Fundamentally, however, the price of a bond is the sum of the present values of all expected coupon payments plus the present value of the par value at maturity. Calculating bond price is simple: all we are doing is discounting the known future cash flows. Remember, to calculate present value--which is based on the assumption that each payment is re-invested at some interest rate once it is received--we have to know the interest rate that would earn us a known future value. For bond pricing, this interest rate is the required yield. (If the concepts of present and future value are new to you or you are unfamiliar with their calculations, refer to our article “Understanding the Time Value of Money” for a quick brush-up.)
Here is the formula for calculating a bond's price, which uses the basic present value (PV) formula:
C = coupon payment
n = number of payments
i = interest rate, or required yield
M = value at maturity, or par value
The succession of coupon payments to be received in the future is referred to as an ordinary annuity, which is a series of fixed payments at set intervals over a fixed period of time. (Coupons on a straight bond are paid at ordinary annuity.) The first payment of an ordinary annuity occurs one interval from the time at which the debt security is acquired (the calculation assumes this time is the present).
You may have guessed that the bond pricing formula shown above may be tedious to calculate since it requires us to add the present value of each future coupon payment. But since these payments are paid at an ordinary annuity, we can use the shorter PV-of-ordinary-annuity formula that is mathematically equivalent to the summation of all the PVs of future cash flows. This PV-of-ordinary-annuity formula replaces the need to add all the present values of the future coupon. The following diagram illustrates how present value is calculated for an ordinary annuity:
By incorporating the annuity model into the bond pricing formula, which requires us to include also the present value of the par value received at maturity, we arrive at the following formula:
Let's now go through a basic example to find the price of a plain vanilla bond.
Example 1 Calculate the price of a bond with a par value of $1000 to be paid in ten years, a coupon rate of 10%, and a required yield of 12%. In our example we'll assume that coupon payments are made semi-annually to bond holders, and that the next coupon payment is expected in six months. Here are the steps we have to take to calculate the price:
1. Determine the number of coupon payments: Since two coupon payments will be made each year for ten years, we will have a total of 20 coupon payments.Not too hard was it? From the above calculation, we have determined that the bond is selling at a discount: the bond price is less than its par value because the required yield of the bond is greater than the coupon rate. The bond must sell at a discount to attract investors, who could find higher interest elsewhere in the prevailing rates. In other words, because investors can make a larger return in the market, they need an extra incentive to invest in the bonds.
2. Determine the value of each coupon payment: Since the coupon payments are semi-annual, divide the coupon rate in half. The coupon rate is the percentage off the bond's par value. As a result, each semi-annual coupon payment will be $50 ($1000 X 0.05).
3. Determine the semi-annual yield: Like the coupon rate, the required yield of 12% must be divided by two because the number of periods used in the calculation has doubled. (If we left the required yield at 12%, our bond price would be very low and inaccurate.) Therefore, the required semi-annual yield is 6% (0.12/2).
4. Plug the amounts into the formula:
Accounting for Different Payment Frequencies
In the example above coupons were paid semi-annually, so we divided the interest rate and coupon payments in half to represent the two payments per year. You may be now wondering whether there is a formula that does not require steps two and three outlined above (which are required if the coupon payments occur more than once a year). A simple modification of the above formula will allow you to adjust interest rates and coupon payments to calculate a bond price for any payment frequency:
Notice that the only modification to the original formula is the addition of “F,” which represents the frequency of coupon payments, or the number of times a year the coupon is paid. Therefore, for bonds paying annual coupons, F would have a value of 1. Should a bond pay quarterly payments, F would equal 4, and, if the bond paid semi-annual coupons, F would equal 2.
Pricing Zero-Coupon Bonds
So what happens when there are no coupon payments? For the aptly-named zero-coupon bond, there is no coupon payment until maturity. Because of this, the present value of annuity formula is unnecessary. You simply calculate the present value of the par value at maturity. Here's a simple example:
Example 2(a) Let's look at how to calculate the price of a zero-coupon bond that is maturing in five years, has a par value of $1000, and a required yield of 6%.
1. Determine the number of periods: Unless otherwise indicated, the required yield of most zero-coupon bonds is based on a “semi-annual coupon payment.” Here's why: the interest on a zero-coupon bond is equal to the difference between purchase price and maturity value, but we need a way to compare a zero-coupon bond to a coupon bond, so the 6% required yield must be adjusted to the equivalent of its semi-annual coupon rate. Therefore, the number of periods for zero-coupon bonds will be doubled, so the zero coupon bond maturing in five years would have ten periods (5 x 2).You should note that zero-coupon bonds are always priced at a discount: if zero-coupon bonds were sold at par, investors would have no way of making money from them and therefore no incentive to buy them.
2. Determine the yield: The required yield of 6% must also be divided by two since the number of periods used in the calculation has doubled. The yield for this bond is 3% (6% / 2).
