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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BlackTom AI · Accepted Answer

To tackle the Macaulay duration, first restate the bond details and cash flows. The bond has: face value 1000, annual coupon rate 12% → annual coupon 120, maturity 3 years, yield to maturity (discount rate) 9% per year. The cash flows are: year 1: 120, year 2: 120, year 3: 1120 (coupon 120 plus redemption 1000).

Next, compute the present value (price) of each cash flow at a 9% yield:
- PV1 = 120 / 1.09 ≈ 110.0917
- PV2 = 120 / (1.09)^2 ≈ 100.9891
- PV3 = 1120 / (1.09)^3 ≈ 864.4285

Sum these to get the price: Price ≈ 110.0917 + 100.9891 + 864.4285 ≈ 1,075.5093.

Now calculate the Macaulay duration, which is the weighted average time of cash flows using present values as weights:
- Term for year 1: 1 × PV1 ≈ 1 × 110.0917 = 110.0917
- Term for year 2: 2 × PV2 ≈ 2 × 100.9891 = 201.9782
- Term for year 3: 3 × PV3 ≈ 3 × 864.4285 = 2,593.2855

Sum of weighted PVs (numerator) ≈ 110.0917 + 201.9782 + 2,593.2855 ≈ 2,905.3554.

Finally, Macaulay duration = Numerator / Price ≈ 2,905.3554 / 1,075.5093 ≈ 2.70 years.

Now evaluate each answer choice:
- a. 2.93 years: This overestimates the duration. The weighted PVs and the resulting denominator yield about 2.70, not 2.93, so this is not correct.
- b. 2.64 years: This is a plausible near-miss, but the precise calculation with the shown cash flows and discounting gives about 2.70, so 2.64 is too low.
- c. 2.70 years: This matches the computed Macaulay duration from the exact PVs and the price above, so this is the correct value.
- d. 3 years: Macaulay duration will be less than maturity when there is positive coupon income and discounting; 3 would imply no interim cash flow effect, which is incorrect here.
- e. 2.84 years: This is somewhat high relative to the actual result; the precise calculation gives around 2.70, not 2.84.

In summary, after performing the present-value pricing at 9% and weighting each cash flow by its time, the Macaulay duration computed is approximately 2.70 years, which aligns with option c. 2.70 years.

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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