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What is Bond Duration ?


Bond Duration: It is expressed in number of years. It tells the time an investment in a debt security will take to repay its value.  There is an inverse relationship between bond price and interest rate. It means when the interest rate increases then there will be the decrease in bond price. There are two types of duration Macaulay duration and modified duration. Macaulay Duration is a weighted average maturity period of all cash flows of a bond.

Formula:
D = Ʃ t t=1 PV CF * t / Ʃ t t=1 PV CF
Where,
D = Duration
PV CF = Present Value of all cash flows
T = Time period

Factors that affects the duration of a bond:
Maturity: There is a direct relationship between maturity and duration.  If the maturity of a bond increases then there is also increase in duration of bond.
Coupon rate: There is an inverse relationship between coupon rate and duration. It means if the coupon rate increases then there will be increase in duration. The duration of zero coupon bonds is same as its maturity.
Time remaining for maturity: There is direct relation between time remaining for maturity and duration. As shorter time remaining for maturity then there will be shorter duration of a bond.
Yield to maturity: The other inverse relationship of duration is with yield to maturity. The higher yield to maturity the lower will be the duration.
Frequency of coupon payments: The one more factor that affects the duration of a bond is frequency of coupon payments. It means interest paid annually, semi-annually or quarterly can also help to determine the duration. There is an inverse relationship between frequency of coupon payments and duration. If coupon payments are made quarterly then the duration will be decrease.

Example: Find out the duration of a bond if investor invests on 5 year bond whose face value and market price of bond is Rs.10, 000 and Rs.8, 900 respectively. The coupon rate is 6% annually. The required return is 5.8%.

Solution: 
Years
Cash flows
Discount factor @ 5.8%
Present value of cash flow
Present value of cash flow * time
1
600
1.058
567.107
567.107
2
600
1.119
536.193
1, 072.386
3
600
1.184
506.756
1, 520.268
4
600
1.252
479.233
1, 916.932
5
10, 600
1.325
8, 000
40, 000
Total


10, 089.289
45, 076

D = Ʃ t t=1 PV CF * t / PV CF
= 45, 076 / 10, 089
= 4.467 years

Modified Duration: It is an extension of Macaulay Duration. It measures the percentage changes in bond price with respect to percentage changes in interest rate. It exposes the interest rate risk. It is calculate with the help of Macaulay duration.

Formula:
MD = Mac. Duration / 1 + (YTM /n)
Where,
MD = Modified duration
YTM = Yield to maturity
N = number of coupon payment in a year

Example: A bond whose face value is Rs. 10, 000 and interest paid in quarterly. The required return rate is rate is 9% and Macaulay duration is 8 years. Find out the modified duration.

Solution:  MD = Mac. Duration / 1 + (YTM /n)
= 8 / 1+ (0.09 / 4)
= 8 / 1+ 0.0225
= 8 / 1.0225
= 7.82 years.

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