基于对数先验的协方差矩阵的参数估计

Parameter Estimation of Covariance Matrix Based on Logarithmic Priors

摘要: 为了更好地估计股票收益协方差矩阵，提出了一种基于Black-Litterman思想的协方差矩阵估计方法。不同于传统方法对协方差矩阵添加Wishart先验分布的方法，考虑将先验信息概念应用在协方差矩阵的对数上，通过贝叶斯方法，得到协方差矩阵的参数估计。

Abstract: In order to better estimate the covariance matrix of stock return, a method of covariance matrix estimation based on Black-Litterman idea is proposed. Different from the traditional method of adding Wishart prior distribution to the covariance matrix, the concept of prior information is applied to the logarithm of the covariance matrix, and the parameter estimation of the covariance matrix is obtained by Bayesian method.

1. 引言

投资组合优化的基础是由Markowitz [1] 在20世纪50年代提出的均值方差模型，该模型根据投资组合的预期收益和波动率来计算投资组合的资产配置。假设市场上有n种资产，这些资产收益的均值 $\mu \in {ℝ}^n$ ，协方差矩阵 $\Sigma \in {ℝ}^{n\times n}$ ，经典的投资组合优化问题可描述为：

$\begin{array}{ll}\underset{w}{\max }\hfill & w^{\text{T}}\mu -\frac{\lambda }{2}{w}^{\text{T}}\Sigma w\hfill \\ \text{s}.\text{t}.\hfill & w^{\text{T}}1=1,\hfill \end{array}$ (1.1)

其中 $\lambda$ 为投资者的风险规避系数。

Black-Litterman模型(以下简称BL模型)是均值方差模型(1.1)的延伸。考虑市场上的n项资产，这些资产的收益 $r={\left(r_1,r_2,\cdots ,r_m\right)}^{\text{T}}$ 服从多元正态分布，即： $r_1,r_2,\cdots ,r_m~N_n\left(\mu ,\Sigma \right)$ 。根据资本资产定价模型(CAPM)，当市场处于均衡状态时，即当对资产组合的需求等于供给时，所有投资者都持有最优市场投资组合，此时资产配置的权重为市场均衡权重 $w_{eq}$ 。BL模型想要得到市场均衡收益率为 $\pi$ ，则需要反向优化均值方差模型(1.1)，对模型求一阶导，并令其等于零，可得方程 $\pi =\lambda \Sigma w_{eq}$ 。BL模型以这个市场均衡收益率为客观的市场先验收益分布， $\mu ~N_m\left(\pi ,\tau \Sigma \right)$ ，这里 $\tau$ 是反映对市场均衡不确定性的参数， $\tau$ 越小，表示先验 $\mu$ 越接近市场均衡收益 $\pi$ ，反之 $\tau$ 越大，则表示先验 $\mu$ 越背离市场均衡收益 $\pi$ 。另外再加入投资者主观观点，利用贝叶斯方法得到后验资产收益率，进而得出最优资产配置方案。假设投资者对收益均值向量 $\mu$ 有k个观点，他的观点可以用下面的公式表示：

$q=P\mu +\epsilon ,$ (1.2)

其中P是 $k\times n$ 维矩阵，即对n个资产有k个观点，例如考虑三种资产A，B和C，当

$P=\left(\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill -1& \hfill 1\end{array}\right),q=\left(\begin{array}{l}4\%\hfill \\ 6\%\hfill \end{array}\right),$ (1.3)

则P的第一行也就是投资者的第一个观点，是一个绝对观点，表示投资者认为资产A的预期收益率为4%；P的第二行为相对观点，表示投资者认为资产B与资产C的平均预期收益率的差为6%。绝对观点行和为1，相对观点行和为0。 $\epsilon$ 是观点的误差项， $\epsilon ~N\left(0,\Omega \right)$ ，Ω是 $k\times k$ 的对角矩阵，为观点误差的协方差矩阵，表示投资者对单个预测的信心水平。因此投资者观点的分布为： $q|\mu ~N_k\left(P\mu ,\Omega \right)$ 。

结合上述信息可知整个BL模型由以下三个分布构成：

$\begin{array}{r}\hfill r_1,r_2,\cdots ,r_m~N_n\left(\mu ,\Sigma \right),\\ \hfill \mu ~N_n\left(\pi ,\tau \Sigma \right),\\ \hfill q|\mu ~N_k\left(P\mu ,\Omega \right).\end{array}$ (1.4)

通过结合(1.4)中后两个先验分布，并以此作为收益均值 $\mu$ 的新的先验分布，根据文献 [2] 中给出的证明可以计算出收益的后验分布：

$r~N\left({\mu }_{BL},{\Sigma }_{BL}\right).$ (1.5)

