基于绩效正则化法的均值-下偏矩投资组合选择

doi:10.16381/j.cnki.issn1003-207x.2022.0262

摘要/Abstract

摘要：

Ban等^［1］提出了绩效正则化法（performance-based regularization，PBR），通过向均值-方差模型、均值-CVaR模型中加入PBR约束条件，从而减小参数估计误差，提高模型的样本外表现。本文则将他们的PBR约束条件推广到更一般化的均值-下偏矩模型中。具体地，本文证明了含PBR约束条件的均值-下偏矩模型的凸性，从而保证优化问题的解是全局最优解。进一步地，证明了该优化问题解的渐近一致性。此外，改造了标准的k-折交叉检验法，并通过线性回溯法实时更新PBR约束条件的最优参数取值。最后，将该模型置于震荡的市场中进行实证检验。结果显示，相较于传统的均值-下偏矩模型，加入两个PBR约束条件的均值-下偏矩模型具有更高的夏普比率、索提诺比率、Omega比率、平均收益率和累计收益率，从而更好地在收益与风险之间取得平衡；此外，提出的带PBR约束的均值-下偏矩模型的表现优于Ban等^［1］提出的带PBR约束的均值-方差模型。

关键词: 均值-下偏矩, k-折交叉检验, 线性回溯法, 绩效正则化法

Abstract:

The core of portfolio optimization problem is how to balance return and risk in the process of investment. The mean-variance model is a classical model to solve this problem. However， with the deepening of research， the variance as a risk measure cannot adapt to the real market situation， and the mean lower partial moment model came into being. At the same time， the mean-risk model often confronted the problem of parameter estimation error. Parameter uncertainty and parameter estimation error will have a negative impact on investment decision-making. Therefore， solving this problem has become a hot research direction.It is refered to Ban et al. ［Management Science， 2016， 63， 1136-1154］ that proposed performance-based regularization （PBR）. By adding PBR constraints to the mean variance model and the mean CVaR model， they reduced the parameter estimation error and improved the out of sample performance of the model. The core idea of PBR method is to improve the performance of investment strategy outside the sample by punishing the variance term of risk measurement and expected return estimate.In this paper， their PBR constraints are extended to a more general model， which is mean-lower partial moment （LPM） model. Specifically， the convexity of the mean-LPM model with PBR constraints is proved， so as to ensure that the solution of the optimization problem is globally optimal. Furthermore， the asymptotic consistency of the solution of the optimization problem is proved. In addition， the standard k-fold cross test method is modified， and the optimal parameter values of PBR constraint conditions in real time are updated by using machine learning method and linear backtracking method. Finally， the model in the downward market is put for empirical test.As a result， it is found out that compared with the traditional mean-LPM model， the mean-LPM model with two PBR constraints has higher Sharpe ratio， Sortino ratio， Omega ratio， average rate of return and cumulative rate of return， so as to achieve a better balance between return and risk. In addition， the mean-LPM model combined with PBR constraint performs better than the mean variance model combined with PBR constraint in Ban et al. （2016）.In this paper， the PBR method is extended to the general mean-LPM portfolio model， and the effectiveness of the PBR method for solving the parameter estimation error is confirmed. Regularization method has a wide range of applications， and the research of PBR method still has broad prospects. In the next step， the PBR method can be applied to the index tracking problem， or further the nature and other applicable fields of the PBR method can be explored.

Key words: mean-lower partial moment, k-fold cross test, linear back tracking method, performance-based regularization

中图分类号:

F830

姚海祥,张逸雪,李仲飞. 基于绩效正则化法的均值-下偏矩投资组合选择[J]. 中国管理科学, 2026, 34(10): 1-13.

Haixiang Yao,Yixue Zhang,Zhongfei Li. Mean-lower Partial Moment Portfolio Selection Based on Performance-based Regularization Method[J]. Chinese Journal of Management Science, 2026, 34(10): 1-13.

