1

Topic 9 (Ch. 18) Equity Valuation Models

Intrinsic value versus market price

Dividend discount models (DDM)

The constant-growth DDM

Stock prices and investment opportunities

Life cycles and multistage growth models

Price-earnings (P/E) ratio Free cash flow valuation approaches Inflation and equity valuation

2

 Intrinsic Value versus Market Price

The expected holding-period return:

Assume a one-year holding period.

Suppose that ABC stock has:• an expected dividend per share: E(D1) = $4.

• the current price of a share: P0 = $48

• the expected price at the end of a year: E(P1) = $52.

Question:

Is the stock attractively priced today given your forecast of next year’s price?

3

The expected holding-period return (HPR):

Expected HPR =

The stock’s expected HPR is the sum of:• the expected dividend yield: E(D1)/P0

• the expected capital gains yield (i.e. the expected rate of price appreciation): [E(P1) - P0]/P0.

0

011 ])([)()(

PPPEDE

rE

%7.16167.048

]4852[4or

4

The required rate of return for ABC stock:

From the CAPM, when stock market prices are at equilibrium levels, the rate of return that investors can expect to earn on a security is:

This is the return that investors will require of a security given its risk as measured by beta.

])([ fMf rrEr

5

Suppose:

rf = 6%

E(rM) – rf = 5%

the beta of ABC: 1.2.

The required rate of return for ABC stock:

The rate of return the investor expects exceeds the required rate based on ABC’s risk by a margin of 4.7% (= 16.7% - 12%).

Thus, the investor will want to buy ABC stock.

%12%52.1%6 k

6

Another way:

Compare the intrinsic value of a share of stock to its market price.

The intrinsic value (V0) of a share of stock:

The present value of all expected cash payments to the investor in the stock (including expected dividends as well as the proceeds expected from the ultimate sale of the stock) discounted at the appropriate risk-adjusted interest rate, k.

7

Whenever the intrinsic value (i.e. the investor’s own estimate of what the stock is really worth) exceeds the market price, the stock is considered undervalued and a good investment.

In case of ABC stock:

Because intrinsic value ($50) exceeds current price ($48), the stock is under valued in the market.

Thus, investors will want to buy ABC.

50$%12152$4$

1)()( 11

0

kPEDE

V

8

In market equilibrium:

market price = intrinsic value. (i.e. P0 = V0)

required return = expected return. (i.e. k = E(r))

Note: k is the discount rate, the required rate of return, or the market capitalization rate.

9

We will use the simpler notation Pt and Dt instead of E(Pt) and E(Dt) to avoid clutter.

Notations:

Dt: expected dividend to be received at the end of year t.

(D0: most recent dividend, has just been paid.)

Pt: expected stock price at end of year t.

Keep in mind that future prices and dividends are unknown, and we are dealing with expected values, notcertain values.

 Dividend Discount Models (DDM)

10

kP

kD

V

11

110

kkPD

kD

11

1

22

1

kV

kD

1111

22

221

)1()1(1 k

P

k

Dk

D

Suppose the investor holds stock for 1 year, then sell.

11

2

33

221

)1(

1

)1(1 k

kPD

k

Dk

D

...)1()1()1(1 4

43

32

21

k

D

k

D

k

Dk

D

10

)1(tt

t

k

DVi.e.

22

221

)1()1(1 k

V

k

Dk

D

12

Suppose the investor plans to hold the stock for H years, then sell.

HH

HH

k

P

k

Dk

DV

)1()1(...

11

0

HH

HH

k

V

k

Dk

D

)1()1(...

11

H

HHHH

HH

k

k

PDk

D

k

Dk

D

)1(

)1(...

)1(

)1(...

1

221

1

13

HHH

HH

HH

k

PD

k

D

k

Dk

D2

221

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)1(...

)1()1(...

1

HHH

HH

HH

k

VD

k

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k

Dk

D2

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)1(...

)1()1(...

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...)1(

...)1()1(

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21

11

HH

HH

HH

k

D

k

D

k

Dk

D

i.e.

10

)1(tt

t

k

DV

14

Suppose the investor plans to hold the stock forever.

