I need help finding a coupon rate

Question

Ashes Divide Corporation has bonds on the market with 19 years to maturity, a YTM of 10.0 percent (APR), and a current price of $1,236.50. The bonds make semiannual payments. What must the coupon rate be on these bonds (as an APR)? Note: first find the semi-annual payment. Then convert it into an annual payment and use this annual payment to find the coupon rate as an APR. (Do not round your intermediate calculations.)

 

25.65%
10.36%
12.80%
20.75%
12.90%

Answer 1

The correct answer is 12.8%

 

You can diagram this problem as follows:

 

PV=1,236.50 - - -? - - -? - - -? - - -   - - -? - - -?=$1,000

 

The question marks represent the semiannual interest payments, and there are 38 of them (2*19). $1,236 is the present value of the annuity made up of interest payments and the maturity value of the bond, discounted to the present at 5%. A financial calculator solves this easily.

 

By discounting the future $1,000 maturity value to the present at 10% semiannually, over 38 periods, you get $156.6053647. Deduct this from the market value of $1,236.50 which gives you the present value of the interest payments, also discounted at 5% over 38 periods. Now you have the PV of the annuity, the discount rate, and the number of periods, and you have to solve for the annuity payment. That turns out to be $64 per period or 128 per year, which is 12.8 percent of the $1,000 face value.