Valuing Bonds Professor: Burcu Esmer Valuing Bonds A bond is a debt instrument issued by governments or corporations to raise money The successful investor must be able to: • Understand bond structure • Calculate bond rates of return • Understand interest rate risk • Differentiate between real and nominal returns 2 Bond Basics Bond: long-term debt security usually issued by a corporation or government body • Notes Notes are issued in two-, three-, five- and 10-year terms • Bonds Bonds are long-term investments with terms of more than 10 years • Mortgage Bonds These bonds are typically backed by real estate holdings and/or real property such as equipment. • Collateral Trust Bonds A bond that is secured by a financial asset - such as stock or other bonds - that is deposited and held by a trustee for the holders of the bond. • Debentures A type of debt instrument that is not secured by physical asset or collateral. 3 Bond Basics (cont.) Bond indenture: contract between issuer and investor that specifies terms of agreement • face (par) value: principal to be repaid at end of loan • coupon rate: (CR) the amount of the coupon payment (C) as a % of the face value of the bond • coupons: (coupon payment) periodic interest payments made over the life of the bond • maturity date: when bond’s face value is paid • frequency of payments: usually semiannually for U.S. corporate bonds http://0.tqn.com/d/beginnersinvest/1/0/W/H/investing_in_bonds_ bond_certificate.jpg 4 Straight Bonds An annual bond pays the holder a coupon payment, C, each year and returns the “face” or “par” value, FV, at maturity. C=(Coupon rate)x(Face Value) Coupon rate is a stated rate written onto the bond. It does not change! C C C C+FV 0 1 2 3 r N 4 Decompose into a $C, N-period annuity + a lump sum of $FV received in N- periods. Pricing Bonds • value of any financial asset: depends on amount, timing, and riskiness of cash flows • => use Discounted Cash Flow (DCF) valuation: Find PV of cash flows! 6 Bond Pricing: Example What is the price of a 9% annual coupon bond with a par value of $1,000 that matures in 3 years? Assume a required rate of return of 4%. 7 Bond Pricing A bond is a package of two investments: an annuity and a final repayment. 푐표푢푝표푛 푝푎푦푚푒푛푡 1 푝푎푟 푣푎푙푢푒 Bond price= x 1 − + 푟 (1+푟)푛 (1+푟)푛 PVBond PV Coupons PV ParValue PVBond coupon ()() Annuity Factor par value Discount Factor 1 (1r )t where Annuity Factor r 1 8 and Discount Factor (1 r )t Bond Pricing: Example What is the value of a 3-year annuity that pays $90 each year and an additional $1,000 at the date of the final repayment? Assume a discount rate of 4%. 1 (1 .04)3 1 PV $90 $1,000 Bond .04 (1 .04)3 $1,138.75 9 Semiannual Coupons • Most bonds in the U.S. pay interest twice a year (1/2 of the annual coupon). • coupon rates and yield (YTM)s quoted on annual basis • Adjustment needed: • divide coupon payment and yield (YTM) by 2 • multiply n by 2. 10 Example • PK Inc. issues 10% bonds with 20 years to maturity. Similar bonds have a YTM of 11%. What is the price of the PK Inc. bond if coupons payments occur annually? What is the price of the PK Inc. bond if coupons payments occur semi-annually • Annual coupon pmt: $100, N=20. FV=1000, YTM=11% PV of annual coupon payments: 796.33 Annual pmt: . FV=1000, N=20, YTM=11% PV of face value : 124.03 Total price= 796.33 + 124.03 =920.36 • Semi-Annual coupon pmt: $50, N=40. FV=1000, YTM=5.5% PV of semi-coupon payments: 802.31 PV of face value : 117.46 Total price= 919.77 11 Sample Treasury bond quotes for May 14, 2010 Bid-ask spread 12 Bond Yields To calculate how much we earn on a bond investment, we can calculate two types of bond yields: • Current Yield • Annual coupon payments divided by bond price. • Yield to Maturity • Interest rate for which the present value of the bond’s payments equals the price 13 Current Yield: Example Suppose you spend $1,150 for a $1,000 face value bond that pays a $60 annual coupon payment for 3 years. What is the bond’s current yield? Your income as a proportion of the initial outlay. 