Yield-Based Bond Duration Measures and Properties

Fixed Income. Worth 11 to 14 percent of the exam. One session: the lesson, the rules, the method, then the questions.

Fixed IncomeYield-Based Bond Duration Measures and Properties
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The lesson

Runtime 14 minutes 52 seconds, measured from the published video.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to calculate Macaulay and modified duration and convert between them, calculate the approximate percentage and dollar price change from a given change in yield, describe when effective duration replaces modified duration, and describe how maturity, coupon and yield level each drive duration.

Macaulay duration is the present-value-weighted average time until a bond's cash flows arrive, measured in years: picture every cash flow lined up on a timeline, each one weighted by its present value, and Macaulay duration is the point where that timeline balances, like the fulcrum of a seesaw. It tells you when money arrives. It does not, by itself, tell you how much a bond's price will move.

Modified duration answers that second question instead, and it is not a time measure at all despite being expressed in years. It converts Macaulay duration into a direct price-sensitivity figure: modified duration = Macaulay duration / (1 + y/m), where y is the annual yield and m is the number of compounding periods per year. Because the divisor always exceeds one for a positive yield, modified duration is always smaller than Macaulay duration for the same bond. The approximate percentage price change for a given yield move then follows directly: %change in price is approximately -(modified duration) x (change in yield), with the yield change entered in decimal form, 75 basis points as 0.0075, never 0.75 or 75. The negative sign is not optional: a yield increase produces a price decrease and a yield decrease produces a price increase, every time. Money duration translates that same sensitivity into dollars instead of a percentage: money duration = modified duration x the bond's full price, giving the approximate dollar move for a 100 basis point yield change. The price value of a basis point, PVBP, is simply money duration rescaled by a factor of 10,000, the dollar move for a single basis point.

Effective duration exists for exactly one reason: an option-free bond's future cash flows never change no matter what happens to yields, but a callable, putable or mortgage-backed security's cash flows do change, because the embedded option can be exercised. Effective duration captures that reality using full repricing instead of a formula: shift yields down to get a higher price, shift yields up to get a lower price, and effective duration = (P-down - P-up) / (2 x P0 x change in yield). For a callable bond, effective duration always comes out lower than what modified duration alone would predict, because the call option caps how far the price can rise when rates fall, and less movement in that direction is exactly what a lower duration number means.

Three factors drive duration in a predictable, additive way. Longer maturity raises duration, because more of the bond's value sits further out in time and is discounted more heavily. Lower coupon raises duration, because a larger share of total value sits in the single final redemption payment rather than being returned earlier through coupons. Lower yield raises duration too, because a smaller discount rate lets distant cash flows retain more of their present-value weight. A bond that is simultaneously longer, lower-coupon and lower-yielding than another dominates it in interest rate risk on every dimension at once.

Duration as a seesaw of maturity and coupon short maturity high coupon long maturity low coupon heavier side = more price sensitivity to yield
Duration tips like a seesaw. Longer maturity and lower coupon both add weight on the same side: more price movement for the same change in yield.

Worked in full

A bond has a Macaulay duration of 6.20 years and a yield to maturity of 5.5 percent, compounded semi-annually. What is its modified duration, and what is the approximate percentage price change if yields rise by 75 basis points? Modified duration = Macaulay duration / (1 + y/m) = 6.20 / (1 + 0.055/2) = 6.20 / 1.0275 = 6.03 years. Percentage price change = -(modified duration) x (change in yield) = -6.03 x 0.0075 = -4.53 percent. The bond's price is expected to fall by about 4.53 percent.

The same problem, one step removed

Same bond: Macaulay duration 6.20 years, YTM 5.5 percent semi-annual, yields rise 75 basis points. Compute modified duration from Macaulay first, then apply the price-change formula yourself.

The trap

Modified duration and Macaulay duration are close enough in size to swap under time pressure, but only modified duration belongs in the percentage price-change formula; reporting the Macaulay figure instead produces a plausible-looking answer built on the wrong number.

What this unit turns on

Read these before the questions, not after them. Everything here traces to this module's own lesson and to the 2026 outline.

Modified duration converts the time-based Macaulay duration into a direct price-sensitivity measure

Modified duration equals Macaulay duration divided by one plus the periodic yield (1 + y/m, where m is the number of compounding periods per year), and unlike Macaulay duration it is not a time measure at all, it is the approximate percentage change in a bond's price for a one-unit (100 basis point) change in yield. Because the divisor (1 + y/m) always exceeds one for a positive yield, modified duration is always somewhat smaller than Macaulay duration for the same bond.

