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Bond Prices and Yields
CHAPTER 10
Bond Prices and Yields
Objectives:
1. Analyze the relationship between bond prices and bond
yields.
2. Calculate how bond prices will change over time for a
given interest-rate projection.
3. Identify the determinants of bond safety and rating.
4. Analyze how callable, convertible, and sinking fund
provisions will affect a bond's equilibrium yield to
maturity.
5. Define the yield curve and study its properties
Bond Characteristics








Long-term debt contract
Fixed interest payment is paid throughout the life of bond
Entire principal payment is paid at maturity date
Coupon rate: determines the fixed interest payment
Yield to maturity: the average return per year that the
investors (or the market) require on the bond if they buy
and hold the bond until maturity
Coupon rate is fixed, determined by the issuing firm
YTM can fluctuate, depending on the investors in the
market
Zero-coupon bond


zero coupon payment
par at maturity date
Treasury Notes and Bonds





T Note maturities range up to 10 years
T bond maturities range from 10 – 30 years
Bid and ask price
 Quoted in points and as a percent of par
Accrued interest
 Quoted price does not include interest accrued
Example: if the coupon payments are made on May 1
and Nov 1, you buy a bond on June 11. Assume the
price on June 11 is 990, how much you have to pay in
order to buy the bond. (assume there are 40 days from
May 1-June 11
Figure 10.1 Listing of Treasury Issues
Corporate Bonds




Most bonds are traded over the counter
Registered
Bearer bonds
Secured and unsecured

Secured:



Unsecured



Collateral
Mortgage
Debentures
Notes
Call provisions: allows issuer to buy back bond before
maturity date at a specific call price


Why the company wants to call the bond?
Call provision is in favor of the issuer. So if everything else is the
same, callable bond would have to give higher yield, higher
coupon to investors than regular bonds
Corporate Bonds

Convertible bond: a bond with option allowing the
bondholder to exchange the bond for a specific number
of shares of common stock in the firm


Puttable bond: gives the option to bondholder to either
exchange the bond for par value at some date or to
extend for a given number of year



When bondholders want to exchange for par value before
maturity
When bondholders want to extend the bond for a given number
of year after maturity
Floating rate bond


When bondholder want to convert bond into stocks?
coupon rate is tied to current market rates
Preferred stocks
Figure 10.2 Investment Grade
Bonds
Other Domestic Issuers
Federal Home Loan Bank Board
 Farm Credit Agencies
 Ginnie Mae
 Fannie Mae
 Freddie Mac

Innovations in the Bond Market

Reverse floaters


Asset-backed bonds


backed by assets of the firm
Pay-in-kind bonds


Reverse of floating rate bond
issuers may choose to pay interest either in cash or in additional bon
Catastrophe bonds

issued by insurance company, give high yield
 In the event of catastrophe, the obligation to pay interest and principal
can be delayed or forgiven

Indexed bonds

payments are tied to a general price index or price of a particular
commodity

TIPS (Treasury Inflation Protected Securities)
Innovations in the Bond Market

TIPS: adjust for inflation
Example: n = 3 years, annual coupon, par 1000, coupon
4%
Time Inflation
0
1
2
3
2%
3%
1%
Par
1000
1020
1050.60
1061.11
coupon
40.80
42.02
42.44
principal
payment
total
payment
0
0
1061.11
40.80
42.02
1103.55
Bond Pricing

T
P
B =
t =1
Ct T
(1+ r )
1

1  (1 + r )T
PB = C 
r


+
Par Value
T
T
(1+ r )


par
+
T
(
1
+
r
)


Price of bond = present value of all future coupon payments +
present value of the par value
PB =
Price of the bond
Ct =
interest or coupon payments
T = number of periods to maturity
r = semi-annual discount rate or the semi-annual yield to maturity
Example
1) 8% coupon, pay annually, 10 years to maturity, par = 1000,
YTM = 6%
•Using formula
•Using calculator
•PMT = 80, FV = 1000, n = 10, I/Y = 6
2) The same information, but the bond is paying interest semiannually. What is the price of the bond?
Yield to Maturity
Yield to maturity is a measure of the average rate of return
that will be earned if the bond is held to maturity.
YTM is the discount rate that makes the present value of a
bond’s payments equal to its price
YTM is the solution of :
T
coupont
par
Bond Price = 
+
t
T
(
1
+
YTM
)
(
1
+
YTM
)
t =1
8%
coupon, 30-year bond selling at $1,276.76, what is the yield to
maturity?
Bond Prices and Yields


