Question

A two-year coupon-paying bond has a face value of $100 000, yield of 7.5% p.a. and coupon rate of 7.5% p.a. The interest rates are paid half yearly.

a) Calculate the price of the bond

b) Calculate the duration of a bond

c) Calculate the convexity of the bond.

d) The yield on the bond instantaneously increases from 7.5% to 7.7%.

*Please write down the formula instead of Excel format

Answer 1

a

K = Nx2

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

k=1

K =2x2

Bond Price =∑ [(7.5*100000/200)/(1 + 7.5/200)^k]     +   100000/(1 + 7.5/200)^2x2

k=1

Bond Price = 100000

b

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($100,000.00)	=(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	3,750.00	1.04	3,614.46	3,614.46	6,715.79
2	3,750.00	1.08	3,483.81	6,967.63	19,419.14
3	3,750.00	1.12	3,357.89	10,073.68	37,434.50
4	103,750.00	1.16	89,543.83	358,175.33	1,663,755.36
			Total	378,831.10	1,727,324.79

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

=378831.1/(100000*2)

=1.894156

Modified duration = Macaulay duration/(1+YTM)

=1.89/(1+0.075)

=1.825692

c

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

=1727324.79/(100000*2^2)

=4.32

d

Using only modified duration

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

=-1.83*0.002*100000

=-365.14

Using convexity adjustment to modified duration

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

0.5*4.32*0.002^2*100000

=0.86

New bond price = bond price+Mod.duration pred.+convex. Adj.

=100000-365.14+0.86

=99635.73