Question

The Frush Corporation has two different bonds currently outstanding. Bond M has a face value of $30,000 and matures in 20 years. The bond makes no payments for the first six years, then pays $1,900 every six months over the subsequent eight years, and finally pays $2,200 every six months over the last six years. Bond N also has a face value of $30,000 and a maturity of 20 years; it makes no coupon payments over the life of the bond. The required return on both these bonds is 12 percent compounded semiannually. What is the current price of Bond M and Bond N?

Answer 1

Required return on both these bonds is 12 percent compounded semiannually

Hence, dicount rate per period = semi annual discount rate, R = 12% / 2 = 6%

Bond M's stream of payments:

• Stream 1: An annuity A = $ 1,900 every period of six month over next 8 years i.e. over N = 2 x 8 = 16 periods, payment start from period = 2 x 6.5 = 13.
  • Hence PV of stream 1 at the end of 12 periods, PV_{1, 12} = A / R x [1 - (1 + R)^-N] = 1,900 / 6% x [1 - (1 + 6%)^-16] = $ 19,201.20
  • Hence, PV of stream 1 payment at t = 0 i.e. today = PV_{1, 0} = PV_{1, 12} / (1 + R)¹² = 19,201.20 / (1 + 6%)¹² = $ 9,542.41
• Stream 2: An annuity A = $ 2,200 every period of six month over next 6 years i.e. over N = 2 x 6 = 12 periods, payment start from period = 2 x 8.5 = 17.
  • Hence PV of stream 2 at the end of 16 + 12 = 28 periods, PV_{2, 28} = A / R x [1 - (1 + R)^-N] = 2,200 / 6% x [1 - (1 + 6%)^-12] = $18,444.46
  • Hence, PV of stream 2 payment at t = 0 i.e. today = PV_{2, 0} = PV_{2, 28} / (1 + R)²⁸ = 18,444.46 / (1 + 6%)²⁸ = $ 3,608.29

Hence, price of bond M = PV_{1, 0} + PV_{2 ,0} = 9,542.41 + 3,608.29 = $ 13,150.70

Price of bond N = Present value of all the future payment = Present value of face value at the end of 20 years = Face Value / (1 + R)^{(2 x 20)} = 30,000 / (1 + 6%)⁴⁰ = $ 2,916.67