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Time and Bond Prices Holding all other things constant, a bond’s yield to maturitywill not change over time. Holding all other things constant, the price of discount orpremium bond will move toward par value over time. If a bond’s yield to maturity has not changed, then the IRRof an investment in the bond equals its yield to maturityeven if you sell the bond early.Copyright 2020 Pearson Education Ltd. All Rights Reserved.35

Textbook Example 6.6 (1 of 4)The Effect of Time on the Price of a Coupon BondProblemConsider a 30-year bond with a 10% coupon rate (annualpayments) and a 100 face value. What is the initial price ofthis bond if it has a 5% yield to maturity? If the yield tomaturity is unchanged, what will the price be immediatelybefore and after the first coupon is paid?Copyright 2020 Pearson Education Ltd. All Rights Reserved.36

Textbook Example 6.6 (2 of 4)SolutionWe computed the price of this bond with 30 years to maturityin Example 6.5:1 1 100P 10 1 176.86 30 300.05 1.05 1.05Now consider the cash flows of this bond in one year,immediately before the first coupon is paid. The bond nowhas 29 years until it matures, and the timeline is as follows:Copyright 2020 Pearson Education Ltd. All Rights Reserved.37

Textbook Example 6.6 (3 of 4)Again, we compute the price by discounting the cash flows by theyield to maturity. Note that there is a cash flow of 10 at date zero,the coupon that is about to be paid. In this case, we can treat thefirst coupon separately and value the remaining cash flows as inEq. 6.5:P(just before first coupon) 10 10 1 1 1001 185.71 29 290.05 1.05 1.05Note that the bond price is higher than it was initially. It will makethe same total number of coupon payments, but an investor doesnot need to wait as long to receive the first one. We could alsocompute the price by noting that because the yield to maturityremains at 5% for the bond, investors in the bond should earn areturn of 5% over the year: 176.86 1.05 185.71.Copyright 2020 Pearson Education Ltd. All Rights Reserved.38

Textbook Example 6.6 (4 of 4)What happens to the price of the bond just after the first coupon is paid?The timeline is the same as that given earlier, except the new owner ofthe bond will not receive the coupon at date zero. Thus, just after thecoupon is paid, the price of the bond (given the same yield to maturity)will beP (just after first coupon) 10 1 1 1001 175.71 29 290.05 1.05 1.05The price of the bond will drop by the amount of the coupon ( 10)immediately after the coupon is paid, reflecting the fact that the ownerwill no longer receive the coupon. In this case, the price is lower than theinitial price of the bond. Because there are fewer coupon paymentsremaining, the premium investors will pay for the bond declines. Still, aninvestor who buys the bond initially, receives the first coupon, and thensells it earns a 5% return if the bond’s yield does not change:(10 175.71) 1.05.176.86Copyright 2020 Pearson Education Ltd. All Rights Reserved.39

Figure 6.1 The Effect of Time on BondPricesCopyright 2020 Pearson Education Ltd. All Rights Reserved.40

Accrued Interest (Stückzinsen)Accrued InterestTime (Coupon Periods)Clean Price Dirty Price - Accrued Interest days since last coupon payment Accrued Int. Coupon Amount days in current coupon period Tage seit letzer Kuponzahlung Stückzinsen Kuponbetrag Tage in aktueller Kuponperiode Copyright 2020 Pearson Education Ltd. All Rights Reserved.41

Interest Rate Changes and Bond Prices(1 of 2) There is an inverse relationship between interest rates andbond prices.– As interest rates and bond yields rise, bond prices fall.– As interest rates and bond yields fall, bond prices rise.Copyright 2020 Pearson Education Ltd. All Rights Reserved.42

Interest Rate Changes and Bond Prices(2 of 2) The sensitivity of a bond’s price to changes in interestrates is measured by the bond’s duration.– Bonds with high durations are highly sensitive tointerest rate changes.– Bonds with low durations are less sensitive to interestrate changes.Copyright 2020 Pearson Education Ltd. All Rights Reserved.43

Textbook Example 6.7 (1 of 3)The Interest Rate Sensitivity of BondsProblemConsider a 15-year zero-coupon bond and a 30-year couponbond with 10% annual coupons. By what percentage will theprice of each bond change if its yield to maturity increasesfrom 5% to 6%?Copyright 2020 Pearson Education Ltd. All Rights Reserved.44

Textbook Example 6.7 (2 of 3)SolutionFirst, we compute the price of each bond for each yield tomaturity:Yield to Maturity15-Year, Zero-CouponBond30-Year, 10% Annual CouponBond5%100 48.10151.0510 1 1 1001 176.86 0.05 1.0530 1.05306%100 41.73151.0610 1 1 1001 155.06 0.06 1.0630 1.0630Copyright 2020 Pearson Education Ltd. All Rights Reserved.45

