Professor XXXXX Course Name / Number Valuing Bonds Professor XXXXX Course Name / Number

Valuation Fundamentals Present Value of Future Cash Flows Link Risk & Return Expected Return on Assets Valuation

The Basic Valuation Model P0 = Price of asset at time 0 (today) CFt = Cash flow expected at time t r = Discount rate (reflecting asset’s risk) n = Number of discounting periods (usually years) This model can express the price of any asset at t = 0 mathematically. Value of any financial asset is the PV of future cash flows Bonds: PV of promised interest & principal payments Stocks: PV of all future dividends Patents, trademarks: PV of future royalties Valuation is the process linking risk & return Output of process is asset’s expected market price A key input is the required [expected] return on an asset Defined as the return an arms-length investor would require for an asset of equivalent risk Debt securities: risk-free rate plus risk premium(s) Required return for stocks found using CAPM or other asset pricing model Beta determines risk premium: higher beta, higher reqd return Marginal benefit of owning the asset: right to receive the cash flows Marginal cost: opportunity cost of owning the asset

Valuation Fundamentals: Example Company issues a 5% coupon interest rate, 10‑year instrument with a $1,000 par value Assume annual interest payments Investors in company’s financial instrument receive the contractual rights $50 coupon interest paid at the end of each year $1,000 principal at the end of the 10th year The simplest debt instruments to value are U.S. Treasury securities since there is no default risk. Instead, the discount rate to use, r, is the pure cost of borrowing. Assume you are asked to value two Treasury securities, when rf is 1.75 percent (r = 0.0175): A (pure discount) Treasury bill with a $1,000 face value that matures in three months, and A 1.75% coupon rate Treasury note, also with a $1,000 face value, that matures in three years. For the T-Bill, three months is one-quarter year (n=0.25) For 3-year bond, n = 3 Most U.S. corporate bonds: Pay interest at a fixed coupon interest rate Have an initial maturity of 10 to 30 years, and Have a par value (also called face or principal value) of $1,000 that must be repaid at maturity. Bond’s value has two separable parts: (1) PV of stream of annual interest payments, t=1 to t=10 (2) PV of principal repayment at end of year 10. Can thus also value bond as the PV of an annuity plus the PV of a single cash flow using PVFA and PVF from tables. P0 = C x (PVFA5%,10yr) + Par x (PVF5%,10yr) = $50 (7.7220) + $1,000 (0.6139) = $1,000.00 Bonds with a few cash flows can be valued with Eq 4.1; for bonds with many cash flows, use PVFA/PVF factors, calculator or Excel.

Yield to Maturity (YTM) Estimate of return investors earn if they buy the bond at P0 and hold it until maturity The YTM on a bond selling at par will always equal the coupon rate. -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 YTM is the discount rate that equates the PV of a bond’s cash flows with its price.

Bond Premiums and Discounts What happens to bond values if required return is not equal to the coupon rate? The bond's price will differ from its par value P0 < par value r > Coupon Interest Rate DISCOUNT = When r is greater than the coupon interest rate, P0 will be less than par value, and the bond will sell at a discount For Sun, if r >5%, P0 will be less than $1,000 For practice: Value Sun Company, 10-year, 5% coupon rate bond if required return, r =6% and again if r = 4%. Premiums & discounts change systematically as r changes. P0 > par value r < Coupon Interest Rate PREMIUM =

Semi-Annual Interest Payments An example.... Value a T-Bond Par value = $1,000 Maturity = 2 years Coupon rate = 4% r = 4.4% per year = $992.43 Most bonds pay interest semi-annually rather than annually Can easily modify basic valuation formula; divide both coupon payment (C) and discount rate (r) by 2, as in Eq 4.3: In Eq 4.3, C is the annual coupon payment, so C/2 is the semi-annual payment. r is the annual required return, so r/2 is the semi-annual discount rate. n is the number of years, so there are 2n semi-annual payments.

Factors that Affect Bond Prices Time to maturity: bond prices converge to par value (plus final coupon) with passage of time. Interest rates: bond prices and interest rates move in opposite directions. -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 Changes in interest rates have larger impact on long-term bonds than on short-term bonds.

Interest Rate Risk What does this tell you about the relationship Whenever r is different from the coupon interest rate, the time to maturity affects bond value (even if the required return remains constant until maturity) The shorter is n, the less responsive is P0 to changes in r. Assume r falls from 5% to 4% For n=8 years, P0 rises from $1,000 to $1,067.33, or 6.73% For n=3 years, P0 rises from $1,000 to $1,027.75, or 2.775% Same relationship if r rises from 5% to 6%, though percentage declines in price less than increases (maximum decline is 100%, increase unlimited) For n=8 years, P0 falls from $1,000 to $937.89, or 6.21% For n=3 years, P0 falls from $1,000 to $973.25, or 2.675% Even if r doesn’t change, premiums and discounts will decline towards par as bond nears maturity. What does this tell you about the relationship between bond prices and yields for bonds with different maturities?

