Chapter 2:
Valuation of Stocks and Bonds
1
What is Value?
v
v
In general, the value of an asset is the price
that a willing and able buyer pays to a willing
and able seller
Note that if either the buyer or seller is not
both willing and able, then an offer does not
establish the value of the asset
2
Several Kinds of “Value”
v
There are several types of value, of which we
are concerned with three:
• Book Value - The asset’s historical cost less
its accumulated depreciation
• Market Value - The price of an asset as
determined in a competitive marketplace
• Intrinsic Value - The present value of the
expected future cash flows discounted at
the decision maker’s required rate of return
3
Determinants of Intrinsic Value
v
There are two primary determinants of the intrinsic value
of an asset to an individual:
• The size and timing of the expected future cash flows
• The individual’s required rate of return (this is determined by a
number of other factors such as risk/return preferences, returns on
competing investments, expected inflation, etc.)
v
Note that the intrinsic value of an asset can be, and often
is, different for each individual (that’s what makes
markets work)
4
Chapter 2:
Valuation of Stocks and Bonds
2.3 Valuation of Preferred
Stocks
5
Preferred Stock Features
v
v
v
v
v
v
v
v
v
Preferred stock differs from common stock because it has preference over
common stock on payment of dividends and in the distribution of
corporation assets in the event of liquidation.
Preferred stock is a form of equity from a legal, tax, and regulatory
standpoint.
Holders of preferred stock generally have no voting privileges.
However, holders of preferred stock are often granted voting and other
rights if preferred dividends have not been paid for some time.
Preferred stock have a stated liquidating value.
The cash dividend is described in dollars per share.
A preferred dividend is not like bond interest
Dividends on preferred stock are either cumulative or non-cumulative.
Dividends not declared on cumulative preferred stock are carried
forward and must be paid before common shareholders can receive
anything
6
Features of preferred stock
v
v
v
v
v
v
v
v
v
v
A hybrid security
May be perpetuity or redeemable
Paid before common dividends.
Cumulative or Non-cumulative dividends
Dividends not a liability
Protective provisions (voting)
Call provisions & sinking funds
Convertible or Non-convertible
Usually non-voting and non-participative.
Priority-lower than debt, higher than stock.
7
Preferred Stock Valuation
Preferred stocks can usually be valued like a
perpetuity:
Vp =
D
kp
8
Example:
Xerox preferred that pays $4.125 dividend
per year. Suppose our required rate of
return on Xerox preferred is 9.5%
Vp
=
4.125
.095
=
$43.42
9
Expected Rate of Return on Preferred
Just adjust the valuation model:
D
kp =
Vp
kp =
D
Po
10
Example
If we know the preferred stock price is $40,
and the preferred dividend is $4.125, the
expected return is:
kp =
D
Po
=
4.125
=
.1031
40
11
Valuation of redeemable preferred stock
v
The value of a preferred stock equals the present value of all
future dividends
n
D
M
Vp  

