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Bond Duration Analysis for CFP – 1

In the case of bonds with a fixed term to maturity, the tenor of the bond is a simple measure of the time until the bond’s maturity. However, if the bond is coupon paying, the investor receives some cash flows prior to the maturity of the bond. Therefore it may be useful to understand what the ‘average’ maturity of a bond, with intermittent cash flows.

Since the coupons accrue at various points in time, it would be appropriate to use the present value of the cash flows as weights, so that they are comparable. Therefore we can arrive at an alternate measure of the tenor of a bond, accounting for all the intermittent cash flows, by finding out the weighted average maturity of the bond, the present value of cash flows being the weightage used. This technical measure of the tenor of a bond is called duration of the bond.

• Duration is defined as a weighted average of the maturities of the individual payments
• ™It is an important measure for investors to consider, as bonds with higher durations carry more risk and have higher price volatility than bonds with lower durations
• Duration is also a point where the investor faces no interest rate risk

Macaulay Duration
The formula usually used to calculate a bond’s basic duration is the Macaulay duration

 

• n = number of cash flows
• t = time to maturity
• C = cash flow
• r = required yield (YTM)
• M = maturity (par) value

Simplified Equation:

Example:

Consider a 12.5% bond with annual coupons, redeemable after 5 years at a premium of 5%. If the current interest rate is 15%, calculate duration of the bond.

Year	Cash Flow	P/V	Year*P/V
1	12.5	10.87	10.87
2	12.5	9.45	18.90
3	12.5	8.22	24.66
4	12.5	7.15	28.59
5	117.5	58.42	292.09
Total		94.11	375.11

Hence Duration of the bond = 375.11/94.11 = 3.99 Years

Modified Duration

• Modified duration is a modified version of the Macaulay model that accounts for changing interest rates
• Modified formula shows how much the duration changes for each percentage change in yield
• There is an inverse relationship between modified duration and an approximate 1% change in yield

Example:

Calculate the modified duration of a bond with FV Rs.- 100,Coupon Rate – 9%, Term to Maturity – 8 Yrs & Mkt Price – Rs.92

Step -1: Calculate YTM as discount rate used in duration computation is YTM. In this case it is 10.53%

Step-2: Calculate Duration

N	CF	PV at YTM	PV*N
1	9	8.14	8.14
2	9	7.37	14.73
3	9	6.67	20.00
4	9	6.03	24.12
5	9	5.46	27.28
6	9	4.94	29.62
7	9	4.47	31.26
8	109	48.93	391.45
		91.99	546.59
		Duration	5.94
		1+YTM	1.1053
step 3		Dmod	5.38

Duration of a Bond and Price Sensitivity:

Change in yield lead to change in the duration of a bond. This sensitivity can be used to identify approximate change in the price of a bond.

% Change in price = -Dmod * % change in yield

Example:

Consider a bond with YTM of 12% and duration is 5 years. If the interest rate increases by 50 basis points, change in the price of the bond will be…?

Modified duration of the bond is 4.46 years

% change in price = -4.46 * 0.5% =-2.23

Which indicates that approximate change in price will be

• -2.23% with 0.50% increase in YTM
• +2.23% with 0.50% decrease in YTM

Remember:

• It is an approximate measure to find change in price with change in yield.
• Used more for immunization of portfolios.
• Also used to find changes in value of assets and liabilities with changes in interest.

CFPs are not required to capture greater depth of duration concepts hence I have introduced just concepts.

Reference:

Investment Valuation by Aswath Damodaran