## 3 Preliminaries  
### Classical Black-Litterman Model  
The MV model can be mathematically defined by (1) [1]  
\[\text{minimize}\ \ W^{T}\Sigma W\] \[\text{maximize}\ \ W^{T}R\] \[\sum_{i=1}^{N}w_{i}=1 \tag{1}\]  
where \(N\) is the number of assets available, \(R=(r_{1},r_{2},\ldots,r_{N})^{T}\) is the expected return vector of assets in the portfolio, \(W=(w_{1},w_{2},\ldots,w_{N})^{T}\) defines the proportion of capital invested in each asset in the portfolio, and \(\Sigma=(\sigma^{\prime})_{N\times N}\) denotes the covariance matrix of the asset returns.  
It is obvious that the MV model uses the variance as a risk measure, targeting at obtaining the weight vector that maximizes the expected return of the portfolio but minimizes the risk. However, from a practical perspective, there exist some drawbacks in the model (1), such as highly-concentrated portfolio [5] and sensitive input [34]. To overcome the above problems, the BL model is proposed by using the investor views of the asset returns to revise \(R\) and \(\Sigma\) in the model (1).  
The cornerstone of the BL model is to introduce investor preferences from the perspective of Bayesian analysis. With the help of the views, a realistic estimate of expected return is no longer demanded. Instead, what is needed is the prior expected return obtained by historical data, so that a posterior estimate of expected returns can be produced. It is assumed that investors will assign their wealth to \(N\) assets and \(\hat{E}(R)\) denotes the posterior estimate of expected returns, which is composed of two parts. The first part is the market equilibrium return \(\Pi\) obtained by the capital asset pricing model, and the second one comes from views on certain assets. Therefore, \(\hat{E}(R)\) can be formulated as follows:  
\[\hat{E}(R)=[(\tau\Sigma)^{-1}+P^{T}\Omega^{-1}P]^{-1}[(\tau\Sigma)^{-1}\Pi+P^{T }\Omega^{-1}Q] \tag{2}\]  
where  
* \(Q\) is a \(K\)-dimensional view vector which maintains \(K\) subjective returns of certain assets.
* \(\Omega\) refers to a \(K\times K\) matrix that reflects the confidence in views.
* \(P\) is a \(K\times N\) mapping matrix, representing the correspondence between \(K\) views and corresponding assets.
* \(\Sigma\) is the \(N\times N\) covariance matrix of asset returns.
* \(\Pi\) is the market equilibrium return calculated by the Formula (3), where \(w_{\textit{mkt}}\) is the market-cap weights, and \(\delta\) denotes the risk aversion coefficient which is obtained by dividing the market excess return \(R-R_{f}\) by its variance \(\sigma^{2}\).  
\[\Pi=\delta\Sigma w_{\textit{mkt}},\ \ \ \delta=\frac{R-R_{f}}{\sigma^{2}} \tag{3}\]  
Similarly, the posterior estimate of the covariance matrix is shown in (4).  
\[\hat{\Sigma}=\Sigma+[(\tau\Sigma)^{-1}+P^{T}\Omega^{-1}P]^{-1} \tag{4}\]  
Combining the view vector \(Q\) with the mapping matrix \(P\) can either express relative or absolute opinions on the expected returns. A relative view usually describes statements about the connection between asset returns, for example, 'A will outperform B by X%'. On the other hand, an absolute view directly provides specific return forecasts for certain assets, such as 'C will climb to 8%'. \(\Omega\) is another parameter that is critical to BL model. Based on it, each view can be allocated an uncertainty level. In practice, it is believed to be proportional to the variance of asset returns, and can be obtained by the following expression:  
\[\Omega=\tau P\Sigma P^{T} \tag{5}\]where \(\tau\) represents the relative proportion of the variance of asset returns.