Bond Duration Calculator

Bond duration measures how much a bond's price moves when interest rates change. Macaulay duration is the weighted average time until the bondholder receives the cash flows, expressed in years. Modified duration converts that into a price estimate: the approximate percentage change in price for a one-percentage-point move in yield. For a 10-year bond paying a 5% coupon semi-annually at a 5% market yield, Macaulay duration is 7.99 years and modified duration is 7.79, so a one-point rise in yield points to a price fall of roughly 7.79%.

Quick example

With an annual coupon rate of 5% and a market yield of 5%, over 10 years to maturity with two payments a year, the result is 7.99 years.

The estimate is linear; the real curve is not

Modified duration draws a straight line through the price-yield curve at the current yield, and the curve bends away from that line in both directions. Repricing the sample bond exactly, a rise from 5% to 6% costs 7.44%, not 7.79%, and a fall from 5% to 4% gains 8.18%. The estimate overstates the loss and understates the gain: the gap is 0.36 of a percentage point on the rise, about 4.8% of that move, and 0.38 of a point on the fall. That asymmetry is convexity, and it works in the holder's favour: prices rise more on a rate cut than they fall on an equivalent rate rise. The gap grows with the square of the yield change: 0.02 of a point at a quarter-point rise, 0.36 at one point, 1.38 at two and 7.82 at five. At five points that correction is about a fifth of the linear estimate: material, though the duration term is still the larger of the two.

Which inputs move duration most

Measured at the sample figures with one-percentage-point moves, the coupon is the larger lever at 2.9 times the yield lever on a downward move, and 2.4 times on an upward one:

• Coupon one point lower (4%) raises Macaulay duration to 8.26 years, a move of +0.27
• Coupon one point higher (6%) lowers it to 7.76 years, a move of −0.23
• Yield one point lower (4%) raises it to 8.08 years, a move of +0.09
• Yield one point higher (6%) lowers it to 7.89 years, a move of −0.09

The direction rule behind the first two is consistent: a lower coupon and a lower yield each raise duration. A lower coupon leaves more of the total value sitting in the final redemption, which pushes the weighted average time outward. A lower yield discounts distant cash flows less heavily, doing the same thing. Maturity is less straightforward. For a coupon-paying bond, duration converges towards (1+i)/(i·m) years as the term grows, where i is the periodic yield. That limit is 20.50 years at the 5% default yield and 8.83 years at a 12% one. A par or premium bond approaches that limit from below, so a longer term raises duration across the whole range this tool accepts. A discount bond can overshoot the limit and then fall back towards it. At a 5% coupon the turn first appears inside the 50-year cap near an 8% yield, with the peak at 47.5 years, moving in to 24.5 years at a 12% yield. At a 2% coupon against a 12% yield the peak sits near 22.5 years at 12.00, and a 30-year bond on the same terms comes in lower, at 11.39. A 1% coupon against a 4% yield is a far deeper discount and never turns inside the range at all. The zero-coupon case stands outside this entirely. With no interim cash flows, duration equals maturity exactly at any yield, rising without limit rather than converging.

Payment frequency pulls the two figures apart

The fourth input moves the two outputs in opposite directions, which is easy to miss. Holding the sample bond at a 5% coupon and 5% yield over 10 years:

• Annual payments: Macaulay 8.11 years, modified 7.72
• Semi-annual: Macaulay 7.99, modified 7.79
• Quarterly: Macaulay 7.93, modified 7.83
• Monthly: Macaulay 7.89, modified 7.86

More frequent coupons pull cash forward, which shortens Macaulay duration. Modified duration divides by one plus the periodic yield, and that divisor shrinks as the period gets shorter, which lengthens the result. The two converge as frequency rises.

How maturity and coupon interact

Quoting a duration range for a given maturity does not work, because the coupon and yield move it too much. Across coupons from 0% to 12% and yields from 1% to 12%, a 10-year semi-annual bond spans 6.08 to 10.00 years of Macaulay duration, and a 30-year bond spans 8.57 to 30.00. At the sample 5% coupon and 5% yield the figures are 7.99 and 15.84. The maturity sets a ceiling that duration can never exceed, and the coupon and yield decide how far below it the bond sits.

What duration matching describes

Duration matching is the practice of setting a portfolio's duration close to the holding period. The intention is that price and reinvestment effects move in opposite directions: if rates rise, prices fall but coupons are reinvested at higher yields, and the two roughly offset over a horizon equal to the duration. Pension funds and insurers work with the technique because their liabilities have a known timing profile. Bond ladders reach a similar effect through a different route, by spreading maturities across a range of dates.

What the calculation assumes

The model prices a plain fixed-coupon bond held to maturity at a single flat yield, with the redemption falling exactly at the maturity entered and any short first period absorbing the remainder when the term does not divide evenly into whole coupon periods. It assumes no default, no early redemption and no change in yield across the bond's life, and it excludes transaction costs, tax and liquidity effects. Where the term produces a stub period, the price row is a dirty price rather than a clean one: the next coupon is discounted in full from its payment date, which is the mechanics a buyer meets, so the figure carries the accrued interest settled alongside it. On the 10.5-year annual-coupon case at a 5% coupon and 5% yield the row reads 102.47, of which 2.50 is half a period of accrued interest, leaving a clean price of 99.97. The price row is quoted per 100 of face value on the same basis.