7/27/2019 A Note on Duration(1)

1/3

A Note on Duration:

When an investor considers investment in Bonds the relationship between the time to maturity,

yield and price is very clear in case of a zero coupon bond. A zero coupon bond has no coupons and

thus a zero coupon bond that has a face value of Rs. 100, has a maturity period of 5 years and a yield

of 5% (could also be called a required rate of return) would be priced as follows:

Price = Face Value/ (1+ y%)t= 100/(1+5%)

5= 78.15

If an investor invests in a bond that a face value of Rs. 100, has a maturity of 10 years and an

identical yield of 5%, the bond would be priced at:

100/(1+5%)10

= 61.39

Similarly if the bond was 15 years the price would be 48.10

A sensitivity table of these bonds looks as follows:

5 Yr Zero 10 Yr Zero 15 Year Zero

1% 95.15 90.53 86.13

2% 90.57 82.03 74.30

3% 86.26 74.41 64.19

4% 82.19 67.56 55.53

5% 78.35 61.39 48.10

6% 74.73 55.84 41.73

7% 71.30 50.83 36.24

8% 68.06 46.32 31.52

-

20.00

40.00

60.00

80.00

100.00

120.00

0% 2% 4% 6% 8% 10% 12% 14%

Price Sensitivity of a zero coupon bond

5 yr Bond 10 Yr Bond 15 Yr Bond

7/27/2019 A Note on Duration(1)

2/3

When one deals with a zero coupon bond the relationship between the price and the maturity is

very clear. However when the bonds have a coupon two bonds cannot be compared as easily as the

zero coupon bond. In a zero coupon bond the effective maturity of the a zero coupon bond is the

same as its years to maturity. By effective maturity what I mean is the period in which the original

investment of the bond is recovered

Understanding the concept of weights:

Let us assume that an investor has two Zero coupon Bonds as follows:

Bond 1 Bond 2

Face Value 100 100

Years to maturity 5 10

Yield 5% 5%

Now it is obvious that the effective period for which he is invested is 7.5 ( (5+10)/2= 7.5) ) so if thatsame investor purchased a zero coupon bond of face value 200 and invested it for 7.5 years at 5%

yield the price should be the price of the above put together ????

Let us do the math:

Bond 1 Bond 2 Bond 3

Face Value 100 100 200

Years to maturity 5 10 7.5

Yield 5% 5% 5%

Price 100/(1+5%)^5 100/(1+5%)^10 200/(1+5%)^7.5

Price 78.35 61.39 138.71

You can see that the first two bonds add up to 139.74 which is very close to the third bond !!!!

You can see from the above it is the TIME element that enables the investor to replicate the

investment of two bonds into a single bond.

If in the above example I extend the period of the first bond to 7.5 years and I reduce the second

bond to 7.5 years the math would be as follows:

Bond 1 Bond 2 Bond 3

Face Value 100 100 200

Years to maturity 7.5 7.5 7.5

Yield 5% 5% 5%

Price 100/(1+5%)^7.5 100/(1+5%)^7.5 200/(1+5%)^7.5

Price 69.36 69.36 138.71

Thus we find that the time element unifies the investments !!!!

Thus when we want to find in WHAT TIME was an investment in Bond that gives coupons is

recovered the TIME element would be the best weight.

7/27/2019 A Note on Duration(1)

3/3

Macaulay did exactly the same thing, he took every discounted coupon amount and multiplied it

with the weight of TIME to find the recovery period of the Bond.

Let us take an actual example !!!!

Let us take a sample bond:

Face Value 100

Coupon 5%

Years to Maturity 10

Frequency of the coupon 2

Yield of comparable bond in the

market

5%

A B C D

Time Cash flows

Discounting Factor:

Cash flow X

1/(1+yield/f)^t

Cash Flow X

Discount Factor

Amount recovered of

investment in

%=Amount/total of

column C Column D X T

-

1 2.5 0.9756 2.44 0.0244 0.02439

2 2.5 0.9518 2.38 0.0238 0.04759

3 2.5 0.9286 2.32 0.0232 0.06964

4 2.5 0.9060 2.26 0.0226 0.09060

5 2.5 0.8839 2.21 0.0221 0.11048

6 2.5 0.8623 2.16 0.0216 0.12934

7 2.5 0.8413 2.10 0.0210 0.14722

8 2.5 0.8207 2.05 0.0205 0.16415

9 2.5 0.8007 2.00 0.0200 0.18016

10 2.5 0.7812 1.95 0.0195 0.19530

11 2.5 0.7621 1.91 0.0191 0.20959

12 2.5 0.7436 1.86 0.0186 0.22307

13 2.5 0.7254 1.81 0.0181 0.23576

14 2.5 0.7077 1.77 0.0177 0.24770

15 2.5 0.6905 1.73 0.0173 0.25892

16 2.5 0.6736 1.68 0.0168 0.26945

17 2.5 0.6572 1.64 0.0164 0.27931

18 2.5 0.6412 1.60 0.0160 0.28852

19 2.5 0.6255 1.56 0.0156 0.29713

20 102.5 0.6103 62.55 0.6255 12.51055

0 Total: Price of Bond 100.00

total of column

divided by 2 is the

Duration 7.99

The duration is calculated in the last column as the total of all weighted cash flow (after discounting)

where TIME is the weight !!!!