Question

Consider Bond ABC. It’s a bit of an odd bond that has a step-up clause in it. Specifically, the bond has a CR of 5% for the next five years, but then it increases to 6% for the remaining 10 years of the bond. The YTM on the bond is 5.4%, and payments are made semi-annually. The face value of the bond is $1,000 and payments are made semiannually.  

a.   What is the Modified Duration of this bond?       

b.   What is the Convexity of the Bond?        

c.   What is the predicted price of the bond according to both Duration and Convexity, if the YTM increases by 1%?

Answer 1

We need to formulate the table as shown below. Payment frequency is semi annual. Hence, number of periods = nos. of half years in 15 years = 2 x 15 = 30

Cash flow per period over first 5 years i.e 10 periods = Semi annual coupon = 5%/2 = 2.5% x 1,000 = 25

Cash flow per period over next 10 years i.e 20 periods = Semi annual coupon = 6%/2 = 3% x 1,000 = 30

Cash flow in the last period = semi annual coupon + Face value repayment = 30 + 1,000 = 1,030

Yield = 5.4% per annum

Yield per period = Semi annual yield = y = 5.4% / 2 = 2.7%

Discount factor = DF = (1 + y)^-n

Please see the second row of the table below. That will help you understand the mathematics.

Period	Cash flows	Discount factor	PVn of Cn	n x PVn of Cn	(n2 + n) x PV of Cn
n	Cn	Dn = (1 + 5.4%/2)-n	PVn = Dn x Cn
1	25	0.973709834	24.34	24.34	48.69
2	25	0.948110842	23.70	47.41	142.22
3	25	0.923184851	23.08	69.24	276.96
4	25	0.898914168	22.47	89.89	449.46
5	25	0.875281566	21.88	109.41	656.46
6	25	0.852270269	21.31	127.84	894.88
7	25	0.829863942	20.75	145.23	1,161.81
8	25	0.808046682	20.20	161.61	1,454.48
9	25	0.786803001	19.67	177.03	1,770.31
10	25	0.76611782	19.15	191.53	2,106.82
11	30	0.745976455	22.38	246.17	2,954.07
12	30	0.726364611	21.79	261.49	3,399.39
13	30	0.707268365	21.22	275.83	3,861.69
14	30	0.688674163	20.66	289.24	4,338.65
15	30	0.670568805	20.12	301.76	4,828.10
16	30	0.65293944	19.59	313.41	5,327.99
17	30	0.635773554	19.07	324.24	5,836.40
18	30	0.619058962	18.57	334.29	6,351.54
19	30	0.6027838	18.08	343.59	6,871.74
20	30	0.586936514	17.61	352.16	7,395.40
21	30	0.571505856	17.15	360.05	7,921.07
22	30	0.556480872	16.69	367.28	8,447.38
23	30	0.541850898	16.26	373.88	8,973.05
24	30	0.527605548	15.83	379.88	9,496.90
25	30	0.513734711	15.41	385.30	10,017.83
26	30	0.50022854	15.01	390.18	10,534.81
27	30	0.487077449	14.61	394.53	11,046.92
28	30	0.474272102	14.23	398.39	11,553.27
29	30	0.46180341	13.85	401.77	12,053.07
30	1030	0.449662522	463.15	13,894.57	430,731.73
Total			Price, P = 1,017.84	21,531.54	580,903.06

Part (a)

Duration = Sum of [n x PV_n of C_n] / Sum of [PV_n of C_n] = 21,531.54 / 1,017.84 = 21.15 period = 21.15 /2 years =10.58 years

Modified duration, MD = Duration / (1 + y) = 10.58 / (1 + 2.7%) = 10.30

Part (b)

Convexity, C = [1 / (4 x (1 + y)²] x Sum of [(n² + n) x PV of C_n] / Sum of [PV_n of C_n]

= [1 / 4 x (1 + 2.7%)²] x 580,903.06 / 1,017.84 = 135.28

Please note the factor of 1/4 is because we are converting the semi annual payment driven number to annual convexity.

Part (c)

%age change in price of the bond predicted by the duration and convexity rule = - MD x %age change in yield + ½ x C x (%age change in yield)²

Hence, if YTM increases by 1%, %age change in price of the bond predicted by the duration and convexity rule = - MD x %age change in yield + ½ x C x (%age change in yield)² = - 10.30 x (+ 1%) + ½ x 135.28 x (+ 1%)² = - 9.62%

Hence, the predicted price = Old Price x (1 - 9.62%) = 1,017.84 x (1 - 9.62%) = 919.89