1 Introduction
Today in many countries around the globe, life expectancy from birth is well over 80 years. In 1989, after the Department of Housing and Urban Development (HUD) introduced the Home Equity Conversion Mortgage (HECM) program, reverse mortgages became widely available in the United States. A reverse mortgage is a new financial product that allows retirees to convert a proportion of the equity in their home into cash until they die. Homeowners 62 and older who have paid off their mortgages or have only small mortgage balances remaining are eligible to participate in HUD's reverse mortgage program.
A reverse mortgage is a loan against the equity in your home that you don‟t need to pay back for as long as you live in the home. Thus, the reverse mortgage program enables seniors that may be "real estate rich and cash poor" to unlock the financial potential in their homes, and their homes work for them. In general, the reverse mortgage does not become payable until the senior homeowner no longer occupies the property as his or her primary residence. Homeowners can receive payments in a lump sum, annuity income, on a monthly basis (for a fixed term or for as long as they live in the home), or on an occasional basis as a line of credit. Homeowners whose circumstances change can restructure their payment options. The size of reverse mortgage loans is determined by the borrower's age, the interest rate, and the home's value.
In order to protect lenders of reverse mortgages from possible losses, the Federal Housing Administration (FHA), which is one part of HUD‟s Office of Housing, which is charges insurances premiums from borrowers, and pays insurance claims to lenders if the loan balance exceeds the home equity value. That is, according to the actuarial equivalence principle, the present value of expected premiums should be equal to the present value of expected losses.
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Over the past two decades, most of the existing literature on risk modeling in the HECM program for pricing insurance premium. Szymanoski (1994) provided the method of building actuarial model of HECM. Tse (1995b) incorporate the different risks when modeling interest rates via Cox-Ingersoll-Ross (1985) model. Rodda, Lam, and Youn (2004) analyzed the HECM program using stochastic models of interest rates and housing prices and a new model of termination rates. Chia and Tsui (2004) conducted research of reverse mortgages for Singaporeans. Ma, Kim and Lew (2006; 2007) using stochastic models to estimating reverse mortgage insurer‟s risk and confirmed that the present values of expected losses were very sensitive to the processes of housing prices and interest rates.
However, the loan balance depends not only on the age and sex of the homeowners, but also on the appraised value of the property, the projected rate of house price appreciation and the levels of interest rates. The most important factor in the size of reverse mortgage loan is the value of home. As a result, it‟s important to understand how the fluctuation of home prices affect reverse mortgage financing, so that borrowers can make an optimal decision. Empirical work by Kutty (1998) indicates that the use of home equity conversion mortgage products could possibly raise about 29% of the poor elderly homeowners in the US above the poverty line. In order to model the house price risk, Szymanoski (1994), Ma, et al. (2007) and Valdez, et al. (2007) assume house prices are driven by a geometric Brownian motion. However, the dynamics of house prices, like those of any asset, are vital to understand for proper risk and portfolio management.
Volatility is a key aspect of such dynamics. Crawford and Fratantoni (2003), Dolde and Tirtiroglu (1997), Miller and Peng (2006) have investigated whether house price volatility is time-varying; that is, house prices exhibit the volatility clustering or GARCH (generalized autoregressive conditional heteroskedasticity) effects, such as those for stocks and bonds. Chen, et al. (2010) propose a generalized Lee-Carter model with permanent jump effects to fit the actually mortality data, and model the house price index
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via an ARMA-GARCH (Autoregressive moving average - generalized autoregressive conditional heteroskedasticity) process and employ the conditional Esscher transform to price the non-recourse provision of reverse mortgages. Therefore, we follow Chen, et al.
(2010) works by using an ARMA-GARCH model to fit house price returns.
A major difference between the cash flows of a traditional home purchase, or forward, mortgage and a reverse mortgage is in the pattern of equity and debt over time.
Reverse mortgages loan balance grows due to principal advances, interest accruals, and other loan charges over the life of the loan. The key risk factors affecting the cash flows and pricing of HECM insurance, are (1) borrower mortality rates and voluntary loan terminations, which determine the timing of lump sum or other types repayments; If a borrower lives a longer time than the expected lifespan that may lead the loan balance above the sale proceed of the property. In other words, lenders of reverse mortgages are faced with longevity risk. (2) Interest rate changes, which affect the rate at which the debt rises; Reverse mortgages can generally be categorized as either fixed-rate or variable-rate.
Fixed-rate reverse mortgages accrue interest at the same (fixed) rate for their entire duration, whereas the rate associated with variable rate mortgages rises and falls in accordance with a stated benchmark rate, such as the 1 year T-bill, 1-year LIBOR, and other obscure rate indexes. The rise of interest rates increases the possibility of non-repayment when the loan eventually terminates. In this study, we choose a fixed interest rate with a risk adjustment. (3) The future property values, which affects the net proceeds from a sale. If the house price grows at a lower rate than expected, the loan balance may exceed the home value. Lenders may suffer from the losses.
The difference between interest rate risk and house price risk are that the interest rate risk could not be diversified, while the house price risk can be partially diversified by holding a large portfolio of loans across areas. Therefore, we will focus on house price risk.
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In general, almost all of previous studies ignore longevity risk. We employ Lee-Carter (1992) model to model the mortality rates for pricing the present value of claim losses and calculating mortgage insurance premiums. Moreover, it is well known that house price returns (quarterly) empirical distributions are closer to the Gaussian case.
Unfortunately, house price returns (monthly) in our study are potentially non-Gaussian.
In this paper, we want to construct the house price model via ARMA-GARCH with Normal Inverse Gaussian distribution (ARMA-GARCH-NIG) option pricing model via local risk-neutral valuation relationship (LRNVR) and conditional Esscher transform.
Therefore, the proposed method proves to play an important role in pricing reverse mortgage.
The remainder of this paper is organized as follows. In Section 2, we introduce the reverse mortgage pricing framework. Section 0 model the longevity risk via Lee-Carter model in the reverse mortgage. Modeling the house price risk and introduce the normal inverse Gaussian distribution in section 3. In section 5, we introduce the LRNVR and the conditional Esscher transform to in order to valuation. Section 6 use empirical data to show the result of this analysis. The final section concludes this study.
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