Introduction Modelling A two-decrement model for the valuation and framework Derivation of risk measurement of a guaranteed annuity GAO prices Risk mea- option surement of GAO Conclusion Yixing Zhao Department of Statistical & Actuarial Sciences The University of Western Ontario London, Ontario, Canada Joint work with Rogemar Mamon ( Western ), Huan Gao ( Bank of Montreal ) 52nd Actuarial Research Conference 26 – 29 July 2017, Atlanta, Georgia, USA A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 1 / 41 Yixing Zhao
Outline Introduction Introduction 1 Modelling framework Derivation of GAO prices Modelling framework 2 Risk mea- surement of GAO 3 Derivation of GAO prices Conclusion 4 Risk measurement of GAO 5 Conclusion A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 2 / 41 Yixing Zhao
Section Outline Introduction Modelling framework Derivation of GAO prices Risk mea- surement of Introduction GAO 1 Conclusion A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 3 / 41 Yixing Zhao
Research motivation Introduction Financial innovations – response to increased longevity and Modelling framework ageing population. Derivation of GAO prices Risk mea- Insurance market is becoming an investment hub. surement of GAO Conclusion Interest and mortality risks – primary factors in valuation and risk management of longevity products. But, lapse risk is also very important. Lapse risk – possibility that policyholders terminate their policies early ... for various reasons. Dire consequences from policy lapses – huge losses and liquidity problem for insurance companies. A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 4 / 41 Yixing Zhao
Research motivation (cont’d) Introduction Modelling framework Derivation of In current practice, lapse rate is assumed constant or GAO prices deterministic in actuarial valuation. Risk mea- surement of GAO Conclusion Research advances on lapse risk modelling are rather slow, unlike those for interest and mortality dynamics. Policyholders’ decision to surrender is directly affected by economic circumstances. A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 5 / 41 Yixing Zhao
Objectives Introduction Develop an integrated approach that addresses simultaneously Modelling framework guaranteed annuity option (GAO)’s pricing and capital Derivation of requirement calculation. GAO prices Risk mea- surement of Construct a two-decrement stochastic model in which death GAO and policy lapse occurrences with their correlations to the Conclusion financial risk are explicitly modelled. Apply series of probability measure changes resulting to forward, survival, and risk-endowment measures. Determine risk measures using moment-based density method and results benchmarked with the Monte-Carlo simulation. Our forumulation highlights the link between pricing and capital requirement. A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 6 / 41 Yixing Zhao
Section Outline Introduction Modelling framework Interest rate model Mortality model Lapse rate model Modelling framework 2 Valuation framework Interest rate model Derivation of GAO prices Mortality model Risk mea- surement of GAO Lapse rate model Conclusion Valuation framework A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 7 / 41 Yixing Zhao
Interest rate model Introduction Modelling framework Interest rate model Mortality model Lapse rate model Valuation framework We assume short-interest rate r t follows the Vasiček model via the Derivation of SDE GAO prices dr t = a ( b − r t ) dt + σdX t , Risk mea- surement of GAO where a, b, and σ are positive constants and X t is a standard Conclusion one-dimensional Brownian motion. A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 8 / 41 Yixing Zhao
Interest rate model (cont’d) Introduction Modelling framework Price B ( t,T ) of a T -maturity zero-coupon bond at time t < T is Interest rate model known to be Mortality model Lapse rate model Valuation � T framework B ( t,T ) = E Q [ e − t r u du |F t ] = e − A ( t,T ) r t + D ( t,T ) , Derivation of GAO prices Risk mea- surement of where GAO A ( t,T ) = 1 − e − a ( T − t ) Conclusion a and � � b − σ 2 [ A ( t,T ) − ( T − t )] − σ 2 A ( t,T ) 2 D ( t,T ) = . 2 a 2 4 a A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 9 / 41 Yixing Zhao
