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OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME

Year 2015, Volume: 3 Issue: 2, 219 - 244, 01.10.2015

Abstract

In this paper, we study the problem of simultaneous maximization of the value of expected terminal surplus and, minimization of risks associated with the terminal surplus in a de ned contribution (DC) pension scheme. The surplus, which is discounted, is solved with dynamic programming techniques. The pension plan member (PPM) makes a ow of contributions from his or her stochastic salary into the scheme. The ow of contributions are invested into a market that is characterized by a cash account, an index bond and a stock. The ecient frontier for the discounted and real surplus are obtained. Optimal consumption of the PPM was found to depend on the terminal wealth, random evolution of minimum pension bene t and "variance minimizing" parameter. It was found that as the variance minimizing parameter, tends to zero, the op- timal consumption tends to negative in nity. The optimal expected discounted and real surplus, optimal total expected pension bene ts and expected min- imum pension bene ts were obtained. We found that the optimal portfolio depends linearly on the random evolution of PPM's minimum bene ts. Some numerical examples of the results are established.

References

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  • [2] Battocchio, P. and Menoncin, F. (2004). Optimal pension management in a stochastic framework. Insurance: Mathematics and Economics, 34, 79-95.
  • [3] Bielecky, T., Jim, H., Pliska, S. and Zhou, X. (2005). Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Mathematical Finance, 15, 213-244.
  • [4] Blake, D., Wright, D. and Zhang, Y. (2008). Optimal funding and investment strategies in de ned contribution pension plans under Epstein-Zin utility. Discussion paper, the pensions Institute, Cass Business School, City University, UK.
  • [5] Boulier, J. F., Huang, S. J. and Taillard, G. (2001). Optimal management under stochas- tic interest rates: the case of a protected de ned contribution pension fund. Insurance: Mathematics and Economics, 28, 173-189.
  • [6] Cairns, A. J. G., Blake, D. and Dowd, K. (2006). Stochastic lifestyling: Optimal dynamic asset allocation for de ned contribution pension plans. Journal of Economic Dynamic and Control, 30, 843-377.
  • [7] Chiu, M. and Li, D. (2006). Asset and liability management under a continuous-time mean- variance optimization framework. Insurance: Mathematics and Economics, 39, 330-355.
  • [8] Colombo, L. and Haberman, S. (2005). Optimal contributions in a de ned bene t pension scheme with stochastic new entrants. Insurance: Mathematics and Economics 37, 335354.
  • [9] Da Cunha, N.O., Polak, E., (1967). Constrained minimization under vector-valued crite- ria in nite dimensional spaces. Journal of Mathematical Analysis and Applications 19, 103124.
  • [10] Deelstra, G., Grasselli, M. and Koehl, P. (2000). Optimal investment strategies in a CIR framework. Journal of Applied Probability, 37, 936-946.
  • [11] Devolder, P. Bosch Princep, M.and Fabian, I. D. (2003). Stochastic optimal control of annuity contracts. Insurance: Mathematics and Economics, 33, 227-238.
  • [12] Di Giacinto, M., Federico, S. and Gozzi, F. (2010). Pension funds with a minimum guar- antee: a stochastic control approach. Finance and Stochastic.
  • [13] Gao, J. (2008). Stochastic optimal control of DC pension funds. Insurance: Mathematics and Economics, 42, pp. 1159-1164.
  • [14] Gerrard, R., Haberman S. and Vigna, E. (2004). Optimal investment choices post retire- ment in a de ned contribution pension scheme. Insurance: Mathematics and Economics, 35, 321-342.
  • [15] Haberman, S., Sung, J.H., 1994. Dynamics approaches to pension funding. Insurance: Mathematics and Economics 15, 151162.
