1. BOND PRICES AND INTEREST RATE RISK CHAPTER 5

2. The Time Value of Money: Copyright© 2006 John Wiley & Sons, Inc. 2 Time value of money is based on the belief that people have positive time preference for consumption; that is people prefer to consume goods today rather than consume similar goods in the future. A dollar today worth more than a dollar received at some future date. Because if you have a dollar today, you can invest it and earn interest.

3. Future Value or Compound Value Copyright© 2006 John Wiley & Sons, Inc. 3 The future value (FV) of a sum (PV) is FV = PV (1+i) n where i is the periodic interest rate and n is the number of compounding periods.

4. Present Value Copyright© 2006 John Wiley & Sons, Inc. 4 The value now of a sum expected at a future time is given by With risk present, a premium may be added to the risk- free rate. The higher the discount rate, the lower the present value.

5. Bond Pricing: What is a bond? Copyright© 2006 John Wiley & Sons, Inc. 5 A form of loan—a debt security obligating a borrower to pay a lender principal and interest. Borrower (issuer) promises contractually to make periodic payments to lender (investor or bondholder) over given number of years At maturity, holder receives principal (or face value or par value). Periodically before maturity, holder receives interest (coupon) payments determined by coupon rate, original interest rate promised as percentage of par on face of bond.

6. Copyright© 2006 John Wiley & Sons, Inc. 6 Two types of contractual cash flows: 1. The principal, face value or par value (upon the maturity) 2. The coupon payment (C), which is determined by the coupon rate (c). Coupon rate = coupon payment / face rate c = C / F so, C = c * F and F = C / c

7. What is a bond? Example Copyright© 2006 John Wiley & Sons, Inc. 7 Par value$1,000 Coupon Rate 5% Issued Today Matures30 years from today Scheduled Payments:$50/year interest for 30 years $1,000 par at end of year 30

8. Bond Pricing: bond cash flows Copyright© 2006 John Wiley & Sons, Inc. 8 Bondholder thus owns right to a stream of cash flows: Ordinary annuity of interest payments and Future lump sum in return of par value, Discountable to a present value at any time while bond is outstanding.

9. Bond Pricing: Present Value Copyright© 2006 John Wiley & Sons, Inc. 9 The value (price) of a bond is the present value of the future cash flows promised, discounted at the market rate of interest (the required rate of return on this risk class in today’s market rate)

10. PV of bond cash flows Copyright© 2006 John Wiley & Sons, Inc. 10 WherePB = price of bond or present value of promised payments; Ct = coupon payment in period t, where t = 1, 2, 3,…, n; Fn = par value (principal amount) due at maturity; i = market interest rate (discount rate or market yield); and n = number of periods to maturity.

11. Notes: 1.The coupon rate and the market rate of interest may differ. 2. The coupon rate, par value and term to maturity are fixed throughout the life of a bond. 3. The market interest rate (yield) on a bond varies with changes in: A.the demand and supply of credit and / or B. the issuer’s risk. 11

12. Bond pricing: principles Copyright© 2006 John Wiley & Sons, Inc. 12  Cash flows are assumed to flow at end of the period and to be reinvested at i. Bonds typically pay interest semiannually.  Increasing i decreases price (PB); decreasing i increases price; thus bond prices and interest rates move inversely.  If market rate equals coupon rate, bond trades at par. Par bond, if the price of the bond = the par value.  If market rate is lower than the coupon rate, the bond trades above par—at a premium. Premium, if the price of the bond > the par value.  If market rate exceeds coupon rate, bond trades below par—at a discount. Discount, if the price of the bond < the par value.

13. NOT annual Compounding Where m is the number of times coupon payments are made each year. In case of semi-annual coupon payments, m = 2. In case of quarterly coupon payments, m = 4. In case of monthly coupon payments, m= 12. 5 13

14. Example 1: If a bond has an 5% coupon rate (semiannually), a 3 –year maturity and similar bonds are selling for an 6 % yield, what is the price of the bond if the face value is BD 1000? = 972.91 14

15. Zero coupon bonds are “pure discount” instruments. Copyright© 2006 John Wiley & Sons, Inc. 15  No periodic coupon payments.  Issued at discount from par.  Single payment of par value at maturity.  The interest rate = the price paid for the security – the amount received upon maturity.  The example of zero coupon bonds are US treasury bills and US savings bonds.

