Optimal investment strategies and intergenerational risk sharing for - PowerPoint PPT Presentation

Outline Optimal investment strategies and intergenerational risk sharing for target benefit pension plans Barbara Sanders with Suxin Wang and Yi Lu Department of Statistics and Actuarial Science Simon Fraser University July 29, 2017

Outline Outline Introduction 1 TBP Model 2 Model formulation Solution to the optimization problem Illustrations 3 Conclusion 4 Barbara Sanders Simon Fraser University July 2017 2 / 26

Introduction TBP model Illustrations Conclusion Target Benefit Plans Key features: Predefined contribution level Sponsor liability limited to contributions Target benefit level Actual benefits vary Collective asset pool Members bear risk collectively Barbara Sanders Simon Fraser University July 2017 3 / 26

Introduction TBP model Illustrations Conclusion Target Benefit Plans Key features: Predefined contribution level Sponsor liability limited to contributions Target benefit level Actual benefits vary Collective asset pool Members bear risk collectively Barbara Sanders Simon Fraser University July 2017 3 / 26

Introduction TBP model Illustrations Conclusion Target Benefit Plans Practical objectives: Provide adequate benefits Maintain stability Respect intergenerational equity Key question: Given some starting asset value and contribution commitment, how should assets be invested and benefits be paid out to achieve these goals? Barbara Sanders Simon Fraser University July 2017 4 / 26

Introduction TBP model Illustrations Conclusion Target Benefit Plans Practical objectives: Provide adequate benefits Maintain stability Respect intergenerational equity Key question: Given some starting asset value and contribution commitment, how should assets be invested and benefits be paid out to achieve these goals? Barbara Sanders Simon Fraser University July 2017 4 / 26