  • [16] Haberman, S., Butt, Z., Megaloudi, C., (2000). Contribution and solvency risk in a de ned bene t pension scheme. Insurance: Mathematics and Economics 27, 237259.
  • [17] Haberman, S. and Vigna, E. (2002). Optimal investment strategies and risk measures in de ned contribution pension schemes. Insurance: Mathematics and Economics, 31, 35-69.
  • [18] Hjgaard, B. and Vigna, E. (2007). Mean-variance portfolio selection and ecient fron- tier for de ned contribution pension schemes. technical report R-2007-13, Department of Mathematical Sciences, Aalborg University.
  • [19] Huang, H.C., Cairns, A.J.G., (2005). On the control of de ned-bene t pension plans. Insurance: Mathematics and Economics 38, 113131.
  • [20] Jensen, B.A. and Srensen, C. (1999). Paying for minimum interest guarantees. Who should compensate who? European Financial Management 7: 183-211.
  • [21] Josa-Fombellida, R., Rincon-Zapatero, J.P., (2001). Minimization of risks in pension fund- ing by means of contribution and portfolio selection. Insurance: Mathematics and Eco- nomics 29, 3545.
  • [22] Josa-Fombellida, R., Rincon-Zapatero, J.P., (2004). Optimal risk management in de ned bene t stochastic pension funds. Insurance: Mathematics and Economics 34, 489503.
  • [23] Josa-Fombellida, R. and Rincon-Zapatero, J. (2008). Mean-variance portfolio and contri- bution selection in stochastic pension funding. European Journal of Operational Research, 187, 120-137.
  • [24] Korn, R, and Krekel, M. (2001). Optimal portfolios with xed consumption or income streams. Working paper, University of Kaiserslautern.
  • [25] Li, D. and Ng, W. -L. (2000). Optimal dynamic portfolio selection: multiperiod mean- variance formulation. Mathematical Finance, 10, 387-406.
  • [26] Markowitz, H. (1952). Portfolio selection. Journal of nance, 7, 77-91.
  • [27] Markowitz, H. (1959). Portfolio selection: ecient diversi cation of investments, New York, Wiley.
  • [28] Merton, R.C. (1971). Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3, 373413.
  • [29] Nkeki, C. I. (2011). On optimal portfolio management of the accumulation phase of a de ned contributory pension scheme. Ph.D thesis, Department of Mathematics, University of Ibadan, Ibadan, Nigeria.
  • [30] Nkeki, C. I. and Nwozo, C. R. (2012), Variational form of classical portfolio strategy and expected wealth for a de ned contributory pension scheme. Journal of Mathematical Finance 2(1): 132-139.
  • [31] Nkeki, C. I, (2013). Optimal portfolio strategy with discounted stochastic cash in ows. Journal of Mathematical Finance, 3: 130-137.
  • [32] Nkeki, C. I. (2012). Mean-variance portfolio selection with in ation hedging strategy: a case of a de ned contributory pension scheme. Theory and Applications of Mathematics and Computer Science, 2(2), 67-82.
  • [33] Nkeki, C. I. and Nwozo, C. R. (2013). Optimal investment under in ation protection and optimal portfolios with stochastic cash ows strategy. To appear in IAENG Journal of Applied Mathematics.
  • [34] Nwozo, C. R. and Nkeki, C. I. (2011) Optimal portfolio and strategic consumption planning in a life-cycle of a pension plan member in a de ned contributory pension scheme, IAENG International Journal of Applied Mathematics, 41(4), 299-309.
  • [35] Vigna, E. (2010). On eciency of mean-variance based portfolio selection in DC pension schemes, Collegio Carlo Alberto Notebook, 154.
  • [36] Richardson, H. (1989). A minimum variance result in continuous trading portfolio opti- mization. Management Science, 35, 1045-1055.
  • [37] Zhou, X and Li, D. (2000). Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42, 19-33.
  • [38] Zhang, A., Korn, R. and Ewald, C. O. (2007). Optimal management and in ation protec- tion for de ned contribution pension plans, Working paper, University of St. Andrews.
Year 2015, Volume: 3 Issue: 2, 219 - 244, 01.10.2015

