Question

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1. Consider a bond that pays $50 semi-annual interest and has a remaining life of 10 years. The bond currently sells for $987 and has a yield to maturity of 10.20%.
   1. What is the bond’s duration?

2. What is its modified duration?

3. What is its convexity?

4. What percentage change in bond price would be expected for one percent decrease in interest rate?

Answer 1

a

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($987.00)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	50.00	1.05	47.57	47.57
2	50.00	1.10	45.27	90.53
3	50.00	1.16	43.07	129.21
4	50.00	1.22	40.98	163.92
5	50.00	1.28	38.99	194.95
6	50.00	1.35	37.10	222.59
7	50.00	1.42	35.30	247.09
8	50.00	1.49	33.59	268.68
9	50.00	1.56	31.96	287.60
10	50.00	1.64	30.40	304.05
11	50.00	1.73	28.93	318.22
12	50.00	1.82	27.53	330.31
13	50.00	1.91	26.19	340.47
14	50.00	2.01	24.92	348.87
15	50.00	2.11	23.71	355.65
16	50.00	2.22	22.56	360.95
17	50.00	2.33	21.46	364.90
18	50.00	2.45	20.42	367.62
19	50.00	2.57	19.43	369.21
20	1,050.00	2.70	388.27	7,765.42
			Total	12,877.79

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

=12877.79/(987*2)

=6.523704

b

Modified duration = Macaulay duration/(1+YTM)

=6.52/(1+0.102)

=6.20714

c

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($987.00)	=(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	50.00	1.05	47.57	47.57	86.14
2	50.00	1.10	45.27	90.53	245.87
3	50.00	1.16	43.07	129.21	467.88
4	50.00	1.22	40.98	163.92	741.97
5	50.00	1.28	38.99	194.95	1,058.94
6	50.00	1.35	37.10	222.59	1,410.58
7	50.00	1.42	35.30	247.09	1,789.51
8	50.00	1.49	33.59	268.68	2,189.15
9	50.00	1.56	31.96	287.60	2,603.65
10	50.00	1.64	30.40	304.05	3,027.82
11	50.00	1.73	28.93	318.22	3,457.07
12	50.00	1.82	27.53	330.31	3,887.38
13	50.00	1.91	26.19	340.47	4,315.20
14	50.00	2.01	24.92	348.87	4,737.46
15	50.00	2.11	23.71	355.65	5,151.52
16	50.00	2.22	22.56	360.95	5,555.08
17	50.00	2.33	21.46	364.90	5,946.20
18	50.00	2.45	20.42	367.62	6,323.27
19	50.00	2.57	19.43	369.21	6,684.93
20	1,050.00	2.70	388.27	7,765.42	147,631.43
			Total	12,877.79	207,311.04

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

=207311.04/(987*2^2)

=52.51

d

Using only modified duration

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

=-6.21*-0.01*987

=61.26

Using convexity adjustment to modified duration

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

0.5*52.51*-0.01^2*987

=2.59

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

=(61.26+2.59)/987

=6.47%