Question
Suppose you bought a five-year zero-coupon Treasury bond for $800 per $1000 face value. Assume the yield to maturity on comparable bonds increases to 7% after you purchase the bond and remains there. Calculate your holding period return (annual return) if you sell the bond after one year.
Answers
GPT-4o mini
To calculate the holding period return (HPR) of a zero-coupon Treasury bond after one year, we need to follow these steps:
1. **Understand the initial purchase**: You bought a zero-coupon bond for $800 with a face value of $1,000, which means you will receive $1,000 at maturity in five years.
2. **Calculate the value of the bond after one year**: After one year, your bond has 4 years remaining until maturity. With the new yield to maturity (YTM) set at 7%, we can find the price of the bond at that time.
The price of a zero-coupon bond can be calculated using the present value formula:
\[
P = \frac{F}{(1 + r)^n}
\]
Where:
- \(P\) = price of the bond
- \(F\) = face value of the bond ($1,000)
- \(r\) = yield to maturity (0.07 for 7%)
- \(n\) = number of years until maturity (4 years)
Plug in the values:
\[
P = \frac{1000}{(1 + 0.07)^4}
\]
\[
P = \frac{1000}{(1.3107961)} \quad (\text{using } (1.07)^4 \approx 1.3107961)
\]
\[
P \approx 763.19
\]
3. **Calculate the holding period return**: The holding period return is calculated based on the initial investment and the selling price after one year.
\[
\text{HPR} = \frac{\text{Ending Value} - \text{Beginning Value}}{\text{Beginning Value}}
\]
Where:
- Ending Value is the price after one year ($763.19).
- Beginning Value is the initial purchase price ($800).
Now, plug in the values:
\[
\text{HPR} = \frac{763.19 - 800}{800}
\]
\[
\text{HPR} = \frac{-36.81}{800} = -0.0460125 \quad (\text{approximately})
\]
\[
\text{HPR} \approx -0.0460 \text{ or } -4.60\%
\]
4. **Annualize the return**: Since we are calculating a holding period return for one year, the HPR itself serves as the annual return for this specific period.
Therefore, if you sell the bond after one year, your holding period return is approximately **-4.60%**.
1. **Understand the initial purchase**: You bought a zero-coupon bond for $800 with a face value of $1,000, which means you will receive $1,000 at maturity in five years.
2. **Calculate the value of the bond after one year**: After one year, your bond has 4 years remaining until maturity. With the new yield to maturity (YTM) set at 7%, we can find the price of the bond at that time.
The price of a zero-coupon bond can be calculated using the present value formula:
\[
P = \frac{F}{(1 + r)^n}
\]
Where:
- \(P\) = price of the bond
- \(F\) = face value of the bond ($1,000)
- \(r\) = yield to maturity (0.07 for 7%)
- \(n\) = number of years until maturity (4 years)
Plug in the values:
\[
P = \frac{1000}{(1 + 0.07)^4}
\]
\[
P = \frac{1000}{(1.3107961)} \quad (\text{using } (1.07)^4 \approx 1.3107961)
\]
\[
P \approx 763.19
\]
3. **Calculate the holding period return**: The holding period return is calculated based on the initial investment and the selling price after one year.
\[
\text{HPR} = \frac{\text{Ending Value} - \text{Beginning Value}}{\text{Beginning Value}}
\]
Where:
- Ending Value is the price after one year ($763.19).
- Beginning Value is the initial purchase price ($800).
Now, plug in the values:
\[
\text{HPR} = \frac{763.19 - 800}{800}
\]
\[
\text{HPR} = \frac{-36.81}{800} = -0.0460125 \quad (\text{approximately})
\]
\[
\text{HPR} \approx -0.0460 \text{ or } -4.60\%
\]
4. **Annualize the return**: Since we are calculating a holding period return for one year, the HPR itself serves as the annual return for this specific period.
Therefore, if you sell the bond after one year, your holding period return is approximately **-4.60%**.