Abstract

References

  • [1] Bajeux-Besnainou I. and Portait, R. (1998). Dynamic asset allocation in a mean-variance framework. Management Science, 44, S79-S95.
  • [2] Battocchio, P. and Menoncin, F. (2004). Optimal pension management in a stochastic framework. Insurance: Mathematics and Economics, 34, 79-95.
  • [3] Bielecky, T., Jim, H., Pliska, S. and Zhou, X. (2005). Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Mathematical Finance, 15, 213-244.
  • [4] Blake, D., Wright, D. and Zhang, Y. (2008). Optimal funding and investment strategies in de ned contribution pension plans under Epstein-Zin utility. Discussion paper, the pensions Institute, Cass Business School, City University, UK.
  • [5] Boulier, J. F., Huang, S. J. and Taillard, G. (2001). Optimal management under stochas- tic interest rates: the case of a protected de ned contribution pension fund. Insurance: Mathematics and Economics, 28, 173-189.
  • [6] Cairns, A. J. G., Blake, D. and Dowd, K. (2006). Stochastic lifestyling: Optimal dynamic asset allocation for de ned contribution pension plans. Journal of Economic Dynamic and Control, 30, 843-377.
  • [7] Chiu, M. and Li, D. (2006). Asset and liability management under a continuous-time mean- variance optimization framework. Insurance: Mathematics and Economics, 39, 330-355.
  • [8] Colombo, L. and Haberman, S. (2005). Optimal contributions in a de ned bene t pension scheme with stochastic new entrants. Insurance: Mathematics and Economics 37, 335354.
  • [9] Da Cunha, N.O., Polak, E., (1967). Constrained minimization under vector-valued crite- ria in nite dimensional spaces. Journal of Mathematical Analysis and Applications 19, 103124.
  • [10] Deelstra, G., Grasselli, M. and Koehl, P. (2000). Optimal investment strategies in a CIR framework. Journal of Applied Probability, 37, 936-946.
  • [11] Devolder, P. Bosch Princep, M.and Fabian, I. D. (2003). Stochastic optimal control of annuity contracts. Insurance: Mathematics and Economics, 33, 227-238.
  • [12] Di Giacinto, M., Federico, S. and Gozzi, F. (2010). Pension funds with a minimum guar- antee: a stochastic control approach. Finance and Stochastic.
  • [13] Gao, J. (2008). Stochastic optimal control of DC pension funds. Insurance: Mathematics and Economics, 42, pp. 1159-1164.
  • [14] Gerrard, R., Haberman S. and Vigna, E. (2004). Optimal investment choices post retire- ment in a de ned contribution pension scheme. Insurance: Mathematics and Economics, 35, 321-342.
  • [15] Haberman, S., Sung, J.H., 1994. Dynamics approaches to pension funding. Insurance: Mathematics and Economics 15, 151162.
  • [16] Haberman, S., Butt, Z., Megaloudi, C., (2000). Contribution and solvency risk in a de ned bene t pension scheme. Insurance: Mathematics and Economics 27, 237259.
  • [17] Haberman, S. and Vigna, E. (2002). Optimal investment strategies and risk measures in de ned contribution pension schemes. Insurance: Mathematics and Economics, 31, 35-69.
  • [18] Hjgaard, B. and Vigna, E. (2007). Mean-variance portfolio selection and ecient fron- tier for de ned contribution pension schemes. technical report R-2007-13, Department of Mathematical Sciences, Aalborg University.
  • [19] Huang, H.C., Cairns, A.J.G., (2005). On the control of de ned-bene t pension plans. Insurance: Mathematics and Economics 38, 113131.
  • [20] Jensen, B.A. and Srensen, C. (1999). Paying for minimum interest guarantees. Who should compensate who? European Financial Management 7: 183-211.
  • [21] Josa-Fombellida, R., Rincon-Zapatero, J.P., (2001). Minimization of risks in pension fund- ing by means of contribution and portfolio selection. Insurance: Mathematics and Eco- nomics 29, 3545.
  • [22] Josa-Fombellida, R., Rincon-Zapatero, J.P., (2004). Optimal risk management in de ned bene t stochastic pension funds. Insurance: Mathematics and Economics 34, 489503.
  • [23] Josa-Fombellida, R. and Rincon-Zapatero, J. (2008). Mean-variance portfolio and contri- bution selection in stochastic pension funding. European Journal of Operational Research, 187, 120-137.
  • [24] Korn, R, and Krekel, M. (2001). Optimal portfolios with xed consumption or income streams. Working paper, University of Kaiserslautern.
  • [25] Li, D. and Ng, W. -L. (2000). Optimal dynamic portfolio selection: multiperiod mean- variance formulation. Mathematical Finance, 10, 387-406.
  • [26] Markowitz, H. (1952). Portfolio selection. Journal of nance, 7, 77-91.
  • [27] Markowitz, H. (1959). Portfolio selection: ecient diversi cation of investments, New York, Wiley.
  • [28] Merton, R.C. (1971). Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3, 373413.
  • [29] Nkeki, C. I. (2011). On optimal portfolio management of the accumulation phase of a de ned contributory pension scheme. Ph.D thesis, Department of Mathematics, University of Ibadan, Ibadan, Nigeria.
  • [30] Nkeki, C. I. and Nwozo, C. R. (2012), Variational form of classical portfolio strategy and expected wealth for a de ned contributory pension scheme. Journal of Mathematical Finance 2(1): 132-139.
  • [31] Nkeki, C. I, (2013). Optimal portfolio strategy with discounted stochastic cash in ows. Journal of Mathematical Finance, 3: 130-137.
  • [32] Nkeki, C. I. (2012). Mean-variance portfolio selection with in ation hedging strategy: a case of a de ned contributory pension scheme. Theory and Applications of Mathematics and Computer Science, 2(2), 67-82.
  • [33] Nkeki, C. I. and Nwozo, C. R. (2013). Optimal investment under in ation protection and optimal portfolios with stochastic cash ows strategy. To appear in IAENG Journal of Applied Mathematics.
  • [34] Nwozo, C. R. and Nkeki, C. I. (2011) Optimal portfolio and strategic consumption planning in a life-cycle of a pension plan member in a de ned contributory pension scheme, IAENG International Journal of Applied Mathematics, 41(4), 299-309.
  • [35] Vigna, E. (2010). On eciency of mean-variance based portfolio selection in DC pension schemes, Collegio Carlo Alberto Notebook, 154.
  • [36] Richardson, H. (1989). A minimum variance result in continuous trading portfolio opti- mization. Management Science, 35, 1045-1055.
  • [37] Zhou, X and Li, D. (2000). Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42, 19-33.
  • [38] Zhang, A., Korn, R. and Ewald, C. O. (2007). Optimal management and in ation protec- tion for de ned contribution pension plans, Working paper, University of St. Andrews.
There are 38 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Charles İ. Nkekı This is me

