Question

Suppose a five-year, $1,000 bond with annual coupons has a price of $895.17 and a yield to maturity of 5.6%. What is the bond's coupon rate? The bond's coupon rate is %. (Round to three decimal places.)

Answer 1

The coupon rate is the annual coupon payment divided by the bond's face value. To find it for a $1,000 bond with a price of $895.17 and a yield to maturity of 5.6%, we must solve for the annual coupon payment using the present value of annuity formula and then divide by the face value.

Step-by-step explanation:

To calculate the coupon rate of a five-year, $1,000 bond priced at $895.17 with a yield to maturity of 5.6%, we need to establish the annual coupon payment that makes the present value of the bond equal to its price. This calculation involves finding the sum of the present value of the bond's future cash flows, which are the annual coupon payments and the lump sum repayment of the face value at maturity. We can denote the coupon payment as C and set up the equation like this:

Present Value = C / (1+0.056) + C / (1+0.056)^2 + C / (1+0.056)^3 + C / (1+0.056)^4 + (C+1000) / (1+0.056)^5

Simplifying and solving this equation for C, we would eventually find the annual coupon payment. We would then divide this annual coupon payment by the face value of the bond, which is $1,000, to obtain the coupon rate. This calculation often requires financial calculators or spreadsheet functions as it involves the iterative process of solving for the coupon rate. Once found, the coupon rate is expressed as a percentage with three decimal places, as per the question's requirement.

Answer 2

To find the bond's coupon rate, we can use the formula for yield to maturity (YTM) and solve for the coupon rate. In this case, the YTM is 5.6% and the face value is $1,000. So, the coupon rate would be $56.

Step-by-step explanation:

Calculating the coupon rate for a bond with given price and yield to maturity involves an iterative process that typically requires a financial calculator or software, as it is not straightforward to solve it with a direct formula. The coupon rate determines the periodic coupon payments for the bond, which, along with the face value at maturity, must equal the present value represented by the bond's current price.

To find the bond's coupon rate, we can use the formula for yield to maturity (YTM) and solve for the coupon rate:

Coupon Rate = YTM * Face Value

In this case, the YTM is 5.6% and the face value is $1,000. So, the coupon rate would be:

Coupon Rate = 0.056 * $1,000 = $56