Bring 4 cups chicken stock to a boil. Add rice, reduce heat to medium-low, cover and simmer until tender, about 17 minutes. Meanwhile, in another saucepan, cook black beans, remaining 3/4 cup chicken stock, coriander and cider vinegar over medium until heated through, 3 to 4 minutes. Stir cilantro into rice. Serve with sliced avocado and liimes.