The percentage price change formula uses modified duration directly, with a negative sign reflecting the inverse price-yield relationship

The approximate percentage price change for a given yield change is %delta-P is approximately equal to -(Modified Duration) x delta-Yield, expressed with the yield change in decimal form (75 basis points enters as 0.0075, not 0.75 and not 75). The negative sign is not optional: a yield increase produces a price decrease and a yield decrease produces a price increase, and the single most common calculation error on this measure is dropping that sign or mishandling the basis-point-to-decimal conversion.

Money duration and PVBP translate the same percentage sensitivity into dollar terms, at two different scales

Money duration equals modified duration multiplied by the full price of the bond (or portfolio), giving the approximate dollar price change for a 100 basis point (1%) yield move; the price value of a basis point (PVBP), also called dollar duration per basis point, is simply money duration scaled down by a factor of 10,000 (or equivalently, repriced at plus and minus one basis point and averaged), giving the dollar price change for a single one-basis-point move, the unit typically used to describe portfolio-level risk in trading and risk-management contexts.

Interest rate risk rises with longer maturity, lower coupon, and lower yield, and these effects compound when they point the same direction

Holding all else equal, a longer maturity increases a bond's duration and therefore its interest rate risk, because more of its value is discounted from further in the future; a lower coupon rate increases duration because a larger share of total value sits in the single final redemption payment rather than in earlier coupons; and a lower yield level increases duration because a lower discount rate gives relatively more present-value weight to the most distant cash flows. A bond that is simultaneously longer-maturity, lower-coupon, and priced at a lower yield than another bond dominates it in interest rate risk on all three dimensions at once.

The trick

Modified duration, not Macaulay duration, goes into the price-change formula

The two numbers are close in size and easy to swap under time pressure; only modified duration is a direct measure of price sensitivity, Macaulay duration is a time-weighted average of when cash flows arrive.

Basis-point conversion is the number-one arithmetic trap on this module

75 basis points is 0.0075 in the formula, never 0.75 and never 75; misplacing the decimal produces an answer off by a factor of 100 or 10,000 that often still matches a wrong answer choice exactly.

PVBP is money duration divided by 10,000, nothing more exotic

Money duration gives the dollar move for a full 100 basis points; PVBP gives the dollar move for one basis point, the same number simply rescaled, so a portfolio's total dollar loss for any number of basis points is PVBP times that number.

The method

The order to work a question of this type in, every time, before you touch the numbers.

  1. Identify whether the question asks for a percentage price change (use modified duration) or a dollar price change (use money duration or PVBP).
  2. Convert the yield change to decimal form correctly before applying any formula: basis points divided by 10,000.
  3. Apply %delta-P is approximately -(Modified Duration) x delta-Yield for a percentage estimate, keeping the negative sign for a yield increase.
  4. For a dollar estimate, multiply money duration by the yield change in percent, or multiply PVBP by the number of basis points directly.
  5. For a comparative interest-rate-risk question, check maturity, coupon, and yield level together; a bond that is longer, lower-coupon, and lower-yield than another dominates it in risk on every dimension.
  6. [BA II Plus: duration itself is not a single keystroke output; compute bond price at the given yield, then at yield plus and minus a small shift, to approximate duration numerically if the exam question requires derivation rather than a stated duration figure]

Two worked examples, then you are on your own

The first is worked in full. The second gives you the setup and stops. After that the questions give you nothing, which is the point: the help fades on purpose, so the last thing you practise is the thing the exam actually asks of you.

Worked in full

A bond has a Macaulay duration of 5.40 years and a yield to maturity of 6.00% (semi-annual compounding). The bond's modified duration is closest to:

Answer A. Modified Duration = Macaulay Duration / (1 + y/m) = 5.40 / (1 + 0.06/2) = 5.40 / 1.03 = 5.24 years. The exam trap here is dividing by (1 + 0.06) = 1.06, which gives answer A and uses annual compounding instead of semi-annual. Always divide YTM by the number of compounding periods per year (m=2 for semi-annual).

Your turn, setup given

A bond has a modified duration of 7.5 years. If the yield to maturity increases by 50 basis points, the approximate percentage price change is closest to:

Identify whether the question asks for a percentage price change (use modified duration) or a dollar price change (use money duration or PVBP).

