Purpose – To develop a new theory of portfolio and risk based on incremental entropy and Markowitz's theory. Design/methodology/approach – Replacing arithmetic, the mean return adopted by M.H. Markowitz, with geometric mean return as a criterion for assessing a portfolio, one gets incremental entropy: one of the generalized entropies. It indicates that the incremental speed of capital is a more objective and testable criterion. Findings – The difference between the new theory based on incremental entropy and Markowitz's theory is that the new theory emphasizes that there is an objectively optimal portfolio for given probability of returns. Originality/value – This paper provides some formulas for optimizing portfolio allocations. Based on the new portfolio theory, this paper also presents a new measure of information value, analyzes the differences and similarities between this measure and K.J. Arrow's measure of information value, and discusses how to optimize forecasts with the new measure.