图/表 6

图1

表1

表2

沪深300指数中30只成分股关于投资绩效评价指标的实证结果"

下偏矩阶数	期望收益率R	$0.9 ⋅ E π$					$1.0 ⋅ E π$					$1.1 ⋅ E π$
下偏矩阶数	期望收益率R	MV- PBR	SAA	PBR-m	PBR-z	PBR- both	MV- PBR	SAA	PBR-m	PBR-z	PBR- both	MV- PBR	SAA	PBR-m	PBR-z	PBR- both
1	平均回报率	-0.0004	-0.0003	-0.0003	-0.0003	0.0004	-0.0003	-0.0002	-0.0002	-0.0002	0.0005	-0.0003	-0.0001	-0.0001	-0.0003	0.0006
	夏普比率	-0.0383	-0.0292	-0.0284	-0.0287	0.0327	-0.0342	-0.0238	-0.0216	-0.0180	0.0405	-0.0303	-0.0164	-0.0149	-0.0255	0.0479
	索提诺比率	-0.0436	-0.0321	-0.0310	-0.0314	0.0521	-0.0386	-0.0252	-0.0222	-0.0174	0.0630	-0.0336	-0.0155	-0.0136	-0.0274	0.0737
	Omega比率	0.9155	0.9363	0.9385	0.9390	1.1081	0.9253	0.9504	0.9561	0.9658	1.1305	0.9350	0.9693	0.9732	0.9466	1.1519
	换手率	0.1168	0.1476	0.1431	0.1539	0.1420	0.1262	0.1536	0.1514	0.1822	0.1485	0.1347	0.1567	0.1494	0.1732	0.1498
	累计收益率	0.9583	0.9668	0.9677	0.9673	1.0354	0.9616	0.9722	0.9747	0.9783	1.0448	0.9649	0.9800	0.9816	0.9690	1.0540
2	平均回报率	-0.0004	-0.0004	-0.0004	-0.0003	0.0004	-0.0003	-0.0003	-0.0003	-0.0002	0.0005	-0.0003	-0.0003	-0.0003	-0.0002	0.0006
	夏普比率	-0.0383	-0.0357	-0.0371	-0.0276	0.0338	-0.0342	-0.0313	-0.0321	-0.0238	0.0422	-0.0303	-0.0262	-0.0268	-0.0203	0.0500
	索提诺比率	-0.0436	-0.0403	-0.0418	-0.0298	0.0566	-0.0386	-0.0347	-0.0359	-0.0250	0.0689	-0.0336	-0.0283	-0.0291	-0.0206	0.0803
	Omega比率	0.9155	0.9214	0.9182	0.9418	1.1067	0.9253	0.9323	0.9305	0.9510	1.1302	0.9350	0.9447	0.9434	0.9598	1.1524
	换手率	0.1168	0.1203	0.1785	0.1420	0.1201	0.1262	0.1294	0.1880	0.1468	0.1292	0.1347	0.1376	0.2105	0.1569	0.1374
	累计收益率	0.9583	0.9604	0.9599	0.9688	1.0352	0.9616	0.9642	0.9638	0.9721	1.0448	0.9649	0.9689	0.9684	0.9753	1.0542
3	平均回报率	-0.0004	-0.0002	-0.0003	-0.0001	0.0006	-0.0003	-0.0001	-0.0003	-0.0001	0.0007	-0.0003	-0.0001	-0.0003	0.0000	0.0008
	夏普比率	-0.0383	-0.0191	-0.0348	-0.0169	0.0505	-0.0342	-0.0142	-0.0309	-0.0097	0.0586	-0.0303	-0.0097	-0.0320	-0.0043	0.0663
	索提诺比率	-0.0436	-0.0189	-0.0390	-0.0159	0.0835	-0.0386	-0.0126	-0.0344	-0.0066	0.0958	-0.0336	-0.0066	-0.0359	0.0005	0.1076
	Omega比率	0.9155	0.9630	0.9243	0.9689	1.1515	0.9253	0.9754	0.9336	0.9871	1.1743	0.9350	0.9871	0.9306	1.0009	1.1966
	换手率	0.1168	0.1183	0.2192	0.1323	0.1174	0.1262	0.1281	0.2394	0.1384	0.1285	0.1347	0.1364	0.2172	0.1458	0.1364
	累计收益率	0.9583	0.9777	0.9624	0.9801	1.0528	0.9616	0.9825	0.9652	0.9875	1.0627	0.9649	0.9872	0.9626	0.9933	1.0727
4	平均回报率	-0.0004	0.0000	-0.0004	0.0000	0.0007	-0.0003	0.0000	-0.0003	0.0000	0.0008	-0.0003	0.0001	-0.0002	0.0001	0.0009
	夏普比率	-0.0383	-0.0064	-0.0369	-0.0064	0.0607	-0.0342	-0.0030	-0.0336	-0.0030	0.0675	-0.0303	0.0004	-0.0247	0.0003	0.0772
	索提诺比率	-0.0436	-0.0021	-0.0416	-0.0021	0.1005	-0.0386	0.0024	-0.0378	0.0024	0.1110	-0.0336	0.0068	-0.0260	0.0067	0.1263
	Omega比率	0.9155	0.9958	0.9192	0.9958	1.1804	0.9253	1.0047	0.9271	1.0047	1.2000	0.9350	1.0135	0.9491	1.0131	1.2281
	换手率	0.1168	0.1174	0.2349	0.1174	0.1191	0.1262	0.1266	0.2625	0.1267	0.1300	0.1347	0.1364	0.2716	0.1365	0.1384
	累计收益率	0.9583	0.9915	0.9599	0.9915	1.0643	0.9616	0.9951	0.9623	0.9951	1.0731	0.9649	0.9988	0.9709	0.9986	1.0858