The practice run

Pick an answer, say how sure you are, then reveal. Being sure and wrong is the most useful thing that can happen in a session, so answer honestly: it sends the unit back to learning and puts it at the front of your revision queue.

Question 1Exam level

Which of the following bond types most likely REQUIRES the use of effective duration rather than modified duration to estimate price sensitivity?

How sure are you?

Correct: B. Effective duration must be used whenever a bond has embedded options (callable, putable, or convertible) because the cash flows change when interest rates change. The issuer may call the bond if rates fall. Modified duration assumes cash flows are fixed regardless of rate changes, making it inappropriate for option-embedded bonds. Zero-coupon bonds (A) and fixed-rate Treasuries (D) have fixed, deterministic cash flows so modified duration is appropriate. Floating-rate bonds (B) have near-zero duration to the next reset date, but modified duration can still be applied.
A. You might be tempted to choose a floating-rate bond because it seems to have complex cash flows, but floating-rate bonds adjust their coupon payments regularly, making their duration very short and predictable, thus modified duration is appropriate here, unlike for callable bonds where cash flows are uncertain due to the option.
C. You might be tempted to choose a 30-year fixed-rate Treasury bond because it has a long maturity, but remember, fixed-rate Treasuries have predictable cash flows, making modified duration suitable, unlike the callable bond where cash flows are uncertain due to the issuer's option to call the bond.

Unit: yield-based-bond-duration-measures-and-properties

Question 2Exam level

A bond has the following characteristics: 3-year maturity, 5% annual coupon, YTM = 5%. Its Macaulay duration is closest to:

How sure are you?

Correct: A. For a 3-year, 5% annual coupon bond priced at par (YTM = coupon rate = 5%, price = 100): CF1 = 5, PV1 = 5/1.05 = 4.762, weight1 = 4.762/100 = 0.04762, contribution = 1 × 0.04762 = 0.04762. CF2 = 5, PV2 = 5/1.05^2 = 4.535, weight2 = 0.04535, contribution = 2 × 0.04535 = 0.09070. CF3 = 105, PV3 = 105/1.05^3 = 90.703, weight3 = 0.90703, contribution = 3 × 0.90703 = 2.72109. Macaulay Duration = 0.04762 + 0.09070 + 2.72109 = 2.8594 ≈ 2.86 years. A zero-coupon bond would have duration = 3.00 (answer C. The most common wrong choice).
B. Selecting 3.00 years because the bond matures in 3 years; duration equals maturity ONLY for zero-coupon bonds, not coupon bonds
C. 2.59 years understates the true weighted-average time to receive this bond's cash flows; it does not correctly weight each year's present value against the total price. The correct Macaulay duration, weighting each year's cash flow by its own present-value share of the $100 price, comes to 2.86 years, not 2.59.

Unit: yield-based-bond-duration-measures-and-properties

Question 3Exam level

A portfolio manager holds a $100 million bond portfolio with a modified duration of 8.0. If yields rise by 25 basis points across all maturities, the approximate dollar change in portfolio value is closest to:

How sure are you?

Correct: A. Dollar change ≈ -Modified Duration × Portfolio Value × ΔYield = -8.0 × $100,000,000 × 0.0025 = -$2,000,000. The negative sign reflects the inverse price-yield relationship. Answer C uses 0.025 (250 bps) instead of 0.0025 (25 bps). Answer D divides by 10. Answer B has wrong sign.
B. Choosing B (+$2,000,000) might seem logical if you mistakenly think that rising yields would increase the portfolio value, but this ignores the inverse relationship between bond prices and yields, where rising yields actually decrease bond prices.
C. You might be tempted by choice C if you mistakenly used 250 basis points instead of 25 basis points for the yield change, leading to an incorrect calculation that violates the precise definition of basis points and the formula for dollar change in portfolio value.

Unit: yield-based-bond-duration-measures-and-properties

Question 4Exam level

The effective duration of a callable bond will most likely be LOWER than its modified duration because:

How sure are you?

