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1. Bond valuation The application of the present value concept Financial management: lecture 4

2. Today’s plan • Review of what we have learned in the last lecture • Interest rates and compounding • Some terminology about bonds • Value bonds • The yield curve • Default risk Financial management: lecture 4

3. What have we learned in the last lecture? • The present value formulas of perpetuity and annuity • The application of the PV of annuity Financial management: lecture 4

4. My solution Ending balance Total payment Interest payment Principle payment year Beginning balance 0 \$20,000 \$1,500 \$6,191 \$7,691 \$13,809 1 7,154 13,809 1,036 6,655 7,691 2 7,154 7,691 0 7,154 537 3 Financial management: lecture 4

5. A problem • John is 65 years old and wants to retire next year. After retirement, he wants to have an annual income of \$24,000 for 20 years from his retirement fund, which has an annual interest rate of 6%. Suppose John will get the first retirement income one year from now. Then • How much money should John have in his retirement fund in the end of this year? • Suppose John started to work 19 years ago and put the same amount of money every year in his retirement fund. How much should he put every year? ( including this year, there will be a total of 20 years) Financial management: lecture 4

6. Nominal and real interest rates • Nominal interest rate • What is it? • Real interest rate • What is it? • Inflation • What is it? • Their relationship • 1+real rate =(1+nominal rate)/(1+inflation) Financial management: lecture 4

7. Inflation rule • Be consistent in how you handle inflation!! • Use nominal interest rates to discount nominal cash flows. • Use real interest rates to discount real cash flows. • You will get the same results, whether you use nominal or real figures Financial management: lecture 4

8. Example You own a lease that will cost you \$8,000 next year, increasing at 3% a year (the forecasted inflation rate) for 3 additional years (4 years total). If discount rates are 10% what is the present value cost of the lease? Financial management: lecture 4

9. Inflation Example - nominal figures Financial management: lecture 4

10. Inflation Example - real figures Financial management: lecture 4

11. Interest • Simple interest - Interest earned only on the original investment. • Compounding interest - Interest earned on interest. • In Bus 785, we consider compounding interest rates Financial management: lecture 4

12. Simple interest Example Simple interest is earned at a rate of 6% for five years on a principal balance of \$100. Financial management: lecture 4

13. Simple interest Today Future Years 12345 Interest Earned 6 6 6 6 6 Value 100 106 112 118 124 130 Value at the end of Year 5 = \$130 Financial management: lecture 4

14. Compound interest Example Compound interest is earned at a rate of 6% for five years on \$100. Today Future Years 1 2 34 5 Interest Earned 6.00 6.36 6.74 7.15 7.57 Value 100 106.00 112.36 119.10 126.25 133.82 Value at the end of Year 5 = \$133.82 Financial management: lecture 4

15. Interest compounding • The interest rate is often quoted as APR, the annual percentage rate. • If the interest rate is compounded m times in each year and the APR is r, the effective annual interest rate is Financial management: lecture 4

16. Compound Interest i ii iii iv v Periods Interest Value Annually per per APR after compounded year period(i x ii) one year interest rate 1 6% 6% 1.06 6.000% 2 3 6 1.032 = 1.0609 6.090 4 1.5 6 1.0154 = 1.06136 6.136 12 .5 6 1.00512 = 1.06168 6.168 52 .1154 6 1.00115452 = 1.06180 6.180 365 .0164 6 1.000164365 = 1.06183 6.183 Financial management: lecture 4

17. Compound Interest Financial management: lecture 4

18. Interest Rates Example Given a monthly rate of 1% (interest is compounded monthly), what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? Financial management: lecture 4

19. Solution Financial management: lecture 4

20. Interest Rates Example If the interest rate 12% annually and interest is compounded semi-annually, what is the Effective Annual Rate (EAR)? What is the Annual Percentage Rate (APR)? Financial management: lecture 4

21. Solution • APR=12% • EAR=(1+0.06)2-1=12.36% Financial management: lecture 4

22. Bonds • Bond – a security or a financial instrument that obligates the issuer (borrower) to make specified payments to the bondholder during a time horizon. • Coupon - The interest payments made to the bondholder. • Face Value (Par Value, Face Value, Principal or Maturity Value) - Payment at the maturity of the bond. • Coupon Rate - Annual interest payment, as a percentage of face value. Financial management: lecture 4

23. Bonds • A bond also has (legal) rights attached to it: • if the borrower doesn’t make the required payments, bondholders can force bankruptcy proceedings • in the event of bankruptcy, bond holders get paid before equity holders Financial management: lecture 4

24. An example of a bond • A coupon bond that pays coupon of 10% annually, with a face value of \$1000, has a discount rate of 8% and matures in three years. • The coupon payment is \$100 annually • The discount rate is different from the coupon rate. • In the third year, the bondholder is supposed to get \$100 coupon payment plus the face value of \$1000. • Can you visualize the cash flows pattern? Financial management: lecture 4

