On Risk-Neutral Valuation of
Reverse Mortgages
Department of Applied Finance
and Actuarial Studies
Macquarie University
Jackie Li PhD, PhD, FIAA
RSFAS Summer Camp
2 December 2015
Reverse Mortgages
life expectancy and dependency ratio
continue to rise
 many retirees are ‘asset-rich-cash-poor’
 reverse mortgages can unlock home
equity and provide supplementary
retirement funding
 market is growing in Australia and AsiaPacific

2
Reverse Mortgages
it allows homeowner to borrow against
home property value
 loan is repaid with interest from sales
proceeds of home property when
borrower dies
 it is often non-recourse, i.e. lender cannot
have a claim on borrower’s other assets
 borrower can stay in the same home
 types / features are many and varied

3
Industry Bodies
Senior Australians Equity Release
(SEQUAL)
 Safe Home Income Plans (SHIP) in UK
 Home Equity Conversion Mortgage
(HECM) in US

4
Underlying Risks
longevity risk
 house price risk
 interest rate risk
 mis-selling
 fraud
 legal risk
 operational risk

5
Current Regulations
Solvency II : make use of and be
consistent with information provided by
financial markets and generally available
data on underwriting risks
 Prudential Standard APS 111 : maximise
use of relevant observable inputs and
minimise use of unobservable inputs; only
mark-to-model when mark-to-market is
not possible

6
Two Valuation Approaches
7
Risk-Neutral Valuation
market is complete if there are many
securities being traded
 any cash flow can be replicated by
dynamic hedging strategies
 no-arbitrage principle indicates that there
is only one risk-neutral measure and
there is only one price
 expected return is equal to risk-free rate

8
Current Life Market
there has been significant development in
securitisation of insurance liabilities in
recent years
 e.g. LLMA, catastrophe bond, mortality
bond, survivor bond, q-forward, survivor
swap
 but current market is far from having
sufficient liquidity
 market is incomplete

9
Risk-Neutral Valuation
if market is incomplete, there are infinitely
many risk-neutral measures
 it is necessary to choose one that is
relatively suitable to particular problem
 e.g. Esscher transform (Gerber and Shiu

x
X






f
x

e
f
x

e
1994)
 e.g. Wang transform (Wang 2000)

1
F x     F x   



10
Risk-Neutral Measures
these two transforms have decent
properties
 subjective decisions are usually needed
when number of parameters is different
to number of security prices available
 it is not straightforward to allow for
multiple risks

11
Current Literature
Authors
Mortality
House Price
Interest Rate
Wang et al. 2008
GLM + Wang
GBM + no-arbitrage
Vasicek
Hosty et al. 2008
fixed P-spline
lognormal
fixed
Chen et al. 2010
LC with jumps
ARIMA-GARCH +
conditional Esscher
fixed
Li et al. 2010
LC
ARMA-EGARCH +
conditional Esscher
fixed
Ji et al. 2012
fixed
GBM + no-arbitrage
fixed
Lee et al. 2012
LC + Wang
jump diffusion +
conditional Esscher
risk-neutral CIR
Kogure et al. 2014
LC + entropy
ARIMA-GARCH +
entropy
fixed
12
Maximum Entropy Principle
minimise Kullback-Leibler information
f  x

criterion  f x  ln f  x  dx
 subject to m constraints of m securities



f  x  dx  vi
(hi = discounted
 hi xpayoff
vi = market price)
 subject to constraint


 f x  dx  1



f x   f x  exp 0  i 1 i hi x 

m
13
Maximum Entropy Principle
discrete version straightforward and
flexible to apply in practice
 find πj* that minimise Kullback-Leibler
information criterion Σπj*ln(πj*/πj)
for j = 1 , 2 , … , n paths
 subject to m constraints of m securities
Σhi,jπj* = vi
 subject to constraint Σπj* = 1
 Lagrange multiplier

14
Maximum Entropy Principle
any number of security prices can be
incorporated
 different simulation methods can be used
 it can be applied to different risks
consistently and coherently
 there are some theoretical and empirical
support

15
Maximum Entropy Principle
Frittelli (2000) proves equivalence between
maximisation of expected exponential utility
and minimisation of Kullback-Leibler
information criterion
 Stutzer (1996) reports that it produces
prices close to Black-Scholes prices
 Gray and Newman (2005) show that it
outperforms historic-volatility-based BlackScholes estimator
 it has been applied to other futures options