Correct: A. When interest rates fall, the issuer becomes more likely to call the bond (refinance at lower rates). This caps the bond's price appreciation. A phenomenon called 'negative convexity' or 'price compression.' Effective duration captures this real-world behavior by using actual repriced values at shifted yields. Because the price rise is capped (call option kicks in), the computed price sensitivity is lower than for a comparable non-callable bond. Modified duration ignores this by assuming cash flows are fixed.
B. You might be thinking that call options imply shorter time horizons, but callable bonds can have any maturity length; the call feature does not dictate the bond's maturity, unlike how it affects price sensitivity to interest rate changes.
C. A callable bond's credit rating has nothing to do with why its effective duration runs below its modified duration, and there is no general rule that callable issuers carry higher credit ratings in the first place. The actual cause is the call option itself: when rates fall enough for the issuer to profit from refinancing, the option caps how much the bond's price can rise, and that capped upside is exactly what effective duration measures and modified duration, which assumes fixed cash flows, does not.

Unit: yield-based-bond-duration-measures-and-properties

Question 5Exam level

Which of the following relationships is most likely ALWAYS true for a standard fixed-rate coupon bond (no embedded options)?

How sure are you?

Correct: A. Modified Duration = Macaulay Duration / (1 + y/m). Since (1 + y/m) > 1 for any positive yield, dividing always produces a smaller number. Therefore Macaulay Duration > Modified Duration for all positive yields. For a bond with no embedded options, Effective Duration ≈ Modified Duration (answer D would be false). This is a pure definitional question testing whether candidates understand the mathematical relationship.
B. Choosing B might seem correct if you confuse the definitions, but remember that modified duration is always a fraction of Macaulay duration due to the division by (1 + y/m), so they can never be equal for any bond with a positive yield.
C. You might be tempted to choose C because you think effective duration is always smaller due to its calculation method, but for a standard fixed-rate bond without embedded options, effective duration is approximately equal to modified duration, not less, as effective duration accounts for the same price-yield curve as modified duration in this case.

Unit: yield-based-bond-duration-measures-and-properties

Question 6Exam level

A zero-coupon bond with 8 years to maturity has a yield to maturity of 4.0% (annual compounding). Its modified duration is closest to:

How sure are you?

Correct: A. For a zero-coupon bond, Macaulay Duration = maturity = 8.00 years. Modified Duration = Macaulay Duration / (1 + y/m) = 8.00 / (1 + 0.04/1) = 8.00 / 1.04 = 7.692 ≈ 7.69 years. Answer A is the Macaulay duration (the most common wrong choice. Candidates stop after noting it equals maturity). Answer C uses 8/(1.08) incorrectly. The key rule: zero-coupon Macaulay = maturity, then still divide by (1+y) to get modified.
B. Choosing 7.40 years might tempt you if you incorrectly apply a formula meant for coupon bonds, where the duration is less than the maturity; however, for a zero-coupon bond, the Macaulay duration equals the maturity, and the modified duration is derived by dividing this by (1 + yield), making 7.40 years inconsistent with the bond's characteristics.
C. Choosing 6.00 years might tempt you if you mistakenly subtracted the yield from the maturity, but this ignores the proper calculation of dividing the Macaulay duration by one plus the yield to find the modified duration.

Unit: yield-based-bond-duration-measures-and-properties

Question 7Harder

Among the following formulas for a bond's price sensitivity to a change in yield, the one that most likely defines effective duration is:

How sure are you?

Correct: A. Effective Duration = (P- − P+) / (2 × P0 × Δy), where P- is the bond price when yield decreases by Δy, P+ is the bond price when yield increases by Δy, P0 is the initial bond price, and Δy is the yield shift. This formula uses full repricing (including changes in embedded option value) rather than a mathematical derivative of the price function. Answer B is the formula for modified duration. Answer C is the theoretical definition of modified duration via calculus. Answer D is Macaulay duration.
B. You might be tempted by choice B because it resembles a derivative-based approach, which is intuitive for measuring sensitivity. However, this choice represents modified duration, which uses a mathematical derivative and does not account for the full repricing effect, unlike effective duration which recalculates the bond price with actual changes in yield.
C. You might be tempted by choice C because it resembles the formula for Macaulay duration, but it fails to account for the change in bond price due to yield shifts, which is crucial for effective duration as calculated in choice A.

Unit: yield-based-bond-duration-measures-and-properties

Question 8Exam level

A bond portfolio manager wants to immunize a single liability due in 6 years. Which portfolio characteristic is MOST critical for immunization?

How sure are you?