25. Bonds WARNING The coupon rate IS NOT the discount rate used in the Present Value calculations. The coupon rate merely tells us what cash flow the bond will produce. Since the coupon rate is listed as a %, this misconception is quite common. Financial management: lecture 4

26. Bond Valuation The price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return. Financial management: lecture 4

27. Zero coupon bonds • Zero coupon bonds are the simplest type of bond (also called stripped bonds, discount bonds) • You buy a zero coupon bond today (cash outflow) and you get paid back the bond’s face value at some point in the future (called the bond’s maturity ) • How much is a 10-yr zero coupon bond worth today if the face value is \$1,000 and the effective annual rate is 8% ? Face value PV Time=0 Time=t Financial management: lecture 4

28. Zero coupon bonds (continue) • P0=1000/1.0810=\$463.2 • So for the zero-coupon bond, the price is just the present value of the face value paid at the maturity of the bond • Do you know why it is also called a discount bond? Financial management: lecture 4

29. Coupon bond The price of a coupon bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return. Financial management: lecture 4

30. Bond Pricing Example What is the price of a 6 % annual coupon bond, with a \$1,000 face value, which matures in 3 years? Assume a required return of 5.6%. Financial management: lecture 4

31. Bond Pricing Example What is the price of a 6 % annual coupon bond, with a \$1,000 face value, which matures in 3 years? Assume a required return of 5.6%. Financial management: lecture 4

32. Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 6 %? Financial management: lecture 4

33. Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 15 %? Financial management: lecture 4

34. Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually? Financial management: lecture 4

35. Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually? Financial management: lecture 4

36. Bond Pricing Example (continued) Q: How did the calculation change, given semi-annual coupons versus annual coupon payments? Financial management: lecture 4

37. Bond Pricing Example (continued) Q: How did the calculation change, given semi-annual coupons versus annual coupon payments? Time Periods Paying coupons twice a year, instead of once doubles the total number of cash flows to be discounted in the PV formula. Financial management: lecture 4

38. Bond Pricing Example (continued) Q: How did the calculation change, given semi-annual coupons versus annual coupon payments? Time Periods Paying coupons twice a year, instead of once doubles the total number of cash flows to be discounted in the PV formula. Discount Rate Since the time periods are now half years, the discount rate is also changed from the annual rate to the half year rate. Financial management: lecture 4

39. Bond Yields • Current Yield - Annual coupon payments divided by bond price. • Yield To Maturity (YTM)- Interest rate for which the present value of the bond’s payments equal the market price of the bond. Financial management: lecture 4

40. An example of a bond • A coupon bond that pays coupon of 10% annually, with a face value of \$1000, has a discount rate of 8% and matures in three years. It is assumed that the market price of the bond is the same as the present value of the bond. • What is the current yield? • What is the yield to maturity. Financial management: lecture 4

41. My solution • First, calculate the bond price • P=100/1.08+100/1.082+1100/1.083 • =\$1,051.54 • Current yield=100/1051.54=9.5% • YTM=8% Financial management: lecture 4

42. Bond Yields Calculating Yield to Maturity (YTM=r) If you are given the market price of a bond (P) and the coupon rate, the yield to maturity can be found by solving for r. Financial management: lecture 4

43. Bond Yields Example What is the YTM of a 6 % annual coupon bond, with a \$1,000 face value, which matures in 3 years? The market price of the bond is \$1,010.77 Financial management: lecture 4

44. Bond Yields • In general, there is no simple formula that can be used to calculate YTM unless for zero coupon bonds • Calculating YTM by hand can be very tedious. We don’t have this kind of problems in the quiz or exam • You may use the trial by errors approach get it. Financial management: lecture 4

45. Bond Yields (3) • Can you guess which one is the solution in the previous example? • 6.6% • 7.1% • 6.0% • 5.6% Financial management: lecture 4

46. The bond price, coupon rates and discount rates • If the coupon rate is larger than the discount rate, the bond price is larger than the face value. • If the coupon rate is smaller than the discount rate, the bond price is smaller than the face value. Financial management: lecture 4

47. The rate of return on a bond Example: An 8 percent coupon bond has a price of \$110 dollars with maturity of 5 years and a face value of \$100. Next year, the expected bond price will be \$105. If you hold this bond this year, what is the rate of return? Financial management: lecture 4

48. My solution • The expected rate of return for holing the bond this year is (8-5)/110=2.73% • Price change =105-110=-\$5 • Coupon payment=100*8%=\$8 • The investment or the initial price=\$110 Financial management: lecture 4

49. The Yield Curve Term Structure of Interest Rates - A listing of bond maturity dates and the interest rates that correspond with each date. Yield Curve - Graph of the term structure. Financial management: lecture 4

50. The term structure of interest rates (Yield curve) Financial management: lecture 4