Demographic risk sharing in overlapping generations: the case of - PDF document

Demographic risk sharing in overlapping generations: the case of DC pension funds Daniel Gabay, CNRS, EHESS-CAMS and ESILV Martino Grasselli, Univ. Padova and ESILV on the occasion of the 65th birthday of Wolfgang Runggaldier Brixen July 2007

With Wolfgang: • Professor and much more • Pa-Pa no-crossing principle • Pension funds: back to the future

Outline 1) Defined Benefit, Defined Contribution pen- sion funds 2) The classic actuarial approach: no market interaction and stationarity 3) The modern approaches for DB and DC 4) Demographic risk and generational overlap- ping: a delayed optimization problem 5) Optimal design of a DC pension scheme without market interaction 6) Adding the market: how can pension funds beat the market?

MOTIVATIONS 1) reduction of the birth rate + 2) longer average life + 3) expanded school period and consequent 4) delay in beginning the working-life ⇓ Crisis of the pay-as-you-go system ⇓ 3 PILLARS system:

• 1 st PILLAR : pay-as-you-go pension (France: ∼ 88% of total pension, about 68% in 2020) • 2 nd PILLAR: collective pension by capi- talization (France: ∼ 7,8% of total pension, should be 25% in 2020) • 3 rd PILLAR: individual insurance contracts (France: ∼ 0,9% of total pension, should be 3,3% in 2020)

2 nd PILLAR: PENSION FUNDS • Organisms established in order to assure benefits to workers, when they mature the rights provided for by the regulation. • How much money do they manage? 10,000 BILLIONS OF DOLLARS!!!! (WITH 10% INCREASE/YEAR, see Boulier and Dupr´ e1999)

DB and DC pension funds • Defined benefit plans: solution preferred by workers, while the spon- soring employers will face the “ contribution rate risk ” • Defined contribution plans: solution preferred by the corporate, because the investment’s risk is totally charged to the beneficiary (“ solvency risk ”) NOWADAYS: • Defined Contribution with minimum guar- antee

Actuarial approach: no mar- ket interaction • It is not possible for the trustee to directly allocate the fund wealth into the financial market. • Asset Liability Management: equilibrium between present value of contributions and present value of liabilities

Standard stationarity assumptions on : • exogenous fund returns (i.i.d. or Wilkie 1987), • mortality rate • demographic variables (growth of popula- tion, salary..) ↓ stability of the pension fund wealth. Haberman (1993a) and (1993b), Haberman (1994) and Haberman and Zimbidis (1993), Dufresne (1989), Cairns (1995), Cairns (1996) and Cairns and Parker (1996)

Modern approach: interplay between Finance and Insur- ance • If the trustees of the fund have the possi- bility to invest in a portfolio, fund returns depend on the funding method adopted (Boulier, Florens, Trussant 1995). • When market fall, also interest rates de- crease, then fund wealth decreases and li- ability increase, so that ALM has strong interdependence with the allocation strat- egy (Martellini 2003)

The DC scheme with minimum guarantee What is the role of the guarantee? 0 ≤ E Q ( G T ) ≤ interest rate • It is a way to shift the risk from the con- tributor to the manager of the fund • It is a way to transform a DC in a DB • It is a way to introduce attractive payoffs

Literature: • Boulier, Huang and Taillard (2001): opti- mal dynamic allocation with deterministic guarantee and Vasicek interest rates (finite horizon and CRRA utility) • Deelstra, Grasselli and Koehl (2003): op- timal dynamic allocation with determinis- tic guarantee and CIR interest rates (finite horizon and CRRA utility) • Di Giacinto, Gozzi (2006): optimal dy- namic allocation with deterministic guar- antee (infinite time span and general util- ity) • CPPI-OBPI based strategies: Pringent (2003), El Karoui, Jeanblanc, Lacoste (2003) (also in the American guarantee)

Guarantee on the entire wealth path: Boyle and Imai (2000), Gerber and Pafumi (2000), El Karoui, Jeanblanc and Lacoste (2001) Separation result: Optimal strategy = strategy without guarantee +continuum of American put options • Superreplication constraint is too strong (the Lagrange multiplier is a process: contin- uous time almost sure constraint) Quantile approach is perhaps better: Pr( F t ≥ target) ≥ 1 − α

