The optimal portfolios based on a modified safety-first rule with risk-free saving

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January 2016, 12(1): 83-102. doi: 10.3934/jimo.2016.12.83

The optimal portfolios based on a modified safety-first rule with risk-free saving

Yuanyao Ding ^1, and Zudi Lu ^2,

1.	College of Science & Technology, and Faculty of Business, Ningbo University, Ningbo 315211, China
2.	Southampton Statistical Sciences Research Institute, and School of Mathematical Sciences, University of Southampton, SO17 1BJ, United Kingdom

Received July 2013 Revised November 2014 Published April 2015

Abstract
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How to manage the social security trust funds is a topic of wide interests both academically and professionally. In the setting of portfolio selection with social security funds investment, we propose a modified safety-first (MSF) rule with portfolio selection including risk-free saving. We first demonstrate under some mild assumptions that the solution to the MSF model for an individual investor can be expressed by explicitly analytical formula and the necessary and sufficient conditions for their existence are obtained. We then derive the safety-first efficient frontiers in both the space of expected return and insured return level and the space of standard deviation and expected return, with numerical examples illustrated. By comparing the results of the MSF model with those of the mean-variance (M-V) model, some novel insights into the differences between them are further gained.

Keywords: MSF model, safety-first efficient, security return, risk-free saving., Optimal portfolio, social security fund.

Mathematics Subject Classification: Primary: 91B28; Secondary: 91G1.

Citation: Yuanyao Ding, Zudi Lu. The optimal portfolios based on a modified safety-first rule with risk-free saving. Journal of Industrial and Management Optimization, 2016, 12 (1) : 83-102. doi: 10.3934/jimo.2016.12.83

References:

[1]	F. R. Arzac and V. S. Bawa, Portfolio choice and equilibrium in capital markets with safety first investors, Journal of Financial Economics, 4 (1977), 277-288. doi: 10.1016/0304-405X(77)90003-4.
[2]	V. S. Bawa, Optimal rules for ordering uncertain prospects, Journal of Financial Economics, 2 (1975), 95-121. doi: 10.1016/0304-405X(75)90025-2.
[3]	M. C. Chiu and D. Li, Asset-liability management under the safety-first principle, Optimization Theory and Applications, 143 (2009), 455-478. doi: 10.1007/s10957-009-9576-6.
[4]	S. Das, H. Markowitz, J. Scheid and M. Statman, Portfolio optimization with mental accounts, Journal of Financial and Quantitative Analysis, 45 (2010), 311-334. doi: 10.1017/S0022109010000141.
[5]	Y. Ding and B. Zhang, Risky asset pricing based on safety first fund management, Quantitative Finance, 9 (2009), 353-361. doi: 10.1080/14697680802392488.
[6]	Y. Ding and B. Zhang, Optimal portfolio of safety-first models, Journal of Statistical Planning and Inference, 139 (2009), 2952-2962. doi: 10.1016/j.jspi.2009.01.018.
[7]	M. Engels, Portfolio Optimization: Beyond Markowitz, Master's thesis, Leiden University,Netherlands, 2004.
[8]	P. C. Fishburn, Mean-risk analysis with risk associated with below-target returns, American Economical Review, 67 (1977), 116-126.
[9]	S. Kataoka, A stochastic programming model, Econometrica, 31 (1963), 181-196. doi: 10.2307/1910956.
[10]	K. Boda, J. A. Filar, Y. Lin and L. Spanjers, Stochastic target hitting time and the problem of early retirement, IEEE Transactions on Automatic Control, 49 (2004), 409-419. doi: 10.1109/TAC.2004.824469.
[11]	K. Boda and J. A. Filar, Time consistent dynamic risk measures, Mathematical Methods of Operations Research, 63 (2006), 169-186. doi: 10.1007/s00186-005-0045-1.
[12]	H. Levy and M. Levy, The safety first expected utility model: Experimental evidence and economic implications, Journal of Banking & Finance, 33 (2009), 1494-1506. doi: 10.1016/j.jbankfin.2009.02.014.
[13]	D. Li, T. F. Chan and W. L. Ng, Safety-first dynamic portfolio selection, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 585-600.
[14]	Z. F. Li, J. Yao and D. Li, Behavior patterns of investment strategies under Roy's safety-first principle, The Quarterly Review of Economics and Finance, 50 (2010), 167-179. doi: 10.1016/j.qref.2009.11.004.
[15]	H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 79-91.
[16]	R. C. Merton, An analytic derivation of the efficient portfolio frontier, The Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872. doi: 10.2307/2329621.
[17]	V. I. Norkin and S. V. Boyko, Safety-First Portfolio Selection, Cybernetics and Systems Analysis, 48 (2012), 180-191. doi: 10.1007/s10559-012-9396-9.
[18]	L. S. Ortobelli and S. T. Rachev, Safety-first analysis and stable paretian approach to portfolio choice theory, Mathematical and Computer Modelling, 34 (2001), 1037-1072. doi: 10.1016/S0895-7177(01)00116-9.
[19]	L. S. Ortobelli and F. Pellerey, Market stochastic bounds with elliptical distributions, Journal of Concrete and Applicable Mathematics, 6 (2008), 293-314.
[20]	A. D. Roy, Safety-first and the holding of assets, Econometrica, 20 (1952), 431-449. doi: 10.2307/1907413.
[21]	H. Shefrin and M. Statman, Behavioral portfolio theory, Journal of Financial and Quantitative Analysis, 35 (2000), 127-151. doi: 10.2307/2676187.
[22]	N. Signer, Safety-first portfolio optimization: Fixed versus random target, Thuenen-Series of Applied Economic Theory, No. 113, 2010. Available from: http://www.econstor.eu/bitstream/10419/74652/1/747604916.pdf.
[23]	L. G. Telser, Safety first and hedging, Review of Economic Studies, 23 (1955), 1-16. doi: 10.2307/2296146.
[24]	S. M. Zhang, S. Y. Wang and X. T. Deng, Portfolio Selection Theory with Different Interest Rates for Borrowing and Lending, Journal of Global Optimization, 28 (2004), 67-95. doi: 10.1023/B:JOGO.0000006719.64826.55.

