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Bonds The Discounted Cash Flow Valuation Model Basics of Bonds and their Valuation Ref. Ch 7 1 Main Points important bond features and bond types bond values and why they fluctuate bond ratings and what they mean the impact of inflation on interest rates the term structure of interest rates and the determinants of bond yields 2 2 Bond Definitions Outset 1st Year 2nd Year 3rd Year Bond CB -$1000 $50 $50 $50+1000 Bond: A bond is a security that represents a loan made by investors to the issuer. Par value (face value): The issuer promises to pay the face/par/maturity value of the bond when it matures. Coupon payment: The issuer may promise to pay the investor a regular coupon payments every period until the bond matures. Coupon rate: Percentage of face value. Maturity date: Duration of the contract. Yield or Yield to maturity: Average rate of return. 3 Valuing Coupon Bonds Value of a Level-coupon bond= PV of coupon payment annuity + PV of face value $C $C $C $C $F 0 1 2 T 1 T C 1 F PV 1 T r (1 r ) (1 r ) T 4 Present Value of Cash Flows as Rates Change Bond Value = PV of coupons + PV of par Bond Value = PV annuity + PV of lump sum Remember, as interest rates increase present values of future cash flows decrease So, as interest rates increase, bond prices decrease and vice versa 5 Valuing a Par Bond with Annual Coupons Consider a bond with a coupon rate of 10% and annual coupons. The face value is $1000 and the bond has 5 years to maturity. The yield to maturity is 10%. What is the value of the bond? Using the formula: B = PV of annuity + PV of lump sum B = 100[1 – 1/(1.10)5] / .10 + 1000 / (1.10)5 B = 379.08 + 620.92 = 1000 6 Valuing a Discount Bond with Annual Coupons Consider a bond with a coupon rate of 10% and annual coupons. The par value is $1000 and the bond has 5 years to maturity. The yield to maturity is 11%. What is the value of the bond? Using the formula: B = PV of annuity + PV of lump sum B = 100[1 – 1/(1.11)5] / .11 + 1000 / (1.11)5 B = 369.59 + 593.45 = 963.04 7 Valuing a Premium Bond with Annual Coupons Suppose you are looking at a bond that has a 10% annual coupon and a face value of $1000. There are 5 years to maturity and the yield to maturity is 8%. What is the price of this bond? Using the formula: B = PV of annuity + PV of lump sum B = 100[1 – 1/(1.08)5] / .08 + 1000 / (1.08)5 B = 399.27 + 680.27 = 1079.54 8 Coupon Bond Principles Let us summarize the findings from the previous examples: #1: For par bonds: yield-to-maturity = coupon rate. #2: for premium bonds (price > face value): ytm < coupon rate. #3: for discount bonds (price < face value): ytm > coupon rate 9 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 (same coupon rates) Low coupon rate bonds have more price risk than high coupon rate bonds (same maturity) Reinvestment Rate Risk Uncertainty concerning rates at which cash flows can be reinvested Short-term bonds have more reinvestment rate risk than long-term bonds High coupon rate bonds have more reinvestment rate risk than low coupon rate bonds 10 Price Risk 11 Computing Yield-to-Maturity The yield-to-maturity is the discount rate that makes the present value of the cash flows from the bond equal to the current price of the bond. Finding the YTM requires trial and error if you do not have a financial calculator. Example: What is the yield-to-maturity of a $1,000 par value, 10% coupon rate bond coming due in 3 years that currently sells for $1076? $1076 = $100(PVAFr%,T) + ($1,000)(1+r)-3 => r = YTM = 7.10% 12 Yield-to-Maturity, what does it tell? They allow you to compare different kinds of bonds – those with dissimilar coupons, different market prices, and different maturities. YTM will equal your total earnings if: You hold the bond to maturity, Coupons are reinvested at an interest rate equal to YTM. So, It is a promised annual rate of return. Why? 13 Current Yield and YTM The current yield of a coupon bond = annual coupon payment / current price. Example: The current yield of the previous bond is: = 100 / 1076 = 0.09293 or 9.293% YTM = Interest return + Capital Gain (Loss) Current yield: the portion of an investor’s return that comes in the form of interest income. 