To calculate the holding period return (HPR) on the five-year zero-coupon Treasury bond, we first need to determine the bond's price after three years, given that the yield to maturity (YTM) has declined to 3%.
Step 1: Calculate the price of the bond after 3 years.
The bond initially has a maturity of 5 years, and after 3 years, it has 2 years remaining until maturity (5 - 3 = 2).
The price of a zero-coupon bond can be calculated using the formula:
\[ P = \frac{F}{(1 + r)^n} \]
where:
- \(P\) is the price of the bond
- \(F\) is the face value of the bond (which is $1000)
- \(r\) is the yield to maturity (3% or 0.03)
- \(n\) is the number of years until maturity (2 years)
Substituting in the values:
\[ P = \frac{1000}{(1 + 0.03)^2} \] \[ P = \frac{1000}{(1.03)^2} \] \[ P = \frac{1000}{1.0609} \] \[ P \approx 941.76 \]
Step 2: Calculate the holding period return (HPR).
The holding period return is calculated as follows:
\[ HPR = \frac{Selling\ Price - Purchase\ Price}{Purchase\ Price} \]
Where:
- Selling Price = Price of the bond after 3 years = $941.76
- Purchase Price = $800 (the price at which the bond was initially bought)
Substituting in the values:
\[ HPR = \frac{941.76 - 800}{800} \] \[ HPR = \frac{141.76}{800} \] \[ HPR \approx 0.1772 \text{ or } 17.72% \]
To express this in decimal form (which is often used in financial contexts), the HPR is approximately:
17.72 (without the percent sign or any currency symbol).