Question1.10 Consider the following four bonds: [table]
Bond | Term to Maturity (year) | Coupon Rate | YTM I | 5 | 5% | 8% II | 5 | 5% | 6% III | 10 | 3% | 6% IV | 5 | 3% | 6%
[/table] If the yield-to-maturity for all four bonds changes by 1%, rank the bonds from largest percentage change in price to the smallest percentage change in price? III, IV, II, I II, I, IV, III None of the options is correct III, I, II, IV III, IV, II, I ResetMaximum marks: 2.5 Flag question undefined

Question

Question1.10 Consider the following four bonds: [table]
Bond | Term to Maturity (year) | Coupon Rate | YTM I | 5 | 5% | 8% II | 5 | 5% | 6% III | 10 | 3% | 6% IV | 5 | 3% | 6%
[/table] If the yield-to-maturity for all four bonds changes by 1%, rank the bonds from largest percentage change in price to the smallest percentage change in price? III, IV, II, I  II, I, IV, III None of the options is correct III, I, II, IV III, IV, II, I  ResetMaximum marks: 2.5 Flag question undefined

BlackTom AI · Accepted Answer

Duration (modified duration) gives the approximate percentage price change ≈ −(modified duration) × Δy for a small change in yield. Lower coupons, longer maturities, and lower yields increase duration, so we compute modified durations for each bond to rank percentage sensitivity.

1) Key formulas and concepts
- Price P = sum_{t=1..n} C/(1+y)^t + 100/(1+y)^n (C = annual coupon).
- Macaulay duration = (1/P) × sum_{t=1..n} t × CF_t/(1+y)^t.
- Modified duration = Macaulay / (1+y).
- Percentage price change for Δy = 1% ≈ modified duration × 1%.

2) Complete step-by-step calculations (intermediate values shown)

Set face = 100.

Bond II (n=5, C=5, y=6%):
- PV coupons = 5 × [1 − (1.06)^−5]/0.06 = 21.06182
- PV principal = 100/(1.06)^5 = 74.72582
- Price P = 95.78764
- Sum t·PV(CF) = 434.36364 → Macaulay = 434.36364/95.78764 = 4.534
- Modified = 4.534/1.06 = 4.276 → % change ≈ 4.276%

Bond I (n=5, C=5, y=8%):
- Price P = 88.02187 (coupons PV 19.96355 + principal 68.05832)
- Sum t·PV(CF) = 397.11727 → Macaulay = 397.11727/88.02187 = 4.511
- Modified = 4.511/1.08 = 4.177 → % change ≈ 4.177%

Bond IV (n=5, C=3, y=6%):
- PV coupons = 12.63710; PV principal = 74.72582; P = 87.36292
- Sum t·PV(CF) = 410.06982 → Macaulay = 410.06982/87.36292 = 4.694
- Modified = 4.694/1.06 = 4.430 → % change ≈ 4.430%

Bond III (n=10, C=3, y=6%):
- PV coupons = 3 × [1 − (1.06)^−10]/0.06 = 22.08024
- PV principal = 100/(1.06)^10 = 55.83950; P = 77.91974
- Use sum S = sum_{t=1..10} t x^t with x=1/1.06; S = x(1 − (n+1)x^n + n x^{n+1})/(1−x)^2 ≈ 37.391
- Sum t·PV(CF) = 3×S + 10×55.83950 = 670.568 → Macaulay = 670.568/77.91974 = 8.606
- Modified = 8.606/1.06 = 8.116 → % change ≈ 8.116%

3) Why the correct answer is correct
Bond III has the largest modified duration (long maturity + low coupon), so it shows the largest percentage price change. Among 5‑year bonds, the low‑coupon Bond IV (3%) is more sensitive than the 5% coupon Bond II; Bond I has the highest yield (8%), which reduces its duration, so it is least sensitive. Thus ordering by descending percentage change: III (≈8.12%), IV (≈4.43%), II (≈4.28%), I (≈4.18%).

4) Why other options are incorrect
- Any order placing a 5‑year bond ahead of the 10‑year (III) ignores the dominant effect of maturity on duration. Orders swapping II and IV ignore the coupon effect (3% > 5% sensitivity). Orders placing I above II ignore the yield effect (higher y reduces duration).

Conclusion: The ranking from largest to smallest percentage price change is III, IV, II, I, which matches the provided answer.

类似问题

Consider two bonds with the same maturity and yield - Bond M: 10-year zero-coupon bond, and Bond N: 10-year bond with an 8% annual coupon. If market interest rates increase unexpectedly, which bond will experience a larger percentage price decline?[Fill in the blank]

Consider the following statements about the duration of loans and bonds. Which of these statements is correct?

Choose the most correct answer. Ceteris paribus, the duration of a bond is positively correlated with the bond's

Consider the following two Treasury securities: Bond A is sold at $120 with modified duration = 5, and Bond B is sold at $80 with modified duration = 7. Which bond will have greater dollar price change for a 50-basis-point change in interest rates?

Question at position 3 For bonds, price sensitivity to a given change in interest rates is generally greater the longer before the bond matures.TrueFalse

If an investor is convinced that all YTMs in the market will decrease by 1% in the next month, then, they should invest in the bond with the longest maturity available, as the price of this bond will increase the most due to the change in the YTM.

Question at position 18 All else equal, the the coupon rate on a bond, the the bondʹs duration.greater; longerhigher; shorterhigher; longerlower; shorter

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单