A Note on Duration(1)
Content
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 95.15 90.57 86.26 82.19 78.35 74.73 71.30 68.06 10 Yr Zero 90.53 82.03 74.41 67.56 61.39 55.84 50.83 46.32 15 Year Zero 86.13 74.30 64.19 55.53 48.10 41.73 36.24 31.52
1% 2% 3% 4% 5% 6% 7% 8%
Price Sensitivity of a zero coupon bond
120.00 100.00 80.00 60.00 40.00 20.00 0% 2% 4% 5 yr Bond 6% 10 Yr Bond 8% 10% 15 Yr Bond 12% 14%
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 1 100 5 5% Bond 2 100 10 5%
Face Value Years to maturity Yield
Now it is obvious that the effective period for which he is invested is 7.5 ( (5+10)/2= 7.5) ) so if that same 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 1 100 5 5% 100/(1+5%)^5 78.35 Bond 2 100 10 5% 100/(1+5%)^10 61.39 Bond 3 200 7.5 5% 200/(1+5%)^7.5 138.71
Face Value Years to maturity Yield Price Price
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 1 100 7.5 5% 100/(1+5%)^7.5 69.36 Bond 2 100 7.5 5% 100/(1+5%)^7.5 69.36 Bond 3 200 7.5 5% 200/(1+5%)^7.5 138.71
Face Value Years to maturity Yield Price Price
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.
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 Coupon Years to Maturity Frequency of the coupon Yield of comparable bond in the market
A B Discounting Factor: Cash flow X 1/(1+yield/f)^t
100 5% 10 2 5%
C D Amount recovered of investment in %=Amount/total of column C
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Cash flows
Cash Flow X Discount Factor
Column D X T
2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 102.5
0.9756 0.9518 0.9286 0.9060 0.8839 0.8623 0.8413 0.8207 0.8007 0.7812 0.7621 0.7436 0.7254 0.7077 0.6905 0.6736 0.6572 0.6412 0.6255 0.6103
2.44 2.38 2.32 2.26 2.21 2.16 2.10 2.05 2.00 1.95 1.91 1.86 1.81 1.77 1.73 1.68 1.64 1.60 1.56 62.55
0.0244 0.0238 0.0232 0.0226 0.0221 0.0216 0.0210 0.0205 0.0200 0.0195 0.0191 0.0186 0.0181 0.0177 0.0173 0.0168 0.0164 0.0160 0.0156 0.6255 total of column divided by 2 is the Duration
0.02439 0.04759 0.06964 0.09060 0.11048 0.12934 0.14722 0.16415 0.18016 0.19530 0.20959 0.22307 0.23576 0.24770 0.25892 0.26945 0.27931 0.28852 0.29713 12.51055
0
Total: Price of Bond
100.00
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 !!!!
1% 2% 3% 4% 5% 6% 7% 8%
Price Sensitivity of a zero coupon bond
120.00 100.00 80.00 60.00 40.00 20.00 0% 2% 4% 5 yr Bond 6% 10 Yr Bond 8% 10% 15 Yr Bond 12% 14%
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 1 100 5 5% Bond 2 100 10 5%
Face Value Years to maturity Yield
Now it is obvious that the effective period for which he is invested is 7.5 ( (5+10)/2= 7.5) ) so if that same 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 1 100 5 5% 100/(1+5%)^5 78.35 Bond 2 100 10 5% 100/(1+5%)^10 61.39 Bond 3 200 7.5 5% 200/(1+5%)^7.5 138.71
Face Value Years to maturity Yield Price Price
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 1 100 7.5 5% 100/(1+5%)^7.5 69.36 Bond 2 100 7.5 5% 100/(1+5%)^7.5 69.36 Bond 3 200 7.5 5% 200/(1+5%)^7.5 138.71
Face Value Years to maturity Yield Price Price
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.
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 Coupon Years to Maturity Frequency of the coupon Yield of comparable bond in the market
A B Discounting Factor: Cash flow X 1/(1+yield/f)^t
100 5% 10 2 5%
C D Amount recovered of investment in %=Amount/total of column C
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Cash flows
Cash Flow X Discount Factor
Column D X T
2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 102.5
0.9756 0.9518 0.9286 0.9060 0.8839 0.8623 0.8413 0.8207 0.8007 0.7812 0.7621 0.7436 0.7254 0.7077 0.6905 0.6736 0.6572 0.6412 0.6255 0.6103
2.44 2.38 2.32 2.26 2.21 2.16 2.10 2.05 2.00 1.95 1.91 1.86 1.81 1.77 1.73 1.68 1.64 1.60 1.56 62.55
0.0244 0.0238 0.0232 0.0226 0.0221 0.0216 0.0210 0.0205 0.0200 0.0195 0.0191 0.0186 0.0181 0.0177 0.0173 0.0168 0.0164 0.0160 0.0156 0.6255 total of column divided by 2 is the Duration
0.02439 0.04759 0.06964 0.09060 0.11048 0.12934 0.14722 0.16415 0.18016 0.19530 0.20959 0.22307 0.23576 0.24770 0.25892 0.26945 0.27931 0.28852 0.29713 12.51055
0
Total: Price of Bond
100.00
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 !!!!
