Question
Question 5
Suppose you bought a five-year zero-coupon Treasury bond for $800 per $1000 face value. Assuming yields to maturity on comparable bonds remain at 7%, calculate your holding period return if you sell the bond after two years.
*Make sure to input all currency answers without any currency symbols or commas, and use two decimal places of precision.
Suppose you bought a five-year zero-coupon Treasury bond for $800 per $1000 face value. Assuming yields to maturity on comparable bonds remain at 7%, calculate your holding period return if you sell the bond after two years.
*Make sure to input all currency answers without any currency symbols or commas, and use two decimal places of precision.
Answers
GPT-4o mini
To calculate the holding period return of the zero-coupon Treasury bond, we need to follow these steps:
1. **Determine the future value of the bond at maturity:**
The bond has a face value of $1000 and matures in 5 years. Given the yield to maturity (YTM) is 7%, we calculate the price of the bond at the time of purchase (which is already given as $800):
\[
P = \frac{F}{(1 + r)^n}
\]
where:
- \( P \) = price of the bond today ($800)
- \( F \) = face value of the bond ($1000)
- \( r \) = yield to maturity (7% or 0.07)
- \( n \) = years to maturity (5)
This confirms that the bond is priced correctly based on the given yield.
2. **Calculate the price of the bond after 2 years:**
After 2 years, there will be 3 years left until maturity. We will calculate the price of the bond 2 years later using the formula for present value:
\[
P_{2} = \frac{F}{(1 + r)^{n-2}} = \frac{1000}{(1 + 0.07)^{3}}
\]
Calculating that:
\[
P_{2} = \frac{1000}{(1.07)^{3}} = \frac{1000}{1.225043} \approx 816.30
\]
3. **Calculate the holding period return (HPR):**
The holding period return is calculated using the formula:
\[
HPR = \frac{P_{end} - P_{begin}}{P_{begin}}
\]
Where:
- \( P_{end} = P_{2} \) (price after 2 years)
- \( P_{begin} = 800 \) (initial investment)
So we plug in the values:
\[
HPR = \frac{816.30 - 800}{800} = \frac{16.30}{800} \approx 0.020375
\]
To express this as a percentage (holding period return percentage):
\[
HPR \times 100 \approx 2.04\%
\]
Thus, the holding period return expressed as a decimal for your calculations is approximately **0.02038**.
However, since the problem asks for the return more typically is represented in a percentage form of holding period return return, you can also express it as:
In decimal form:
**0.02038**
This is the holding period return expressed as a decimal value.
1. **Determine the future value of the bond at maturity:**
The bond has a face value of $1000 and matures in 5 years. Given the yield to maturity (YTM) is 7%, we calculate the price of the bond at the time of purchase (which is already given as $800):
\[
P = \frac{F}{(1 + r)^n}
\]
where:
- \( P \) = price of the bond today ($800)
- \( F \) = face value of the bond ($1000)
- \( r \) = yield to maturity (7% or 0.07)
- \( n \) = years to maturity (5)
This confirms that the bond is priced correctly based on the given yield.
2. **Calculate the price of the bond after 2 years:**
After 2 years, there will be 3 years left until maturity. We will calculate the price of the bond 2 years later using the formula for present value:
\[
P_{2} = \frac{F}{(1 + r)^{n-2}} = \frac{1000}{(1 + 0.07)^{3}}
\]
Calculating that:
\[
P_{2} = \frac{1000}{(1.07)^{3}} = \frac{1000}{1.225043} \approx 816.30
\]
3. **Calculate the holding period return (HPR):**
The holding period return is calculated using the formula:
\[
HPR = \frac{P_{end} - P_{begin}}{P_{begin}}
\]
Where:
- \( P_{end} = P_{2} \) (price after 2 years)
- \( P_{begin} = 800 \) (initial investment)
So we plug in the values:
\[
HPR = \frac{816.30 - 800}{800} = \frac{16.30}{800} \approx 0.020375
\]
To express this as a percentage (holding period return percentage):
\[
HPR \times 100 \approx 2.04\%
\]
Thus, the holding period return expressed as a decimal for your calculations is approximately **0.02038**.
However, since the problem asks for the return more typically is represented in a percentage form of holding period return return, you can also express it as:
In decimal form:
**0.02038**
This is the holding period return expressed as a decimal value.