Fixed income terminology
Types of Fixed income
Bonds vs Debentures
Government Securities and Corporate Debentures
Fixed Income Market Structure
How are the prices decided in debt market
Fixed income prices
When talking about bonds (especially when buying them) you have to take into consideration the yield of the bond. Simply put, a bond’s yield shows the relationship of the investment and the income. For the sake of comparability yield is always shown as an annualized (yearly) percentage.
Although we may know the coupon rate of a bond that only shows the real yield of the bond if the bond was bought at par value and will be held till maturity. Current yield compares the yearly interest income with the current market price. If the bond was bought with a discount the current yield will be higher than the coupon rate, if it was bought with a premium it will be lower. Current yield is calculated with the following formula:
CY=
This yield is used when the investor knows when he will sell his bonds. In this case we use an estimated selling price instead of the face value.
Yield to maturity is the rate at which if we discount the incomes (cash-flows) of the bond, we get the par value exactly. This yield is used most because it takes into consideration all future incomes and also their change over time.
Yield can be calculated many ways. There are bond-tables from which you can calculate yield fairly easily. Some more serious calculators are also capable of doing the calculations. Without these we have two choices. We can either use a method called the trial and error method, or we can use simplified yield calculated method, aptly named SYTM or simplified yield to maturity. Both these choices are approximations, although I consider the trial and error method to be inferior because doing it in reality would be much longer and most probably less exact. The formula for calculating it is:
Because there is no interest payment here, only one cash-flow, the repayment of the face value, there is no need to use approximations because we know all the information we need. We know already that:
Rearranged:
Please note that we can also use PVIF tables to solve these problems because:
By substituting this we get:
Calculating yield here is very simple, I will only mention it because no further explanation is needed.
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:
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:
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.