Advanced Bond Concepts: Bond Pricing
It is important for prospective bond buyers to know how to determine the price of a bond because it will indicate the yield received should the bond be purchased. In this section, we will run through some bond price calculations for various types of bond instruments.
Bonds can be priced at a premium, discount, or at par. If the bond's price is higher than its par value, it will sell at a premium because its interest rate is higher than current prevailing rates. If the bond's price is lower than its par value, the bond will sell at a discount because its interest rate is lower than current prevailing interest rates. When you calculate the price of a bond, you are calculating the maximum price you would want to pay for the bond, given the bond's coupon rate in comparison to the average rate most investors are currently receiving in the bond market. Required yield or required rate of return is the interest rate that a security needs to offer in order to encourage investors to purchase it. Usually the required yield on a bond is equal to or greater than the current prevailing interest rates.
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 that to calculate present value (PV)  which is based on the assumption that each payment is reinvested at some interest rate once it is receivedwe 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 the calculations, refer to Understanding the Time Value of Money.)
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, as it requires adding the present value of each future coupon payment. Because these payments are paid at an ordinary annuity, however, we can use the shorter PVofordinaryannuity formula that is mathematically equivalent to the summation of all the PVs of future cash flows. This PVofordinaryannuity 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:
Each full moneybag on the top right represents the fixed coupon payments (future value) received in periods one, two and three. Notice how the present value decreases for those coupon payments that are further into the future the present value of the second coupon payment is worth less than the first coupon and the third coupon is worth the lowest amount today. The farther into the future a payment is to be received, the less it is worth today  is the fundamental concept for which the PVofordinaryannuity formula accounts. It calculates the sum of the present values of all future cash flows, but unlike the bondpricing formula we saw earlier, it doesn't require that we add the value of each coupon payment. (For more on calculating the time value of annuities, see Anything but Ordinary: Calculating the Present and Future Value of Annuities and Understanding the Time Value of Money. )
By incorporating the annuity model into the bond pricing formula, which requires us to also include the present value of the par value received at maturity, we arrive at the following formula:
Let's 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 $1,000 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 semiannually 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: Because two coupon payments will be made each year for ten years, we will have a total of 20 coupon payments.
2. Determine the Value of Each Coupon Payment: Because the coupon payments are semiannual, divide the coupon rate in half. The coupon rate is the percentage off the bond's par value. As a result, each semiannual coupon payment will be $50 ($1,000 X 0.05).
3. Determine the SemiAnnual 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 semiannual yield is 6% (0.12/2).
4. Plug the Amounts Into the Formula:
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.
Accounting for Different Payment Frequencies
In the example above coupons were paid semiannually, 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 one. Should a bond pay quarterly payments, F would equal four, and if the bond paid semiannual coupons, F would be two.
Pricing ZeroCoupon Bonds
So what happens when there are no coupon payments? For the aptlynamed zerocoupon 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 zerocoupon bond that is maturing in five years, has a par value of $1,000 and a required yield of 6%.
1. Determine the Number of Periods: Unless otherwise indicated, the required yield of most zerocoupon bonds is based on a semiannual coupon payment. This is because the interest on a zerocoupon bond is equal to the difference between the purchase price and maturity value, but we need a way to compare a zerocoupon bond to a coupon bond, so the 6% required yield must be adjusted to the equivalent of its semiannual coupon rate. Therefore, the number of periods for zerocoupon bonds will be doubled, so the zero coupon bond maturing in five years would have ten periods (5 x 2).
2. Determine the Yield: The required yield of 6% must also be divided by two because 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:
You should note that zerocoupon bonds are always priced at a discount: if zerocoupon bonds were sold at par, investors would have no way of making money from them and therefore no incentive to buy them.
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 paymentfrequency pattern. So far, if we were to price a bond that pays semiannual 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, because 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 semiannual 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 daycount 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 semiannual Treasury bond on March 1, 2003, and its next coupon payment is in four months (July 1, 2003), the next coupon payment would be in 122 days:
Time Period = Days Counted
March 131 = 31 days
April 130 = 30 days
May 131 = 31 days
June 130 = 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 sixmonth 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 sixmonth 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 semiannual corporate bond. In this case, the next coupon payment would be in 120 days.
Time Period = Days Counted
March 130 = 30 days
April 130 = 30 days
May 130 = 30 days
June 130 = 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 daycount conventions tell us that 60 days have passed into the payment period.
Determining Interest Accrued
Accrued interest is the fraction of the coupon payment that 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, the bond prices quoted are often clean prices.
However, because many of the 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. The following example will make this concept more clear.
Example 3: On March 1, 2003, Francesca is selling a corporate bond with a face value of $1,000 and a 7% coupon paid semiannually. 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 SemiAnnual Coupon Payment: Because the coupon payments are semiannual, divide the coupon rate in half, which gives a rate of 3.5% (7% / 2). Each semiannual coupon payment will then be $35 ($1,000 X 0.035).
2. Determine the Number of Days Remaining in the Coupon Period: Because it is a corporate bond, we will use the 30/360 daycount convention.
Time Period = Days Counted
March 130 = 30 days
April 130 = 30 days
May 130 = 30 days
June 130 = 30 days
Total Days = 120 days
There are 120 days remaining before the next coupon payment, but because the coupons are paid semiannually (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 (180120).
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 180day coupon period.
Now you know how to calculate the price of a bond, regardless of when its next coupon will be paid. Bond price quotes are typically the clean prices, but buyers of bonds pay the dirty, or full price. As a result, both buyers and sellers should understand the amount for which 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 as well as the reasons for which bond prices change in the market.
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