One Measures Time, the Other Measures Price Sensitivity

Macaulay duration is the weighted average time, in years, until a bond’s cash flows are received, with each payment weighted by its present value. Modified duration adjusts Macaulay duration for the bond’s yield and estimates how much the bond’s price will change, in percent, for a 1 percentage point change in interest rates.

The two are closely linked. Modified duration equals Macaulay duration divided by one plus the bond’s yield per period, so for most bonds the two numbers are close, with modified duration slightly lower.

When people say a bond has a duration of 7 and will lose about 7 percent if rates rise by 1 percentage point, they are using modified duration. For the full route from the exam to a career applying these ideas, our complete guide to how to become an investment adviser representative covers every step.

Get ahead of the January rush with the FRC Series 65 playlists on YouTube

What Duration Tells You

Duration is the most widely used measure of a bond’s interest rate risk. The longer the duration, the more a bond’s price moves when interest rates change.

Duration depends on three things: maturity, coupon and yield. Longer maturities increase duration, while higher coupons and higher yields reduce it, because more of the bond’s value is received sooner.

A zero-coupon bond pays nothing until maturity, so its Macaulay duration equals its maturity. The Series 65 Exam Preparation course at FRC covers these relationships, which the exam tests far more often than the calculation itself.

How Macaulay Duration Is Calculated

To calculate Macaulay duration, find the present value of each cash flow, multiply it by the time until it is received, add those figures together and divide by the bond’s price. The answer is expressed in years.

It is named after Frederick Macaulay, who introduced it in 1938 as a better measure of a bond’s effective life than its maturity. A 10-year bond paying coupons has a duration shorter than 10 years, because part of its value arrives before the final payment.

Macaulay duration is most useful for matching the timing of investments to future needs. An investor who must pay a known sum in seven years can reduce interest rate risk by holding bonds with a Macaulay duration of about seven years.

How Modified Duration Is Calculated

Modified duration equals Macaulay duration divided by one plus the yield per period. For a bond paying annual coupons with a 5 percent yield, modified duration is Macaulay duration divided by 1.05.

It is used to estimate price changes. The approximate percentage price change equals minus modified duration times the change in yield, in percentage points.

A bond with a modified duration of 6 would be expected to fall about 6 percent if yields rose by 1 percentage point, and to rise about 6 percent if yields fell by 1 percentage point. That simple rule makes modified duration the everyday tool for measuring interest rate risk.

Macaulay and Modified Duration Compared

Worked Example: Calculating Both Measures

A three-year bond pays a 5 percent annual coupon, or $50 a year, and its yield is 5 percent, so it trades at $1,000. The present values of its cash flows are about $47.62 in year one, $45.35 in year two and $907.03 in year three.

Step one: multiply each present value by its year, giving $47.62, $90.70 and $2,721.09, which add up to $2,859.41. Step two: divide by the price of $1,000, giving a Macaulay duration of about 2.86 years.

Step three: divide by 1.05 to get a modified duration of about 2.72. If yields rose by 1 percentage point, the bond’s price would be expected to fall by about 2.72 percent, or about $27.

Worked Example: Estimating a Price Change

A 10-year bond pays a 5 percent annual coupon and yields 5 percent, so it trades at $1,000. Its Macaulay duration is about 8.11 years and its modified duration is about 7.72.

Step one: if yields rise by 1 percentage point to 6 percent, modified duration predicts a price fall of about 7.72 percent, to about $922.80. Step two: the actual new price is about $926.40, a fall of about 7.36 percent.

Step three: if yields fall by 1 percentage point to 4 percent, modified duration predicts a rise of about 7.72 percent, to $1,077.20, while the actual price rises to about $1,081.10, a gain of about 8.11 percent. In both cases the bond does better than duration predicts, which is the effect of convexity.

Study Bond Risk With FRC

Duration questions on the exam usually test the relationships: which bond has the longer duration, which will fall more when rates rise, and how coupons and maturities change the answer. Our Series 65 course builds those relationships with clear examples and practice questions.

Consider studying with FRC for every exam on your route. Explore our USA exam courses, from the SIE through the Series 7, Series 63 and Series 65.

