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Chapter 6
Valuing Bond
The Application of the
Present Value Concept
1
Bond Characteristics

Bond - Security that obligates the issuer to make
specified payments to the bondholder.

Coupon - The interest payments made to the
bondholder.

Face Value - (Par Value, Principal or Maturity
Value) - Payment at the maturity of the bond.
 Coupon Rate - Annual interest payment as a
percentage of face value.
3
Bond Characteristics

A bond also has (legal) rights attached to
it:

If the borrower doesn’t make the required
payments, bondholders can force
bankruptcy proceedings

In the event of bankruptcy, bond holders get
paid before equity holders
4
An Example of A Bond
Example

A coupon bond that pays coupon of 5% annually, with
a face value of $1000, has a discount rate of 2.15%
and matures in three years.



The coupon payment is $50 annually
In the third year, the bondholder is supposed to get $50
coupon payment plus the face value of $1000.
The discount rate is different from the coupon rate.
5
Bond Cash Flows
6
Coupon Rate vs. Discount Rate
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.
7
Bond Pricing – Zero Coupon Bonds
Example
How much is a 10-yr zero coupon bond worth today if
the face value is $1,000 and the effective annual rate
is 8% ?



P0=1000/1.0810=$463.2
present value of the face value paid at the maturity
Zero coupon bonds are also called zeros or
stripped bonds.
8
Bond Pricing – Coupon Bonds

The price of a coupon bond is the Present
Value of all cash flows generated by the bond
(i.e. coupons and face value) discounted at the
required rate of return.
cpn
cpn
(cpn  par )
PV 

....
1
2
t
(1  r ) (1  r )
(1  r )
9
Bond Pricing
Example
What is the price of a 5 % 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
2
(1.0215) (1.0215) (1.0215)3
PV  $1,081.95
10
Bond Pricing

Another way to think of bond pricing
PV = PV (coupons) + PV (face value)
PV of An Annuity
 1

1
1
PV  $50  


$1,000

3
3
0.0215
0.0215(1.0
215)
(1.0215)


PV  $143.77  $938.18  $1,081.95
Financial calculator: n=3, i=2.15, PMT=50, FV= 1,000  PV=(-)1,081.95
11
Bond Pricing
Example
What is the price of the 5% coupon bond if the
required rate of return is 2.15% AND the coupons
are paid semi-annually?
12
Bond Pricing
Example
What is the price of the 5% coupon bond if the
required rate of return is 2.15% AND the coupons
are paid semi-annually?
25
25
25
25
25
1,025





(1.01075)1 (1.01075) 2 (1.01075)3 (1.01075) 4 (1.01075)5 (1.01075)6
PV  $1,082.37
PV 
Financial calculator: n=6, i=1.075, PMT=25, FV= 1,000  PV=(-)1,082.37
13
Interest Rates and Bond Prices
Example
What is the price of the 5% annual coupon bond if
the required rate of return is 5 %?
50
50
1,050
PV 


1
2
(1.05) (1.05) (1.05)3
PV  $1,000
Financial calculator: n=3, i=5, PMT=50, FV= 1,000  PV=(-)1,000
14
Interest Rates and Bond Prices
Example (continued)
What is the price of the bond if the required rate of
return is 15 %?
50
50
1,050


(1.15)1 (1.15) 2 (1.15)3
PV  $771.68
PV 
Financial calculator: n=3, i=15, PMT=50, FV= 1,000  PV=(-)771.68
15
Interest Rates and Bond Prices
Example (continued)
Q: How do bond prices vary with interest rates?

Market interest rate > Coupon rate


Bond price < Face value : discount bond
Market interest rate < Coupon rate

Bond price > Face value: premium bond
16
Bond Prices Over Time
17
Interest Rates and Bond Prices
18
Interest Rate Risk
19
Bond Yields

Current Yield - Annual coupon payments
divided by bond price.

