Download Introduction to Financial Management

Bond valuation
The application of the
present value concept
Fin351: lecture 3
Today’s plan
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Interest rates and compounding
Some terminology about bonds
Value bonds
The yield curve
Default risk
Interest

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Simple interest - Interest earned only on
the original investment.
Compounding interest - Interest earned
on interest.
In Fin 351, we consider compounding
interest rates
Simple interest
Example
Simple interest is earned at a rate of 6% for
five years on a principal balance of $100.
Simple interest
Today
Future Years
1
2
3
4
5
Interest Earned
6
6
6
6
6
Value
100 106 112 118 124 130
Value at the end of Year 5 = $130
Compound interest
Example
Compound interest is earned at a rate of 6% for
five years on $100.
Today
Interest Earned
Value
100
Future Years
1
2
3
4
5
6.00 6.36 6.74 7.15 7.57
106.00 112.36 119.10 126.25 133.82
Value at the end of Year 5 = $133.82
Interest compounding
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The interest rate is often quoted as APR, the
annual percentage rate.
If the interest rate is compounded m times in
each year and the APR is r, the effective
annual interest rate is
m
1  r   1


 m
Compound Interest
i
ii
Periods Interest
per
per
year
period
iii
APR
(i x ii)
iv
Value
after
one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
6.000%
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
Compound Interest
Interest Rates
Example
Given a monthly rate of 1% (interest is
compounded monthly), what is the Effective
Annual Rate(EAR)? What is the Annual
Percentage Rate (APR)?
Solution
EAR = (1 + .01)12 - 1 = .1268 or 12.68%
APR = .01 x 12 = .12 or 12.00%
Interest Rates
Example
If the interest rate 12% annually and interest
is compounded semi-annually, what is the
Effective Annual Rate (EAR)? What is the
Annual Percentage Rate (APR)?
Solution
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APR=12%
EAR=(1+0.06)2-1=12.36%
Nominal and real interest rates
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Nominal interest rate
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Real interest rate
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Inflation
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Their relationship
• What is it?
• What is it?
• What is it?
• 1+real rate =(1+nominal rate)/(1+inflation)
Bonds
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Bond – a security or a financial instrument that
obligates the issuer (borrower) to make specified
payments to the bondholder during some time horizon.
Coupon - The interest payments made to the
bondholder.
Face Value (Par Value, Face Value, Principal or
Maturity Value) - Payment at the maturity of the bond.
Coupon Rate - Annual interest payment, as a
percentage of face value.
Bonds

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
An example of a bond
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A coupon bond that pays coupon of 10%
annually, with a face value of $1000, has a
discount rate of 8% and matures in three
years.
•
•
•
•
The coupon payment is $100 annually
The discount rate is different from the coupon rate.
In the third year, the bondholder is supposed to get
$100 coupon payment plus the face value of $1000.
Can you visualize the cash flows pattern?
Bonds
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.
Bond Valuation
The price of a 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.
PV 
cpn
(1  r )1

cpn
(1  r ) 2
 ... 
1,000  cpn
(1  r ) N
Zero coupon bonds
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Zero coupon bonds are the simplest type of bond
(also called stripped bonds, discount bonds)
You buy a zero coupon bond today (cash outflow)
and you get paid back the bond’s face value at
some point in the future (called the bond’s maturity )
How much is a 10-yr zero coupon bond worth today
if the face value is $1,000 and the effective annual Face
value
rate is 8% ?
PV
Time=0
Time=t
Zero coupon bonds (continue)
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P0=1000/1.0810=$463.2
So for the zero-coupon bond, the price is
just the present value of the face value
paid at the maturity of the bond
Do you know why it is also called a
discount bond?
Coupon bond
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.
PV 
cpn
(1  r )1

cpn
(1  r ) 2
 .... 
(cpn  par )
(1  r )t
1

1

  par  PV (annuity)  PV ( par )
 cpn 
 r r (1  r )t  (1  r )t


Bond Pricing
Example
What is the price of a 6 % annual coupon
bond, with a $1,000 face value, which matures
in 3 years? Assume a required return of 5.6%.
Bond Pricing
Example
What is the price of a 6 % annual coupon
bond, with a $1,000 face value, which matures
in 3 years? Assume a required return of 5.6%.
60
60
1,060
PV 