To take advantage of the membership, explore FRC Professional Membership, which gives you a verified Digital Profile that shows employers your real-time course progress, together with the Video Resume. Current offers are on our membership deals page.

Convexity: Why Duration Is Only an Estimate

Duration assumes the relationship between price and yield is a straight line, but in reality it is curved. That curvature is called convexity.

For an ordinary bond, convexity means prices rise more when yields fall than they drop when yields rise by the same amount. Duration therefore overstates losses and understates gains for large rate changes, as the worked example shows.

For small rate changes, duration alone is accurate enough. For large changes, professionals add a convexity adjustment to improve the estimate.

What Raises or Lowers Duration

Longer maturity means higher duration, because cash flows are spread further into the future. A 30-year bond is far more sensitive to interest rates than a 2-year bond.

A higher coupon means lower duration, because more of the bond’s value is paid out early. For example, a 10-year bond yielding 5 percent has a Macaulay duration of about 8.11 years with a 5 percent coupon, but only about 7.27 years with a 10 percent coupon.

Higher yields also mean lower duration, because distant cash flows are discounted more heavily. That is why the same bond’s duration shortens slightly when market rates rise.

Duration for Callable Bonds

Macaulay and modified duration assume a bond’s cash flows are fixed. For a callable bond, that is not true, because the issuer may redeem the bond early if rates fall.

Professionals use effective duration for bonds with embedded options. It is calculated by modelling how the bond’s price would change if rates moved up and down, taking into account the chance of a call.

When rates fall, a callable bond’s price rises less than a similar non-callable bond, because investors expect it to be called, an effect known as negative convexity. Our comparison of callable vs non-callable bonds explains why call features limit price gains, and our article on yield to call vs yield to maturity shows how they change a bond’s yield.

Portfolio Duration and Immunization

A bond portfolio’s duration is approximately the weighted average of the durations of the bonds in it. A portfolio holding equal amounts of bonds with durations of 2 and 8 has a duration of about 5.

Matching the duration of a bond portfolio to the timing of a future liability is called immunization. When rates rise, the portfolio’s value falls but its coupons can be reinvested at higher rates, and when rates fall, the reverse happens, so the two effects roughly offset at the liability date.

Worked example: a client needs $200,000 in six years for a child’s education. Step one: the adviser builds a bond portfolio with a Macaulay duration of about six years. Step two: whether rates rise or fall modestly, the portfolio’s value at the six-year mark should stay close to the target, as long as the portfolio is rebalanced as its duration drifts.

Dollar Duration and the Value of a Basis Point

Modified duration gives a percentage change, but clients and portfolio managers often want to know the change in dollars. Dollar duration multiplies modified duration by the bond’s or portfolio’s market value.

Worked example: a client holds a $500,000 bond portfolio with a modified duration of 6. Step one: a 1 percentage point rise in yields would reduce its value by about 6 percent, or $30,000. Step two: dividing by 100 gives the price value of a basis point, about $300 for every 0.01 percentage point move in yields.

Professional bond managers use the price value of a basis point to compare the risk of different positions and to size hedges. For individual clients, the dollar figure simply makes the risk easier to picture.

Duration of Bond Funds and Individual Bonds

An individual bond’s duration falls steadily as it approaches maturity, and at maturity the investor receives face value regardless of what rates have done. A bond fund, by contrast, usually keeps its duration roughly constant by buying new bonds as old ones mature.

That means a bond fund never matures, and its price sensitivity stays similar over time. An investor in a fund with a duration of 6 faces about the same interest rate risk next year as this year, while an investor holding a single 6-year bond faces less risk each year.

Fund factsheets report the fund’s average duration, usually effective duration, which makes it easy to compare funds. Matching that figure to the client’s time horizon is one of the simplest ways to keep a bond allocation suitable.

Using Duration in Client Advice

Duration translates interest rate risk into a number clients can understand. Telling a client that a bond fund with a duration of 6 could lose about 6 percent if rates rose by 1 percentage point is clearer than talking about maturities alone.

