高维数据下的均值-方差-CVaR稳定投资组合优化策略
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本文《Stable Portfolio Selection Strategy for Mean-Variance-CVaR》探讨了在高维数据背景下，如何应用稳定策略来优化投资组合选择，特别是在考虑均值、方差和条件价值-at-risk (CVaR) 的情况下。研究的主要目标是解决现代金融市场中面临的复杂性，特别是当资产数量众多时，如何有效地进行风险管理。
作者Yu Shi、Xia Zhao、Fengwei Jiang和Yipin Zhu来自上海对外经贸大学统计与信息学院，他们提出了一种创新的方法论，结合了不同协方差矩阵估计器、CVaR的计算方法以及正则化技术。通过这三种关键元素，他们构建了五个允许卖空的渐进优化问题。这些优化模型旨在平衡风险和收益，同时在高维数据环境中提供稳健的决策。
在高维数据条件下，传统的均值-方差分析可能受到多重共线性和噪声数据的影响，因此引入CVaR作为衡量尾部风险的工具，有助于投资者更好地了解极端损失的可能性。论文中提到的不同协方差矩阵估计方法，如样本协方差、 shrinkage估计（如Ledoit-Wolf和Oracle方法）或使用稀疏性假设的估计，能够提高估计的稳定性和准确性。
优化过程中的正则化技术有助于减少过拟合，并在数据稀疏或者存在异常值的情况下保持模型的鲁棒性。通过比较这些优化模型在实际样本上的表现，研究者揭示了不同方法对投资组合稳定性、风险控制和收益期望的实际效果。
文章的结论指出，对于高维数据的稳定投资组合选择，使用具有良好条件数的优化模型，结合合适的协方差估计和CVaR处理，能够在控制风险的同时实现更优的性能。此外，允许卖空的设定使得策略更具灵活性，适应动态市场环境。
这篇研究为金融从业者和投资者提供了一个在高维数据环境下进行稳定投资组合优化的实用框架，强调了CVaR在风险管理中的重要性，并展示了如何通过综合运用多种技术手段来提升投资组合的选择效率和稳定性。这对于理解和应对现代金融市场中的复杂挑战具有显著的实践意义。
where Σ
−
� Σ − Σ
+
and Σ
+
denote a diagonal matrix with the
same diagonal as Σ, σ
i
(Σ) is the ith largest eigenvalue of
matrix Σ, and
σ
Σ
is the mean of eigenvalues of Σ, and the way
to identify the best pair (λ
1
, τ
1
) can also be found in Maurya
[23].
2.2. CVaR. For financial assets or portfolio, VaR refers to the
greatest possible loss over specific holding period, at a
certain confidence level 100(1 − α)%, α ∈ (0, 1). If the cu-
mulative distribution function for return Y is F, VaR can be
defined as
VaR
1−α
� −F
−1
(α).
(3)
As we know, compared with VaR, a key advantage of
CVaR is that CVaR satisfies the four coherence axioms of
Artzner et al. [35], whereas VaR is not coherent. e def-
inition of CVaR is as follows:
CVaR
1−α
� E −Y | − Y ≥VaR
1−α
. (4)
Evidently, CVaR
1−α
, the conditional expectation of losses
exceeding VaR
1−α
, is more informative about the tail of the
distribution than VaR
1−α
. Two popular ways to compute
CVaR without distribution constraints have been employed
recently. e first one is a linear programming model for
optimizing CVaR proposed by Rockafellar and Uryasev
[25, 26], generally called as Rockafellar–Uryasev’s approx-
imation. ey have proved that CVaR can be calculated by
solving a convex optimization problem. In other words,
CVaR can be formulated as
CVaR
1
1−α
� min
β,ξ
ξ + α
−1
E −X
′
β − ξ
+
,
(5)
where ξ is the αth quantile of Y. is method has been widely
used in many literatures (e.g., Roman et al. [12], Lim et al.
[28], and Gao and Wu [31]).
e second way is quantile regression method proposed
in Bassett et al. [27]. Xu et al. [7] have proved that it has faster
computational speed compared with Rockafellar–Uryasev’s
approximation method. According to Bassett et al. [27], the
CVaR of a portfolio return Y could be calculated by
CVaR
2
1−α
� α
−1
min
ξ
E ρ
α
(Y − ξ)
− E[Y],
(6)
where ξ is the αth quantile of Y and ρ
α
(u) � u(α − I(u <0))
is the check function, in which I(·) is an indicator function
that takes on the value one whenever its argument is true and
zero otherwise.
Since Y � X
′
β, we can have Y � X
1
−
p
j�2
(X
1
− X
j
)β
j
under the constraint 1
′
β � 1. Let
X
j
� X
1
− X
j
(j �
2, 3, . . . , p), we can convert equation (6) into
CVaR
2
1−α
� α
− 1
min
β,ξ
E ρ
α
X
1
−
p
j�2
X
j
β
j
− ξ
⎛
⎝
⎞
⎠
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
− E X
′
β
.
(7)
2.3. Regularization Method. To encourage a stable and
sparse solution in optimization problem, penalty item Pen(·)
is frequently used, which is regarded as regularization
method. Pen(·) is a general penalty function that allows
shrinking the components in β to zero. In this paper, we
focus on smoothly clipped absolute deviation (SCAD)
penalty proposed by Fan et al. [36] and reweighted L
1
norm
penalty proposed by Emmanuel et al. [37], since they have
oracle properties.
e SCAD penalty is formulated as follows:
λ
p
k�1
Pen
1
β
k
�
p
k�1
λ β
k
I β
k
≤λ
+
−β
2
k
+ 2ζλ β
k
− λ
2
2(ζ − 1)
I λ < β
k
≤ζλ
+
(ζ + 1)λ
2
2
I β
k
>ζλ
,
(8)
where ζ(ζ <2) and λ are tuning parameters. It is well known
that SCAD is continuous and singular at the origin, penalizes
large coefficients equally, and has no bias, and the
reweighted L
1
norm is as follows:
λ
p
k�1
Pen
2
β
k
� λ
p
k�1
ω
k
β
k
,
(9)
where λ is a tuning parameters and the value of ω
k
can be
identified by iteration method based on the value of |β
k
|.
Reweighed L
1
norm has a better out-of-sample performance
compared with L
1
norm penalty [34], since it gives every
component in β a corresponding weight so that the pe-
nalization is more accurate.
3. Model Formation
Motivated by Roman et al. [12], we consider the following
mean-variance-CVaR model:
P
mvc
:
min
β
ηV X
′
β
+(1 − η)CVaR
1−α
s.t.
1
′
β � 1
E X
′
β
� μ
0
,
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
(10)
where 0 ≤η ≤1 is a weighting parameter, which represents
the attitude that investors hold towards the two risk mea-
sures and hence balances the importance of them.
Mathematical Problems in Engineering 3
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