To calculate the holding period return, we first need to determine the selling price of the bond after 3 years, when the yield to maturity (YTM) is 3%.
A zero-coupon bond pays no interest until maturity, thus its price 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 ($1000)
- \(r\) = yield to maturity (0.03 for 3%)
- \(n\) = number of years to maturity at the time of sale
Since the bond was purchased for a 5-year term and 3 years have passed, there are \(n = 2\) years remaining before maturity.
Now substituting the known values into the formula:
\[ P = \frac{1000}{(1 + 0.03)^2} \]
Calculating it step by step:
- Calculate \(1 + r\):
\[ 1 + 0.03 = 1.03 \]
- Raise it to the power of \(n = 2\):
\[ (1.03)^2 = 1.0609 \]
- Now calculate the price \(P\):
\[ P = \frac{1000}{1.0609} \approx 941.76 \]
Now we know that the selling price of the bond after 3 years is approximately $941.76.
Next, we calculate the holding period return (HPR):
\[ HPR = \frac{Selling\ Price - Purchase\ Price}{Purchase\ Price} \]
Here:
- Selling Price = 941.76
- Purchase Price = 800
So, substituting the values:
\[ HPR = \frac{941.76 - 800}{800} \] \[ HPR = \frac{141.76}{800} \approx 0.1772 \]
To express the holding period return as a percentage, we multiply by 100:
\[ HPR\ (as\ a\ percentage) = 0.1772 \times 100 \approx 17.72% \]
Therefore, the holding period return if you sell the bond at that time is approximately:
17.72