Chapter 6: Bonds, Bond Prices and the Determination of Interest Rates, Part I
There is a lot going on in chapter 6. Understanding measurement is the first step towards analyzing the behavior of anything. This chapter is substantially quantitative so take the necessary time to review the formulas and the examples.
Bond Prices
Here we look at four basic types of bonds: Interest rates apply to four types of credit market instruments:
zero coupon bond or discount bond
- purchased at some price below its face value (or at a discount)
- entitles the owner to a face value payment at the maturity date.
- There are no interest payments, hence the name "zero coupon bond."
- U.S. Treasury Bills are an example of a zero coupon bond.
- provides the borrower with an amount of principal
- the principal and interest are repaid with equal monthly payments for a certain period
- each monthly payment is a combination of principal and interest
- mortgages and car loans are fixed-payments loans
- purchased at some price
- entitles the owner to fixed interest payments annually (coupon payments) until maturity and a face value payment (or par value) at maturity
- characterized by the issuer, the maturity, and the coupon rate, which is multiplied by the face value to determine the coupon payment
- Note: your textbook focuses on annual payments, but in fact, almost all coupon bonds issued in the United States have semi-annual payments.
- purchased at some price
- entitles the owner to interest payments forever
- the principal is never repaid
Zero-coupon Bonds
Because discount bonds have only one payment at maturity, it yield to maturity is easy to calculate and is similar to that of a simple loan. Most discount bonds have a maturity of LESS than one year, so the example below looks at such a case:
example 1: Consider a Treasury bill with 90 days to maturity, a price of $9875, and a face value of $10,000.
The current value is $9850, and the only future payment is $10,000 at maturity. However, we do not wait a year for this payment but only 90 days so we need to adjust the discounting for this.
The yield to maturity solves the following equation:
Solving for i,
This method is the convention in financial markets, known as the bond equivalent basis. If you use a financial calculator you may come up with a different answer. Here's why.
In general, the yield to maturity is found by the formula
where F is the face value, P is the bond price, and d is the days to maturity.
Fixed-Payment Loan
This case is more complicated due to the multiple payments through the life of the loan. Your textbook example uses a loan with annual payments on page 71. However, the most common forms of this type of loan are for monthly payments, like a mortgage, student loans or an auto loan. Loans with multiple payments during the year are a bit more complicated, as shown in the example below:
example 2: Suppose you take out a $15,000 car loan for 5 years, with monthly payments of $300.
The value of the loan today is $15,000. The future payments are $300 payments over the next 60 months.
The yield to maturity is the i that solves the following equation:
Note that since payments are monthly for 5 years, there are a total of 5 x 12 = 60 payment periods. Also, the yield to maturity, i, is expressed on an annual basis, so i/12 represents the monthly discount rate (note 1)
.
So how do we solve this for i? Well it is not easy, since there is no way to isolate i in this equation. It could be done by trial and error (trying values of i until the right-hand side of the equation is $15,000), but that is too time consuming. This problem is solved with the aid of a table, financial calculator, or spreadsheet programs that do this automatically. A financial calculator is not required for this course, so I provide loan or bond table when needed.
Consider the following loan table:
We are looking for a 5 year loan (shaded yellow), and a monthly payment of $300. Looking at the table above we see that at 7.5% yield to maturity, the payment is $300.57. So the yield to maturity is slightly under 7.5% (7.42% to be more precise).
Click below for the "high tech" ways to solve this example:
| Financial Calculator: TI BA II+ | Excel Spreadsheet |
With the multiple interest payments involved, this case is similar to the fixed payment loan in its complexity. Again, your textbook example uses a coupon bond with annual coupon payments on page 121. However, all bonds issued in the United States have coupon payments semi-annually, or every 6 months, including Treasury notes, Treasury bonds, and corporate bonds. So the example below also uses 6-month payments.
example 3: Consider a 2-year Treasury note with a face value of $10,000, a coupon rate of 6%, and a price of $9750.
So the yield to maturity will solve the equation:
bond price = PV(future bond payments)
What are the future payments?
There are coupon payments every 6 months, and a face value payment at maturity.
What are the coupon payments?
The coupon payments are [face value x coupon rate]/2 = $10,000 x .06 x .5 = $300. Note that we divide by 2 because there are 2 coupon payments in a year. So the payment schedule is
| 6 months | $300 |
| 1 year | $300 |
| 18 months | $300 |
| 2 years | $10,300 |
Note that since payments are every 6 months for 2 years, there are a total of 2 x 2 = 4 payment periods. Also, the yield to maturity, i, is expressed on an annual basis, so i/2 represents the 6 month discount rate (note 2)
Like the fixed payment loan, this problem is solved with the aid of a table, financial calculator, or spreadsheet programs that do the trial-and-error calculations automatically. A financial calculator is not required for this course, so I provide loan or bond table when needed.
