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?

View Explanation

Log in for full answers

Similar Questions

Select ALL of the below statements that are TRUE.

What is a reasonable way to define Duration for an asset with cash flows CF over times t and a present value function PV? We have 4 possible answers Answer A Answer B Answer C Answer D

Consider a security with annual cash flows of $100,000 and a time to maturity of 9 years. The current interest rate is 9% p.a. What is the security’s duration? 考虑一种年现金流为 100,000 美元、到期时间为 9 年的证券。目前的利率为每年 9%。证券的期限是多久？

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?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler