Chapter 8. Valuation of Known Cash Flows: Bonds

1. Chapter 8. Valuation of Known Cash Flows: Bonds Objective Explain the principles of bond pricing Understand the features that affect bond prices

2. Using Present Value Formulas to Value Known Flows The Basic Building Blocks: Pure Discount Bonds Coupon Bonds, Current Yield, and Yield-To-Maturity Reading Bond Listings Why Yields for the Same Maturity Differ The Behavior of Bond Prices over Time Chapter 8 Contents

3. Essence of valuation process To estimate an asset’s market value using information about the prices of comparable assets. Valuation models A quantitative method used to infer an asset’s value from market information about the prices of other assets and market interest rates. Fixed-income securities and other contracts promising a stream of known future cash payments Bonds Mortgages Pension annuities Valuation and Fixed-Income Securities

4. To have an agreed-upon valuation procedure in setting the terms of the contracts at the outset. To revaluate the securities when they are sold before maturity. Reasons for Valuing Fixed-Income Securities

5. A fixed-income security that promises to pay $100 each year for the next three years. The appropriate discount rate is 6% per year. An hour after you buy the security, the risk-free interest rate rises from 6% to 7% per year. Using Present Value Formulas to Value Known Cash Flows

6. Bond Prices Fall as the Interest Rates Rise • Write the PV of the fixed-income security as the sum terms

7. The Difficulty of Valuation of Known Cash Flows • We do not know usually which discount rate to use in the present value formula. • Is it correct to use the interest rate corresponding to a three-year maturity in valuing the three-year annuity in the previous example?

9. The difficulties of finding equivalent fixed-income securities, or comparables and making adjustments for differences. Any fixed-income security can be decomposed into a series of known payments at different time points in the future. Pure discount bonds (zero-coupon bonds): Promising a single payment of cash at the maturity date (in the future). The Basic Building Blocks: Pure Discount Bonds

10. Pure Discount Bonds • The pure discount bond is an example of the present value of a lump sum equation we analyzed in Chapter 4. • Solving this, the yield-to-maturity on a pure discount bond is given by the relationship:

11. Pure Discount Bonds • In this equation, • P is the present value or price of the bond • F is the face or future value • n is the investment period • i is the yield-to-maturity

12. Pure Discount Bonds • A two-year pure discount bond with a face value of $1,000 and a price of $880

13. A 3-year bond with a face value of $1,000 that makes annual coupon payments at a coupon rate 10% Prices of pure discount bonds Pricing a Coupon Bond

14. First Solution Method

15. Second Solution Method

16. The YTM of the Coupon Bond • It would be a mistake to discount all three cash flows using the same three-year yield of 7.28%. • The single discount rate that we can use to discount all three cash flows is the yield-to-maturity (YTM). • However, can we get it?

17. Coupon Stripping • You would like to create a 2-year synthetic zero-coupon bond. • Assume you are aware of the following information: • 1-year zero-coupon bonds are trading for $0.93 per dollar of face value, and • 2-year 7% coupon bonds (annual payments) are selling at $985.30 (Face value = $1,000). • Assume you can purchase the 2-year coupon bond and unbundle the two cash flows and sell them. • You would receive .93×$70 = $65.10 from the sale of the first payment. • To break even, you would need to receive $985.30- $65.10 = $920.20 from the sale of the 2-year strip.

18. Output Security 6-month zeros Input Security Decompose the CFs n-year coupon treasure bond 1-year zeros Investment Bank Term Intermediation  n-year zeros Compensation for reinvestment risk Single term Low value No compensation for reinvestment risk Multiple terms Secondary market High value The Principle of STRIPs

19. The Development of STRIPs • In 1982, Merrill Lynch: TIGRs—Treasury Investment Growth Receipts. • Follow up: Salomon Brother’s Certificates of Accrual on Treasury Security (CATs)、Lehman Investment Opportunity Notes (LIONs) —‘Animals’. • In 1984, American government: STRIPS—Separate Trading of Registered Interest and Principal of Securities. • In 1985, the outstanding face value is over 100 billion dollars.

