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)13 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.
total income Rate of return = investment
Coupon income + price change Rate of return = investment
37 Do not confuse the bond’s rate of return over a particular investment period with its YTM! Rate of Return
38 Rate of Return: Example
Suppose you purchase a 5% coupon bond, par value $1,000, with 5 years until maturity, for $975.00 today. After one year you sell the bond for $965.00.
What was the rate of return during the period?
39 What is the YTM when you bought the bond? Lower or higher than 4.10% ? What happened to YTM after 1 year when you sold the bond? Interest Rate Risk
• 30-year maturity, 6% coupon PREMIUM bond with fixed 4% YTM and • 30-year maturity, 2% coupon DISCOUNT bond with fixed 4% YTM 1,400 Price path for 1,300 Premium Bond
1,200
1,100
1,000 Bond Price Bond
900 Maturity 800 Today
700 Price path for Discount Bond 40 600 0 5 10 15 20 25 30 Time to Maturity Time and Bond Prices
• Holding all other things constant, the price of discount or premium bond will move towards par value over time. • If a bond’s yield to maturity has not changed, then the rate of return of an investment in the bond equals its yield to maturity even if you sell the bond early.
41 Example
• One bond has a coupon rate of 8%, another a coupon rate of 12%. Both bonds have 10-year maturities and sell at a yield to maturity of 10%. If their yields to maturity next year are still 10%, what is the rate of return on each bond? Does the higher coupon bond give a higher rate of return?
42 Answer Bond 1 1 1$1,000 PV$80 $877.11 Year 0: 10 10 0.100.10(1.10)1.10
Year 1: 1 1$1,000 PV$80 99$884.82 0.100.10(1.10)1.10 $80($884.82$877.11) Rate of return = 0.10010.0% $877.11
Bond 2 Year 0: 11$1,000 PV$120 1010$1,122.89 0.100.10(1.10)1.10
11$1,000 PV$120 $1,115.18 Year 1: 9 9 0.100.10(1.10)1.10
$120 ($1115.18 $1122.89) 43 • Rate of return = 0.100 10.0% $1122.89 The Yield Curve
Term Structure of Interest Rates - A listing of bond maturity dates and the interest rates that correspond with each date.
Yield Curve - Graph of the term structure.
44 The Yield Curve
• Treasury strips are bonds that make a single payment. The yields on Treasury strips in February 2008 show that investors received a higher yield on longer term bonds. • Why do some people prefer short-term bonds then?
45 Nominal and Real Rates of Interest
• TIPS (treasury inflation protected securities) • The real cash flows are fixed but the nominal cash flows (interest and principle) are increased as the CPI increases.
• E.g. In 2008, 10- year TIPS offered a yield of 1.5% (Real interest rate). The yield on nominal 10-year Treasury bonds was 3.8%.
46 Example • The US treasury issues 3% coupon, 2-year TIPS. Assume 5% inflation in the first year and further 4% in the second year.
Year 1 Year 2 Real Cash flows 30 1030
Year 1 Year 2 Nominal Cash flows 30*1.05=31.5 1030*1.05*1.04=1,124.76
47 Real vs. Nominal Yields
Red line – Real yield on long-term UK indexed bonds 48 Blue line – Nominal yield on long-term UK bonds Corporate Bonds and Default Risk (a.k.a. Credit Risk)
• Default premium • The difference between the promised yield on a corporate bond and the yield on a U.S. Treasury Bond with the same coupon and maturity. • Investment grade vs. Junk bonds • Investors pay less for bonds with credit risk than they would for an otherwise identical default-free bond. • The yield of bonds with credit risk will be higher than that of otherwise identical default-free bonds. 49 Bond Ratings
Standard Moody' s & Poor's Safety
Aaa AAA The strongest rating; ability to repay interest and principal is very strong. Aa AA Very strong likelihood that interest and principal will be repaid Investment A A Strong ability to repay, but some vulnerability to changes in grade circumstances Baa BBB Adequate capacity to repay; more vulnerability to changes in economic circumstances Ba BB Considerable uncertainty about ability to repay. B B Likelihood of interest and principal payments over Junk bonds sustained periods is questionable. Caa CCC Bonds in the Caa/CCC and Ca/CC classes may already be Ca CC in default or in danger of imminent default C C C-rated bonds offer little prospect for interest or principal on the debt ever to be repaid. 50 Corporate Yield Curves for Various Ratings, February 2009
51 Source: Reuters Yield Spreads and the Financial Crisis
Source: Bloomberg.com 52 Corporate Bonds
• Zero coupons • no periodic interest payments • issued at a substantial discount from par • Floating rate bonds • Coupon rate change over time • E.g. Treasury rate plus 2% • Convertible bonds • Can be exchanged for a specified number of common stock shares.
53 Zero-Coupon Bonds
• Zero-Coupon Bond • Does not make coupon payments • Always sells at a discount (a price lower than face value), so they are also called pure discount bonds • Treasury Bills are U.S. government zero-coupon bonds with a maturity of up to one year.
54 Zero-Coupon Bonds (cont'd)
• Suppose that a one-year, risk-free, zero-coupon bond with a $100,000 face value has an initial price of $96,618.36. The cash flows would be:
• Although the bond pays no “interest,” your compensation is the difference between the initial price and the face value.
55 Zero-Coupon Bonds (cont'd)
• Yield to Maturity • The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond. • Price of a Zero-Coupon bond
FV P n (1 ) YTM n
56 Zero-Coupon Bonds (cont'd)
• Yield to Maturity • For the one-year zero coupon bond:
100,000 96,618.36 (1 ) YTM1
100,000 1 YTM 1.035 1 96,618.36
• Thus, the YTM is 3.5%.
57 Zero-Coupon Bonds (cont'd)
• Yield to Maturity • Yield to Maturity of an n-Year Zero-Coupon Bond
1 FV n YTMn 1 P
58