16. Where: PB = price of the zero coupon bond. Fn = amount of cash payments at maturity. i = interest rate (yield) for n periods. n = number of years until the payment is due. m = number of times interest is compounded each year Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 16

17. Example 1: A zero coupon bond with a BD 2,000 face value, 10 – year maturity, annually compounding. What is the price if the market interest rate is 12 %? PB=643.94 Example 2: I n example 1, assume it is a semiannually compounding. Calculate the price. PB= 623.60 Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 17

18. Bond yields: risks rewarded Copyright© 2006 John Wiley & Sons, Inc. 18 Yield rewards the bondholder for at least 3 risks:  Credit or default risk: chance that issuer may be unable or unwilling to pay as agreed.  Reinvestment risk: It is the risk caused by the changes in the market interest rates, causing the lender to reinvest coupon payments at interest rates different from the interest rate at the time the bond is purchased.  Price risk: It is the risk caused by the changes in the market interest rates,causing the market value of a bond to change, resulting in capital gains or losses to the investor.

19. Common yield measures Copyright© 2006 John Wiley & Sons, Inc. 19 Yield to Maturity Realized Yield Expected Yield Total Return

20. Yield to maturity Copyright© 2006 John Wiley & Sons, Inc. 20 Investor's expected yield if bond is held to maturity,all coupon and principal payments are made as promised and all coupons payments are reinvested at promised yield for the remaining term to maturity. If the coupon payments are reinvestment at lower rate, what is this risk called? Normally determined by trial and error—try different discount rates until PB=present value of future payments. Similar to IRR of a capital project.

21. Computing yield to maturity Copyright© 2006 John Wiley & Sons, Inc. 21 Investor buys 5% percent coupon (semiannual payments) bond for $951.90; bond matures in 3 years. Solve the bond pricing equation for the interest rate (i) such that price paid for the bond equals PV of remaining payments due under the bond.

22. The formula is as follows: Where PB = price of bond or present value of promised payments; C = coupon payment, Fc = Face value (principal amount) due at maturity; n = number of periods to maturity. Note: If its compounded not annually, you have to multiply n by m. (Fc + PB) / 2 is called average investment value Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 22

23. Computing yield to maturity, cont. Copyright© 2006 John Wiley & Sons, Inc. 23 Solving either by trial and error or with a financial calculator or the formula in the previous slide, results in yield to maturity of 3.4% semiannually, or 6.8% annually. If Bond price is $951.90, face value is $1000,maturity is 3 years,5% coupon rate (compounded semiannually),what is the yield to maturity ?

24. Examples 1: Textbook page 134 No. 4 What is the yield – to maturity of a corporate bond with a 3 – year maturity, 5% coupon (semiannual payments), and BD 1000 face value if the bond sold for BD 978.30? Yield to maturity = 2.89 % semiannual or 5.78% annual. 2:What is the yield – to maturity of a corporate bond with a 8 – year maturity, 6% coupon (annual payments), and BD 1000 face value if the bond sold for BD 978? Yield to maturity = 6.34% annual. Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 24

25. Summary The yield to maturity is what return the investor earn on a bond if: 1.The borrower makes all cash payments as promised, 2.Interest rates do not change over the bond’s maturity, and 3.The investor holds the bond to maturity. Not occur of one or more of the above will lead to the actual return on a bond will be different from the promised yield. The actual return on a bond called realized yield. Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 25

26. Realized Yield Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 26 Investor pays $1,000 for 10-year 8% coupon bond; after 3 years, he sells the bond. At the time he sells the bond, 7 years bonds with similar characteristics sell at yield of 10%. What is the realized yield? Steps for solution: 1- calculate the PB for the bond BUT till 7 years. 2- calculate the yield to maturity BUT for 3 years (period to maturity) and by using the PB from (1) as principal (face value) will be received ay maturity. The calculation of yield is done by using the yield to maturity formula.

27. Calculation the PB results (BD 902.63). Step 1 : = BD 902.63 (This price is the selling price). Step 2 : (BD1000 is The purchase price ) = 4.91 % Capital losses = Selling price – Purchase price=BD 902.63 – BD 1000 = BD - 97.37 Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 27

28. Expected yield Copyright© 2006 John Wiley & Sons, Inc. 28 Expected yield is calculated exante before the event/fact. Predicted yield for a given holding period (same procedure as YTM, but for some holding period shorter than maturity) Must forecast— Expected interest rate(s) Bond price at end of holding period Plug forecast results into bond pricing formula

29. Example : Suppose you purchase a 10-year,8-percent coupon(annual payments)bond at par and you plan to sell it at the end of 2 years at the prevailing market price. When you purchase the bond, your investment adviser predicts that similar bonds with 8 years to maturity will yield 6 per cent at the end of 2 years. Copyright© 2006 John Wiley & Sons, Inc. 29

30. Dr. Hisham Abdelbaki - FIN 221 - Chapter 5 30 Steps: 1- calculate the price of the bond for 8 years by using the forecasted yield (6%). 2- calculate the yield for 2 years using the price from the first step. The calculation of yield is done by using the yield to maturity formula. = 1124.20 (This is the selling price) (1000 is the purchase price) The expected yield is 13.81%.

