PurposeThe purpose of this paper is to study a fractional grey model FAGM(1,1,tα) based on the GM(1,1,tα) model and the fractional accumulated generating operation, and then predict the national health expenditure, the government health expenditure and the out-of-pocket health expenditure of China.Design/methodology/approachThe presented univariate grey model is systematically studied by using the grey modelling technique, the fractional accumulated generating operation and the trapezoid approximation formula of definite integral. The optimal system parametersrandαare evaluated by the particle swarm optimisation algorithm.FindingsThe expressions of the time response function and the restored values of this model are derived. The GM(1,1), NGM(1,1,k,c) and GM(1,1,tα) models are particular cases of the FAGM(1,1,tα) model with deterministicrandα. Compared with other forecasting models, the results of the FAGM(1,1,tα) model have higher precision.Practical implicationsThe superiority of the new model has high potential to be used in the medicine and health fields and others. Results can provide a guideline for government decision making.Originality/valueThe univariate fractional grey model FAGM (1,1,tα) successfully studies the China’s health expenditure.