20. Coupon Rate • Coupon rate is the interest rate applied to the face value to compute the coupon payment. • A bond with a face value of $1,000 and a coupon rate of 10% • An annuity component of $100 per year and a “balloon” or “bullet” payment at maturity

21. Current Yield and Yield-to-maturity • Current yield is the annual coupon divided by the bond’s price. • Yield-to-maturity is the discount rate that makes the present value of a bond’s stream of promised cash payments equal to its price.

22. A 20-year-maturity bond with a face value of $1,000 and a coupon rate of 10% was originally issued 19 years ago. At that time, the yield curve was flat at 10% per year. Now the interest rate on one-year bonds is 5% per year. Example 1 • Its market price will now be • Its current yield is • Its yield-to-maturity is

23. A bond with a face value of $1,000 and a coupon rate of 4% will mature in two years. Its market price is $950. Its current yield is Its yield-to-maturity Example 2

24. Bonds Trading at Par • Bond Pricing Principle #1: (Par Bonds) • If a bond’s price equals its face value, then its yield-to-maturity = current yield = coupon rate. • Proof:

25. Bonds Trading at Premium or Discount • Bond Pricing Principle #2: (Premium Bonds) • If a bond has a price higher than its face value, then its yield-to-maturity < current yield < coupon rate. • Bond Pricing Principle #3: (Discount Bonds) • If a bond has a price lower than its face value, then its yield-to-maturity > current yield > coupon rate.

26. Proof:

28. You have $10,000 to invest for one year. You are deciding between: Putting your money in a one-year government-insured bank CD offering an interest rate of 5%; Investing in the shares of a U.S. Treasure bond fund that holds one-year bonds with a coupon rate of 8%. The bonds are selling at a premium: you must pay $10,285.71 for $10,000 of face value. The fund advertises a yield of 7.78%. The fund charges a 1% annual fee for their services. Beware of “High-Yield” US Treasure Bond Funds

29. Beware of “High-Yield” US Treasure Bond Funds

30. Two different two-year coupon bonds—one with a coupon rate of 5% and the other with a coupon rate of 10%. The current market prices and yields of one- and two-year pure discount bonds: The Effect of Coupon Rate

31. The market prices of the two coupon bonds should be For the 5%-coupon bond: For the 10%-coupon bond: The yields to maturity on the coupon bonds should be For the 5%-coupon bond, the YTM is 5.9500% For the 10%-coupon bond, the YTM is 5.9064% When the yield curve is not flat, bonds of the same maturity with different coupon rates have different yields to maturity. The Effect of Coupon Rate

32. Default risk Taxes Callability Convertibility Other Effects on Bond Yields

33. The Effect of the Passage of Time • A 20-year pure discount bond with a face value of $1,000 and a constant yield of 6% should be priced at • After one year goes by, its price should be • If the yield curve were flat and interest rates did not change, any default-free discount bond’s price would rise with the passage of time, and any premium bond’s price would fall.

34. Movement of a Pure discount Bond’s Price over Time

35. Interest-Rate Risk • The concept • The sensitivity of bond prices to interest rates • The prices of 30-year 8% coupon par bond would fall by roughly 10% if the level of interest rates were to rise from 8% to 9%. • The prices of 30-year pure discount bond would fall by roughly 23% if the level of interest rates were to rise from 8% to 9%. • Why?

36. Sensitivity of Bond Price to Interest Rates

37. Duration and Modified Duration

38. The Duration of A Bond Portfolio

39. An Illustration • An pension fund is selling a new insurance policy (pension annuity), which promises an annual payment of $100 for 15 years. • At the discount rate of 10%, the PV of the liability is $760.61, and the modified duration is 5.708.

40. Continued…… • The pension fund will invest the $760.61, requiring at least a return of 10%. • There are two instruments: A 30-year treasure bond paying an interest rate of 12% and selling at par, and a 6-month treasure bill offering an interest rate of 8% per year. The duration for the two securities are 8.080 and 0.481 respectively. • Consider investing in a portfolio of the two treasure securities: • The rate of return on the portfolio = 10.75%. • When the interest rate increases by 0.1%, the change of the liability value = -4.32, and the value of the 30-year bond and 6-month bill will change by -4.2 and -0.12 respectively, and the total of both changes accounts for -4.32.