14 Current Yield vs. Yield to Maturity Previous example: 10% coupon bond, face value of 1000, 3 years to maturity, $1076 price Current yield = 100 / 1076 = .0929 = 9.293% Price in one year = 1052.36 USD, assuming no change in YTM. Capital gain yield = (1052.36–1076) / 1076 = -.02197 = -2.197% Yield to maturity = current yield + capital gains yield YTM = 9.293 – 2.197 = 7.09%, 15 Bond Values with Semiannual Compounding C 2N F P0 2 t 2N t 1 r r 1 1 2 2 16 Semi-annual bonds -Example Suppose that the Genesco 15 year, 15% bond paid interest semi-annually rather than annually. What would be its price upon issue if current rates are 15% on similar bonds? C = $1,000 x 0.15 = 150 C/2 = 150/2=75 N = 15 2N = 30 F = $1,000 r = 15% r / 2 = 15/2 = 7.5% $PV = $75(PVAF7.5%,30) + ($1,000)(1+0.075)-30 PV = $1,000 17 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 $1,197.93. Is the YTM more or less than 10%? What is the semiannual coupon payment? How many periods are there? 1197.93 = $50(PVAFYTM/2%,40) + ($1,000)(1+0.05)-40 YTM/2 = 4% YTM=4%*2 = 8% Simple annualization is a convention! 18 18 Example: Valuing Bonds w. Semi-Annual Payments Find the present value (as of January 1, 2002), of a 6.375% coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent. $31.875 $31.875 $31.875 $1,031.875 1 / 1 / 02 6 / 30 / 02 12 / 31 / 02 6 / 30 / 09 12 / 31 / 09 $31.875 1 $1,000 PV 1 (1.025)16 (1.025)16 $1,049.30 .05 2 19 Yield to Call Some bonds are calleable, they can be called back by the issuer before the maturity. Condional upon the market interest rates, the issuer may prefer to use this option. Why? ..... In this case, we compute the yield to maturity of the bond as if you receive the call price and the bond is called on its earliest date. 20 Yield to Call - Example Suppose that AZ Inc has a 10 year 8% coupon bond outstanding that can be called at the end of year 5 for a 5% premium. Further suppose that its current market price is 112.42% (assume face value = $100) and that it has been outstanding for 2 years. If you buy this bond, what yield you would probably obtain out of this investment (assume annual coupon payments)? 21 Yield to call - Example Without call the yield is: PV=$112.42, C=$8, N=8, F=$100 r% = YTM = 6%, Since the bond has been outstanding for 2 years, the bond can be called in 3 years. Since the bond is selling for premium, it is most likely that the firm will call the bonds in 3 years. Why? Since the call premium is 5%, then its maturity value will be $100 x 1.05 = $105 if called. So, what is the yield for: PV=$112.42, C=$8, N=3, F=$105 The YTM between 5.5% and 5.75%. Find out the number yourself. Conclusion: .. 22 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 23 Differences Between Debt and Equity Debt Equity Not an ownership interest Ownership interest Creditors do not have Common stockholders vote voting rights for the board of directors Interest is considered a and other issues cost of doing business and Dividends are not is tax deductible considered a cost of doing Creditors have legal business and are not tax deductible recourse if interest or principal payments are Dividends are not a liability missed of the firm and stockholders Excess debt can lead to have no legal recourse if dividends are not paid financial distress and bankruptcy An all equity firm can not go bankrupt 24 The Bond Indenture Contract between the company and the bondholders and includes The basic terms of the bonds The total amount of bonds issued A description of property used as security, if applicable Sinking fund provisions Call provisions Details of protective covenants 25 Bond Characteristics and Required Returns The coupon rate depends on the risk characteristics of the bond when issued