How do actuaries earn bonuses in their total compensation? do they sell insurance to their clients?

Actuaries do not sell insurance. Ignoring the fact actuaries couldn't find the time to sell insurance, it would probably be a conflict of interest for an actuary to sell the product of an insurer they represent as the actuary has a vested interest in keeping the insurer's rates low. Plus no insurer would hire an actuary who is selling some other insurer's as the actuary has a vested interest in keeping their competitor's rates as high as possible. Unless the actuary has negotiated a bonus based on an insurer's results an actuary wouldn't automatically earn a bonus. Despite what other people have incorrectly posted, insurance brokers along with insurance agents and sales staff (used by insurance companies that sell directly to the public) sell a company's insurance products to the public.

I am an insurance agent who studied actuaries mathematics during college and practice the art for several years until the economy turned down. Actuaries do calculations for Insurance firm to give them a formula for determining life spans or predictability of accidents. This is considered to be an art form, not science. A group of actuaries can all develop a statistical worksheet where no two are the same, yet all can be considered correct and functional. Second part of your statement is also incorrect. An Agent Sales Insurance, the broker works between the agent and the insurance companies to find the best quotes. Both the Agent and Broker usually receives a commission from the insurance company for binding coverage with them. Rarely does the Broker ever work directly with the client or insured. Hope this help.

actuaries seldom receive a bonus it is their job to deal with facts not make up stuff to get better results