What Is Bond Duration? Macaulay vs Modified Duration, With a Worked Example

Bond duration measures how much a bond's price moves when interest rates change, usually in years; for most bonds it grows with maturity and a lower coupon.

Duration is the number fund fact sheets print next to "interest rate risk", and the reason two bonds can lose very different amounts in the same week. This page covers the two main versions, Macaulay and modified duration, works one example through by hand and then checks it against an exact repricing, adds convexity, and shows how to use duration with a move in the Treasury curve. It is descriptive throughout.

The idea in one paragraph

A bond is a series of payments: coupons along the way and the face value at the end (see what a bond is). When yields change, every one of those payments is discounted at the new rate, and payments far in the future are hit hardest. Duration condenses that into one number. FINRA's investor guide puts the working rule this way: for every 1 percentage point change in interest rates, a bond's price moves in the opposite direction by roughly its duration in percent, so a bond with a duration of 10 loses about 10 percent if rates rise 1 point.

Macaulay duration and modified duration

The two versions measure related things.

• Macaulay duration, named after the economist Frederick Macaulay, who set it out in a 1938 study for the National Bureau of Economic Research, is the weighted average time until the bond's cash is received. Each payment's time is weighted by that payment's present value as a share of the bond's price. It is measured in years.
• Modified duration turns that into price sensitivity. It equals Macaulay duration divided by (1 + yield per period), and it gives the approximate percentage change in price for a 1 point change in yield. When people use FINRA's rule of thumb, this is the number they are using.

For a bond paying coupons twice a year, the period yield is half the annual yield, so the divisor is 1 plus half the yield.

A worked example

Take a 10-year bond with a 5 percent coupon, paid as $2.50 every six months per $100 of face value, priced at a yield of 5 percent. At that yield it sells at exactly $100.

Step 1, the payments. There are 20 half-year payments: nineteen coupons of $2.50 and a final payment of $102.50 at year 10.

Step 2, present values. Discounting each payment at 2.5 percent per half-year, the final payment is worth $62.55 today and the nineteen coupons $37.45 together, which adds up to the $100 price. Most of the bond's value sits in the last payment, but more than a third arrives earlier.

Step 3, Macaulay duration. Multiply each payment's time in years by its present value, add the results, and divide by the price. The answer is 7.99 years, about two years shorter than the 10-year maturity, because the coupons pull the average forward.

Step 4, modified duration. Divide by 1 plus the half-year yield: 7.99 / 1.025 = 7.79. The bond's price would move by about 7.79 percent for a 1 point move in its yield.

Step 5, check it by repricing. Discount the same twenty payments at the new yields and compare.

For a 0.10 point move, duration alone matches the exact repricing to two decimals. For a full point it is off by about 0.4 of a point, and in a consistent direction: it overstates the loss when yields rise and understates the gain when they fall.

Duration also converts into money. A modified duration of 7.79 on a $100 price means a 0.01 point move in yield, one basis point, changes the price by about 7.8 cents per $100, or about $779 per $1 million of face value. Traders call this the DV01, the dollar value of a basis point.

Convexity: why the estimate bends

Duration draws a straight line through a curved relationship. The real price and yield relationship curves upward (it is convex), so prices rise more than the straight line implies when yields fall and fall less than it implies when yields rise. That curvature is called convexity.

For the example bond, convexity is about 73.6. Adding half of convexity times the squared yield change to the duration estimate brings the 1 point estimates to minus 7.43 percent and plus 8.16 percent, within two hundredths of a point of the exact figures in the table above. For small moves the convexity term barely matters; for large moves it does, and more so for long bonds. Take the 30-year 4 percent bond in the table below. For a 1 point rise in yield, its duration gives an estimate of about minus 17.4 percent, while exact repricing gives about minus 15.5 percent. That is a fall to about 845 dollars per 1,000 of face value, in line with the 846 dollars in what a bond is; the small gap comes from how often the coupon is assumed to be paid.

What makes duration longer or shorter

Three features of a bond set its duration. All of the figures below come from the same calculation, with payments twice a year.