Textbook Example 6.7 (3 of 3)The price of the 15-year zero-coupon bond changes by(41.73 48.10) 13.2% if its yield to maturity increases from48.105% to 6%. For the 30-year bond with 10% annual coupons,the price change is (155.06 176.86) 12.3%.176.86Even though the 30-year bond has a longer maturity,because of its high coupon rate, its sensitivity to a change inyield is actually less than that of the 15-year zero couponbond.Copyright 2020 Pearson Education Ltd. All Rights Reserved.46

Figure 6.2 Yield to Maturity and BondPrice Fluctuations over TimeCopyright 2020 Pearson Education Ltd. All Rights Reserved.47

6.3 The Yield Curve and Bond Arbitrage Using the Law of One Price and the yields of default-freezero-coupon bonds, one can determine the price and yieldof any other default-free bond. The yield curve provides sufficient information to evaluateall such bonds.Copyright 2020 Pearson Education Ltd. All Rights Reserved.48

Replicating a Coupon Bond (1 of 3) Replicating a three-year 1000 bond that pays 10% annualcoupon using three zero-coupon bonds:Copyright 2020 Pearson Education Ltd. All Rights Reserved.49

Replicating a Coupon Bond (2 of 3)Table 6.2 Yields and Prices (per 100 Face Value) for ZeroCoupon BondsMaturity1 year2years3 years4 yearsYT M3.50%4.00%4.50%4.75%Price 96.62 92.45 87.63 83.06%Copyright 2020 Pearson Education Ltd. All Rights Reserved.50

Replicating a Coupon Bond (3 of 3) By the Law of One Price, the three-year coupon bond musttrade for a price of 1153.Copyright 2020 Pearson Education Ltd. All Rights Reserved.51

Valuing a Coupon Bond Using ZeroCoupon Yields The price of a coupon bond must equal the present valueof its coupon payments and face value.– Price of a Coupon BondV PV (Bond Cash Flows)CPNCPNCPN FV 21 YTM 1 (1 YTM 2 )(1 YTM n ) n100100 100 1000P 1153231.035 1.041.045Copyright 2020 Pearson Education Ltd. All Rights Reserved.52

Coupon Bond Yields Given the yields for zero-coupon bonds, we can price acoupon bond100100100 1000P 1153 2(1 y ) (1 y )(1 y )3100100100 1000P 1153231.0444 1.04441.0444BlankGivenSolve for RateN P E R RAT E3blankblankPVPM TFVExcel Formula 1,1531001,000blankblankblank RATE(3, 100, 1153, 1000)4.44% blankCopyright 2020 Pearson Education Ltd. All Rights Reserved.53

Textbook Example 6.8 (1 of 3)Yields on Bonds with the Same MaturityProblemGiven the following zero-coupon yields, compare the yield tomaturity for a three-year, zero-coupon bond; a three-yearcoupon bond with 4% annual coupons; and a three-yearcoupon bond with 10% annual coupons. All of these bondsare default free.Maturity1 year2 years3 years4 yearsZero- coupon Y T M3.50%4.00%4.50%4.75%Copyright 2020 Pearson Education Ltd. All Rights Reserved.54

Textbook Example 6.8 (2 of 3)SolutionFrom the information provided, the yield to maturity of the threeyear, zero-coupon bond is 4.50%. Also, because the yields matchthose in Table 6.2, we already calculated the yield to maturity forthe 10% coupon bond as 4.44%. To compute the yield for the 4%coupon bond, we first need to calculate its price. Using Eq. 6.6,we haveP 404040 1000 986.98231.035 1.041.045The price of the bond with a 4% coupon is 986.98. From Eq. 6.5,its yield to maturity solves the following equation: 986.98 404040 1000 (1 y ) (1 y ) 2(1 y )3Copyright 2020 Pearson Education Ltd. All Rights Reserved.55

Textbook Example 6.8 (3 of 3)We can calculate the yield to maturity using the annuityspreadsheet:BlankGivenSolve for RateN PE RRAT EPVPM TFVExcel Formula3blank 986.981001,000blankblank4.47% blankblank Blank RATE(3, 40, 986.98, 1000)To summarize, for the three-year bonds consideredCoupon rate0%4%10%YT M4.50%4.47%4.44%Copyright 2020 Pearson Education Ltd. All Rights Reserved.56

Treasury Yield Curves Treasury Coupon-Paying Yield Curve– Often referred to as “the yield curve” On-the-Run Bonds– Most recently issued bonds– The yield curve is often a plot of the yields on thesebonds.Copyright 2020 Pearson Education Ltd. All Rights Reserved.57

