Longevity 8 ＠University of Waterloo, Canada
Session III:11:00-12:30, 8 September, 2012
Pricing Reverse Mortgages in Japan
Using
a Multivariate Bayesian Risk-neutral Method
by
Atsuyuki Kogure
Keio University, Japan
Jackie Li
Nanyang Technological University, Singapore
Shinichi Kamiya
Nanyang Technological University, Singapore
The financial support from SCOR is greatly acknowledged.
1
Introduction
2
What is a reverse mortgage?
3
Non-recourse Provision
payoff of put option on house price Ht
with strike price Lt
4
Present Value of a Reverse Mortgage
r=discount rate
Maximum life
years
Risk
neutral
expectation
5
Bayesian approach
6
What is the Risk-neutral Pricing?
r＝discount rate
State price density
7
Risk-neutral Density
state price density
(weight function)
8
Bayesian Risk-neutral Pricing
9
Risk-neutral Predictive Density
state price density
10
Risk-neutral Density based on Cross-entropy (1)
リスク中立性
の条件
11
Risk Neutral Density based on Cross-entropy (2)
12
Bivariate Risk-neutral Density based on Cross-entropy (1)
13
Bivariate Risk-neutral Density based on Cross-entropy (2)
14
Present value of reverse mortgage
15
Evaluation of E*[It]: Modeling Mortality Risk
16
Lee-Carter method
17
One-factor Lee-Carter model
temporal changes
18
Bayesian estimation of Lee-Carter model
19
Evaluation of E*[It]: MCMC sampling results
male
female
20
MCMC sampling results（basic statistics）
male
 65
65
1970
 2005
Posterior mean
Posterior sd
95%HPD
Geweke
-4.0067
0.0105
(-4.0270, -3.9858)
0.97
0.0329
0.0015
(0.0299, 0.0358)
0.42
12.302
0.3299
(11.6891, 12.9763)
0.09
-9.7325
0.3253
(-10.3521, -9.0798)
0.58
Posterior mean
Posterior sd
95%HPD
Geweke
-4.7484
0.0092
(-4.7669, -4.7310)
0.18
0.0305
0.0008
(0.0289, 0.0321)
0.14
17.7812
0.2966
(17.1937, 18.3509)
0.13
-16.2597
0.2938
(-16.8414, -15.6887)
0.42
female
 65
65
1970
 2005
21
Risk-neutralization of Predictive Distribution of tpx
market value of the
annuity (given)
expectation of
ax(j) under π*
22
Bayesian Risk-neutral Density of tpx
market value of the
annuity (given)
23
E*[It]: cohort of age 65
male
female
24
Evaluation of of E*[max(Lt-Ht, 0)]
25
House Price data: Case-Shiller Indices
US
Japan
26
Modeling house prices
27
Risk-neutralization of predictive distribution for Ht
the current house price
28
Evaluation of E*[max(Lt-Ht,0)]
29
E*[max(Lt-Ht,0)]:
30
Evaluation of Reverse Mortgages in Japan
31
Evaluation of Reverse Mortgages in Japan (u=4%)
32
Evaluation of Reverse Mortgages in Japan (u=5%)
33
Evaluation of Reverse Mortgages in Japan (u=6%)
34
Two-factor Lee-Carter model
cyclical changes （|Φ|<1)
35
Estimated Parameters of Two-factor Lee-Carter Model (male)
36
Evaluation of E*[It]: age 65 male
one-factor model
two-factor model
Little difference between the models
37
Evaluation of E*[It]: age 65 female
one-factor model
two-factor model
Some difference between the models !
38
One-factor vs. Two-factor (u=4%)
no change
large increase
39
One-factor vs. Two-factor (u=5%)
no change
large increase
40
One-factor vs. Two-factor (u=6%)
no change
large increase
41
Conclusions
42
43
44