Example: Find out
the convexity of a 6 year bond whose face value is Rs. 10, 000. It bears 6%
coupon rate p.a. The current market price of a bond is Rs. 9, 880. If the yield
rate increases from 6% to 8% then the price will be Rs. 9, 860. And if the
yield rate decreases from 6% to 4% then the price will increases up to Rs. 10,
100.
Solution: P (interest
decreases) + P (interest increases) – 2P0 / 2* P0 * ∆y 2
= 10,100 + 9, 860 – 2*9, 880 / 2*9, 880*(0.02) 2
= 19, 960 – 19, 760 / 19, 760* 0.0004
= 200/ 7.904
= 25.30
Example: Find out
which of these bonds are less risky with the help of convexity of bond.
Both bond A and B has 3 years of maturity and
bears 7% coupon rate compounded annually. The face value of both the bond is
Rs. 10, 000. The yield rate of Bond A is 8.5% and bond B is 10% compounded
annually.
Solution: Bond A:
Period
|
Cash flow
(Coupon rate 7%)
|
Discounted factor @8.2%
|
Present value
|
Pv*t
|
Convexity (Pv*t(t+1))
|
1
|
700
|
0.9242
|
646.94
|
646.94
|
1, 293.88
|
2
|
700
|
0.8541
|
597.87
|
1, 195.74
|
3, 587.22
|
3
|
700
|
0.7894
|
552.58
|
1, 657.74
|
6, 630.96
|
4
|
700
|
0.7296
|
510.72
|
2, 042.88
|
10, 214.4
|
5
|
10, 700
|
0.6743
|
7, 215.01
|
36, 075.05
|
2, 16, 450.3
|
Total
|
9, 523.12
|
41, 618.35
|
2, 38, 176.76
|
Macaulay Duration =
41, 618.35 / 9523.12
= 4.37 years
Convexity = PV*t(t+t)
/ Pv*(1+y) 2
= 2, 38, 176.76 / 9, 523.12* (1.082)2
= 2, 38, 176.76 / 11, 148.71
= 25.01
Bond B:
Period
|
Cash flow
(Coupon rate 7%)
|
Discounted factor @10%
|
Present value
|
Pv*t
|
Convexity (Pv*t(t+1))
|
1
|
700
|
0.9090
|
636.3
|
636.3
|
1, 272.6
|
2
|
700
|
0.8264
|
578.48
|
1, 156.96
|
3, 470.88
|
3
|
700
|
0.7513
|
525.91
|
1, 577.73
|
6, 310.92
|
4
|
700
|
0.6830
|
478.1
|
1, 912.4
|
9, 562
|
5
|
10, 700
|
0.6209
|
6, 643.63
|
33, 218.15
|
1, 99,308.9
|
Total
|
8, 862.42
|
38, 501.54
|
2, 19, 925.3
|
Macaulay Duration =
38, 501.54 / 8, 862.42
= 4.34 years
Convexity = PV*t(t+t)
/ Pv*(1+y) 2
= 2, 19, 925.3 / 8, 862.42*(1.1) 2
= 20.50
Bond B has lower convexity than bond A so, bond B is best
for investment.
Example: Calculate
the convexity and duration of a bond if the bond bears 6% semi-annual coupon
rate for 4 years. The yield rate of a bond is 7.5% compounded semi-annually. The
face value of bond is Rs. 10, 000.
Solution:
Period
|
Cash flow
(Coupon rate 3%)
|
Discounted factor @3.75%
|
Present value
|
Pv*t
|
Convexity
(Pv*t(t+0.5))
|
||
0.5
|
300
|
0.9638
|
289.14
|
144.57
|
144.57
|
||
1.0
|
300
|
0.9290
|
278.7
|
278.7
|
418.05
|
||
1.5
|
300
|
0.8954
|
268.62
|
402.93
|
805.86
|
||
2.0
|
300
|
0.8631
|
258.93
|
517.86
|
1, 294.65
|
||
2.5
|
300
|
0.8319
|
249.57
|
623.925
|
1, 871.775
|
||
3.0
|
300
|
0.8018
|
240.54
|
721.62
|
2, 525.67
|
||
3.5
|
300
|
0.7728
|
231.84
|
811.44
|
3, 245.76
|
||
4.0
|
10, 300
|
0.7449
|
7, 672.47
|
30, 689.88
|
1, 38, 104.46
|
||
Total
|
9, 489.81
|
34, 190.925
|
1, 48,
410.795
|
||||
Macaulay Duration = 34,
190.925 / 9, 489.81
= 3.60 years
Modified duration =
3.60 / (1 + 0.0375)
= 3.46 years
Convexity = PV*t(t+t)
/ Pv*(1+y) 2
= 1, 48, 410.795 / 9, 489.81*(1+0.0375) 2
= 1, 48, 410.795 / 10, 214.89
= 14.528
Example: Calculate
the percentage change in bond price if the yield decreases by 3%. The coupon
rate of a bond is 8% compounded quarterly. The present value of a bond is Rs.
10,200. The modified duration of a bond is 4.36 years.
Solution: ∆P /P= D*∆y
= 4.36*0.03
= 13.08% increase in
price of a bond
Example: If the
yield rate is increases by 4%.The current yield rate is 5% p.a. The current
market price of a bond is Rs. 8, 870. Find out the change in price of a bond if
the modified duration is 2.64 years and convexity is 5.48.
Solution: ∆P = -D*P*∆y
+ 0.5*C*P*∆y 2
= -2.64*8,
870*0.04 + 0.5*5.48* 8, 870*(0.04) 2
= -936.672 +
38.88608
=Rs.-897.785 fall in
price of a bond.
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