Question

Suppose there is a Treasury bond with exactly 15 years left to maturity, a 5% coupon, par of 100, and a 2.6% yield-to-maturity? (Recall that Treasury notes pay semi-annual coupons.) What is the modified duration of this bond?

A 3.41

B 7.99

C 11.17

D 14.23

Answer 1

K = Nx2

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

k=1

K =15x2

Bond Price =∑ [(5*100/200)/(1 + 2.6/200)^k]     +   100/(1 + 2.6/200)^15x2

k=1

Bond Price = 129.65

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($129.65)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	2.50	1.01	2.47	2.47
2	2.50	1.03	2.44	4.87
3	2.50	1.04	2.40	7.21
4	2.50	1.05	2.37	9.50
5	2.50	1.07	2.34	11.72
6	2.50	1.08	2.31	13.88
7	2.50	1.09	2.28	15.99
8	2.50	1.11	2.25	18.04
9	2.50	1.12	2.23	20.03
10	2.50	1.14	2.20	21.97
11	2.50	1.15	2.17	23.86
12	2.50	1.17	2.14	25.69
13	2.50	1.18	2.11	27.48
14	2.50	1.20	2.09	29.21
15	2.50	1.21	2.06	30.90
16	2.50	1.23	2.03	32.53
17	2.50	1.25	2.01	34.12
18	2.50	1.26	1.98	35.67
19	2.50	1.28	1.96	37.16
20	2.50	1.29	1.93	38.62
21	2.50	1.31	1.91	40.03
22	2.50	1.33	1.88	41.40
23	2.50	1.35	1.86	42.72
24	2.50	1.36	1.83	44.01
25	2.50	1.38	1.81	45.25
26	2.50	1.40	1.79	46.46
27	2.50	1.42	1.76	47.63
28	2.50	1.44	1.74	48.76
29	2.50	1.45	1.72	49.85
30	102.50	1.47	69.57	2,087.19
			Total	2,934.19

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

=2934.19/(129.65*2)

=11.315826

Modified duration = Macaulay duration/(1+YTM)

=11.32/(1+0.026)

=11.17