You buy a zero coupon bond with a face value of $13,000 that matures in 11 years for $7,000. What is your annual compound rate of return? %. Round to the nearest thousandths of a percent (3 decimal places).
You buy a zero coupon bond with a face value of $13,000 that matures in 11 years for $7,000. What is your annual compound rate of return? %. Round to the nearest thousandths of a percent (3 decimal places).
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Sequences And Series
Section8.4: Mathematics Of Finance
Problem 21E
Related questions
Question
f11
![### Understanding Zero Coupon Bonds
**Question:**
You buy a zero coupon bond with a face value of $13,000 that matures in 11 years for $7,000. What is your annual compound rate of return?
__________%. Round to the nearest thousandths of a percent (3 decimal places).
**Explanation:**
Zero coupon bonds are a type of bond that does not pay interest (a coupon) periodically. Instead, they are sold at a discount to their face value and the bondholder receives the face value upon maturity. The difference between the purchase price and the face value is the bondholder's return.
In this case, to determine the annual compound rate of return, we use the formula for compound interest:
\[ FV = PV \times (1 + r)^n \]
Where:
- \( FV \) = Future Value (face value of the bond)
- \( PV \) = Present Value (purchase price of the bond)
- \( r \) = annual compound rate of return
- \( n \) = number of years
Given:
- \( FV = \$13,000 \)
- \( PV = \$7,000 \)
- \( n = 11 \)
We need to solve for \( r \):
\[ 13,000 = 7,000 \times (1 + r)^{11} \]
To isolate \( r \), we follow these steps:
1. Divide both sides by 7,000:
\[
\frac{13,000}{7,000} = (1 + r)^{11}
\]
\[
1.8571 = (1 + r)^{11}
\]
2. Take the 11th root of both sides:
\[
(1.8571)^{1/11} = 1 + r
\]
3. Subtract 1 from both sides:
\[
r = (1.8571)^{1/11} - 1
\]
Using a calculator, we find:
\[
r \approx 0.05703
\]
To express this as a percentage, we multiply by 100:
\[
r \approx 5.703\%
\]
Therefore, the annual compound rate of return is approximately **5.703%**, rounded to the nearest thousandth of a percent.
**Note for Students:**
It's crucial to understand the relationship between the present value, the face value, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d929fa4-2fdc-40c7-801d-1fbf9b4298bd%2F0c40160d-9ea5-4383-8180-dc573850b94a%2F1z8v25b_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Zero Coupon Bonds
**Question:**
You buy a zero coupon bond with a face value of $13,000 that matures in 11 years for $7,000. What is your annual compound rate of return?
__________%. Round to the nearest thousandths of a percent (3 decimal places).
**Explanation:**
Zero coupon bonds are a type of bond that does not pay interest (a coupon) periodically. Instead, they are sold at a discount to their face value and the bondholder receives the face value upon maturity. The difference between the purchase price and the face value is the bondholder's return.
In this case, to determine the annual compound rate of return, we use the formula for compound interest:
\[ FV = PV \times (1 + r)^n \]
Where:
- \( FV \) = Future Value (face value of the bond)
- \( PV \) = Present Value (purchase price of the bond)
- \( r \) = annual compound rate of return
- \( n \) = number of years
Given:
- \( FV = \$13,000 \)
- \( PV = \$7,000 \)
- \( n = 11 \)
We need to solve for \( r \):
\[ 13,000 = 7,000 \times (1 + r)^{11} \]
To isolate \( r \), we follow these steps:
1. Divide both sides by 7,000:
\[
\frac{13,000}{7,000} = (1 + r)^{11}
\]
\[
1.8571 = (1 + r)^{11}
\]
2. Take the 11th root of both sides:
\[
(1.8571)^{1/11} = 1 + r
\]
3. Subtract 1 from both sides:
\[
r = (1.8571)^{1/11} - 1
\]
Using a calculator, we find:
\[
r \approx 0.05703
\]
To express this as a percentage, we multiply by 100:
\[
r \approx 5.703\%
\]
Therefore, the annual compound rate of return is approximately **5.703%**, rounded to the nearest thousandth of a percent.
**Note for Students:**
It's crucial to understand the relationship between the present value, the face value, the
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill