Question
A zero coupon bond with face value of $1,000 and 5.00 years to maturity sells for $858.00. What is the discount rate for the bond?
Answers
GPT-4o mini
To calculate the discount rate for a zero coupon bond, we can use the present value formula for a single future cash flow:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where:
- \( PV \) = Present Value (current selling price of the bond, $858.00)
- \( FV \) = Future Value (face value of the bond, $1,000)
- \( r \) = discount rate (annual)
- \( n \) = number of years to maturity (5.00 years)
Rearranging the formula to solve for \( r \) gives:
\[
r = \left(\frac{FV}{PV}\right)^{\frac{1}{n}} - 1
\]
Plugging in the values:
\[
r = \left(\frac{1000}{858}\right)^{\frac{1}{5}} - 1
\]
Calculating it step by step:
1. Divide \( 1000 \) by \( 858 \):
\[
\frac{1000}{858} \approx 1.1655
\]
2. Raise this result to the power of \( \frac{1}{5} \):
\[
(1.1655)^{\frac{1}{5}} \approx 1.0315
\]
3. Subtract 1:
\[
1.0315 - 1 \approx 0.0315
\]
4. Convert to percentage:
\[
r \approx 0.0315 \times 100 \approx 3.15\%
\]
So, the discount rate for the bond is approximately **3.15%**.
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where:
- \( PV \) = Present Value (current selling price of the bond, $858.00)
- \( FV \) = Future Value (face value of the bond, $1,000)
- \( r \) = discount rate (annual)
- \( n \) = number of years to maturity (5.00 years)
Rearranging the formula to solve for \( r \) gives:
\[
r = \left(\frac{FV}{PV}\right)^{\frac{1}{n}} - 1
\]
Plugging in the values:
\[
r = \left(\frac{1000}{858}\right)^{\frac{1}{5}} - 1
\]
Calculating it step by step:
1. Divide \( 1000 \) by \( 858 \):
\[
\frac{1000}{858} \approx 1.1655
\]
2. Raise this result to the power of \( \frac{1}{5} \):
\[
(1.1655)^{\frac{1}{5}} \approx 1.0315
\]
3. Subtract 1:
\[
1.0315 - 1 \approx 0.0315
\]
4. Convert to percentage:
\[
r \approx 0.0315 \times 100 \approx 3.15\%
\]
So, the discount rate for the bond is approximately **3.15%**.