We investigate an insurer's optimal investment and liability problem by maximizing the expected terminal wealth under different utility functions. The insurer's aggregate claim payments are modeled by a Levy risk process. We assume that the financial market consists of a riskless and a risky assets. It is also assumed that the insurer's liability is negatively correlated with the return of the risky asset. The closed-form solution for the optimal investment and liability ratio is obtained using Pontryagin's Maximum Principle. Moreover, the solutions of the optimal control problems are examined and compared to the findings where the jump sizes are assumed to be constant.