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.

Question

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.

BlackTom AI · Accepted Answer

A zero-coupon bond has all of its cash flow at maturity, so its Macaulay duration equals its time to maturity (5 years). That makes it more price sensitive than a comparable coupon bond whose cash flows are spread earlier.

1) Relevant concepts
- Macaulay duration measures the weighted average time to receive cash flows; higher duration → greater price sensitivity to yield changes.
- For a zero-coupon bond, Macaulay duration = maturity.
- For coupon bonds, duration < maturity because coupons are paid earlier.
- Here both instruments have semiannual compounding and the same nominal yield (10% annually → 5% per half-year), and bond X is a par coupon bond (coupon = yield), so price = par.

2) Step-by-step reasoning and calculations
- Periods: 5 years × 2 = 10 semiannual periods. Periodic rate i = 10%/2 = 5% = 0.05.
- Bond X cash flows: 5 each period for t = 1..9, and 105 at t = 10 (per $100 par). Since coupon = yield, price = $100.
- Present value factors: 1.05^10 ≈ 1.628894, so PV(principal) = 100/1.628894 = 61.3913208.
- PV of coupons = 5 * [1 − 1/1.628894]/0.05 ≈ 38.6086792. Total price = 100.

Compute Macaulay duration (in semiannual periods):
- Sum t * PV(CF_t) = 5 * Σ_{t=1}^{10} t/(1.05^t) + 10*100/1.05^10.
- Numerically Σ t/(1.05^t) ≈ 39.37379 → coupon-weighted sum = 5 * 39.37379 ≈ 196.86895.
- Principal-weighted term = 10 * 61.3913208 ≈ 613.913208.
- Numerator ≈ 810.782158; Macaulay duration in periods = 810.782158 / 100 = 8.10782158 periods.
- Convert to years: 8.10782158 / 2 ≈ 4.05391 years.

3) Why the correct answer is correct
- Bond Y (zero coupon) has duration = 5.00 years, which is larger than Bond X’s Macaulay duration ≈ 4.054 years. Larger duration means greater price sensitivity to interest-rate changes, so Bond Y is more sensitive.

4) Why the other choices are incorrect
- "Bond X, because of the higher coupon rate": Higher coupons reduce duration and thus reduce sensitivity, so incorrect.
- "Bond Y, because of the higher yield to maturity": Both bonds have the same yield (10%); the reason Y is more sensitive is longer duration, not higher yield.
- "Both have the same sensitivity because both have the same yield to maturity": Sensitivity depends on cash-flow timing (duration) as well as yield; same yield does not imply same sensitivity.

5) Conclusion
Bond Y is more price sensitive due to its longer duration. This matches the provided answer: "Bond Y, because of the longer duration."

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.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.)

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

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:

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