Duration

Duration measures how sensitive a bond's price is to changes in interest rates. It is the headline risk number for fixed income, and it has two related meanings.

Two readings of duration

• Macaulay duration — the weighted-average time, in years, until you receive the bond's cash flows, where each time is weighted by the present value of that flow. A longer wait means more rate sensitivity.

• Modified duration — derived from Macaulay duration, it directly estimates the percentage price change for a 1% change in yield:

  Modified duration = Macaulay duration / (1 + y)

The price-change estimate

The key practical formula approximates how much a bond's price moves:

Percent price change is approximately -Modified duration * change in yield

The minus sign encodes the seesaw: yields up, price down.

Worked example

Take the 3-year, 5% coupon bond from the pricing lesson, priced at 973.27 with a 6% yield. Computing its Macaulay duration weights each year by the present value of its flow:

Year 1 weight: 47.17 / 973.27 = 0.04847,  x 1 = 0.04847
Year 2 weight: 44.50 / 973.27 = 0.04572,  x 2 = 0.09144
Year 3 weight: (41.98 + 839.62) / 973.27 = 0.90581,  x 3 = 2.71743

Macaulay duration = 0.04847 + 0.09144 + 2.71743 = 2.857 years

Modified duration = 2.857 / 1.06 = 2.696

Putting it to work

If yields rise by 1% (0.01), the estimated price change is:

Percent change is approximately -2.696 * 0.01 = -0.02696 = -2.70%

So the bond's price falls about 2.70%, from 973.27 to roughly 947.0. Duration gives you, in one number, the answer to "if rates move, how much do I lose or gain?" — but it is only a first approximation, as the final lesson shows.