevity risk on the pricing of reverse mortgage loans. The Wills-Sherris model has been applied in some prior studies to analyse the pricing and risk profiles of financial products that are exposed to longevity risk (see, e.g. Hanewald et al., 2013; Meyricke and Sherris, 2014; Ngai and Sherris, 2011; Wills and Sherris, 2010). In this thesis, the two-factor CBD model (Cairns et al., 2006) and the Wills- Sherris model (Wills and Sherris, 2008) are employed to assess longevity risk of reverse mortgage loans in Chapter 4. The impacts of the two stochastic mortality models are also compared. Other Risks In addition to the above described risks, maintenance risk and expense risk are two important risk factors in the pricing and risk analysis of reverse mortgage loans (Wang et al., 2008). Maintenance risk refers to the risk that reverse mortgage borrowers lack incentives to properly maintain the house as agreed (Davidoff and Welke, 2004; Szymanoski, 1994). Maintenance risk therefore results in larger crossover risk for the product provider. Miceli and Sirmans (1994) develop a theoretical model to analyse maintenance risk and suggest two feasible solutions for reverse mortgage providers to take into account maintenance risk: by limiting the amount of issued loans or by charging an additional premium to fund the expected maintenance 16

40 risk. Expense risk is related to the possibility that future expenses in servicing reverse mortgage loans are higher than expected. Inflation risk is a main factor driving the uncertainties of future expenses (Wang et al., 2008) Pricing Models of Reverse Mortgages NNEGs are in essence options on house values with uncertain terms to maturity and uncertain strike prices. Many prior studies price reverse mortgage loans using option pricing models (e.g., see Alai et al., 2014; Chen et al., 2010; Li et al., 2010; Wang et al., 2008). Different stochastic processes are employed in the literature for modelling the underlying house price dynamics. Wang et al. (2008) assume a deterministic appreciation rate of house values. Chen et al. (2010) employ an ARIMA-GARCH process to model the house price index and then price NNEGs, or non-recourse provisions as called in Chen et al. (2010), using option pricing models with an ARIMA-GARCH underlying process. To account for the leverage effect found in the U.K. nationwide house price data, Li et al. (2010) model the underlying house price index as an ARIMA-EGARCH process. Prices of NNEGs calculated under the ARIMA- GARCH model are also compared in Li et al. (2010) with results calculated using the Black-Scholes model where the house price index is assumed to follow a geometric Brownian motion. They conclude that prices of NNEGs calculated with an ARIMA-EGARCH house price process are higher than those calculated with the Black-Scholes model. A Vector Auto-Regression (VAR) model is employed in Alai et al. (2014) to describe the dynamics of state variables including house price growth rates, GDP growth rates, zero-coupon bond yield rates, and rental yield rates. Prices of NNEGs are calculated in Alai et al. (2014) using simulated values in the state variables based on the estimated VAR model. The uncertain term to maturity of NNEGs embedded reverse mortgage loans 17