31. Total Return Copyright© 2006 John Wiley & Sons, Inc. 31 It is the return received on a bond considering the capital gains or losses and changes in the reinvestment rate.

32. Bond price volatility (price risk) Copyright© 2006 John Wiley & Sons, Inc. 32 Percentage change in price for given change in interest rates: where %∆PB = percentage change in price P t = new price in period t P t – 1 = bond’s price one period earlier

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35. Bond theorems Copyright© 2006 John Wiley & Sons, Inc. 35 Bond prices are inversely related to bond yields. The price volatility of a long-term bond is greater than that of a short-term bond, holding the coupon rate constant.(Positive relationship between price volatility and maturity). The price volatility of a low-coupon bond is greater than that of a high-coupon, bond, holding maturity constant.(Negative relationship between price volatility and coupon rate).

36. Interest rate risk Copyright© 2006 John Wiley & Sons, Inc. 36 Interest rate risk is the risk related to changes in interest rates that cause a bond’s total return to differ from the promised yield or the yield to maturity. Interest rate risk comprises price risk and reinvestment risk. Price risk is the variability in bond prices caused by their inverse relationship with interest rates. Reinvestment risk It is the risk caused by the changes in the market interest rates, causing the lender to reinvest coupon payments at interest rates different from the interest rate at the time the bond is purchased. Price risk and reinvestment risk work against each other. As interest rates fall — Bond prices rise but Coupons are reinvested at lower return. As interest rates rise— Bond prices fall but Coupons are reinvested at higher return.

37. Duration Duration is a measure of interest rate risk that considers both coupon rate and term to maturity. Duration is also the weighted average of the number of years until each of the bond’s cash flows is received. Copyright© 2006 John Wiley & Sons, Inc. 37

38. Copyright© 2006 John Wiley & Sons, Inc. 38 Duration - a measure of interest rate risk where:D = duration of the bond CFt = interest or principal payment at time t t = time period in which payment is made n = number of periods to maturity i = the yield to maturity (interest rate)

39. Duration Example #1 Copyright© 2006 John Wiley & Sons, Inc. 39 Suppose we have a bond with a 3-year term to maturity, an 8% coupon paid annually, and a market yield of 10%. Duration is:

40. Duration Example #2 Copyright© 2006 John Wiley & Sons, Inc. 40 If the yield increases to 15%:

41. Properties of Duration Copyright© 2006 John Wiley & Sons, Inc. 41

42. Duration concepts (all else equal): Copyright© 2006 John Wiley & Sons, Inc. 42 Higher coupon rates mean shorter duration and less price volatility. Duration equals term to maturity for zero coupon securities. Longer maturities mean longer durations and greater price volatility. The higher the market rate of interest, the shorter the duration.

43. Duration can be calculated for an entire portfolio Copyright© 2006 John Wiley & Sons, Inc. 43 where:w i = proportion of bond i in portfolio and D i = duration of bond i.

44. Copyright© 2006 John Wiley & Sons, Inc. 44 Duration is directly related to bond price volatility…. Using the 3-year, 4% coupon bond in Exhibit 5.6— If yield increases to 12%:

45. The previous equation does not work well for large changes in interest rate because based on duration the price-yield relationship is a linear straight line which is not true in reality. Copyright© 2006 John Wiley & Sons, Inc. 45

46. Using Duration to Measure & Manage Interest Rate Risk Example: You wish to plan for a vacation 2 years from now. Therefore, you have a 2-year investment horizon over which you must accumulate enough funds to take your vacation. You consider investing in bonds, but concerned about the interest rate risk that can affect your returns and hence your ability to afford the vacation. Copyright© 2006 John Wiley & Sons, Inc. 46

47. : Three possible approaches to deal with interest rate risk based on the previous example Copyright© 2006 John Wiley & Sons, Inc. 47 Zero-coupon approach: zero-coupon bonds have no reinvestment risk, because there are no coupons payments to reinvest. The duration of a “zero” equals its term to maturity. Buy a “zero” with the desired holding/investment period and must hold to maturity to avoid price risk. Maturity matching: Selecting coupon bonds with maturity( in this example invest in a coupon bond with maturity of 2 years) equal to the desired holding/investment period(in this example 2 years)this eliminates the price risk, because you hold the bond to maturity, but not the reinvestment risk because coupons are received. Duration matching: Selecting the bonds with the duration equals to your holding/investment period. In this case, both the reinvestment and price risks are eliminated. As the capital gains or losses from interest rate changes are exactly offset by the reinvestment income.