Question

Consider a 2-year bond with a principal of $100 that provides coupons at the rate of 3.8% per annum semiannually. Suppose the yield on this bond is 6.1% per annum with continuous compounding.

(a) What is the duration of this bond?

(b) Suppose the yield on this bond increases by 0.1%.

i. Calculate the new bond price exactly.

ii. Estimate the new bond price approximately using duration.

Answer 1

a

K = Nx2

Bond Price =∑ [( Coupon)/(1 + YTM/2)^k]     +   Par value/(1 + YTM/2)^Nx2

k=1

K =2x2

Bond Price =∑ [(3.8*100/200)/(1 + 6.1/200)^k]     +   100/(1 + 6.1/200)^2x2

k=1

Bond Price = 95.73

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($95.73)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	1.90	1.03	1.84	1.84
2	1.90	1.06	1.79	3.58
3	1.90	1.09	1.74	5.21
4	101.90	1.13	90.36	361.44
			Total	372.08

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

=372.08/(95.73*2)

=1.943361

Modified duration = Macaulay duration/(1+YTM)

=1.94/(1+0.061)

=1.885843

b

i

Actual bond price change

K = Nx2

Bond Price =∑ [( Coupon)/(1 + YTM/2)^k]     +   Par value/(1 + YTM/2)^Nx2

k=1

K =2x2

Bond Price =∑ [(3.8*100/200)/(1 + 6.2/200)^k]     +   100/(1 + 6.2/200)^2x2

k=1

Bond Price = 95.55

ii

Using only modified duration

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

=-1.89*0.001*95.73

=-0.18

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

=-0.18/95.73

=-0.19%

New bond price = bond price+Modified duration prediction

=95.73-0.18

=95.55