Bay and Brown are actually a modification of Black. In other words, bays and browns are black horses with a dominant modifying gene called AGOUTI added.
Our first and most common color modifier in horse genetics is Agouti, also known as the “bay” gene.
Agouti acts to restrict the black of the black horse to just specific points, which are the mane, tail and lower legs. This allows the red that is “uncovered” by agouti to show through on the body of the horse in varying degrees, depending on the type of Agouti present.
Chestnuts can carry agouti, but it won’t show on them as they have no black hair to restrict to the points. They can, of course, pass it on to their offspring.
Black horses cannot have agouti, because if they did, they’d be bay, not black.
A black and chestnut cross might produce a bay foal. This happened because the agouti, or bay gene, actually came from the chestnut parent, and the black base color came from the black parent.
It is theorized that there are three different types of agouti. The Agouti which causes “regular” bay is written as an “A.” The “wild type” Agouti, which results in a light bay horse with very little black on the legs, (usually just around the fetlocks) is written as “A+.”
Brown, or what is sometimes called “seal brown,” (a nearly black horse with reddish areas on the muzzle and flanks) is written “At.” A horse will have some combination of the three types, one of the three types plus one non-agouti gene (one non-agouti gene is written as “a”), or have no agouti at all (written as “aa”).