13

32

210

)1(...

)1()1(1 tt

t

k

D

k

D

k

Dk

DV

The dividend discount model (DDM):The current value of a share of common stock is the sum of all future expected dividend payments discounted to the present, no matter what the investor’s holding horizon is.

15

Note:

It is incorrect to conclude that the DDM focuses exclusively on dividends and ignores capital gains as a motive for investing in stock.

Indeed, capital gains (as reflected in the expected sales price) are part of the stock’s value.

The price at which you can sell a stock in the future depends on dividend forecasts at that time.

16

The constant-growth DDM

Assume that dividends are trending upward at a stable growth rate (g).

For ABC stock:

If g = 0.05, and the most recently paid dividend was D0 = 3.81, then expected future dividends are:

...)1(

)1(0)1(

)1(1

)1(3

3

2

200

0

k

gD

k

gDk

gDV

00.405.181.3)1(01 gDD

20.4)05.1(81.3)1( 2202 gDD

etc. gDD 41.4)05.1(81.3)1( 3303

17

(the constant-growth DDM)

gkD

gkgD

V

10

0)1(

14.57$%5%12

4$0

V

Notes: If g = 0, then .

The model requires: k > g (otherwise, the value of the stock would be infinite).

kD

V 10

18

Implications:

A stock’s value will be greater: The larger its expected dividend per share. The lower the market capitalization rate, k. The higher the expected growth rate of dividends.

The stock price is expected to grow at the same rate (g) as dividends.

Suppose a stock is selling at its intrinsic value, i.e.

00 VP

gkD

P

10

19

i.e. in the case of constant growth of dividends, the rate of price appreciation in any year will equal that constant-growth rate, g.

)1()1(

012

1 gPgk

gDgk

DP

201

232 )1()1(

)1(gPgP

gkgD

gkD

P

.

.

.t

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20

For a stock whose market price equals its intrinsic value (P0 = V0), the expected holding-period return

will be:

E(r) = k = dividend yield + capital gains yield

The discounted cash flow (DCF) formula:

i.e. by observing the dividend yield (D1/P0) and

estimating the growth rate of dividends (g), we can

compute k.

gPD

0

1

gPD

k 0

1

21

Consider a company, X, with expected earnings in the coming year of $5 per share.

X pays out all of its earnings as dividends, maintaining a perpetual dividend flow of $5 per share.

If k = 12.5%, X would then be valued at:

D1/k = $5/12.5% = $40 per share.

X would not grow in value, because with all earnings paid out as dividends, and no earnings reinvested in the firm, capital stock and earnings capacity would remain unchanged over time; earnings and dividends would not grow.

Stock prices and investment opportunities

22

Now suppose X engages in projects that generate a return on investment of 15%, which is greater than the required rate of return, k = 12.5%.

It would be foolish for such a company to pay out all of its earnings as dividends.

If X retains or plows back some of its earnings into its highly profitable projects, it can earn a 15% rate of return for its shareholders, whereas if it pays out all earnings as dividends, it forgoes the projects, leaving shareholders to invest the dividends in other opportunities at a fair market rate of only 12.5%.

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Suppose that X chooses a lower dividend payout ratio (the fraction of earnings paid out as dividends), reducing payout from 100% to 40%, maintaining a plowback ratio (the fraction of earnings reinvested in the firm) at 60%. The plowback ratio is also referred to as the earnings retention ratio.

The dividend of the company will be $2 (= 40% $5 earnings) instead of $5.

Although dividends initially fall under the earnings reinvestment policy, subsequent growth in the assets of the firm because of reinvested profits will generate growth in future dividends, which will be reflected in today’s share price.

24

25

How much growth will be generated?

Suppose X starts with plant and equipment of $100 million and is all equity financed.

With a return on investment or equity (ROE) of 15%:

total earnings = ROE $100 million

= 15% $100 million = $15 million.

There are 3 million shares of stock outstanding, so earnings per share are $5 (= $15 million/3 million).

If 60% of the $15 million in this year’s earnings is reinvested, then the value of the firm’s capital stock will increase by:

0.60 $15 million = $9 million, or by 9%.