The current yield does not measure the
bond’s total rate of return.
It overstates the return of premium bonds
 It understates the return of discount bonds

20
Bond Yields

Yield To Maturity (YTM) - Interest rate for
which the present value of the bond’s
payments equal the price.
21
Bond Yields
Calculating Yield to Maturity (YTM=r)
If you are given the price of a bond (PV) and
the coupon rate, the yield to maturity can be
found by solving for r.
cpn
cpn
(cpn  par )
PV 

....
1
2
t
(1  r ) (1  r )
(1  r )
22
Bond Yields
Example
What is the YTM of a 5% annual coupon bond, with a $1,000 face
value, which matures in 3 years? The market price of the bond is
$1,081.95.
50
50
1, 050
$1081.95 


1
2
(1  r ) (1  r ) (1  r )3
r  2.15%
Financial calculator: n=3, PV=(-)1,081.95, PMT=50, FV= 1,000  i=2.15
23
Bond Yields
WARNING
Calculating YTM by hand can be very
tedious.
It is highly recommended that you learn
to use the “IRR” or “YTM” or “i” functions
on a financial calculator.
24
Bond Yields
Example
In the previous example, what is the YTM if the coupons are paid
semiannually?
25
25
25
25
25
1,025





(1  r )1 (1  r ) 2 (1  r )3 (1  r ) 4 (1  r ) 5 (1  r ) 6
PV  $1,081.95
PV 
r  1.082%
Financial calculator: n=6, PV=(-)1,081.95, PMT=25, FV=1,000  i=1.082
Quoted Annual Yield (Yield to Maturity)  1.082  2  2.164%
Effective Annual Yield  (1  0.01082) 2  1  2.18%
25
Bond Yields
Example
A 4-year maturity bond with a 14 percent coupon rate
can be bought for $1,200. What is the YTM if the
coupon is paid annually? What if it is paid
semiannually?

If coupon is paid annually
Financial calculator: n=4, PV=(-)1,200, PMT=140, FV= 1,000  i=7.97

If coupon is paid semiannually
Financial calculator: n=8, PV=(-)1,200, PMT=70, FV= 1,000
 i=4.026 per 6 month. It would be reported in the financial press as 8.05
percent annual yield (yield to maturity).
26
Bond Rates of Return
Rate of Return – Total income per period per
dollar invested.
total income
Rate of return =
investment
Coupon income + price change
Rate of return =
investment
27
Bond Rates of Return

Rate of Return versus Yield to Maturity

The yield to maturity



defined as the discount rate that equates the bond’s price
to the present value of all its promised cash flows.
a measure of the average rate of return you will earn over
the bond’s life if you hold it to maturity.
The rate of return

can be calculated for any particular holding period

based on the actual income and the capital gain or loss on
the bond over that period.
28
Bond Rates of Return
Example
Our 5.5 percent annual coupon bond currently has 3 years left
until maturity and sells today for $1,056.03. Its yield to maturity is
3.5 percent. Suppose that by the end of the year (Note: at this
time, the bond will have only 2 years to maturity), interest rates
have fallen and the bond’s yield to maturity is now only 2.0
percent. What will be the bond’s rate of return?
$55
$1,055

 $1,067.95
1
2
(1.02)
(1.02)
$55  ($1,067.95  $1,056.03)
Rate of Return 
 0.0634, or 6.34%
$1,056.03
PV at 2.0% 
29
Bond Rates of Return
Example (Continued)
Suppose that the bond’s yield to maturity had risen to 5 percent
during the year. What will be the bond’s rate of return?
$55
$1,055

 $1,009.30
1
2
(1.05)
(1.05)
$55  ($1,009.30  $1,056.03)
Rate of Return 
 0.0078, or 0.78%
$1,056.03
PV at 5% 
30
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.
The term structure of interest
rates (Yield curve)
YTM for corporate and
government bonds
The YTM of corporate bonds is larger
than the YTM of government bonds
 Why does this occur?

Default Risk

Default (or credit) risk


Default premium


The additional yield on a bond investors require for
bearing credit risk.
Investment grade bonds


The risk that a bond issuer may default on its
bonds.
Bonds rated Baa or above by Moody’s or BBB or
above by S&P’s.
Junk bonds

Bonds with a rating below Baa or BBB
34
Default Risk
Moody' s
Standard
& Poor's
Aaa
AAA
Aa
AA
A
A
Baa
BBB
Ba
B
BB
B
Caa
Ca
C
CCC
CC
C
Safety
The strongest rating; ability to repay interest and principal
is very strong.
Very strong likelihood that interest and principal will be
repaid
Strong ability to repay, but some vulnerability to changes in
circumstances
Adequate capacity to repay; more vulnerability to changes
in economic circumstances
Considerable uncertainty about ability to repay.
Likelihood of interest and principal payments over
sustained periods is questionable.
Bonds in the Caa/CCC and Ca/CC classes may already be
in default or in danger of imminent default
C-rated bonds offer little prospect for interest or principal
on the debt ever to be repaid.
35
Default Risk
36
Corporate Bonds
Zero coupons
 Floating rate bonds
 Convertible bonds
 Callable bonds

37