Prices and Yields (required rates of return) have an
inverse relationship
If YTM increase, then the price decreases and vice versa
Maturity
Value
Coupon
Rate
Market
Interest
Rate
Bond
Price
$ 1,000
8%
8%
$1,000.00
$ 1,000
8%
10%
$ 810.71
$ 1,000
8%
6%
$1,276.75
Figure 10.3 The Inverse Relationship Between
Bond Prices and Yields
Bond Prices and Yields
Longer time to maturity, higher change in price when interest
rate changes (or higher interest rate risk)
Interest rate risk: change in rd causes
bond’s price to change.
rd
1-year
Change 10-year Change
5%
$1,048
$1,386
10%
1,000
4.8%
15%
956
4.4%
1,000
38.6%
749
25.1%
Bond Prices and Yields
Value
1,500
10-year
1-year
1,000
500
rd
0
0%
5%
10%
15%
Alternative Measures of Yield

Yield to Call
 Call price replaces par
 Call date replaces maturity
 Example: 8% coupon, semi-annual, 30 years to
maturity, current price = 1150, callable in 10 years,
call price = 1100. What is the yield to call and yield to
maturity
Holding Period Return
interest income + capital gain (loss)
HPR =
beginnning price
Coupon t + (Pt +1 - Pt )
=
Pt
Coupon t (Pt +1 - Pt )
=
+
Pt
Pt
= current yield + capital gain yield
Where
Pt = Bond Price at time t
Pt+1= Bond Price at time t+1
Definitions
Annual
coupon
pmt
Current yield =
Current price
Change
in
price
Capital gains yield =
Beginning price
Expected total
=
return
YTM =
Expected
Curr yld
+
Expected cap
gains yld
Find current yield and capital gains yield
for a 9%, 10-year bond when the bond
sells for $887 and YTM = 10.91%.
$90
Current yield = $887
= 0.1015 = 10.15%.
YTM = Current yield + Capital gains yield.
Cap gains yield = YTM - Current yield
= 10.91% - 10.15%
= 0.76%.
Could also find values in Years 1 and 2,
get difference, and divide by value in
Year 1. Same answer.
BOND PRICES OVER TIME
Premium and Discount Bonds



Premium Bond
 price > par
 Coupon rate exceeds yield to maturity
 Bond price will decline to par over its maturity
Discount Bond
 price < par
 Yield to maturity exceeds coupon rate
 Bond price will increase to par over its maturity
Bond selling at par



price = par
Yield to maturity = coupon rate
bond price is constant throughout the life of bond
BOND PRICES OVER TIME
Suppose the bond was issued 20
years ago and now has 10 years to
maturity. What would happen to its
value over time if the required rate of
return or the YTM remained at 10%, or
at 13%, or at 7%?
Bond Value ($)
1,372
1,211
rd = 7%.
rd = 10%.
1,000
M
837
rd = 13%.
775
30
25
20
15
10
5
0
Years remaining to Maturity
BOND PRICES OVER TIME
At maturity, the value of any bond
must equal its par value.
 The value of a premium bond would
decrease to $1,000.
 The value of a discount bond would
increase to $1,000.
 A par bond stays at $1,000 if rd (YTM)
remains constant.

Figure 10.7 The Price of a Zero-Coupon Bond
over Time
DEFAULT RISK AND BOND PRICING
Default Risk and Ratings

Rating companies
 Moody’s
Investor Service
 Standard & Poor’s
 Fitch

Rating Categories
 Investment
grade
 Speculative grade
Figure 10.8 Definitions of Each Bond Rating Class
Bond Ratings Provide One Measure
of Default Risk
Investment Grade
Junk Bonds
Moody’s Aaa
Aa
A
Baa
Ba
B
S&P
AA
A
BBB
BB
B CCC D
AAA
Caa
C
Factors Used by Rating Companies
Coverage ratios
 Leverage ratios
 Liquidity ratios
 Profitability ratios
 Cash flow to debt

Protection Against Default

Sinking funds


Subordination of future debt


Restrictions on additional borrowing that stipulates that senior
bondholders will be paid first in the event of bankruptcy
Dividend restrictions


A bond that calls for the issuer to periodically repurchase some
proportion of the outstanding bonds prior to maturity
Limit dividend payout to protect bondholders
Collateral



Uses assets to back up bonds: mortgage bond, collateral trust
bond, equipment obligation bond.
Collaterals are secured bonds
Unsecured bond: debentures
THE YIELD CURVE
Term Structure of Interest Rates
Relationship between yields to maturity
and maturity
 Yield curve - a graph of the yields on
bonds relative to the number of years to
maturity

 Usually
Treasury Bonds
 Have to be similar risk or other factors
would be influencing yields
Figure 10.11 Treasury Yield Curves
Theories of Term Structure