Mortality model Introduction Modelling framework The dynamics of the force of mortality process µ t is given by Interest rate model Mortality model Lapse rate model dµ t = cµ t dt + ξd Y t , Valuation framework Derivation of where c and ξ are positive constants, and Y t is a standard GAO prices Brownian motion correlated with X t , Risk mea- surement of GAO dX t d Y t = ρ 12 dt. Conclusion The survival function is defined by S ( t,T ) = E Q � t µ u du � � � T � e − � F t . A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 10 / 41 Yixing Zhao
Lapse rate framework Introduction Modelling framework Interest rate model For the lapse rate process l t , we adopt the dynamics Mortality model Lapse rate model Valuation framework dl t = h ( m − l t ) dt + ζdZ t , Derivation of GAO prices Risk mea- surement of where h , m and ζ are positive constants and Z t is a standard GAO BM correlated with both X t and Y t . Conclusion In particular, dX t dZ t = ρ 13 dt and d Y t dZ t = ρ 23 dt. A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 11 / 41 Yixing Zhao
Valuation framework Introduction Modelling framework Interest rate model Let M d ( t,T ) be the fair value at time t of a pure endowment Mortality model Lapse rate model of $1 at maturity T when mortality is the only decrement, i.e., Valuation framework t µ u du � M d ( t,T ) = E Q � � � T � T � t r u du e − Derivation of e − � F t . GAO prices Risk mea- surement of GAO Let M τ ( t,T ) be the fair value at time t of a $1 pure Conclusion endowment at maturity T under a two-decrement model (both mortality and lapse rates are considered), i.e., M τ ( t,T ) = E Q � t l u du � � � T � T � T � e − t r u du e − t µ u du e − � F t . A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 12 / 41 Yixing Zhao
Valuation framework(cont’d) Introduction Modelling framework Interest rate model Mortality model Lapse rate model Define a x ( T ) as the annuity rate. A life annuity is a contract that Valuation framework pays $1 to an insured annually conditional on his/her survival at the Derivation of moment of payments. GAO prices Risk mea- surement of That is, GAO E Q � µ u du � � Conclusion ∞ ∞ � T + n � T + n � r u du e − M d ( T,T + n ) . ∑ e − ∑ a x ( T ) = � F T = T T n =0 n =0 A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 13 / 41 Yixing Zhao
Valuation framework (cont’d) Introduction GAO is a contract that gives the policyholder the right to Modelling framework convert a survival benefit into an annuity at a pre-specified Interest rate model guaranteed conversion rate g . Mortality model Lapse rate model Valuation framework GAO’s loss function L is the payoff ‘discounted’ by mortality Derivation of GAO prices and lapse factors, i.e., Risk mea- surement of � T � T 0 l u du ( a x ( T ) − K ) + , GAO L = ge − 0 µ u du e − Conclusion where K = 1 /g . The fair value of GAO at time 0 , by risk-neutral pricing, is P GAO = g E Q � 0 l u du ( a x ( T ) − K ) + � � � T � T � T � 0 r u du e − 0 µ u du e − e − � F 0 . A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 14 / 41 Yixing Zhao
Section Outline Introduction Modelling framework Derivation of GAO prices Derivation of GAO prices 3 The forward measure The survival The forward measure measure The endowment- risk-adjusted The survival measure measure Numerical implementation The endowment-risk-adjusted measure Risk mea- surement of Numerical implementation GAO Conclusion A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 15 / 41 Yixing Zhao
The forward measure Introduction Modelling framework Derivation of The forward measure � GAO prices Q is constructed with the aid of the The forward measure Radon-Nikodˆ ym derivative and Girsanov’s theorem, and in The survival measure particular � � T The endowment- � d � 0 r u du B ( T,T ) risk-adjusted T := e − Q � measure = Λ 1 � . Numerical � dQ B (0 ,T ) implementation F T Risk mea- surement of GAO The dynamics of µ t under � Q is given by Conclusion dµ t = [ − ρ 12 σξA ( t,T )+ cµ t ] dt + ξd � Y t . A two-decrement model for the valuation and risk measurement of a guaranteed annuity option 16 / 41 Yixing Zhao
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