Before revision:  The pension manager makes investment decisions for the benefit and on behalf of the representative pension member whose objective is to maximize his expected utility of the terminal surplus between the final fund level and the minimum guarantee . To solve the initial constrained non-self-financing optimization problem , we transform it to an unconstrained self-financing problem by replicating the future contributions and the minimum guarantee .
After revision:  The pension scheme invests in the risky assets available in the market, including the longevity bond, by using the contributions from a representative scheme member to ensure a minimum guarantee such that the member is able to purchase a lifetime annuity upon retirement. We transform this constrained optimal investment problem into an unconstrained problem by replicating the future contributions and the minimum guarantee .