# Chapter 8 Interest Rates and Bond Valuation

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1 University of Science and Technology Beijing Dongling School of Economics and management Chapter 8 Interest Rates and Bond Valuation Oct Dr. Xiao Ming USTB 1

2 Key Concepts and Skills Know the important bond features and bond types Understand bond values and why they fluctuate Understand bond ratings and what they mean Understand the impact of inflation on interest rates Understand the term structure of interest rates and the determinants of bond yields Dr. Xiao Ming USTB 2

3 Chapter Outline 8.1 Bonds and Bond Valuation 8.2 Government and Corporate Bonds 8.3 Bond Markets 8.4 Inflation and Interest Rates 8.5 Determinants of Bond Yields Dr. Xiao Ming USTB 3

4 8.1 Bonds and Bond Valuation A bond is a legally binding agreement between a borrower and a lender that specifies the: Par (face) value Coupon rate Coupon payment Maturity Date The yield to maturity is the required market interest rate on the bond. Dr. Xiao Ming USTB 4

5 Bond Valuation Primary Principle: Value of financial securities = PV of expected future cash flows Bond value is, therefore, determined by the present value of the coupon payments and par value. Interest rates are inversely related to present (i.e., bond) values. Dr. Xiao Ming USTB 5

6 The Bond-Pricing Equation Bond Value = 1- C 1 (1+ r r) T + F (1+ r) T Dr. Xiao Ming USTB 6

7 Bond Example Consider a U.S. government bond with as 6 3/8% coupon that expires in December The Par Value of the bond is \$1,000. Coupon payments are made semiannually (June 30 and December 31 for this particular bond). Since the coupon rate is 6 3/8%, the payment is \$ On January 1, 2009 the size and timing of cash flows are: \$ \$ \$ \$1, L 1/1/ 09 6 / 30 / / 31/ 09 6 / 30 /13 12 / 31/13 Dr. Xiao Ming USTB 7

8 Bond Example On January 1, 2009, the required yield is 5%. The current value is: PV = \$ (1.025) \$1,000 (1.025) + = \$1, Dr. Xiao Ming USTB 8

9 Bond Example: Calculator Find the present value (as of January 1, 2009), of a 6 3/8% coupon bond with semi-annual payments, and a maturity date of December 2013 if the YTM is 5%. N I/Y PV PMT FV , = 1,000 1, Dr. Xiao Ming USTB 9

10 Bond Example Now assume that the required yield is 11%. How does this change the bond s price? PV = \$ (1.055) \$1,000 (1.055) + = \$ Dr. Xiao Ming USTB 10

11 YTM and Bond Value 1300 When the YTM < coupon, the bond trades at a premium. Bond Value When the YTM = coupon, the bond trades at par /8 Discount Rate When the YTM > coupon, the bond trades at a discount. Dr. Xiao Ming USTB 11

12 Bond Concepts Bond prices and market interest rates move in opposite directions. When coupon rate = YTM, price = par value When coupon rate > YTM, price > par value (premium bond) When coupon rate < YTM, price < par value (discount bond) Dr. Xiao Ming USTB 12

13 Interest Rate Risk Price Risk Change in price due to changes in interest rates Long-term bonds have more price risk than short-term bonds Low coupon rate bonds have more price risk than high coupon rate bonds. Reinvestment Rate Risk Uncertainty concerning rates at which cash flows can be reinvested Short-term bonds have more reinvestment rate risk than longterm bonds. High coupon rate bonds have more reinvestment rate risk than low coupon rate bonds. Dr. Xiao Ming USTB 13

14 Maturity and Bond Price Volatility Bond Value Consider two otherwise identical bonds. The long-maturity bond will have much more volatility with respect to changes in the discount rate. Par Short Maturity Bond C Discount Rate Long Maturity Bond Dr. Xiao Ming USTB 14

15 Coupon Rates and Bond Prices Bond Value Consider two otherwise identical bonds. The low-coupon bond will have much more volatility with respect to changes in the discount rate. Par High Coupon Bond C Low Coupon Bond Discount Rate Dr. Xiao Ming USTB 15

