Bond Prices and Yield to Maturity
We have already seen that the price of a bond or any other financial security should equal the present value of the payments the owner receives from the bond. We can apply this concept to determine the price of a coupon bond. Bond Prices Consider a five-year coupon bond with a coupon rate of 6% and a face value of $1,000. The coupon rate of 6% tells us that the seller of the bond will pay the buyer of the bond $60 per year for five years, as well as make a final payment of $1,000 at the end of the fifth year. (Note that, in practice, coupons are typically paid twice per year, so a 6% bond will pay $30 after six months and another $30 at the end of the year.
a security are received at the end of the year.) Therefore, the expression for the price, P, of the bond is the sum of the present values of the six payments the investor will receive:
We have already seen that the price of a bond or any other financial security should equal the present value of the payments the owner receives from the bond. We can apply this concept to determine the price of a coupon bond. Bond Prices Consider a five-year coupon bond with a coupon rate of 6% and a face value of $1,000. The coupon rate of 6% tells us that the seller of the bond will pay the buyer of the bond $60 per year for five years, as well as make a final payment of $1,000 at the end of the fifth year. (Note that, in practice, coupons are typically paid twice per year, so a 6% bond will pay $30 after six months and another $30 at the end of the year.
a security are received at the end of the year.) Therefore, the expression for the price, P, of the bond is the sum of the present values of the six payments the investor will receive:
We can use this reasoning to arrive at a general expression for a bond that makes coupon payments, C, has a face value, FV, and matures in n years
The dots (ellipsis) indicate that we have omitted the terms representing the years between the third year and the nth year—which could be the tenth, twentieth, thirtieth, or other, year
Yield to Maturity
To use the expression for the price of a bond, we need information on the future payments to be received and the interest rate. Often, we know the price of a bond and the future payments, but we don’t always know the interest rate. Suppose you face a decision like this one: Which is a better investment, (1) a three-year, $1,000 face value coupon bond with a price of $1,050 and a coupon rate of 8% or (2) a two-year, $1,000 face value coupon bond with a price of $980 and a coupon rate of 6%? One important factor in making a choice between these two investments is determining the interest rate you will receive on each investment. Because we know the prices and the payments for the two bonds, we can use the present value calculation to find the interest rate on each investment:
Yield to Maturity
To use the expression for the price of a bond, we need information on the future payments to be received and the interest rate. Often, we know the price of a bond and the future payments, but we don’t always know the interest rate. Suppose you face a decision like this one: Which is a better investment, (1) a three-year, $1,000 face value coupon bond with a price of $1,050 and a coupon rate of 8% or (2) a two-year, $1,000 face value coupon bond with a price of $980 and a coupon rate of 6%? One important factor in making a choice between these two investments is determining the interest rate you will receive on each investment. Because we know the prices and the payments for the two bonds, we can use the present value calculation to find the interest rate on each investment:
Using a financial calculator, an online calculator, or a spreadsheet program, we can solve this equation for i. The solution is i = 0.061, or 6.1%
These calculations show us that even though Bond 1 may appear to be a better investment because it has a higher coupon rate than Bond 2, Bond1’s higher price means that it has a significantly lower interest rate than Bond 2. So, if you wanted to earn the highest interest rate on your investment, you would choose Bond 2. The interest rate we have just calculated is called the yield to maturity. The yield to maturity equates the present value of the payments from an asset with the asset’s price today. The yield to maturity is based on the concept of present value and is the interest rate measure that participants in financial markets use most often. In fact, it is important to note that unless they indicate otherwise, whenever participants in financial markets refer to the interest rate on a financial asset, the interest rate is the yield to maturity. Calculating yields to maturity for alternative investments allows savers to compare different types of debt instruments.
It’s useful to keep in mind the close relationship between discounting and compounding. We just calculated the yield to maturity by using a discounting formula. We can also think of the yield to maturity in terms of compounding. To do so, we need to ask, “If I pay a price, P, today for a bond with a particular set of future payments, what is the interest rate at which I could invest P and get the same set of future payments?” For example, instead of calculating the present value of the payments to be received on a 30-year Treasury bond, we can calculate the interest rate at which the money paid for the bond could be invested for 30 years to get the same present value.
