What is the approximate yield to maturity for the following bonds? Assume these are bonds issued in the United States.
a. 10 years to maturity, 6 percent coupon rate, current price is $950.
b. 16 years to maturity, 0 percent coupon rate, current price is $339.
c. 25 years to maturity, 8.5 percent coupon rate, current price is $1030.
To calculate the approximate yield to maturity (YTM) for each bond, we need to use the formula for YTM and solve for the annual interest rate. The formula is as follows:
Current price = (Coupon payment / (1 + YTM)^1) + (Coupon payment / (1 + YTM)^2) + ... + (Coupon payment + Face value / (1 + YTM)^n)
Where:
- Current price is the present value of the bond,
- Coupon payment is the annual interest payment,
- YTM is the annual yield to maturity rate, and
- n is the number of years to maturity.
Let's apply this formula to each bond:
a. 10 years to maturity, 6 percent coupon rate, current price is $950.
To find the approximate YTM, we need to find the interest rate that makes the sum of present values of future cash flows equal to the current price of $950.
For this bond, the coupon payment per year is 6% * $100 face value = $6.
Using the formula above, we have:
$950 = ($6 / (1 + YTM)^1) + ($6 / (1 + YTM)^2) + ... + ($6 / (1 + YTM)^10) + ($100 / (1 + YTM)^10)
To solve this equation, we can use trial and error or a financial calculator or software to find the YTM.
b. 16 years to maturity, 0 percent coupon rate, current price is $339.
Since this bond has a 0 percent coupon rate, all the cash flows come from the face value at maturity. The current price is $339, which is the present value of the face value received in 16 years.
So, we have the equation:
$339 = ($100 / (1 + YTM)^16)
To find the YTM, we can follow the same steps as in the previous example.
c. 25 years to maturity, 8.5 percent coupon rate, current price is $1030.
For this bond, we have an annual coupon payment of 8.5% * $100 face value = $8.5.
So, the equation becomes:
$1030 = ($8.5 / (1 + YTM)^1) + ($8.5 / (1 + YTM)^2) + ... + ($8.5 / (1 + YTM)^25) + ($100 / (1 + YTM)^25)
Again, we can use trial and error or a financial calculator or software to find the YTM.
Please note that these calculations are approximate, as they assume constant interest rates over the life of the bond, whereas actual market interest rates may fluctuate.