bond valuation.ppt

Jacoby, Stangeland and Wajeeh, 2000 1
Valuation of Bonds and Stock
First Principles:
 Value of financial securities = PV of expected
future cash flows
To value bonds and stocks we need to:
 Estimate future cash flows:
size (how much) and timing (when)
 Discount future cash flows at an appropriate
rate
Chapter 5

Bond Features
 What is a bond -
 debt issued by a corporation or a governmental body.
 A bond represents a loan made by investors to the issuer.
 In return for his/her money, the investor receives a legal claim on
future cash flows of the borrower.
 The issuer promises to:
 make regular coupon payments every period until the bond
matures, and
 pay the face (par) value of the bond when it matures.
 Default
 an issuer who fails to pay is subject to legal action on behalf of the
lenders (bondholders).

3
Pure-Discount (Zero-Coupon) Bonds
Information needed for valuing pure discount bonds:
 Time to maturity (T):
T = Maturity date - today’s date
 Face value (F)
 Discount rate (r)
0 1 2 … T
|-------------------|-------------------|------ … ------|
F
Value of a pure discount bond:
PV = F / (1 + r)T

4
Examples - Pure Discount Bonds
Q1. Consider a zero-coupon bond, with a face value of $1,000,
maturing in 5 years. Suppose that the appropriate discount rate
is 8%. What is the current value of the bond?
A1. This is a simple TVM problem:
Use the above PV equation to solve:
PV = F / (1 + r)T = 1,000 / (1.08)5 = $
Q2. Suppose 6 months have past. What is the bond value now?
A1. Again, use the above PV equation to solve:
PV = F / (1 + r)T = 1,000 / (1.08)4.5 = $
Note: As we get closer to maturity(T), the z.c. bond value increases
(PVm), since we have to wait less time to receive $1,000
0 1 2 3 4 5
Year:
(r = 8%)
1,000
PV0

5
Level-Coupon Bonds
Information needed to value level-coupon bonds:
 Coupon payment dates and Time to maturity (T)
 Coupon (C) per payment period and Face value (F)
 Discount rate
0 1 2 … T
|----------------|------------------|------- … ------|
Coupon Coupon Coupon + F
Value of a Level-coupon bond:
PV= C/(1+r) + C/(1+r)2 + .. + C/(1+r)T + F/(1+r)T
= C (1/r){1 - [1 / (1 + r)T]} + F/(1 + r)T
= PV of coupon payments + PV of face value

6
Example - Coupon Bonds
Q1. Consider a coupon bond paying a 4% coupon rate annually, with a
face value of $1,000, maturing in 10 years. Suppose that the
appropriate discount rate is 6%. What is the current value of the bond?
A1. The time line:
Define:
c = annual coupon rate (%)
C = dollar periodic coupon payment = c%F
In the above example:
c = % C = c%F = = $
F = $ T = years r = %
PV= C (1/r){1 - [1 / (1 + r)T]} + F/(1 + r)T
= 40(1/0.06){1 - [1 / (1.06)10]} + 1,000/(1.06)10 = $
0 1 2 9 10 (Years)
(r = 6%)
…

7
First, clear previous data, and check that your calculator is set to 1 P/YR:
The display should show: 1 P_Yr
Input data (based on above bond example)
PV of a Bond in your HP 10B Calculator
PMT
I/YR
N
PV
Key in coupon
payment
Key in discount
rate
Key in number of
periods to maturity
Compute PV of
the bond
Display should show:
-852.79825897
40
6
10
FV
Key in face value
(paid at maturity)
1,000
Yellow
C
C ALL

8
Example - Discount, Premium and Par Bonds
Q2. For the above coupon bond: when discount rate is 6% and
coupon rate is 4% (c < r), the value of the bond is $852.80, less
than its face value (PV < F). In this case we say that the bond is
priced at discount. Recalculate the PV of the above bond with
discount rates of 2% and 4%.
A2. r = 2%
We have: r = 2% < 4% = c.
PV= C (1/r){1 - [1 / (1 + r)T]} + F/(1 + r)T
= 40(1/0.02){1 - [1 / (1.02)10]} + 1,000/(1.02)10 = $1,179.65
We see that when c > r, the bond is priced at premium (PV > F).
r = 4%
We have: r = c = 4%.
PV= 40(1/0.04){1 - [1 / (1.04)10]} + 1,000/(1.04)10 = $1,000
We say that when c = r, the bond is priced at par (PV = F).

Some Tips on Bond Pricing
Bond prices and market interest rates move in
opposite directions.