Is there an optimal guarantee? Optimal for whom? ACTORS: • 1 manager Jensen and Sørensen (2000): the presence of the guarantee is an obstacle = ⇒ no guarantee!! • 1 manager + 1 client Deelstra, Grasselli and Koehl (2003): necessity to specify how the fund surplus will be shared!! ( F π T − G T ) = Fund wealth - Guarantee (1 − β ) ( F π T − G T ) + β ( F π = T − G T )

Mortality risk Insurance companies face this risk due to huge changes in mortality tables together with low interest rates (then increase in the liabilities) • Haberman et al. (2005-2007): convert an- nuity into lumpsum and vice versa (pension funds typically hedge mortality risk by del- egating insurance companies) • Battocchio, Menoncin, Scaillet (2004), Menoncin (2005), Menoncin, Scaillet (2005) • 1 ”representative” client, who works and contributes during [0, a ] (Accumulation Phase) and keeps pension till his death [ a , τ ] (De- cumulation Phase)

• Mortality risk modelled through a deter- ministic distribution function for τ (Gom- pertz) ⇒ classic Merton trading strategy (see also PhD dissertation of Nicolas Rousseau, 1999)

IN FACT: El Karoui, Martellini (2001), Bouchard and Pham (2004), Zitkovic (2005),Blanchet-Scalliet, El Karoui, Jeanblanc and Martellini (2003) Utility maximization strategies with random hori- zon can be different from Merton’s strategies only if the random time distribution is corre- lated with the market!!

Demographic risk in a DC scheme • Colombo, Haberman (2005): Demographic risk due to stochastic entry process (opti- mal contribution rate in a DB scheme with- out market!!) • Menoncin (2005) ”Cyclical risk exposure of pension funds: a theoretical framework”: Demographic risk for a PAYG pension fund (no generational overlapping!!) • Gollier (2002), Demange and Rochet (2001) repartition vs. capitalization!! ⇓ Almost no literature!

Why? • DC schemes have been introduced to re- move demographic risk typical of PAYG systems!!

OPEN QUESTIONS AND MOTIVATIONS • 1) Is there a ”representative” client in a pension fund? (overlapping generations..) • 2) What are the Accumulation and Decu- mulation Phases for a pension fund? • 3) How to distinguish the ”mortality” (longevity) and ”demographic” (fluctuations of global contribution) risk? • 5) Which is the advantage to enter a pen- sion fund w.r.t. invest directly into the market? (apart from taxes, legal and ad- ministrative incentives..)

A SIMPLE DEMOGRAPHIC RISK FRAMEWORK: = − a t = 0 t = T t • At time t = − a the first clients (paying contributions) enter the fund • The number of clients entering the fund is described by a stochastic process c t includ- ing inflation and demographic fluctuations • At time 0 the first clients receive the pen- sion (lumpsum!! annuity ⇔ longevity risk!!) ⇒ [ − a, 0] = FUND ”APh”

• At (a random) time T there are no more clients paying contributions ( T could be correlated with the market and fund per- formance!!) • At time T + a the last clients (entered at time T ) receive the (last lumpsum) pension ⇒ [ T, T + a ] = FUND ”DPh” • We focus on the FUND TRANSITORY PHASE [0 , T ]: at each time there are new entries and pensions to be paid out!!

THE FUND MANAGER PROBLEM: • Optimize fund performance • Design a suitable (socially fair) contract for the fund clients allowing for demographic fluctuations ⇓ STOCHASTIC CONTROL PROBLEM WITH DELAY (pensions depends on past contributions!!)

Delayed stochastic control • Typically difficult problem.. • Make it Markovian by adding suitable state variables by keeping finite the dimension of the problem (Oksendal-Sulem 2002): it works in very special cases • Embed the problem into an infinite dimen- sional Markovian setting where state vari- ables belong to a Hilbert space (Gozzi-Marinelli 2004, Federico 2007): difficult to obtain explicit solutions even in simple cases • OTHER APPROACHES?..

WARM UP: the delayed model without market • Participants entry at time t , pay a lumpsum contribution c t and will receive a lumpsum pension at time t + a (No mortality risk for the participants!! and r = 0) • F t fund wealth dF t = ( c t − c t − a f t ) dt F 0 = x 0 , so that � T � T F T = x 0 + 0 c s ds − 0 c s − a f s ds. • No possibility to invest fund wealth at a rate greater than zero (not restrictive as- sumption)

Aims of the manager 1. grant (almost surely) a minimal benefit: f t ≥ g a.s. 2. maximize the expected utility function of the clients receiving c t − a f t at (current) time t ; 3. find the solvency admissibility conditions for the fund (i.e. manage the ruin prob- ability)