It is demonstrated that the duration of every bond declines linearly between payment dates by an amount that exactly equals the time between such dates. Once the interest or coupon payment is received, duration rises abruptly to a recovery point that depends on the coupon, the yield, and the years to maturity of the individual bond. The duration of every bond follows the same downward path between payment dates, and every bond, except zero coupons, describes a sawtoothed pattern as its maturity shortens. This phenomenon is illustrated for the simple case of a 10-year bond with an annual coupon of $100. The time between payments is one year, and duration is shown to decrease by that amount in each and every year. The linear relation is both surprising and significant. The linearity stems from the fact that the compounding bond price is offset by a compounding interest payment. The relation is significant for the way in which the duration of fixed income portfolios is calculated and managed.