Presenter: Sharon S. Yang National Central University, Taiwan

Transcription

1 Pricing Non-Recourse Provisions and Mortgage Insurance for Joint-Life Reverse Mortgages Considering Mortality Dependence: a Copula Approach Presenter: Sharon S. Yang National Central University, Taiwan 1

2 Outline Introduction and Literature Review The Valuation Framework for HECM Program Non-recourse provision Mortgage insurance Modeling Joint-Life Mortality Copula model Numerical Analysis Conclusion 2

3 Introduction 3

4 What are Reverse Mortgage Products? A kind of home equity conversion that allows the elder persons to borrow money with their home as the collateral. The loans accrue interest are only repaid once the people is died or leave the house. No Fixed Maturity Date Loan value is determined by borrower s age, property value and interest rate For example: a rolled-up mortgage (Lump-Sum) vt Loan Value: K --- K Ke at time Property Value: H t ---> H t t Age x Loan Period Die(x+s) 4 4

5 The Risk from Lender Prospective Key risk Longevity risk Property risk Interest rate risk If a borrower lives a longer time than the expected lifespan that may lead the loan balance above the sale proceed of the property. 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 rise of interest rates increases the possibility of nonrepayment when the loan eventually terminates. $ Ke v Home Value Loan balance H v Ke time H 5

6 The RM Market: in the U.S. Home Equity Conversion Mortgage (HECM) program first introduced in After financial crisis, the private reverse mortgage market has evaporated so that HECM loans represent nearly 1% of newly originated reverse mortgages.(shen, REE 211) The HECM loan is a non-recourse debt. Some factors drive the need for RM Globally Increasing Life Expectancies Many elders are considered to be Cash Poor & Equity Rich. 6

7 HECM loan Thousands Billions 2 5 Number of Loans (left axis) Property Value (right axis) Initial Principal Limit (right axis) 7

8 7% 6% 5% 4% 3% 2% 1% % HECM Loan 8 Percentages Type of Borrower ` Single Female Single Male Multiple***

9 Literature Most of the existing literatures of RMs focus on the following issues: Single-life RMs Except for Chia and Tsui(24) work on joint-life RMs Assume mortality is Independent. Lump-Sum payment Except for Lee et al.(213) work on tenure RMs Underlying House Price Dynamic Szymamoski(1994) uses the GBM model. Li et al.(21) propose the ARMA-EGARCH model Chen et al. (21) propose ARMA-GARCH model. 9

10 Motivation of this research Joint life RMs are getting more and more popular in HECM loans. Shemyakin and Youn (1999) point out three possible sources of association between husbands' and wives' mortality. *common lifestyle *common disaster *broken-heart factor 1

11 Purpose of this research To capture the mortality dependence for pricing joint-life RMs Lump Sum vs. Tenure Pricing Non-Recourse Provisions (NRP) and Mortgage Insurance Premiums (MIP) Investigate the effect of mortality dependence on NRP and MIP 11

12 Pricing Framework for Jointlife Revere Mortgages 12

13 Payoff of the non-recourse provision The cash flow of the borrower can be written as follows: repayment = { -H t H t < L t where L t is the outstanding balance of the loan and H t is the value of the mortgaged property at a time t. Claim loss function at time t : CL t = max (L t -H t, ) -L t H t L t = -L t + max(l t -H t, ) 13

14 Payoff of the non-recourse provision The fair value of the non-recourse provision (NRP) written on a cohort group aged x can be expressed as the present value of total expected claim losses on a cohort group aged x as follows: where x1 rt PV ECL= q. E [ e max( L H,)] t t1 xy Q t t r is the risk-free interest rate q t1 xy is the probability that both a male aged x and a female aged y will survive another t year, but die before next year. E Q is the expectation under the risk-adjusted measure Q. 14

15 Outstanding Balance At time t, the house value and the accumulated loan balance are Yt H=H t e t t ut u( t j) u( t j) j j1 j L =.2H e e.5l A e t where A is annuity value(unknown). 15

16 Mortgage insurance premium The present value of mortgage insurance premium of reverse mortgage can be calculated as rt PV MIP =.2H p. e (.5 Lt ) t t1 Initial premium: 2% of the property value Continuous premium:.5% of outstanding balance What does the ratio of PVMIP/NRP mean? xy 16

17 Modeling Joint-Life Mortality 17

18 Independent Joint-life Mortality We denote the marginal survival functions by m f S x and S, so far all t y m m m t x x p S Pr[ T ( x) t] f f f t py Sy Pr[ T ( y) t] Joint survival probability notation (, ) Pr[ m ( ) and f ( ) ] indep m ( ) f t pxy Sxy t t T x t T y t Sx t Sy ( t) m f m f p 1 q q p p p t xy t x t y t x t y t xy q p p t 1 xy t xy t1 xy 18