其中

$\begin{array}{l}{\mu }_{BL}={\left({\left(\tau \Sigma \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\tau \Sigma \right)}^{-1}\pi +P^{\text{T}}{\Omega }^{-1}q\right),\\ {\Sigma }_{BL}=M^{-1}+\Sigma ,M^{-1}={\left({\left(\tau \Sigma \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}.\end{array}$

2. 模型建立

不同于传统BL模型对协方差矩阵引入逆Wishart分布的方法 [3] ，我们基于Leonard和Hsu [4] 关于协方差矩阵的不同先验的想法，将协方差矩阵的先验信息应用在协方差矩阵的对数，即 $\log \left(\Sigma \right)$ 上，这种先验分布不仅融合了市场参与者的实际观点，还增强了模型对于复杂金融环境的适应性。假设考虑市场上的n项资产，这些资产在一段时间m内的独立收益为 $r={\left(r_1,r_2,\cdots ,r_m\right)}^{\text{T}}$ ，每个收益服从多元正态分布，其均值为 $\mu$ ，协方差矩阵为 $\Sigma$ ，即：

(2.1)

我们假设收益的均值 $\mu$ 是已知的，无不确定性，则 $\Sigma$ 的似然函数可以表示为：

$L\left(\Sigma |r_1,\cdots ,r_m\right)={\left(2\pi \right)}^{-\frac{nm}{2}}{\left|\Sigma \right|}^{-\frac{m}{2}}\exp \left\{-\frac{1}{2}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\left(r_i-\mu \right)}^{\text{T}}}{\Sigma }^{-1}\left(r_i-\mu \right)\right\}.$ (2.2)

令 $A=\log \left(\Sigma \right)$ ， ${\lambda }_{Ai}$ 和 ${\lambda }_{{\Sigma }_i}$ 为A与 $\Sigma$ 的特征值，因此 ${\lambda }_{Ai}=\log \left({\lambda }_{{\Sigma }_i}\right)\Rightarrow {\lambda }_{{\Sigma }_i}={\text{e}}^{{\lambda }_{Ai}}$ ，利用矩阵行列式的性质可以得到 $det\left(\Sigma \right)={\displaystyle \underset{i=1}{\overset{n}{\prod }}{\lambda }_{\Sigma i}}={\displaystyle \underset{i=1}{\overset{n}{\prod }}{\text{e}}^{{\lambda }_{Ai}}}={\text{e}}^{Tr\left(A\right)}$ 。那么 $\Sigma$ 的似然函数可以写为：

$\begin{array}{c}L\left(A|r_1,\cdots ,r_m,\Sigma \right)={\left(2\pi \right)}^{-\frac{mn}{2}}\exp \left\{-\frac{1}{2}Tr\left({\displaystyle \underset{i=1}{\overset{m}{\sum }}{\left(r_i-\mu \right)}^{\text{T}}}{\Sigma }^{-1}\left(r_i-\mu \right)\right)\right\}\exp \left\{-\frac{m}{2}Tr\left(A\right)\right\}\\ ={\left(2\pi \right)}^{-\frac{mn}{2}}\exp \left\{-\frac{1}{2}Tr\left({\displaystyle \underset{i=1}{\overset{m}{\sum }}\left(r_i-\mu \right)}{\left(r_i-\mu \right)}^{\text{T}}{\Sigma }^{-1}\right)\right\}\exp \left\{-\frac{m}{2}Tr\left(A\right)\right\}\\ ={\left(2\pi \right)}^{-\frac{mn}{2}}\exp \left\{-\frac{m}{2}Tr\left(A+S{\text{e}}^{-A}\right)\right\}.\end{array}$ (2.3)

其中 $S=\frac{1}{m}{\displaystyle \underset{i=1}{\overset{m}{\sum }}{\left(r_i-\mu \right)}^{\text{T}}}\left(r_i-\mu \right)$ 。Leonard和Hsu [4] 的想法是通过近似 ${\text{e}}^{-A}$ 来近似 $\Sigma$ 的似然函数。Bellman [5]

在他书中展示了一个更一般的结果：

$\begin{array}{l}X\left(t\right)={\text{e}}^{\left(A_0+c{B}_0\right)t}={\text{e}}^{A_{0}t}+c{\displaystyle {\int }_0^t{\text{e}}^{A_0\left(t-s\right)}}{B}_{0}X\left(s\right)\text{d}s,0<t<\infty ,\\ A_0=-\Lambda ,B_0=\Lambda -A,c=1,\Lambda =\log \left(S\right),\\ X\left(t\right)={\text{e}}^{-At}=S^{-t}-{\displaystyle {\int }_0^t{S}^{s-t}}\left(A-\Lambda \right)X\left(s\right)\text{d}s,0<t<\infty .\end{array}$ (2.4)