• Maturity. The longer the bond, the longer its duration: compare the 2-year and the 30-year. The 30-year figure is the "about 18 years" quoted in what a bond is.
• Coupon. A lower coupon pushes more of the value into the final payment and lengthens duration. A zero-coupon bond has a single payment, so its Macaulay duration equals its maturity exactly.
• Yield. Higher yields discount distant payments more heavily, which shortens duration a little.

Bonds that can be repaid early, such as callable corporate bonds or mortgage-backed securities, need a different measure called effective duration, because their payments change when rates change; their price can even curve the other way, which is called negative convexity. FINRA notes that a bond fund's duration can be found on its fact sheet, which is where most readers meet the number.

Using duration with the Treasury curve

Duration turns a move in yields into an approximate move in price, and the yield move can be read off the curve. Kresmion's free yield curve tool shows the current US Treasury curve at eleven maturities, from 1 month to 30 years, against the curve a year earlier. With a free account it adds a comparison against any day in its archive, which starts in July 1969, with the change in basis points at each maturity published on both days.

Reading the tool this way, a 25 basis point rise at the 10-year point, applied to a 10-year bond with a modified duration of 7.79, implies a price fall of about 7.79 x 0.25 = 1.95 percent, before convexity. A curve that steepens or flattens moves different maturities by different amounts, so each bond takes the change at its own remaining maturity, not a single headline number, and a bond that is not a Treasury also moves with its credit spread. Treasury bills, notes and bonds shows real auction prices for a bill and for 2-year, 10-year and 30-year securities, and what the yield curve shows covers the curve's shapes.

Honest limitations

Duration is a first-order approximation. It assumes a small, one-time change in the bond's own yield, so it drifts from the exact answer as the move gets bigger, which is what convexity partly corrects. Using one duration for a whole portfolio assumes every yield moves by the same amount, and real curves twist. Duration measures sensitivity to yields only: it says nothing about a change in the issuer's credit, which moves its spread over government bonds (see what a credit spread is). Duration also changes as time passes and as yields move, so a figure on a fact sheet is a reading on a date. The examples on this page are hypothetical bonds priced from a formula, not quotes for real securities.

Key takeaways

Frequently asked questions

Is duration the same as maturity?

No. Maturity is the date the face value is repaid. Duration is a weighted average of when all the payments arrive, so for any bond that pays coupons it is shorter than the maturity. Only a zero-coupon bond, which pays once, has a Macaulay duration equal to its maturity.

What does a duration of 8 mean?

That the bond's price would move by about 8 percent in the opposite direction for a 1 percentage point change in its yield, and about 0.08 percent for a 1 basis point change. Strictly, that reading uses modified duration; Macaulay duration is the time measure it is derived from.

What is the difference between Macaulay and modified duration?

Macaulay duration is a time, the weighted average years until the bond's cash arrives. Modified duration divides it by 1 plus the yield per period and is used as a price sensitivity. For ordinary yields the two are close: 7.99 and 7.79 in this page's example.

Does a higher duration mean a bond is riskier?

It means its price is more sensitive to changes in interest rates. That is one risk among several: FINRA lists credit risk, inflation risk and call risk alongside it, and a bond with a short duration can still carry the others.

Does duration predict bond returns?

No. Duration describes how a bond's price responds if yields move. It says nothing about which way yields will move or by how much.

This page is information, not investment advice.

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Source: FINRA, "Brush Up on Bonds: Interest Rate Changes and Duration" (19 September 2024), https://www.finra.org/investors/insights/bonds-interest-rate-changes-duration ; Frederick R. Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856 (NBER, 1938), https://www.nber.org/books-and-chapters/some-theoretical-problems-suggested-movements-interest-rates-bond-yields-and-stock-prices-united ; durations, convexity, prices and DV01 computed by Kresmion for hypothetical bonds with semiannual coupons ; Kresmion yield curve tool (US Treasury constant-maturity yields, Federal Reserve H.15 via FRED).

Kresmion Research.