CORPORATION FINANCE (EXECUTIVES) CHAPTER 3 BONDS VALUATION Professor Ronen Israel Ó This manuscript was prepared by Professor Ronen Israel (risrael101@gmail.com). Don't distribute or reproduce any part of it without a written consent of the author © 2021 Ronen Israel Chapter 3 Outline • Topics of discussion § Zero-coupon bonds and their pricing § Yield to Maturity (YTM) § The yield curve § Coupon bonds and their pricing § Relations between bond prices and interest rates changes § Interest rates risk § Bond duration • Readings § Berk & DeMarzo chapters 5.3, 6.1-6.3 Corporation Finance - Chapter 3 © 2021 Ronen Israel 2 The First Principle of Security Valuation • The price of a security today is equal to the present value of its cash flows, where the interest or discount rate is that which can be earned on alternative investments with the same characteristics (similar cash flows) § This principle combines two important ideas that we discussed a) PV of a stream of cash flows represents its market value b) Value additivity- the PV of a stream of future cash flows equals to the sum of the PVs of the individual cash flows Corporation Finance - Chapter 3 © 2021 Ronen Israel 3 Bonds • A bond is a security issued by governments, agencies, and corporations § The bond certificate specifies the amounts, the frequency, and the final date of the bond payments Ø The bond pays its last payment on its maturity date Ø The bond’s principal or face value is a single payment that is typically paid on the maturity date ² The face value is used to calculate the bond coupon payments Ø The interest rate on a bond is quoted on an APR basis and is known as the coupon rate Ø The bond interest payments are known as coupons Ø Different bonds pay a different number of coupons per year, e.g., annual coupons, semi-annual coupons, quarterly coupons Corporation Finance - Chapter 3 © 2021 Ronen Israel 4 Bonds • Since coupon rates are quoted on APR basis, the coupon payment is πͺπππππ πΉπππ × ππππ π½ππππ πͺπ·π΅ = π΅πππππ ππ πͺπππππ π·πππππππ πππ ππππ • Yield to Maturity – YTM § In bond pricing, the internal rate of return (IRR) is called the yield to maturity or YTM Ø YTM is the discount rate that equates the PV of the bond payments to its market price • While default is a possibility, we are now only going to look at riskless bonds, where we can be sure of the payout Corporation Finance - Chapter 3 © 2021 Ronen Israel 5 Zero-Coupon Bonds • The simplest kind of bond is called a zero-coupon bond or a discount bond § This kind of bond has only one cash payment, equal to the face value of the bond when the bond matures Ø A Treasury Bill is an example of this kind of bond Ø A commercial paper is a short-term zero-coupon bond issued by corporations Corporation Finance - Chapter 3 © 2021 Ronen Israel 6 Zero-Coupon Bonds • Applying the principle of securities valuation, the price of a zero-coupon bond is ππππ π½ππππ π·= (π + π)π Corporation Finance - Chapter 3 © 2021 Ronen Israel 7 Example: Zero-Coupon Bond • Suppose that you want to find the price of a zero-coupon bond that pays $100,000 in exactly two years. If the yield to maturity on similar bonds is 7%, then the bond price should be 0 2 P=? $100,000 πΉπππ ππππ’π $100,000 π= = = $87,344 " # (1 + π) (1.07) Corporation Finance - Chapter 3 © 2021 Ronen Israel 8 Yield to Maturity (YTM) • For a zero-coupon bond, the YTM is obtained by inversion of the pricing formula § YTM on a risk-free zero-coupon bond is also called spot interest rate πΉπππ ππππ’π πππππ = (1 + πππ)" β β β Corporation Finance - Chapter 3 (1 + πππ)" = (1 + πππ) = ππ»π΄ = $%&' (%)*' +,-&' ! " $%&' (%)*' +,-&' ππππ π½ππππ π·ππππ π π © 2021 Ronen Israel −π 9 Example: Computing YTM of a ZeroCoupon Bond § Suppose that we are considering buying a 20-year zero-coupon bond with a face value of $1,000,000 and a current price of $195,616.39. What is the YTM on this bond? 0 20 -$195,616.39 $1,000,000 πππ = = πΉπππ ππππ’π πππππ 1,000,000 195,616.39 ! " ! #$ −1 −1 = 8.5% Corporation Finance - Chapter 3 © 2021 Ronen Israel 10 Example: Zero-Coupon Prices and YTMs • Suppose the following zero-coupon bonds are selling at prices shown below per $100 face value Maturity 1 year 2 years 3 years 4 years Price $98.04 $95.18 $91.51 $87.14 § Determine the yield to maturity for each bond Ø πππ! = !$$⁄ %&.$( Ø πππ# = +⁄ !$$⁄ , %).!& − 1 = 2.5% Ø πππ- = +⁄ !$$⁄ . %!.)! − 1 = 3.0% Ø πππ( = +⁄ !$$⁄ 0 &/.!