Ay¸ se Arık Introduction Motivation Pricing of Pension Bulk Annuities Pricing Pension Buy-outs Integrating the Models Mortality Models Ays ¸e Arık, S ¸ule S ¸ahin and Yeliz Yolcu Okur Interest Rate Model Application Case Study I Hacettepe University, Ankara/Turkey Case Study II Numerical Results 07 September-09 September 2015 Conclusion
Content Introduction Ay¸ se Arık 1 Introduction Motivation 2 Motivation Pricing Pension Buy-outs 3 Pricing Pension Integrating the Models Buy-outs Integrating the Mortality Models Models Mortality Models Interest Rate Model Interest Rate Model Application Application 4 Case Study I Case Study II Case Study I Numerical Results Case Study II Conclusion Numerical Results 5 Conclusion 6
Introduction Ay¸ se Arık Introduction Motivation Difficulties to cope with the unprecedented levels of DB Pricing pension liabilities Pension Buy-outs Asset-liability management or De-risking strategies Integrating the Models Mortality Models Pension buy-in and buy-out deals Interest Rate Model Application A benchmark pricing model based on the risk neutral Case Study I market framework and independence assumption for Case Study II Numerical pension bulk annuities (Lin et al., 2014) Results Conclusion
Motivation Ay¸ se Arık Introduction Investigating the general set-up and assumptions to Motivation enhance the pricing mechanism of the pension buy-outs Pricing Pension Buy-outs Integrating the Models Mortality Models Interest Rate Model Application Case Study I Case Study II Numerical Results Conclusion
Motivation Ay¸ se Arık Introduction Investigating the general set-up and assumptions to Motivation enhance the pricing mechanism of the pension buy-outs Pricing Pension Discussion on the independence assumption between the Buy-outs biometric events and the financial market by Miltersen and Integrating the Models Persson (2006); Bauer et al. (2008); Jalen and Mamon Mortality Models Interest Rate Model (2009); Hoem et al.(2009) and Neyer et al. (2012) . Application Case Study I Case Study II Numerical Results Conclusion
Motivation Ay¸ se Arık Introduction Investigating the general set-up and assumptions to Motivation enhance the pricing mechanism of the pension buy-outs Pricing Pension Discussion on the independence assumption between the Buy-outs biometric events and the financial market by Miltersen and Integrating the Models Persson (2006); Bauer et al. (2008); Jalen and Mamon Mortality Models Interest Rate Model (2009); Hoem et al.(2009) and Neyer et al. (2012) . Application Case Study I Aim of the Study Case Study II Numerical Pricing pension buy-outs under dependence assumption of Results financial and insurance markets Conclusion
A Proposed Model for Pricing of the Pension Buy-outs under the Dependence Assumption Ay¸ se Arık A combined modelling framework (Ω , G , ( G t ) , P ) s.t. Introduction G t = M t ∨ F t Motivation M t is the filtration of mortality process µ and F t is the Pricing filtration of short rate process r . Pension Buy-outs Integrating the Models Mortality Models Interest Rate Model Application Case Study I Case Study II Numerical Results Conclusion
A Proposed Model for Pricing of the Pension Buy-outs under the Dependence Assumption Ay¸ se Arık A combined modelling framework (Ω , G , ( G t ) , P ) s.t. Introduction G t = M t ∨ F t Motivation M t is the filtration of mortality process µ and F t is the Pricing filtration of short rate process r . Pension Buy-outs The deal guarantees to pay the pension liabilities and Integrating the Models Mortality Models compensate any potential asset-liability mismatching. Interest Rate Model Application Case Study I Case Study II Numerical Results Conclusion
A Proposed Model for Pricing of the Pension Buy-outs under the Dependence Assumption Ay¸ se Arık A combined modelling framework (Ω , G , ( G t ) , P ) s.t. Introduction G t = M t ∨ F t Motivation M t is the filtration of mortality process µ and F t is the Pricing filtration of short rate process r . Pension Buy-outs The deal guarantees to pay the pension liabilities and Integrating the Models Mortality Models compensate any potential asset-liability mismatching. Interest Rate Model Adopt the suggested model by Jalen and Mamon (2009) to Application Case Study I state the liability process of a hypothetical pension scheme Case Study II Numerical Results Conclusion
