An individual investor has to decide how to allocate his/her savings from a retirement perspective. This problem covers a long-term horizon. In this paper we consider a 40-year horizon formulating a multi-criteria multistage program with stochastic dominance constraints in an intermediate stage and in the final stage. As we are dealing with a real problem and we have formulated the model in cooperation with a commercial Italian bank, the intermediate stage corresponds to a possible withdrawal allowed by the Italian pension system. The sources of uncertainty considered are: the financial returns, the interest rate evolution, the investor’s salary process and a considerable withdrawal event. We include a set of portfolio constraints according to the pension plan regulation. The objective of the model is to minimize the Average Value at Risk Deviation measure and to satisfy wealth goals. Three different wealth target formulations are considered: a deterministic wealth target (i.e. a comparison between the accumulated average wealth and a fixed threshold) and two stochastic dominance relations—the first order and the second order—introducing a benchmark portfolio and then requiring the optimal portfolio to dominate the benchmark. In particular, we prove that solutions obtained under stochastic dominance constraints ensure a safer allocation while still guaranteeing good returns. Moreover, we show how the withdrawal event affects the solution in terms of allocation in each of the three frameworks. Finally, the sensitivity and convergence of the stochastic solutions and computational issues are investigated.
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Berger, A. J., & Mulvey, J. M. (1998). The home account advisor: Asset and liability management for individual investors (pp. 634–665). Cambridge: Cambridge University Press.
Bertocchi, M., Schwartz, S. L., & Ziemba, W. T. (2010). Optimizing the aging, retirement, and pensions dilemma. Hoboken: Wiley.
Black, F., & Karasinski, P. (1991). Bond and option pricing when short rates are lognormal. Financial Analysts Journal, 47(4), 52–59.
Black, F., Derman, E., & Toy, W. (1990). A one-factor model of interest rates and its application to treasury bond options. Financial Analysts Journal, 46(1), 33–39.
Blake, D., Wright, D., & Zhang, Y. (2013). Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion. Journal of Economic Dynamics and Control, 37(1), 195–209.
Branda, M., & Kopa, M. (2016). Dea models equivalent to general n-th order stochastic dominance efficiency tests. Operations Research Letters, 44, 285–289.
Brunel, J. L. P. (2003). Revisiting the asset allocation challenge through a behavioral finance lens. The Journal of Wealth Management, 6(2), 10–20.
Cai, J., & Ge, C. (2012). Multi-objective private wealth allocation without subportfolios. Economic Modelling, 29(3), 900–907.
Chhabra, A. B. (2005). Beyond Markowitz: A comprehensive wealth allocation framework for individual investors. The Journal of Wealth Management, 7(4), 8–34.
Consigli, G. (2007). Individual asset liability management for individual investors. In S. A. Zenios & W. T. Ziemba (Eds.), Handbook of asset and liability management: Applications and case studies (pp. 752–827). North-Holland Finance Handbook Series: Elsevier.
Consigli, G., Iaquinta, G., Moriggia, V., di Tria, M., & Musitelli, D. (2012). Retirement planning in individual asset-liability management. IMA Journal of Management Mathematics, 23(4), 365–396.
Consiglio, A., Cocco, F., & Zenios, S. A. (2004). www.Personal_Asset_Allocation. Interfaces, 34(4), 287–302.
Consiglio, A., Cocco, F., & Zenios, S. A. (2007). Scenario optimization asset and liability modelling for individual investors. Annals of Operations Research, 152(1), 167–191.
Cox, J. C., Ingersoll, J. E. Jr., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica: Journal of the Econometric Society, 53(2), 385–408.
DeMiguel, V., Garlappi, L., & Uppal, R. (2009). Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, 22(5), 1915–1953.
Dentcheva, D., & Ruszczynski, A. (2003). Optimization with stochastic dominance constraints. SIAM Journal on Optimization, 14(2), 548–566.
Dentcheva, D., & Ruszczyński, A. (2004). Semi-infinite probabilistic optimization: first-order stochastic dominance constrain. Optimization, 53(5–6), 583–601.
Dentcheva, D., & Ruszczynski, A. (2010). Robust stochastic dominance and its application to risk-averse optimization. Mathematical Programming, Series B, 123, 85–100.
Dupačová, J., & Kopa, M. (2012). Robustness in stochastic programs with risk constraints. Annals of Operations Research, 200(1), 55–74.
Dupačová, J., & Kopa, M. (2014). Robustness of optimal portfolios under risk and stochastic dominance constraints. European Journal of Operational Research, 234(2), 434–441.
