What is the coupon rate of a two-year, $10,000 bond with semiannual coupons and a price of $9,543.45, if it has yield to maturity of 6.8%?
To find the coupon rate of a bond, you need to know the bond's price, face value, and the yield to maturity (YTM).
In this case, we are given the following information:
- Face value (or par value) of the bond: $10,000
- Price of the bond: $9,543.45
- Yield to maturity (YTM) of the bond: 6.8%
First, let's calculate the annual coupon payment:
The bond pays semiannual coupons, so in one year, there will be 2 coupon payments. Therefore, the number of years is 2.
The YTM is the effective interest rate the bondholder will earn annually if held until maturity. So, to find the annual coupon payment, we can use the following formula:
Bond Price = (Coupon Payment / (1 + YTM/2)^1) + (Coupon Payment / (1 + YTM/2)^2) + ... + (Coupon Payment + Face Value) / (1 + YTM/2)^(n * 2)
Where:
Coupon Payment = Annual coupon payment
YTM = Yield to Maturity
n = Number of years
In this case, since the bond has 2 years until maturity, we can simplify the formula to:
Bond Price = (Coupon Payment / (1 + YTM/2)^1) + (Coupon Payment + Face Value) / (1 + YTM/2)^2
Rearranging the formula to solve for the Coupon Payment:
Bond Price - (Coupon Payment + Face Value) / (1 + YTM/2)^2 = Coupon Payment / (1 + YTM/2)^1
Bond Price * (1 + YTM/2)^2 - (Coupon Payment + Face Value) = Coupon Payment
Now, let's substitute the given values into the equation:
$9,543.45 * (1 + (6.8%/2))^2 - (Coupon Payment + $10,000) = Coupon Payment
Simplifying the equation:
$9,543.45 * (1 + 0.034)^2 - Coupon Payment - $10,000 = Coupon Payment
$9,543.45 * (1.034)^2 - Coupon Payment - $10,000 = Coupon Payment
$9,543.45 * (1.034)^2 - $10,000 = 2 * Coupon Payment
$9,543.45 * 1.0682128 - $10,000 = 2 * Coupon Payment
$10,193.75 - $10,000 = 2 * Coupon Payment
$193.75 = 2 * Coupon Payment
Now let's solve for the coupon payment by dividing both sides by 2:
Coupon Payment = $193.75 / 2
Coupon Payment = $96.875
Finally, to calculate the coupon rate, divide the annual coupon payment by the face value of the bond:
Coupon Rate = (Coupon Payment / Face Value) * 100
Coupon Rate = ($96.875 / $10,000) * 100
Coupon Rate ≈ 0.96875%
Therefore, the coupon rate of the bond is approximately 0.96875%.