Question

A newly issued bond has a maturity of 10 years and pays a 5.5% coupon rate (with coupon payments coming once annually). The bond sells at par value.
a. What are the convexity and the duration of the bond?
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 5.5% to 6.5% (with maturity still 10 years). Assume a par value of 100.
c. What price would be predicted by the modified duration rule?
d. What is the percentage error of that rule? What price would be predicted by the modified duration-with-convexity rule? What is the percentage error of that rule?

Answer 1

a.

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($1,000.00)	=(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	55.00	1.06	52.13	52.13	93.68
2	55.00	1.11	49.41	98.83	266.38
3	55.00	1.17	46.84	140.52	504.99
4	55.00	1.24	44.40	177.59	797.77
5	55.00	1.31	42.08	210.41	1,134.27
6	55.00	1.38	39.89	239.33	1,505.19
7	55.00	1.45	37.81	264.66	1,902.30
8	55.00	1.53	35.84	286.70	2,318.31
9	55.00	1.62	33.97	305.73	2,746.81
10	1,055.00	1.71	617.63	6,176.29	61,040.16
			Total	7,952.20	72,309.85

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

=7952.2/(1000*1)

=7.95

Modified duration = Macaulay duration/(1+YTM)

=7.95/(1+0.055)

=7.54

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

=72309.85/(1000*1^2)

=72.31

b.

Actual bond price change

K = N

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

k=1

K =10

Bond Price =∑ [(5.5*1000/100)/(1 + 6.5/100)^k]     +   1000/(1 + 6.5/100)^10

k=1

Bond Price = 928.11

c.

Using only modified duration

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

=-7.54*0.01*1000

=-75.38

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

=-75.38/1000

=-7.54%

New bond price = bond price+Modified duration prediction

=1000-75.38

=924.62

d.

Difference in price predicted and actual

=predicted price-actual price

=924.62-928.11

=-3.49

%age difference = difference/actual-1

=-3.49/928.11

=-0.3756%

Using convexity adjustment to modified duration

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

0.5*72.31*0.01^2*1000

=3.62

%age change in bond price=(Mod.duration pred.+convex. Adj.)/bond price

=(-75.38+3.62)/1000

=-7.18%

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

=1000-75.38+3.62

=928.24

Difference in price predicted and actual

=predicted price-actual price

=928.24-928.11

=0.129

%age difference = difference/actual-1

=0.13/928.11

=0.0139%