For example, a bond with a par value of $1,000 might be priced at par. This means that it costs $1,000 to purchase the bond. Alternately, a bond with a par value of $1,000 might be purchased at a discount for $980 or at a premium for $1,050. Discounted bonds are generally those that provide relatively low, or zero, interest payments. Bonds sold at a premium, however, might pay very high interest payments. The discount or premium is based upon the bond’s coupon rate versus the current interest paid for bonds of similar quality and term.
For example, a bond with a par value of $1,000 might be priced at par. This means that it costs $1,000 to purchase the bond. Alternately, a bond with a par value of $1,000 might be purchased at a discount for $980 or at a premium for $1,050. Discounted bonds are generally those that provide relatively low, or zero, interest payments. Bonds sold at a premium, however, might pay very high interest payments. The discount or premium is based upon the bond’s coupon rate versus the current interest paid for bonds of similar quality and term.
For example, the $1,000 bond mentioned above might pay an annual coupon payment at 3 percent. This would result in a payment of $1000*0. 03, or $30. Keep in mind that some bonds do not pay interest at all. These “zero-coupon” bonds are sold at a deep discount to par when issued, but can be sold at their full par value when they mature.
For example, a bond that makes annual payments for three years would have three total payments.
YTM will be expressed as a percentage. For the purpose of later calculations, you will need to convert this percentage to a decimal. To do this, divide the percentage by 100. For example, 3 percent would be 3/100, or 0. 03. The example bond would have a YTM of 3 percent.
t{\displaystyle t} is the time in years until maturity (from the payment being calculated). c{\displaystyle c} is the coupon payment amount in dollars. i{\displaystyle i} is the interest rate (the YTM). n{\displaystyle n} is the number of coupon payments made. m{\displaystyle m} is the par value (paid at maturity). P{\displaystyle P} is the bond’s current market price. [3] X Research source
The t{\displaystyle t} variable represents the number of years to maturity. For example, the first payment on the example bond from the “gathering your variables” part would be made three years before maturity. This part of the equation would be represented as: (3∗$30(1+0. 03)3){\displaystyle \left({\frac {3*$30}{(1+0. 03)^{3}}}\right)} The next payment would be: (2∗$30(1+0. 03)2){\displaystyle \left({\frac {2*$30}{(1+0. 03)^{2}}}\right)}. In total, this part of the equation would be: (3∗$30(1+0. 03)3)+(2∗$30(1+0. 03)2)+(1∗$30(1+0. 03)1){\displaystyle \left({\frac {3*$30}{(1+0. 03)^{3}}}\right)+\left({\frac {2*$30}{(1+0. 03)^{2}}}\right)+\left({\frac {1*$30}{(1+0. 03)^{1}}}\right)}
This gives: duration=(3∗$30(1. 03)3)+(2∗$30(1. 03)2)+(1∗$30(1. 03)1)+3∗$1,000(1. 03)3$1,000{\displaystyle {\text{duration}}={\frac {\left({\dfrac {3*$30}{(1. 03)^{3}}}\right)+\left({\dfrac {2*$30}{(1. 03)^{2}}}\right)+\left({\dfrac {1*$30}{(1. 03)^{1}}}\right)+{\dfrac {3*$1,000}{(1. 03)^{3}}}}{$1,000}}}
Note that the result 1. 0927 is rounded to three decimal places to make calculation easier. Leaving more decimal places in your calculations will make your answer more accurate.
These results have been rounded to two decimal places, as they are dollar amounts.
For example, a 1 percent decrease in interest rates would lead to an increase in the example bond’s price of 1 percent*2. 914, or 2. 914 percent. An increase in interest rates would have the opposite effect. [4] X Research source
For the example bond described in the other parts of this article, the modifier would be 1+0. 031{\displaystyle 1+{\frac {0. 03}{1}}}, or 1. 03.
