Bond Valuation, Yields, and Duration

1. Bond Valuation, Yields, and Duration Judson W. Russell, Ph.D., CFA University of North Carolina-Charlotte

2. Valuation Review • Value is simply the present value of expected future cash flows • Finding the Present Value of Cash Flows (Discounting)

3. Valuation Review • “Time value of money” says: A dollar in the future is worth less than a dollar today • How much less depends on: • Current required rate of return based on the risk of receiving the cash flow • How long we wait for the dollar

4. Valuation Review • If the current rate is 7%, then the PDV of $100 in: • Present discounted value says: A dollar in the future with more risk is worth less than a dollar in the future with less risk

5. Bond Pricing Basics • The price (value) of a bond is just the present value of all of its future cash flows. • Pricing Pure Discount (Zero-Coupon) Bonds A discount bond is an obligation that is issued at a price lower than its face (or maturity) value and that pays no coupon. The interest or income component is the difference between the bond's issue price and the bond's maturity value. Pure discount bonds promise to pay their face value upon maturity. A U.S. savings bond and a U.S. Treasury Strip are examples of discount bonds.

6. Bond Pricing Basics • The formula for pricing a pure discount bond is given below. P = where: • Cm = cash flow at maturity (face value) • P = price of the bond • t = number of years to maturity • r = market rate of interest (yield)

7. Bond Pricing Basics Pricing Coupon Bonds Coupon bonds make regular coupon payments and pay their face value at maturity • where: • Ct= cash flow in period t • m = the periods until maturity

8. Definitions: • Premium Bond - Market price exceeds face value • Discount Bond - Face value exceeds market price • Par Bond - Market price equals face value

9. Relationship between the Bond yield and the Bond price A bond's yield and the price of the bond are inversely related. For example, a five-year bond selling for $1,000 with an 8 percent coupon offers a current yield of 8 percent. If the general level of interest rates should rise to 10 percent, newly issued bonds will have coupons that are approximately 10 percent. The 8 percent bond will be less attractive to investors who can earn a higher return simply by purchasing the newly issued bonds offering the higher coupon. As investors unload the older issue, the price of the 8 percent bond will fall until its yield is approximately equal to that of the 10 percent bond.

10. Relationship between the Bond yield and the Bond price • If interest rates fall to 6 percent, then newly issued bonds will have coupons that are approximately 6 percent. The 8 percent bond will be more attractive, and as more investors demand the 8 percent bond, its price will rise and its yield will fall until it is approximately equal to that of the 6 percent bond. The example below illustrates the relationship between bond prices and bond yields under different interest rate assumptions.

11. Example of the Effect of Market Interest Rates on Bond Prices: Principal amount: $1,000 Coupon: 8% Payment frequency: Semiannual Time to maturity: Five years Market interest rateBond price 6% $1,085.30 8% $1,000.00 10% $922.78

12. Bond Pricing Principles • A decrease in interest rates creates an increase in the price of any bond, and any increase in interest rates will cause the price of any bond to fall, but the amount of the price change depends on the particular features of the bond. • The longer the maturity of a bond, the more sensitive is its price to a change in interest rates, holding other factors constant.

13. Bond Pricing Principles • The price sensitivity of any bond increases with its maturity, but the increase occurs at a decreasing rate. • The lower the coupon rate on a bond, the more sensitive is its price to a change in interest rates, holding other factors constant.

14. Duration: A measure of the price sensitivity of a bond • Duration is a measure of the bond’s sensitivity to changes in interest rates. It reflects the bond’s maturity, coupon rate and yield. Duration is calculated as follows • Using the duration computation, we can then approximate the price change of a bond for a given change in interest rate using the following formula Change in Price =

15. Prices for Bonds A & B for Various Yields • (Bond A: 30-Year, 10% Coupon; Bond B: 5-Year, 10% Coupon)

16. Duration: A measure of the price sensitivity of a bond • Duration Example: 3 Year 8% Annual Coupon Bond Yield = 11% Face Value = $1,000 • Duration = Sum of the Time Weighted PV of Cash Flows / Price = 2571.00/926.69 = 2.7745

17. Duration: A measure of the price sensitivity of a bond • The Duration Calculationrepresents the approximate % change in a bond price for a 1% change (100 basis points) in interest rates: • For Example: • Assume a 1% increase in interest rates • The 20-year zero coupon bond with a duration of 20 will decrease in price by approximately 20%. • The three-year coupon bond with a duration of 2.7745 will decrease in price by approximately 2.7745%.

