Pricing No-Negative-Equity-Guarantee for Equity Release Products under a Jump ARMA-GARCH Model

1. Pricing No-Negative-Equity-Guarantee for Equity Release Products under a Jump ARMA-GARCH Model Presenter: Sharon Yang Co-authors:Chuang-Chang Chang Jr-Wei Huang National Central University, Taiwan

2. Outline • Introduction. • Investigation of House Price Return Dynamics With Jumps. • Valuation Framework for No-Negative-Equity-Guarantee. • Numerical Analysis. • Conclusion.

3. Introduction

4. What are Equity Releasing 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. • Such products are needed for “equity rich and cash poor” persons. • For example: a rolled-up mortgage Age x Die(x+s) Loan Period 4

5. The Risk from Lender Prospective • The loan value may exceed the value of the property. • How to deal with such risk? • Using Insurance. Ex: HECM program in the united states. • Securitization • Writing a no-negative-equity-guarantee(NNEG) • Payoffs:  an European put option on the mortgaged property

6. Purpose of this study • Can Black & Sholes option pricing formula apply to value NNEG? No! • We built up a general framework which considers the dynamics of the house price return with jumps.

7. Purpose of this study-Con’t • Li et al . (2010) conclude thatthe Nationwide House Price Index has the following statistical properties: • there is a strong positive autocorrelation effect among the log-returns • the volatility of the log-returns varies with time; • a leverage effect is present in the log-return series ARMA-EGARCH Model • Chen et al.(2010) use the ARMA-GARCH model to price reverse mortgage for the HECM program in the U.S..

8. Purpose of this study-Con’t • We consider a jump model that incorporate both autocorrelation effect and volatility cluster.  a Jump ARMA-GARCH Model

9. An Investigation of House Price Return Dynamics with Jumps

10. Jumps in House Price Returns? • According to the quarterly data from 1952 to 2008, it can show that the quarterly housing price changed more than three standard deviations.

11. Jumps in HousePrice or Equity Returns ? • Chen et al. (2009) study U.S. mortgage insurance premium using Merton jump diffusion process for house price returns. • Merton (1976) build a jump diffusion model with a continuous-time basis.

12. Jumps in House Price or Equity Returns ? • Kou (2002) also considers the leptokurtic feature and proposes a double exponential jump-diffusion model. The return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution.

13. Jumps in House Price or Equity Returns ? • Chan and Maheu (2002) and Duan et al. (2006, 2007) both examine the jump effect with equity returns under a GARCH model • Dynamic jumps in return v.s. Constant jumps in both returns and volatility.

14. Jumps in House Price or Equity Returns ? • Chan and Maheu (2002) • Dynamic jumps in return

15. Jumps in House Price or Equity Returns ? • Duan et al. (2006, 2007) • Constant jumps in both returns and volatility.

16. Jumps in House Price or Equity Returns ? • We extend Chan and Maheu (2002) to consider the dynamic jump effect with house price returns under an ARMA-GARCH model and develop a framework for pricing the NNEG.

17. ARMA-GARCH Model follows an ARMA process. follows a GARCH process.

18. Dynamic Jump ARMA-GARCH Model The case for a dynamic jump:

19. A Comparison of Model Fitting Model Selection, 1953Q4~2008Q4

20. A Comparison of Model Fitting Model Selection, 1958Q4~2008Q4

21. A Comparison of Model Fitting Model Selection, 1968Q4~2008Q4

22. The Valuation Framework for No-Negative-Equity-Guarantee

23. Pricing No Negative Equity Guarantee Let us define the following notation: • K : the amount of loan advanced at time zero; • : the value of the mortgaged property at time t; • r : the constant risk-free interest rate; • v: the roll-up interest rate; • g : the rental yield; • : the average delay in time from the point of home exit until the actual sale of the property.

24. Pricing No Negative Equity Guarantee • Assuming the person dies in the middle of the year • Considering the delaying time • Payoff • Valuation

25. Pricing No Negative Equity Guarantee The value of P under measure Q can be obtained using conditional Esscher transform.

26. Pricing No Negative Equity Guarantee • Under the risk-neutral measure Q, the return processes of and to characterize the jump ARMA(s,m)-GARCH(p,q) model become • Special Case: Constant Jump

27. Pricing No Negative Equity Guarantee • Black and Sholes • Merton Jump

28. Making Numerical Analysis

29. Numerical Analysis

30. Numerical Analysis

31. Numerical Analysis

32. Conclusion • This article contributes to the literature in the following ways. • Dynamic Jump ARMA-GARCH model can better capture the dynamics of house price return. • The estimation of the proposed jump ARMA-GARCH model is carried out and presents a better fitting result compared with various house price return models proposed in the literature.

33. Conclusion • This article contributes to the literature in the following ways. • The risk neutral pricing framework for the jump ARMA-GARCH model is derived using the conditional Esscher transform technique. • Numerical result shows that incorporating the jump effect in house price returns is important for pricing NNEG.

34. The End.Thanks!