Advisers match duration to the client’s time horizon and tolerance for price swings. A client who may need money soon should usually hold short-duration bonds, while one with a long horizon and known future liabilities may accept longer duration.

Duration only measures interest rate risk. It says nothing about credit risk, which is covered in our article on systematic vs unsystematic risk, so a full bond review looks at both.

The Memory Aid

Remember “Macaulay measures the clock, modified measures the shock”: Macaulay duration is in years, modified duration estimates the price change.

For the link, remember “modified is Macaulay over one plus yield”.

For what changes duration, remember “longer up, coupon down, yield down”: longer maturity raises duration, while higher coupons and higher yields lower it.

Exam Traps to Avoid

The first trap is using Macaulay duration to estimate a price change. Modified duration is the price-sensitivity measure, although the two are close.

The second trap is assuming a higher coupon means higher duration. A higher coupon lowers duration, because more value is received earlier.

The third trap is forgetting that a zero-coupon bond’s Macaulay duration equals its maturity. The fourth trap is relying on duration alone for large rate changes; convexity makes actual price changes differ from the estimate.

A Client Scenario

A client nearing retirement holds a bond fund with a duration of 9 and is surprised that it fell 8 percent in a year when interest rates rose by about 1 percentage point. She will need part of the money for living costs over the next three years.

The adviser explains that the fund’s duration meant it was highly sensitive to interest rates, and that the fall was close to what duration predicted. Because she needs some of the money soon, holding all of it in a long-duration fund exposes her to more price risk than her time horizon allows.

The adviser recommends moving the money she needs over the next three years into short-term bonds and a Treasury ladder, keeping a smaller long-duration holding for later years. The recommendation, including the duration of each part of the portfolio, is documented in her file.

The Dictionary as a Video Library

The FRC dictionary is a study tool, not just a glossary. Every entry pairs a written explanation with deep-dive video, on-page quizzes and interactive flash cards, so you can read a concept, watch it explained and test yourself in one place.

Rules are covered in the same depth: open a rule and you will find the full explanation alongside video walk-throughs of how it applies in practice. Open the duration entry to see the calculation worked through on video and test yourself with the quizzes.

Practice Questions to Test Yourself

Question one: A bond has a Macaulay duration of 6.3 years and a yield of 5 percent with annual coupons. What is its modified duration? About 6.0, because 6.3 divided by 1.05 is 6.0.

Question two: A bond has a modified duration of 4.5. Roughly how much will its price change if yields fall by 1 percentage point? It will rise by about 4.5 percent.

Question three: What is the Macaulay duration of a 12-year zero-coupon bond? 12 years.

Question four: Two 10-year bonds have the same yield, but one has a 3 percent coupon and the other an 8 percent coupon. Which has the longer duration? The 3 percent coupon bond.

Question five: Why does a bond’s price rise more when yields fall than it falls when yields rise by the same amount? Because of convexity.

Turn Fixed Income Knowledge Into Career Evidence

Explaining interest rate risk clearly helps clients stay calm when bond prices fall, and firms value advisers who can do it with confidence. FRC’s Digital Profile shows employers the courses you are actively studying and the progress you have earned through assessed learning, verified by FRC, and a QR code on your resume takes them straight to it.

The FRC Video Resume lets a hiring manager hear you before the interview. Walking through how duration matches a portfolio to a client’s future needs shows the technical skill fixed income and advisory teams look for.

Related Questions

Bonds without call features, such as most Treasuries, have fixed cash flows, so standard duration measures describe their price behaviour well, as our comparison of agency bonds vs Treasury bonds explains.

Key Points to Retain

Macaulay duration is the present-value-weighted average time until a bond’s cash flows are received, measured in years, while modified duration equals Macaulay duration divided by one plus the yield per period and estimates the percentage price change for a 1 percentage point change in yield. A zero-coupon bond’s Macaulay duration equals its maturity.

Duration rises with longer maturity and falls with higher coupons and yields, and convexity means actual price changes differ from duration estimates for large rate moves. Callable bonds are measured with effective duration, portfolio duration is a weighted average, and matching duration to a future liability is called immunization.

For the role the exam leads to, read our guide to what an investment advisor representative is.