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How to calculate Convexity Duration with example?


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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