Question

What is the Macaulay duration of a semiannual-pay 7.62 percent coupon bond with 11 years to maturity and a yield to maturity of 6.06 percent?

Solve with Macaulay duration formula

Answer 1

K = Nx2

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

k=1

K =11x2

Bond Price =∑ [(7.62*1000/200)/(1 + 6.06/200)^k]     +   1000/(1 + 6.06/200)^11x2

k=1

Bond Price = 1123.94

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
1	38.10	1.0303	36.98	36.98
2	38.10	1.0615	35.89	71.78
3	38.10	1.0937	34.84	104.51
4	38.10	1.1268	33.81	135.25
5	38.10	1.1610	32.82	164.09
6	38.10	1.1961	31.85	191.11
7	38.10	1.2324	30.92	216.41
8	38.10	1.2697	30.01	240.05
9	38.10	1.3082	29.12	262.12
10	38.10	1.3478	28.27	282.68
11	38.10	1.3887	27.44	301.80
12	38.10	1.4308	26.63	319.55
13	38.10	1.4741	25.85	336.00
14	38.10	1.5188	25.09	351.21
15	38.10	1.5648	24.35	365.23
16	38.10	1.6122	23.63	378.12
17	38.10	1.6611	22.94	389.93
18	38.10	1.7114	22.26	400.73
19	38.10	1.7632	21.61	410.55
20	38.10	1.8167	20.97	419.45
21	38.10	1.8717	20.36	427.47
22	1,038.10	1.9284	538.32	11,842.97
				17,647.98

as an example : for period 2

CF = coupon*par value/(number of coupons per year*100) = (7.62*1000)/(2*100) = 38.1

Discounting factor = (1+YTM/(number of coupon per year))^corresponding period

=(1+0.0606/2)^2

=1.0615

PV cash flow = cash flow/discounting factor = 38.1/1.0615 = 35.89

Duration calc: = PV cash flow*corresponding period = 35.89*2 = 71.78

Note: in last period : cash flow = coupon + par value

= sum of all duration calcduration calc = PV cash flow*n
PV cash flow = cash flow/(1+ytm)^n

V_b= bond price

macaulay duration = sum of all duration calc/bond price /number of coupons per year= 17647.98/1123.94/2=7.85

N CF

Where Macaulay duration = 1+i 1-1 Mac V.