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November  2020, 16(6): 2703-2721. doi: 10.3934/jimo.2019076

How's the performance of the optimized portfolios by safety-first rules: Theory with empirical comparisons

Yuanyao Ding 1, and  Zudi Lu 2,

1. 

Faculty of Business, Ningbo University, Ningbo 315211, China
2. 

Southampton Statistical Sciences Research Institute, and Mathematical Sciences, University of Southampton, SO17 1BJ, UK

Received  July 2018 Revised  February 2019 Published  November 2020 Early access  July 2019

Safety-first (SF) rules have been increasingly useful in particular for construction of optimal portfolios related to pension and other social insurance funds. How's the performance of the optimal portfolios constructed by different SF rules is an interesting practical question but yet less investigated theoretically. In this paper, we therefore analytically investigate the properties of the risky portfolios constructed by the three popular SF rules, denoted by the RSF, TSF and KSF, which are suggested and developed by A. D. Roy, L. G. Telser and S. Kataoka, respectively. Using Sharpe ratio as a measure of portfolio performance, we theoretically derive that the performance of an optimal portfolio constructed by the KSF approach depends on an acceptable level of extreme risk tolerance. The unique solution where the performance of the KSF portfolio is the same as that of the other two SF portfolios is found. By this we interestingly find that except this special case, under the finite optimal portfolios existent, the KSF portfolio always dominates the TSF portfolio in terms of the Sharpe ratio. In addition, in some market scenarios, even when the RSF and TSF portfolios do not exist in finite forms, the KSF rule can still apply to get a finite optimal portfolio. Moreover, in comparison with the RSF rule, a series of finite KSF portfolios can be interestingly constructed with their Sharpe ratios approaching to the maximum Sharpe ratio, which however cannot be reached by any corresponding finite RSF portfolio. Numerical comparisons of these rules by using a set of real data are further empirically demonstrated.

Keywords: Optimal portfolio, safety-first rule, Sharpe ratio, portfolio choice, extreme loss risk.
Mathematics Subject Classification: Primary: 91B28; Secondary: 90C90.
Citation: Yuanyao Ding, Zudi Lu. How's the performance of the optimized portfolios by safety-first rules: Theory with empirical comparisons. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2703-2721. doi: 10.3934/jimo.2019076
References:
[1]

V. S. Bawa, Safety-first, stochastic dominance, and optimal portfolio choice, Journal of Financial and Quantitative Analysis, 13 (1978), 255-271.  doi: 10.2307/2330386.

[2]

T. Bodnar and T. Zabolotskyy, How risky is the optimal portfolio which maximizes the sharpe ratio?, Asta Advances in Statistical Analysis, 101 (2017), 1-28.  doi: 10.1007/s10182-016-0270-3.

[3]

Y. Ding and B. Zhang, Risky asset pricing based on safety first fund management, Quantitative Finance, 9 (2009a), 353-361.  doi: 10.1080/14697680802392488.

[4]

Y. Ding and B. Zhang, Optimal portfolio of safety-first models, Journal of Statistical Planning and Inference, 139 (2009b), 2952-2962.  doi: 10.1016/j.jspi.2009.01.018.

[5]

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

[6]

R. B. Durand, H. Jafarpour, C. Kluppelberg and R. Maller, Maximize the Sharpe Ratio and Minimize a VaR, The Journal of Wealth Management, 13 (2010), 91-102. 

[7]

M. Engels, Portfolio Optimization: Beyond Markowitz, Master's thesis, Leiden University, Netherlands, 2004.

[8]

N. Gressis and W. A. Remaley, Comment: ``Safety first – an expected utility principle", Jounal of Financial and Quantitative Analysis, 9 (1974), 1057-1061. 

[9]

H. Hagigi and B. Kluger, Assessing return and risk of pension funds–portfolios by the Telser safety-first approach, Journal of Business Finance and Accounting, 14 (1987), 241-253. 

