Practice: Yield-Based Bond Duration Measures and Properties
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Fixed IncomeYield-Based Bond Duration Measures and Properties
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Question 1Exam level
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:
How sure are you?
Correct: 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).
B. Choosing 5.40 years might seem logical if you confuse Macaulay duration with modified duration, but modified duration adjusts Macaulay duration for the yield to maturity, making it a more accurate measure of interest rate sensitivity.
C. Choosing 5.71 years might tempt you if you mistakenly add the yield to the Macaulay duration, but this violates the rule for calculating modified duration, which requires dividing the Macaulay duration by (1 + y/m), not adding the yield.
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:
How sure are you?
Correct: A. %ΔPrice ≈ -Modified Duration × ΔYield = -7.5 × (+0.0050) = -0.0375 = -3.75%. Bond prices move INVERSELY to yields, hence the negative sign. The 50 basis points must be expressed as 0.0050 in decimal form. Answer B has the wrong sign (prices fall when yields rise). Answer C incorrectly uses the full duration as the percentage change. Answer D divides by 10 rather than using the correct formula.
B. Choosing B (+3.75%) might seem logical if you mistakenly think bond prices move in the same direction as yields, but bond prices actually move inversely to yields, so the percentage price change should be negative.
C. You might be tempted to use the full duration value as the percentage change, but this violates the rule that the percentage price change is the product of modified duration and the change in yield, not the duration itself.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.