Question

Bond X has exactly three years until it matures. Bond X is NOT a standard bond. It pays a yearly coupon of $120 every year. The yield-to-maturity (YTM) on similar debt instruments is 9.2 percent per year.

a) If yields on such instruments should fall to 9.0 percent per year, estimate the new value of Bond X using concepts of duration.

b) Calculate the "convexity correction" for this case.

Answer 1

a)

K = N

Bond Price =∑ [( Coupon)/(1 + YTM)^k]     +   Par value/(1 + YTM)^N

k=1

K =3

Bond Price =∑ [(12*1000/100)/(1 + 9.2/100)^k]     +   1000/(1 + 9.2/100)^3

k=1

Bond Price = 1070.62

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($1,070.62)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period	=duration calc*(1+period)/(1+YTM/N)^2
1	120.00	1.09	109.89	109.89	184.31
2	120.00	1.19	100.63	201.26	506.34
3	1,120.00	1.30	860.10	2,580.31	8,655.38
			Total	2,891.46	9,346.03

Macaulay duration =(∑ Duration calc)/(bond price*number of coupon per year)

=2891.46/(1070.62*1)

=2.700735

Modified duration = Macaulay duration/(1+YTM)

=2.7/(1+0.092)

=2.473201

Using only modified duration

Mod.duration prediction = -Mod. Duration*Yield_Change*Bond_Price

=-2.47*-0.002*1070.62

=5.3

%age change in bond price=Mod.duration prediction/bond price

=5.3/1070.62

=0.49%

New bond price = bond price+Modified duration prediction

=1070.62+5.3

=1075.92

b

Convexity =(∑ convexity calc)/(bond price*number of coupon per year^2)

=9346.03/(1070.62*1^2)

=8.73

Using convexity adjustment to modified duration

Convexity adjustment = 0.5*convexity*Yield_Change^2*Bond_Price

0.5*8.73*-0.002^2*1070.62

=0.02