# An ADMM Based Parallel Approach for Fund of Fund Construction
## 2 Problem Formulation  
The core of FOF funds is to diversify investment, so it is no surprise that asset allocation has become an important part of FOF construction. Based on the asset allocation, we can improve the return-risk feature of a fund portfolio. For simplicity, we limit the fund type to equity funds, so asset allocation is beyond the scope of this paper. Furthermore, another determining factor for the performance of an FOF strategy is the quality of the fund pool. It is naturally an extremely important thing for fund managers to screen out their own fund pools from a huge number of funds and build fund portfolios accordingly. There are many fund selection criteria such as risk/return parameters and the professionalism of the management team. In this paper, we take return, standard deviation and fund size as the key factors of focus to screen out funds. It is assumed that investors with initial capital \(C\) will assign their wealth to \(n\) equity funds in the following \(T\) periods. Let \(x_{i}^{n}\) be the proportion on fund \(i\) included in the portfolio at the previous period. \(S_{r}\), \(S_{s}\), \(S_{m}\), \(S_{b}\) and \(S_{c}\) are the set of the funds with high risk and high return, with medium to high risk and medium to high return, with medium risk and medium return, with low to medium risk and low to medium return, and with low return, included in the portfolio, respectively. A new FOF optimization model is proposed as follows:  
\[ \begin{array}{ll}\min\limits_{x\in\mathbb{R}^{n}}&\frac{1}{2}x^{T}Px+q^{T}x +\sum_{i=1}^{n}f_{i}(x_{i})\\ \text{subject to}&\sum\limits_{i\in S_{r}}x_{i}\leq 0.2\\ &\sum\limits_{i\in S_{c}}x_{i}\leq 0.15\\ &0.95\sum\limits_{i\in S_{n}}x_{i}+0.6\sum\limits_{j\in S_{m}}x_{j}\leq 0.4\\ &\sum\limits_{i\in S_{b}}x_{i}+0.3\sum\limits_{j\in S_{m}}x_{j}\geq 0.4\\ &\mathbf{1}^{T}x=1\\ &l_{i}\leq x_{i}\leq u_{i},i=1,\ldots,n\end{array}  \tag{1}\]  
where the transaction cost \(f_{i}(x)=\exp(-(\frac{c\max\{x_{i}-x_{i}^{0},0\}+u_{i}}{b_{i}})^{2})\), \(a_{i}\in\mathbb{R}\) and \(b_{i}\in\mathbb{R}\) are the given parameters, which are various from different funds. \(P\) denotes the covariance matrix of the fund returns, and \(q\in\mathbb{R}^{n}\) refers to the fund returns vector. \(l_{i}\) and \(u_{i}(i=1,\ldots,n)\) are the lower and upper bound of investment proportion on fund \(i\). Without loss of generality, we can abstract the new FOF optimization model into a general form. Notation. Let \(\mathbb{R}\) denote the set of real numbers, \(\mathbb{R}^{n}\) the \(n\)-dimensional real space, and \(\mathbb{S}^{n}_{++}(\mathbb{S}^{n}_{++})\) the set of real \(n\)-by-\(n\) symmetric positive (semi)definite matrices. We denote by \(\mathbb{R}^{m\times n}\) the set of real \(m\)-by-\(n\) matrices, \(I_{n}\in\mathbb{R}^{n\times n}\) is the identity matrix and \(\mathbf{0}\in\mathbb{R}^{n\times n}\) is the zero matrix, and \(\mathbf{1}_{n}\) is the \(n\)-dimensional vector with all the entries being \(1\). \(\mathcal{I}_{\mathcal{C}}\) is the indicator function over the affine constraints of (5), i.e.,  
\[I_{\mathcal{C}}(x)=\left\{ \begin{array}{ll}0&x\in\mathcal{C}\\ \infty&\text{otherwise}\end{array} \right. \tag{2}\]  
For \(a,b\in\mathbb{R}\), the projection of \(z\) onto \([a,b]\) is given by \(\Pi[a,b](z)=\min(\max(z,a),b)\). The FOF optimization model that considered transaction cost is given by:  
\[ \begin{array}{ll}\min\limits_{x\in\mathbb{R}^{n}}&\frac{1}{2}x^{T}Px+q^{T}x +\sum_{i=1}^{n}f_{i}(x_{i})\\ \text{subject to}&Ax\leq b,\\ &l_{i}\leq x_{i}\leq u_{i},i=1,\ldots,n\\ &\mathbf{1}^{T}x=1\end{array}  \tag{3}\]  
where \(x\in\mathbb{R}^{n}\) is the optimization variable. \(P\in\mathbb{S}^{n}_{+}\) is a symmetric semidefinite covariance matrix and \(q\in\mathbb{R}^{n}\). \(A\in\mathbb{R}^{m\times n}\) and \(b\in\mathbb{R}^{\in m}\). \(l_{i}\) and \(u_{i}(i=1,\ldots,n)\) are the lower and upper bound. \(f_{i}(x),x\in\mathbb{R},i=1,\ldots,n\) is the transaction cost function. The function \(f_{i}(x)\) is the subscription and redemption cost for adjusting the \(i\)th fund.