Andrew Jacobson Financial

What are the differences between the bonds coupon rate current yield and yield to maturity

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A bond is a long-term debt of a government or corporation. When you own a bond, you receive a fixed interest payment each year until the bond matures. This payment is known as the coupon. The coupon rate is the annual coupon payment expressed as a fraction of the bond's face value. At maturity the bond's face value is repaid. In the United States most bonds have a face value of $1,000. The current yield is the annual coupon payment expressed as a fraction of the bond's price. The yield to maturity measures the average rate of return to an investor who purchases the bond and holds it until maturity, accounting for coupon income as well as the difference between purchase price and face value.

How can one find the market price of a bond given its yield to maturity and find a bond's yield given its price? Why do prices and yields vary inversely?

Bonds are valued by discounting the coupon payments and the final repayment by the yield to maturity on comparable bonds. The bond payments discounted at the bond's yield to maturity equal the bond price. You may also start with the bond price and ask what interest rate the bond offers. This interest rate that equates the present value of bond payments to the bond price is the yield to maturity. Because present values are lower when discount rates are higher, price and yield to maturity vary inversely.

Why do bonds exhibit interest rate risk?

Bond prices are subject to interest rate risk, rising when market interest rates fall and falling when market rates rise. Long-term bonds exhibit greater interest rate risk than short-term bonds.

Why do investors pay attention to bond ratings and demand a higher interest rate for bonds with low ratings?

Investors demand higher promised yields if there is a high probability that the borrower will run into trouble and default. Credit risk implies that the promised yield to maturity on the bond is higher than the expected yield. The additional yield investors require for bearing credit risk is called the default premium. Bond ratings measure the bond's credit risk.

Related Web Links

Key Terms

www.finpipe.com/ The Financial Pipeline is an Internet site dedicated to financial education; see the page on Bonds www.investinginbonds.com/ All about bond pricing

www.bloomberg.com/markets/C13.html A look at the yield curve, updated daily www.bondmarkets.com/publications/IGCORP/what.htm A guide to corporate bonds www.moodys.com The Web site of the bond rating agency

www.standardandpoors.com/ratings/ Standard & Poor's Corporation provides information on how it rates securities bond coupon face value, par value, maturity value coupon rate current yield yield to maturity rate of return yield curve default premium investment grade junk bond credit risk default risk interest rate risk

1. Bond Yields. A 30-year Treasury bond is issued with par value of $1,000, paying interest of $80 per year. If market yields increase shortly after the T-bond is issued, what happens to the bond's:

a. coupon rate b. price c. yield to maturity d. current yield

2. Bond Yields. If a bond with par value of $1,000 and a coupon rate of 8 percent is selling at a price of $970, is the bond's yield to maturity more or less than 8 percent? What about the current yield?

3. Bond Yields. A bond with par value $1,000 has a current yield of 7.5 percent and a coupon rate of 8 percent. What is the bond's price?

4. Bond Pricing. A 6-year Circular File bond pays interest of $80 annually and sells for $950. What is its coupon rate, current yield, and yield to maturity?

5. Bond Pricing. If Circular File (see question 4) wants to issue a new 6-year bond at face value, what coupon rate must the bond offer?

6. Bond Yields. An AT&T bond has 10 years until maturity, a coupon rate of 8 percent, and sells for $1,050.

a. What is the current yield on the bond?

b. What is the yield to maturity?

7. Coupon Rate. General Matter's outstanding bond issue has a coupon rate of 10 percent and a current yield of 9.6 percent, and it sells at a yield to maturity of 9.25 percent. The firm wishes to issue additional bonds to the public at par value. What coupon rate must the new bonds offer in order to sell at par?

8. Financial Pages. Refer to Figure 3.2. What is the current yield of the 6V4 percent, August 2002 maturity bond? What was the closing ask price of the bond on the previous day?

Practice Problems

9. Bond Prices and Returns. One bond has a coupon rate of 8 percent, another a coupon rate of 12 percent. Both bonds have 10-year maturities and sell at a yield to maturity of 10 percent. If their yields to maturity next year are still 10 percent, what is the rate of return on each bond? Does the higher coupon bond give a higher rate of return?

10. Bond Returns.

a. If the AT&T bond in problem 6 has a yield to maturity of 8 percent 1 year from now, what will its price be?

b. What will be the rate of return on the bond?

c. If the inflation rate during the year is 3 percent, what is the real rate of return on the bond?

11. Bond Pricing. A General Motors bond carries a coupon rate of 8 percent, has 9 years until maturity, and sells at a yield to maturity of 9 percent.

a. What interest payments do bondholders receive each year?

b. At what price does the bond sell? (Assume annual interest payments.)

c. What will happen to the bond price if the yield to maturity falls to 7 percent?

12. Bond Pricing. A 30-year maturity bond with face value $1,000 makes annual coupon payments and has a coupon rate of 8 percent. What is the bond's yield to maturity if the bond is selling for a. $900

13. Bond Pricing. Repeat the previous problem if the bond makes semiannual coupon payments.

14. Bond Pricing. Fill in the table below for the following zero-coupon bonds. The face value of each bond is $1,000.

Price Maturity (Years) Yield to Maturity

15. Consol Bonds. Perpetual Life Corp. has issued consol bonds with coupon payments of $80. (Consols pay interest forever, and never mature. They are perpetuities.) If the required rate of return on these bonds at the time they were issued was 8 percent, at what price were they sold to the public? If the required return today is 12 percent, at what price do the consols sell?

16. Bond Pricing. Sure Tea Co. has issued 9 percent annual coupon bonds which are now selling at a yield to maturity of 10 percent and current yield of 9.8375 percent. What is the remaining maturity of these bonds?