19 Correlated Joint-life Mortality Shemyakin and Youn (1999) utilized copula functions to calculate joint survival probability for pricing joint life insurance and list three possible sources of association between husbands' and wives' mortality. *common lifestyle *common disaster *broken-heart factor 19

20 Correlated Joint-life Mortality The joint survival probability can be written as Pr[ m ( ) and f ( ) ] dep ( m ( ), f t pxy T x t T y t C Sx t Sy ( t); ) where is correlated parameter. The joint survival probability which the couple are survival exceeding one year can be written as p S (1,1) C( p p ; ) xt: yt ( xt)( yt) xt, yt t Combing above equation p p p... p C( p p ; ) t xy xy x1: y1 x t 1: y t 1 xi, yi t i t 1 2

21 Copula Functions Gaussian copula(normal copula): Cu u u u 1 1 ( 1, 2; ) G ( ( 1), ( 2); ) Student-t copula: C( u, u ;, ) t { t ( u ), t ( u )} , 1 2 Clayton copula: Frank copula: Gumbel copula: ( 1, 2; ) ( 1 2 1) Cu u u u θu1 1 ( e 1)( e C( u1, u2; θ) = ln[1+ θ θ e 1 1) ] 1 ( 1, 2; ) exp[ {( log 1) ( log 2)} Cu u u u θu 2 21

22 Copula Functions Gaussian Copula ( =.5) Student-t Copula ( =.5, = 2 ) u2 Clayton Copula ( = 2 ) u u2 Gumbel Copula ( = 2 ) 1.5 u1 Frank Copula ( = 4 ) u2 u1 u2 u1 u2 u1 22

23 Lee-Carter Model with Cohort Effect Renshaw and Haberman (26) made cohortbased extension to the LC model. ln( m ) ( t) ( c) m xt x x t * c x,t * xtc x x t x c xt : central death rate for a person aged x at time t : describes the average age-specific mortality : decline in mortality at age x : represents the general mortality level : cohort effect : deviation of the model from the observed log-central death rates 23 23

24 Modeling the House Price Dynamics 24

25 Risk-adjust Probability House Price Dynamic Esscher transform (Buhlmann et al.(1996) Lee et al.(21), Li et al.(21), Chen et al.(21) Mortality Dynamic Wang Transform Lin and Cox (25), Lee et al.(21), Yang et al.(213) 25

26 26

27 Empirical Study 27

28 Fitting House price dynamics The house price data *Quarterly House Price Index (HPI) *1991Q1~211Q3 ARMA(2,2)-GARCH (1, 1) 28

29 Mortality Dynamics The mortality rate data *U.S., Male and Female Data range *Age: 6~99 *Year: 195~27 Source *Human Mortality Database(HMD) * 29

30 Fitting model (Male) Alpha Age-Groups 1 Beta Age-Groups Beta Age-Groups Kappa Year 4 Gamma Year Figure: Lee-Carter model with cohort estimated parameters 3

31 Fitting model (Female) Alpha Age-Groups 5 Beta Age-Groups Beta Age-Groups Kappa Year 2 Gamma Year Figure: Lee-Carter model with cohort estimated parameters 31

32 Selection of Copula Models Comparing with AIC and BIC, the Clayton and Gumbel copulas are selected for difference ages. Clayton copula is selected for 6~72 age and Gumbel copula is selected for 73~95 age. Trivedi and Zimmer (25) suggested, when correlation between spouses' age at death is strongest in the left tail of the joint distribution, Clayton is an appropriate modeling choice. 32

33 Joint-life Probability at same age(xx) 33

34 Assumptions for Calculating NRP and MIP Assume house price independent mortality rate. Male and Female are at the same age. H =3, r =1.7523%. u =2.451%. g =2%. =6%. =99. We use Monte Carlo simulation 1, times and utilize Antithetic variance reduction to reduce the variation of pricing the reverse mortgage. 34

35 NRP and MIP : lump sum and tenure payment 35

36 MIP/NRP : lump sum and tenure payment 36

37 Initial house price value 37

38 Initial house price value 38

39 Conclusion Ignoring the dependence between joint-life mortality overestimates the NRP and MIP Such effect is more significant for lump sum RMs The insurance company can use their own mortality experience to replace the empirical analysis. Frees et al.(1996) Luciano, et al. (28) model the mortality risk of couples of individuals, according to the stochastic intensity approach. 39

40 Thank you! Q&A 4