Question

A 15-year bond with a coupon of $X payable every 6 months has a face (and redemption) value of $10,000. At the nominal annual interest rate, convertible semi-annually, of 6.5%, the price of the bond is $8,576.36. What is X?

ANSWER: $250

Answer 1

price of bond = [present value of annuity factor * coupon payments] + [present value factor * face value]

here,

price = $8,576.36.

present value of annuity factor = [1 -(1+r)^(-n)] / r

where, r= 6.5% per annum=>3.25% for six months => 0.0325.........(since we have semi annual coupon payments,6 months rate is relevant).

n = 15 years* 2 payments each year=.30 periods

=> present value of annuity factor = [ 1- (1.0325)^(-30)]/0.0325

=>[ 1- 0.38308768]/0.0325

=>0.6169123/0.0325

=>18.9819169

present value factor = 1 /(1+r)^n

=>1/(1.0325)^30

=.>0.38308768

face value =$10,000.

coupon =$x ...(to be found out)

now,

substituting the terms in the above mentioned equation.

$8,576.36 = [18.9819169 * $x] + [0.38308768*$10,000]

=>$8,576.36 = [18.9819169*$x] + $3,830.87

=>$4,745.49 =18.9819169x

=> x = $4,745.49 / 18.9819169

=>x=$250.00.........(rounded to two decimals).

coupon payment every six months is $250.