t
(1  k p ) n
t 1 (1  k p )
Vp  Current value of preferencestock
D  periodicaldividend
n  life of thepreferencestock
k p  required rateof returnon preferencestock
M  maturity value
Since thestreamof dividends is an ordinaryannuity,
Vp  D  P VIFAk p ,n  M  P VIFk p ,n
12
Chapter 2:
Valuation of Stocks and Bonds
2.4 Valuation of Common
Stocks
13
Features of common stock
v
v
v
v
Residual income and asset claimants
• Unlimited upside
• First to suffer
Priority
1. Debt
2. Preferred Stock
3. Common Stock
A firm cannot go bankrupt for not declaring dividends
Dividends and Taxes
• Dividend payments are not considered a business
expense, therefore, they are not tax deductible
• Dividends received by individuals are taxed as ordinary
income
14
Features of Common Stock
v
v
v
The term common stock usually implies the shareholder
has no special preference either in dividends or in
bankruptcy.
Shareholders, however, control the corporation through
their right to elect the directors. The directors in turn hire
management to carry out their directives.
Directors are elected at an annual shareholders’ meeting by
a vote of the holding of a majority of shares present and
entitled to vote.
15
Common Stock Features
v
Shareholders usually have the following rights
also:
1. The right to share proportionally in dividends
paid.
2. The right to share proportionally in assets
remaining after liabilities and preferred
shareholders have been paid in a liquidation.
3. The right to vote on stockholder matters of great
importance, such as a merger or new share
issuance.
16
Common Stock Features
Dividends
v
v
v
Dividend payments are at the discretion of the BoD.
Dividends are not a liability of the corporation until
declared by the BoD.
Dividends are not tax deductible for the issuing
corporation.
17
Common Stock Features
Classes of Stock
v
v
v
Some firms have more than one class of common
stock; often, the classes are created with unequal
voting rights.
Non-voting shares must receive dividends no lower
than dividends on voting shares.
A primary reason for creating dual classes of stock
has to do with control of the firm.
18
Common Stock Valuation
v
v
Just like with bonds, the first step in valuing
common stocks is to determine the cash flows
For a stock, there are two:
• Dividend payments
• The future selling price
v
Again, finding the present values of these
cash flows and adding them together will
give us the value
19
Cash flows for stockholders
v
If you buy a share of stock, you can receive
cash in two ways
• The company pays dividends
• You sell your shares, either to another investor in
the market or back to the company
v
As with bonds, the price of the stock is the
present value of these expected cash flows
20
Common Stock Valuation
Unlike bonds, valuing common stock is more difficult.
Why?
1.
2.
3.
The timing and amount of future cash flows is not
known.
The life of the investment is essentially forever.
There is no way to observe the rate of return that
the market requires.
21
Some Notes About Common Stock
v
In valuing the common stock, we have to make two
assumptions:
• We know the dividends that will be paid in the future
• We know how much you will be able to sell the stock for in
the future
v
v
Both of these assumptions are unrealistic, especially
knowledge of the future selling price
Furthermore, suppose that you intend on holding on
to the stock for twenty years, the calculations would
be very tedious!
22
Common Stock: Some Assumptions
v
v
v
We cannot value common stock without making
some simplifying assumptions
If we make the following assumptions, we can derive
a simple model for common stock valuation:
Assume:
• Your holding period is infinite (i.e., you will never sell the
stock)
• The dividends will grow at a constant rate forever (this is
only one example of assumption that simplifies the problem)
v
Note that the second assumption allows us to predict
every future dividend, as long as we know the most
recent dividend
23
Common Stock Valuation:
Dividend Discount Model
24
Single-Period Valuation Model
v
v
Suppose you are thinking of purchasing the stock of Moore Oil, Inc.
and you expect it to pay a $2 dividend in one year and you believe
that you can sell the stock for $14 at that time. If you require a return
of 20% on investments of this risk, what is the maximum you would
be willing to pay?
Remember, the cash flows to the stockholder is simply the dividends
received + the future sales price
D1
P1
Vc 

(1  k c ) (1  k c )
25
Single Holding Period
You expect XYZ stock to pay a $5.50 dividend at the end of
the year. The stock price is expected to be $120 at that
time. If you require a 15% rate of return, what would you
pay for the stock now?
?
5.50 + 120
0
1
Ans: $ 109.13
26
What happens if ?
The price of common stock is expected to grow
at the rate of g % annually ?
The current price P0 becomes Po(1+g) a year
hence.
Po (1  g)
D1
D1
Po 


(1  k c ) (1  k c ) (k c  g)
27
Example
The expected dividend per share on the equity share of
Roadking Limited is Rs 2.00. The dividend per share of
Roadking Limited has grown over the past five years at
the rate of 5 % per year. This growth rate will continue in
future. Further, the market price of the equity share of
Roadking Limited, too, is expected to grow at the same
rate. What is a fair istimate of the intrinsic value of the
equity share of Roadking Limited if the required rate is
15% ?
D1
2
Po 

 Rs 20.00
(k c  g) (0.15  0.05)
28
Expected Rate of Return
What rate of return can the investor expect, given the
current market price and forecasted values of dividend
and share price ?
Kc = (D1 / Po)+ g
29
Multi-period Valuation Model
v
The value of a stock today (its current price) is in
theory equal to the present value of all future
dividends plus that of the selling price.
D3
D1
D2
D4
Dn
Pn
P0 



 ........... 

