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.
*Make sure to input all currency answers without any currency symbols or commas, and use two decimal places of precision.
*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 (HPR) after one year, we need to determine the selling price of the bond after one year based on the new yield to maturity (YTM) and the bond's pricing formula.
### Step 1: Calculate the selling price of the bond after one year.
The bond is a zero-coupon bond that pays $1000 at maturity (in 5 years) and does not make any periodic interest payments. The price of a zero-coupon bond is calculated by the formula:
\[ P = \frac{F}{(1 + YTM)^n} \]
Where:
- \( P \) = price of the bond
- \( F \) = face value of the bond ($1000)
- \( YTM \) = yield to maturity (7% or 0.07)
- \( n \) = number of years to maturity remaining
After one year, the bond will have 4 years remaining until maturity.
Substituting the values into the formula:
\[ P = \frac{1000}{(1 + 0.07)^4} \]
Calculating \( (1 + 0.07)^4 \):
\[ (1.07)^4 = 1.3107961 \]
Now calculate the price:
\[ P = \frac{1000}{1.3107961} \approx 763.28 \]
### Step 2: Calculate the holding period return (HPR).
The HPR is calculated using the formula:
\[ HPR = \frac{Selling \, Price - Purchase \, Price}{Purchase \, Price} \]
Substituting the values:
- Selling Price (after one year) = 763.28
- Purchase Price = 800
\[ HPR = \frac{763.28 - 800}{800} \]
Calculating:
\[ HPR = \frac{-36.72}{800} \approx -0.0459 \]
### Step 3: Convert HPR to an annual return.
Since the HPR is for one year, we can express this as a percentage:
\[ HPR \text{ (as a percentage) } = -0.0459 \times 100 \approx -4.59\% \]
### Final answer in required format:
-4.59
### Step 1: Calculate the selling price of the bond after one year.
The bond is a zero-coupon bond that pays $1000 at maturity (in 5 years) and does not make any periodic interest payments. The price of a zero-coupon bond is calculated by the formula:
\[ P = \frac{F}{(1 + YTM)^n} \]
Where:
- \( P \) = price of the bond
- \( F \) = face value of the bond ($1000)
- \( YTM \) = yield to maturity (7% or 0.07)
- \( n \) = number of years to maturity remaining
After one year, the bond will have 4 years remaining until maturity.
Substituting the values into the formula:
\[ P = \frac{1000}{(1 + 0.07)^4} \]
Calculating \( (1 + 0.07)^4 \):
\[ (1.07)^4 = 1.3107961 \]
Now calculate the price:
\[ P = \frac{1000}{1.3107961} \approx 763.28 \]
### Step 2: Calculate the holding period return (HPR).
The HPR is calculated using the formula:
\[ HPR = \frac{Selling \, Price - Purchase \, Price}{Purchase \, Price} \]
Substituting the values:
- Selling Price (after one year) = 763.28
- Purchase Price = 800
\[ HPR = \frac{763.28 - 800}{800} \]
Calculating:
\[ HPR = \frac{-36.72}{800} \approx -0.0459 \]
### Step 3: Convert HPR to an annual return.
Since the HPR is for one year, we can express this as a percentage:
\[ HPR \text{ (as a percentage) } = -0.0459 \times 100 \approx -4.59\% \]
### Final answer in required format:
-4.59