Question

I. 10-year zero coupon government bond, par value $1000, current price = $613.91

What is the convexity of Bond I? If Bond I’s yield increases by 1%, what is the price of Bond I based on duration-with-convexity rule?

Answer 1

K = N

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

k=1

K =10

613.91 =∑ [(0*1000/100)/(1 + YTM/100)^k]     +   1000/(1 + YTM/100)^10

k=1

YTM% = 5

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($613.91)	=(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	-	1.05	-	-	-
2	-	1.10	-	-	-
3	-	1.16	-	-	-
4	-	1.22	-	-	-
5	-	1.28	-	-	-
6	-	1.34	-	-	-
7	-	1.41	-	-	-
8	-	1.48	-	-	-
9	-	1.55	-	-	-
10	1,000.00	1.63	613.91	6,139.13	61,252.12
			Total	6,139.13	61,252.12

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

=6139.13/(613.91*1)

=10.000053

Modified duration = Macaulay duration/(1+YTM)

=10/(1+0.05)

=9.52386

Using only modified duration

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

=-9.52*0.01*613.91

=-58.47

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

=61252.12/(613.91*1^2)

=99.77

Using convexity adjustment to modified duration

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

0.5*99.77*0.01^2*613.91

=3.06

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

=613.91-58.47+3.06

=558.5