Publication Date October 1, 2015
Submission Date July 10, 2014
Published in Issue Year 2015 Volume: 3 Issue: 2

Cite

APA Nkekı, C. İ. (2015). OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME. Konuralp Journal of Mathematics, 3(2), 219-244.
AMA Nkekı Cİ. OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME. Konuralp J. Math. October 2015;3(2):219-244.
Chicago Nkekı, Charles İ. “OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME”. Konuralp Journal of Mathematics 3, no. 2 (October 2015): 219-44.
EndNote Nkekı Cİ (October 1, 2015) OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME. Konuralp Journal of Mathematics 3 2 219–244.
IEEE C. İ. Nkekı, “OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME”, Konuralp J. Math., vol. 3, no. 2, pp. 219–244, 2015.
ISNAD Nkekı, Charles İ. “OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME”. Konuralp Journal of Mathematics 3/2 (October 2015), 219-244.
JAMA Nkekı Cİ. OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME. Konuralp J. Math. 2015;3:219–244.
MLA Nkekı, Charles İ. “OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME”. Konuralp Journal of Mathematics, vol. 3, no. 2, 2015, pp. 219-44.
Vancouver Nkekı Cİ. OPTIMAL SURPLUS, MINIMUM PENSION BENEFITS AND CONSUMPTION PLANS IN A MEAN-VARIANCE PORTFOLIO APPROACH FOR A DEFINED CONTRIBUTION PENSION SCHEME. Konuralp J. Math. 2015;3(2):219-44.
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