A risk minimization problem is considered in a continuous-time Markovian regime-switching financial model modulated by a continuous-time, finite-state, Markov chain. We interpret the states of the chain as different states of an economy. A particular form of convex risk measure, which includes the entropic risk measure as a particular case, as a measure of risk and an optimal portfolio is determined by minimizing the convex risk measure of the terminal wealth. We explore the state of the art of the stochastic differential game to formulate the problem as a Markovian regime-switching version of a two-player, zero-sum, stochastic differential game. A novel feature of our model is that we provide the flexibility of controlling both the diffusion risk and the regime-switching risk. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution of the game is provided.