In this work, we tackled an optimal investment strategy problem of an insurance investor, who had logarithmic utility preference and invested in two assets: 1) a riskless bond with a constant rate of return and 2) a risky asset (stock) whose price dynamics followed modified constant elasticity of variance (M-CEV) model. We focused on getting an optimal investment strategy that will maximize his returns and pays policy holders their claims whenever they occur. We derived formulae that allowed us to analyze the impact of the models parameters of the coefficient of correlation of the Brownian motions and transaction cost. It was found, among others, that if the Brownian motions increase or decrease together, the investor will need less funds to be in business than when the Brownian motions do not increase or decrease together.