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References:

[1]	F. R. Arzac and V. S. Bawa, Portfolio choice and equilibrium in capital markets with safety first investors, Journal of Financial Economics, 4 (1977), 277-288. doi: 10.1016/0304-405X(77)90003-4.
[2]	V. S. Bawa, Optimal rules for ordering uncertain prospects, Journal of Financial Economics, 2 (1975), 95-121. doi: 10.1016/0304-405X(75)90025-2.
[3]	M. C. Chiu and D. Li, Asset-liability management under the safety-first principle, Optimization Theory and Applications, 143 (2009), 455-478. doi: 10.1007/s10957-009-9576-6.
[4]	S. Das, H. Markowitz, J. Scheid and M. Statman, Portfolio optimization with mental accounts, Journal of Financial and Quantitative Analysis, 45 (2010), 311-334. doi: 10.1017/S0022109010000141.
[5]	Y. Ding and B. Zhang, Risky asset pricing based on safety first fund management, Quantitative Finance, 9 (2009), 353-361. doi: 10.1080/14697680802392488.
[6]	Y. Ding and B. Zhang, Optimal portfolio of safety-first models, Journal of Statistical Planning and Inference, 139 (2009), 2952-2962. doi: 10.1016/j.jspi.2009.01.018.
[7]	M. Engels, Portfolio Optimization: Beyond Markowitz, Master's thesis, Leiden University,Netherlands, 2004.
[8]	P. C. Fishburn, Mean-risk analysis with risk associated with below-target returns, American Economical Review, 67 (1977), 116-126.
[9]	S. Kataoka, A stochastic programming model, Econometrica, 31 (1963), 181-196. doi: 10.2307/1910956.
[10]	K. Boda, J. A. Filar, Y. Lin and L. Spanjers, Stochastic target hitting time and the problem of early retirement, IEEE Transactions on Automatic Control, 49 (2004), 409-419. doi: 10.1109/TAC.2004.824469.
[11]	K. Boda and J. A. Filar, Time consistent dynamic risk measures, Mathematical Methods of Operations Research, 63 (2006), 169-186. doi: 10.1007/s00186-005-0045-1.
[12]	H. Levy and M. Levy, The safety first expected utility model: Experimental evidence and economic implications, Journal of Banking & Finance, 33 (2009), 1494-1506. doi: 10.1016/j.jbankfin.2009.02.014.
[13]	D. Li, T. F. Chan and W. L. Ng, Safety-first dynamic portfolio selection, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 585-600.
[14]	Z. F. Li, J. Yao and D. Li, Behavior patterns of investment strategies under Roy's safety-first principle, The Quarterly Review of Economics and Finance, 50 (2010), 167-179. doi: 10.1016/j.qref.2009.11.004.
[15]	H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 79-91.
[16]	R. C. Merton, An analytic derivation of the efficient portfolio frontier, The Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872. doi: 10.2307/2329621.
[17]	V. I. Norkin and S. V. Boyko, Safety-First Portfolio Selection, Cybernetics and Systems Analysis, 48 (2012), 180-191. doi: 10.1007/s10559-012-9396-9.
[18]	L. S. Ortobelli and S. T. Rachev, Safety-first analysis and stable paretian approach to portfolio choice theory, Mathematical and Computer Modelling, 34 (2001), 1037-1072. doi: 10.1016/S0895-7177(01)00116-9.
[19]	L. S. Ortobelli and F. Pellerey, Market stochastic bounds with elliptical distributions, Journal of Concrete and Applicable Mathematics, 6 (2008), 293-314.
[20]	A. D. Roy, Safety-first and the holding of assets, Econometrica, 20 (1952), 431-449. doi: 10.2307/1907413.
[21]	H. Shefrin and M. Statman, Behavioral portfolio theory, Journal of Financial and Quantitative Analysis, 35 (2000), 127-151. doi: 10.2307/2676187.
[22]	N. Signer, Safety-first portfolio optimization: Fixed versus random target, Thuenen-Series of Applied Economic Theory, No. 113, 2010. Available from: http://www.econstor.eu/bitstream/10419/74652/1/747604916.pdf.
[23]	L. G. Telser, Safety first and hedging, Review of Economic Studies, 23 (1955), 1-16. doi: 10.2307/2296146.
[24]	S. M. Zhang, S. Y. Wang and X. T. Deng, Portfolio Selection Theory with Different Interest Rates for Borrowing and Lending, Journal of Global Optimization, 28 (2004), 67-95. doi: 10.1023/B:JOGO.0000006719.64826.55.

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