CHAPTER 5
Bonds and Their Valuation
6-1
Bond Overview





Issuers & Key Characteristics of Bonds
Bond Valuation
Bond Yields
Bond Risks
Bond Markets
6-2
What is a bond?


Long-term contract between Issuer & Buyer
(Bondholder) with agreement to pay
Bondholder principal & usually interest on
specific dates
A long-term debt instrument in which a
borrower agrees to make payments of
principal and interest, on specific dates, to
the holders of the bond
6-3
Bond Issuers

Treasury Bonds (government bonds)


Have no default risk
Corporate Bonds (Private Companies)

Exposed to default risk (if the issuing company
gets into trouble, it may be unable to make the
promised interest and principal payments)


Default risk often is referred as “credit risk”
The larger the default or credit risk, the higher the
interest rate the issuer must pay
6-4
Bond Issuers

Municipal Bonds (State & Local Government)



Have default risk
Interest rates on municipal bonds are lower than those
on corporate bonds with the same default risk
Foreign Bonds (Foreign governments & Companies)


Foreign corporate bonds are exposed to default risk,
and so are some foreign government bonds
An additional risk exists if the bonds are denominated in
a currency other than that of the investor’s home
currency
6-5
Key Characteristics of Bonds

Par Value: Stated face value of the bond


Coupon Interest Rate: Stated interest rate (if any)
paid by the issuer




Face amount of the bond, which is paid at maturity (usually
$1,000)
(multiply by par value to get Coupon Payment)
Maturity date: Years until bond must be repaid
(maturity declines over time)
Issue date: Date bond was issued
Yield to maturity: rate of return earned on a bond
held until maturity (also called the “promised
yield”)
6-6
Call Provision

Issuer can “buy-back” if rates decline. That
helps the issuer but hurts the investor



Company would Call if rd is below the coupon rate and
bond sells at a premium. Use open market purchase if
rd is above coupon rate and bond sells at a discount
Therefore, borrowers are willing to pay more,
and lenders require more, on callable bonds
(higher interest rate)
Most bonds have a deferred call and a declining
call premium
6-7
Sinking Fund

Provision to pay off a loan periodically over
its life rather than all at maturity



Similar to amortization on a loan
Reduces risk to investor, shortens average
maturity
3 Ways to Implement



Call x % of Bonds each year
Buy-back x % of Bonds on Open Market
Put aside x % of Funds into a Trust Account
6-8
Basic Determinants of
Market Interest Rates

r = Nominal (market) Interest Rate
= (r* + IP) + MRP + DRP + LP
= rrf + MRP + DRP + LP






r* = Real Risk-free Rate
IP = Inflation Risk Premium
rrf = Nominal Risk-free Rate = r* + IP
MRP = Maturity (Interest Rate) Risk Premium
DRP = Default Risk Premium
LP = Liquidity Risk Premium (Marketability)
6-9
Bond Valuation - Coupon Bond
VB
INT
1 rd
N
t 1
1
INT
1 rd
1
INT
INT
1 rd
t
2
M
1 rd
1
1 rd
rd
N
...
INT
1 rd
N
M
1 rd
N
N
M
1 rd
N
6-10
Bond Valuation - Coupon Bond
rd
VB
INT PVIFArd , N
VB
INT IVrdN
M PVIFrd , N
M IIrNd
the bond's market rate of interest (yield to maturity)
N number of years before the bond matures
INT dollars of interest paid each year Coupon rate Par value
M the par value, face or maturity v alue of the bond
(this amount must be paid off at maturity)
6-11
Required Returns & Bond
Value

If coupon rate < rd bond sells at a “Discount”

If coupon rate = rd bond sells at ”Par”

If coupon rate > rd bond sells at a ”Premium”

If rd rises Price falls
6-12
What is the value of a 10-year, 10%
annual coupon bond, if kd = rd=10%?
0
1
2
k
VB = ?
VB
VB
VB
n
...
100
100
100 + 1,000
$100
$100
$1,000
...
1
(1.10)
(1.10)10 (1.10)10
$90.91 ...
$38.55
$385.54
$1,000
6-13
Using a financial calculator to
value a bond

This bond has a $1,000 lump sum due at t = 10, and
annual $100 coupon payments beginning at t = 1 and
continuing through t = 10, the price of the bond can be
found by solving for the PV of these cash flows.
INPUTS
OUTPUT
10
10
N
I/YR
PV
100
1000
PMT
FV
-1000
6-14
An example:
Increasing inflation and kd (rd)

