Your horse would be considered an asset and must be depreciated.  

Broodmares, stallions, horses older than 12 years of age, and racehorses depreciate over three years; broodmares, stallions, show horses, riding horses, or any other horse 12 years or younger depreciate over seven years. 

Depreciation begins when the horse is ‘placed in service, and that doesn’t always mean the year in which the horse is purchased. A Thoroughbred yearling bought for racing might be considered “placed in service” when training starts in the fall of the yearling year or when the horse begins racing as a 2-year-old. The same yearling bought as a show hunter prospect might not be “placed in service” for tax purposes until much later.