Duration Explained: How Much a Bond Falls When Rates Rise

Frederick Macaulay published the idea in 1938, buried in a National Bureau of Economic Research monograph on the movement of interest rates and bond yields since 1856, and then very little happened with it for about thirty years. The insight was that the stated maturity of a bond is a poor description of how long the lender’s money is actually committed, because a bond that pays coupons hands some of the money back along the way. Macaulay weighted each payment date by the present value of the payment arriving on it. He called the resulting average the duration. For three decades that was a curiosity, since interest rates in the middle of the twentieth century moved slowly enough that nobody needed a precise measure of what happened when they did not. The 1970s changed the incentives. Duration went from a footnote to the number every fixed income desk quotes before it quotes anything else.

It answers one question with one number. If yields move a percentage point, how far does this bond move? A bond with a modified duration of 7.72 loses roughly 7.72% of its value. That is what one point on the yield does. That is why professionals talk about duration all day and about maturity almost never.

The word covers two related measures. One is a weighted average of time. The other is a sensitivity. Both fall out of the same table of cash flows.

Macaulay’s weighted average wait

Work it for a ten-year bond with a 5% coupon. It is priced at par, because the market yield is also 5%. Discount each payment at 5%, multiply every present value by the year it arrives in, add the column and divide by the price, remembering that each year brings $50 and year ten brings $1,050.

Year	Cash flow	Present value at 5%	Year x present value
1	$50	$47.62	$47.62
5	$50	$39.18	$195.88
9	$50	$32.23	$290.08
10	$1,050	$644.61	$6,446.09
All years		$1,000.00	$8,107.83

Macaulay duration = $8,107.83 / $1,000 = 8.11 years

The bond runs for ten years. The weighted average wait is 8.11 years, because the coupons pull money forward out of the final repayment. A zero-coupon bond has no coupons to pull anything forward, so its Macaulay duration equals its maturity exactly, which is why zeros are the most rate-sensitive instrument available at any given maturity and why they are the tool of choice for anyone deliberately taking a view on rates.

Modified duration, and the check against an exact price

Modified duration converts the average wait into a price sensitivity:

Modified duration = Macaulay duration / (1 + yield per period)

For this bond, 8.11 / 1.05 = 7.72. The estimated price change is then:

Price change = -modified duration x change in yield

so a one point rise in yields predicts -7.72 x 1% = -7.72%.

Now check it against the arithmetic done the long way. In how bonds work this same bond is priced exactly at a 6% yield. It comes out at $926.40, a fall of 7.36%. Duration predicted 7.72%. The gap of 0.36 percentage points is curvature. A straight line cannot capture it, and it grows with the size of the move.

What maturity and coupon do to it

Duration rises with maturity. It falls as the coupon rises, because a larger coupon returns more of the money earlier. Below are 5% coupon bonds priced at a 5% yield. The duration estimate is set against the exact repricing at a 6% yield.

Maturity	Macaulay	Modified	Duration estimate	Exact price change
2 years	1.95	1.86	-1.86%	-1.83%
5 years	4.55	4.33	-4.33%	-4.21%
10 years	8.11	7.72	-7.72%	-7.36%
30 years	16.14	15.37	-15.37%	-13.77%

Two things fall out of that table. The estimate is accurate at the short end. It drifts badly at the long end, because curvature compounds with duration. And the thirty-year bond is roughly eight times as sensitive as the two-year to an identical move in yields, which is the entire argument for thinking in duration, whatever the maturity column says. Two bonds described as government paper can differ in risk by a factor of eight, and the word government tells you nothing about which is which.

Convexity, and why the error runs one way

The relationship between price and yield is a curve, and it is steeper on the side where yields fall than on the side where they rise. The thirty-year bond above loses 13.77% when yields rise a point. When yields fall a point instead, it gains 17.29%. The straight line of duration sits below the true curve on both sides. It overstates the loss and understates the gain every time.

That asymmetry is convexity. On a conventional bond it amounts to a small gift to the holder, and it grows with maturity and shrinks as coupons rise, which means the instruments carrying the most duration also carry the most convexity.

Callable bonds break the arrangement. When yields fall far enough, the issuer redeems the bond, so the price stops rising somewhere near the call price while the downside remains entirely intact. The holder keeps the loss and gives away the gain, which is negative convexity, and it is the structural reason callable corporates and mortgage-backed securities have to pay extra yield to find buyers, and the measures that price it, yield to call and yield to worst, are in bond yields explained.

What 2022 demonstrated, and what it repeated

2022 was the worst calendar year for the Bloomberg US Aggregate index since its inception. No credit event caused it. Yields rose across the curve from very low starting levels and the arithmetic on this page did the rest, which is a duller explanation than the one usually offered and a complete one.

What made it painful was the starting point. When a bond yields 1.5%, a year of coupon income covers a yield rise of less than a quarter of a point on a portfolio with a duration of 6 years, and when the same portfolio yields 5%, it can absorb a rise of over three quarters of a point before the year turns negative. Low yields leave no buffer, and duration is at its most dangerous precisely when income is at its smallest, which is the opposite of the intuition most people bring to safe assets.

That breakeven is worth calculating before buying anything:

Breakeven yield rise = yield / modified duration

A fund yielding 4.5% with a duration of 6 breaks even on a rise of 4.5 / 6 = 0.75 percentage points over a year.

The institutional version of the same lesson arrived in March 2023, when a California bank that had funded long-dated securities with deposits capable of leaving in a single afternoon discovered that the losses duration had been quietly accumulating became real the moment it had to sell. Nothing in its portfolio had defaulted. The bond mathematics had simply been left to run for two years. The funding base could not wait.

Where duration stops telling the truth

Duration assumes the entire yield curve shifts by the same amount at once. Real curves twist. The two-year can rise while the thirty-year falls, which is what makes the shapes described in the yield curve explained worth watching in the first place. Desks handle this with key rate durations, which measure sensitivity to each segment of the curve separately, and the fact that they bother tells you how often the simple version is wrong.

Duration also assumes the yield change comes from interest rates. For a corporate bond a great deal of the move comes from credit spreads widening instead, which duration will describe correctly in size and entirely wrongly in cause. When spreads blow out, Treasuries of matching duration often rise while the corporate falls. That divergence is set out in corporate bonds and credit spreads.

And the estimate degrades on large moves. For a yield change of three percentage points on a long bond, duration alone can be off by several percent of value, at which point the convexity term stops being an academic refinement and starts being the difference between a bad year and a ruinous one.

Choosing the duration you hold

The practical use of all this is to select duration deliberately. Ask first what you are being paid for the extra duration. That means looking at the slope of the curve on the yield curve tool. A flat curve pays nothing for extending maturity, and a long bond bought in that situation is a position taken for a reason that has to be stated out loud, usually as a hedge against a recession.

Then check what the position does to the portfolio as a whole, since duration is one of the very few risks that can be sized precisely in advance, before anything has happened. Price individual bonds with the bond yield calculator, think about how the bond side sits against the equity side in asset allocation, and test the mechanics with the bonds and yields quiz.