Suppose inflation rises by 3%, causing kd = rd=13%.
When kd (rd) rises above the coupon rate, the bond’s
value falls below par, and sells at a discount.
INPUTS
OUTPUT
10
13
N
I/YR
PV
100
1000
PMT
FV
-837.21
6-15
An example:
Decreasing inflation and kd (rd)

Suppose inflation falls by 3%, causing kd = rd = 7%.
When kd (rd) falls below the coupon rate, the bond’s
value rises above par, and sells at a premium.
INPUTS
OUTPUT
10
7
N
I/YR
PV
100
1000
PMT
FV
-1210.71
6-16


Suppose the bond was issued 20 years
ago and now has 10 years to maturity
What would happen to its value over
time if the required rate of return
remained at 10%, or at 13%, or at
7%?
6-17
Bond Value ($)
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 values over time


At maturity, the value of any bond must equal
its par value.
If rd remains constant:



The value of a premium bond would decrease
over time, until it reached $1,000.
The value of a discount bond would increase
over time, until it reached $1,000.
A value of a par bond stays at $1,000.
6-19
Bond Valuation – Zero Coupon
Bond
VB
VB
VB
M
1
1 rd
N
M PVIFrd , N
M II
N
rd
Always sells at a “Discount”
6-20
Bond Yields

YTM is the rate earned on a bond held to
maturity
VB
INT
1 rd
INT
1 rd
1
2
...
INT
1 rd
N
M
1 rd

Internal rate of return of the bond

Approximation – Gabriel’s formula
y
M VB
INT
T
0 ,6VB 0 ,4M
N
6-21
Bond Yields
Current yield (CY)
Annual coupon payment
Current price
Capital gains yield (CGY)
Change in price
Beginning price
YTM CY
Expected total return YTM
Expected
Expected
CY
CGY
6-22
Bond Yields

Yield to call (YTC) is the rate earned on a bond “called”
before maturity
N
INT
1 rc
VB
t 1
t
Call price
N
1 rc
N
time to call
rc
Yield to call
6-23
What is the YTM on a 10-year, 9%
annual coupon, $1,000 par value bond,
selling for $887?

Must find the rd that solves this model.
VB
INT
1
(1 rd )
90
$887
(1 rd )1
INT
...
N
(1 rd )
90
...
(1 rd )10
M
N
(1 rd )
1,000
(1 rd )10
6-24
Using a financial calculator to
find YTM

Solving for I/YR, the YTM of this bond is 10.91%.
This bond sells at a discount, because YTM >
coupon rate.
INPUTS
10
N
OUTPUT
I/YR
- 887
90
1000
PV
PMT
FV
10.91
6-25
Current and capital gains yield

Find the current yield and the capital gains
yield for a 10-year, 9% annual coupon bond
that sells for $887, and has a face value of
$1,000.
Current yield = $90 / $887
= 0.1015 = 10.15%
6-26
Calculating capital gains yield
YTM = Current yield + Capital gains yield
CGY = YTM – CY
= 10.91% - 10.15%
= 0.76%
Could also find the expected price one year
from now and divide the change in price by the
beginning price, which gives the same answer.
6-27
YTM vs YTC
 A 10-year, 10%, $1,000 par bond is selling for
$1,134.20. The bond can be called after 5 years
at $1,050 ($50 Call Premium)
 What is YTM?
YTM = 8.00%
 What is YTC?
YTC = 7.50%
6-28
Are you more likely to earn
YTM or YTC?


Coupon rate = 10% vs. YTM = rd = 8.00%.
Corporation could sell new bonds which pay 8.00%
Could replace bonds which pay $100/year with bonds
that pay only $80.00/year



Corporation likely to “Call” bonds
Investors should expect a call, hence YTC = 7.50%, not YTM
= 8.00%
In general, if a bond sells at a premium, then coupon
> rd, so a call is more likely

So, expect to earn:
 YTC on premium bonds
 YTM on par & discount bonds
6-29
Semiannual Coupon
Payments
1.
2.
3.
Multiply years by 2 : number of periods = 2n.
Divide nominal rate by 2 : periodic rate (I/YR) =
rd / 2.
Divide annual coupon by 2 : PMT = ann cpn / 2.
INPUTS
2n
rd / 2
OK
cpn / 2
OK
N
I/YR
PV
PMT
FV
OUTPUT
6-30
Semiannual Coupon
Payments
1.
2.
3.
Multiply years by 2 : number of periods = 2n.
Divide nominal rate by 2 : periodic rate (I/YR) =
rd / 2.
Divide annual coupon by 2 : PMT = ann cpn / 2.
2N
VB
t 1
INT / 2
t
1 rd / 2
M
1 rd / 2
2N
6-31
What is the value of a 10-year, 10%
semiannual coupon bond, if kd = 13%?
Adjust Number of payments, Interest Rate per
period, Payment per period as necessary

