chapter 2: valuation of stocks and bonds

1

Chapter 2: Chapter 2: Valuation of Stocks and BondsValuation of Stocks and Bonds

2

What is Value?What is Value?

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

3

Several Kinds of “Value”Several Kinds of “Value”

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

4

Determinants of Intrinsic ValueDeterminants of Intrinsic Value

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.)

Note that the intrinsic value of an asset can be, and often is, different for each individual (that’s what makes markets work)

5


2.3 Valuation of Preferred Stocks

6

Preferred Stock FeaturesPreferred Stock Features 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

7

Features of preferred stockFeatures of preferred stock

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.

8

Preferred Stock ValuationPreferred Stock Valuation

Preferred stocks can usually be valued like a perpetuity:

V =Dk

pp

9

Example: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

10

Expected Rate of Return on Expected Rate of Return on PreferredPreferred

Just adjust the valuation model:

=D

pp

kV

=D

pkPo

11

ExampleExample

If we know the preferred stock price is $40, and the preferred dividend is $4.125, the expected return is:

=D

Popk =

4.125

40= .1031

12

Valuation of redeemable Valuation of redeemable preferred stockpreferred stock

The value of a preferred stock equals the present value of all future dividends

n,kn,kp

p

p

n

1tn

pt

pp

ppPVIFMPVIFADV

annuity,ordinary an is dividends of stream theSince

aluematurity v M

stock preferenceon return of rate required k

stock preference theof life n

dividend periodical D

stock preference of lueCurrent va V

)k1(

M

)k1(

DV

13


2.4 Valuation of Common Stocks

14

Features of common stockFeatures of common stock

Residual income and asset claimants• Unlimited upside• First to suffer

Priority1. Debt2. Preferred Stock3. 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

15

Features of Common StockFeatures of Common Stock

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.

16

Common Stock FeaturesCommon Stock Features

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.

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Dividends

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.

18


Classes of Stock

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.

19

Common Stock ValuationCommon Stock Valuation

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

Again, finding the present values of these cash flows and adding them together will give us the value

20

Cash flows for stockholdersCash flows for stockholders

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

As with bonds, the price of the stock is the present value of these expected cash flows

21

Common Stock ValuationCommon Stock Valuation

Unlike bonds, valuing common stock is more difficult.

Why?

1. The timing and amount of future cash flows is not known.

2. The life of the investment is essentially forever.

3. There is no way to observe the rate of return that the market requires.

22

Some Notes About Common Some Notes About Common StockStock

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 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!

23

Common Stock: Some Common Stock: Some AssumptionsAssumptions

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)

Note that the second assumption allows us to predict every future dividend, as long as we know the most recent dividend

24

Common Stock Valuation: Common Stock Valuation: Dividend Discount ModelDividend Discount Model

25

Single-Period Valuation ModelSingle-Period Valuation Model

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

)k1(

P

)k1(

DV

c

1

c

1c

26

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?

0 1

? 5.50 + 120

Single Holding PeriodSingle Holding Period

Ans: $ 109.13

27

What happens if ?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.

)gk(

D

)k1(

)g1(P

)k1(

DP

c

1

c

o

c

1o

28

ExampleExample

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% ?

20.00 Rs)05.015.0(

2

)gk(

DP

c

1o

29

Expected Rate of ReturnExpected 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

30

Multi-period Valuation ModelMulti-period Valuation Model

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.

nc

nn

1tt

c

t

nc

nn

c

n4

c

43

c

32

c

2

c

10

)k1(

P

)k1(

D

)k1(

P

)k1(

D...........

)k1(

D

)k1(

D

)k1(

D

k1

DP

31


But common shares have no maturity period – they may be expected to bring a dividend stream of infinite duration

1tt

c

t

c4

c

43

c

32

c

2

c

10

)k1(

D

)k1(

D...........

)k1(

D

)k1(

D

)k1(

D

k1

DP

32


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.

33

Commonly used assumptions Commonly used assumptions types:types:

1. The dividend per share remains constant forever, implying that the growth rate is nil (THE ZERO GROWTH MODEL)

2. The dividend per share grows at a constant rate per year forever (THE CONSTANT GROWTH MODEL)

3. 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)

4. The dividend per share, currently growing at an above-normal rate, experiences a gradually declining rate of growth for a while. Thereafter it grows at a constant normal rate (THE “H” MODEL)

34

Zero Growth Model 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;

c1tt

c

c4

c3

c2

cc0

k

D

)k1(

D

)k1(

D...........

)k1(

D

)k1(

D

)k1(

D

k1

DP

35

ExampleExample

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?

20.00 $025.0

5.0

k

DP

c0

36

Constant Growth modelConstant Growth model

Assumes that the dividend per share grows at a constant rate (g)

g-k

D

g-k

g)1(DP

c

1

c

00

.......)k1(

)g1(D...........

)k1(

)g1(D

)k1(

)g1(D

)k1(

)g1(D

k1

DP

1nc

n1

4c

31

3c

21

2c

1

c

10

With a little algebra, this reduces to:

37

Example 1Example 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?

23.39 $0.13

5.10

0.02-0.15

0.02)1(5P0

38

Example 2Example 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

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Example 3Example 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?

42$)05.10(.

)05.1(20

P

40

Example 4Example 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%?

)10.01(

)05.01(1.2

)10.01(

)05.01(1.2

)10.01(

)05.01(1.2

10.01

1.2P

4

3

3

2

20

41

What drives growth ?What drives growth ?

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 Ratiob) Return on Equity (ROE)

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

42

Financials of Omega LimitedFinancials 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 ?