Correct: A. Classical immunization theory (Redington, 1952, as cited in CFA curriculum) requires that the Macaulay duration of the asset portfolio equals the investment horizon (liability due date). When Macaulay duration = horizon, the price risk and reinvestment risk perfectly offset each other over the horizon. Modified duration (answer A) is not the correct measure for immunization. It is a price sensitivity measure, not a time measure. Answer D (market value matching) addresses the funding ratio but not the interest rate risk. Answer C is insufficient alone.
B. Choosing B might seem logical if you think matching yields ensures the portfolio will cover the liability, but this overlooks the critical role of duration in immunization against interest rate risk, which is not addressed by simply matching yields.
C. Matching the portfolio's market value to the present value of the liability might seem like a straightforward way to ensure funds are available, but this approach overlooks the critical need to align the portfolio's duration with the liability's time horizon to manage interest rate risk effectively.

Unit: yield-based-bond-duration-measures-and-properties

Question 9Harder

Two bonds have the same modified duration of 5.0. Bond X is a 5% coupon bond and Bond Y is a 2% coupon bond. Which bond most likely has a HIGHER Macaulay duration?

How sure are you?

Correct: B. If two bonds have the same modified duration and are priced with the same YTM, then Macaulay Duration = Modified Duration × (1 + y/m) is identical for both. Both bonds have Macaulay Duration = 5.0 × (1 + y/m). While a lower-coupon bond generally has a HIGHER Macaulay duration (for the same maturity and yield), the question states they have the SAME modified duration, which means their Macaulay durations are also equal. This is a critical conceptual trap: the question tests whether candidates conflate 'lower coupon = longer duration' (true when comparing same maturity) with this scenario where modified duration is already equalized.
A. You might be thinking that a lower coupon bond always has a higher duration, but this overlooks that the modified durations are the same for both bonds, meaning their Macaulay durations must also be equal despite the coupon difference.
C. Maturity does not need to be known separately here because Macaulay duration is computed directly from modified duration through Macaulay Duration equals Modified Duration times (1 plus yield over compounding frequency). Since both bonds already share the same modified duration and, implicitly, the same yield and compounding frequency, their Macaulay durations are pinned down as equal without ever needing either bond's maturity date.

Unit: yield-based-bond-duration-measures-and-properties

Question 10Above the exam

A bond has a Macaulay duration of 8.0 years and a yield to maturity of 6%, with annual coupon payments. Combining the relationship between Macaulay duration and modified duration, modified duration is closest to:

How sure are you?

Correct: B. Modified duration = Macaulay duration / (1 + YTM per period) = 8.0 / 1.06 = 7.5472, closest to 7.55. Modified duration is always slightly LESS than Macaulay duration (for a positive yield), since dividing by (1 + yield) shrinks the figure; this relationship is a direct algebraic derivative of the price-yield relationship, not a separate independent measure.
A. 8.48 comes from MULTIPLYING by (1 + YTM) instead of dividing (8.0 x 1.06), reversing the correct direction of the Macaulay-to-modified-duration conversion; modified duration is always smaller than Macaulay duration for a positive yield, not larger.
C. 8.00 simply repeats the Macaulay duration figure unchanged, as though modified duration equaled Macaulay duration exactly; the two are related but distinct measures, and the conversion always requires dividing by (1 + yield per period).

Unit: yield-based-bond-duration-measures-and-properties

Question 11Above the exam

Bond X has a modified duration of 7.0 and Bond Y has a modified duration of 7.0 as well, but Bond X is option-free while Bond Y is callable and currently trading near the price at which the call becomes attractive to the issuer. Combining modified duration with the concept of effective duration, an analyst estimating each bond's price sensitivity to a large parallel shift in rates should most likely:

How sure are you?

Correct: B. Modified duration assumes a bond's cash flows do NOT change as yields change, which is a reasonable assumption for an option-free bond like Bond X. For a callable bond like Bond Y, especially one trading near the price where the issuer's call becomes attractive, actual cash flows CAN change with rates (the bond may be called away), so EFFECTIVE duration, which explicitly accounts for cash flows changing with yield via a valuation model, is the appropriate measure for Bond Y, even though its modified duration happens to show the same number as Bond X's.
A. Trusting modified duration equally for both bonds ignores that modified duration's core assumption (cash flows fixed regardless of yield changes) is violated for a callable bond near its call-triggering price; the identical modified duration figure is misleading precisely because it cannot capture Bond Y's option-driven cash flow uncertainty.
C. Two bonds reporting the same modified duration will NOT necessarily show the same actual price change for a given rate shift if one of them has an embedded option whose behavior changes with rates; that is exactly why effective duration exists as a separate, more appropriate measure for bonds with embedded options.

Unit: yield-based-bond-duration-measures-and-properties