表2

表3

上证50指数中50只成分股的关于投资绩效评价指标的实证结果"

下偏矩阶数	期望收益率R	$0.9 · E π$					$1.0 · E π$					$1.1 · E π$
下偏矩阶数	期望收益率R	MV- PBR	SAA	PBR-m	PBR-z	PBR- both	MV- PBR	SAA	PBR-m	PBR-z	PBR- both	MV- PBR	SAA	PBR-m	PBR-z	PBR- both
1	平均回报率	-0.0008	-0.0009	-0.0010	-0.0007	-0.0005	-0.0008	-0.0009	-0.0010	-0.0008	-0.0005	-0.0008	-0.0009	-0.0009	-0.0009	-0.0004
	夏普比率	-0.0954	-0.1006	-0.1028	-0.0830	-0.0592	-0.0947	-0.1019	-0.1035	-0.0917	-0.0545	-0.0939	-0.0994	-0.1006	-0.0960	-0.0530
	索提诺比率	-0.1076	-0.0097	-0.0098	-0.0094	-0.0092	-0.1067	-0.0098	-0.0098	-0.0094	-0.0092	-0.1059	-0.0098	-0.0098	-0.0094	-0.0092
	Omega比率	0.7716	0.7573	0.7517	0.7971	0.8509	0.7739	0.7555	0.7514	0.7781	0.8645	0.7763	0.7618	0.7586	0.7716	0.8693
	换手率	0.1419	0.1910	0.1903	0.3696	0.1827	0.1451	0.1961	0.1920	0.3748	0.1884	0.1482	0.1987	0.1997	0.3423	0.1890
	累计收益率	0.9177	0.9092	0.9068	0.9262	0.9486	0.9180	0.9077	0.9062	0.9176	0.9525	0.9184	0.9097	0.9085	0.9131	0.9534
2	平均回报率	-0.0008	-0.0009	-0.0009	-0.0006	-0.0003	-0.0008	-0.0009	-0.0008	-0.0007	-0.0004	-0.0008	-0.0009	-0.0009	-0.0007	-0.0003
	夏普比率	-0.0954	-0.0977	-0.1005	-0.0649	-0.0437	-0.0947	-0.0987	-0.0946	-0.0773	-0.0502	-0.0939	-0.0993	-0.0969	-0.0750	-0.0428
	索提诺比率	-0.1076	-0.0094	-0.0094	-0.0092	-0.0074	-0.1067	-0.0094	-0.0092	-0.0092	-0.0078	-0.1059	-0.0094	-0.0093	-0.0092	-0.0074
	Omega比率	0.7716	0.7663	0.7590	0.8402	0.9016	0.7739	0.7647	0.7762	0.8114	0.8839	0.7763	0.7641	0.7708	0.8172	0.9042
	换手率	0.1419	0.1462	0.2134	0.2336	0.1954	0.1451	0.1510	0.2437	0.2189	0.1933	0.1482	0.1558	0.2350	0.2424	0.2161
