Ann Logue (rhymes with vogue) is a writer specializing in business and finance. She is the author of five books on investing, including Hedge Funds for Dummies and Day Trading for Dummies, and publishes a Substack newsletter called “The Whatever Years.”
She has worked for several fintech startups as a consultant, employee, and investor. She has also served as a Fulbright grantee and holds the Chartered Financial Analyst designation.
Doug is a Chartered Alternative Investment Analyst who spent more than 20 years as a derivatives market maker and asset manager before “reincarnating” as a financial media professional a decade ago.
Before joining Britannica, Doug spent nearly six years managing content marketing projects for a dozen clients, including The Ticker Tape, TD Ameritrade’s market news and financial education site for retail investors. He has been a CAIA charter holder since 2006, and also held a Series 3 license during his years as a derivatives specialist.
Doug previously served as Regional Director for the Chicago region of PRMIA, the Professional Risk Managers’ International Association, and he also served as editor of Intelligent Risk, PRMIA’s quarterly member newsletter. He holds a BS from the University of Illinois at Urbana-Champaign and an MBA from Illinois Institute of Technology, Stuart School of Business.
Bond duration estimates how long it takes, on a weighted basis, to get a bondâs value back through interest payments and principal. It also serves as the foundation for measuring how sensitive a bondâs price is to changes in interest rates.
What is the difference between Macaulay duration and modified duration?
Macaulay duration measures the weighted average time it takes to receive a bondâs cash flows. Modified duration adjusts that figure to estimate how much a bondâs price will change for a 1% move in interest rates, making it the more practical measure of interest rate risk.
How does duration affect bond prices when interest rates change?
When interest rates rise, bond prices tend to fall, and when rates fall, bond prices tend to rise. Duration estimates how large that price change may be. Longer-duration bonds generally experience larger price swings than shorter-duration bonds for the same change in rates.
Time changes everything. If youâre invested in bonds or other fixed-income securities, you know that youâve taken on some interest rate risk. And the longer it takes to receive the total value of a bondâs coupons and principal payments, the greater its sensitivity to interest rate risk. (If you need a primer on how bonds work, start here.)
But how much interest rate sensitivity? In bond lingo, the answer is called duration. Time changes constantly, so pinning down its effects takes a little bit of math. Duration uses the perspective of time to assess the sensitivity of bond prices to changes in interest rates.
What is duration?
Bond duration is a fundamental concept in fixed-income investing. Bond duration estimates how long it takes, on a weighted basis, to recoup a bondâs value through its interest payments (âcouponsâ) and final principal repaymentâwhich is why it also measures price sensitivity to rate changes. The longer the duration, the greater the interest sensitivity.
Bond prices change with interest rates. When rates go up, bond prices go down. When rates go down, bond prices go up. But the magnitude of the effect varies with the amount of time left until the bond matures, the size of the coupon payments, and the amount of the principal. If a bond matures next week, a change in interest rates tomorrow will have very little effect on its price. If it matures in 30 years, then even a small change in rates will have a big impact. See figure 1.
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Figure 1: TIME SENSITIVE? If a bond pays a coupon thatâs higher than current bond yields, the bond will trade at a premium to par. If current bond yields are higher than your coupon, your bond will trade at a discount. But that variability is more pronounced when thereâs more time to maturity.
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Duration is a crucial tool for managing bond portfolios. It helps investors assess bond risk, make investment decisions, and implement strategies to optimize returns. Because itâs difficult to calculate, individual investors donât use it as much as they should. However, there are several online duration calculators that do the math for you. (And if your fixed-income dollars are invested in bond funds, their management teams include duration information in their fund disclosuresâwhich is helpful if you know how to interpret it.)
The term duration applies to two related concepts:
Macaulay duration measures the sensitivity of a bondâs price to interest rate changes based on all the factors of time, number of payments, and so on (see the formula below). Itâs named for Frederick Macaulay, the economist who developed the formula back in 1938.
Modified duration takes that sensitivity metric and estimates how much a bondâs price will actually change as rates change; itâs expressed as a percentage.
You might hear the two abbreviated as âMac durationâ and âmod duration.â
Macaulay duration: Relative sensitivity to interest rate changes
The Mac duration is the sum of the present value of each cash flow (each coupon, plus the return of principal at maturity), multiplied by the time it takes to receive each cash flow.