当 $t=1$ 时，通过迭代可得：

$\begin{array}{c}{\text{e}}^{-A}=X\left(1\right)=S^{-1}-{\displaystyle {\int }_0^1{S}^{s-1}}\left(A-\Lambda \right)S^{-s}\text{d}s\\ =S^{-1}-{\displaystyle {\int }_0^1{S}^{s-1}}\left(A-\Lambda \right)\left(S^{-s}-{\displaystyle {\int }_0^s{S}^{u-s}}\left(A-\lambda \right)X\left(u\right)\text{d}u\right)\text{d}s\\ =S^{-1}-{\displaystyle {\int }_0^1{S}^{s-1}}\left(A-\Lambda \right)S^{-s}\text{d}s+{\displaystyle {\int }_0^1{{\displaystyle {\int }_0^{s}S}}^{s-1}}\left(A-\Lambda \right)S^{u-s}\left(A-\Lambda \right)\cdot \left(S^{-u}-{\displaystyle {\int }_0^u{S}^{v-u}}\left(A-\lambda \right)X\left(v\right)\text{d}v\right)\text{d}u\text{d}s\\ \approx S^{-1}-{\displaystyle {\int }_0^1{S}^{s-1}}\left(A-\Lambda \right)S^{-s}\text{d}s+{\displaystyle {\int }_0^1{{\displaystyle {\int }_0^{s}S}}^{s-1}}\left(A-\Lambda \right)S^{u-s}\left(A-\Lambda \right)S^{-u}\text{d}u\text{d}s.\end{array}$ (2.5)

这是一个近似因为我们忽略了三重及更高阶的积分。已知 $\Sigma$ 的似然函数形式为：

$L\left(A|r_1,\cdots ,r_m,\Sigma \right)={\left(2\pi \right)}^{-\frac{mn}{2}}\exp \left\{-\frac{m}{2}Tr\left(A+S{\text{e}}^{-A}\right)\right\}.$ (2.6)

因此我们计算

$\begin{array}{c}Tr\left(S{\text{e}}^{-A}\right)\approx Tr\left(S{S}^{-1}\right)-{\displaystyle {\int }_0^{1}T}r\left(S^s\left(A-\Lambda \right)S^{-s}\right)\text{d}s+{\displaystyle {\int }_0^1{\displaystyle {\int }_0^{s}T}}r\left(S^s\left(A-\Lambda \right)S^{u-s}\left(A-\Lambda \right)S^{-u}\text{d}u\text{d}s\right)\\ =n-Tr\left(A-\Lambda \right)\text{d}s+{\displaystyle {\int }_0^1{\displaystyle {\int }_0^{s}T}}r\left(\left(A-\Lambda \right)S^{u-s}\left(A-\Lambda \right)S^{-\left(u-s\right)}\text{d}u\text{d}s\right).\end{array}$ (2.7)

对第一项积分有： ${\displaystyle {\int }_0^{1}T}r\left(A-\Lambda \right)\text{d}s=Tr\left(A-\Lambda \right)$ 。

对第二项积分首先设 $S$ 有谱分解 $S=ED{E}^{\text{T}}$ ，若log由泰勒展开定义，则有：

$\begin{array}{c}\log \left(S\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n}{n+1}}{\left(I_n-S\right)}^{n+1}\\ ={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n}{n+1}}{\left(E{E}^{\text{T}}-ED{E}^{\text{T}}\right)}^{n+1}\\ ={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n}{n+1}}E{\left(I_n-D\right)}^{n+1}{E}^{\text{T}}\\ =E\log \left(D\right)E^{\text{T}}.\end{array}$ (2.8)

记 $B=E^{\text{T}}\left(A-\Lambda \right)E$ ，即 $EB{E}^{\text{T}}=A-\Lambda$ ，则第二项积分内部：

$\begin{array}{c}Tr\left(\left(A-\Lambda \right)S^{u-s}\left(A-\Lambda \right)S^{-\left(u-s\right)}\right)=Tr\left[EB{E}^{\text{T}}{\left(ED{E}^{\text{T}}\right)}^{u-s}EB{E}^{\text{T}}{\left(ED{E}^{\text{T}}\right)}^{-\left(u-s\right)}\right]\\ =Tr\left(EB{D}^{u-s}B{D}^{-\left(u-s\right)}{E}^{\text{T}}\right)\\ =Tr\left(B{D}^{u-s}B{D}^{-\left(u-s\right)}\right).\end{array}$ (2.9)

不妨设 $B{D}^{u-s}$ 的标量形势为：

$B{D}^{u-s}=\left[\begin{array}{cccc}cccc{b}_{11}{d}_1^{u-s}& b_{12}{d}_2^{u-s}& \mathrm{...}& b_{1n}{d}_n^{u-s}\\ ⋮& ⋮& \mathrm{...}& ⋮\\ b_{n1}{d}_1^{u-s}& b_{n2}{d}_2^{u-s}& \mathrm{...}& b_{nn}{d}_n^{u-s}\end{array}\right]$ (2.10)

