Pension funds with a minimum guarantee: a stochastic control approach

In this paper we propose and study a continuous-time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon. In our model the dynamics of wealth takes directly into account the flows of contributions and benefits, and the level of wealth is constrained to stay above a “solvency level.” The fund manager can invest in a riskless asset and in a risky asset, but borrowing and short selling are prohibited. We concentrate the analysis on the effect of the solvency constraint, analyzing in particular what happens when the fund wealth reaches the allowed minimum value represented by the solvency level.The model is naturally formulated as an optimal stochastic control problem with state constraints and is treated by the dynamic programming approach. We show that the value function of the problem is a regular solution of the associated Hamilton–Jacobi–Bellman equation. Then we apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties. We finally give a special example with explicit solution.

[1]  Paola Loreti,et al.  A class of stochastic optimal control problems with state constraint , 2002 .

[2]  Xun Yu Zhou,et al.  A Diffusion Model for Optimal Dividend Distribution for a Company with Constraints on Risk Control , 2002, SIAM J. Control. Optim..

[3]  David Blake,et al.  Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans , 2006 .

[4]  Giuseppe Da Prato,et al.  Stochastic Partial Differential Equations and Applications - VII , 2005 .

[5]  William N. Goetzmann,et al.  High Water Marks , 1998, Machinery.

[6]  Suresh P. Sethi,et al.  Infinite-horizon investment consumption model with a nonterminal bankruptcy , 1986 .

[7]  Jean Boulier,et al.  Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund , 2001 .

[8]  Suresh P. Sethi,et al.  Consumption-Investment Problem with Subsistence Consumption, Bankruptcy, and Random Market Coefficients , 1997 .

[9]  Huyên Pham,et al.  Wealth-path dependent utility maximization in incomplete markets , 2004, Finance Stochastics.

[10]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[11]  N. Karoui,et al.  Optimal investment decisions when time-horizon is uncertain , 2008 .

[12]  Marco Fuhrman,et al.  Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces , 2004 .

[13]  Salvatore Federico,et al.  A Pension Fund Model with Surplus: an Infinite Dimensional Stochastic Control Approach , 2007 .

[14]  Alex W. H. Chan.  Merton, Robert C. , 2010 .

[15]  Steven Haberman,et al.  Optimal investment strategies and risk measures in defined contribution pension schemes , 2002 .

[16]  G. Kallianpur.  Stochastic differential equations and diffusion processes , 1981 .

[17]  R. C. Merton,et al.  Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case , 1969 .

[18]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[19]  Carlo Marinelli,et al.  Stochastic optimal control of delay equations arising in advertising models , 2004, math/0412403.

[20]  W. Feller.  THE PARABOLIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED SEMI-GROUPS OF TRANSFORMATIONS , 1952 .

[21]  Steven Haberman,et al.  Optimal investment strategy for defined contribution pension schemes , 2001 .

[22]  Massimo Bernaschi,et al.  A MODEL FOR THE OPTIMAL ASSET-LIABILITY MANAGEMENT FOR INSURANCE COMPANIES , 2003 .

[23]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[24]  Suresh P. Sethi,et al.  Explicit Solution of a General Consumption/Investment Problem , 1986, Math. Oper. Res..

[25]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[26]  Olivier Scaillet,et al.  Optimal asset allocation for pension funds under mortality risk during the accumulation and decumulation phases , 2007, Ann. Oper. Res..

[27]  William N. Goetzmann,et al.  High-Water Marks and Hedge Fund Management Contracts , 2001 .

[28]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[29]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[30]  W. Fleming,et al.  Hedging in incomplete markets with HARA utility , 1997 .

[31]  Nicole El Karoui,et al.  APPLICATION TO AMERICAN OPTIONS AND PORTFOLIO INSURANCE By , 2008 .

[32]  Fausto Gozzi,et al.  Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: approach , 2006 .

[33]  Salvatore Federico,et al.  A pension fund in the accumulation phase: a stochastic control approach , 2008 .

[34]  Salvatore Federico,et al.  A stochastic control problem with delay arising in a pension fund model , 2011, Finance Stochastics.

[35]  Paolo Battocchio,et al.  Optimal pension management in a stochastic framework , 2004 .

[36]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[37]  Claudio Tebaldi,et al.  Illiquid Assets and Optimal Portfolio Choice , 2004 .

[38]  N. Karoui,et al.  Optimal portfolio management with American capital guarantee , 2005 .

[39]  G. Deelstra,et al.  Optimal investment strategies in the presence of a minimum guarantee , 2003 .

[40]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[41]  H. Mete Soner.  Stochastic optimal control in finance , 2007 .

[42]  Martino Grasselli,et al.  Optimal design of the guarantee for defined contribution funds , 2004 .

[43]  T. Zariphopoulou.  Consumption-Investment Models with Constraints , 1994 .

[44]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[45]  Laura T. Starks,et al.  Performance Incentive Fees: An Agency Theoretic Approach , 1987, Journal of Financial and Quantitative Analysis.

[46]  Elisabeth Rouy,et al.  Second Order Hamilton--Jacobi Equations in Hilbert Spaces and Stochastic Boundary Control , 2000, SIAM J. Control. Optim..

[47]  S. Sethi,et al.  Explicit Solution of a General Consumption/Portfolio Problem with Subsistence Consumption and Bankruptcy , 1992 .

[48]  W. Feller.  Diffusion processes in one dimension , 1954 .

[49]  Giambattista Giacomin.  Second Order Hamilton-Jacobi Equations in Hilbert Spaces and Stochastic Optimal Control , 2002 .