Addressing climate change requires a transition to low-carbon, climate-resilient investments, with green bonds gaining traction among institutional investors focused on sustainability objectives. This study models their decision-making by integrating green investment targets into utility functions, combining a constant relative risk aversion (CRRA) component with a goal-oriented utility modification. Employing a Lagrangian approach, we derive the optimal terminal wealth and portfolio composition, including green bond allocation. Contrary to common intuition, more ambitious green investment targets do not always yield greater green bond allocations. The outcome significantly hinges on investors' risk aversion. This intricate interaction between risk aversion and goal-oriented utility reveals the complexity of green investment decisions.