2
3
4
n
1  k c (1  k c ) (1  k c ) (1  k c )
(1  k c )
(1  k c ) n
n
Dt
Pn


t
(1  k c ) n
t 1 (1  k c )
30
Multi-period Valuation Model
But common shares have no maturity period –
they may be expected to bring a dividend stream
of infinite duration
D3
D1
D2
D4
D
P0 



 ........... 
2
3
4
1  k c (1  k c ) (1  k c ) (1  k c )
(1  k c ) 

Dt

t
(
1

k
)
t 1
c
31
Multi-period Valuation Model
v
v
That was the generalized multi-period
valuation formula – which is general enough
to permit any dividend pattern – constant,
rising, declining or randomly fluctuating.
For practical applications, it is helpful to
make simplifying assumptions about the
pattern of dividend growth.
32
Commonly used assumptions types:
1.
2.
3.
4.
The dividend per share remains constant forever, implying
that the growth rate is nil (THE ZERO GROWTH MODEL)
The dividend per share grows at a constant rate per year
forever (THE CONSTANT GROWTH MODEL)
The dividend per share grows at a constant rate for a finite
period, followed by a constant normal rate of growth
forever thereafter (THE TWO STAGE MODEL)
The dividend per share, currently growing at an abovenormal rate, experiences a gradually declining rate of
growth for a while. Thereafter it grows at a constant
normal rate (THE “H” MODEL)
33
Zero Growth Model
Assuming that the dividend per share remains
constant year after year, at a value of D, the
valuation model becomes as that of the
perpetual preference stock;
D
D
D
D
D
P0 



 ........... 
2
3
4
1  k c (1  k c ) (1  k c ) (1  k c )
(1  k c ) 

D
t 1
(1  k c ) t


D
kc
34
Example
Suppose stock is expected to pay a $0.50
dividend every quarter and the required
return is 10% with quarterly compounding.
What is the price?
D
0.5
P0 

 $ 20.00
k c 0.025
35
Constant Growth model
Assumes that the dividend per share grows at
a constant rate (g)
D1
D1 (1  g) D1 (1  g) 2 D1 (1  g)3
D1 (1  g) n
P0 



 ........... 
 .......
2
3
4
n 1
1  k c (1  k c )
(1  k c )
(1  k c )
(1  k c )
With a little algebra, this reduces to:
D0 (1  g)
D1
P0 

kc - g
kc - g
36
Example 1
Suppose Big K, Inc. just paid a dividend of
$5. It is expected to increase its dividend by
2% per year. If the market requires a return
of 15% on assets of this risk, how much
should the stock be selling for?
5(1  0.02) 5.10
P0 

 $ 39.23
0.15 - 0.02 0.13
37
Example 2
Suppose Comolli, Inc. is expected to pay a $2
dividend in one year. If the dividend is
expected to grow at 5% per year and the
required return is 20%, what is the price?
Ans: $ 13.33
38
Example 3
Griggs Inc. last dividend (D0) was $2. The dividend
growth rate (g) is a constant 5%. If the required
return (kc) = 10%, what is P0?
2(1.05)
P0 
 $42
(.10  .05)
39
Example 4
Overton Corp. just paid a $2 dividend. If
the dividends will grow at a constant rate of
5% in the future, what is the stock price in 4
years (at t = 4) assuming a required rate of
return = 10%?
2.1
2.1(1  0.05) 2.1(1  0.05)
2.1(1  0.05)
P0 



