No Zakah is to be paid by the Muslim on his slave or horse

Zakah is founded on justice and equality, hence, Allah has made it obligatory upon all types of properties, growing or apt to growth, such as harvested crops and merchandise for sale.
As for properties that do not grow, those intended for possession or personal use, no Zakah is due on its owners, because it is a personal item. This category includes what one rides, be that a horse, a camel, or a car. The same rule applies to the slave used for service, one's furniture and utensils made for personal use.
However, the exception here is Zakat al-Fitr on the slave, which is obligatory because it is associated with people not properties.