Suppose a zero-coupon bond is selling for $614.00 today. It promises to pay $1,000 in exactly 10 years with annual compounding. Its annual rate of return would be about
1000=614(1+i)^10
1000/614=(1+i)^10
taking log of each side
.2188= 10 log(1+i)
.02188=log(1+i)
1-i= 10^ .2188=1.05167
i= 5.167percent
Double check all that, it is slightly off
To calculate the annual rate of return of a zero-coupon bond, you can use the formula for compound interest:
\[ P = \frac{F}{(1 + r)^n} \]
Where:
P = Present value of the bond (selling price)
F = Future value of the bond (face value or amount it promises to pay at maturity)
r = Annual interest rate
n = Number of compounding periods
In this case, we know the selling price (P = $614.00), the future value (F = $1,000), and the number of compounding periods (n = 10 years). We need to solve for the annual interest rate (r).
\[ 614 = \frac{1000}{(1 + r)^{10}} \]
To find the annual rate of return, we can rearrange the equation:
\[ (1 + r)^{10} = \frac{1000}{614} \]
Taking the 10th root of both sides gives us:
\[ 1 + r = \left( \frac{1000}{614} \right)^{\frac{1}{10}} \]
Now, subtract 1 from both sides to isolate r:
\[ r = \left( \frac{1000}{614} \right)^{\frac{1}{10}} - 1 \]
Solving this equation will give us the annual rate of return for the zero-coupon bond. Plugging in the values:
\[ r = \left( \frac{1000}{614} \right)^{\frac{1}{10}} - 1 \approx 0.0433 \]
So, the annual rate of return would be approximately 4.33%.