3. Plug the amounts into the formula:
Pricing Bonds between Payment Periods
Up to this point we have assumed that we are purchasing bonds whose next coupon payment occurs one payment period away, according to the regular payment-frequency pattern. So far, if we were to price a bond that pays semi-annual coupons and we purchased the bond today, our calculations would assume that we would receive the next coupon payment in exactly six months. Of course, since you won't always be buying a bond on its coupon payment date, it's important you know how to calculate price if, say, a semi-annual bond is paying its next coupon in three months, one month, or 21 days.
Determining Day Count
To price a bond between payment periods, we must use the appropriate day-count convention. Day count is a way of measuring the appropriate interest rate for a specific period of time. There is actual/actual day count, which is used mainly for Treasury securities. This method counts the exact number of days until the next payment. For example, if you purchased a semi-annual Treasury bond on March 1, 2003, and its next coupon payment is in four months (July 1st, 2003), the next coupon payment would be in 122 days:
Time Period = Days Counted
March 1-31 = 31 days
April 1-30 = 30 days
May 1-31 = 31 days
June 1-30 = 30 days
July 1 = 0 days
Total Days = 122 days
To determine the day count, we must also know the number of days in the six-month period of the regular payment cycle. In these six months there are exactly 182 days, so the day count of the Treasury bond would be 122/182, which means that out of the 182 days in the six-month period, the bond still has 122 days before the next coupon payment. In other words, 60 days of the payment period (182 - 122) have already passed. If the bondholder sold the bond today, he or she must be compensated for the interest accrued on the bond over these 60 days.
(Note that if it is a leap year, the total number of days in a year is 366 rather than 365.)
For municipal and corporate bonds, you would use the 30/360 day count convention, which is much simpler as there is no need to remember the actual number of days in each year and month. This count convention assumes that a year consists of 360 days and each month consists of 30 days. As an example, assume the above Treasury bond was actually a semi-annual corporate bond. In this case, the next coupon payment would be in 120 days.
Time Period = Days Counted
March 1-30 = 30 days
April 1-30 = 30 days
May 1-30 = 30 days
June 1-30 = 30 days
July 1 = 0 days
Total Days = 120 days
As a result, the day count convention would be 120/180, which means that 66.7% of the coupon period remains. Notice that we end up with almost the same answer as the actual/actual day count convention above: both day-count conventions tell us that 60 days have passed into the payment period.
Determining Interest Accrued
Accrued interest is the fraction of the coupon payment the bond seller earns for holding the bond for a period of time between bond payments. The bond price's inclusion of any interest accrued since the last payment period determines whether the bond's price is “dirty” or “clean.” Dirty bond prices include any accrued interest that has accumulated since the last coupon payment while clean bond prices do not. In newspapers, bond prices quoted are often their clean prices.
Since, however, many bonds traded in the secondary market are often traded in between coupon payment dates, the bond seller must be compensated for the portion of the coupon payment he or she earns for holding the bond since the last payment. The amount of the coupon payment that the buyer should receive is the coupon payment minus accrued interest.
Let's go through a simple example:
On March 1, 2003, Francesca is selling a corporate bond with a face value of $1000 and a 7% coupon paid semi-annually. The next coupon payment after March 1, 2003, is expected on June 30, 2003. What is the interest accrued on the bond?
1. Determine the semi-annual coupon payment: Since the coupon payments are semi-annual, divide the coupon rate in half, which gives a rate of 3.5% (7% / 2). Each semi-annual coupon payment will then be $35 ($1000 X 0.035).
2. Determine the number of days remaining in the coupon period: Since it is a corporate bond, we will use the 30/360 day-count convention.
Time Period = Days Counted
March 1-30 = 30 days
April 1-30 = 30 days
May 1-30 = 30 days
June 1-30 = 30 days
Total Days = 120 days
There are 120 days remaining before the next coupon payment, but, since the coupons are paid semi-annually (two times a year), the regular payment period if the bond is 180 days, which, according to the 30/360 day count, is equal to six months. The seller, therefore, has accumulated 60 days worth of interest (180-120).
3. Calculate the accrued interest: Accrued interest is the fraction of the coupon payment that the original holder (in this case Francesca) has earned. It is calculated by the following formula:
In this example, the interest accrued by Francesca is $11.67. If the buyer only paid her the clean price, she would not receive the $11.67 to which she is entitled for holding the bond for those 60 days of the 180-day coupon period.
Now you know how to calculate the price of a bond, regardless of when its next coupon will be paid. Since bond price quotes are typically their clean prices but buyers of bonds pay the dirty, or full price, both buyers and sellers should understand for what amount a bond should be sold or purchased. In addition, the tools you learned in this section will better enable you to learn the relationship between coupon rate, required yield, and price, and the reasons why bond prices change in the market.
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