Hereâs the formula:
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Where:
n = number of periods until each cash flow is received, usually in years
i = number of cash flows
tt = time until the ith cash flow is received
Ci = cash flow at time i
tiy = yield to maturity
M = amount of the final principal payment
The final cash flow is broken out separately because it usually includes both the principal (the return of the bondâs face value) and the final coupon payment.
Unlike a standard present value equation, which places time in the denominator, the duration equation places time in the numerator. The result is the weighted average amount of time it takes to receive the total cash generated by the bond.
For zero-coupon bonds, duration="time" to maturity
A zero-coupon bond is sold at a discount to its face value. It pays no interest coupons, but it does give investors the full face value at maturity. Because thereâs only one payment, a zero-coupon bondâs duration is the same as its time to maturity.
In contrast, a bond that pays coupons in addition to its face value will always have a lower duration than a zero-coupon bondâbecause the holder wonât have to wait as long to receive the present value of the cash.
Macaulay duration example
Suppose you pay $1,000 for a 5-year bond with a 4% coupon.
In year 1, you get $40
Year 2, another $40
Year 3, another $40
Year 4, another $40
In year 5, you receive the final $40 coupon plus your $1,000 principal
You donât wait five years to start getting your money backâpart of it comes back every year through coupon payments. Macaulay duration answers a simple question: When do you get your $1,000 back, on average, once you account for all those payments? Because those cash flows arrive over timeânot all at maturityâthe weighted average âpaybackâ date comes earlier than year five (4.63 years, to be precise).
The bond in this example would have a Mac duration of 4.63. The higher a bondâs Mac duration, the longer it will take to receive the total value of the cash flows, and the more the bond will be affected by interest rate changes.
You can find bond duration calculators online to save you the trouble of doing the math, or check with your brokerage firm or fund company.
Modified duration: The percentage change in bond price as rates change
Modified duration is the Macaulay duration divided by the yield to maturity (YTM) adjusted by the number of coupon payments each year. The result is a percentage, which estimates the change in a bondâs price for each 1% change in yield. The formula for modified duration is:
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Modified duration example
Consider the same bondâa 5-year, $1,000 bond with a 4% coupon, with a Macaulay duration of 4.63 years. For simplicity, letâs assume the bond is currently trading at its par value of $1,000 (in other words, its YTM is its coupon rate). To get modified duration, divide the Macaulay duration by (1 + YTM), keeping in mind that, in our example, the bond pays one coupon per year.
Duration matters, but itâs not the only bond risk
Duration measures rate sensitivity, but itâs only one piece of the bond risk puzzle. When you invest in bonds and other fixed-income assets, you also face reinvestment risk, credit risk, liquidity risk, and structural risks (such as call features, prepayment, and collateral quality). Learn more about bond risks.
4.63 Ă· 1.04 = 4.45
That means if interest rates rise from 4% to 5%, the bondâs price would theoretically fall by about 4.45%âfrom roughly $1,000 to $955. If rates fall from 4% to 3%, the price would rise by a similar amount, to about $1,045. This is why modified duration matters: it turns cash-flow timing into a practical estimate of price risk.
Keep in mind that modified duration is only an estimate. It assumes a linear relationship between the yield and the price, but in reality, that relationship is curved. (This concept is known as convexity. Look back at figure 1 to see how much more convexity, or curvature, is apparent in the 30-year maturity compared to the 2-year.)
Still, the duration estimate is reliable enough to help you choose among different bonds.
Using duration to make investment decisions
Macaulay duration allows investors to compare the risk of interest rate changes. Modified duration allows them to estimate the effects. Both equations come into play for portfolio management.
Remember that bonds with a long duration are more sensitive to changes in interest rates than bonds with shorter durations. So:
If you expect rates to rise in the future, look for short-duration bonds, as theyâll generally lose less value than long-duration bonds.
If you expect rates to fall, look for long-duration bonds, as their prices will likely rise more than short-duration bonds.
Understanding this relationship can help you increase your risk-adjusted returns. As mentioned above, many mutual fund and ETF managers include information about target portfolio durations to help you decide whatâs best for your goalâno math required.
Professionals also use duration in other ways. For example, the manager of an insurance companyâs portfolio may choose to reduce risk by matching the duration of the portfolio to the expected duration of the companyâs liabilities. The same goes for pension funds, and even banks, who need to balance out their short- and long-term assets and liabilities.
The bottom line
Duration is a tool for evaluating interest rate risk. The shorter the duration, the less interest rate risk is associated with a given bond. Keep in mind that the risk works in a bondholderâs favor when rates are going down, so consider long-duration bonds if you expect rates to fall.
