What is an agency (wakalah)? And is the owner of the money sinful if the agent (wakeel) is unable to fulfill the condition of investing the money with a specified increase, or must he accept the loss?

Agency (wikalah) is the authorization by one person for another to perform on their behalf an act that is legally permissible to delegate during their lifetime. As for giving money to someone to trade with it, with the profit shared between them according to an agreement, this is called Mudarabah (commenda). For Mudarabah to be permissible, it is stipulated that the owner does not stipulate a fixed profit for the agent, nor does he guarantee the principal. If either of these is stipulated, the contract becomes void, and the agent is entitled to the wage of their like (ujrat al-mithl), while the owner receives their principal along with its profit or loss. Therefore, the described scenario is not an agency, and it is impermissible due to the stipulation of a fixed profit.