16 Computing Yield to Maturity Yield to maturity is the rate implied by the current bond price. Finding the YTM requires trial and error if you do not have a financial calculator and is similar to the process for finding r with an annuity. If you have a financial calculator, enter N, PV, PMT, and FV, remembering the sign convention (PMT and FV need to have the same sign, PV the opposite sign). Dr. Xiao Ming USTB 16

17 YTM with Annual Coupons Consider a bond with a 10% annual coupon rate, 15 years to maturity, and a par value of \$1,000. The current price is \$ Will the yield be more or less than 10%? N = 15; PV = ; FV = 1,000; PMT = 100 CPT I/Y = 11% Dr. Xiao Ming USTB 17

18 YTM with Semiannual Coupons Suppose a bond with a 10% coupon rate and semiannual coupons has a face value of \$1,000, 20 years to maturity, and is selling for \$1, Is the YTM more or less than 10%? What is the semi-annual coupon payment? How many periods are there? N = 40; PV = -1,197.93; PMT = 50; FV = 1,000; CPT I/Y = 4% (Is this the YTM?) YTM = 4%*2 = 8% Dr. Xiao Ming USTB 18

19 Current Yield vs. Yield to Maturity Current Yield = annual coupon / price Yield to maturity = current yield + capital gains yield Example: 10% coupon bond, with semi-annual coupons, face value of 1,000, 20 years to maturity, \$1, price Current yield = 100 / =.0835 = 8.35% Price in one year, assuming no change in YTM = 1, Capital gain yield = ( ) / = = -.35% YTM = = 8%, which is the same YTM computed earlier Dr. Xiao Ming USTB 19

20 Bond Pricing Theorems Bonds of similar risk (and maturity) will be priced to yield about the same return, regardless of the coupon rate. If you know the price of one bond, you can estimate its YTM and use that to find the price of the second bond. This is a useful concept that can be transferred to valuing assets other than bonds. Dr. Xiao Ming USTB 20

21 Zero Coupon Bonds Make no periodic interest payments (coupon rate = 0%) The entire yield to maturity comes from the difference between the purchase price and the par value Cannot sell for more than par value Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs) Treasury Bills and principal-only Treasury strips are good examples of zeroes Dr. Xiao Ming USTB 21

22 Pure Discount Bonds Information needed for valuing pure discount bonds: Time to maturity (T) = Maturity date - today s date Face value (F) Discount rate (r) \$ 0 \$ 0 L \$ 0 \$ F T 1 T Present value of a pure discount bond at time 0: PV = F ( 1+ r ) T Dr. Xiao Ming USTB 22

23 \$ 0,1\$ L Pure Discount Bonds: Example Find the value of a 15-year zero-coupon bond with a \$1,000 par value and a YTM of 12%. \$0 \$0 L \$0 \$1, PV = F (1 + r) T = \$1,000 (1.06) 30 = \$ Dr. Xiao Ming USTB 23

24 Bond Pricing with a Spreadsheet There are specific formulas for finding bond prices and yields on a spreadsheet. PRICE(Settlement,Maturity,Rate,Yld,Redemption, Frequency,Basis) YIELD(Settlement,Maturity,Rate,Pr,Redemption, Frequency,Basis) Settlement and maturity need to be actual dates The redemption and Pr need to given as % of par value Click on the Excel icon for an example. Dr. Xiao Ming USTB 24

25 8.2 Government and Corporate Bonds Treasury Securities Federal government debt T-bills pure discount bonds with original maturity less than one year T-notes coupon debt with original maturity between one and ten years T-bonds coupon debt with original maturity greater than ten years Municipal Securities Debt of state and local governments Varying degrees of default risk, rated similar to corporate debt Interest received is tax-exempt at the federal level Dr. Xiao Ming USTB 25

26 After-tax Yields A taxable bond has a yield of 8%, and a municipal bond has a yield of 6%. If you are in a 40% tax bracket, which bond do you prefer? 8%(1 -.4) = 4.8% The after-tax return on the corporate bond is 4.8%, compared to a 6% return on the municipal At what tax rate would you be indifferent between the two bonds? 8%(1 T) = 6% T = 25% Dr. Xiao Ming USTB 26