则同样的 $B{D}^{-(u-s)}$ 的标量形势为：

$B{D}^{u-s}=\left[\begin{array}{cccc}cccc{b}_{11}\frac{1}{d_1^{u-s}}& b_{12}\frac{1}{d_2^{u-s}}& \mathrm{...}& b_{1n}\frac{1}{d_n^{u-s}}\\ ⋮& ⋮& \mathrm{...}& ⋮\\ b_{n1}\frac{1}{d_1^{u-s}}& b_{n2}\frac{1}{d_2^{u-s}}& \mathrm{...}& b_{nn}\frac{1}{d_n^{u-s}}\end{array}\right]$ (2.11)

下面计算 $B{D}^{u-s}B{D}^{-\left(u-s\right)}$ 的对角元素：

$diag\left(B{D}^{u-s}B{D}^{-\left(u-s\right)}\right)=\left[\begin{array}{c}{b}_{11}^2+b_{12}{b}_{21}{\left(\frac{d_2}{d_1}\right)}^{u-s}+\cdots +b_{1n}{b}_{n1}{\left(\frac{d_n}{d_1}\right)}^{u-s}\\ b_{21}{b}_{12}{\left(\frac{d_1}{d_2}\right)}^{u-s}+b_{22}^2+\cdots +b_{2n}{b}_{n2}{\left(\frac{d_n}{d_2}\right)}^{u-s}\\ ⋮\\ b_{n1}{b}_{1n}{\left(\frac{d_1}{d_n}\right)}^{u-s}+b_{n2}{b}_{2n}{\left(\frac{d_2}{d_n}\right)}^{u-s}+\cdots +b_{nn}^2\end{array}\right]$ (2.12)

由于 $B$ 是对称阵，因此

${\displaystyle {\int }_0^1{\displaystyle {\int }_0^{s}T}}r\left(B{D}^{u-s}B{D}^{-\left(u-s\right)}\right)\text{d}u\text{d}s={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^s{b}_{ij}^2}}}\text{d}u\text{d}s+{\displaystyle \underset{i\ne j}{\overset{n}{\sum }}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^s{b}_{ij}^2}}}{\left(\frac{d_i}{d_j}\right)}^{u-s}\text{d}u\text{d}s$ (2.13)

其中第一项积分

${\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^s{b}_{ii}^2}}}\text{d}u\text{d}s={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^s{b}_{ii}^2}}}\text{d}u\text{d}s={\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{b_{ii}^2}{2}},$ (2.14)

第二项积分

$\begin{array}{c}{\displaystyle \underset{i\ne j}{\overset{n}{\sum }}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^s{b}_{ij}^2}}}{\left(\frac{d_i}{d_j}\right)}^{u-s}\text{d}u\text{d}s={\displaystyle \underset{i\ne j}{\overset{n}{\sum }}{\displaystyle {\int }_0^1{b}_{ij}^2}}{\left(\frac{d_i}{d_j}\right)}^{u-s}{\frac{1}{\log \left(d_i\right)-\log \left(d_j\right)}|}_0^s\text{d}s\\ ={\displaystyle \underset{i\ne j}{\sum }\frac{b_{ij}^2}{\log \left(d_i\right)-\log \left(d_j\right)}}{\displaystyle {\int }_0^{1}1}-{\left(\frac{d_i}{d_j}\right)}^{-s}\text{d}s\\ ={{\displaystyle \underset{i\ne j}{\sum }\frac{b_{ij}^2}{\log \left(d_i\right)-\log \left(d_j\right)}}\left(1-{\left(\frac{d_j}{d_i}\right)}^s\frac{1}{\log \left(d_j\right)-\log \left(d_i\right)}\right)|}_0^1\\ ={\displaystyle \underset{i\ne j}{\sum }\frac{b_{ij}^2}{\log \left(d_i\right)-\log \left(d_j\right)}}+{\displaystyle \underset{i\ne j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}-1}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}\\ ={\displaystyle \underset{i<j}{\sum }\left(\frac{b_{ij}^2}{\log \left(d_i\right)-\log \left(d_j\right)}+\frac{b_{ji}^2}{\log \left(d_j\right)-\log \left(d_i\right)}\right)}+{\displaystyle \underset{i<j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}+\frac{d_i}{d_j}-2}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}\\ =0+{\displaystyle \underset{i<j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}+\frac{d_i}{d_j}-2}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}.\end{array}$ (2.15)

综上可得

$Tr\left(S{\text{e}}^{-A}\right)=n-Tr\left(A-\Lambda \right)+\frac{1}{2}{\displaystyle \underset{i=0}{\overset{n}{\sum }}{b}_{ii}^2}+{\displaystyle \underset{i<j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}+\frac{d_i}{d_j}-2}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}.$ (2.16)

