在线投资组合策略及算法研究
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摘要
投资决策的核心问题是如何在不确定环境下选择最优的资产组合进行投资。由于金融市场是一个极其复杂的系统，投资者进行投资活动时面临着环境的不断变化。以实现收益最大化为目的的投资者需在对未来信息一无所知的情形下，根据当前环境不断地调整投资策略。因此，投资组合选择是一个在线决策问题。以Markowitz提出的均值-方差模型为基础的现代投资组合理论主要在静态情形下研究，而有关动态情形下的研究成果相对较少。近年来，随着在线学习算法的广泛应用，在线算法逐渐运用到投资组合选择问题的研究中，使得以泛证券投资组合为基础的在线投资组合理论研究得以快速发展。该方法的主要思想是不考虑证券价格所遵循的任何模型，对未来信息不做任何统计假设前提下，通过在线学习的方法提出序贯决策方法。由于其不考虑或甚少考虑证券市场运动规律，从而导致它的实验效果不佳和应用价值不高。针对泛证券投资组合研究中存在的不足，本文从市场“异象”的角度出发，借助在线学习算法对投资组合问题进行深入系统地研究。本文主要研究工作和创新如下：
1.综合利用反转和动量现象建立了相关关系指标，来度量策略转移比例，借助启发式算法得到反转在线投资策略。由于已有启发法式算法仅考虑证券的反转现象，完全忽视动量现象的存在。实践中收益不佳且严重依赖于历史窗口大小的取值，为消除窗口大小的影响要进行大量的复合运算，使得其方法失去一定的时效性。针对上述问题，本文综合考虑反转和动量两种现象，构建相关关系指标。进一步地，分析相关程度的强弱对决策转移的影响，借助启发式算法更全面地把握市场运动规律，进而得出在线投资组合策略。利用国外不同金融市场的六个数据集和国内两市的四个数据集进行实证分析。证结果表明在国外金融市场的收益显著提高，而在国内证券市场上累计收益不显著依赖窗口大小。这就表明所得的策略是稳定的。
2.提出了利用均值回归的非对称性构建多分段损失函数，借助PA分类学习算法优化投资组合策略，得到了在线投资组合策略PACS。建立在PA分类算法基础上的在线投资组合策略，简单利用了均值回归理论，难以准确刻画市场运动规律。针对此问题，依据均值回归的非对称性，本文设计了旨在更准确地捕捉金融市场波动特征的多分段损失函数。在损失最小约束条件下，利用所构建的多分段损失函数构建优化模型，以寻求变化最小的投资比例。进一步地，利用最优化原理推导出在线算法，得到适用于不同市场情形的反转策略。在允许卖空情形下还得到了算法的损失界，同时不允许卖空情形下考察了交易费用对策略的影响。理论上，算法具有线性时间复杂度；通过实证分析发现该算法在多数市场上的收益有显著提高。这表明算法具有一定的实用性和可操作性。
3.构建了权重函数来研究多期在线投资组合模型，充分利用利用历史价格信息，得到了收益更好的在线投资策略。现有的有关单期在线投资问题的研究在进行策略的调整时会给算法带来较大的盲目性，导致错失绝大部分历史数据信息。本文以既充分利用历史数据信息又不带来过大计算量为研究的出发点，展开多期的研究。以离当期时间越近赋权越大为原则构建权重函数，得到移动窗口下的加权平均价格序列。在此基础上构造损失函数，并从两个方面建立优化模型。一方面直接优化投资组合策略；另一方面，建立在预测未来价格基础上，利用PA分类算法和优化原理推导出加权移动平均的在线投资组合算法，并得到了不同形式的反转策略。进一步地，将行为金融学中反转的时间区间作为边侧信息，来确定历史窗口大小，以降低算法的时间复杂度。算法具有线性时间复杂度，利于大规模计算，实证研究中也获得了优异的收益。因此，带有权重的损失函数的构造方法扩展了利用分类算法研究在线投资组合问题的研究方法。
4.利用动量效应和投资者心理预期构建了不敏感损失函数，以此建立了优化模型，得到了动量在线投资策略。上述方法多利用反转特征，无法有效解决动量效应市场的投资决策问题。本文提出根据证券市场“异象”特征灵活设计投资策略的研究思路，将动量效应和投资主体主观态度相结合来构建不敏感损失函数，利用PA分类算法建立优化模型，得到保守策略和动量策略的转换机制，算法具有线性时间复杂度。该策略运用于国内外不同证券市场，进行比较分析，发现该策略在动量效应金融市场上收益很好，而对于反转效应的金融市场收益几乎为零。结果表明该策略适用于动量效应的金融市场。因此，它是在线反转策略的一个有益补充。
The core problem of investment decision is how to allocate financial assets to achieve anoptimal portfolio under uncertain circumstance. Because financial market is an extremelycomplex system, the investors are faced with an endlessly changing situation when they areengaging in investment practices. In the case where the information about future isunavailable, the investors who try to maximize the returns of their investments need to updatetheir investment strategy according to the current circumstance. Consequently, portfolioselection problem actually is an online one which sequentially determines portfolios usingavailable public information. The modern portfolio selection theory based on Markowitz’smean-variance model has been extensively studied by many researchers and lots of valuableresults have been obtained in the static case; however, people have seldom dealt with it in thedynamic case. In recent years, online algorithm has been applied to portfolio selection as wellas other fields, and it then helps develop the theory of online portfolio selection which isbased on universal portfolio. The main idea of this methodology is that proposingsequential-decision steps by online learning techniques, without making any statisticalhypotheses about the behavior of the asset price and the future information. However, since itdoes not figure out how the equity market behaves, the empirical results show that thismethod does not perform well and the potential application is negative. Taking thedisadvantages of the existing researches on universal portfolio into account, this thesis, fromthe respective of market’s anomaly, tries to explore the portfolio selection problemsystematically and thoroughly by means of online learning algorithm. The main contributionsof this thesis are listed in the following four aspects:
