This research discussed on finding the optimal investment strategy for an insurance investor who had logarithmic utility preference and traded two assets; (i) a risk-free bond which rate of return was constant and (ii) a risky stock which rate of return was modeled to follow the Modified Constant Elasticity of Variance (M-CEV) model. The dynamic programming principle and the Ito’s lemma were employed to obtain Hamilton-Jacobi-Bellman (HJB) equation from which the optimal investment strategy was got. The relationships between the optimal investment strategies got when the Brownian motions did not correlate and when the Brownian motions correlated were looked into and it was found that, among others, when the Brownian motions have a negative correlation the investor required less amount of money to be in business when the Brownian motions did not correlate than when the Brownian motions correlated