[10]

M. R. Haley and M. K. McGee, Tilting safety first and the Sharpe portfolio, Finance Research Letters, 3 (2006), 173-180. 

[11]

S. Kataoka, A stochastic programming model, Econometrica, 31 (1963), 181-196.  doi: 10.2307/1910956.

[12]

H. Levy and M. Sarnat, Safety first–an expected utility principle, Journal of Financial and Quantitative Analysis, 7 (1972), 1829-1834. 

[13]

Z. F. Li and G. J. Chen, Some discussions on Telser's safety-first model for portfolio selection (in Chinese), Theory and Practice of System Engineering, 36 (2005), 8-14. 

[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]

R. A. Maller and D. A. Turkington, New light on the portfolio allocation problem, Mathematical Methods of Operational Research, 56 (2002), 501-511.  doi: 10.1007/s001860200211.

[16]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. 

[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]

Y. Okhrin and W. Schmid, Distributional properties of portfolio weights, Journal of Econometrics, 134 (2006), 235-256.  doi: 10.1016/j.jeconom.2005.06.022.

[19]

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.

[20]

D. H. Pyle and S. J. Turnovsky, Safety-first and expected utility maximization in mean-standard deviation portfolio analysis, The Review of Economics and Statistics, 52 (1970), 75-81. 

[21]

A. D. Roy, Safety-first and the holding of assets, Econometrica, 20 (1952), 431-449.  doi: 10.2307/1907413.

[22]

W. F. Sharpe, The Sharpe Ratio, Journal of Portfolio Management, 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.

[23]

L. G. Telser, Safety first and hedging, Review of Economic Studies, 23 (1955), 1-16.  doi: 10.2307/2296146.

[24]

S. Wang and Y. Xia, Portfolio Selection and Asset Pricing, Springer-Verlag, Berlin Heidelberg New York, Printed in Germany, 2002. doi: 10.1007/978-3-642-55934-1.

show all references

References:
[1]

V. S. Bawa, Safety-first, stochastic dominance, and optimal portfolio choice, Journal of Financial and Quantitative Analysis, 13 (1978), 255-271.  doi: 10.2307/2330386.

[2]

T. Bodnar and T. Zabolotskyy, How risky is the optimal portfolio which maximizes the sharpe ratio?, Asta Advances in Statistical Analysis, 101 (2017), 1-28.  doi: 10.1007/s10182-016-0270-3.

[3]

Y. Ding and B. Zhang, Risky asset pricing based on safety first fund management, Quantitative Finance, 9 (2009a), 353-361.  doi: 10.1080/14697680802392488.

[4]

Y. Ding and B. Zhang, Optimal portfolio of safety-first models, Journal of Statistical Planning and Inference, 139 (2009b), 2952-2962.  doi: 10.1016/j.jspi.2009.01.018.

[5]

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

[6]

R. B. Durand, H. Jafarpour, C. Kluppelberg and R. Maller, Maximize the Sharpe Ratio and Minimize a VaR, The Journal of Wealth Management, 13 (2010), 91-102. 

[7]

M. Engels, Portfolio Optimization: Beyond Markowitz, Master's thesis, Leiden University, Netherlands, 2004.

[8]

N. Gressis and W. A. Remaley, Comment: ``Safety first – an expected utility principle", Jounal of Financial and Quantitative Analysis, 9 (1974), 1057-1061. 

[9]

H. Hagigi and B. Kluger, Assessing return and risk of pension funds–portfolios by the Telser safety-first approach, Journal of Business Finance and Accounting, 14 (1987), 241-253. 

[10]

M. R. Haley and M. K. McGee, Tilting safety first and the Sharpe portfolio, Finance Research Letters, 3 (2006), 173-180. 

[11]

S. Kataoka, A stochastic programming model, Econometrica, 31 (1963), 181-196.  doi: 10.2307/1910956.

[12]

H. Levy and M. Sarnat, Safety first–an expected utility principle, Journal of Financial and Quantitative Analysis, 7 (1972), 1829-1834. 

