Question

There is a par 30 year, 6% bond. What is the % alteration, if the ytm alters by 2%? Do it with duration alone. Then, find the change, if you include convexity also. The convexity is 40.

Please show formula and show all work.

Answer 1

Duration

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	60.00	1.06	56.60	56.60
2	60.00	1.12	53.40	106.80
3	60.00	1.19	50.38	151.13
4	60.00	1.26	47.53	190.10
5	60.00	1.34	44.84	224.18
6	60.00	1.42	42.30	253.79
7	60.00	1.50	39.90	279.32
8	60.00	1.59	37.64	301.16
9	60.00	1.69	35.51	319.63
10	60.00	1.79	33.50	335.04
11	60.00	1.90	31.61	347.68
12	60.00	2.01	29.82	357.82
13	60.00	2.13	28.13	365.69
14	60.00	2.26	26.54	371.53
15	60.00	2.40	25.04	375.54
16	60.00	2.54	23.62	377.90
17	60.00	2.69	22.28	378.79
18	60.00	2.85	21.02	378.37
19	60.00	3.03	19.83	376.78
20	60.00	3.21	18.71	374.17
21	60.00	3.40	17.65	370.64
22	60.00	3.60	16.65	366.31
23	60.00	3.82	15.71	361.28
24	60.00	4.05	14.82	355.65
25	60.00	4.29	13.98	349.50
26	60.00	4.55	13.19	342.90
27	60.00	4.82	12.44	335.94
28	60.00	5.11	11.74	328.66
29	60.00	5.42	11.07	321.13
30	1,060.00	5.74	184.56	5,536.70
			Total	14,590.72

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

=14590.72/(1000*1)

=14.590721

Modified duration = Macaulay duration/(1+YTM)

=14.59/(1+0.06)

=13.764831

Using only modified duration

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

=-13.76*0.02*1000

=-275.3

%age change in bond price=Mod.duration prediction/bond price

=-275.3/1000

=-27.53%

New bond price = bond price+Modified duration prediction

=1000-275.3

=724.7

Using convexity adjustment to modified duration

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

0.5*40*0.02^2*1000

=8

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

=(-275.3+8)/1000

=-26.73%

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

=1000-275.3+8

=732.7