27 Corporate Bonds Greater default risk relative to government bonds The promised yield (YTM) may be higher than the expected return due to this added default risk Dr. Xiao Ming USTB 27

28 Bond Ratings Investment Quality High Grade Moody s Aaa and S&P AAA capacity to pay is extremely strong Moody s Aa and S&P AA capacity to pay is very strong Medium Grade Moody s A and S&P A capacity to pay is strong, but more susceptible to changes in circumstances Moody s Baa and S&P BBB capacity to pay is adequate, adverse conditions will have more impact on the firm s ability to pay Dr. Xiao Ming USTB 28

29 Bond Ratings - Speculative Low Grade Moody s Ba and B S&P BB and B Considered speculative with respect to capacity to pay. Very Low Grade Moody s C S&P C & D Highly uncertain repayment and, in many cases, already in default, with principal and interest in arrears. Dr. Xiao Ming USTB 29

30 8.3 Bond Markets Primarily over-the-counter transactions with dealers connected electronically Extremely large number of bond issues, but generally low daily volume in single issues Makes getting up-to-date prices difficult, particularly on a small company or municipal issues Treasury securities are an exception Dr. Xiao Ming USTB 30

31 Treasury Quotations 8 Nov :23 132: What is the coupon rate on the bond? When does the bond mature? What is the bid price? What does this mean? What is the ask price? What does this mean? How much did the price change from the previous day? What is the yield based on the ask price? Dr. Xiao Ming USTB 31

32 Clean versus Dirty Prices Clean price: quoted price Dirty price: price actually paid = quoted price plus accrued interest Example: Consider T-bond in previous slide, assume today is July 15, 2009 Number of days since last coupon = 61 Number of days in the coupon period = 184 Accrued interest = (61/184)(.04*1,000) = Prices (based on ask): Clean price = 1, Dirty price = 1, = 1, So, you would actually pay \$1, for the bond. Dr. Xiao Ming USTB 32

33 8.4 Inflation and Interest Rates Real rate of interest change in purchasing power Nominal rate of interest quoted rate of interest, change in purchasing power and inflation The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation. Dr. Xiao Ming USTB 33

34 Real versus Nominal Rates (1 + R) = (1 + r)(1 + h), where R = nominal rate r = real rate h = expected inflation rate Approximation R = r + h Dr. Xiao Ming USTB 34

35 Inflation-Linked Bonds Most government bonds face inflation risk TIPS (Treasury Inflation-Protected Securities), however, eliminate this risk by providing promised payments specified in real, rather than nominal, terms Dr. Xiao Ming USTB 35

36 The Fisher Effect: Example If we require a 10% real return and we expect inflation to be 8%, what is the nominal rate? R = (1.1)(1.08) 1 =.188 = 18.8% Approximation: R = 10% + 8% = 18% Because the real return and expected inflation are relatively high, there is a significant difference between the actual Fisher Effect and the approximation. Dr. Xiao Ming USTB 36

37 8.5 Determinants of Bond Yields Term structure is the relationship between time to maturity and yields, all else equal. It is important to recognize that we pull out the effect of default risk, different coupons, etc. Yield curve graphical representation of the term structure Normal upward-sloping, long-term yields are higher than short-term yields Inverted downward-sloping, long-term yields are lower than short-term yields Dr. Xiao Ming USTB 37

38 Factors Affecting Required Return Default risk premium remember bond ratings Taxability premium remember municipal versus taxable Liquidity premium bonds that have more frequent trading will generally have lower required returns (remember bid-ask spreads) Anything else that affects the risk of the cash flows to the bondholders will affect the required returns. Dr. Xiao Ming USTB 38

39 Quick Quiz How do you find the value of a bond, and why do bond prices change? What are bond ratings, and why are they important? How does inflation affect interest rates? What is the term structure of interest rates? What factors determine the required return on bonds? Dr. Xiao Ming USTB 39

40 Dr. Xiao Ming USTB 40

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