因此 $\Sigma$ 的似然函数可以改写为：

$\begin{array}{l}L\left(\alpha |r_1,\cdots ,r_m\right)={\left(2\pi \right)}^{-\frac{mn}{2}}\exp \left\{-\frac{m}{2}Tr\left(A+S{\text{e}}^{-A}\right)\right\}\\ ={\left(2\pi \right)}^{-\frac{mn}{2}}\exp \left\{-\frac{m}{2}\left(Tr\left(A\right)+n-Tr\left(A-\Lambda \right)+\frac{1}{2}{\displaystyle \underset{i=0}{\overset{n}{\sum }}{b}_{ii}^2}+{\displaystyle \underset{i<j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}+\frac{d_i}{d_j}-2}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}\right)\right\}\\ ={\left(2\pi e\right)}^{-\frac{mn}{2}}{\text{e}}^{\left(-\frac{m}{2}Tr\left(A\right)\right)}\exp \left\{-\frac{m}{2}\left(\frac{1}{2}{\displaystyle \underset{i=0}{\overset{n}{\sum }}{b}_{ii}^2}+{\displaystyle \underset{i<j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}+\frac{d_i}{d_j}-2}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}\right)\right\}\\ ={\left(2\pi e\right)}^{-\frac{mn}{2}}det{\left(S\right)}^{-\frac{m}{2}}\exp \left\{-\frac{m}{2}\left(\frac{1}{2}{\displaystyle \underset{i=0}{\overset{n}{\sum }}{b}_{ii}^2}+{\displaystyle \underset{i<j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}+\frac{d_i}{d_j}-2}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}\right)\right\}.\end{array}$ (2.17)

定义：

$\begin{array}{l}M=\left\{\begin{array}{c}\frac{m}{2},i=j,\\ \frac{m{\left(d_i-d_j\right)}^2}{d_i{d}_j{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2},i\ne j.\end{array}\right\},\\ Ve{c}^{*}\left(A\right)={\left[a_{11},a_{21},\cdots ,a_{n1},a_{12},a_{22},\cdots ,a_{n2},\cdots ,a_{1n},a_{2n},\cdots ,a_{nn}\right]}^T.\end{array}$ (2.18)

则 $\Sigma$ 似然函数指数内部：

$-\frac{m}{2}\left(\frac{1}{2}{\displaystyle \underset{i=0}{\overset{n}{\sum }}{b}_{ii}^2}+{\displaystyle \underset{i<j}{\sum }{b}_{ij}^2}\frac{\frac{d_j}{d_i}+\frac{d_i}{d_j}-2}{{\left(\log \left(d_i\right)-\log \left(d_j\right)\right)}^2}\right)=-\frac{1}{2}\left({\left(Vec\left(B\right)\right)}^{\text{T}}Vec\left(B\odot M\right)\right),$ (2.19)

由 $B=E^{\text{T}}\left(A-\Lambda \right)E$ ，得

$Vec\left(B\right)=Vec\left[E^{\text{T}}\left(A-\Lambda \right)E\right]=\left(E^{\text{T}}\otimes E^{\text{T}}\right)Vec\left(A-\Lambda \right)=\left(E^{\text{T}}\otimes E^{\text{T}}\right)\left(\alpha -\lambda \right).$ (2.20)

因此指数内部：

$\begin{array}{l}-\frac{1}{2}\left({\left(Vec\left(B\right)\right)}^{\text{T}}Vec\left(B\odot M\right)\right)=-\frac{1}{2}{\left(Vec\left(B\right)\right)}^{\text{T}}diag\left(Vec\left(M\right)\right)Vec\left(B\right)\\ =-\frac{1}{2}\left({\left(\alpha -\lambda \right)}^{\text{T}}{\left(E^{\text{T}}\otimes E^{\text{T}}\right)}^{\text{T}}diag\left(Vec\left(M\right)\right)\left(E^{\text{T}}\otimes E^{\text{T}}\right)\left(\alpha -\lambda \right)\right).\end{array}$ (2.21)

令 $Q={\left({\left(E^{\text{T}}\otimes E^{\text{T}}\right)}^{\text{T}}diag\left(Vec\left(M\right)\right)\left(E^{\text{T}}\otimes E^{\text{T}}\right)\right)}^{-1}$ ，则 $\Sigma$ 的似然函数可以写为：

$L\left(\alpha |r_1,\cdots ,r_m\right)={\left(2\pi e\right)}^{-\frac{mn}{2}}det{\left(S\right)}^{-\frac{m}{2}}\exp \left\{-\frac{1}{2}\left({\left(\alpha -\lambda \right)}^{\text{T}}Q\left(\alpha -\lambda \right)\right)\right\}.$ (2.22)