2
3
4
1  0.10 (1  0.10)
(1  0.10)
(1  0.10)
2
3
40
What drives growth ?
v
v
v
Most stock valuation models are based on the assumption
that dividends grow over time.
What drives this growth ?
The two major drivers of growth are :
a) Plough-back or Retention Ratio
b) Return on Equity (ROE)
v
v
Growth = Retention Ratio x Return on Equity
Illustration:
Omega limited has an equity (net worth) base of 100 at
the beginning of year 1. It earns a ROE of 20 %. It pays
out 40 % of its equity earnings and ploughs back 60 % of
its equity earnings
41
Financials of Omega Limited
Year 1
Year 2
Year 3
Beginning Equity
ROE
Equity Earnings
Dividend Payout Ratio
Dividends
Retention Ratio
Retained earnings
What is the growth Rate of Dividend ?
42
Financials of Omega Limited
Year 1
Year 2
Year 3
Beginning Equity
100
112
125.44
ROE
Equity Earnings
20%
20
20%
22.4
20%
25.1
Dividend Payout Ratio
0.40
0.40
0.40
Dividends
Retention Ratio
8
0.60
8.96
0.60
10.04
0.60
12
13.44
15.06
Retained earnings
Growth Rate = RE x ROE = 0.60 x 20 %= 12 %
43
What is this growth actually ?
Sustainable growth rate =ROE  Retention ratio
Return on equity (ROE) = Net income / Equity
Retention ratio = 1 – Payout ratio
44
Estimation of Growth
v
The growth rate in dividends
(g) can be estimated in a
number of ways.
 Using
the
company’s
historical average growth
rate.
 Using an industry median or
average growth rate.
 Using
the
sustainable
growth rate.
45
Two Stage Growth Model
The simplest extension of the constant growth model
assumes that the extraordinary growth will continue for a
finite number of years and thereafter the normal growth
rate will prevail indefinitely.
 D1
D1 (1  g1 ) D1 (1  g1 ) 2
D1 (1  g1 ) n 1 
Pn
P0  



......



2
3
n
n
1

k
(
1

k
)
(
1

k
)
(
1

k
)
(
1

k
)
c
c
c
c
c


where, P0  current priceof theequity share
D1  dividend expecteda year hence
g1  extraordinary growth rateapplicablefor n years
Pn  priceof theequity share at theend of the year n
46
Two Stage Growth Model (contd….)
The first term on the right hand side of above
equation is the PV of a growing annuity, and
its value is equal to:
47
Reminder: Present Value of a Growing annuity
If ,
A(1 g)  cash flow at t heend of 1st year
A(1 g) 2  cash flow at t heend of 2nd year
A(1 g) n  cash flow at t heend of nt h year
 (1  r)n  (1  g ) n 
P V of growing annuit y A(1 g) 

n
(
r

g
)
(1

r)


T hisis t rue for g  r and g  r but not for g  r in t he
case of which, P V shall be only nA.
48
Two Stage Growth Model (contd….)
The first term on the right hand side of above
equation is the PV of a growing annuity, and its
value is equal to:
  1  g n 
1
 
1  
 1 kc  
D1 
k c  g1 




49
Two Stage Growth Model (contd….)
Hence,
  1 g 
1

1  
 1 kc 
P0  D1 
k c  g1


n


Pn


 (1  k ) n
c


50
Two Stage Growth Model (contd….)
Since the two-stage growth model assumes that the
growth rate after n years remains constant at g2, Pn will be
equal to:
D n 1
Pn 
kc  g2
where, D n 1  dividend for year (n  1)  D1 (1  g1 ) n 1 (1  g 2 )
g 2  growt h rat ein t hesecond period
51
Two Stage Growth Model (contd….)
Substituting the above expressions, we have:
  1  g n 
1
 
1  
n 1

1
  1  k c    D1 (1  g1 ) (1  g 2 ) 

P0  D1 

n 




k c  g1
k c  g1 
(1  k c ) 