 When coupon rate = market rate (r) => price = par value.
(par bond)
 When coupon rate > market rate (r) => price > par value
(premium bond)
 When coupon rate < market rate (r) => price < par value
(discount bond)

PV
r (%)
1,000 = F
4 = c
2 6
($)
Premium Bond
(r < c , and PV>F)
Discount Bond
(r > c , and PV<F)
Par Bond
(r = c , and PV=F)
Discount, Premium, and Par Bonds

11
The bond’s indenture provides: F, c, and T
The bond price (B) is set by the market
Given F, c, T, and B, what return (y) does the
market demand for holding the bond?
To find y, solve the following equation:
There is no analytical solution (use calculator)
   
  T
T
T
T
y
F
y
y
y
F
y
C
y
C
y
C
C
B
B
)
1
(
)
1
(
1
1
)
1
(
)
1
(
)
1
(
)
1
(
1
or
,
2
1













 
Yield-To-Maturity

Example - Bond’s YTM
Q. Consider a coupon bond paying a 7% coupon rate annually, with a
face value of $1,000, maturing in 20 years. The current market price
of the bond is $1,072.93. What is the yield to maturity (YTM) of the
bond?
A. We have:
c = % C = c%F = =$
F = $ T = B = $
Use your HP 10B Financial Calculator:
PMT
PV
N
I/YR
1) Key in coupon
payment
3) Key in the
bond price (PV)
4) Key in number of
periods to maturity
5) Compute YTM Display should show:
6.346178%
70
1,072.93
20
FV
2) Key in face value 1,000
+/-
Yellow
C
C ALL

13
Canadian Bonds
Canadian bonds usually pay coupons every six months (semiannually)
Q. Consider a GofC bond paying semiannual coupons at an annual rate of
6%, with a face value of $1,000, maturing in 8 years. The bond’s YTM is
7% per year compounded semiannually. What is the value of the bond?
A. The time line:
Define:
c = (stated) annual coupon rate (%) = 6%
C = dollar periodic coupon payment = (c/2)%F= = $
We also have:
F = 1,000 N = s.a. periods y1/2 = 7% per year comp. s.a.
Use the following PV equation to solve:
0 1 2 15 16 (6-month Periods)
(y1/2 = 7% per year comp. s.a.)
…
53
.
939
$
1
30
)
1
(
)
1
(
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16
2
07
.
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16
2
07
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0
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07
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1
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F
c
PV

14
First, clear previous data, and check that your calculator is set to 1 P/YR:
The display should show: 1 P_Yr
Input data (based on above bond example)
PV of a S.A. Coupon Bond in your HP 10B Calculator
PMT
I/YR
N
PV
Key in the s.a.
coupon payment
Key in the effective
s.a. discount rate
Key in number of 6-month
periods to maturity
Compute PV of
the bond
-939.52941596
30
3.5
16
FV
Key in face value
(paid at maturity)
1,000
Yellow
C
C ALL

15
Finding the YTM of Canadian Bonds
Q. Consider a GofC bond paying semiannual coupons at an annual rate of
12%, with a face value of $1,000, maturing in 25 years. The bond’s market
value is $1,057.98. What is the yield to maturity (YTM) of the bond per
year compounded semiannually?
A. We have: c = 12% C = (c/2)%F= = $
F = 1,000 N = s.a. periods B= 1,057.98
Since (y1/2/2) = 5.649995%, the YTM of the bond per year compounded
semiannually is given by: y1/2 = 2%5.649995% = 11.299990%
PMT
PV
N
I/YR
1) Key in s.a. coupon
payment
3) Key in the
bond price (PV)
4) Key in number of s.a.
periods to maturity
5) Compute the
effective YTM
PER 6 MONTHS
5.649995%
60
1,057.98
50
FV
+/-
Yellow
C
C ALL

16
Finding the Maturity of Canadian Bonds
Q. Consider a GofC bond paying semiannual coupons at an annual rate of 7%,
with a face value of $1,000. The bond’s market value is $1,026.82, with a
YTM of 6.6% per year compounded semiannually. What is the time to
maturity of this bond in years?
A. We have: c = 7% C = (c/2)%F= = $
F = 1,000 (y1/2 /2) = = % B=$
Thus, T = 0.5%(# of 6-month periods to maturity)
= 0.5% 17.99806470 = 8.99903235 years to maturity
PMT
PV
I/YR
N
1) Key in s.a. coupon
payment
3) Key in the
bond price (PV)
4) Key in the effective
YTM per 6 months
5) Compute the number
of 6 MONTHS
PERIODS to maturity
17.99806470
35
1,026.82
3.3
FV
+/-
Yellow
C
C ALL

