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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### 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
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