Question

A bond has a 6% coupon with a face value of 100. Calculate its price for a yield-to-maturity of 5%, for each of the following maturities: a) 10 years b) 5 years c) 0 years Use these results to answer the following questions: Suppose the bond is sold today with ten years to maturity at a y-t-m of 5%. Assuming its y-t-m is unchanged, what would its price be five years from today? What about five years after that?

Answer 1

a)

Formula for Bond Price = C x [1-{1/ (1+r)ⁿ}/r ] +M/(1+r)ⁿ

M = Face Value = $100

C = Coupon amount = (Face Value x Coupon rate) / No. of coupon payments annually

= ($ 100 x 6 %)/1 = $ 6

r = Rate of interest = 5 % or 0.05 p.a.

n = No of periods to maturity = 10

[As no. of coupon payments in a year is not mentioned, let it be 1]

Bond Price = $ 6 x [1-{1/ (1+0.05)¹⁰}/0.05 ] + $ 100/ (1+0.05)¹⁰

                    = $ 6 x [1-{1/ (1.05)¹⁰}/0.05 ] + $ 100/ (1.05)¹⁰

                = $ 6 x [1-{1/ 1.628895}/0.05] + $ 100/1.628895

                = $ 6 x [(1- 0.613913)/0.05] + $ 61.39133

                = $ 6 x [0.386087/0.05] + $ 61.39133

                = $ 6 x 7.721735 + $ 61.39133

               = $ 46.33041 + $ 61.39133

               = $ 107.72

b)

If n = 5 years

Bond Price = $ 6 x [1-{1/ (1+0.05)⁵}/0.05 ] + $ 100/ (1+0.05)⁵

                    = $ 6 x [1-{1/ (1.05)⁵}/0.05 ] + $ 100/ (1.05)⁵

                = $ 6 x [1-{1/ 1.276281563}/0.05] + $ 100/ 1.276281563

                = $ 6 x [(1- 0.783526166)/0.05] + $ 78.35261665

                = $ 6 x [0.216473834/0.05] + $ 78.35261665

                = $ 6 x 4.329476671 + $ 78.35261665

               = $ 25.97686002 + $ 61.39133

               = $ 104.33

c)

If n = 0

Bond Price = $ 6 x [1-{1/ (1+0.05)⁰}/0.05 ] + $ 100/ (1+0.05)⁰

                    = $ 6 x [1-(1/1)/0.05] + $ 100/1

                   = $ 6 x [(1 – 1)/0.05] + $ 100

                  = $ 6 x 0 + $ 100 = $ 100

Price of bond will be $ 107.72 if sold today with 10 years to maturity at YTM of 5 %.

Price of bond will be $ 104.33 if sold today with 5 years from today.

Price of bond will be $ 100 if sold 5 years after that.