If necessary to keep the tenderloin in a compact piece, tie meat at several intervals with cotton string.
Melt butter in an ovenproof 10- to 12-inch nonstick frying pan over high heat. Add pork tenderloin and turn as needed to brown well on all sides, 5 to 6 minutes total.
Add apricot chutney and 1/3 cup water to pan and stir to incorporate meat drippings. Baste tenderloin with some of the sauce, and then put it in a 450° oven.
Roast pork, basting often with sauce, until meat is 150° to 155° in center of thickest part, 12 to 14 minutes. If sauce starts to scorch, stir in water, 2 tablespoons at a time.
Transfer the tenderloin to a platter. Pour the apricot chutney sauce over pork, or serve the sauce in a small bowl and add to taste. Slice tenderloin and season to taste with salt and pepper.