( − 1 = 3.5% Corporation Finance - Chapter 3 −1 = 2% © 2021 Ronen Israel 11 The Yield Curve • We just saw that sport interest rates may be different for different investment terms • The relationship between the investment terms and riskfree interest rates is called the term structure of interest rates § The graph representing the term structure of interest rates is called the yield curve Corporation Finance - Chapter 3 © 2021 Ronen Israel 12 Example: Yield Curve • In the previous example we calculated the spot rates for years 1, 2, 3, and 4 § Here is the corresponding yield curve Corporation Finance - Chapter 3 © 2021 Ronen Israel 13 Yield Curve • Click here to go to U.S. treasury yield curve Current Month Treasury Yields - US Treasury • Click here to go to dynamic yield curve http://stockcharts.com/freecharts/yieldcurve.php • Notice how the stock market and interest rates co-move § Interest rates changes reflect economic activities combined with policy makers strategy and intervention • We can use yield curve data to discount future cash flows and find future values Corporation Finance - Chapter 3 © 2021 Ronen Israel 14 Example: Yield Curve 2006, 2007, 2008 Ø The FV of $100 invested for one year in November 2008 is ² 100×1.0091 = $100.91 Ø The FV of $100 invested for 15 years in November 2008 is ² 100×(1.0386)!" = $176.49 Ø The PV on November 2007 of $100 to be obtained in 2017 is ² ππ = Corporation Finance - Chapter 3 !## (!.#&!')!" = $66.40 © 2021 Ronen Israel 15 Discounting and the Yield Curve • With a non-flat yield curve where discount rates are different for different terms we must match corresponding spot interest rates and CFs πͺπ π·π½ = π + ππ π • For a stream of cash flows we have πͺπ πͺπ π·π½ = + π + ππ π + ππ Corporation Finance - Chapter 3 πͺπ + β―+ π π + ππ © 2021 Ronen Israel π π πͺπ = C π + ππ π πUπ 16 Example: Discounting with Different Spot Interest Rates • What’s the value in November 2007 of a $1,000 threeyear annuity starting on November 2008, given the yield curve for November 2007? § The spot rates from the table are πV = 3.16%, π# = 3.16%, πW = 3.12% § Then, the PV of the annuity is ππ = V,XXX V,XXX + V.XWVY V.XWVY Z + V,XXX V.XWV# [ = $2,821 Ø Note that with different spot interest rates we can’t use the shortcut formulae to calculate the PV of an annuity Corporation Finance - Chapter 3 © 2021 Ronen Israel 17 Coupon Bonds • Like zero-coupon bonds, coupon bonds pay their face value at maturity. In addition, these bonds pay periodic coupons § There are two types of government coupon bonds Ø Treasury Notes – are issued with maturities from 1 to 10 years Ø Treasury bonds – are issued with maturities longer than 10 years § Corporations and agencies also issue coupon bonds Corporation Finance - Chapter 3 © 2021 Ronen Israel 18 Pricing Coupon Bonds π· = π·π½ πππππππ πππππππ + π·π½ ππππ πππππ πͺπ·π΅ π = π × π− π Qπ π + Qπ π×π ππ½ + π π + Qπ π×π Where • FV = the bond's face value or the bond’s principal • k = number of coupon payments per year § For a bond with semi-annual coupon payment, π = 2 • πΆππ = (&_*`_" ,%a')×$( c • y = yield on similar bonds on an APR basis • n = number of years to maturity Corporation Finance - Chapter 3 © 2021 Ronen Israel 19 Example: Pricing a Coupon Bond • Consider a $1,000 default free government bond with 6% coupon rate paid semi-annually and 13 years to maturity. The yield on similar bonds is 2% APR, given semi-annual compounding • For this bond Ø FV = $1,000 Øk=2 Ø coupon rate = 6% Ø πΆππ = !,$$$ ×3% # = $30 Ø y = 2% β 5⁄# = 1% Ø n = 13 β π×π = 13×2 = 26 30 1 P= × 1− .01 1.01 Corporation Finance - Chapter 3 #Y 1000 + = $1,456 #Y 1.01 © 2021 Ronen Israel 20 Notes 1. If the price of the bond exceeds $1456, no one will buy it because they can do better elsewhere. If the price is less than this value, then investors will bid up the price 2. If the first coupon is not exactly six months away, you will need to be a bit more careful about discounting the coupons and face value for the appropriate amount of time. However, the general principle of P = PV works 3. You may apply different periodical spot interest rates to different bond payments πͺπ·π΅ πͺπ·π΅ π·= + π + ππ π + ππ Corporation Finance - Chapter 3 πͺπ·π΅ + ππ½ π + β―+ π + π π π © 2021 Ronen Israel 21 Notes 4. The yield to maturity on a coupon bond is simply the IRR of the bond § The periodical IRR is multiplied by the number of coupons per year Ø For a semi-annual bond, a semi-annual IRR