Abstract: This study will discuss the problems of quadratic investment portfolio without a risk-free asset based on value-at-risk. It is assumed that the risk of an investment portfolio measured by value-at-risk. The resolution of problems that do include: first, formulate models the trade-off problem. Secondly, formulate expectation maximization model of the problem. Third, formulate model minimization of value-at-risk problem. Based on the results of the discussions can be concluded that the trade-off between risk and expected return does not only depend on the type of investor but also on the size of the investment. In a realistic investment situation, it is likely that more constraints, e.g., restriction on short-selling, need to be considered.