A Proposed Model for Pricing of the Pension Buy-outs under the Dependence Assumption Ay¸ se Arık A combined modelling framework (Ω , G , ( G t ) , P ) s.t. Introduction G t = M t ∨ F t Motivation M t is the filtration of mortality process µ and F t is the Pricing filtration of short rate process r . Pension Buy-outs The deal guarantees to pay the pension liabilities and Integrating the Models Mortality Models compensate any potential asset-liability mismatching. Interest Rate Model Adopt the suggested model by Jalen and Mamon (2009) to Application Case Study I state the liability process of a hypothetical pension scheme Case Study II Numerical Consider the difference between asset and liability Results processes as one year put option spreads where the strike Conclusion prices are defined according to the pension liabilities on the valuation dates as offered by Lin et al. (2014)
Model Framework Ay¸ se Arık A new model to derive the price of a pure endowment Introduction policy under the dependence assumption using the Motivation change of measure technique by Jalen and Mamon (2009) Pricing Pension The price of the pure endowment policy Buy-outs Integrating the Models Mortality Models � T Interest Rate Model B S ( t , T , C T ) = E [ exp ( − r ( s ) ds ) 1 τ> T C T |G t ] Application t Case Study I � T Case Study II = 1 τ> t E [ exp ( − ( r ( s ) + µ ( s , x + s )) ds ) C T |G t ] Numerical Results t Conclusion turns into the following formula:
Model Framework (Continued) Ay¸ se Arık Introduction p ( t , T , x )˜ E T [ C T |G t ] B S ( t , T , C T ) = 1 τ> t B ( t , T )˜ (1) Motivation Pricing Pension where Buy-outs Integrating the B S ( t , T , C T ) represents the present value of the contract Models 1 Mortality Models with a variable survival benefit C T at the end of the Interest Rate Model Application maturity T . Case Study I � T Case Study II B ( t , T ) = E Q [ exp ( − t r ( s ) ds )] 2 Numerical � T Results p ( t , T , x ) = E T [ exp ( − ˜ 0 µ ( s , x + s ) ds )] 3 Conclusion From Bayes’ rule E T [ H |G T ] is the expectation under the 4 forward measure P T , for a contingent claim H
Model Framework (Continued) Ay¸ se Arık E T [ H |G T ] could be written more explicitly as below: Introduction Motivation � T E T [ H |G T ] = E [ exp ( − t r ( s ) ds ) H |G T ] Pricing Pension B ( t , T ) Buy-outs Integrating the Models The Radon-Nikodym derivative of P T with respect to the Mortality Models Interest Rate Model risk-neutral measure Q as Application Case Study I Case Study II Numerical � T d P T Results d Q |G T = Λ 0 , T = exp ( − 0 r ( s ) ds ) Conclusion B (0 , T )
Model Framework (Continued) Adopt B S ( t , T , C T ) to derive the liability process of the Ay¸ se Arık pension scheme for a constant benefit amount C , i.e. Introduction Motivation B S ( t , t i , C ) = 1 t i > t B ( t , t i , r t ) E t i [ C × exp ( − � t i Pricing t µ ( s , x + s ) ds ) Pension |G t i ] Buy-outs Integrating the Models Mortality Models where Interest Rate Model Application Case Study I Case Study II � t i d P t i d Q = Λ 0 , t i = exp ( − 0 r ( s ) ds ) Numerical Results B (0 , t i , r t ) Conclusion a ( t , C ) = � T t i = t +1 B S ( t , t i , C ) L t = N ( t ) × a ( t , C )
Model Framework (Continued) Ay¸ se Arık The similar asset process suggested by Lin et al. (2014) Introduction may be applied, i.e. Motivation Pricing Pension Buy-outs 3 t = ( r − 1 Integrating the dlogPA ∗ 2 σ 2 � W ( t )) dt + π i ( t ) σ i dW it (2) Models Mortality Models i =1 Interest Rate Model Application Case Study I r is the risk free rate. 1 Case Study II W ( t ) = � 3 σ 2 i , j =1 π i ( t ) π j ( t ) ρ ij σ i σ j where ρ ij is the 2 Numerical Results correlation coefficient between asset i and j ′ are the weights of the Conclusion π ( t ) = ( π 1 ( t ) , π 2 ( t ) , π 3 ( t )) 3 portfolio at time t ( Fernholz (2002) ).
Model Framework (Continued) Ay¸ se Arık The risk-neutral price of the payoff associated with both Introduction risks, conditioning on N ( t ), is given by Motivation Pricing Pension Buy-outs T Integrating the � e − rt E Q [( L t − PA t ) + | N ( t )] − v ( τ N +1) E Q [ PA τ N +1 ](3) V (0 , T ) = Models Mortality Models t =1 Interest Rate Model Application Case Study I The buy-out price under the dependence assumption Case Study II Numerical Results P buyout = E M [ V (0 , T )] Conclusion L 0
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