Question

1. Consider a bond that has a 30-year maturity, an 8% coupon rate, and sells at an initial yield to maturity of 8%. Because the coupon rate equals the yield to maturity, the bond sells at par value: P = $1,000.00. Calculate the duration and the modified duration. If we assume the convexity of the bond is 212.4 and the bond’s yield increases from 8% to 10%, how much should the bond price decline?

Answer 1

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($1,000.00)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	80.00	1.08	74.07	74.07
2	80.00	1.17	68.59	137.17
3	80.00	1.26	63.51	190.52
4	80.00	1.36	58.80	235.21
5	80.00	1.47	54.45	272.23
6	80.00	1.59	50.41	302.48
7	80.00	1.71	46.68	326.75
8	80.00	1.85	43.22	345.77
9	80.00	2.00	40.02	360.18
10	80.00	2.16	37.06	370.55
11	80.00	2.33	34.31	377.42
12	80.00	2.52	31.77	381.23
13	80.00	2.72	29.42	382.41
14	80.00	2.94	27.24	381.32
15	80.00	3.17	25.22	378.29
16	80.00	3.43	23.35	373.62
17	80.00	3.70	21.62	367.57
18	80.00	4.00	20.02	360.36
19	80.00	4.32	18.54	352.20
20	80.00	4.66	17.16	343.28
21	80.00	5.03	15.89	333.74
22	80.00	5.44	14.72	323.74
23	80.00	5.87	13.63	313.38
24	80.00	6.34	12.62	302.78
25	80.00	6.85	11.68	292.04
26	80.00	7.40	10.82	281.22
27	80.00	7.99	10.01	270.40
28	80.00	8.63	9.27	259.65
29	80.00	9.32	8.59	249.00
30	1,080.00	10.06	107.33	3,219.83
			Total	12,158.41

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

=12158.41/(1000*1)

=12.158406

Modified duration = Macaulay duration/(1+YTM)

=12.16/(1+0.08)

=11.257783

New bond price @ YTM =10 using duration

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

=-11.2577*0.02*1000

=-225.154

New bond price = bond price+Modified duration prediction

=1000+-225.154

=774.85

New bond price @ YTM =10 using duration and convexity

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

=0.5*212.4*0.02^2*1000

=42.48

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

=1000+-225.15+21.24

=817.33

Price declined by 1000-817.33=182.67