On Risk-Neutral Valuation On Risk-Neutral Valuation of Reverse Mortgagesof Reverse Mortgages

Department of Applied Finance and Actuarial Studies

Macquarie University

Jackie Li PhD, PhD, FIAA

RSFAS Summer Camp2 December 2015

Reverse MortgagesReverse Mortgageslife expectancy and dependency

ratio continue to risemany retirees are ‘asset-rich-

cash-poor’reverse mortgages can unlock

home equity and provide supplementary retirement funding

market is growing in Australia and Asia-Pacific

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Reverse MortgagesReverse Mortgagesit allows homeowner to borrow

against home property valueloan 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 hometypes / features are many and varied

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Industry BodiesIndustry BodiesSenior Australians Equity Release

(SEQUAL)Safe Home Income Plans (SHIP)

in UKHome Equity Conversion

Mortgage (HECM) in US

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Underlying RisksUnderlying Riskslongevity riskhouse price riskinterest rate riskmis-sellingfraud legal riskoperational risk

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Current RegulationsCurrent RegulationsSolvency 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

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Two Valuation ApproachesTwo Valuation Approaches

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Risk-Neutral ValuationRisk-Neutral Valuationmarket is complete if there are

many securities being tradedany cash flow can be replicated by

dynamic hedging strategiesno-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

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Current Life MarketCurrent Life Marketthere 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

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Risk-Neutral ValuationRisk-Neutral Valuationif 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 1994)

e.g. Wang transform (Wang 2000)

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Xx exfexf

xFxF 1

Risk-Neutral MeasuresRisk-Neutral Measuresthese two transforms have

decent propertiessubjective 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

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Current LiteratureCurrent LiteratureAuthors 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

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Maximum Entropy Maximum Entropy PrinciplePrincipleminimise Kullback-Leibler

information criterion subject to m constraints of m

securities (hi = discounted payoff

vi = market price)subject to constraint

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dxxf xfxfln

ii vdxxfxh

1 dxxf

m

i ii xhxfxf10 exp

Maximum Entropy Maximum Entropy PrinciplePrinciplediscrete 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 pathssubject to m constraints of m

securities Σhi,jπj* = vi

subject to constraint Σπj* = 1

Lagrange multiplier

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Maximum Entropy Maximum Entropy PrinciplePrincipleany number of security prices

can be incorporateddifferent simulation methods can

be usedit can be applied to different risks

consistently and coherentlythere are some theoretical and

empirical support

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Maximum Entropy Maximum Entropy PrinciplePrincipleFrittelli (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 Black-Scholes estimator

it has been applied to other futures options

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Longevity RiskLongevity RiskLee and Carter (1992) modelln(mx,t) = ax + bx kt mx,t = central death rateax = mortality schedulebx = age-specific sensitivity kt = mortality indexsimple model structurehighly linear kt empiricallyfit ARIMA(0,1,0) to kt for projection

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Financial RisksFinancial Riskshouse price risk and interest rate riskvector autoregressive (VAR) modele.g. VAR(1) model

rtH = c1 + A1,1 rt-1

H + A1,2 rt-1I + e1,t

rtI = c2 + A2,1 rt-1

H + A2,2 rt-1I + e2,t

simple model structure for autoregressive effects

assume real-world independence between longevity risk and financial risks

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Reverse Mortgage Reverse Mortgage ExampleExamplesuppose 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 rateHt = house priceIt = proportion who dies within (t-1 , t)d(t) = discount factorEPV must be larger than L0 to make it

financially viable19

Two Valuation ApproachesTwo Valuation Approaches

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Historical DataHistorical Datafor fitting LC and VAR modelsAustralian 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

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Market Prices DataMarket Prices Datafor setting constraintsAustralian 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

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Preliminary ResultsPreliminary Results

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Preliminary ResultsPreliminary Results

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Preliminary ResultsPreliminary Results

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ReferencesReferences 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

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ReferencesReferences 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

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ReferencesReferences Li J.S.H., Hardy M.R., and Tan K.S., 2010, On pricing and

hedging the no-negative-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

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