Rinse the rice well. Place in a bowl and cover with water. Let soak 10 minutes. Rinse again. Allow the rice to drain in the strainer.
Mix the achiote paste in the vegetable broth, pushing with the back of the spoon until the paste is completely dissolved. Set aside.
Heat the oil in a heavy saucepan over medium flame. Add the onions and cook and stir a few minutes. Stir in the rinsed rice and diced carrot. Cook, stirring frequently, about 10 minutes, until the rice is coated with oil and turns light golden brown.
Add the achiote broth and the vegetable juice.
Bring the rice to a boil, cover, and lower the heat to the lowest setting possible. Cook without stirring, 35 to 45 minutes. (To check if the liquid has been absorbed or not, tilt the pan to the side.) Remove from heat and let stand 5 minutes before serving.