Bond Duration Analysis for CFP – 1
Posted by Prashant Shah on January 16, 2013
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%
Step2: 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
R Varadarajan said
Dear Prashant Sir,
Thanks for the fantastic and higly educative post – perhaps in the best manner to explain the concept of Duration. Thanks again. There appears to be some error on the calculation w r t cnange in price v/s change in ytm I think they should read as follows
“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”
Am I right or have I missed out something ?
I would like to add another problem for your advice
“The Bonds of XYZ Ltd are presently selling at a premium of 9% against their FV and MV of Rs.100. The current Yield on these bonds are7.34%. The coupons are paid semi annually. If the Bonds are to mature 3 years hence what should be the annualised yield to the investor today ? ”
I wonder whether the problem is complete . Please could you advise me with your solutions. Thanks
With regards,
R Varadarajan
Manish said
Dear R varadarajan coupon rate is missing in the question.
Prashant Shah said
Dear Varadarajan,
You are right. There was an error which is now corrected.
Thanks for observation.
Regards,
Prashant V Shah.
Prashant Shah said
Dear Varadrajan,
In the stated question
You need to find the coupon first which is 109*7.34% = 8%
YTM of the bond
PMT = 4
FV = 100
PV = 109
N = 6
YTM = ? = 2.37
Hence annualised equivalent yield is 4.75%
Regards,
Prashant V Shah
Manish said
Dear sir thanks for updating this post in bond theorems u have mentioned that “Price of the bond increases more than it declines when equal change in interest rate is given” but in the above post u have written that change in price will be equal with 1% change in ytm and sir i m also waiting for the illustrations of remaining 2 theorems.
Prashant Shah said
Dear Manish,
Fantastic question to ask. Duration is an approximate measure to find change in price. We need to add convexity adjustment to get the exact change in price. Which goes out of CFP course. For a single bond TVM is a better way to calculate change in price but when it comes to a portfolio, duration and convexity is more relavent.
Superb observation.
Regards,
Prashant V Shah.
Manish said
Dear sir thanks for the superb clarification and thanks for the solution of the above stated question i forgot that sometimes investor may be interested in knowing the interest received by him as a % of the bond’s market value.so for this purpose we calculate current yield on market value.thanks for reminding me this with ur solution.
udaycfp said
Dear Sir
how u get that ans=5.38
i dont understand pls clear my doubt
Regards,
Uday
vidisha said
=(duration/(1+ytm/1)
i.e.
=(5.94/(1.1052/1)
=5.38