Which bonds will have the higher coupon, all else equal? Secured debt versus a debenture Subordinated debenture versus senior debt A bond with a sinking fund versus one without A callable bond versus a non-callable bond 26 Examples of Credit Ratings •Moody's S&P’s Fitch’s DCR’s Definition •Aaa AAA AAA AAA Prime. Maximum Safety •Aa1 AA+ AA+ AA+ High Grade High Quality •Aa2 AA AA AA •Aa3 AA- AA- AA- •A1 A+ A+ A+ Upper Medium Grade •A2 A A A •A3 A- A- A- •Baa1 BBB+ BBB+ BBB+ Lower Medium Grade •Baa2 BBB BBB BBB •Baa3 BBB- BBB- BBB- •Ba1 BB+ BB+ BB+ Non Investment Grade •Ba2 BB BB BB Speculative •Ba3 BB- BB- BB- •B1 B+ B+ B+ Highly Speculative •B2 B B B •B3 B- B- B- •Caa1 CCC+ CCC CCC Substantial Risk •Caa2 CCC - - In Poor Standing •Caa3 CCC- - - •Ca - - - Extremely Speculative •C - - - May be in Default •- - DDD - Default •- - DD DD Default •- D D - Default •- - - DP Default 27 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 28 28 Bond Ratings - Speculative Low Grade Moody’s Ba, B, Caa and Ca S&P BB, B, CCC, CC Considered speculative with respect to capacity to pay. The “B” ratings are the lowest degree of speculation. Very Low Grade Moody’s C and S&P C – income bonds with no interest being paid Moody’s D and S&P D – in default with principal and interest in arrears 29 29 Issuer: Government and Agencies Treasury Securities Federal government debt T-bills – pure discount bonds with original maturity of one year or less 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 30 Example 7.4 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% 31 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, or deep discount bonds. Treasury Bills and Treasury strips are good examples of zeroes. 32 Pure Discount Bonds: Example At the 1-year treasury auction the issue sells for a price of 90. What is the annual rate you would earn if you purchased this bond on the issue and held it until maturity? Rate = (P1 – P0) / P0 Rate = (100/90 – 1) = 0.11 (effective rate!) 33 Pure Discount Bonds: Example 2 At the 6 month treasury auction the issue sells for a price of 95. What is the (effective) annual rate if you purchase and hold this bond until maturity? For 6 months: r= (100/95 – 1)=0.05263 Annualization: 5.263% x 2 = 10.526% (Simple yield) 1.052631/m -1 = 0.108 => 10.8% (Compound Yield) Where m=0,5 34 Pure Discount Bonds: Example 3 At the 2-year treasury auction the issue sells for a price of 81. What is the annual rate you would earn if you purchased this bond on the issue and held it until maturity? HPR for 2 years: r= (100/81 – 1)=0.23456 Annualization: HPR/n = 23.456 / 2 = 11.728% (Simple yield) (1+EAR)2= (1+HPR) and EAR=(1+HPR)1/2 -1 EAR = (1.23456)1/2 - 1 = 0.1111 or 11.11% (Compound Yield) Compare simple yield vs compound yield? 35 Floating-Rate Bonds Coupon rate floats depending on some index value Examples – adjustable rate mortgages and inflation- linked Treasuries There is less price risk with floating rate bonds The coupon floats, so it is less likely to differ substantially from the yield-to-maturity Coupons may have a “collar” – the rate cannot go above a specified “ceiling” or below a specified “floor” 36 36 Other Bond Types Disaster bonds Income bonds Convertible bonds Put bonds There are many other types of provisions that can be added to a bond and many bonds have several provisions – it is important to recognize how these provisions affect required returns 37 37 Bond Markets (Second Hand) 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 small company or municipal issues Treasury securities are an exception 38 Treasury Quotations Quotation: 8.00 Nov 21 128:07 128:08 5 5.31% Coupon rate 8% and Matures in November 2021 Bid price is 128 and 7/32% of par value. 7/32= 0.21875 of 1 percent 128 7/32 = 128.21875%. Bid price is 1.2821875*1000= $1282.1875. Ask price is 128 and 8/32% of par value or 128.25%. Ask price is 1.2825*1000=$1282.50 . The difference between the bid and ask prices ($0.3125) is called the bid-ask spread and it is how the dealer makes money. Ask price changed by 5/32 of 1 percent from the previous day: 0.15625 of 1% or $1.5625 for a $1000 worth of T-bond. The yield is 5.31% based on the ask price. 