	累计收益率	0.9177	0.9144	0.9132	0.9399	0.9672	0.9180	0.9133	0.9180	0.9288	0.9609	0.9184	0.9125	0.9158	0.9302	0.9675
3	平均回报率	-0.0008	-0.0005	-0.0008	-0.0004	-0.0004	-0.0008	-0.0005	-0.0005	-0.0004	-0.0002	-0.0008	-0.0005	-0.0008	-0.0004	-0.0002
	夏普比率	-0.0954	-0.0568	-0.0889	-0.0417	-0.0493	-0.0947	-0.0579	-0.0637	-0.0415	-0.0341	-0.0939	-0.0584	-0.0875	-0.0435	-0.0341
	索提诺比率	-0.1076	-0.0090	-0.0091	-0.0090	-0.0075	-0.1067	-0.0090	-0.0089	-0.0090	-0.0072	-0.1059	-0.0091	-0.0091	-0.0091	-0.0074
	Omega比率	0.7716	0.8623	0.7842	0.8980	0.8872	0.7739	0.8597	0.8467	0.8976	0.9263	0.7763	0.8587	0.7909	0.8929	0.9259
	换手率	0.1419	0.1388	0.4629	0.1928	0.1641	0.1451	0.1419	0.4285	0.1862	0.1615	0.1482	0.1429	0.4026	0.1821	0.1980
	累计收益率	0.9177	0.9468	0.9246	0.9578	0.9602	0.9180	0.9456	0.9439	0.9570	0.9728	0.9184	0.9449	0.9241	0.9554	0.9728
4	平均回报率	-0.0008	-0.0004	-0.0006	-0.0004	-0.0003	-0.0008	-0.0004	-0.0006	-0.0004	-0.0003	-0.0008	-0.0004	-0.0006	-0.0004	-0.0003
	夏普比率	-0.0954	-0.0442	-0.0680	-0.0451	-0.0390	-0.0947	-0.0438	-0.0686	-0.0443	-0.0362	-0.0939	-0.0430	-0.0628	-0.0428	-0.0372
	索提诺比率	-0.1076	-0.0089	-0.0089	-0.0089	-0.0074	-0.1067	-0.0089	-0.0090	-0.0090	-0.0074	-0.1059	-0.0089	-0.0091	-0.0089	-0.0074
	Omega比率	0.7716	0.8919	0.8332	0.8893	0.9117	0.7739	0.8931	0.8337	0.8918	0.9192	0.7763	0.8954	0.8500	0.8959	0.9166
	换手率	0.1419	0.1451	0.3491	0.1532	0.1479	0.1451	0.1482	0.3856	0.1630	0.1555	0.1482	0.1514	0.3662	0.1522	0.1555
	累计收益率	0.9177	0.9557	0.9388	0.9547	0.9670	0.9180	0.9559	0.9374	0.9553	0.9694	0.9184	0.9565	0.9416	0.9568	0.9684