14 How about capital gain return? What will happen to the price of the bond after 3 years? Yield to Maturity Yield to Maturity: coupon coupon (coupon par) PV .... (1 r)1 (1 r)2 (1 r)t 15 Yield to Maturity: Example Suppose you spend $1,150 for a $1,000 face value bond that pays a $60 annual coupon payment for 3 years. What is the bond’s yield to maturity? $60 $60 ($60 $1,000) $1,150 (1 r)1 (1 r)2 (1 r)3 16 Pricing Bonds • To price a bond: discount the coupon payments and face value at appropriate market rate • Yield to Maturity (YTM): the required market interest rate that makes the discounted cash flows of the bond equal to the bond’s price 17 WARNING The coupon rate is NOT the discount rate used in the Present Value calculations. The coupon rate merely tells us what cash flow the bond will produce. Since the coupon rate is listed as a %, this misconception is quite common. 18 Pricing Bonds In general, Bond Value= PV of coupons + PV of par = PVA(r,n,pmt=coupon) + PV(r,n,FV=par) • r = YTM per coupon period for this type of bond • n = # of coupon periods until maturity 푐표푢푝표푛 푝푎푦푚푒푛푡 1 푝푎푟 푣푎푙푢푒 Bond price= x 1 − + 푟 (1+푟)푛 (1+푟)푛 19 Treasury Yields The interest rate on 10-year U.S. Treasury bonds 20 Bond Prices & Interest Rates As interest rates change, so do bond prices. What is the present value of a 4% coupon bond with face value $1,000 that matures in 3 years? Assume a discount rate of 5%. What is the present value of this same bond at a discount rate of 2%? 21 Bond Pricing Example What is the price of a 5.0 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 2.15%. 50 50 1,050 PV (1.0215)1 (1.0215)2 (1.0215)3 PV $1,081.95 22 Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 5 %? 50 50 1,050 PV (1.050)1 (1.050)2 (1.050)3 PV $1,000 23 Bond Pricing Example (continued) What is the price of the bond if the required rate of return is 8 %? 50 50 1,050 PV (1.08)1 (1.08)2 (1.08)3 PV $922.69 24 Dynamic Behavior of Bond Prices • Discount • A bond is selling at a discount if the price is less than the face value. • Par • A bond is selling at par if the price is equal to the face value. • Premium • A bond is selling at a premium if the price is greater than the face value. 25 Discounts and Premiums • If a coupon bond trades at a discount, an investor will earn a return both from receiving the coupons and from receiving a face value that exceeds the price paid for the bond. • If a bond trades at a discount, its yield to maturity will exceed its coupon rate. • What is the relationship between current yield and the return on bonds in this case? • If a coupon bond trades at a premium it will earn a return from receiving the coupons but this return will be diminished by receiving a face value less than the price paid for the bond. • Most coupon bonds have a coupon rate so 26 that the bonds will initially trade at, or very close to, par. Discounts and Premiums (cont'd) Bond Prices Immediately After a Coupon Payment 27 Example 28 Example (cont'd) 29 Interest Rate Risk Definition: changes in bond prices arising from fluctuating market interest rates 1,200 1,100 1,000 900 Bond price ($) 800 700 0 2 4 6 8 10 12 14 16 Interest rate (%) Note: The value of the 5% bond falls as interest rates rise 30 Fixed vs. variable components of a bond: •WARNING!!! • fixed: coupon, face value, maturity date • variable: time to maturity, YTM 31 Interest rate sensitivity Bond A: 8% Coupon, FV=$1,000, and matures in 5 years Bond B: 8% Coupon, FV=$1,000, and matures in 10 years Which bond is more sensitive to interest rates? Why? 32 Interest Rate Risk 3,000 2,500 2,000 30 yr bond When the interest rate equals the 5.0% coupon rate, both 1,500 bonds sell at face value $ Bond Price Bond $ 1,000 3 yr bond 500 33 - 0 2 4 6 8 10 YTM Interest rate sensitivity (Cont’d) Bond A: 0% Coupon, FV=$1,000, and matures in 10 years Bond B: 8% Coupon, FV=$1,000, and matures in 10 years Which bond is more sensitive to interest rates? Why? 34 Yield to Maturity - YTM What rate of return would you earn if pay $935.82 for a $1000 face value bond that pays an 8% coupon and that has 10 years to maturity? P0=935.82 80 80 80 80+1,000 0 1 2 3 10 r=YTM=? SOLVE: 935.82 = 80(PVIFAYTM=?,10) + 1000(PVIFYTM=?,10) ITERATE (1st try 10%, then 9%!) ( or use your financial calculator… N=10, PMT=80, FV=1,000, PV=-935.82, I=?=YTM=9% ) 35 What happens to the price of the bond if interest rates change causing your required return to increase to 12%? To decrease to 4%? P0= 80(PVIFA12%,10) + 1000(PVIF12%,10) =$773.99 Bond sells at a “discount” P0= 80(PVIFA4%,10) + 1000(PVIF4%,10) =$1,324.44 Bond sells at a “premium” If you buy this bond today and hold it to maturity your return will 36 be the yield to maturity! Bond Rate of Return • Rate of Return - Earnings per period per dollar invested.