1.
2.
3.
Multiply years by 2 : N = 2 * 10 = 20.
Divide nominal rate by 2 : I/YR = 13 / 2 = 6.5.
Divide annual coupon by 2 : PMT = 100 / 2 = 50.
INPUTS
OUTPUT
20
6.5
N
I/YR
PV
- 834.72
50
1000
PMT
FV
6-32
Would you prefer to buy a 10-year, 10%
annual coupon bond or a 10-year, 10%
semiannual coupon bond, all else equal?
The semiannual bond’s effective rate is:
EFF%
iNom
1
m
m
1
0.10
1
2
2
1 10.25%
10.25% > 10% (the annual bond’s effective
rate), so you would prefer the semiannual
bond.
6-33
If the proper price for this semiannual
bond is $1,000, what would be the proper
price for the annual coupon bond?


The semiannual coupon bond has an
effective rate of 10.25%, and the annual
coupon bond should earn the same EAR.
At these prices, the annual and semiannual
coupon bonds are in equilibrium, as they
earn the same effective return.
INPUTS
OUTPUT
10
10.25
N
I/YR
PV
100
1000
PMT
FV
- 984.80
6-34
Interest Rate (or Price) Risk
 Interest rates go up and down over
time
 An increase in interest rates leads to a
decline in the value of outstanding bonds
 Interest rate risk is the concern that
rising rd will cause the value (price) of a
bond to fall
6-35
Interest Rate (or Price) Risk
 Does a 1-year or 10-year 10% bond have more
Interest Rate (Price) Risk?
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%
 the longer the maturity of the bond, the more its
price changes in response to a given change in
interest rates
6-36
What is reinvestment rate risk?

Reinvestment rate risk is the concern that rd
will fall, and future CFs will have to be
reinvested at lower rates, hence reducing
income.
EXAMPLE: Suppose you just won
$500,000 playing the lottery. You
intend to invest the money and
live off the interest.
6-37
Reinvestment rate risk example


You may invest in either a 10-year bond or a
series of ten 1-year bonds. Both 10-year and 1year bonds currently yield 10%
If you choose the 1-year bond strategy:


After Year 1, you receive $50,000 in income and
have $500,000 to reinvest. But, if 1-year rates
fall to 3%, your annual income would fall to
$15,000
If you choose the 10-year bond strategy:

You can lock in a 10% interest rate, and $50,000
annual income
6-38
Conclusions about interest rate and
reinvestment rate risk
Short-term AND/OR
High coupon bonds
Long-term AND/OR Low
coupon bonds
Interest
rate risk
Low
High
Reinvestment
rate risk
High
Low

CONCLUSION: Nothing is riskless!
6-39
Default risk


If an issuer defaults, investors receive less
than the promised return. Therefore, the
expected return on corporate and municipal
bonds is less than the promised return
Influenced by the issuer’s financial strength
and the terms of the bond contract,
especially whether collateral has been
pledged to secure the bond
6-40
Bond Ratings Provide One
Measure of Default Risk (DRP)
Investment Grade
Junk Bonds
Moody’s Aaa
Aa
A
Baa
Ba
B
S&P
AA
A
BBB
BB
B CCC D

AAA
Caa
Bond ratings are designed to reflect the
probability of a bond issue going into default
C
6-41
Other types (features) of
bonds




Convertible bond – may be exchanged for
common stock of the firm, at a fixed price, at
the option of the bondholder
Warrant – long-term option to buy a stated
number of shares of common stock at a
specified price
Putable bond – allows holder to sell the bond
back to the company prior to maturity
Income bond – pays interest only when interest
is earned by the firm
6-42
Other types (features) of
bonds

Mortgage bonds (secured)


The corporation pledges certain assets as security
for the bond
Debentures (unsecured)

Debentures are quite risky if they are issued by
weak companies
6-43
Bond Markets

Bonds are generally traded (bought &
sold) in “Over-the-counter” markets


Large-block Transactions
Most Bonds are traded among large
financial institutions



Life Insurance Firms
Mutual Funds
Pension Funds
6-44