表3

图2

图3

参考文献 26

[1]	Ban G Y， El Karoui N， Lim A E B. Machine learning and portfolio optimization［J］.Management Science， 2016， 64（3）： 1136-1154.
[2]	Markowitz H. Portfolio selection［J］. The Journal of Finance， 1952， 7（1）： 77-91.
[3]	Markowitz H M. Portfolio selection： Efficient diversification of investments［M］.New York：John Wiley， 1959.
[4]	Fishburn P C. Mean-risk analysis with risk associated with below-target returns［J］. American Economic Review， 1977， 67（2）： 116-126.
[5]	Bawa V S， Lindenberg E B. Capital market equilibrium in a mean-lower partial moment framework［J］. Journal of Financial Economics， 1977， 5（2）： 189-200.
[6]	Mansini R， Speranza M G. Heuristic algorithms for the portfolio selection problem with minimum transaction lots［J］. European Journal of Operational Research， 1999， 114（2）： 219-233.
[7]	Cai J， Jiao Z， Mao T. Worst-case values of target semi-variances with applications to robust portfolio selection［J］. European Journal of Operational Research， 2025， 327（3）： 905-921.
[8]	Mei X， Wang Y， Zhu W. Bayesian nonparametric portfolio selection with rolling maximum drawdown control［J］. Quantitative Finance， 2023， 23（10）： 1497-1510.
[9]	Yang X， Chen J， Liu W， et al. A multi-period fuzzy portfolio optimization model with investors’ loss aversion［J］. Soft Computing， 2023， 27（24）： 18829-18842.
[10]	黄金波，李仲飞，邹新月. 考虑下偏矩约束的增强指数模型［J］. 管理科学学报， 2019， 22（12）： 56-69.
	Huang J B， Li Z F， Zou X Y. Enhanced indexation model with lower partial moment constraint［J］. Journal of Management Sciences in China，2019，22（12）： 56-69.
[11]	Lassance N， Martín-Utrera A， Simaan M. The risk of expected utility under parameter uncertainty［J］. Management Science， 2024， 70（11）： 7644-7663.
[12]	Shi F， Shu L， He F， et al. Improving minimum-variance portfolio through shrinkage of large covariance matrices［J］.Economic Modelling， 2025， 144： 106981.
[13]	Tu J， Zhou G. Markowitz meets talmud： A combination of sophisticated and naive diversification strategies［J］.Journal of Financial Economics，2011，99（1）： 204-215.
[14]	Wu Z， Xie G， Ge Z， et al. Nonconvex multi-period mean-variance portfolio optimization［J］. Annals of Operations Research， 2024， 332（1）： 617-644.
[15]	Zhao H， Kong L. Penalty method for the sparse portfolio optimization problem［J］. Journal of Industrial and Management Optimization， 2024， 20（9）： 2864-2884.
[16]	Bian Z， Liao Y， O’Neill M， et al. Large-scale minimum variance portfolio allocation using double regularization［J］. Journal of Economic Dynamics and Control， 2020， 116： 103939.
[17]	宋鹏，刘程程，胡永宏. 稳健高维协方差矩阵估计及其投资组合应用——基于中心正则化算法［J］. 统计研究， 2020， 37（7）： 116-128.
	Song P， Liu C C， Hu Y H. Robust high-dimensional covariance matrix estimation and the application in portfolio selection—Based on the central-regularized algorithm［J］.Statistical Research，2020， 37（7）： 116-128.
[18]	Li Q， Zhang W. Sparse and risk diversification portfolio selection［J］.Optimization Letters，2023，17（5）： 1181-1200.
[19]	Wu Z， Sun K， Ge Z， et al. Sparse portfolio optimization via ℓ 1 over ℓ 2 regularization［J］. European Journal of Operational Research， 2024， 319（3）： 820-833.
[20]	Kroll Y， Marchioni A， Ben-Horin M. Sortino（γ）： A modified sortino ratio with adjusted threshold［J］. Journal of Accounting and Finance， 2023， 23（6）： 6699.
[21]	Ghaemi Asl M， Rashidi M M， Tavakkoli H R， et al. Does Islamic investing modify portfolio performance？ Time-varying optimization strategies for conventional and Shariah energy-ESG-utilities portfolio［J］.The Quarterly Review of Economics and Finance， 2024， 94： 37-57.
[22]	DeMiguel V， Garlappi L， Nogales F J， et al. A generalized approach to portfolio optimization： Improving performance by constraining portfolio norms［J］. Management Science， 2009， 55（5）： 798-812.
[23]	陈宝林. 最优化理论与算法［M］. 2版. 北京：清华大学出版社， 2005.
	Chen B L. Optimization theory and algorithm［M］. 2nd ed. Beijing： Tsinghua University Press， 2005.
[24]	Boyd S， Vandenberghe L. Convex optimization［M］. Cambridge， UK： Cambridge University Press， 2004.
[25]	Van der Vaart A W. Asymptotic statistics［M］. Cambridge， UK； New York， NY， USA： Cambridge University Press， 1998.
[26]	Yao H， Huang J， Li Y， et al. A general approach to smooth and convex portfolio optimization using lower partial moments［J］. Journal of Banking & Finance， 2021， 129： 106167.