Macaulay Duration

The calculation proceeds in four steps. Discount every scheduled cash flow at the bond's yield to maturity to obtain its present value. Divide each present value by the bond's full price, which includes accrued interest, to obtain that payment's weight; the weights sum to one by construction. Multiply each weight by the time, measured from the settlement date, until that payment arrives. Sum the products.

Because discounting is normally done on a periodic basis, the intermediate result is expressed in coupon periods. Dividing by the number of periods per year converts it to years, the unit in which Macaulay duration is quoted. Two structural results follow immediately: a single-payment security has a duration equal to the time to that payment, and a perpetuity's duration equals one plus the periodic yield divided by that yield, a finite number despite infinite maturity.

Macaulay duration is a time measure, not a sensitivity measure. Converting it into one requires dividing by one plus the yield per period. The result, modified duration, estimates the approximate percentage change in the full price for a one percentage point change in yield, with the sign reversed because price and yield move inversely.

Multiplying modified duration by the full price gives money duration, the currency change in value per unit change in yield, and scaling that to a single basis point produces the price value of a basis point used on trading desks. Each of these is a first-order, linear approximation drawn from the same tangent line, so all inherit the same weakness: they understate the price gain from falling yields and overstate the loss from rising yields, an error that convexity corrects.

Lower coupons push weight toward the final principal payment and lengthen duration, which is why zero-coupon issues sit at the extreme. Higher yields compress the present value of distant cash flows relative to near ones and shorten duration, meaning duration itself changes as the market moves. Longer maturity generally lengthens duration, though for deeply discounted long-dated bonds the relationship can flatten or even reverse at very long tenors.

Duration also drifts within a coupon period, declining steadily as the next payment approaches and stepping back up once that coupon is paid, producing a sawtooth path over the bond's life. Embedded options break the relationship entirely: a callable bond's cash flows are not fixed, so effective duration, computed by repricing under shifted yield curves, replaces the analytical measure.

Macaulay duration has an economic meaning beyond arithmetic. Rising yields depress a bond's price but raise the return on reinvested coupons; these two effects offset at one particular horizon, and that horizon is the Macaulay duration. An investor whose holding period equals the bond's Macaulay duration is therefore approximately insulated from a one-time parallel shift in rates. The difference between the two is the duration gap: positive when duration exceeds the horizon, leaving the investor exposed to rising rates, and negative when the horizon is longer, leaving reinvestment risk dominant.

This logic underpins liability-driven investing and single-liability immunization, where portfolio duration is matched to the timing of an obligation and rebalanced as yields and time shift it. Candidates should be able to compute duration for a short-maturity bond, convert between Macaulay and modified duration in both directions, and reason about the sign of a duration gap.