We show that for a disappointment-averse decision maker, splitting a lottery into several stages reduces its value. To do this, we extend Gul.s (1991) model of disappointment aversion into a dynamic setting while keeping its basic characteristics intact. The result depends solely on the sign of the coefficient of disappointment aversion. It can help explain why people often buy periodic insurance for moderately priced objects, such as electrical appliances and cellular phones, at much more than the actuarially fair rate.