[13]

Z. F. Li and G. J. Chen, Some discussions on Telser's safety-first model for portfolio selection (in Chinese), Theory and Practice of System Engineering, 36 (2005), 8-14. 

[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]

R. A. Maller and D. A. Turkington, New light on the portfolio allocation problem, Mathematical Methods of Operational Research, 56 (2002), 501-511.  doi: 10.1007/s001860200211.

[16]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. 

[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]

Y. Okhrin and W. Schmid, Distributional properties of portfolio weights, Journal of Econometrics, 134 (2006), 235-256.  doi: 10.1016/j.jeconom.2005.06.022.

[19]

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.

[20]

D. H. Pyle and S. J. Turnovsky, Safety-first and expected utility maximization in mean-standard deviation portfolio analysis, The Review of Economics and Statistics, 52 (1970), 75-81. 

[21]

A. D. Roy, Safety-first and the holding of assets, Econometrica, 20 (1952), 431-449.  doi: 10.2307/1907413.

[22]

W. F. Sharpe, The Sharpe Ratio, Journal of Portfolio Management, 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.

[23]

L. G. Telser, Safety first and hedging, Review of Economic Studies, 23 (1955), 1-16.  doi: 10.2307/2296146.

[24]

S. Wang and Y. Xia, Portfolio Selection and Asset Pricing, Springer-Verlag, Berlin Heidelberg New York, Printed in Germany, 2002. doi: 10.1007/978-3-642-55934-1.

