A $1,000 face value annual coupon payment bond with 3 years to maturity with a 12% annual coupon rate has an annualized yield-to-maturity of 9%. What is the Macaulay duration of the bond?

Question

BlackTom AI · Accepted Answer

Question restatement: We are given a 1,000 face value annual coupon bond with 3 years to maturity, a coupon rate of 12% per year, and a yield to maturity of 9% per year. We need to determine the Macaulay duration.

Option analysis:
- a. 2.84 years
  To assess this, compute the present value of each cash flow at the 9% yield and weight by time. The bond pays 120 each year for years 1 and 2, and 1120 at year 3. PVs: 120/1.09 ≈ 109.17, 120/1.09^2 ≈ 100.99, 1120/1.09^3 ≈ 863.86. Price ≈ 1,074.02. Macaulay duration = (1*109.17 + 2*100.99 + 3*863.86) / 1,074.02 ≈ (109.17 + 201.98 + 2,591.58) / 1,074.02 ≈ 2.703 years. This is not 2.84; thus this option is inconsistent with the computed duration.
- b. 2.64 years
  Using the same PVs and weightings, the numerator is 109.17 + 2*100.99 + 3*863.86 ≈ 2,902.74, and dividing by price ≈ 1,074.02 gives ≈ 2.70 years. This value is noticeably different from 2.64, so this option is incorrect.
- c. 3 years
  A Macaulay duration of 3 would imply the weighted average of cash flows has all weight at maturity, which would be the case for a zero-coupon bond. Here we have coupons, so the duration must be less than the maturity. Therefore 3 years is not correct for this coupon bond.
- d. 2.70 years
  This matches the computed result: price ≈ 1,074.02, PVs ≈ {year1: 109.17, year2: 100.99, year3: 863.86}, and the weighted sum 1*109.17 + 2*100.99 + 3*863.86 ≈ 2,902.74. Dividing by price 1,074.02 gives ≈ 2.70 years. This aligns with the standard Macaulay duration formula and the given yield.
- e. 2.93 years
  A duration of 2.93 would require a higher weight on earlier cash flows or a different yield/price structure. Given the calculated PVs and their weights, the resulting duration is around 2.70, not 2.93, so this option is not supported by the math.

Summary of reasoning: By discounting each cash flow at the yield to maturity (9%), computing the present values, and taking the time-weighted average of these PVs relative to the bond price, we obtain a Macaulay duration of approximately 2.70 years. The other options deviate from this computed value due to either incorrect PVs, incorrect weighting, or misunderstanding of how coupons affect duration versus maturity.

A $1,000 face value annual coupon payment bond with 3 years to maturity with a 12% annual coupon rate has an annualized yield-to-maturity of 9%. What is the Macaulay duration of the bond?

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A $1,000 face value annual coupon payment bond with 3 years to maturity with a 12% annual coupon rate has an annualized yield-to-maturity of 9%. What is the Macaulay duration of the bond?

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