17
Finding the Coupon Rate of Canadian Bonds
Q. Consider a 30-year GofC bond paying semiannual coupons, with a face value
of $1,000. The bond’s market value is $912.83, with a YTM of 8.4% per year
compounded semiannually. What is the the bond’s annual coupon rate?
A. We have: F = 1,000 (y1/2/2) = = %
B= 912.83 N = s.a. periods
Since: C = (c/2)%F, we get:
c = (C/F)%2 = (38.00001999/1,000)%2 = 7.600004% per annum
N
PV
I/YR
PMT
1) Key in number of s.a.
periods to maturity
3) Key in the
bond price (PV)
4) Key in the effective
YTM per 6 months
5) Compute the SEMI
ANNUAL DOLLAR
coupon payment
38.00001999
60
912.83
4.2
FV
+/-
Yellow
C
C ALL

From the financial Post, November 28, 1997
Years to Bond
Maturity maturity yields (%)
1-Dec-97 0.00 0.00
1-Nov-98 0.92 4.57
1-Mar-99 1.25 4.70
1-Dec-99 2.00 4.98
1-Mar-00 2.25 5.02
1-Sep-00 2.75 5.09
1-Mar-01 3.25 5.16
1-Dec-01 4.00 5.24
1-Apr-02 4.33 5.25
1-Jun-03 5.50 5.37
1-Dec-03 6.00 5.41
1-Jun-04 6.50 5.42
1-Dec-04 7.01 5.47
1-Dec-05 8.01 5.53
1-Dec-06 9.01 5.56
1-Jun-10 12.51 5.73
1-Mar-11 13.25 5.76
15-Mar-14 16.30 5.84
1-Jun-21 23.52 5.93
The Yield Curve
4
4.5
5
5.5
6
6.5
0 5 10 15 20 25 30
Years to maturity
Bond
yields
(%)

19
Four theories:
I. Expectations theory
e.g. if investors expect next years yield to be 12%, then
the forward rate will also be 12%: 1f2 = 12%
Example: two alternative investments
 B1: a zero coupon bond with: T = 1, YTM: 0r1 = 8%
 B2: a zero coupon bond with: T = 2, YTM: 0r2 = 9%,
The Term Structure of Interest Rates
 
1
1 over year
expected
rate
spot




t
t
t
t
r
E
t
f

The following investments of $1 must be equivalent:
(i) investing in B2 for 2 years. At t = 2, receive: 1.092
(ii) investing in B1 for 1 year. At t = 1, investing in a new
1-year bond at a rate 1f2. At t = 2, receive: 1.08(1+1f2)
The forward rate (1f2) must take a value such that:
The Term structure of Interest Rates
)
1
(
08
.
1
)
09
.
1
( 2
1
2
f




21
This implies the following forward rate for year-2:
The general case:
Note: This formula can be used only for zero-coupon bonds
Example: 3 alternative zero-coupon bonds, with the following spot rates:
0r1 = 8%
0r2 = 10%
0r3 = 12%
Calculating 1f2 and 2f3:
%
01
.
10
1
1 08
.
1
09
.
1
)
1
(
)
1
(
2
1
2
2
1
0
2
0




 

r
r
f
1
)
1
(
)
1
(
1
0
0
1
1







t
t
t
t
r
r
t
t f
%
037
.
12
1
1 08
.
1
10
.
1
)
1
(
)
1
(
2
1
2
1
2
1
0
2
0







r
r
f
%
110
.
16
1
1 2
3
2
3
10
.
1
12
.
1
)
1
(
)
1
(
3
2
2
0
3
0







r
r
f

22
Example - Using the Term Structure
Q1. You observe the above spot rates for GofC zero-coupon bonds for
different maturities: 0r1 = 8%, 0r2 = 10%, and 0r3 = 12%. A zero-
coupon bond has a face value of $1,000 and maturity of 2 years.
What must be its price today?
A1. Since: (1+0r2)2 = (1+0r1)(1+1f2), we can use either spot rates or
forward rates (same result) to find B:
Q2. Assume that the Pure Expectations Hypothesis (PEH) holds, what do
you expect the bond price to be one year from today?
A2. One year from today, the bond will have one year remaining to
maturity. Based on the PEH:
expected spot rate for second year = 1f2
Thus, the expected bond price in a year is:
45
.
826
$
:
:
or
,
45
.
826
$
:
12037
.
1
08
.
1
000
,
1
)
1
)(
1
(
000
,
1
)
1
.
1
(
000
,
1
)
1
(
000
,
1
2
1
1
0
2
0
2
2










