# Price a Bond

The ability to price a bond is essential for anyone interested in investing in, or understanding, bonds. Although the terminology can be a bit intimidating, the actual process of pricing bonds requires some math and a basic understanding of what bonds are and how they work. A bond is a financial instrument that pays a fixed amount of interest until it matures, at which point the investor receives a payment of the bond's face value (an amount printed on the bond). To price a bond (which means to ascertain its present value as opposed to its face value), you must understand the meaning of present value, discount rate and cash flow.

## Contents

## Steps

### Pricing a Bond with Coupon Payments

- 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.
^{[1]}- 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.

- 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 <math>Price = C/(1+i) + C/(1+i)^2</math> …+ <math>C/(1+i)^n + M/(1+i)^n</math>. In this formula, the variables are assigned as follows:
^{[1]}- <math>C</math> is the amount of each coupon payment that you expect to receive.
- <math>i</math> is the interest rate
- <math>M</math> is the face value of the bond at maturity.
- <math>n</math> 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.

- Revise the formula to account for annuity payments. 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:
^{[1]}- <math>Price=C*(1-(1/(1+i)^n)/i +M/(1+i)^n</math>.

- Determine the variables to use in the formula. 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:
^{[1]}- <math>C</math> 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.
- <math>i</math> 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 <math>i</math> for your calculations should be 6%, which you will write as a decimal of 0.06.
- <math>n</math> 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, <math>n</math> will be 20.

- Calculate the bond’s current value. 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.
^{[1]}- Performing the calculations results in a bond price of $885.30.

- Understand the meaning of the bond price. 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.
^{[1]}- 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.

### Pricing a Zero Coupon Bond

- Find out if you have a zero coupon bond. The phrase “zero coupon bond” applies to any bond that will not provide any coupon or interest payments during the life of the bond. As a result, the only payment you expect will be the payment of the face value of the bond when it reaches maturity. To find out, ask the broker who is selling the bonds whether the bond will be making coupon payments or not.
^{[1]}- Although the price of a zero coupon bond may seem obvious -- that it should be the face value -- the price actually takes into account a deduction for the time you must wait until maturity. For example, $1,000 to be paid in five years is not as valuable as $1,000 that you could have today. Therefore, even a zero coupon bond needs to have its price calculated.

- Determine the par value of the bond. The par value is the face value amount of the bond, which you will expect to receive when the bond reaches maturity. The par value should be evident if the bond is being offered for sale. If you have any question, ask the broker who is selling the bonds what its par value is.
^{[1]} - Find out the bond’s required yield. This is the interest value that you are promised to receive when the bond reaches maturity. This value should be made evident if the bond is being offered for sale.
- For example, a bond may be offered upon the terms that it is a $1,000 face value and a 6% yield over five-years.

- Calculate a theoretical number of payment periods. Although you will not be receiving interest payments over the life of the bond, you need to use a theoretical number of payment periods to calculate its value over time. The most common payment schedule is semi-annually. So to compare with these, you would select a number that is equal to the life of your bond, multiplied by two.
- For example, if you are considering a bond that matures in five years, you would used a number of payment periods of 5x2, which is 10.

- Adjust the bond’s yield for the calculation. The bond’s yield is presented as an annual figure, but you will be calculating the bond value based on semi-annual payments. For that reason, you would divide the yield in half.
- If the bond has an advertised yield of 6%, use the value of 3% (or 0.03) for the calculation of the bond’s value. This halving of the yield correlates with the number of theoretical payment periods.

- Use the calculation formula to find the current value of the bond. The formula for finding the current value of a bond with zero coupon payments is <math>P=\frac{M}{(1+i)^n}</math>. The variables in this calculation are the data that you should know about the bond:
- <math>M</math>. This is the face value of the bond at maturity. For the example above, this is 1000.
- <math>i</math>. This is the interest rate, adjusted for the calculation purpose. Therefore, if you calculating a 6% yield, at theoretical semi-annual payments, you would use a value of i=0.03. Be sure to rewrite the percentage figure as the correct decimal value.
- <math>n</math>. This represents the number of theoretical payments for the calculation. It will be the number of payments per year times the life of the bond. For this example, n=10.

