Question

2-assume you have a one year investment horizon and purchase a semiannual coupon bond today that pays 9% coupon anually, had a bar of 1000 matures in 20 years and 10% ytm. If you owned the bond for exactly one year( exactly 19 of maturity left ) and the bond is currently yelding 8% to maturity . What is the rate of return?

Answer 1

coupon rate = 9%

par value, M = $1000

coupon value, C = coupon rate*M = 0.09*1000 = 90

YTM when bond was purchaed = 10% = 0.10

maturity of bond, n = 20 years

no. of semi-annual periods, t = n*2 = 20*2 = 40

semi-annual YTM = 10/2 = 5% = 0.05

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = (C/2)*PVIFA( 5%% , 40)

PVIFA( 5% , 40) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.05)⁴⁰ - 1)/((1.05)⁴⁰*0.05)] = 17.15908635

Present value(PV) of coupons = (C/2)*PVIFA( 5% , 40) = 45*17.15908635 = 772.15888593

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.05)⁴⁰ = 142.04568230

Price of bond when (semi-annual) YTM is 5%, P0 = 772.15888593 + 142.04568230 = $914.20456823

when YTM next year = 8%

semi-annual YTM = 4% = 0.04

years to maturity left = 19

no. of semi-annual periods = 19*2 = 38

price of bond = present value of coupons + present value of maturity amount

Present value of coupons = (C/2)*PVIFA( 4% , 38)

PVIFA( 4% , 38) = present value interest rate factor of annuity

= [((1+YTM)ⁿ - 1)/((1+YTM)ⁿ*YTM)] = [((1.04)³⁸ - 1)/((1.04)³⁸*0.04)] = 19.36786423

Present value(PV) of coupons = (C/2)*PVIFA( 4% , 38) = 45*19.36786423 = 871.55389045

PV of maturity amount = par value/(1+YTM)ⁿ = 1000/(1.04)³⁸ = 225.28543071

Price of bond when (semi-annual) YTM is 4% , P1= 871.55389045 + 225.28543071 = $1096.83932116

value of investment at the end of 1 year(V1) = value of semi-annual coupon at the end of 1 year + value of price after 1 year

value of semi-annual coupon at the end of 1 year = semi-annual coupon*FVIFA

r = semi-annual YTM = 10/2 = 5% = 0.05

n = no. of semi-annual periods in 1 year = 2

FVIFA = future value interest rate factor of annuity = [((1+r)ⁿ-1)/r] = [((1.05)²-1)/0.05] = 2.05

value of semi-annual coupon at the end of 1 year = semi-annual coupon*FVIFA = 45*2.05 = 92.25

= 92.25+ 1096.83932116 = 1189.089321

semi-annual return = (V1/P0)^(1/2) - 1 = (1189.089321/914.20456823)^(1/2) - 1 = 0.140474= 14.0474% or 14.05% ( rounding off to 2 decimal places)

Annual return = semi-annual return*2 = 14.0474*2 = 28.0949% or 28.09%