39 Clean vs. 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, 2007 Number of days since last coupon = 61 Number of days in the coupon period = 182 Accrued interest = (61/182)(.04*1000) = 13.4 Prices (based on ask): Clean price = 1282.50 Dirty price = 1282.50 + 13.4 = 1295.9 So, you would actually pay $ 1295.9 for the bond 40 Inflation and Interest Rates Nominal rate of interest= quoted rate of interest, not adjusted for inflation. Real rate of interest= shows change in purchasing power, adjusted for inflation. The nominal rate of interest includes our desired real rate of return plus a compensation for inflation. 41 The Fisher Effect The Fisher Effect defines the relationship between real rates, nominal rates and inflation (1 + R) = (1 + r*)(1 + h), where R = nominal rate r* = real rate h = expected inflation rate R = r* + h + r*h Approximation R = r* + h 42 Example 7.6 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 significant difference between the actual Fisher Effect and the approximation. 43 Term Structure of Interest Rates 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. For ex. Yields on treasury issues. 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 44 44 Example of Yield Curve Below is the Treasury yield curve for 2/11/03 45 Figure 7.6 – Upward-Sloping Yield Curve 46 The shape of the Treasuries yield curve The yields on Treasuries depend upon: Real rate of interest – opportunity cost of deferred consumption in real terms. Expected inflation – investors must be compensated for anticipated loses in purchasing power. Maturity risk premium – investors demand compensation for their interest rate risk exposure. 47 Decomposition of Yields to Maturity Corporate bonds face additional risks: credit risk + liquidity risk Credit risk: The risk that coupons and the principal may not be paid off. A bond’s credit risk is often captured by its bond rating. Bonds issuers pay credit rating firms to rate their debt. Liquidty risk: The more thinly traded a bond, the wider the bid/ask spread. Thus the most costly it is to trade that bond. 48 Bond Prices with a Spreadsheet There is a specific formula for finding bond prices 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 49 End.. 50 Example - About T-Papers On 15/08/1995, a bond with 9 month maturity (15/05/1996) is bought at a price that would yield 84.70% annual. Six months later, on 13/02/1996 the same T-Bond is sold at a price that would yield 71% annual. What would be your return over the investment period should you sell the bond on 13/02/1996? Purchasing price on 15/08/1995: Number of days to maturity: 274 Price = 100,000 / (1+ 0.8470*274/365) = 61,131 TL Selling price on 13/02/1996: Number of days to maturity: 92 Selling Price = 100,000 / (1+ 0.71*92/365) = 84,820 TL 51 About T-Papers (contn’d) Periodic rate of return (over 274 – 92 = 182 days): (84,820 – 61,131) / 61,131 = 23,689 / 61,131 = 0.3875 or 38.75% Annual rate of return (simple): 0.3875*365/182 = 0.7771 or 77.71% Annual rate of return (compounded): Investment duration is 182 days and the period rate of return is 0.3875. EAR = (1.3875)^365/182 – 1 = 0.9274 or 92.74%. 52 Turkish Treasury Bills - Example Example: Here is a line from Reuter page on March 19, 2007: Value: 19March07 Maturity: 04June07 Average Price: 95 Simple Yield: 24.95 Compounded Yield: 27.52 Now, let us verify how these values are computed: 53 Turkish Treasury Bills (contn’d) Number of days from 19 March 07 to 04 June 07 = 77 days. Period rate of return over (77/365) years : r = (Face / PV) - 1 =(100 / 95) - 1 = 0.05263 Annual Simple Rate = 0.05263 * (365/77) = 0.2495 Compounded yield: EAR = ( 1 + 0.05263)(365/77) – 1 = 0.27523 Conclusion: 24.95% vs 27.5% 54

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