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The Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization) is a factorization of an $n \times n$ nonnegative symmetric matrix $A$ of the form $BCB^T$, where $C$ is a $k \times k$ symmetric matrix, and both $B$ and $C$ are required to be nonnegative. This work introduces the SNT-rank of $A$, as the minimal $k$, for which such factorization exists. After listing basic properties and exploring SNT-rank of low rank matrices, the class of nonnegative symmetric matrices with SNT-rank equal to rank is studied. The paper concludes with a completion problem, that asks for matrices with the smallest possible SNT-rank among all nonnegative symmetric matrices with given diagonal blocks.

In this paper, it is investigated whether government, when promises pension fundʼs members a so-called minimum guaranteed return, to reduce the exposure of members to financial risks , should at the same time hinders portfolio diversification process of pension funds. We provide a detailed analysis of the connection between the requirements for providing a minimum guaranteed return and managing financial risks on the one hand and the investment structure of pension funds on the other. We intend to demonstrate with an illustrative case, using the simulation technique and a combination of actual data and some hypothetical one, that by precisely matching the investments' characteristics to the characteristics of the pension fund's liabilities, some important financial risks can even be hedged entirely. We also intend to demonstrate that with the implementation of a proper policy of risk measurement and management, complemented with stress testing practices, excessive legislative restrictions for investments are no longer necessary. At the very least, governments should avoid implementing legislation that hinders the portfolio diversification process and therefore makes pension fund risk management more difficult.

Game theory : introduction and applications