Question at position 18 Which bond is most sensitive to interest rate changes?All equal5-year, 8% coupon3-year, 10% coupon10-year, 3% coupon

Question

BlackTom AI · Accepted Answer

Lower coupon rates and longer maturities both raise a bond’s interest‑rate sensitivity (duration), so the bond with the longest maturity and the lowest coupon will be most sensitive. Key concepts: price sensitivity is measured by duration (Macaulay duration and modified duration); longer time to receive cash flows and smaller coupon cash flows push duration higher.

Step‑by‑step reasoning and numerical illustration (assume annual coupons and a yield-to-maturity y = 5% to illustrate relative magnitudes):

1) Compute each bond’s price and Macaulay duration. Discount factors for t = 1..10 at 5%: df1 = 0.952381, df2 = 0.907029, df3 = 0.863838, df4 = 0.822702, df5 = 0.783526, df6 = 0.746215, df7 = 0.710681, df8 = 0.676839, df9 = 0.644609, df10 = 0.613913.

2) 10‑year, 3% coupon (coupon = 3 on par 100):
- Price = PV(coupons) + PV(principal) = 3*[1 - (1.05)^-10]/0.05 + 100*(1.05)^-10 = 23.1652 + 61.3913 = 84.5565.
- Numerator for Macaulay D = Σ t·PV(CF_t) = 732.0343 (calculated by summing t × discounted cash flows, see intermediate per‑year breakdown below).
- Macaulay D = 732.0343 / 84.5565 ≈ 8.656 years. Modified D = 8.656 / 1.05 ≈ 8.244.

3) 5‑year, 8% coupon (coupon = 8):
- Price ≈ 34.6358 + 78.3526 = 112.9884.
- Numerator Σ t·PV ≈ 492.2941.
- Macaulay D = 492.2941 / 112.9884 ≈ 4.357 years. Modified D ≈ 4.150.

4) 3‑year, 10% coupon (coupon = 10):
- Price ≈ 27.2324 + 86.3838 = 113.6162.
- Numerator Σ t·PV ≈ 312.7309.
- Macaulay D = 312.7309 / 113.6162 ≈ 2.752 years. Modified D ≈ 2.621.

Why the correct answer is correct: The 10‑year, 3% coupon bond has by far the largest duration (~8.24 modified) because it returns most cash flows later and has a low coupon so intermediate cash flows are small; therefore its price changes most for a given yield change.

Why other options are incorrect:
- 3‑year, 10% coupon: short maturity and high coupons concentrate cash flows early → lowest duration.
- 5‑year, 8% coupon: intermediate maturity and coupon → moderate duration.
- “All equal”: clearly false because differing maturities and coupons change durations.

Conclusion: The 10‑year, 3% coupon bond is most sensitive to interest‑rate changes, confirming the provided answer (10‑year, 3% coupon).

Question at position 18 Which bond is most sensitive to interest rate changes?All equal5-year, 8% coupon3-year, 10% coupon10-year, 3% coupon单项选择题

类似问题

A zero-coupon bond has three years until maturity. The bond’s face value is $100 and the price today is $80. The bond’s duration is:

Consider the following statements. Choose the correct statement.

A 6% coupon bond paying interest annually has a modified duration of 10 years, sells for $800, and is priced at a yield to maturity of 8%. If the YTM increases to 9%, what is the predicted dollar change in price based on the bond’s duration? (Round to 2 decimal places.)

Which of the following two semiannual Treasury bonds is more price sensitive to changes in interest rates? 1) A par value bond X, with 5-year-to-maturity and 10% coupon rate. 2) A zero-coupon bond Y, with 5-year-to-maturity and 10% yield to maturity.

An investor buys a 6% annual payment bond with three years to maturity. The bond has a yield-to-maturity of 8% and is currently priced at 94.845806 per 100 of par. The bond’s Macaulay duration is closest to:

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