is figured and then doubled to give the annual APR, which is the bond YTM 5. For a bond with n periods to maturity, the periodical YTM solves the following equation π πͺπ·π΅ ππ½ π=j +β― + π + ππ»π΄ π + ππ»π΄ π π π7π § Comparing the YTM obtained from this equation to the pricing equation with spot rates πV , π# …π" , it follows that the YTM is a complex average of the relevant spot interest rates Corporation Finance - Chapter 3 © 2021 Ronen Israel 22 Relations Between Bond Prices and Interest Rates • Consider the formula for bond pricing π = ππ πππ’ππππ ππππ’ππ‘π¦ + ππ ππππ π£πππ’π = r+s × u tv 1− V u Vw tv v×x + $( Vwutv v×x • Applying this pricing rule for different interest rates yields important observations and insights regarding the relations between market yields to maturity, y, and bond prices, P Corporation Finance - Chapter 3 © 2021 Ronen Israel 23 Relations Between Bond Prices and Interest Rates 1. Saying that interest rates have risen is equivalent to saying that bond prices have fallen and vice versa 2. Bond prices are convex in interest rates. Therefore, bond prices increase when the uncertainty about interest rates increase Corporation Finance - Chapter 3 © 2021 Ronen Israel 24 Example: Bond Prices and Changes in Market Interest Rates • Recall our 6%, $1,000, 13-year to maturity bond with semi-annual compounding and consider the following table MARKET APR 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% Corporation Finance - Chapter 3 BOND PRICE $1,456 $1,321 $1,201 $1,095 $1,000 $916 $840 $773 $712 $658 $610 $566 $527 © 2021 Ronen Israel % CHANGE IN PRICE -9.46 % -9.27% -9.07% -8.86% -8.66% -8.45% -8.23% -8.02% -7.80% -7.59% -7.37% -7.15% -6.94% 25 EXAMPLE: Bond Prices and Changes in Market Interest Rates Corporation Finance - Chapter 3 © 2021 Ronen Israel 26 Relations Between Bond Prices and Interest Rates 3. A Bond is selling § Above Par or at a Premium π· > ππ½ whenever Coupon Rate > Market Rate § At Par (π· = ππ½) whenever Coupon Rate = Market Rate § Below Par or at a Discount π· < ππ½ whenever Coupon Rate < Market Rate Corporation Finance - Chapter 3 © 2021 Ronen Israel 27 Relations Between Bond Prices and Interest Rates 4. Bonds with longer maturity command larger premiums and deeper discounts § Intuitively, a further maturity date implies that bondholders benefit (suffer) more when the coupon rate exceeds (is less than) the market discount rate Corporation Finance - Chapter 3 © 2021 Ronen Israel 28 Relations Between Bond Prices and Interest Rates 5. A long-term bond is more heavily affected by a change in interest rates than is a short-term bond § This must be true, because with a longer term bond the same rate is compounded more times, leading to a greater effect of price π = 8% π = 7% % price change 3-year 8% $100 bond $100 $103 2.6% 30-year 8% $100 bond $100 $112 12.4% Corporation Finance - Chapter 3 © 2021 Ronen Israel 29 Relations Between Bond Prices and Interest Rates 6. Bonds with lower coupon are more sensitive to changes in interest rates A smaller coupon means that more of the bond payments are coming in the back-end § Ø In other words, the lower coupon bond has "effectively" longer maturity π = 8% π = 7% % price change 30-year 8% $100 bond $100 $112 12.4% 30-year 0% $100 bond $9.9 $13.1 32.2% Corporation Finance - Chapter 3 © 2021 Ronen Israel 30 Interest Rates Risk • The uncertainty concerning bond prices due to interest rates fluctuations is known as the interest rates risk of bonds § Bonds with a longer maturity and a lower coupon have more interest rates risk Corporation Finance - Chapter 3 © 2021 Ronen Israel 31 Chapter Formulas 1. Coupon calculation πΆππ = :;<=;" >?@A × B?CA D?E<A F<GHAI ;J :;<=;" K?5GA"@L =AI MA?I 2. Price of a zero-coupon bond π= B?CA D?E<A (!OI)Q 3. YTM on a zero-coupon bond maturing n periods from now πππ = B?CA D?E<A KIRCA + Q −1 4. The yield curve and discounting a stream of cash flows ππ = :+ !OI+ + :, !OI, + β―+ , :Q !OIQ , = ∑"R7! :S !OIS S where πΆ! , πΆ# …πΆ" and π! , π# …π" are the cash flows and spot interest rates for years i = 1,2 … π, respectively Corporation Finance - Chapter 3 © 2021 Ronen Israel 35 Chapter Formulas 5. Pricing of a coupon bond with face value FV, APR on similar bonds y, k coupon payments per year, n years to maturity, and a coupon CPN π = ππ πππ’ππππ ππππ’ππ‘π¦ + ππ ππππ π£πππ’π = r+s × % t& Corporation Finance - Chapter 3 1− V Vw%t& &×" + $( Vw%t& © 2021 Ronen Israel &×" 36
