Share / Export Citation / Email / Print / Text size:

Statistics in Transition New Series

Polish Statistical Association

Central Statistical Office of Poland

Subject: Economics, Statistics & Probability


ISSN: 1234-7655
eISSN: 2450-0291





Volume / Issue / page

Related articles

VOLUME 16 , ISSUE 1 (March 2015) > List of articles


Radosław Pietrzyk * / Paweł Rokita *

Keywords : financial goals, personal finance, intertemporal choice, financial plan optimization, stochastic goals.

Citation Information : Statistics in Transition New Series. Volume 16, Issue 1, Pages 111-136, DOI:

License : (CC BY 4.0)

Published Online: 31-October-2017



In household financial planning two types of risk are typically being taken into account. These are life-length risk and risk connected with financing. In addition, also various types of events of insurance character, like health deterioration, are sometimes taken into account. There are, however, no models addressing stochastic nature of household financial goals. The last should not be confused with modelling factors that influence performance of financing the goals, which is a popular research topic. The problem of modelling goals themselves is, in turn, not so well explored. There are two main characteristics that describe a goal: magnitude and time. At least for some goals one or both of these characteristics may show a stochastic nature. This article puts forward a proposition of working goal time and magnitude into a household financial plan and taking their distributions into account when optimizing the plan. A model of two-person household is used. The decision variables of the optimization task are consumption-investment proportion and division of household investments between household members.

