BARRA组合优化：现代投资理论的风险与收益策略

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BARRA 组合优化是一种在投资领域广泛应用的方法，它源于现代投资组合理论，特别是哈里·马科维茨的开创性工作。马科维茨模型的核心是通过资产预期收益和协方差矩阵来构建风险与回报的最佳平衡，即所谓的有效前沿。BARRA 的组合优化技术旨在解决实际投资中的挑战，比如马歇尔·米肖（Michaud, 1989）所指出的，仅仅依靠理论上的最优组合可能并不足以确保实践中的成功实施。在实践中，BARRA 方法引入了"Eigen-Adjusted Covariance Matrices"（特征调整协方差矩阵），这是一个关键的改进。Eigen-Adjusted方法考虑了资产间的非线性和动态相关性，通过调整传统的协方差矩阵，减少了估计误差和模型过度拟合的风险。这种方法有助于提高风险预测的准确性，从而在优化投资组合时更为稳健，特别是在复杂市场环境中，如金融危机或经济周期波动时。 Eigen-Adjusted模型的一个具体研究论文由Jose Menchero、Jun Wang和D.J. Orr于2011年发布，他们的工作提供了一个电子版可获取的资源（http://ssrn.com/abstract=1915318）。这篇研究洞察指出，通过特征调整，投资者可以构建出更符合实际市场动态的优化组合，这在投资决策中具有重要意义。然而，尽管BARRA 组合优化提供了强大的工具，投资者在应用时仍需注意以下几点： 1. 数据质量：精确的预期收益和调整后的协方差矩阵依赖于可靠的历史数据，因此数据清洗和预处理至关重要。 2. 市场条件：理论模型假设市场是有效的，但在现实中可能存在信息不对称或行为金融效应，这些都需要考虑。 3. 模型的局限性：特征调整不能消除所有市场异常，投资者应保持一定的灵活性，适时调整策略。 4. 风险管理：除了风险敞口，投资者还需要考虑流动性风险、信用风险等其他类型的风险。总结来说，BARRA 组合优化通过特征调整协方差矩阵提升了投资组合的风险管理能力，使投资者能够在复杂的市场环境中更好地平衡风险与回报。理解和掌握这一技术，对于现代投资经理和寻求最优资产配置的投资者而言，都是至关重要的实践技能。

MSCI Research msci.com

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Research Insight

Eigen-Adjusted Covariance Matrices

May 2011

5of 21

The covariance matrix, however, can always be “rotated” to an alternative basis representing different

linear combinations of the original assets. One basis that is of particular interest is the diagonal basis, in

which the off-diagonal elements are all zero. The “assets” in the diagonal basis are termed eigenfactors,

whose predicted variances are given by the diagonal matrix elements. The off-diagonal elements, now

being zero, indicate that the eigenfactors are mutually uncorrelated.

The return of eigenfactor

on day

is given by

tntnt





, (7)

where

u is the weight of asset

in the eigenfactor portfolio (see Appendix A for further details). Note

that the number of eigenfactors is equal to the dimensionality of the sample covariance matrix

. We

follow the convention of sorting the eigenfactors in ascending order by predicted volatility.

Eigenfactors are not economically intuitive. However, they do play an important role in portfolio

optimization. For instance, the first eigenfactor solves for the minimum variance portfolio subject to the

constraint that the sum of squared weights adds up to 1. Similarly, the last eigenfactor solves the

corresponding maximum variance problem.

This suggests that the problem of underestimation of risk of optimized portfolios may be related to

eigenfactors. To investigate this further, we compute bias statistics for the eigenfactor portfolios defined

by Equation 7. The results are plotted in Figure 1(c): we find a very strong relationship between the bias

statistic of the eigenfactor and the eigenfactor number. More specifically, the smallest eigenfactors (i.e.,

those with the lowest variances) have bias statistics far greater than 1, indicating that

strongly

underestimates the risk of these portfolios. For the largest eigenfactors, by contrast, the sample

covariance matrix slightly overpredicts risk. This should not be too surprising, perhaps, since the largest

eigenfactor solves the maximum variance problem and, naturally, is also subject to estimation error.

Optimized Portfolios

While the eigenfactor biases are certainly intriguing, the biases of optimized portfolios are of more

direct interest for portfolio construction purposes. To study these biases, we generate random alpha

signals (for

1, 100j 

)

jjj

nt n t







 , (8)

Mathematically, we have 51 eigenfactors, which we label

0,...,50k



. It is interesting to consider the scenario in which the stock weights are constant over

the 200-day estimation window. In this case, the smallest eigenfactor

(0)k



would have exactly zero variance. This represents the cap-weighted asset returns

of Equation 2 summing to zero; i.e.,

u 

, with

exactly proportional to the cap weights of the stocks. The



eigenfactor therefore reflects an exact

colinearity in the return structure; it is not an investable portfolio since the market exposure

(0)n



does equal the sum of stock exposures

(1,50)n



. In

reality, stock weights are not constant in time, and these results are only approximate.

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