17. Bond Pricing. Large Industries bonds sell for $1,065.15. The bond life is 9 years, and the yield to maturity is 7 percent. What must be the coupon rate on the bonds?

18. Bond Prices and Yields.

a. Several years ago, Castles in the Sand, Inc., issued bonds at face value at a yield to maturity of 8 percent. Now, with 8 years left until the maturity of the bonds, the company has run into hard times and the yield to maturity on the bonds has increased to 14 percent. What has happened to the price of the bond?

b. Suppose that investors believe that Castles can make good on the promised coupon payments, but that the company will go bankrupt when the bond matures and the principal comes due. The expectation is that investors will receive only 80 percent of face value at maturity. If they buy the bond today, what yield to maturity do they expect to receive?

19. Bond Returns. You buy an 8 percent coupon, 10-year maturity bond for $980. A year later, the bond price is $1,050.

a. What is the new yield to maturity on the bond?

b. What is your rate of return over the year?

20. Bond Returns. You buy an 8 percent coupon, 10-year maturity bond when its yield to maturity is 9 percent. A year later, the yield to maturity is 10 percent. What is your rate of return over the year?

21. Interest Rate Risk. Consider three bonds with 8 percent coupon rates, all selling at face value. The short-term bond has a maturity of 4 years, the intermediate-term bond has maturity 8 years, and the long-term bond has maturity 30 years.

a. What will happen to the price of each bond if their yields increase to 9 percent?

b. What will happen to the price of each bond if their yields decrease to 7 percent?

c. What do you conclude about the relationship between time to maturity and the sensitivity of bond prices to interest rates?

22. Rate of Return. A 2-year maturity bond with face value $1,000 makes annual coupon payments of $80 and is selling at face value. What will be the rate of return on the bond if its yield to maturity at the end of the year is a. 6 percent b. 8 percent c. 10 percent

23. Rate of Return. A bond that pays coupons annually is issued with a coupon rate of 4 percent, maturity of 30 years, and a yield to maturity of 8 percent. What rate of return will be earned by an investor who purchases the bond and holds it for 1 year if the bond's yield to maturity at the end of the year is 9 percent?

24. Bond Risk. A bond's credit rating provides a guide to its risk. Long-term bonds rated Aa currently offer yields to maturity of 8.5 percent. A-rated bonds sell at yields of 8.8 percent. If a 10-year bond with a coupon rate of 8 percent is downgraded by Moody's from Aa to A rating, what is the likely effect on the bond price?

25. Real Returns. Suppose that you buy a 1-year maturity bond for $1,000 that will pay you back $1,000 plus a coupon payment of $60 at the end of the year. What real rate of return will you earn if the inflation rate is a. 2 percent b. 4 percent c. 6 percent d. 8 percent

26. Real Returns. Now suppose that the bond in the previous problem is a TIPS (inflation-indexed) bond with a coupon rate of 4 percent. What will the cash flow provided by the bond be for each of the four inflation rates? What will be the real and nominal rates of return on the bond in each scenario?

27. Real Returns. Now suppose the TIPS bond in the previous problem is a 2-year maturity bond. What will be the bondholder's cash flows in each year in each of the inflation scenarios?

Challenge 28. Interest Rate Risk. Suppose interest rates increase from 8 percent to 9 percent. Which bond will suffer the greater percentage decline in price: a 30-year bond paying annual coupons of PROBLEM 8 percent, or a 30-year zero coupon bond? Can you explain intuitively why the zero exhibits greater interest rate risk even though it has the same maturity as the coupon bond?

Solutions to

Self-Test

Questions

1 a. The ask price is 101 23/32 = 101.71875 percent of face value, or $1,017.1875.

b. The bid price is 101 21/32 = 101.65625 percent of face value, or $1,016.5625.

c. The price increased by 1/32 = .03125 percent of face value, or $.3125.

d. The annual coupon is 6 1/4 percent of face value, or $62.50, paid in two semiannual installments.

e. The yield to maturity, based on the ask price, is given as 5.64 percent.

2 The coupon is 9 percent of $1,000, or $90 a year. First value the 6-year annuity of coupons:

Then value the final payment and add up: PV ^ $1,000 _

$506.63

$876.66

3 The yield to maturity is about 8 percent, because the present value of the bond's cash returns is $1,199 when discounted at 8 percent:

= (coupon x annuity factor) + (face value x discount factor) = $140 x

1.084

4 The 6 percent coupon bond with maturity 2002 starts with 3 years left until maturity and sells for $1,010.77. At the end of the year, the bond has only 2 years to maturity and investors demand an interest rate of 7 percent. Therefore, the value of the bond becomes

You invested $1,010.77. At the end of the year you receive a coupon payment of $60 and have a bond worth $981.92. Your rate of return is therefore

„ . . . $60 + ($981.92 - $1,010.77) „-„„ , „eo.

The yield to maturity at the start of the year was 5.6 percent. However, because interest rates rose during the year, the bond price fell and the rate of return was below the yield to maturity.

5 By the end of this year, the bond will have only 1 year left until maturity. It will make only one more payment of coupon plus face value, so its price will be $1,060/1.056 = $1,003.79. The rate of return is therefore

$60 + ($1,003.79 - $1,007.37) „„ , ,0, -( °1,007.37-^ = .°56, or 5.6%

6 At an interest rate of 5.6 percent, the 3-year bond sells for $1,010.77. If the interest rate jumps to 10 percent, the bond price falls to $900.53, a decline of 10.9 percent. The 30-year bond sells for $1,057.50 when the interest rate is 5.6 percent, but its price falls to $622.92 at an interest rate of 10 percent, a much larger percentage decline of 41.1 percent.

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