Question

A bond with a coupon rate of 2% just paid its semi-annual coupon. It has two more coupon payments to make and will expire in exactly one year. If it is selling at $971.09 currently, what is its yield to maturity? (Hint: the roots of the quadratic equation ax2 + bx + c = 0 are −b ± √ b 2 − 4ac 2a .)

Please show work!

Answer 1

Let annual yield to maturity be r
Hence,
971.09=2%*1000/2*1/(1+r/2)+(1000+2%*1000/2)*1/(1+r/2)^2

Let 1/(1+r/2) be x
Hence,
971.09=10x+1010x^2
=>1010x^2+10x-971.09=0

the roots of the quadratic equation ax2 + bx + c = 0 are −b ± √ b 2 − 4ac 2a

roots=(-10+sqrt(10^2-4*1010*(-971.09)))/(2*1010) and (-10-sqrt(10^2-4*1010*(-971.09)))/(2*1010)

roots=0.975610444, -0.985511434

as r or ytm is positive, we should use positive root

Hence,
1/(1+r/2)=0.975610444
=>r=(1/0.975610444-1)*2
=>r=5.00%