f
r
r
B
Forward
B
Spot
56
.
892
$
]
[ 12037
.
1
000
,
1
)
1
(
000
,
1
1 2
1


  f
B
E
 
2
1r
E

23
II. Liquidity Premium Theory
If you invest for (t+1) years, you commit to reinvest in every
year after the 1st year, and thereby lose liquidity and ask
for a liquidity premium:
III. Augmented Expectations Theory
Combines the pure expectations theory with the liquidity premium theory:
Example - Suppose: 0r1 = 8% and 0r2 = 9%. By the Expectations Theory:
By the Liquidity Premium Theory, when L2 =1%, we get:
1f2= E[1r2] + L2. = E[1r2] + 1%
Both theories together, give:
  1
1
1
1 )
1
(
over year
expected
rate
spot









t
t
t
t
t
t
L
r
E
L
t
f
 
2
1
2
1 r
E
f 

)
1
(
08
.
1
09
.
1 2
1
2
f


  )
01
.
0
1
(
08
.
1
09
.
1 2
1
2


 r
E

Shapes of the Term Structure under the Liquidity Premium Theory & the
Augmented Expectations Theory
The shape of the term structure depends on the magnitude of the premium:
 when there exist constant expectations:
 when there exist decreasing expectations:
%
t
f ``
f `
t
% f
E[trt+1]
L
E[trt+1]

25
IV. Market Segmentation Theory
 Different segments of investors choose to invest in assets with different
investment horizons:
e.g. mutual funds that commit to invest in long term bonds.
 Thus, demand and supply in each segment could set different rates:
t
%
Short
Term
Bonds
Medium
Term
Bonds
Long
Term
Bonds

Common Stocks
 What are stocks -
 legal representation of of ownership in a corporation (equity)
 a stock holder is entitled to receive profit distributions of the
corporation (dividends)
 Dividends:
 cash payments made by the corporation to stockholders
 since stocks have no expiration date, we assume that dividends will
be paid forever
 Valuation
 the value of stocks at any point in time equals the present value of
all future dividends

Common Stock Valuation
 The value of a stock = PV of all expected future cash flows
 Thus, the information needed to value common stocks:
 Common Stock Dividends (Dt)
 Discount rate (r)
 PV0 = D1/(1 + r)1 + D2/(1 + r)2 + D3/(1 + r)3 + . . . forever. .
 We have to estimate future dividends

Case 1: Zero Growth
Assume that dividends will remain at the same
level forever, i.e. D1 = D2 =…= Dt = D
Since future cash flows are constant, the value of a
zero growth stock is the present value of a
perpetuity:
Pt = Dt+1 / r

Example - Valuation of Common Stocks with Zero Growth
Q. ABC Corp. is expected to pay $0.75 dividend per annum, starting a
year from now, in perpetuity. If stocks of similar risk earn 12%
annual return, what is the expected price of a share of ABC stock?
A. The stock price is given by the the present value of the perpetual
stream of dividends:
P0 = D1 / r
= =
0 1 2 3 4
$ $ $ $

Case 2: Constant Growth
Assume that dividends will grow at a constant
rate, g, forever, i. e.,
D1 = D0 x (1+g)
D2 = D1 x (1+g) = D0 x (1+g)2

Dt = D0 x (1+g)t

Since future cash flows grow at a constant rate
forever, the value of a constant growth stock is the
present value of a growing perpetuity:
Pt = Dt+1 / (r - g)

Examples - Valuation of Common Stocks with Constant Growth
Q1. XYZ Corp. has a common stock that paid its annual dividend this
morning. It is expected to pay a $3.60 dividend one year from now,
and following dividends are expected to grow at a rate of 4% per
year into the foreseeable future (forever)in perpetuity. If stocks of
similar risk earn 16% effective annual return, what is the price of a
share of XYZ stock?
A1. The stock price is given by the the present value of the perpetual
stream of growing dividends:
P0 = D1 / (r-g)
= =
0 1 2 3 4
$3.60 $3.60% $3.60% $3.60%
forever . . .

32
Q2. In the above example, assume that XYZ’s common stock that paid its
quarterly dividend two months ago. It is expected to pay a $0.90 dividend
in one month, and following quarterly dividends are expected to grow at
a rate of 1% per quarter into the foreseeable future. Recall that the
effective annual required rate of return on XYZ stock is 16%. What is the
price of a share of XYZ stock now?
A2. Time line of the quarterly dividends:
We first need to calculate EPR1/4 and EPR1/12:
Using: EPRn = (1+EAR)n - 1, we get:
EPR1/4 = 3.780199% and EPR1/12 = 1.244514%
The stock price in one month (after D1 month is paid):
P1 month = D4 months / (EPR1/4 - g)
= (0.90%1.01)/(0.03780199 - 0.01) = $32.69550129
The stock price today:
P0 = (D1 month + P1 month) / (1+EPR1/12)
= (0.90+ 32.69550129) / 1.01244514 = $33.18
0 1 month 4 months 7 months 10 months
$0.90 $0.90% $0.90% $0.90%
forever . . .