Yields to Maturity on Other Debt Instruments
We saw in section 3.2 that there are four categories of debt instruments.We have seen how to calculate the yield to maturity on a coupon bond. Now we can calculate the yield to maturity for each of the other three types of debt instruments. Simple Loans Calculating the yield to maturity for a simple loan is straightforward. We need to find the interest rate that makes the lender indifferent between having the amount of the loan today or the final payment at maturity. Consider again the $10,000 loan to Nate’s Nurseries. The loan requires payment of the $10,000 principal plus $1,000 in interest one year from now. We calculate the yield to maturity as follows:
It’s useful to keep in mind the close relationship between discounting and compounding. We just calculated the yield to maturity by using a discounting formula. We can also think of the yield to maturity in terms of compounding. To do so, we need to ask, “If I pay a price, P, today for a bond with a particular set of future payments, what is the interest rate at which I could invest P and get the same set of future payments?” For example, instead of calculating the present value of the payments to be received on a 30-year Treasury bond, we can calculate the interest rate at which the money paid for the bond could be invested for 30 years to get the same present value.
Yields to Maturity on Other Debt Instruments
We saw in section 3.2 that there are four categories of debt instruments.We have seen how to calculate the yield to maturity on a coupon bond. Now we can calculate the yield to maturity for each of the other three types of debt instruments. Simple Loans Calculating the yield to maturity for a simple loan is straightforward. We need to find the interest rate that makes the lender indifferent between having the amount of the loan today or the final payment at maturity. Consider again the $10,000 loan to Nate’s Nurseries. The loan requires payment of the $10,000 principal plus $1,000 in interest one year from now. We calculate the yield to maturity as follows:
Note that the yield to maturity, 10%, is the same as the simple interest rate. From this example, we can come to the general conclusion that, for a simple loan, the yield to maturity and the interest rate specified on the loan are the same.Discount Bonds Calculating the yield to maturity on a discount bond is similar to calculating the yield to maturity for a simple loan. For example, suppose that Nate’s Nurseries issues a $10,000 one-year discount bond. We can use the same equation to find the yield to maturity on the discount bond that we did in the case of a simple loan. If Nate’s Nurseries receives $9,200 today from selling the bond, we can calculate the yield to maturity by setting the present value of the future payment equal to the value today, or $9,200 = $10,000/(1 + i). Solving for i gives:
From this example, we can write a general equation for a one-year discount bond that sells for price, P, with face value, FV. The yield to maturity is
Fixed-Payment Loans Calculating the yield to maturity for a fixed-payment loan is similar to calculating the yield to maturity for a coupon bond. Recall that fixedpaymentloans require periodic payments that combine interest and principal, butthere is no face value payment at maturity. Suppose that Nate’s Nurseries borrows$100,000 to buy a new warehouse by taking out a mortgage loan from a bank. Nate’shas to make annual payments of $12,731. After making the payments for 20 years,Nate’s will have paid off the $100,000 principal of the loan. Because the loan’s valuetoday is $100,000, the yield to maturity can be calculated as the interest rate thatsolves the equation:
Using a financial calculator, an online calculator, or a spreadsheet program, we can solve this equation to find that i = 0.112, or 11.2%. In general, for a fixed-payment loan with fixed payments, FP, and a maturity of n years, the equation is:
To summarize, if i is the yield to maturity on a fixed-payment loan, the amount of the loan today equals the present value of the loan payments discounted at rate i.
Perpetuities Perpetuities are a special case of coupon bonds. A perpetuity pays a fixed coupon, but unlike a regular coupon bond, a perpetuity does not mature. The main example of a perpetuity is the consol, which was at one time issued by the British government, although it has not issued new perpetuities in decades. Existing consols with a coupon rate of 2.5% are still traded in financial markets. You may think that computing the yield to maturity on a perpetuity is difficult because the coupons are paid forever. Actually, however, the relationship between the price, coupon, and yield to maturity is simple. If your algebra skills are sharp, see if you can derive this equation from the equation for a coupon bond that pays an infinite number of coupons
Perpetuities Perpetuities are a special case of coupon bonds. A perpetuity pays a fixed coupon, but unlike a regular coupon bond, a perpetuity does not mature. The main example of a perpetuity is the consol, which was at one time issued by the British government, although it has not issued new perpetuities in decades. Existing consols with a coupon rate of 2.5% are still traded in financial markets. You may think that computing the yield to maturity on a perpetuity is difficult because the coupons are paid forever. Actually, however, the relationship between the price, coupon, and yield to maturity is simple. If your algebra skills are sharp, see if you can derive this equation from the equation for a coupon bond that pays an infinite number of coupons
So, a perpetuity with a coupon of $25 and a price of $500 has a yield to maturity
of i = $25/$500 = 0.05, or 5%.
of i = $25/$500 = 0.05, or 5%.
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