Yield CurveCopyright 2020 Pearson Education Ltd. All Rights Reserved.58

Historic Yield Curves GermanyVon Henning. H., Thomas Steiner, and Vlado Plaga http://de.wikipedia.org/w/index.php?title Datei:Zinsstrukturkurve.png&filetimestamp 20050219234920&,CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid 30175882Copyright 2020 Pearson Education Ltd. All Rights Reserved.59

6.4 Corporate Bonds Corporate Bonds– Issued by corporations Credit Risk– Risk of defaultCopyright 2020 Pearson Education Ltd. All Rights Reserved.60

Corporate Bond Yields (1 of 9) Investors pay less for bonds with credit risk than theywould for an otherwise identical default-free bond. The yield of bonds with credit risk will be higher than thatof otherwise identical default-free bonds.Copyright 2020 Pearson Education Ltd. All Rights Reserved.61

Corporate Bond Yields (2 of 9) No Default– Consider a one-year, zero-coupon Treasury Bill with aYTM of 4%. What is the price?10001000P 961.541 YTM 1 1.04Copyright 2020 Pearson Education Ltd. All Rights Reserved.62

Corporate Bond Yields (3 of 9) Certain Default– Suppose now bond issuer will pay 90% of theobligation. What is the price?900900P 865.381 YTM 1 1.04Copyright 2020 Pearson Education Ltd. All Rights Reserved.63

Corporate Bond Yields (4 of 9) Certain Default– When computing the yield to maturity for a bond withcertain default, the promised rather than the actualcash flows are used.FV1000YTM 1 1 15.56%P865.38900 1.04865.38Copyright 2020 Pearson Education Ltd. All Rights Reserved.64

Corporate Bond Yields (5 of 9) Certain Default– The yield to maturity of a certain default bond is notequal to the expected return of investing in the bond.– The yield to maturity will always be higher than theexpected return of investing in the bond.Copyright 2020 Pearson Education Ltd. All Rights Reserved.65

Corporate Bond Yields (6 of 9) Risk of Default– Consider a one-year, 1000, zero-coupon bond issued.– Assume that the bond payoffs are uncertain. There is a 50% chance that the bond will repay itsface value in full and a 50% chance that the bondwill default and you will receive 900.– Thus, you would expect to receive 950. Because of the uncertainty, the discount rate is5.1%.Copyright 2020 Pearson Education Ltd. All Rights Reserved.66

Corporate Bond Yields (7 of 9) Risk of Default– The price of the bond will be950P 903.901.051– The yield to maturity will beYTM FV1000 1 1 10.63%P903.90Copyright 2020 Pearson Education Ltd. All Rights Reserved.67

Corporate Bond Yields (8 of 9) Risk of Default– A bond’s expected return will be less than the yield tomaturity if there is a risk of default.– A higher yield to maturity does not necessarily implythat a bond’s expected return is higher.Copyright 2020 Pearson Education Ltd. All Rights Reserved.68

Corporate Bond Yields (9 of 9)Table 6.3 Price, Expected Return, and Yield to Maturity of aOne-Year, Zero-Coupon Avant Bond with DifferentLikelihoods of DefaultAvant Bond (1-year,zero-coupon)Bond PriceYield toMaturityExpectedReturnDefault Free 961.544.00%4%50% Chance of Default 903.9010.63%5.1%Certain Default 865.3815.56%4%Copyright 2020 Pearson Education Ltd. All Rights Reserved.69

Bond Ratings Investment Grade Bonds Speculative Bonds– Also known as Junk Bonds or High-Yield BondsCopyright 2020 Pearson Education Ltd. All Rights Reserved.70

Table 6.4 Bond Ratings (1 of 2)Copyright 2020 Pearson Education Ltd. All Rights Reserved.71

Table 6.4 Bond Ratings (2 of 2)[Table 6.4 continued]*Ratings: Moody’s/Standard & Poor’sSource: www.moodys.comCopyright 2020 Pearson Education Ltd. All Rights Reserved.72

Corporate Yield Curves Default Spread– Also known as Credit Spread– The difference between the yield on corporate bondsand Treasury yieldsCopyright 2020 Pearson Education Ltd. All Rights Reserved.73

Figure 6.3 Corporate Yield Curves forVarious Ratings, February 2018Source: BloombergCopyright 2020 Pearson Education Ltd. All Rights Reserved.74

Figure 6.4 Yield Spreads and theFinancial CrisisSource: Bloomberg.comCopyright 2020 Pearson Education Ltd. All Rights Reserved.75

6.5 Sovereign Bonds Bonds issued by national governments– U.S. Treasury securities are generally considered to bedefault free– All sovereign bonds are not default-free, e.g., Greece defaulted on its outstanding debt in2012– Importance of inflation expectations Potential to “inflate away” the debt– European sovereign debt, the EMU, and the ECBCopyright 2020 Pearson Education Ltd. All Rights Reserved.76