18. Duration: A measure of the price sensitivity of a bond • The Duration Calculationrepresents the approximate % change in a bond price for a 1% change (100 basis points) in interest rates: • For Example: • Assume a 1% decrease in interest rates • The 20-year zero coupon bond with a duration of 20 will increase in price by approximately 20%. • The three-year coupon bond with a duration of 2.7745 will increase in price by approximately 2.7745%.

19. Interest Rate and Yield Curve Analysis:Understanding the Term Structure of Interest Rates Judson W. Russell, Ph.D., CFA University of North Carolina-Charlotte

20. The Spot Yield Curve • The term structure of interest rates is simply the relationship between the yield-to-maturity on similar credit quality bonds and the bond’s time left to maturity. • The yield curve depicts the term structure. The yield curve for Treasury bonds is most often used since Treasury bonds are essentially default risk-free; thus, they serve as a benchmark for yields on other bonds with varying levels of risk. • The shape of the yield curve (term structure) is important because it provides information about the future level of interest rates.

21. Four Hypothetical Yield Curves

22. Taking apart the Yield Curve to Calculate Forward Rates • A spot rate of interest is the yield prevailing on a bond for immediate purchase. • Interest rates for future time periods are known as forward rates. • Current (spot) interest rates on bonds are averages of the current rates and the forward rates of interest. • Forward rates are calculated assuming that expected returns over a given time period should be equal regardless of the maturities of the bonds held over that time.

23. Taking apart the Yield Curve to Calculate Forward Rates

24. Theories of the Term Structure Once forward rates are determined, we can use the theories of the term structure to form our expectations about the level of future interest rates. The theories of the term structure are given below.

25. 1. The Pure Expectations Theory • The pure expectations theory simply sates that forward rates are the market’s expectation of the level of future interest rates. Forward Rate = Expected Future Interest Rate • This means that the shape of the term structure reflects the markets expectation of future short-term rates. • The pure expectations theory is based on the assumption that market participants who buy and sell treasury bonds are willing and able to exploit profit opportunities whenever forward rates differ form expected future rates.

26. 2. The Liquidity Preference (Premium) Theory • Prices of longer term bonds are more sensitive to changes in interest rate--as measured by duration-- therefore they are more susceptible to interest rate risk. • If investors are averse to this risk they will prefer to hold short-term bonds unless they are adequately compensated for bearing the interest rate risk inherent in longer-term bonds. • Forward Rate = Expected Rate + Liquidity Premium • Expected Rate = Forward Rate – Liquidity Premium

27. Market Segmentation Theory • The yield curve reflects the actions and preferences of the major participants in the market. Institutions have strong maturity preferences: a) Commercial banks dominate demand for short-term securities b) Insurance Companies dominate demand for long term securities.

28. Market Segmentation Theory • Insurance demand is stable over time • Commercial bank demand is less stable, • During strong business activity, banks sell short-term bond to accommodate loans--so short-term spot rates rise. • During weak business activity, banks buy short term bond --so short term spot rates decline.

29. Deriving Expected Future Rates: Pure Expectations Theory

30. Deriving Expected Future Rates: Liquidity Preference Theory

31. Liquidity Premium in the Term Structure

32. Liquidity Preference vs. Expectations

33. Market Segmentation

34. The Default Risk Structure of Interest Rates and the Term Structure • The risk structure of interest rates analyzes the difference in risk among varying classes of bonds. • In comparing risk-free treasury bonds with risky bonds -- the yield differential is known as the risk premium. • Bond Interest Rate = Treasury Rate + Risk premium

35. The Default Risk Structure of Interest Rates and the Term Structure • Risk Premium is based on: • Type of issuer • Issuer’s credit worthiness • Bond options callability, convertibility, etc. • Liquidity of the issue • Taxability of the interest

36. The Default Risk Structure of Interest Rates and the Term Structure • Risk premium is a measure of the risk of the bond, especially default risk. The greater the chance of default--the greater the risk premium. The greater the chance of default, the larger is the risk premium that a bond must pay to attract investors. • Rating agencies analyze bond issues to determine default risk. The bond-rating services, such as Moody's and Standard and Poor's attempt to summarize all factors that affect default risk and determine the risk premium in their bond ratings. • Expected return of a risky bond is always less than the promised return because of the chance the bond will default and the holder will receive less than the promised amount.

37. Components of Corporate Yield Spread