注意这里的 $Q$ 不同于Andrei和Hsu [3] 中给出的结构，而是一种显示的表示。可以看到 $\alpha$ 的近似似

然函数是服从均值为 $\lambda$ ，协方差矩阵为 $Q^{-1}$ 的多元正态分布，维数 $d=\frac{1}{2}m\left(m+1\right)$ ，根据这一函数形式我

们可以给出协方差矩阵的共轭先验：

$\alpha |\theta ,\Delta ~N\left(\theta ,\Delta \right),$ (2.23)

其中 $\theta$ 与 $\Delta$ 由历史数据获得。此外在许多实际情况下，专家对投资组合的协方差往往有强烈的意见 [6] ，他们想用这些意见信息来确定他们的最优投资组合，这种信息可以是对个别资产的任何绝对或相对的意见，数学形式表示为 $q=P\alpha +ϵ$ ，因此投资者观点的分布为：

$q|\mu ~N\left(P\alpha ,\Omega \right).$ (2.24)

因此整个模型由以下3个分布表示：

$\begin{array}{c}{r}_1,\cdots ,r_m|\Sigma ~N\left(\mu ,\Sigma \right)\\ \alpha ~N\left(\theta ,\Delta \right)\\ q|\mu ~N\left(P\alpha ,\Omega \right)\end{array}$ (2.25)

3. 模型求解

3.1. α|q的联合先验

首先将 $\alpha$ 的先验分布与投资者观点分布联合得到 $\alpha$ 新的先验分布：

$\begin{array}{c}f\left(\alpha ,q\right)\propto \exp \left\{-\frac{1}{2}\left[{\left(\alpha -\theta \right)}^{\text{T}}{\left(\Delta \right)}^{-1}\left(\alpha -\theta \right)+{\left(q-P\alpha \right)}^{\text{T}}{\Omega }^{-1}\left(q-P\alpha \right)\right]\right\}\\ =\exp \left\{-\frac{1}{2}{\left[\begin{array}{c}\alpha -\theta \\ q-P\alpha \end{array}\right]}^{\text{T}}{\left[\begin{array}{cc}cc\Delta & 0\\ 0& \Omega \end{array}\right]}^{-1}\left[\begin{array}{c}\alpha -\theta \\ q-P\alpha \end{array}\right]\right\}.\end{array}$ (3.1)

令 $V=\left[\begin{array}{cc}\Delta & 0\\ 0& \Omega \end{array}\right]$ ， $b=\left[\begin{array}{c}\alpha -\theta \\ q-P\alpha \end{array}\right]$ ，则 $f\left(\alpha ,q\right)\propto \exp \left\{-\frac{1}{2}{b}^{\text{T}}{V}^{-1}b\right\}$ 。考虑构建矩阵 $A$ ，计算 $b^{\prime }=Ab$ 和 $V^{\prime }=AV{A}^{\text{T}}$ ，得到新的分布 $f\left(\alpha ,q\right)\propto \exp \left\{-\frac{1}{2}{{\alpha }^{\prime }}^{\text{T}}{V^{\prime }}^{-1}{\alpha }^{\prime }\right\}$ 来分隔 $\alpha$ 与 $q$ 。因此令

$\begin{array}{c}A=\left[\begin{array}{cc}{I}_n& -{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{P}^{\text{T}}{\Omega }^{-1}\\ 0& I_k\end{array}\right]\left[\begin{array}{cc}{I}_n& 0\\ P& I_k\end{array}\right]\\ =\left[\begin{array}{cc}{I}_n-{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{P}^{\text{T}}{\Omega }^{-1}P& -{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{P}^{\text{T}}{\Omega }^{-1}\\ P& I_k\end{array}\right].\end{array}$ (3.2)

由于 $I_n={\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)$ ，有

$\begin{array}{l}A=\left[\begin{array}{cc}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{\left(\Delta \right)}^{-1}& -{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{P}^{\text{T}}{\Omega }^{-1}\\ P& I_k\end{array}\right].\\ det\left(A\right)=det\{{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{\left(\Delta \right)}^{-1}+{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{P}^{\text{T}}{\Omega }^{-1}P\}\\ =det\left\{{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)\right\}\\ =det\left(I\right)=1.\end{array}$ (3.3)

得到新的 $b^{\prime }$ 与 $V^{\prime }$ ：

$\begin{array}{l}b\text{'}=Ab=A\left[\begin{array}{c}\alpha -\theta \\ q-P\alpha \end{array}\right]\\ =\left[\begin{array}{c}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{\left(\Delta \right)}^{-1}\left(\alpha -\theta \right)-{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{P}^{\text{T}}{\Omega }^{-1}\left(q-P\alpha \right)\\ P\alpha -P\theta +q-P\alpha \end{array}\right]\\ =\left[\begin{array}{c}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left[\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)\alpha -{\left(\Delta \right)}^{-1}\theta -P^{\text{T}}{\Omega }^{-1}q\right]\\ q-P\theta \end{array}\right]\\ =\left[\begin{array}{c}\alpha -{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\Delta \right)}^{-1}\theta +P^{\text{T}}{\Omega }^{-1}q\right)\\ q-P\theta \end{array}\right]\\ =\left[\begin{array}{c}\left(\alpha -\bar{\alpha }\right)\\ q-P\theta \end{array}\right],\end{array}$ (3.4)