52
Example:
The current dividend on an equity share of Vertigo
Limited is Rs 2.00. Vertigo is expected to enjoy an
above-normal growth rate of 20% for a period of 6
years. Thereafter, the growth rate will fall and
stabilize at 10%. Equity investors require a return of
15 %. What is the intrinsic value of the equity share
of Vertigo ?
g1 = 20 %, g2 = 10 %, n = 6 years, kc = 15%, D0 = Rs 2.00
Ans: Rs 79.597
53
Non-constant growth
v
v
Suppose a firm is expected to increase
dividends by 20% in one year and by 15% in
two years. After that dividends will increase
at a rate of 5% per year indefinitely. If the last
dividend was $1 and the required return is
20%, what is the price of the stock?
Remember that we have to find the PV of all
expected future dividends.
54
Non-constant growth – solution
v
v
v
Compute the dividends until growth levels off
• D1 = 1(1.2) = $1.20
• D2 = 1.20(1.15) = $1.38
• D3 = 1.38(1.05) = $1.449
Find the expected future price (by using the final
dividend calculation)
• P2 = D3 / (k – g) = 1.449 / (.2 - .05) = 9.66
Find the present value of the expected future cash
flows
• P0 = 1.20 / (1.2) + (1.38 + 9.66) / (1.2)2 = 8.67
55
Non-constant growth
v
The Green Shoe Company’s last dividend
paid (D0) was $1.00. Dividends are projected
to grow at a rate of 7% per year for the next 2
years, 5% per year for the 3rd year, and
starting with year 4 they will grow at a
constant rate of 4%, forever. If the required
return on the stock is 12%, what is the price of
the stock today?
56
Non-Constant Growth
v
v
v
v
At times, a new company may pay no dividends
early in its life but start paying dividends that grow
at a constant rate some time in the future.
At other times, a new company may pay small
dividends initially and, at some point in the future,
start paying dividends that grow at a constant rate.
However, as always, the value of the stock is the
present value of all future dividends.
Many cash flow scenarios are possible in this
situation.
57
Non-Constant Growth
Example:
v ABC Company does not plan to pay a dividend
until year 5. ABC’s expects the dividend in year
five to be $1 and dividends in future years to grow
at a constant rate of 5%. If the firm’s risk-adjusted
required rate of return is 13%, what is the value of
a share of stock in the company today?
P4 = 1/(.13 – .05) = $12.50
P0 = 12.50(1.13)-4 = $7.67
58
Components of Required Return
v
v
Thus far, the discount rate or required rate of return
has been given to us.
Later chapters have more to say about this, but for
now, using the dividend growth model, lets analysis
the required rate of return:
Rearranging:
kc = r = D1/P0 + g
where, D1/P0 = the dividend yield
g = the capital gains yield
59
Components of Required Return
v
Hence, Total Return on Common Stock has two
components:
• Dividend Yield
• Capital Gains Yield.
Return = Dividend Yield + Capital Gains Yield
D t Pt  Pt 1
r

