When faced with a new customer, many factors contribute to an insurance firm's decision of what offer to make to that customer. In addition to the expected cost of providing the insurance, the firm must consider the other offers likely to be made to the customer, and how sensitive the customer is to differences in price. Moreover, firms often target a specific portfolio of customers that could depend on, e.g., age, location, and occupation. Given such a target portfolio, firms may choose to modulate an individual customer's offer based on whether the firm desires the customer within their portfolio. We term the problem of modulating offers to achieve a desired target portfolio the portfolio pursuit problem. Having formulated the portfolio pursuit problem as a sequential decision making problem, we devise a novel reinforcement learning algorithm for its solution. We test our method on a complex synthetic market environment, and demonstrate that it outperforms a baseline method which mimics current industry approaches to portfolio pursuit.