Intergenerational Risk Sharing Pension Xiaobai Zhu, Mary Hardy, - PowerPoint PPT Presentation

Intergenerational Risk Sharing Pension Xiaobai Zhu, Mary Hardy, David Saunders University of Waterloo x32zhu@uwaterloo.ca September 8, 2018 Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 1 / 18

Overview Introduction 1 Optimal Pension Design Structure 2 Regulatory Constraint 3 Conclusion and Future Work 4 Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 2 / 18

Traditional Pension Design Defined Benefit (DB) Plan Provides retirees the “defined” monthly income based on their past salary and years of service. Example: Income = Final Salary × Years in Service × Accural Rate Sponsor bears all risks! Defined Contribution (DC) Plan Employee makes contribution into a tax-deferred investment account. Employer matches the employee contribution. Employee bears all risks! Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 3 / 18

Hybrid Pension Plans Risk-sharing between employees and sponsors Cash Balance Pension Plan. Defined Benefit Underpin Plan. Second-Election Option. Risk-sharing between generations Target Benefit. Collective Defined Contribution Plan. More Proposed IRS plan: Cui et al. (2011), Wang et al. (2018), Goecke (2013), Bams et al. (2016), Khorasanee (2012), Gollier (2008), Boes and Siegmann (2016), Bovenberg and Mehlkopf (2014), etc. Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 4 / 18

Assumptions and Problem Formulation Fixed Retirement Date (R) and Fixed Death Time (N). A unit of population for each age. ex. Number of active workers is R. Salary is assumed to be 1. All contribution are made by the employees. The sponsor set a benchmark contribution level c , and the benchmark benefit b and liability L are calculated through actuarial equivalence principle: b = c × ¯ a R R | ¯ a N − R L = b ( N − R ) − cR r Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 5 / 18

Assumptions on Market Market consists of a risk-free asset and a risky asset. Risk-free asset S 0 ( t ): dS 0 ( t ) = rS 0 ( t ) dt Risky asset S 1 ( t ): dS 1 ( t ) = µ S 1 ( t ) dt + σ S 1 ( t ) dB t where B t is the standard Brownian Motion. Pension asset level X t has the SDE: dX t = ( X t ( r + ω t ( µ − r )) + Rc t − ( N − R ) b t ) dt + σω t X t dB t Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 6 / 18

Optimal IRS Structure � � U ( x ) = x 1 − γ Welfare function - sum of individual utility 1 − γ , γ > 1 . �� ∞ � + ( N − R ) b 1 − γ � � R (1 − c t ) 1 − γ t sup E dt 1 − γ 1 − γ ω t , b t , c t 0 Stability of consumption - squared distance from target consumption. �� T � 1 R (1 − c t − c w ) 2 + ( N − R )( b t − c r ) 2 dt ω t , b t , c t lim inf T E T →∞ 0 with c w and c r are the target consumption levels for active employees and retirees. Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 7 / 18

Theoretical Results Welfare Function Stability of Consumption � � � � = µ − r t = µ − r ω W R ω S − 1 + C Portfolio ˆ rX t + 1 ˆ t σ 2 γ σ 2 rX t = c − α W � X t − ψ W � t = c − α S � X t − ψ S � w L w L c W c S Contribution ˆ ˆ t R R = b + β W � X t − ψ W � t = b + β S � X t − ψ S � ˆ r L ˆ r L b W b S Benefit t N − R N − R Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 8 / 18

Optimal IRS Design - Issues Welfare Function Stability of Consumption — Can be negative α β — Can be negative Can be negative Can be negative ψ w Can be negative Can be negative ψ r � X t � lim t →∞ E + ∞ or −∞ Converges to Constant L Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 9 / 18

Optimal IRS design - Linear Risk-Sharing Structure We adopt the linear risk-sharing structure: � X t − ψ L � c t = c − α R � X t − ψ L � b t = b − β N − R The objective function is �� T � 1 �� R (1 − c t − c w ) 2 + ( N − R )( b t − c r ) 2 dt inf inf lim T α,β, c ω t T →∞ 0 Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 10 / 18

Optimal Portfolio Weight - Unconstrained Case Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 11 / 18