Figure 1.  RSF portfolio, KSF portfolio and TSF portfolio
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Figure 2.  Characteristics of Sharpe ratio of KSF portfolio vs. $ \alpha $ when $ R_b <\frac{B}{C} $
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Figure 3.  Comparison of $ P(X'R\le R_b) $ among the SF Portfolios
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Figure 4.  Comparison of $ P(X'R\le R_d) $ between KSF and RSF Portfolios
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Figure 5.  Comparison of Sharpe ratios of RSF, KSF and TSF portfolios
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Figure 6.  Extreme risks of KSF and TSF Portfolios with Equal Sharpe Ratios
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Figure 7.  Characteristics of Sharpe ratio of KSF portfolio vs. $ \alpha $ when $ R_b >\frac{B}{C} $
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Table 1.  Expected Returns and Covariance of Three Risky Assets
Covariance 1B0010 1B0011 1B0006 Returns
1B0010 0.00863 0.00657 0.00830 0.00497
1B0011 0.00657 0.00609 0.00648 0.01214
1B0006 0.00830 0.00648 0.01390 0.00613
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Covariance 1B0010 1B0011 1B0006 Returns
1B0010 0.00863 0.00657 0.00830 0.00497
1B0011 0.00657 0.00609 0.00648 0.01214
1B0006 0.00830 0.00648 0.01390 0.00613
Table 2.  Some results of RSF, TSF and KSF portfolios
Panel Rule $ P(R'X\le R_d) $ $ P(R'X\le R_b) $ $ X_1 $ $ X_2 $ $ X_3 $ $ SR $
RSF 0.40512 0.40512 -2.51889 3.51014 0.00875 0.24011
1 TSF 0.40512 0.40512 -2.51889 3.51014 0.00875 0.24011
KSF 0.40512 0.40512 -2.51889 3.51014 0.00875 0.24011
RSF 0.41019 0.40512 -2.51889 3.51014 0.00875 0.24011
2 TSF 0.41297 0.41000 -4.95741 5.94221 0.01520 0.22754
KSF 0.41000 0.40532 -2.83436 3.82477 0.00959 0.23960
RSF 0.41778 0.40512 -2.51889 3.51014 0.00875 0.24011
3 TSF 0.42074 0.41650 -8.91672 9.89105 0.02567 0.21086
KSF 0.41650 0.40657 -3.51776 4.50637 0.01139 0.23638
RSF 0.42255 0.40512 -2.51889 3.51014 0.00875 0.24011
4 TSF 0.42393 0.42000 -13.26792 14.23075 0.03718 0.20189
KSF 0.42000 0.40813 -4.17332 5.16010 0.01313 0.23235
RSF 0.11226 0.40512 -2.51889 3.51014 0.00875 0.24011
5 TSF N 0.05000 N N N N
KSF 0.05000 0.42963 -0.52073 1.51727 0.00347 0.17730
RSF 0.20922 0.40512 -2.51889 3.51014 0.00875 0.24011
6 TSF N 0.15000 N N N N
KSF 0.15000 0.42565 -0.64983 1.64602 0.00381 0.18745
RSF 0.40016 0.40512 -2.51889 3.51014 0.00875 0.24011
7 TSF N 0.35000 N N N N
KSF 0.35000 0.41119 -1.33611 2.33049 0.00562 0.22449
RSF 0.43132 0.40512 -2.51889 3.51014 0.00875 0.24011
8 TSF 0.42500 0.41307 -6.47843 7.45921 0.01922 0.21966
KSF 0.42500 0.41307 -6.47843 7.45921 0.01922 0.21966
RSF 0.32258 0.40512 -2.51889 3.51014 0.00875 0.24011
9 TSF 0.38805 0.41650 -8.91672 9.89105 0.02567 0.21086
KSF 0.30000 0.41650 -1.01839 2.01361 0.00478 0.21086
RSF 0.24886 0.40512 -2.51889 3.51014 0.00875 0.24011
10 TSF 0.40088 0.42330 -22.61759 23.55569 0.06190 0.19346
KSF 0.20000 0.42330 -0.73281 1.72878 0.00403 0.19346
RSF 0.16529 0.40512 -2.51889 3.51014 0.00875 0.24011
11 TSF 0.42240 0.42768 -160.68449 161.25747 0.42702 0.18229
KSF 0.10000 0.42768 -0.58263 1.57900 0.00363 0.18229
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Panel Rule $ P(R'X\le R_d) $ $ P(R'X\le R_b) $ $ X_1 $ $ X_2 $ $ X_3 $ $ SR $
RSF 0.40512 0.40512 -2.51889 3.51014 0.00875 0.24011
1 TSF 0.40512 0.40512 -2.51889 3.51014 0.00875 0.24011
KSF 0.40512 0.40512 -2.51889 3.51014 0.00875 0.24011
