Question

A three year bond with face value of $1000 pays annual coupons of 4 percent and has a yield- to-maturity of 5 percent. What is the price, duration, and convexity of the bond?
Suppose the yield increases to 6 percent. Use the duration rule to estimate the new price. Use duration and convexity to estimate the new price. Use the bond price equation to compute the exact new price.

Answer 1

K = N

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

k=1

K =3

Bond Price =∑ [(4*1000/100)/(1 + 5/100)^k]     +   1000/(1 + 5/100)^3

k=1

Bond Price = 972.77

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($972.77)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	40.00	1.05	38.10	38.10
2	40.00	1.10	36.28	72.56
3	1,040.00	1.16	898.39	2,695.17
			Total	2,805.83

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

=2805.83/(972.77*1)

=2.884372

Modified duration = Macaulay duration/(1+YTM)

=2.88/(1+0.05)

=2.747021

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($972.77)	=(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	40.00	1.05	38.10	38.10	69.11
2	40.00	1.10	36.28	72.56	197.45
3	1,040.00	1.16	898.39	2,695.17	9,778.41
			Total	2,805.83	10,044.96

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

=10044.96/(972.77*1^2)

=10.33

Using only modified duration

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

=-2.75*0.01*972.77

=-26.72

New bond price = bond price+Modified duration prediction

=972.77-26.72

=946.05

Using convexity adjustment to modified duration

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

0.5*10.33*0.01^2*972.77

=0.5

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

=972.77-26.72+0.5

=946.55

Actual bond price

K = N

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

k=1

K =3

Bond Price =∑ [(4*1000/100)/(1 + 6/100)^k]     +   1000/(1 + 6/100)^3

k=1

Bond Price = 946.54