33
Q3. Manitoba Network Operators (MNO) is expected to pay a dividend
next year of $8.06 per share. Both sales and profits for Pale Hose
are expected to grow at a rate of 2% per year indefinitely. Its
dividend is expected to grow by the same amount. If an investor is
currently willing to pay $62.00 per one MNO share, what is her
required return for this investment?
A3. We have: P0=$62.00, D1=$8.06, and g=0.02. We are looking for r.
The stock price is given by:
P0 = D1/(r-g)
=
Rearranging, we get:
r = =
In general:
r = (D1/P0) + g

34
Q4. Vandalay Industries Corp. (VIC) is expected to pay a dividend
next year of $4.32 per share. Its current stock price is $36. If the
required return for this stock is 15%, what is the constant dividend
growth rate expected for VIC’s stock starting from the second year
forever?
A4. We have: P0=$36.00, D1=$4.32, and r=0.15. We are looking for g.
P0 = D1/(r-g)
=
g = =
In general:
g = r - (D1/P0)

Q5. MT&T Inc. has a common stock that paid its annual dividend this
morning. You expect future annual dividends to grow at a rate of
2% per year into the foreseeable future (forever). The required
return for this stock is 20%, and its current price is $25.50. What is
the dividend that was paid this morning?
A5. We have: P0=$25.50, r=0.20, and g=0.02. We are looking for D0.
P0 = D1/(r-g)
25.5 = D1/(0.20-0.02)
D1 = 25.5(0.20-0.02) = $4.59
We are looking for D0. Since D1=D0(1+g), D0 is given by:
D0 = D1/(1+g) = = $

36
Case 3: Differential Growth
Assume that dividends will grow at different rates
in the foreseeable future and then will grow at a
constant rate thereafter.
To value a Differential Growth Stock, we need to:
 Estimate future dividends in the foreseeable
future.
 Estimate the future stock price when the stock
becomes a Constant Growth Stock (case 2).
 Compute the total present value of the
estimated future dividends and future stock
price at the appropriate discount rate.

37
Examples - Differential Growth
Q1. Whizzkids Inc. is experiencing a period of rapid growth. Earnings and
dividends are expected to grow at a rate of 8 percent during the next three
years, and then at a constant rate of 4% thereafter. Whizzkids’ last dividend,
which has just been paid, was $2 per share. If the required rate of return on
the stock is 12 percent, what is the price of the stock today?
A1. It is given that:
r = 12%, D0 = $2, g1 = g2 = g3 = 8%, and g4 = g* = 4% (forever)
We calculate:
D1=$2% =$ , D2= =$ ,
D3= =$
With g4=g* =4%, we have:
D4= =$
Since constant growth rate applies to D4, we use Case 2 (constant
growth) to compute P3:
P3 = = $

Expected future cash flows of this stock:
0 1 2 3
|----------|---------|---------| (r = 12%)
D1 D2 D3 + P3
2.16 2.33 2.52 + 32.75
The current (time 0) value of the stock:
P0 = D1/(1+r) + D2/(1+r)2 + (D3+P3)/(1+r)3
= + +
= $

39
Q2. An investor has just paid $141.75 for the purchase of one share of UMB
Corp. stock. UMB just paid a $9 dividend per share. Annual dividends paid
at the end of the first, second and third years will grow at a rate of 10% per
annum, and then grow at a constant annual rate of g* forever. Given the risk
inherent in UMB Corp., the investor requires an effective annual rate of
10% on his/her investment. What is the value of g*?
A2. We calculate:
D1=$9% =$ , D2=
=$ ,
D3= =$
With g4=g*, we have:
D4=
UMB’s current stock price is given by:
P0 = D1/(1+r) + D2/(1+r)2 + (D3+P3)/(1+r)3
Where: P3 = D4/(r-g*)

With the above data:
141.75 = 9.9/1.1 + 10.89/(11)2 + 11.979/(11)3 + P3/(1.1)3
Thus, the expected stock price in three years is P3 = 152.73225
Since, this price is given by:
P3 = D4/(r-g*) = [D3(1+g*)]/(r-g*)
We have
152.73225 =
g* = %

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