RSF 0.41019 0.40512 -2.51889 3.51014 0.00875 0.24011
2 TSF 0.41297 0.41000 -4.95741 5.94221 0.01520 0.22754
KSF 0.41000 0.40532 -2.83436 3.82477 0.00959 0.23960
RSF 0.41778 0.40512 -2.51889 3.51014 0.00875 0.24011
3 TSF 0.42074 0.41650 -8.91672 9.89105 0.02567 0.21086
KSF 0.41650 0.40657 -3.51776 4.50637 0.01139 0.23638
RSF 0.42255 0.40512 -2.51889 3.51014 0.00875 0.24011
4 TSF 0.42393 0.42000 -13.26792 14.23075 0.03718 0.20189
KSF 0.42000 0.40813 -4.17332 5.16010 0.01313 0.23235
RSF 0.11226 0.40512 -2.51889 3.51014 0.00875 0.24011
5 TSF N 0.05000 N N N N
KSF 0.05000 0.42963 -0.52073 1.51727 0.00347 0.17730
RSF 0.20922 0.40512 -2.51889 3.51014 0.00875 0.24011
6 TSF N 0.15000 N N N N
KSF 0.15000 0.42565 -0.64983 1.64602 0.00381 0.18745
RSF 0.40016 0.40512 -2.51889 3.51014 0.00875 0.24011
7 TSF N 0.35000 N N N N
KSF 0.35000 0.41119 -1.33611 2.33049 0.00562 0.22449
RSF 0.43132 0.40512 -2.51889 3.51014 0.00875 0.24011
8 TSF 0.42500 0.41307 -6.47843 7.45921 0.01922 0.21966
KSF 0.42500 0.41307 -6.47843 7.45921 0.01922 0.21966
RSF 0.32258 0.40512 -2.51889 3.51014 0.00875 0.24011
9 TSF 0.38805 0.41650 -8.91672 9.89105 0.02567 0.21086
KSF 0.30000 0.41650 -1.01839 2.01361 0.00478 0.21086
RSF 0.24886 0.40512 -2.51889 3.51014 0.00875 0.24011
10 TSF 0.40088 0.42330 -22.61759 23.55569 0.06190 0.19346
KSF 0.20000 0.42330 -0.73281 1.72878 0.00403 0.19346
RSF 0.16529 0.40512 -2.51889 3.51014 0.00875 0.24011
11 TSF 0.42240 0.42768 -160.68449 161.25747 0.42702 0.18229
KSF 0.10000 0.42768 -0.58263 1.57900 0.00363 0.18229
Table 3.  Expected Returns and Covariance of Five Risky Assets
Covariance 1B0012 1B0013 1B0010 1B0011 1B0006 Returns
1B0012 8.62E-05 1.07E-04 -4.22E-05 -5.38E-05 -5.25E-05 0.00246
1B0013 1.07E-04 1.72E-04 -1.62E-04 -1.61E-04 -5.79E-05 0.00351
1B0010 -4.22E-05 -1.62E-04 0.00863 0.00657 0.00830 0.00497
1B0011 -5.38E-05 -1.61E-04 0.00657 0.00609 0.00648 0.01214
1B0006 -5.25E-05 -5.79E-05 0.00830 0.00648 0.01390 0.00613
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Covariance 1B0012 1B0013 1B0010 1B0011 1B0006 Returns
1B0012 8.62E-05 1.07E-04 -4.22E-05 -5.38E-05 -5.25E-05 0.00246
1B0013 1.07E-04 1.72E-04 -1.62E-04 -1.61E-04 -5.79E-05 0.00351
1B0010 -4.22E-05 -1.62E-04 0.00863 0.00657 0.00830 0.00497
1B0011 -5.38E-05 -1.61E-04 0.00657 0.00609 0.00648 0.01214
1B0006 -5.25E-05 -5.79E-05 0.00830 0.00648 0.01390 0.00613
Table 4.  Expected Returns and Covariance of Five Risky Assets
weight $ \alpha $
0.10000 0.20000 0.30000 0.35000 0.36000 0.37000 0.37984
$ X_1 $ 1.21251 1.07523 0.76327 0.20470 -0.11043 -0.81445 -93.92237
$ X_2 $ -0.24424 -0.11734 0.17103 0.68736 0.97865 1.62943 87.69587
$ X_3 $ -0.05544 -0.07397 -0.11610 -0.19151 -0.23406 -0.32912 -12.90054
$ X_4 $ 0.08093 0.11220 0.18327 0.31053 0.38232 0.54272 21.75489
$ X_5 $ 0.00624 0.00388 -0.00148 -0.01107 -0.01648 -0.02858 -1.62785
$ SR $ -0.21947 -0.16942 -0.06622 0.05969 0.10400 0.16515 0.30287
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weight $ \alpha $
0.10000 0.20000 0.30000 0.35000 0.36000 0.37000 0.37984
$ X_1 $ 1.21251 1.07523 0.76327 0.20470 -0.11043 -0.81445 -93.92237
$ X_2 $ -0.24424 -0.11734 0.17103 0.68736 0.97865 1.62943 87.69587
$ X_3 $ -0.05544 -0.07397 -0.11610 -0.19151 -0.23406 -0.32912 -12.90054
$ X_4 $ 0.08093 0.11220 0.18327 0.31053 0.38232 0.54272 21.75489
$ X_5 $ 0.00624 0.00388 -0.00148 -0.01107 -0.01648 -0.02858 -1.62785
$ SR $ -0.21947 -0.16942 -0.06622 0.05969 0.10400 0.16515 0.30287
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