First, this thesis combines the reversal effect and the momentum effect to describestatistical correlations among several stocks and then calculate the proportion of weighttransferred. According to the correlations, a combined mean reversion online strategy ispresented by applying the heuristic algorithm. The existing heuristic algorithms such asAnticor only consider the reversal phenomenon of the price and often produce sub-optimalresults. In practice, they are with depressed earnings. Meanwhile, they depend significantly onthe window size and large-scale computation is required to eliminate the effect, whicheventually loses their timeliness. Thus, this article synthesizes the two phenomena mentionedabove and constructs the statistical correlation indexes to solve these problems. The thesis further analyzes the impact the degree of correlation intensity has on decision transfer,comprehensively grasps the fluctuation of price in stock markets by applying the heuristicalgorithm, and then obtains the online portfolio strategies. Moreover, empirical analysis isbased on six data sets from different foreign financial markets and four sets from twodomestic markets. The results show that they significantly increase the returns in foreignfinancial markets, while the cumulative returns in the domestic stock market are notsignificantly dependent on the window size. In all, they indicate the obtained strategies arestable.
Second, we propose to take use of the asymmetry of mean reversion to constructpiecewise loss function and PA (passive aggressive) algorithm to optimize the strategy ofportfolio selection. Because the strategy of online portfolio selection based on PAclassification algorithm just simply applies mean reversion theory, it is difficult for it todescribe the law of market behavior. To overcome this problem, according to the symmetry ofmean reversion, we construct a multi-piecewise loss function which does a better work indescribing the characteristics of the volatility of financial market. With the loss minimized, weuse the multi-piecewise loss function to build the most optimal model and then find the mostoptimal investment strategy which has the most minimal variance. Moreover, by means ofoptimization method, we get the reversion strategy which is applicable to all kinds of marketconditions. If short selling is allowed in the market, the bounds of the error of this algorithmis derived, and if not, the effect transaction cost has on the strategy is also explored.Theoretically, the algorithm possesses the property of linear time complexity. Besides, it hasbeen shown by empirical analysis that the returns of the selected portfolios have improvedgreatly in most of the markets. Hence the algorithm is applicable and practical, to someextent.
Third, we construct a weighting function in order to study the multiple-period onlineportfolio model. After making full use of the historical price information, we get the onlineinvestment strategy with higher benefits. Now the existing researches on the problem ofsingle-period online investment brings about big algorithm blindness when making strategyadjustment, which leads to the missing of most of the historical data information. In this thesis,we study the multi-period case with keeping a balance between making full use of historicaldata and avoiding large-scale computation. The moving weighted average sequences on atime window is derived by the way in which the shorter it is from current time, the moreweight is assigned to. Taking the above as a basis, we construct the loss function, and thenestablish the optimization model from the following two aspects. One is to optimize the strategy of portfolio model directly. The other is using the PAC classification algorithm andoptimization principle to derive the weighted moving average online portfolio algorithm anddifferent kinds of reversion strategies, based on predicating the future prices. Furthermore, inorder to reduce the time complexity of the algorithm, we determine the history window sizeby using the reverse time interval in the field of behavioral finance as side information. Thealgorithm with linear time complexity is conducive to large-scale computation and theempirical study also show that the strategy receives excellent benefits. Therefore, theweighted loss function method extends the way of the classification algorithm to study onlineportfolio problems of the research methods.