Regulation on Recovery Period When deficit occurs, regulation often require the pension fund to create a recovery plan. Funding level will by automatically recovered in an IRS plan. Here we define the recovery period for our IRS as: � t ∗ = inf { t : t > 0 , X t X 0 � L = f r L < f r } � � dX t = ( rX t − N (1 − Ω) c t + N Ω b t ) dt where all pension assets are invested in risk-free bond. Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 12 / 18

Regulation on Recovery Period The constraint can be expressed as t ∗ ( ξ ), where ξ = α + β , and we are trying to find ξ ∗ such that t ∗ ( ξ ∗ ) − t r = 0, with t r being the required recovery time. To ensure the uniqueness of solution, we define ξ ∗ as � � � ξ ∗ = max � ξ ′ � t ∗ ( ξ ′ ) = t r , ( t ∗ ( ξ ) < t r , ∀ ξ > ξ ′ ) � The constrained parameter domain is simply � ξ ∗ ≤ α + β ≤ 1 , α ≥ 0 , β ≥ 0 � � � ( α, β ) . Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 13 / 18

Optimal Portfolio Weight - Constrained Case Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 14 / 18

Sensitivity Tests for different c r and c w 0.115 0.026 0.114 0.113 0.024 0.112 0.022 0.111 1.2 1.2 1.2 1.1 1.2 1.1 1.1 1.1 1 1 1 1 0.9 0.9 c r 0.9 0.9 c r c w c w p Long Term Funding Level 0.3 1.4 0.25 1.3 1.2 0.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1 1 1 1 0.9 0.9 0.9 0.9 c r c r c w c w Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 15 / 18

Sensitivity Tests for different r and µ p 0.25 0.6 0.5 0.2 0.4 0.15 0.3 0.1 0.1 0.1 0.2 0.05 0.1 0 0.1 0.08 0.05 0.1 0.05 0.06 0.08 0.04 0.06 0.02 0.04 0.02 r r Long Term Funding Level 0.5 1.3 0.4 1.25 0.3 1.2 0.2 0.1 1.15 0.1 0.1 0 1.1 0.1 0.08 0.06 0.04 0.05 0.05 0.1 0.08 0.02 0.06 0.04 0.02 r r Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 16 / 18

Optimality of Pure DB and DC plans (a) c w = 0 . 9, c r = 0 . 9, r = 0 . 03 (b) c w = 1 . 1, c r = 1 . 1, r = 0 . 03 Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 17 / 18

Conclusion and Future Work Conclusion We provide theoretical justification of the linear risk-sharing structure. We illustrate the necessity of incorporating regulatory constraint. Collective DC may not be advantageous against DB or DC plan. Future Work Modelling changes in population structure. Incorporate constraints directly in optimal control problem. The responsibility of sponsor. More realistic pension structure. Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 18 / 18

Bams, D., Schotman, P. C., and Tyagi, M. (2016). Optimal risk sharing in a collective defined contribution pension system. Boes, M.-J. and Siegmann, A. (2016). Intergenerational risk sharing under loss averse preferences. Journal of Banking & Finance . Bovenberg, L. and Mehlkopf, R. (2014). Optimal design of funded pension schemes. Annu. Rev. Econ. , 6(1):445–474. Cui, J., De Jong, F., and Ponds, E. (2011). Intergenerational risk sharing within funded pension schemes. Journal of Pension Economics & Finance , 10(1):1–29. Goecke, O. (2013). Pension saving schemes with return smoothing mechanism. Insurance: Mathematics and Economics , 53(3):678–689. Gollier, C. (2008). Intergenerational risk-sharing and risk-taking of a pension fund. Journal of Public Economics , 92(5-6):1463–1485. Khorasanee, Z. M. (2012). Risk-sharing and benefit smoothing in a hybrid pension plan. North American Actuarial Journal , 16(4):449–461. Wang, S., Lu, Y., and Sanders, B. (2018). Optimal investment strategies and intergenerational risk sharing for target benefit pension plans. Insurance: Mathematics and Economics . Xiaobai Zhu, Mary Hardy, David Saunders (UW) EAJ September 8, 2018 18 / 18