16
Longevity Risk
Lee and Carter (1992) model
 ln(mx,t) = ax + bx kt
 mx,t = central death rate
 ax = mortality schedule
 bx = age-specific sensitivity
 kt = mortality index
 simple model structure
 highly linear kt empirically
 fit ARIMA(0,1,0) to kt for projection

17
Financial Risks
house price risk and interest rate risk
 vector autoregressive (VAR) model
 e.g. VAR(1) model
r tH = c1 + A1,1 r t-1H + A1,2 r t-1I + e1,t
r tI = c2 + A2,1 r t-1H + A2,2 r t-1I + e2,t
 simple model structure for autoregressive
effects
 assume real-world independence between
longevity risk and financial risks

18
Reverse Mortgage Example
suppose an individual borrows L0 and
returns min(Lt , Ht) on death
 EPV = ΣEQ[d(t) min(Lt , Ht) It]
 Lt = L0 eut
 u = fixed loan interest rate
 Ht = house price
 It = proportion who dies within (t-1 , t)
 d(t) = discount factor
 EPV must be larger than L0 to make it
financially viable

19
Two Valuation Approaches
20
Historical Data





for fitting LC and VAR models
Australian female mortality (ages 65 to 99;
period 1968 to 2011) from Human Mortality
Database (HMD)
residential property price index (2003 to 2014)
from Australian Bureau of Statistics (ABS)
90-Day BABs/NCDs yields (2003 to 2014) from
Reserve Bank of Australia (RBA)
mortality data and house price data are proxies
21
Market Prices Data
for setting constraints
 Australian female mortality (ages 65 to
99) from Australian Life Tables 2005-07 by
Australian Government Actuary
 residential property price index
(December 2014) from ABS
 zero-coupon interest rates (31 December
2014) from RBA
 mortality data and house price data are
proxies

22
Preliminary Results
1
0.8
0.6
F*(x)
F(x)
0.4
0.2
0
0.88
0.89
0.9
0.91
0.92
0.93
0.94
0.95
10p65
23
Preliminary Results
1
0.8
0.6
F*(x)
F(x)
0.4
0.2
0
130
150
170
190
210
230
250
270
290
310
330
H10
24
Preliminary Results
1
0.8
0.6
F*(x)
F(x)
0.4
0.2
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
r10
25
References

Australian Prudential Regulation Authority (APRA), 2013, Prudential
Standard APS 111

Chen H., Cox S.H., and Wang S.S., 2010, Is the home equity conversion
mortgage in the United States sustainable? Evidence from pricing mortgage
insurance premiums and non-recourse provisions using the conditional
Esscher transform, Insurance: Mathematics and Economics 46: 371-384

Frittelli M., 2000, The minimal entropy martingale measure and the
valuation problem in incomplete market, Mathematical Finance 10: 39-52

Gerber H.U. and Shiu E.S.W., 1994, Option pricing by Esscher transforms,
Transactions of the Society of Actuaries 46: 99-191

Gray P. and Newman S., 2005, Canonical valuation of options in the
presence of stochastic volatility, Journal of Futures Markets 25: 1-19
26
References

Hosty G.M., Groves S.J., Murray C.A., and Shah M., 2008, Pricing and risk
capital in the equity release market, British Actuarial Journal 14(1): 41-109

Ji M., Hardy M., and Li J.S.H., 2012, A semi-Markov multiple state model for
reverse mortgage terminations, Annals of Actuarial Science 6(2): 235-257

Kogure A., Li J., and Kamiya S., 2014, A Bayesian multivariate risk-neutral
method for pricing reverse mortgages, North American Actuarial Journal
18(1): 242-257

Lee R. and Carter L., 1992, Modeling and forecasting US mortality, Journal of
the American Statistical Association 87: 659-671

Lee Y.T., Wang C.W., and Huang H.C., 2012, On the valuation of reverse
mortgages with regular tenure payments, Insurance: Mathematics and
Economics 51: 430-441
27
References

Li J.S.H., Hardy M.R., and Tan K.S., 2010, On pricing and hedging the nonegative-equity guarantee in equity release mechanisms, Journal of Risk and
Insurance 77(2): 499-522

Solvency II Directive 2009/138/EC, Official Journal of the European Union

Stutzer M., 1996, A simple nonparametric approach to derivative security
valuation, Journal of Finance 51: 1633-1652

Wang L., Valdez E.A., and Piggott J., 2008, Securitization of longevity risk in
reverse mortgages, North American Actuarial Journal 12(4): 345-371

Wang S., 2000, A class of distortion operators for pricing financial and
insurance risks, Journal of Risk and Insurance 67: 15-36
28