Dupačová, J., Hurt, J., & Štěpán, J. (2002). Stochastic modeling in economics and finance. Applied optimization. (Vol.75) New York: Kluwer.
Escudero, L. F., Garín, M. A., Merinoc, M., & Pérez, G. (2016). On timse stochastic dominance induced by mixed integer-linear recourse in multistage stochastic programs. European Journal of Operational Research, 249(1), 164–176.
Gerrard, R., Haberman, S., & Vigna, E. (2004). Optimal investment choices post-retirement in a defined contribution pension scheme. Insurance: Mathematics and Economics, 35(2), 321–342.
Gerrard, R., Haberman, S., & Vigna, E. (2006). The management of decumulation risks in a defined contribution pension plan. North American Actuarial Journal, 10(1), 84–110.
Gerrard, R., Højgaard, B., & Vigna, E. (2012). Choosing the optimal annuitization time post-retirement. Quantitative Finance, 12(7), 1143–1159.
Hadar, J., & Russell, W. R. (1969). Rules for ordering uncertain prospects. The American Economic Review, 59, 25–34.
Hanoch, G., & Levy, H. (1969). The efficiency analysis of choices involving risk. The Review of Economic Studies, 36(3), 335–346.
Ho, T. S., & Lee, S. B. (1986). Term structure movements and pricing interest rate contingent claims. The Journal of Finance, 41(5), 1011–1029.
Horneff, W. J., Maurer, R. H., & Stamos, M. Z. (2008). Optimal gradual annuitization: Quantifying the costs of switching to annuities. Journal of Risk and Insurance, 75(4), 1019–1038.
Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573–592.
Kahneman, D. & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, 47(2), 263–291.
Kilianová, S., & Pflug, G. C. (2009). Optimal pension fund management under multi-period risk minimization. Annals of Operations Research, 166(1), 261–270.
Kopa, M. (2010). Measuring of second-order stochastic dominance portfolio efficiency. Kybernetika, 46(3), 488–500.
Kopa, M., & Post, T. (2015). A general test for ssd portfolio efficiency. OR Spectrum, 37(3), 703–734.
Kuosmanen, T. (2004). Efficient diversification according to stochastic dominance criteria. Management Science, 50(10), 1390–1406.
Levy, H. (2016). Stochastic dominance, investment decision making under uncertainty (3rd ed.). New York: Sprigner.
Luedtke, J. (2008). New formulations for optimization under stochastic dominance constraints. SIAM Journal on Optimization, 19(3), 1433–1450.
Medova, E. A., Murphy, J. K., Owen, A. P., & Rehman, K. (2008). Individual asset liability management. Quantitative Finance, 8(6), 547–560.
Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics, 51(3), 247–257.
Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4), 373–413.
Milevsky, M. A., & Young, V. R. (2007). Annuitization and asset allocation. Journal of Economic Dynamics and Control, 31(9), 3138–3177.
Post, T., & Kopa, M. (2013). General linear formulations of stochastic dominance criteria. European Journal of Operational Research, 230(2), 321–332.
Post, T., & Kopa, M. (2016). Portfolio choice based on third-degree stochastic dominance. Forthcoming in Management Science, 3(4), 373–413.
Post, T., Fang, Y., & Kopa, M. (2015). Linear tests for dara stochastic dominance. Management Science, 61(7), 1615–1629.
Quirk, J. P., & Saposnik, R. (1962). Admissibility and measurable utility functions. The Review of Economic Studies, 29, 140–146.
Rendleman, R. J., & Bartter, B. J. (1980). The pricing of options on debt securities. Journal of Financial and Quantitative Analysis, 15(1), 11–24.
Richard, S. F. (1975). Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics, 2(2), 187–203.
Rockafellar, T. R., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–42.
Rockafellar, T. R., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.
Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188.
Yang, X., Gondzio, J., & Grothey, A. (2010). Asset liability management modelling with risk control by stochastic dominance. Journal of Asset Management, 11(2), 73–93.
The research was partially supported by the Czech Science Foundation under grant 15-02938S and by MIUR-ex60% 2014–2016 sci.resp. Vittorio Moriggia.
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Kopa, M., Moriggia, V. & Vitali, S. Individual optimal pension allocation under stochastic dominance constraints. Ann Oper Res 260, 255–291 (2018). https://doi.org/10.1007/s10479-016-2387-x
- Individual pension problem
- Multistage stochastic programming
- Stochastic dominance constraints
- Average value at risk deviation