In this question, assume all bonds are annual coupon-ed bonds, have the face value of $1,000, have the same credit rating, and are sold at par. Which of the following has the least amount of Interest Rate Risk?

Question

BlackTom AI · Accepted Answer

Interest rate risk is driven by duration (a measure of price sensitivity to yield); higher coupons and shorter maturities lower duration, so they reduce interest rate risk.

1) Relevant knowledge points and concepts
- Macaulay duration measures weighted average time to receive cash flows; lower duration = less interest rate risk.
- For bonds sold at par, coupon rate = market yield.
- Perpetuity Macaulay duration = (1 + y) / y (for yield y).

2) Step-by-step reasoning and 3) intermediate calculations
We compute Macaulay durations for each candidate (face = $1,000, annual coupons, sold at par).

Option C: 6% coupon, 10 years (yield = 6%).
- Coupon = $60. Price = $1,000.
- Sum t·PV(coupons) = 60 · sum_{t=1}^{10} t/(1.06)^t ≈ 2,197.14.
- PV of principal weighted = 10 · 1000/(1.06^10) = 10 · 558.395 = 5,583.95.
- Total numerator = 7,781.09 → Duration = 7,781.09/1,000 = 7.78 years.

Option A: 3% coupon, 10 years (yield = 3%).
- Duration ≈ 8.77 years (calculated similarly).

Option B: 3% coupon, 15 years (yield = 3%).
- Duration ≈ 12.35 years.

Option D: 6% coupon, 15 years (yield = 6%).
- Duration ≈ 10.29 years.

Option E: Perpetuity paying $90 at 9% (sold at par).
- Perpetuity yield y = 9% → Duration = (1 + 0.09)/0.09 ≈ 12.11 years.

4) Why the correct answer is correct
The 6% coupon, 10-year bond has the smallest Macaulay duration (~7.78 years) among all options, so it has the least interest rate risk. Higher coupon (6% vs 3%) brings cash flows earlier; shorter maturity (10 vs 15 or infinite) reduces weight on distant payments.

5) Why other options are incorrect
- 3% coupons (A, B) have lower coupons so greater duration than their 6% counterparts.
- 15-year bonds (B, D) have longer maturities so larger durations than 10-year equivalents.
- The perpetuity (E) has infinite maturity and despite a higher coupon equivalent (9%) its duration (~12.11) is still high.

6) Conclusion / confirmation
Therefore the bond with the least interest rate risk is the 6% coupon, 10-year bond, matching the provided correct answer: "A bond with 6% coupon rate and a maturity of 10 years."

In this question, assume all bonds are annual coupon-ed bonds, have the face value of $1,000, have the same credit rating, and are sold at par. Which of the following has the least amount of Interest Rate Risk?单项选择题

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