Forth, we construct an insensitive loss function by using the momentum effect andinvestor expectations. Then we establish optimization model, and get the momentum onlineportfolio strategy. The above method which using the property of price reversal is unable toeffectively solve the problem of investment decisions in the momentum effect market. In thispaper, we come up with a research approach which flexibly designs the investment strategyreferring to the characteristics of the equity market “anomalies”, and we build the lossfunction by means of combining the momentum effect and investors’ subjective attitude. Byusing PA classification algorithm we build optimization model, and get the transformationmechanism with conservative strategy and momentum strategy. The algorithm is providedwith linear time complexity. By testing and comparatively analyzing the strategy in differentstock markets, we find the strategy on financial market with momentum effect is very good,while the financial market with reversal effect gains almost zero. This suggests that thestrategy is suitable to be applied in the financial markets with momentum effect. Therefore, itis a beneficial supplement of contrarian strategy.
1.综合利用反转和动量现象建立了相关关系指标，来度量策略转移比例，借助启发式算法得到反转在线投资策略。由于已有启发法式算法仅考虑证券的反转现象，完全忽视动量现象的存在。实践中收益不佳且严重依赖于历史窗口大小的取值，为消除窗口大小的影响要进行大量的复合运算，使得其方法失去一定的时效性。针对上述问题，本文综合考虑反转和动量两种现象，构建相关关系指标。进一步地，分析相关程度的强弱对决策转移的影响，借助启发式算法更全面地把握市场运动规律，进而得出在线投资组合策略。利用国外不同金融市场的六个数据集和国内两市的四个数据集进行实证分析。证结果表明在国外金融市场的收益显著提高，而在国内证券市场上累计收益不显著依赖窗口大小。这就表明所得的策略是稳定的。
2.提出了利用均值回归的非对称性构建多分段损失函数，借助PA分类学习算法优化投资组合策略，得到了在线投资组合策略PACS。建立在PA分类算法基础上的在线投资组合策略，简单利用了均值回归理论，难以准确刻画市场运动规律。针对此问题，依据均值回归的非对称性，本文设计了旨在更准确地捕捉金融市场波动特征的多分段损失函数。在损失最小约束条件下，利用所构建的多分段损失函数构建优化模型，以寻求变化最小的投资比例。进一步地，利用最优化原理推导出在线算法，得到适用于不同市场情形的反转策略。在允许卖空情形下还得到了算法的损失界，同时不允许卖空情形下考察了交易费用对策略的影响。理论上，算法具有线性时间复杂度；通过实证分析发现该算法在多数市场上的收益有显著提高。这表明算法具有一定的实用性和可操作性。
3.构建了权重函数来研究多期在线投资组合模型，充分利用利用历史价格信息，得到了收益更好的在线投资策略。现有的有关单期在线投资问题的研究在进行策略的调整时会给算法带来较大的盲目性，导致错失绝大部分历史数据信息。本文以既充分利用历史数据信息又不带来过大计算量为研究的出发点，展开多期的研究。以离当期时间越近赋权越大为原则构建权重函数，得到移动窗口下的加权平均价格序列。在此基础上构造损失函数，并从两个方面建立优化模型。一方面直接优化投资组合策略；另一方面，建立在预测未来价格基础上，利用PA分类算法和优化原理推导出加权移动平均的在线投资组合算法，并得到了不同形式的反转策略。进一步地，将行为金融学中反转的时间区间作为边侧信息，来确定历史窗口大小，以降低算法的时间复杂度。算法具有线性时间复杂度，利于大规模计算，实证研究中也获得了优异的收益。因此，带有权重的损失函数的构造方法扩展了利用分类算法研究在线投资组合问题的研究方法。
4.利用动量效应和投资者心理预期构建了不敏感损失函数，以此建立了优化模型，得到了动量在线投资策略。上述方法多利用反转特征，无法有效解决动量效应市场的投资决策问题。本文提出根据证券市场“异象”特征灵活设计投资策略的研究思路，将动量效应和投资主体主观态度相结合来构建不敏感损失函数，利用PA分类算法建立优化模型，得到保守策略和动量策略的转换机制，算法具有线性时间复杂度。该策略运用于国内外不同证券市场，进行比较分析，发现该策略在动量效应金融市场上收益很好，而对于反转效应的金融市场收益几乎为零。结果表明该策略适用于动量效应的金融市场。因此，它是在线反转策略的一个有益补充。
The core problem of investment decision is how to allocate financial assets to achieve anoptimal portfolio under uncertain circumstance. Because financial market is an extremelycomplex system, the investors are faced with an endlessly changing situation when they areengaging in investment practices. In the case where the information about future isunavailable, the investors who try to maximize the returns of their investments need to updatetheir investment strategy according to the current circumstance. Consequently, portfolioselection problem actually is an online one which sequentially determines portfolios usingavailable public information. The modern portfolio selection theory based on Markowitz’smean-variance model has been extensively studied by many researchers and lots of valuableresults have been obtained in the static case; however, people have seldom dealt with it in thedynamic case. In recent years, online algorithm has been applied to portfolio selection as wellas other fields, and it then helps develop the theory of online portfolio selection which isbased on universal portfolio. The main idea of this methodology is that proposingsequential-decision steps by online learning techniques, without making any statisticalhypotheses about the behavior of the asset price and the future information. However, since itdoes not figure out how the equity market behaves, the empirical results show that thismethod does not perform well and the potential application is negative. Taking thedisadvantages of the existing researches on universal portfolio into account, this thesis, fromthe respective of market’s anomaly, tries to explore the portfolio selection problemsystematically and thoroughly by means of online learning algorithm. The main contributionsof this thesis are listed in the following four aspects:
First, this thesis combines the reversal effect and the momentum effect to describestatistical correlations among several stocks and then calculate the proportion of weighttransferred. According to the correlations, a combined mean reversion online strategy ispresented by applying the heuristic algorithm. The existing heuristic algorithms such asAnticor only consider the reversal phenomenon of the price and often produce sub-optimalresults. In practice, they are with depressed earnings. Meanwhile, they depend significantly onthe window size and large-scale computation is required to eliminate the effect, whicheventually loses their timeliness. Thus, this article synthesizes the two phenomena mentionedabove and constructs the statistical correlation indexes to solve these problems. The thesis further analyzes the impact the degree of correlation intensity has on decision transfer,comprehensively grasps the fluctuation of price in stock markets by applying the heuristicalgorithm, and then obtains the online portfolio strategies. Moreover, empirical analysis isbased on six data sets from different foreign financial markets and four sets from twodomestic markets. The results show that they significantly increase the returns in foreignfinancial markets, while the cumulative returns in the domestic stock market are notsignificantly dependent on the window size. In all, they indicate the obtained strategies arestable.
Second, we propose to take use of the asymmetry of mean reversion to constructpiecewise loss function and PA (passive aggressive) algorithm to optimize the strategy ofportfolio selection. Because the strategy of online portfolio selection based on PAclassification algorithm just simply applies mean reversion theory, it is difficult for it todescribe the law of market behavior. To overcome this problem, according to the symmetry ofmean reversion, we construct a multi-piecewise loss function which does a better work indescribing the characteristics of the volatility of financial market. With the loss minimized, weuse the multi-piecewise loss function to build the most optimal model and then find the mostoptimal investment strategy which has the most minimal variance. Moreover, by means ofoptimization method, we get the reversion strategy which is applicable to all kinds of marketconditions. If short selling is allowed in the market, the bounds of the error of this algorithmis derived, and if not, the effect transaction cost has on the strategy is also explored.Theoretically, the algorithm possesses the property of linear time complexity. Besides, it hasbeen shown by empirical analysis that the returns of the selected portfolios have improvedgreatly in most of the markets. Hence the algorithm is applicable and practical, to someextent.
Third, we construct a weighting function in order to study the multiple-period onlineportfolio model. After making full use of the historical price information, we get the onlineinvestment strategy with higher benefits. Now the existing researches on the problem ofsingle-period online investment brings about big algorithm blindness when making strategyadjustment, which leads to the missing of most of the historical data information. In this thesis,we study the multi-period case with keeping a balance between making full use of historicaldata and avoiding large-scale computation. The moving weighted average sequences on atime window is derived by the way in which the shorter it is from current time, the moreweight is assigned to. Taking the above as a basis, we construct the loss function, and thenestablish the optimization model from the following two aspects. One is to optimize the strategy of portfolio model directly. The other is using the PAC classification algorithm andoptimization principle to derive the weighted moving average online portfolio algorithm anddifferent kinds of reversion strategies, based on predicating the future prices. Furthermore, inorder to reduce the time complexity of the algorithm, we determine the history window sizeby using the reverse time interval in the field of behavioral finance as side information. Thealgorithm with linear time complexity is conducive to large-scale computation and theempirical study also show that the strategy receives excellent benefits. Therefore, theweighted loss function method extends the way of the classification algorithm to study onlineportfolio problems of the research methods.
Forth, we construct an insensitive loss function by using the momentum effect andinvestor expectations. Then we establish optimization model, and get the momentum onlineportfolio strategy. The above method which using the property of price reversal is unable toeffectively solve the problem of investment decisions in the momentum effect market. In thispaper, we come up with a research approach which flexibly designs the investment strategyreferring to the characteristics of the equity market “anomalies”, and we build the lossfunction by means of combining the momentum effect and investors’ subjective attitude. Byusing PA classification algorithm we build optimization model, and get the transformationmechanism with conservative strategy and momentum strategy. The algorithm is providedwith linear time complexity. By testing and comparatively analyzing the strategy in differentstock markets, we find the strategy on financial market with momentum effect is very good,while the financial market with reversal effect gains almost zero. This suggests that thestrategy is suitable to be applied in the financial markets with momentum effect. Therefore, itis a beneficial supplement of contrarian strategy.
引文
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