School of Mathematics and Computational Science, Sun Yat-sen University

In this paper, we investigate efficient portfolio in general M-V model with singular
covariance matrix. This paper not only establishes the necessary and sufficient condition for existing efficient portfolio in the stock market, but derives the general solutions of efficient portfolio and some properties of efficent frontier. Finally we make no-arbitrage analysis for the stock market with singular covariance matrix, obtain the necessary and sufficient condition for not existing abritarge portfolio, which proves the conjecture proposed by Szeg\"{o}.