Figure 6.5 Percent of Debtor Countries inDefault or Restructuring Debt, 1800–2006Source: Data from This Time Is Different, Carmen Reinhart andKenneth Rogoff, Princeton University Press, 2009.Copyright 2020 Pearson Education Ltd. All Rights Reserved.77

Figure 6.6 European Government BondYields, 1976–2018Source: Federal Reserve Economic Data, research.stlouisfed.org/fred2.Copyright 2020 Pearson Education Ltd. All Rights Reserved.78

Chapter Quiz1. What is the relationship between a bond’s price and itsyield to maturity?2. If a bond’s yield to maturity does not change, how doesits cash price change between coupon payments?3. How does a bond’s coupon rate affect its duration—thebond price’s sensitivity to interest rate changes?4. Explain why two coupon bonds with the same maturitymay each have a different yield to maturity.5. There are two reasons the yield of a defaultable bondexceeds the yield of an otherwise identical default-freebond. What are they?Copyright 2020 Pearson Education Ltd. All Rights Reserved.79

Corporate Finance (2 of 2)Fifth Edition, Global EditionChapter 6AppendixCopyright 2020 Pearson Education Ltd. All Rights Reserved.

Forward Interest Rates 6A.1 Computing Forward Rates– A forward interest rate (or forward rate) is an interestrate that we can guarantee today for a loan orinvestment that will occur in the future.– In this chapter, we consider interest rate forwardcontracts for one-year investments, so the forward ratefor year 5 means the rate available today on a oneyear investment that begins four years from today.Copyright 2020 Pearson Education Ltd. All Rights Reserved.81

Computing Forward Rates (1 of 5) By the Law of One price, the forward rate for year 1 isequivalent to an investment in a one-year, zero-couponbond.f1 YTM 1Copyright 2020 Pearson Education Ltd. All Rights Reserved.82

Computing Forward Rates (2 of 5) Consider a two-year forward rate. Suppose the one-year, zero-coupon yield is 5.5% and thetwo-year, zero-coupon yield is 7.0%. We can invest in the two-year, zero-coupon bond at 7.0%and earn (1.07) 2 after two years. Or, we can invest in the one-year bond and earn 1.055 atthe end of the year.– We can simultaneously enter into a one-year interestrate forward contract for year 2 at a rate of f2.Copyright 2020 Pearson Education Ltd. All Rights Reserved.83

Computing Forward Rates (3 of 5) At then end of two years, we will have (1.055)(1 f2). Since both strategies are risk free, by the Law of OnePrice, they should have the same return:(1.07) 2 (1.055)(1 f 2 )Copyright 2020 Pearson Education Ltd. All Rights Reserved.84

Computing Forward Rates (4 of 5) Rearranging, we have1.07 2(1 f 2 ) 1.08521.055 The forward rate for year 2 is f2 8.52%.Copyright 2020 Pearson Education Ltd. All Rights Reserved.85

Computing Forward Rates (5 of 5) In general,(1 YTM n ) n (1 YTM n 1 ) n 1 (1 f n ) Rearranging, we get the general formula for the forwardinterest rate:(1 YTM n ) n 1fn n 1(1 YTM n 1 )Copyright 2020 Pearson Education Ltd. All Rights Reserved.86

Textbook Example 6A.1 (1 of 2)Computing Forward RatesProblemCalculate the forward rates for years 1 through 5 from thefollowing zero-coupon yields:MaturityYT M12345.00%6.00%6.00%5.75%Copyright 2020 Pearson Education Ltd. All Rights Reserved.87

Textbook Example 6A.1 (2 of 2)SolutionUsing Eqs. 6A.1 and 6A.2:f1 YTM 1 5.00%(1 YTM 2 ) 21.062f2 1 1 7.01%(1 YTM 1 )1.05(1 YTM 3 )31.063f3 1 1 6.00%22(1 YTM 2 )106(1 YTM 4 ) 41.05754f4 1 1 5.00%33(1 YTM 3 )1.06Copyright 2020 Pearson Education Ltd. All Rights Reserved.88

Alternative Example 6.A1 (1 of 2) Problem– At the end of 2015, yields on Canadian governmentbonds wereMaturityYield1 Year 2 Years 3 Years 4 Years 5 Years 6 Years 7 Years 8 Years 9 Years 10 Years0.51%0.47% 0.54% 0.67% 0.82% 0.97%1.13% 1.28% 1.42%1.55%– Based on this yield curve, what is f2? f10?Copyright 2020 Pearson Education Lt

Fifth Edition, Global Edition Chapter 6 Valuing Bonds . –A default-free zero-coupon bond that matures on date n provides a risk-free return over the same period. –Thus, the Law of One Price guarantees that the risk- . –U.S. Treasury coupon security