其中 $\bar{\alpha }={\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\Delta \right)}^{-1}\theta +P^{\text{T}}{\Omega }^{-1}q\right)$ 。下面计算 $V^{\prime }$ ：

$V\text{'}=AV{A}^{\text{T}}=\left[\begin{array}{cc}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}& -{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}{P}^{\text{T}}\\ P\Delta & \Omega \end{array}\right]A^{\text{T}}.$ (3.5)

已知 $\Delta$ 与 $\Sigma$ 均为对称阵，因此

$\begin{array}{c}{A}^T=\left[\begin{array}{cc}{\left(\Delta \right)}^{-1}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}& P^{\text{T}}\\ -{\Omega }^{-1}P{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}& I_k\end{array}\right],\\ V^{\prime }=\left[\begin{array}{cc}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right){\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}& 0\\ 0& \Omega +P\Delta P^{\text{T}}\end{array}\right]\\ =\left[\begin{array}{cc}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}& 0\\ 0& \Omega +P\Delta P^{\text{T}}\end{array}\right].\end{array}$ (3.6)

可将 $\alpha$ 与 $q$ 的联合分布分离为： $f\left(\alpha ,q\right)=f\left(\alpha |q\right)f\left(q\right)$ ，得到 $\alpha$ 基于投资者观点的新的先验：

$\begin{array}{l}\alpha |q~N\left(\bar{\alpha },{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\right),\\ \bar{\alpha }={\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\Delta \right)}^{-1}\theta +P^{\text{T}}{\Omega }^{-1}q\right).\end{array}$ (3.7)

3.2. α的后验分布

在上一节中我们得到了 $\alpha$ 基于投资者观点的新的先验，结合第一部分 $\alpha$ 的近似似然函数，我们可以得到 $\alpha$ 的后验分布密度函数

$\begin{array}{c}P\left(\alpha |r_1,\cdots ,r_m,q\right)=\frac{P\left(r_1,\cdots ,r_m,q,\alpha \right)}{P\left(r_1,\cdots ,r_m,q\right)}\\ =\frac{P\left(r_1,\cdots ,r_m|q,\alpha \right)P\left(q|\alpha \right)P\left(\alpha \right)}{P\left(r_1,\cdots ,r_m,q\right)}\\ \propto P\left(r_1,\cdots ,r_m|q,\alpha \right)P\left(\alpha |q\right)\\ =L\left(\alpha |r_1,\cdots ,r_m\right)P\left(\alpha |q\right)\\ \propto \exp \left\{-\frac{1}{2}\left({\left(\alpha -\lambda \right)}^{\text{T}}Q\left(\alpha -\lambda \right)+{\left(\alpha -\bar{\alpha }\right)}^{\text{T}}{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\right)\left(\alpha -\bar{\alpha }\right)\right\},\end{array}$ (3.8)

其中 $\bar{\alpha }={\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\left({\left(\Delta \right)}^{-1}\theta +P^{\text{T}}{\Omega }^{-1}q\right)$ 。在进行下一步计算之前首先引入定理3.1。

定理3.1. 对任意正定矩阵 $A\in {ℝ}^{p\times p}$ ，半正定矩阵 $B\in {ℝ}^{p\times p}$ 与 $a,b\in {ℝ}^p$ 有

${\left(y-a\right)}^{\text{T}}A\left(y-a\right)+{\left(y-b\right)}^{\text{T}}B\left(y-b\right)={\left(y-y^{*}\right)}^{\text{T}}\left(A+B\right)\left(y-y^{*}\right)+{\left(a-b\right)}^{\text{T}}H\left(a-b\right),$

其中 $y^{*}={\left(A+B\right)}^{-1}\left(Aa+Bb\right)$ 且 $H=A{\left(A+B\right)}^{-1}B$ ，若 $B$ 为正定矩阵，则 $H={\left(A^{-1}+B^{-1}\right)}^{-1}$ 。

证明：以一元二次函数配方 $a{x}^2+bx+c=a\left(x-x^{*}\right)+d$ 为例，首先计算左式最小值点，对左式求导可得 $2A\left(y-a\right)+2B\left(y-b\right)=0$ ， $y^{*}={\left(A+B\right)}^{-1}\left(Aa+Bb\right)$ 。由于左右两式二次项系数相同，接下来只要验证常数项是否相等。将 $y=y^{*}$ 代入左式可得：

