Final answer:
The implied interest rate of the zero-coupon bond is approximately 10.45%. This rate is obtained using the formula for compound interest and solving for the annual interest rate. Interest rates changes influence bond prices; if market rates increase beyond the bond's rate, the bond price decreases to attract investors.
Step-by-step explanation:
To calculate the implied interest rate of a zero-coupon bond, you can use the formula for compound interest:
FV = PV (1 + r)^n
Where:
- FV is the future value of the bond (its par value at maturity), which is $1,000.
- PV is the present value of the bond (its current price), which is $500.
- r is the annual interest rate.
- n is the number of years until maturity, which is 7 years.
Plugging the values into the formula:
$1,000 = $500 (1 + r)^7
To find r, we solve for it algebraically:
2 = (1 + r)^7
Take the 7th root of both sides to isolate (1 + r):
2^(1/7) = 1 + r
Subtract 1 from both sides to find r:
r = 2^(1/7) - 1
Calculating this value gives:
r ≈ 0.1045 or 10.45%
The implicit interest rate for the zero-coupon bond is approximately 10.45%.
If there were a change in interest rates, one would expect to pay less for a bond if the market interest rate rises above the bond's implicit interest rate as it will then appear less attractive compared to new issues offering higher rates.