Chapter 9: Bond Prices and Yields
Outline:
I. Bond Characteristics
II. Discounted Cash Flow Techniques
III. Bond Pricing
IV. Bond Yields
V. Default Risk and Bond Pricing
VI. The Yield Curve
Motivation:
- In addition to comparing required rate of
return to expected rate of return, we can also
compare market price to fundamental value.
- Obtaining the estimate of fundamental value
for bonds.
I. Bond Characteristics
Bonds: (1) debt securities, (2) long-term (longer
than 1 year).
Treasury bonds (>10 years) and notes (1-10
years) are quoted in points plus fractions of 1/32
of a point. So, in WSJ when you read 109:08,
this means 109 and 8/32 percent of par value
($1,000), i.e., $1,092.50.
Individual investors can purchase government
debts directly from the Treasury at:
http://www.treasurydirect.gov/
Senior secured bonds: the most senior bonds in a
firm’s capital structure. Example: mortgage
bonds: backed by liens on specific assets, such
as land and buildings.
Debenture (> 10 years) and note (< 10 years): an
unsecured debt.
Callable bonds: bonds that may be repurchased
by the issuer at a specified call price during the
call period.
Zero coupon bond: a bond that makes no coupon
payments, thus initially priced at a deep
discount.
Floating-rate bond: a bond whose coupon
payments are periodically reset.
Convertible bond: a bond that can be swapped
for a fixed number of shares of stock anytime
before maturity at he holder’s option.
Put bond: a bond that allows the holder to force
the issuer to buy the bond back at a stated price.
Indexed bonds: make payments that are tied to a
general price index or the price of a particular
commodity, e.g., inflation-indexed bonds.
Eurobond: An international bond denominated
in a currency not native to the country where it
is issued.
Yankee bonds: bonds sold in the U.S.,
denominated in U.S. $, but issued by foreign
corporations or governments.
II. Discounted Cash Flow Techniques
The fundamental value of an asset is the present
value of its expected future cash flows:
T
Vi =  (1CF
 k)
t 1
t
t
where, Vi = value of security i
T = life of the security
CFt = cash flow in period t
k = the investor’s required rate of return
for security i (the discount rate)
This formula is always true. The problems are:
(1) cash flows are uncertain; but we usually plug
in a series of estimates as if they were certain,
(2) the required rate of return is unknown and
time unstable, (3) the life of the security is
frequently unknown.
Typically, equity evaluation is more problematic
because the above three parameters are
estimated with potentially large errors. In
contrast, bond valuation is less problematic
using discounted cash flow techniques. The
reason is that the three parameters are easier to
obtain.
III. Bond Pricing
Coupon, typically C = C1 = C2 = … = CT: the
stated interest payment made on a bond.
Par (face) value, FV: the principal value of a
bond.
Coupon rate: annual coupon/par value.
Maturity, T: specified date on which the
principal is paid.
Yield to maturity (YTM): the rate required in the
market on a bond
- Market decides.
- Time varying.
- Related to the default risk.
- If coupons are paid out annually, then k =
YTM. If coupons are paid out
semiannually, then k = YTM/2. That is,
YTM is defined as “stated rate” (quoted
rate) in this class.
The pricing model for bonds is:
T
Vbond =  (1CF
=
 k)
t
t 1
=C
1
C1
(1  k )
t
1
(1  k ) T
k
+
+
C2
(1  k ) 2
+…+
CT
(1  k ) T
+
FV
(1  k ) T
FV
(1  k ) T
Example: Suppose that 4Kids Entertainment
Inc. is going to issue a bond. The maturity is 25
years. The average YTM on similar issues is
10%. A series of $120 as coupons is paid out
annually. The face value is $1,000. What is the
fair price of the bond?
V4Kids bond = C 
= 120 
1
1
1
(1  10%) 25
10%
1
(1  k ) T
k
+
+
FV
(1  k ) T
1000
(1  10%) 25
=$1,181.54
If you use a Texas Instruments BAII Plus
calculator, the procedures are:
120
1000
10
25
PV
CPT
PMT
FV
I/Y
N
-1181.54
Most corporate bonds pay coupons
semiannually. The principle of calculation is the
same.
Example: Suppose that 4Kids Entertainment Inc.
is going to issue a bond. The maturity is 25
years. The average YTM on similar issues is
10%. A series of $60 as coupons is paid out
semiannually. The face value is $1,000. What
is the fair price of the bond?
V4Kids bond = C 
= 60 
1
1
(1  5%) 50
5%
1
+
1
(1  k ) T
k
+
FV
(1  k ) T
1000
(1  5%) 50
=$1182.56
Note that this semiannual bond’s value is
somewhat different from the annual bond’s
value, $1181.54. If you use a Texas Instruments
BAII Plus calculator, the procedures are:
CPT
60
1000
5
50
PV
PMT
FV
I/Y
N
-1182.56
IV. Bond Yields
YTM is the internal rate of return on an
investment in the bond. You may need a
financial calculator to do the calculation.
Example: The maturity on 4Kids bonds is 25
years. A series of $120 as coupons is paid out
annually. The face value is $1,000. The market
price of the bonds is $1,181.54. What is the
YTM?
If you use a Texas Instruments BAII Plus
calculator, the procedures are:
1181.54
+/−
PV
CPT
1000
120
25
I/Y
FV
PMT
N
10.0000
When coupons are paid out annually, YTM =
I/Y = 10%.
Example: The maturity on 4Kids bonds is 25
years. A series of $ 60 coupons is paid out
semiannually. The face value is $1,000. The
market price of the bonds is $1,181.54. What is
the YTM?
If you use a Texas Instruments BAII Plus
calculator, the procedures are:
1181.54
CPT
+/−
1000
60
50
I/Y
PV
FV
PMT
N
5.0048
When coupons are paid out annually, YTM =
2×I/Y = 10.0096%.
Current Yield: annual coupon divided by bond
price. For this example, the current yield is
120/1181.54 = 10.16%.
This is also an issue of premium bonds: bonds
selling above par value.
Discount bonds: bonds selling below par value.
V. Default Risk and Bond Pricing
Bond Ratings:
Investment
Quality
Bond
Rating
Low
Quality
(Junk
Bond)
High
Grade
Medium
Grade
Low
Grade
Very
Low
Grade
S&P
AAA
AA
A
BBB
BB
B
CCC
CC
C
D
Moody’s
Aaa
Aa
A
Baa
Ba
B
Caa
Ca
C
D
Bond rating agencies base their ratings largely
on the issuers’ financial ratios. These ratios are
used because they appear to capture the default
risk of the issues:
1. Coverage ratios: ratios of company earnings to
fixed costs.
2. Leverage ratios: debt-to-equity ratios.
3. Liquidity ratios: ratios that measure the firm’s
ability to pay bills coming due with cash
currently being collected.
4. Profitability ratios: measures of rates of return
on assets or equity.
5. Cash flow-to-debt ratio: the ratio of total cash
flow to outstanding debt.
VI. The Yield Curve
Yield curve (term structure of interest rates): a
graph of YTM as a function of term to maturity.
Bonds with shorter maturities generally offer
lower YTMs than longer term bonds. Why so?
Two Theories of Term Structure:
1. The expectations theory: YTMs are
determined solely by expectations of future
short-term interest rates.
2. The liquidity preference theory: investors
demand a risk premium on long-term bonds
because shorter term bonds have more
liquidity.
Note that the two theories are not mutually
exclusive.
End-of-chapter problem sets: #2, #4, #5, #7