$\begin{array}{c}{\left(y^{*}-a\right)}^{\text{T}}A\left(y^{*}-a\right)+{\left(y^{*}-b\right)}^{\text{T}}B\left(y^{*}-b\right)\\ ={\left[{\left(A+B\right)}^{-1}\left(Aa+Bb\right)-{\left(A+B\right)}^{-1}\left(Aa+Ba\right)\right]}^{\text{T}}A\left(y^{*}-a\right)\\ +{\left[{\left(A+B\right)}^{-1}\left(Aa+Bb\right)-{\left(A+B\right)}^{-1}\left(Ab+Bb\right)\right]}^{\text{T}}B\left(y^{*}-b\right)\\ ={\left(b-a\right)}^{\text{T}}B{\left(A+B\right)}^{-1}A{\left(A+B\right)}^{-1}B\left(b-a\right)+{\left(a-b\right)}^{T}A{\left(A+B\right)}^{-1}B{\left(A+B\right)}^{-1}A\left(a-b\right)\\ ={\left(a-b\right)}^{\text{T}}\left[\left(A+B-A\right){\left(A+B\right)}^{-1}A{\left(A+B\right)}^{-1}B+A{\left(A+B\right)}^{-1}B{\left(A+B\right)}^{-1}A\right]\left(a-b\right)\end{array}$

$\begin{array}{c}={\left(a-b\right)}^{\text{T}}A{\left(A+B\right)}^{-1}B+A{\left(A+B\right)}^{-1}\left[B{\left(A+B\right)}^{-1}A-A{\left(A+B\right)}^{-1}B\right]\left(a-b\right)\\ ={\left(a-b\right)}^{\text{T}}A{\left(A+B\right)}^{-1}B+A{\left(A+B\right)}^{-1}\left[\left(A+B-A\right){\left(A+B\right)}^{-1}A-A{\left(A+B\right)}^{-1}B\right]\left(a-b\right)\\ ={\left(a-b\right)}^{\text{T}}A{\left(A+B\right)}^{-1}B+A{\left(A+B\right)}^{-1}\left[A-A{\left(A+B\right)}^{-1}\left(A+B\right)\right]\left(a-b\right)\\ ={\left(a-b\right)}^{\text{T}}\left[A{\left(A+B\right)}^{-1}B\right]\left(a-b\right)\\ ={\left(a-b\right)}^{\text{T}}H\left(a-b\right).\end{array}$

左右两式相等。当 $B$ 为正定矩阵时：

$H=A{\left(A+B\right)}^{-1}B={\left[B^{-1}\left(A+B\right)A^{-1}\right]}^{-1}={\left(A^{-1}+B^{-1}\right)}^{-1}.$

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因此在定理3.1中代入 $y=\alpha ,a=\bar{\alpha },b=\lambda ,A={\left({\left(\Delta \right)}^{-1}+P^T{\Omega }^{-1}P\right)}^{-1},B=Q$ ，可将 $\alpha$ 的后验分布密度函数改写为：

$\begin{array}{c}P\left(\alpha |r_1,\cdots ,r_m,q\right)\propto \exp \left\{-\frac{1}{2}\left({\left(\alpha -\lambda \right)}^{\text{T}}Q\left(\alpha -\lambda \right)+{\left(\alpha -\bar{\alpha }\right)}^T{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}\right)\left(\alpha -\bar{\alpha }\right)\right\}\\ =\exp \left\{-\frac{1}{2}{\left(\alpha -{\alpha }^{*}\right)}^{\text{T}}\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P+Q^{-1}\right)\left(\alpha -{\alpha }^{*}\right)\right\}\\ \cdot \exp \left\{{\left(\lambda -\bar{\alpha }\right)}^{\text{T}}{\left[{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)}^{-1}+Q^{-1}\right]}^{-1}\left(\lambda -\bar{\alpha }\right)\right\},\end{array}$ (3.9)

其中 ${\alpha }^{*}=\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P+Q\right)\left[\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)+Q^{-1}\bar{\alpha }+Q\lambda \right]$ 。继而可以得到 $\alpha$ 的后验分布参数：

$\alpha ~N\left({\alpha }^{*},{\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P+Q^{-1}\right)}^{-1}\right).$ (3.10)

因此

$\begin{array}{l}\stackrel{⌢}{\mu }=\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P+Q\right)\left[\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P\right)+Q^{-1}\bar{\alpha }+Q\lambda \right]，\\ \stackrel{⌢}{\Sigma }={\left({\left(\Delta \right)}^{-1}+P^{\text{T}}{\Omega }^{-1}P+Q^{-1}\right)}^{-1}.\end{array}$ (3.11)

4. 总结

本文在传统BL分布模型的基础上，进一步深入探讨了协方差矩阵的估计问题，通过引入基于投资者观点的协方差矩阵的对数先验分布，并运用贝叶斯方法，得到了协方差矩阵的后验参数估计，为实际投资决策提供了有力的数据支持。