Suppose a five-year, $1000 bond with annual coupons has a price of $900 and a yield to maturity of 6%. What is the bond’s coupon rate?

Step 1: Understand the problem

We are given a bond with a face value of $1000, a price of$900, a yield to maturity of 6%, and a maturity of 5 years. We are asked to find the coupon rate.

Step 2: Use the formula for the price of a bond

The price of a bond is calculated as the present value of its future cash flows, which include the annual coupon payments and the face value at maturity. The formula for the price of a bond is:

P = C * (1 - (1 + r)^-n) / r + FV / (1 + r)^n

where: P = price of the bond C = annual coupon payment r = yield to maturity n = number of years to maturity FV = face value of the bond

Step 3: Plug in the given values and solve for C

900 = C * (1 - (1 + 0.06)^-5) / 0.06 + 1000 / (1 + 0.06)^5

Solving this equation for C gives us the annual coupon payment.

Step 4: Calculate the coupon rate

The coupon rate is the annual coupon payment divided by the face value of the bond. So, once we have C, we can find the coupon rate by dividing C by 1000.

Let's solve the equation from step 3 to find C:

900 = C * (1 - (1 + 0.06)^-5) / 0.06 + 1000 / (1 + 0.06)^5

Rearranging the equation to isolate C gives:

C = (900 - 1000 / (1 + 0.06)^5) * 0.06 / (1 - (1 + 0.06)^-5)

Solving this equation gives C ≈ $58.35

Step 5: Calculate the coupon rate

The coupon rate is C / 1000 = $58.35 /$1000 = 0.05835, or 5.835%.