Pt 1
Pt 1
60
Illustration:
We observe a stock selling for $ 20 per share. The next
dividend will be $ 1 per share. You think that the dividend
will grow by 10 % per year more or less indefinitely. What
return does this stock offer you if this is correct ?
Return = Dividend Yield + Capital Gains Yield
r=
D1/P0
+
g
=
1 / 20
+
0.10
=
0.05
+
0.10
= 0.15
i.e. 15 %
61
Verification
We can verify this answer by calculating the price in one year P1 , using
15 % as the required return.
P1
v
v
v
v
v
= D1 (1+g) / (r – g)
= $ 1 x 1.10 / (0.15 -0.10)
= $ 22
$ 22 is 10 % more than $ 20, so the stock price has grown by 10 %
If you buy the stock today in $ 20, it’ll pay $ 1 dividend at the end of the
year, and you’ll gain $ 2 by selling it.
Dividend yield is thus $ 1 / 20 = 0.05 i.e. 5%
Capital gains yield is thus $ 2 /20 = 0.10 i.e. 10 %
So your Total return would be 5 % + 10 % = 15 %
62
Impact of growth on Price, Returns and P/E Ratio
v
v
v
The expected growth rates of the companies differ
widely.
Some are expected to remain virtually stagnant or
grow slowly; others are expected to show normal
growth; still others are expected to achieve
supernormal growth rate.
Assuming a constant total required return,
differing expected growth rates mean differing
stock prices, dividend yields, capital gains yields,
and P/E ratios.
63
Illustration (contd….)
Consider three cases of growth rates:
Low growth firm
Normal growth firm
Supernormal growth firm
5%
10 %
15%
The expected earnings per share and dividend per share of
each of the three firms are Rs 3.00 and Rs 2.00 respectively.
Investor’s required total return from equity investments is
20%.
Given the above information, calculate the stock price,
dividend yield, capital gains yield, and P/E ratio for the
three cases
64
Illustration (contd…)
Price
Dividend Yield
P0 = D1 / (r – g)
(D1/P0)
Capital
Gains Yield
(P1-P0)/P0
P/E Ratio
P0/EPS
Low
Growth
Firm
Normal
Growth
Firm
Supernorm
al Growth
Firm
65
Illustration (contd…)
Price
Dividend Yield
P0 = D1 / (r – g)
(D1/P0)
Capital
Gains Yield
(P1-P0)/P0
P/E Ratio
P0/EPS
Low
Growth
Firm
13.33
15 %
5%
4.44
Normal
Growth
Firm
20.00
10 %
10 %
6.67
Supernorm
al Growth
Firm
40.00
5%
15 %
13.33
66
Inference
v
v
v
v
As the expected growth in dividend, increases, other
things remaining constant, the expected return
depends more on capital gains yield and less on the
dividend yield.
As the expected growth rate in dividend increases,
other things remaining constant, the P/E ratio
increases.
High dividend yield and low P/E ratio imply limited
growth prospects.
Low dividend yield and high P/E ratio imply
considerable growth prospects.
67
Valuation of Common Stock
Price Ratio Approach
68
Price Ratio Analysis
v
v
v
Price-earnings ratio (P/E ratio)
• Current stock price divided by annual earnings
per share (EPS).
Earnings yield
• Inverse of the P/E ratio: earnings divided by price
(E/P).
High-P/E stocks are often referred to as growth
stocks, while low-P/E stocks are often referred to as
value stocks.
69
Price Ratio Analysis
v
v
v
Price-cash flow ratio (P/CF ratio)
• Current stock price divided by current cash flow
per share.
• In this context, cash flow is usually taken to be net
income plus depreciation.
Most analysts agree that in examining a company’s
financial performance, cash flow can be more
informative than net income.
Earnings and cash flows that are far from each other
may be a signal of poor quality earnings.
70
Price Ratio Analysis
v
v
Price-sales ratio (P/S ratio)
• Current stock price divided by annual sales per
share.
• A high P/S ratio suggests high sales growth,
while a low P/S ratio suggests sluggish sales
growth.
Price-book ratio (P/B ratio)
• Market value of a company’s common stock
divided by its book (accounting) value of equity.
• A ratio bigger than 1.0 indicates that the firm is
creating value for its stockholders.
71
Price Ratio Analysis
Intel Corp (INTC) - Earnings (P/E) Analysis
Current EPS
$1.35
5-year average P/E ratio
30.4
EPS growth rate
16.5%
expected = historical  projected EPS
stock price
P/E ratio
= 30.4
 ($1.351.165)
= $47.81
72
Price Ratio Analysis
Intel Corp (INTC) - Cash Flow (P/CF) Analysis
Current CFPS
$1.97
5-year average P/CF ratio 21.6
CFPS growth rate
15.3%
expected = historical  projected CFPS
stock price
P/CF ratio
= 21.6
 ($1.971.153)
= $49.06
73
Price Ratio Analysis
Intel Corp (INTC) - Sales (P/S) Analysis
Current SPS
$4.56
5-year average P/S ratio
6.7
SPS growth rate
13.3%
expected = historical  projected SPS
stock price
P/S ratio
=
6.7
 ($4.561.133)
= $34.62
74
P/E Ratio Approach
P0
P0  E1 
E1
where, P0  Est imat edP rice
E1  Est imat edEP S
P0
 Just ified P /E Rat io
E1
75
Determinants of P/E Ratio
Accordingto ConstantGrowt h DividendDiscount Model
D1
E1 (1  b)
P0 

r - g r  ROE  b
P0
(1  b)

E1 r  ROE  b
where, b  ploughbackor retent ionratio
76
Determinants of P/E Ratio
Factors that determine the P/E ratio are:
1.
2.
The dividend payout ratio, (1-b)
The required rate or return, r
a) Interest Rate
b) Risk
3.
The expected growth rate, g = ROE x b
77