Content not available PDF Share



  1. ANDO, A., MODIGLIANI, F., (1957). Tests of the Life Cycle Hypothesis of Saving: Comments and Suggestions. Oxford Institute of Statistics Bulletin, Vol. XIX (May), pp. 99−124.
  2. BLAKE, D., CAIRNS A., DOWD, K., (2001). Pensionmetrics: Stochastic Pension Plan Design During the Accumulation Phase. Insurance: Mathematics and Economics, Vol. 29, Issue 2, pp. 187−215.
  3. BLAKE, D., CAIRNS A., DOWD, K., (2003). Pensionmetrics 2: Stochastic Pension Plan Design During the Distribution Phase. Insurance: Mathematics and Economics, Vol. 33, issue 1, pp. 29−47.
  4. BODIE, Z., TREUSSARD, J., WILLEN, P., (2008). The Theory of Optimal Life- Cycle Saving and Investing. In: Z. Bodie, D. McLeavy, L.B. Siegel, eds., The
    Future of Life-Cycle Saving and Investing. The research Foundation of CFA.
  5. BROCKETT, P. L., (1984). General bivariate Makeham laws. Scandinavian Actuarial Journal, Vol. 1984, Issue 3, pp. 150−156.
  6. BROWN, J. R., POTERBA J. M., (2000). Joint Life Annuities And Annuity Demand By Married Couples, Journal of Risk and Insurance, 2000, 67(4), pp. 527−553.
  7. BRUHN K., STEFFENSEN M., (2010). Household consumption, investment and life insurance. Insurance: Mathematics and Economics, 48, pp. 315−325.
  8. CARRIERE, J. F., (2000). Bivariate Survival Models for Coupled Lives. Scandinavian Actuarial Journal, 2000:1, pp. 17−32.
  9. COX, J.C., HUANG, CH., (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory, Vol. 49, Issue 1, pp. 33−83.
  10. FELDMAN, L., PIETRZYK, R., ROKITA, P., (2014a). A practical method of determining longevity and premature-death risk aversion in households and some proposals of its application. In: M. Spiliopoulou, L. Schmidt-Thieme, R. Janning, eds., Data Analysis, Machine Learning and Knowledge Discovery. Studies in Classification, Data Analysis and Knowledge Organization. Berlin-Heidelberg: Springer, pp. 255−264.
  11. FELDMAN, L., PIETRZYK, R., ROKITA, P., (2014b). Multiobjective optimization of financing household goals with multiple investment programs. Statistics in Transition, New Series, Spring, Vol. 15, No. 2, pp. 243−268.
  12. FELDMAN, L., PIETRZYK, R., ROKITA, P., (2014c). Cumulated Surplus Approach and a New Proposal of Life-Length Risk Aversion Interpretation in Retirement Planning for a Household with Two Decision Makers (November 6, 2014). Available at SSRN: .
  13. GEORGES, P., LAMY, A.-G., NICOLAS, E., QUIBEL, G., RONCALLI, T, (2001). Multivariate Survival Modelling: A Unified Approach with Copulas (May 28, 2001). Available at SSRN: [Version: 28 May 2001. Accessed: 04 Dec. 2014].
  14. GEYER, A., HANKE, M., WEISSENSTEINER, A., (2009). Life-cycle asset allocation and consumption using stochastic linear programming. The Journal of Computational Finance, 12(4), pp. 29−50.
  15. GHYSELS, E., PLAZZI, A., TOROUS, W. N., VALKANOV, R. I., (2012). Forecasting Real Estate Prices. In: G. Elliott, A. Timmermann, eds., Handbook of Economic Forecasting. Vol. II, Elsevier.
  16. GOMPERTZ, B., (1825). On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies. Philosophical Transactions of the Royal Society of London, 115, 513−585.
  17. GUTÍERREZ, R., GUTÍERREZ-SANCHEZ, R., NAFIDI, A., (2008). A bivariate stochastic Gompertz diffusion model: statistical aspects and application to the
    joint modeling of the Gross Domestic Product and CO2 emissions in Spain.
  18. Environmetrix Vol. 19, Issue 6, pp. 643−658. GONG, G., WEBB, A., (2008). Mortality Heterogeneity and the Distributional Consequences of Mandatory Annuitization. The Journal of Risk and Insurance, 75(4), pp. 1055−1079.
  19. HUANG, H., MILEVSKY, M. A., (2011). Longevity Risk Aversion and Tax- Efficient Withdrawals. [online] SSRN. Available at: [Accessed: 22 March2012].
  20. HUANG, H., MILEVSKY, M. A., WANG, J., (2008). Portfolio Choice and Life Insurance: The CRRA Case. Journal of Risk & Insurance, Vol. 75, Issue 4, pp. 847−872.
  21. IBBOTSON, R. G., CHEN, P., MILEVSKY, M. A., ZHU, X., (2005). Human Capital, Asset Allocation, and Life Insurance. Yale ICF Working Paper No. 05-11. Available at SSRN: .
  22. KONICZ, A. K., PISINGER, D., RASMUSSEN, K. M., STEFFENSEN, M., (2014). A Combined Stochastic Programming and Optimal Control Approach to Personal Finance and Pensions. Available at SSRN: [Version: 30 April 2014. Accessed: 24 Nov. 2014].
  23. KOTLIKOFF, L. J., SPIVAK, A., (1981). The Family as an Incomplete Annuities Market. Journal of Political Economy, Vol. 89, No. 2, (April 1981), pp. 372−391.
  24. MERTON, R. C., (1969). Lifetime portfolio selection under uncertainty: The continuous time case. The Review of Economics and Statistics, 51(3), pp. 247−257.
  25. MERTON, R. C., (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4), pp.373−413.
  26. MILEVSKY, M. A., HUANG, H., (2011). Spending Retirement on Planet Vulcan: The Impact of Longevity Risk Aversion on Optimal Withdrawal Rates. Financial Analysts Journal, 67(2), pp. 45−58.
  27. MODIGLIANI, F., BRUMBERG, R. H., (1954). Utility analysis and the consumption function: an interpretation of cross-section data. In: Kenneth K. Kurihara, ed. 1954. Post-Keynesian Economics. New Brunswick, NJ: Rutgers University Press, pp.388−436.
  28. OHNISHI, T., MIZUNO, T., SHIMZU, CH., WATANABE, T., (2011). The Evolution of House Price Distribution. RIETI Discussion Paper Series 11-E- 019. Available at [Accessed: 04 Dec. 2014].
  29. PIETRZYK, R. A., ROKITA, P. A., (2014). Facilitating Household Financial Plan Optimization by Adjusting Time Range of Analysis to Life-Length Risk Aversion (October 22, 2014). Available at SSRN: <>.
  30. 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, pp. 187−203.
  31. RUSZCZYŃSKI, A., SHAPIRO, A., (2003). Stochastic Programming Models. In: Ruszczyński, A., Shapiro, A., eds, Handbooks in Operations Research and Management Science, 10: Stochastic Programming, pp. 1−64.
  32. VICKSON, R. G., ZIEMBA, W. T., eds, (2006). Stochastic Optimization Models in Finance. World Scientific.
  33. WILLCOCKS, G., (2009). UK Housing Market: Time Series Processes with Independent and Identically Distributed Residuals. Journal of Real Estate Finance and Economics, Vol. 39, No. 4, 2009, pp. 403−414.
  34. YAARI, M. E., (1965). Uncertain Lifetime, Life Insurance and Theory of the Consumer. The Review of Economic Studies, 32(2), pp.137−150.
  35. ZALEGA, T., (2007). Gospodarstwa domowe jako podmiot konsumpcji (Households as consuming actors) (in Polish). Materials and Studies, Faculty of Management, University of Warsaw.
  36. ZIEMBA, W. T., (2003). The Stochastic Programming Approach to Asset, Liability, and Wealth Management. The Research Foundation of AIMR™.