Consider the following bond table:
We are looking for a 2 year bond (shaded yellow), and a price of $9750. Looking at the table above we see that at 7.5% yield to maturity, the price is $9726.15. So the yield to maturity is slightly under 7.5% (7.37% to be more precise).
Click below for the "high tech" ways to solve this example:
| Financial Calculator: TI BA II+ | Excel spreadsheet |
Looking at the bond table in example 3, there are 3 important points to be made about the relationship between bond prices, maturity, and the yield to maturity:
- The yield to maturity equals the coupon rate ONLY when the bond price equals the face value of the bond.
- When the bond price is less than the face value (the bond sells at a discount), the yield to maturity is greater than the coupon rate. When the bond price is greater than the face value (the bond sells at a premium), the yield to maturity is less than the coupon rate.
- The yield to maturity is inversely related to the bond price. Bond prices and market interest rates move in opposite directions. Why? As interest rates rise, new bonds will pay higher coupon rates than existing bonds. The prices of existing bonds fall in the secondary market, so the yield to maturity rises. This negative relationship between interest rate and value is true for all debt securities, not just coupon bonds.
Consols promise interest payments forever, but never repay principal. Consols are fairly rare and are issued by governments, since they are the only entities that can realisticly promise interest payments forever. (The U.S. government does not issue consols, but the French government has.) The price of the consol is the present value of future payments, but the number of future payments are infinite. If i <1, then this infinite series converges to a finite amount (your book derives this on page 122 if you are curious):
Bond Yields
We see from examples above that calculating the bond price is based on knowing the yield. It is also true that we can calculate a yield based on the bond price. We use the term yield and interest rate interchangeably.
Yield to Maturity
The yield to maturity is the interest rate that makes the discounted value of the future payments from a debt instrument equal to its current value (market price) today. It is the yield bondholders receive if they hold a bond to its maturity.
Looking at the bond table in example 3, there are 3 important points to be made about the relationship between bond prices, maturity, and the yield to maturity:
- The yield to maturity equals the coupon rate ONLY when the bond price equals the face value of the bond.
- When the bond price is less than the face value (the bond sells at a discount), the yield to maturity is greater than the coupon rate. When the bond price is greater than the face value (the bond sells at a premium), the yield to maturity is less than the coupon rate.
- The yield to maturity is inversely related to the bond price. Bond prices and market interest rates move in opposite directions. Why? As interest rates rise, new bonds will pay higher coupon rates than existing bonds. The prices of existing bonds fall in the secondary market, so the yield to maturity rises. This negative relationship between interest rate and value is true for all debt securities, not just coupon bonds.
The yield to maturity is the truest measure of the interest rate, and very useful in comparing different debt securities. However there are other measure out there developed for their computational convenience. It this day of cheap computing, it is easy to forget that calculators were not available until 1975 (and then cost $200 for one that could just do arithmetic!). Bonds traded long before that, so traders used yield measures that approximated the yield to maturity but were easier to calculate.
The current yield is an approximation used for coupon bonds. It is simply the annual coupon payment divided by the price of the bond:
where C is the annual coupon payment and P is the bond price. This is obviously a lot simpler that the yield to maturity
The current yield is a better approximation
- for longer maturity bonds and
- when the price of the bond is close to its face value.
example 4: Consider a 2-year Treasury note with a face value of $10,000, a coupon rate of 6%, and a price of $9750.
the current yield is
Recall that the true yield to maturity, from example 3, is 7.37%. So in this example, the approximation is lousy because it is only a 2-year bond and it is selling at 25% below its face value.
Holding Period Return
The yield to maturity assumes that the bond is held until maturity. If that is not true, then fluctuations in the bond price (which occur with interest rate fluctuations) will affect the return, or the gain to the investor from holding this security.
The return for holding a bond between periods t and t+1 is
where Pt is the initial price and Pt+1 is the price at the end of the holding period.We can rewrite this formula as
The last term is the rate of capital gain, g, or the change in the bond price relative to the initial bond price. So a bond's return can be rewritten as
A bond's return is identical to the yield to maturity if the holding period is identical to the time left to maturity.
Thanks for sharing about bond,
ReplyDelete3 year fixed rate bond