Expectations
 Long
term rates are a function of expected future
short term rates
 Upward slope means that the market is expecting
higher future short term rates
 Downward slope means that the market is expecting
lower future short term rates

Liquidity Preference
 Upward
bias over expectations
 The observed long-term rate includes a risk premium

Combination (Synthesis)
Expectation hypothesis (when short-term interest
rate is expected to increase)







Current interest on 1-year bond = 8% (r1=8%)
Everyone in the market believes that the interest one 1year bond next year will rise to 10% (E(r2) = 10%)
Investment A: year 1: buy 1-year bond
year 2: buy another 1-year bond
Investment B: year 1: buy 2-year bond
In order for investment B to be competitive with
investment A, B has to offer an average annual
compound return = average annual compound return of A
What is the average annual compound return of B or
YTM of the 2-year bond in B?
You will find the YTM of 2-year bond > YTM of 1-year
bond at time = 0, this is due to E(r2) > r1 ( the expected
short-term rate increases in the future)
Figure 10.12 Returns to Two 2-year
Investment Strategies
Expectation hypothesis (when short-term interest
rate is expected to decrease)







Current interest on 1-year bond = 8% (r1=8%)
Everyone in the market believes that the interest one 1year bond next year will decrease to 6% (E(r2) = 6%)
Investment A: year 1: buy 1-year bond
year 2: buy another 1-year bond
Investment B: year 1: buy 2-year bond
In order for investment B to be competitive with
investment A, B has to offer an average annual
compound return = average annual compound return of A
What is the average annual compound return of B or
YTM of the 2-year bond in B?
You will find the YTM of 2-year bond < YTM of 1-year
bond at time = 0, this is due to E(r2) < r1 ( the expected
short-term rate decreases in the future)
The Expectation Hypothesis
•In practice, we don’t observe directly the expectation of next
year’s rate, but we can observe the yields on bonds of different
maturities. So from that, using the yields on bonds of different
maturities, we can calculate the expected short-term interest rate
(or the expected YTM of 1-year bond) in the future
•According to expectation hypothesis, the forward rate =
expected short term rate.
•Forward rate is the inferred short-term rate of interest for a
future period that makes the expected total return of a long-term
bond = that of rolling over short-term bonds.
Example: Suppose that 1-year bonds offer yields to
maturity of 8%, and 2-year bonds have yields of 8.995%.
What is the expected one-period rate or forward rate for
the third year?
Forward Rates Implied
in the Yield Curve
(1+ y n )
(1 . 07 )
3
n
= (1+
y n 1) (1+ f n )
n 1
2
= (1 . 06)
(1 . 0902 )
Example: Suppose that 2-year bonds offer yields to maturity of 6%,
and 3-year bonds have yields of 7%. What is the expected one-period
rate for the third year?
For example, using a 1-yr and 2-yr rates
Longer term rate, y(n) = 7%
Shorter term rate, y(n-1) = 6%
Forward rate, a one-year rate in one year = 9.02%
Liquidity preference theory




According to expectation theory, long-term yield depends
on expected short-term yield. If expected short-term
increases, then long-term increases, and vice versa
This theory does not take into account risk.
Risk of long-term > risk of short-term
Liquidity preference theory:




investors demand of risk premium (liquidity premium) for holding
long-term bonds
fn = E(rn) + Liquidity premium
Forward rate in year n = expected short-term rate in year n +
liquidity premium
Predict upward sloping yield curve even in the case of expected
short-term rates are unchanged.
Liquidity preference theory



Suppose the short-term rate of interest rate is currently 8%, and investors
expect it to remain at 8% next year.
In the absence of liquidity premium, with no expectation of a change in
yields, the YTM on two-year bonds would be 8% (according to expectation
hypothesis), and the yield curve would be flat, the forward rate would be
8%
However, what if investors demand a risk premium to invest in two-year
rather than one-year bonds? If the liquidity premium = 1%, then according
to liquidity theory, the forward rate would be = 8%+1% = 9% and the
YTM of two-year bond would be
(1+y2)2 = 1.08×1.09 = 1.7772
therefore y2=0.085 = 8.5%

YTM of 2-year bond (8.5%) > YTM of 1-year bond (8%) solely due to the
liquidity premium.
A Synthesis
Figure 10.14 Term Spread
Summary
•Inverse relationship between bond prices and
bond yields
•Premium and discount bonds
•Corporate bonds and default risk
•Term structure of interest rates
Expectations theory
Liquidity preference theory
Synthesis
•Next Class: Managing Bond Portfolios