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How to calculate Macaulay duration & Modified Duration?



Example: Find out the duration of a bond if the interest 9.8% p.a. semi-annually for 4 year bond. The coupon rate is 9% p.a. semi- annually. The face value of bond is Rs. 10, 000.

Solution: The semi annual yield rate = 9.8/200 = 0.049
The semi- annual coupon rate = 9/200 = 0.045
The coupon payment = 0.045*10000 = Rs. 450
Year
Cash flows
Interest rate @ 4.9%
Present value of cash flows (C/ (1+r) n
Present value of cash flows * time
1
450
1.049
428.979
428.979
2
450
1.100
409.090
818.18
3
450
1.154
389.948
1, 169.844
4
450
1.210
371.900
1, 487.6
5
450
1.270
354.330
1, 771.65
6
450
1.332
337.837
2, 027.022
7
450
1.397
322.118
2, 254.826
8
10, 450
1.466
7, 128.240
57, 025.62
Total


9, 742.442
66, 984.021

= 66, 984.021 / 9, 742.442
= 6.875
= 6.875 / 2 = 3.437 years

Example: Calculate the duration of 2 years bond whose face value is Rs. 10, 000. The coupon rate is 7.6% per annum compounded quarterly. The yield rate is 8% per annum compounded quarterly.

Solution:
Year
Cash flows
Interest rate @ 4.9%
Present value of cash flows (C/ (1+r) n
Present value of cash flows * t
1
190
1.02
186.274
186.274
2
190
1.040
182.692
365.384
3
190
1.061
179.076
537.228
4
190
1.082
175.600
702.4
5
190
1.104
172.101
860.505
6
190
1.126
168.738
1, 012.428
7
190
1.148
165.505
1, 158.535
8
10, 190
1.171
8, 701.964
69, 615.712
Total


9, 931.95
74, 438.466
= 74, 438.466 / 9, 931.95
= 7.464
= 7.464 / 4
= 1.873 years

Example: A 7 year bond whose face value is Rs. 10, 000 and bears 9% coupon rate annually. Find out the duration of bond if the yield rate is 9.8%.

Solution:
Year
Cash flows
Discount factor (1/1+r) n @ 4.9%
Present value of cash flows (C*(1/(1+r) n)
Weight (PV of cash flows / Total of PV
Weight*time
1
900
0.910
819
0.085
0.085
2
900
0.829
746.1
0.077
0.154
3
900
0.755
679.5
0.070
0.21
4
900
0.688
619.2
0.064
0.256
5
900
0.626
563.4
0.058
0.29
6
900
0.570
513
0.053
0.318
7
10, 900
0.519
5, 657.1
0.589
4.123
Total


9, 597.3
0.996
5.436 years


Example: A 10 year bond which bears 10% coupon rate compounded half yearly. The face value of the bond is Rs. 10, 000. The yield per annum is 10.8% compounded half yearly. The Macaulay duration is 6.44 years. Find out the modified duration.

Solution: MD = Mac Duration / 1+ (ytm/n)
= 6.44 / 1+ (0.108/2)
= 6.11 years

Example: Mr. Mehta wants to know how much bond price changes if the market rate decreases from 8% to 6%. The modified duration is 6 years and the market price of a bond is Rs. 10, 230.

Solution: There is an inverse relationship between bond price and interest rate. It means if interest rate decreases then the bond price will increase. The modification duration helps to measure the changes in bond price in respect to changes in interest rate.
Modification duration = 6 year
Interest rate decreases by 2%
Then,
= 6*2
 Bond price rises = 12%

Note: We can also calculate the duration of a bond by using the weight of each present value of cash flows. If interest rate compounded semi – annually, quarterly or monthly only time period will change and other things remain the same in calculation. For example interest compounded semi-annually time will be as 0.5, 1, 1.5, 2 etc 

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