Introduction TBP model Illustrations Conclusion Stochastic optimization in pension literature DB optimization: asset mix and contribution rate Cairns (1996, 2000), Haberman and Sung (2004), Josa-Fombellida and Rincon-Zapatero (2001, 2004, 2008), Ngwira and Gerard (2007), etc. DC optimization: asset mix and payout pattern Gerrard et al. (2004), He and Liang (2013, 2015), etc. Gollier (2008): asset mix, benefit payout, dividend policy Cui et al . (2011): asset mix, contribution rate, benefit payout Barbara Sanders Simon Fraser University July 2017 5 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Dynamics of financial market Risk-free asset S 0 ( t ) d S 0 ( t ) = r 0 S 0 ( t ) d t , t ≥ 0 , where r 0 represents the risk-free interest rate. Risky asset S 1 ( t ) d S 1 ( t ) = S 1 ( t )[ µ d t + σ d W ( t )] , t ≥ 0 , where µ is the appreciation rate of the stock, σ is the volatility rate, and W ( t ) is a standard Brownian motion. Barbara Sanders Simon Fraser University July 2017 6 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Plan membership The fundamental elements of demographic model: n ( t ) : density of new entrants aged a at time t , s ( x ) : survival function with s ( a ) = 1 and a ≤ x ≤ ω . The density of those who attain age x at time t is n ( t − ( x − a )) s ( x ) , x > a . Barbara Sanders Simon Fraser University July 2017 7 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Salary process Dynamics of salary rate for a member who retires at time t : � � d L ( t ) = L ( t ) α d t + η d W ( t ) , t ≥ 0 , where α ∈ R + and η ∈ R . W is a standard Brownian motion correlated with W , such that E [ W ( t ) W ( t )] = ρ t . For a retiree age x at time t ( x ≥ r ), define assumed salary at retirement ( x − r years ago) as � L ( x , t ) = L ( t ) e − α ( x − r ) , t ≥ 0 , x ≥ r . Barbara Sanders Simon Fraser University July 2017 8 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion The time-age structure of the pension plan Barbara Sanders Simon Fraser University July 2017 9 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Contribution process Individual contribution rate for active member aged x at time t ≥ 0: C ( x , t ) = c 0 ( x ) e α t , a ≤ x < r . Aggregate contribution rate in respect of all active members at time t : � r C ( t ) = n ( t − x + a ) s ( x ) C ( x , t ) d x , t ≥ 0 . a Barbara Sanders Simon Fraser University July 2017 10 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Benefit payment process Individual pension payment rate at time t : for a new retiree aged r : B ( r , t ) = f ( t ) L ( t ) for an existing retiree aged x > r : B ( x , t ) = f ( t ) � L ( x , t ) e ζ ( x − r ) = f ( t ) L ( t ) e − ( α − ζ )( x − r ) Barbara Sanders Simon Fraser University July 2017 11 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Benefit payment process Individual pension payment rate at time t : for a new retiree aged r : B ( r , t ) = f ( t ) L ( t ) for an existing retiree aged x > r : B ( x , t ) = f ( t ) � L ( x , t ) e ζ ( x − r ) = f ( t ) L ( t ) e − ( α − ζ )( x − r ) Barbara Sanders Simon Fraser University July 2017 11 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Benefit payment process Aggregate pension benefit rate for all retirees at time t : � ω B ( t ) = n ( t − x + a ) s ( x ) B ( x , t ) d x = I ( t ) f ( t ) L ( t ) , t ≥ 0 . r The updated aggregate target benefit is B ∗ e β t . Barbara Sanders Simon Fraser University July 2017 12 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Fund dynamics The pension fund dynamic can be described as � d X ( t ) = π ( t ) d S 1 ( t ) S 1 ( t ) + ( X ( t ) − π ( t )) d S 0 ( t ) S 0 ( t ) + ( C ( t ) − B ( t )) d t , X ( 0 ) = x 0 . Barbara Sanders Simon Fraser University July 2017 13 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion The objective function Let J ( t , x , l ) be the objective function at time t with the fund value and the salary level being x and l . It is defined as � � T  �� B ( s ) − B ∗ e β s �� B ( s ) − B ∗ e β s � 2 − λ 1 �  e − r 0 s d s  J ( t , x , l ) = E π, f  t  � � X ( T ) − x 0 e r 0 T � 2 e − r 0 T + λ 2 ,     � X ( T ) − x 0 e r 0 T � 2 e − r 0 T . J ( T , x , l ) = λ 2 The value function is defined as φ ( t , x , l ) := ( π, f ) ∈ Π J ( t , x , l ) , t , x , l > 0 . min See Ngwira and Gerrard (2007), He and Liang (2015). Barbara Sanders Simon Fraser University July 2017 14 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Using variational methods and It ˆ o ’s formula, we get the following HJB equation satisfied by the value function φ ( t , x , l ) : � � � r 0 x + ( µ − r 0 ) π + C 1 ( t ) e α t − fl · I ( t ) φ t + φ x + α l φ l min π, f � � fl · I ( t ) − B ∗ e β t � 2 + 1 2 π 2 σ 2 φ xx + 1 2 η 2 l 2 φ ll + ρση l πφ xl + fl · I ( t ) − B ∗ e β t � � � � e − r 0 t − λ 1 = 0 . Barbara Sanders Simon Fraser University July 2017 15 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion Solution to the optimization problem Optimal strategies are π ∗ ( t , x , l ) = − δ 2 σ [ 2 x + Q ( t )] , � λ 1 � 1 2 + λ 2 2 ( 2 x + Q ( t )) P ( t ) + B ∗ e β t f ∗ ( t , x , l ) = , l · I ( t ) where δ = ( µ − r 0 ) /σ is the Sharp Ratio. The corresponding value function is given by φ ( t , x , l ) = λ 2 e − r 0 t P ( t )[ x 2 + xQ ( t )] + K ( t ) . Barbara Sanders Simon Fraser University July 2017 16 / 26

Introduction TBP model Model formulation Illustrations Solution to the optimization problem Conclusion   1 r 0 = δ 2 , λ 2 ( T − t )+ 1 , P ( t ) = r 0 − δ 2 r 0 � = δ 2 ,  λ 2 +( r 0 − δ 2 − λ ) e − ( r 0 − δ 2 )( T − t ) ,  2 e r 0 t �� T �  t C 1 ( s ) e ( α − r 0 ) s d s − B ∗ ( T − t ) − x 0 , β = r 0 ,  �� T � Q ( t ) = t C 1 ( s ) e ( α − r 0 ) s d s − B ∗ ( e ( β − r 0 ) T − e ( β − r 0 ) t )  2 e r 0 t − x 0 , β � = r 0 ,  β − r 0 � � � T C 1 ( s ) e α s − B ∗ e β s e − r 0 t K ( t ) = λ 2 P ( s ) Q ( s ) t � � � � − λ 2 − 1 δ 2 + λ 2 P ( s ) 1 Q ( s ) d s . 4 4 Barbara Sanders Simon Fraser University July 2017 17 / 26