1
2
3
(1.056) (1.056) (1.056)
PV  $1,010.77
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 6 %?
60
60
1,060
PV 


1
2
3
(1.06) (1.06) (1.06)
PV  $1,000
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 15 %?
60
60
1,060
PV 


1
2
(1.15) (1.15) (1.15)3
PV  $794.51
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 5.6% AND the coupons are
paid semi-annually?
Bond Pricing
Example (continued)
What is the price of the bond if the required
rate of return is 5.6% AND the coupons are
paid semi-annually?
30
30
30
1,030
PV 

 ... 

1
2
5
(1.028) (1.028)
(1.028) (1.028)6
PV  $1,010.91
Bond Pricing
Example (continued)
Q: How did the calculation change, given semiannual coupons versus annual coupon
payments?
Bond Pricing
Example (continued)
Q: How did the calculation change, given semiannual coupons versus annual coupon
payments?
Time Periods
Paying coupons twice a
year, instead of once
doubles the total number of
cash flows to be discounted
in the PV formula.
Bond Pricing
Example (continued)
Q: How did the calculation change, given semiannual coupons versus annual coupon
payments?
Time Periods
Discount Rate
Paying coupons twice a
year, instead of once
doubles the total number of
cash flows to be discounted
in the PV formula.
Since the time periods are
now half years, the
discount rate is also
changed from the annual
rate to the half year rate.
Bond Yields
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Current Yield - Annual coupon
payments divided by bond price.
Yield To Maturity (YTM)- Interest rate
for which the present value of the
bond’s payments equal the market
price of the bond.
P
cpn

cpn
(1  y )1 (1  y ) 2
 .... 
(cpn  par )
(1  y )t
An example of a bond

A coupon bond that pays coupon of 10%
annually, with a face value of $1000, has
a discount rate of 8% and matures in
three years. It is assumed that the
market price of the bond is the same as
the present value of the bond.
• What is the current yield?
• What is the yield to maturity.
My solution
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First, calculate the bond price
P=100/1.08+100/1.082+1100/1.083
=$1,051.54
Current yield=100/1051.54=9.5%
YTM=8%
Bond Yields
Calculating Yield to Maturity (YTM=r)
If you are given the market price of a
bond (P) and the coupon rate, the yield
to maturity can be found by solving for r.
P
cpn

cpn
(1  y )1 (1  y ) 2
 .... 
( cpn  par )
(1  y )t
Bond Yields
Example
What is the YTM of a 6 % annual coupon
bond, with a $1,000 face value, which matures
in 3 years? The market price of the bond is
$1,010.77
60
60
1,060
PV 


1
2
3
(1  r ) (1  r ) (1  r )
PV  $1,010.77
Bond Yields
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In general, there is no simple formula
that can be used to calculate YTM
unless for zero coupon bonds
Calculating YTM by hand can be very
tedious. We don’t have this kind of
problems in the quiz or exam
You may use the trial by errors
approach get it.
Bond Yields (3)
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(a)
(b)
(c)
(d)
Can you guess which one is the
solution in the previous example?
6.6%
7.1%
6.0%
5.6%
The bond price, coupon rates
and discount rates
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If the coupon rate is larger than the
discount rate, the bond price is larger
than the face value.
If the coupon rate is smaller than the
discount rate, the bond price is smaller
than the face value.
The rate of return on a bond
Coupon income + price change
Rate of return =
investment or bond price
Rate of return =
profit
cost of investment
Example: An 8 percent coupon bond has a
price of $110 dollars with maturity of 5 years
and a face value of $100. Next year, the
expected bond price will be $105. If you hold
this bond this year, what is the rate of return?
My solution
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The expected rate of return for holing the
bond this year is (8-5)/110=2.73%
• Price change =105-110=-$5
• Coupon payment=100*8%=$8
• The investment or the initial price=$110
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
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The YTM of corporate bonds is larger
than the YTM of government bonds
Why does this occur?
Default Risk
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Default risk
• The risk associated with the failure of the
borrower to make the promised payments
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Default premium
• The amount of the increase of your discount
rate
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Investment grade bonds
Junk bonds
Ranking bonds
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.