Primary vs. Secondary Markets Primary market: the initial sale of bonds by issuers to large investors or syndicates Secondary market: the market in which investors trade with each other -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 Trades in the secondary market do not raise any capital for issuing firms.

Bonds by Issuer Corporate Bonds Municipal Bonds Treasury Bonds Usually with par $1000 and semi-annual coupon Bonds if maturity > 10 years; notes if maturity < 10 years Municipal Bonds Issued by local and state government Interest on municipal bonds tax-free Treasury Bonds If maturity < 1 year: Treasury Bills If 1 year < maturity < 10 years: Treasury Notes Maturity > 10 years: Treasury Bonds Used to fund budget deficits -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 Agency Bonds Issued by government agencies: FHLB, FNMA (Fannie Mae), GNMA (Ginnie Mae), FHLMC (Freddie Mac)

Bonds by Features Fixed vs. Floating Rates Secured vs. Unsecured Bonds Floating-rate bonds: coupon tied to prime rate, LIBOR, Treasury rate or other interest rate Floating rate = benchmark rate + spread Floating rate can also be tied to the inflation rate: TIPS, for example Secured vs. Unsecured Bonds Unsecured bonds (debentures) are backed only by general faith and credit of issuer Secured bonds are backed by specific assets (collateral) Mortgage bonds, collateral trust bonds, equipment trust certificates -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43

Bonds by Features (Continued) Zero-Coupon Bonds Discount bonds or pure discount bonds Sell below par value Treasury Bills (Tbills) Treasury STRIPs Convertible and Exchangeable Bonds Convertible bonds, in addition to paying coupon, offers the right to convert the bond into common stock of the issuer of the bond Exchangeable bonds are convertible in shares of a company other than the issuer’s -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43

Bonds by Features (Continued) Callable and Putable Bonds Callable bonds: bond issuer has the right to repurchase the bonds at a specified price (call price). Firms could retire and reissue debt if interest rates fall. Putable bonds: the investors have the right to sell the bonds to the issuer at the put price. -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 Protection from Default Risk Sinking fund provisions: the issuer is required to gradually repurchase outstanding bonds. Protective covenants: requirements the bond issuer must meet Positive and negative covenants

Bond Markets The U.S bond market has grown from $250 billion in 1950 to $22 trillion in 2004 -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43

U.S. Treasury Bond Quotations RATE MATURITY MO/YR BID ASKED CHG ASK YLD Government Bonds & Notes 5.500 May 09n 107:13 107:14 3 3.83 Rate Coupon rate of 5.5% Bid prices Ask prices (percentage of par value) Bid price: the price traders receive if they sell a bond to the dealer. Quoted in increments of 32nds of a dollar Ask price: the price traders pay to the dealer to buy a bond Bid-ask spread: difference between ask and bid prices. -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 Ask Yield Yield to maturity on the ask price

Corporate Bond Quotations Company (Ticker) Coupon Maturity Last Price Last Yield Estimated Spread UST Est $ Vol (000s) SBC Comm (SBC) 5.875 Aug 15,2012 107.161 4.836 80 10 73,867 Corporate prices are quoted as percentage of par, without the 32nds of a dollar quoting convention -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 Yield spread: the difference in yield-to-maturities between a corporate bond and a Treasury bond with same maturity The greater the default risk, the higher the yield spread

Investment-grade bonds Bond Ratings Bond ratings: grades assigned to bond issues based on degree of default risk Investment-grade bonds Moody’s Aaa to Baa3 ratings S&P and Fitch AAA to BBB- ratings -Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. When P0  Par, the YTM will differ from the coupon rate. YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 Junk bonds Moody’s Ba1 to Caa1 or lower S&P and Fitch BB to CCC+ or lower

Term Structure of Interest Rates Relationship between yield and maturity is called the Term Structure of Interest Rates Graphical depiction called a Yield Curve Usually, yields on long-term securities are higher than on short-term securities. Generally look at risk-free Treasury debt securities Yield curves normally upwards-sloping Long yields > short yields Can be flat or even inverted during times of financial stress The relationship between nominal (observed) and real (inflation-adjusted) interest rates and expected inflation called the Fisher Effect (or Fisher Equation) Fisher said the nominal rate (r) is approximately equal to the real rate of interest (a) plus a premium for expected inflation (i). If real rate equals 3% (a = 0.03) and expected inflation equals 2% (i = 0.02): r  a + i  0.03 + 0.02  0.05  5% The true Fisher Effect is multiplicative, rather than additive: (1+r) = (1+a)(1+i) = (1.03)(1.02) = 1.0506; so r = 5.06% What do you think a Yield Curve would look like graphically?

Yield Curves U.S. Treasury Securities 16 May 1981 14 12 10 Interest Rate % January 1995 8 August 1996 6 October 1993 4 2 1 3 5 10 15 20 30 Years to Maturity

Bond Valuation Bond price equals present value of its coupons and principal. Bond prices are inversely related to interest rates. Bonds could have a number of features: such as convertibility, callability.