How to Price a Bond
Learn the details of the bond being offered., Understand the present-value formula., Revise the formula to account for annuity payments., Determine the variables to use in the formula., Calculate the bond’s current value., Understand the meaning of...
Step-by-Step Guide
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Step 1: Learn the details of the bond being offered.
Generally, if you are pricing a bond, it is because you are considering buying or selling it.
In either case, there are certain terms of the bond that you will know.
For example, a bond might be offered as a $1,000 bond, to be paid in ten years, with a coupon rate of 10% and a required yield of 12%.
You will use these data to calculate the present value of the bond, incorporating all future payments.Working with the example given above, the face value of the bond is $1,000.
The term is ten years.
The coupon rate is given as 10%, and the yield is provided as 12%.
If the seller of the bond does not provide all this information, then you should ask for it. -
Step 2: Understand the present-value formula.
A basic present-value formula will account for today’s value of money that is to be paid in the future.
Because of interest rates, the money that you could hold today is generally considered more valuable than money that will be paid in the future.
The present-value formula accounts for this difference.
The basic present-value formula is Price=C/(1+i)+C/(1+i)2{\displaystyle Price=C/(1+i)+C/(1+i)^{2}} …+ C/(1+i)n+M/(1+i)n{\displaystyle C/(1+i)^{n}+M/(1+i)^{n}}.
In this formula, the variables are assigned as follows:
C{\displaystyle C} is the amount of each coupon payment that you expect to receive. i{\displaystyle i} is the interest rate M{\displaystyle M} is the face value of the bond at maturity. n{\displaystyle n} is the number of payment periods over the life of the bond.
If you will expect two payments per year, which is standard, then a bond that matures in ten years will have 20 payment periods. , Most bonds make coupon payments on a regular basis.
This allows you to simplify the formula, to avoid an ambiguous added series.
The revised formula may look slightly more complicated but is actually easier to apply.
The revised annuity formula is:
Price=C∗(1−(1/(1+i)n)/i+M/(1+i)n{\displaystyle Price=C*(1-(1/(1+i)^{n})/i+M/(1+i)^{n}}. , You need to apply the information that you know about the bond correctly in order for the formula to work.
Use the example given above, of a $1,000 bond, to be paid in ten years, with a coupon rate of 10% and a required yield of 12%.
With this information, the variables for the formula are as follows:
C{\displaystyle C} is the amount paid on each coupon.
The coupon rate given of 10% is for the year, meaning that you will receive 10% of the face value of the bond, or $100.
This is commonly paid semi-annually, so the value for C is half that, or
50. i{\displaystyle i} is the interest rate given as the required yield of the bond.
In this case, that is 12%.
However, the interest rate given is for the year, but you will be calculating based on semi-annual payments, so use half that figure.
The value of i{\displaystyle i} for your calculations should be 6%, which you will write as a decimal of
0.06. n{\displaystyle n} is the number of payment periods over the life of the bond.
If you are basing your calculation on semi-annual payments, for ten years, n{\displaystyle n} will be
20. , Insert the values into the formula and find the value of the bond.
In this example, applying the values to the formula results in the following:
Price=50*(1-1/(1.06)^20)/0.06)+1000/(1.06)^20.Performing the calculations results in a bond price of $885.30. , The calculated value of $885.30 is less than the face value of $1,000.
This means that the bond should sell at a discount in order to attract investors.
This discount is due to the fact that the coupon payments are only 10% while the required, advertised yield of the bond is 12%.
You would expect to receive less in coupon payments than the promised yield of the bond.If the interest rate were to decrease, then the value of the bond would increase.
Interest rates and bond values operate in an inverse manner. -
Step 3: Revise the formula to account for annuity payments.
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Step 4: Determine the variables to use in the formula.
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Step 5: Calculate the bond’s current value.
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Step 6: Understand the meaning of the bond price.
Detailed Guide
Generally, if you are pricing a bond, it is because you are considering buying or selling it.
In either case, there are certain terms of the bond that you will know.
For example, a bond might be offered as a $1,000 bond, to be paid in ten years, with a coupon rate of 10% and a required yield of 12%.
You will use these data to calculate the present value of the bond, incorporating all future payments.Working with the example given above, the face value of the bond is $1,000.
The term is ten years.
The coupon rate is given as 10%, and the yield is provided as 12%.
If the seller of the bond does not provide all this information, then you should ask for it.
A basic present-value formula will account for today’s value of money that is to be paid in the future.
Because of interest rates, the money that you could hold today is generally considered more valuable than money that will be paid in the future.
The present-value formula accounts for this difference.
The basic present-value formula is Price=C/(1+i)+C/(1+i)2{\displaystyle Price=C/(1+i)+C/(1+i)^{2}} …+ C/(1+i)n+M/(1+i)n{\displaystyle C/(1+i)^{n}+M/(1+i)^{n}}.
In this formula, the variables are assigned as follows:
C{\displaystyle C} is the amount of each coupon payment that you expect to receive. i{\displaystyle i} is the interest rate M{\displaystyle M} is the face value of the bond at maturity. n{\displaystyle n} is the number of payment periods over the life of the bond.
If you will expect two payments per year, which is standard, then a bond that matures in ten years will have 20 payment periods. , Most bonds make coupon payments on a regular basis.
This allows you to simplify the formula, to avoid an ambiguous added series.
The revised formula may look slightly more complicated but is actually easier to apply.
The revised annuity formula is:
Price=C∗(1−(1/(1+i)n)/i+M/(1+i)n{\displaystyle Price=C*(1-(1/(1+i)^{n})/i+M/(1+i)^{n}}. , You need to apply the information that you know about the bond correctly in order for the formula to work.
Use the example given above, of a $1,000 bond, to be paid in ten years, with a coupon rate of 10% and a required yield of 12%.
With this information, the variables for the formula are as follows:
C{\displaystyle C} is the amount paid on each coupon.
The coupon rate given of 10% is for the year, meaning that you will receive 10% of the face value of the bond, or $100.
This is commonly paid semi-annually, so the value for C is half that, or
50. i{\displaystyle i} is the interest rate given as the required yield of the bond.
In this case, that is 12%.
However, the interest rate given is for the year, but you will be calculating based on semi-annual payments, so use half that figure.
The value of i{\displaystyle i} for your calculations should be 6%, which you will write as a decimal of
0.06. n{\displaystyle n} is the number of payment periods over the life of the bond.
If you are basing your calculation on semi-annual payments, for ten years, n{\displaystyle n} will be
20. , Insert the values into the formula and find the value of the bond.
In this example, applying the values to the formula results in the following:
Price=50*(1-1/(1.06)^20)/0.06)+1000/(1.06)^20.Performing the calculations results in a bond price of $885.30. , The calculated value of $885.30 is less than the face value of $1,000.
This means that the bond should sell at a discount in order to attract investors.
This discount is due to the fact that the coupon payments are only 10% while the required, advertised yield of the bond is 12%.
You would expect to receive less in coupon payments than the promised yield of the bond.If the interest rate were to decrease, then the value of the bond would increase.
Interest rates and bond values operate in an inverse manner.
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