43

Financials of Omega LimitedFinancials of Omega Limited

Year 1 Year 2 Year 3

Beginning Equity 100 112 125.44

ROE 20% 20% 20%

Equity Earnings 20 22.4 25.1

Dividend Payout Ratio

0.40 0.40 0.40

Dividends 8 8.96 10.04

Retention Ratio 0.60 0.60 0.60

Retained earnings 12 13.44 15.06

Growth Rate = RE x ROE = 0.60 x 20 %= 12 %

44

What is this growth actually ?What is this growth actually ?

Sustainable growth rate =ROE Retention ratio

Return on equity (ROE) = Net income / Equity Retention ratio = 1 – Payout ratio

45

Estimation of GrowthEstimation of Growth

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.

46

Two Stage Growth Model 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.

nyear theof end at the shareequity theof price P

yearsn for applicable rategrowth ary extraordin g

henceyear a expected dividendD

shareequity theof pricecurrent P where,

)k1(

P

)k1(

)g1(D......

)k1(

)g1(D

)k1(

)g1(D

k1

DP

n

1

1

0

nc

nn

c

1n11

3c

211

2c

11

c

10

47

Two Stage Growth Model Two Stage Growth Model (contd….)(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:

48

Reminder: Present Value of a Growing Reminder: Present Value of a Growing annuityannuity

nA.only be shall PV which,of case

in ther gfor not but r g andr gfor trueis This

r)(1)gr(

)g1(r)(1g)A(1annuity growing of PV

yearnth of end at the flowcash g)A(1

year 2nd of end at the flowcash g)A(1

year1st of end at the flowcash g)A(1

,If

n

nn

n

2

49


The first term on the right hand side of above equation is the PV of a growing annuity, and its value is equal to:

1c

n

c

1

1 gk

k1g1

1

D

50


Hence,

nc

n

1c

n

c

1

10 )k1(

P

gk

k1g1

1

DP

51


Since the two-stage growth model assumes that the growth rate after n years remains constant at g2, Pn will be equal to:

period second in the rategrowth g

)g1()g1(D 1)(nyear for dividend D ,where

gk

DP

2

21n

111n

2c

1nn

52


Substituting the above expressions, we have:

nc1c

21n

11

1c

n

c

1

10 )k1(

1

gk

)g1()g1(D

gk

k1g1

1

DP

53

Example: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

54

Non-constant growth Non-constant growth

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.

55

Non-constant growth – solutionNon-constant growth – solution

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

56

Non-constant growth Non-constant growth

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?

57

Non-Constant GrowthNon-Constant Growth

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.

58

Non-Constant GrowthNon-Constant Growth

Example: 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

59

Components of Required ReturnComponents of Required Return

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

60

Components of Required ReturnComponents of Required Return

Hence, Total Return on Common Stock has two components:• Dividend Yield• Capital Gains Yield.

Return = Dividend Yield + Capital Gains Yield

1t

1tt

1t

t

P

PP

P

Dr

61

Illustration: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 %

62

VerificationVerification

We can verify this answer by calculating the price in one year P1 , using 15 % as the required return.

P1 = 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 %

63

Impact of growth on Price, Returns and Impact of growth on Price, Returns and P/E RatioP/E Ratio

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.

64

Illustration (contd….)Illustration (contd….)

Consider three cases of growth rates:Low growth firm 5 %

Normal growth firm 10 %Supernormal growth firm 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

65

Illustration (contd…)Illustration (contd…)

PriceP0 = D1 / (r –

g)

Dividend Yield

(D1/P0)

Capital Gains Yield

(P1-P0)/P0

P/E RatioP0/EPS

Low Growth

Firm

Normal Growth

Firm

Supernormal

Growth Firm

66

Illustration (contd…)Illustration (contd…)

PriceP0 = D1 / (r –

g)

Dividend Yield

(D1/P0)

Capital Gains Yield

(P1-P0)/P0

P/E RatioP0/EPS

Low Growth

Firm

13.33 15 % 5 % 4.44

Normal Growth

Firm

20.00 10 % 10 % 6.67

Supernormal

Growth Firm

40.00 5 % 15 % 13.33

67

InferenceInference

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.

68

Valuation of Common StockValuation of Common Stock

Price Ratio Approach

69

Price Ratio AnalysisPrice Ratio Analysis

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.

70


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.

71


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.

72


Intel Corp (INTC) - Earnings (P/E) Analysis

Current EPS $1.355-year average P/E ratio 30.4EPS growth rate 16.5%

expected = historical projected EPS stock price P/E ratio

= 30.4 ($1.351.165)= $47.81

73


Intel Corp (INTC) - Cash Flow (P/CF) Analysis

Current CFPS $1.975-year average P/CF ratio 21.6CFPS growth rate 15.3%

expected = historical projected CFPS stock price P/CF ratio

= 21.6 ($1.971.153)= $49.06

74


Intel Corp (INTC) - Sales (P/S) Analysis

Current SPS $4.565-year average P/S ratio 6.7SPS growth rate 13.3%

expected = historical projected SPS stock price P/S ratio

= 6.7 ($4.561.133)= $34.62

75

P/E Ratio ApproachP/E Ratio Approach

Ratio P/E Justified E

P

EPS Estimated E

Price Estimated P where,

E

PEP

1

0

1

0

1

010

76

Determinants of P/E RatioDeterminants of P/E Ratio

ratioretention or ploughback b ,where

bROEr

)b1(

E

P

bROEr

)b1(E

g -r

DP

ModelDiscount DividendGrowth Constant toAccording

1

0

110

77

Determinants of P/E RatioDeterminants of P/E Ratio

Factors that determine the P/E ratio are:

1. The dividend payout ratio, (1-b)2. The required rate or return, r

a) Interest Rateb) Risk

3. The expected growth rate, g = ROE x b

chapter 2: valuation of stocks and bonds

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