- Calculate your bond value. Applying the basic order of operations to the formula, calculating the value of the bond is fairly straightforward. Perform the operations as follows:
- Calculate the base of the denominator by adding <math>1+i</math>. In the given example, this will result in 1.03.
- Apply the exponent to the denominator only. The exponent of <math>n=10</math> means that your base of 1.03 is to be multiplied by itself ten times. This will give the result of 1.34. (You can do this by multiplying 1.03x1.03x1.03… for ten times. Alternately, if you have an advanced calculator with a “^” button, you can just enter “1.03^10” to get the result.)
- Perform the division last. The final step is to divide the face value by the denominator you have calculated. This gives the result of 1000/1.34, which is 746.27. Expressed in monetary terms, this would be a value of $746.27.

- Understand the meaning of the calculated value. A bond that will be worth $1,000 in five years, with no interest payments along the way, is always sold at a discounted rate. In this case, the discount should be the calculated value of $746.27.

### Understanding Bond Pricing Terminology

- Know your bond’s “face value.” The face value (or par value) is the amount that the bond pays at maturity. For instance, a 10-year, $5,000 bond will pay $5,000 when it matures ten years after its date of issue. Therefore, the face value of the bond is $5,000, regardless of any interest or dividend payments you may receive during that time.
^{[2]} - Understand your “coupon payment.” The coupon payment is the amount the bond pays periodically in interest. In most cases, coupon payments occur twice a year. If a $5,000 bond pays 10% annually, then each semiannual coupon will pay $250. That's a total payment of $500 a year (or 10% of $5,000).
^{[3]} - Find out your bond’s “coupon yield.” The coupon yield is the annual coupon payment expressed as a percentage of the bond’s face value. In the example above, the coupon payment is $500, and the face value is $5,000. Therefore, the coupon yield can be expressed as 10%, which is $500/$5,000.
^{[3]} - Measure your bond’s “current yield.” The current yield is the annual coupon payment amount divided by the current bond price. This gives you the coupon payment as a percentage of the current bond price.
^{[4]}- If the coupon payment is, for example, $500 and you calculate the bond's price (value) to be $4,800, then the current yield is $500/$4,800, which would be 10.4%.

- Find the bond’s “yield to maturity.” A bond’s yield to maturity is defined as the discount rate that yields the market price of the bond. This requires some additional calculations. You can read more about this particular calculation at yield to maturity.
^{[4]} - Look up the bond rating. In the financial industry, a few companies research and rate bonds based on their quality, history and expected performance. The primary agencies that provide bond ratings are Standard & Poor, Moody, and Fitch.
^{[5]}- Bond ratings are grades given to bonds so that investors may judge the relative safety of any given bond investment. The highest Standard & Poor’s rating is AAA. AA, A and BBB are medium-quality bonds. BB, B, CCC, CC, C, and D are "junk" bonds. The D rating means the bond has defaulted.

- Consider whether a bond is at a discount, at par, or at a premium. A bond is selling at a discount when its yield to maturity is greater than its current yield and its coupon yield. A bond is selling at par when its yield to maturity is equal to its current yield and its coupon yield. A bond is selling at a premium when its yield to maturity is less than its current yield and its coupon yield.
^{[3]}

## Warnings

- This bond pricing method is the simplest bond valuation method and applies to basic coupon bonds. There are more complicated ways of pricing bonds and more to understand about the bond market than covered here. If you’re serious about bond investing, research other bond valuation methods and consult a financial professional before investing in bonds or any other investment instruments.

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## Sources and Citations

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^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}^{1.6}^{1.7}http://www.investopedia.com/university/advancedbond/advancedbond2.asp?ad=dirN&qo=investopediaSiteSearch&qsrc=0&o=40186 - http://www.investopedia.com/ask/answers/013015/how-does-face-value-differ-price-bond.asp
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^{3.0}^{3.1}^{3.2}http://www.slideshare.net/shekharsharma.12/bond-valuation-6435590 - ↑
^{4.0}^{4.1}http://www.investopedia.com/university/bonds/bonds3.asp?ad=dirN&qo=investopediaSiteSearch&qsrc=0&o=40186 - http://www.investopedia.com/walkthrough/corporate-finance/3/bonds/ratings.aspx