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i.e. The percentage increase in the capital stock:

= ROE the plowback rate (b)

= 15% 60% = 9%.

Now endowed with 9% more capital, the company earns 9% more income, and pays out 9% higher dividends.

The growth rate of the dividends is:

g = ROE b = 15% 60% = 9%.

If the stock price equals its intrinsic value, it should sell at:

14.57$%9%5.12

2$10

gkD

P

27

When X reduces current dividends and reinvest some of its earnings in new investments, its stock price increases.

The increase in the stock price reflects the fact that the planned investments provide an expected rate of return greater than the required rate.

That is, the investment opportunities have positive net present value (NPV). The value of the firm rises by the NPV of these investment opportunities.

This NPV is called the present value of growth opportunities (PVGO).

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the value of the firm

= the value of assets already in place (i.e. the no-

growth value of the firm)

+ the NPV of the future investments the firm will

make (i.e. the PVGO)

i.e. Price = No-growth value per share + PVGO

57.14 = 40 + 17.14

PVGOk

EP 1

0

29

Note:

Growth enhances company value only if it is achieved by investment in projects with attractive profit opportunities (i.e., with ROE > k).

Now suppose that ROE is only 12.5%, just equal to the required rate of return, k.

The growth rate of the dividends is:

g = ROE b = 12.5% 60% = 7.5%.

40$%5.7%5.12

2$10

gkD

P

30

$40 = $40 + PVGO

Þ PVGO = 0.

If the firm’s projects yield only what investors can earn on their own, shareholders cannot be made better off by a high reinvestment rate policy.

To justify reinvestment, the firm must engage in projects with better prospective returns than those shareholders can find elsewhere.

Price = No-growth value per share + PVGO

PVGOk

EP 1

0

31

Firms typically pass through life cycles with very different dividend profiles in different phases.

In early years, there are ample opportunities for profitable reinvestment in the company. Payout ratios are low, and growth is correspondingly rapid.

In later years, the firm matures, attractive opportunities for reinvestment may become harder to find. In this mature phase, the firm may choose to increase the dividend payout ratio, rather than retain earnings. The dividend level increases, but thereafter it grows at a slower rate because the company has fewer growth opportunities.

Life cycles and multistage growth models

32

To compute the intrinsic value, follow the following 3 steps:

Find the present value of the dividends during the nonconstant growth period.

Find the price of the stock at the end of the nonconstant growth period (i.e. at which point it enters its constant-growth phase), and discount this price back to the present.

Add these 2 components.

33

Example:

D2013 = $0.78

D2014 = $0.85

D2015 = $0.92

D2016 = $1.00

Dividends enter their constant-growth phase at the end of 2016.

Suppose: dividend payout ratio = 25%.

ROE = 10%.

Þ The long-term constant steady-state growth rate:

g = ROE b = 10% (1 – 25%) = 7.5%.

34

Find k:

beta = 0.95

risk-free rate = rf = 2%

expected market return = E(rM) = 10%

%6.921095.02

][ fMf rrErk

35

Find V2012:

420162016

32015

220142013

2012)1()1()1()1( k

PD

k

D

k

D

k

DV

42016

32 )096.1(

00.1

)096.1(

92.0

)096.1(

85.0

096.1

78.0 P

19.51$075.0096.0

075.100.1)1(201620172016

gk

gD

gk

DP

29.38$2012 V

36

Price-earnings multiple:

the ratio of price per share to earnings per share, commonly called the P/E ratio.

 Price-Earnings (P/E) Ratio

The P/E ratio and growth opportunities

]/

1[11

111

0kE

PVGOkE

PVGOkE

P

Price = No-growth value per share + PVGO

PVGOk

EP 1

0

37

Implications:

When PVGO = 0:

P0 = E1/k and P/E ratio = 1/k.

PVGO P/E ratio.

The P/E ratio might serve as a useful indicator of expectations of growth opportunities.

A high P/E multiple indicates that a firm enjoys ample growth opportunities.

38

Recall:

The constant growth DDM: .

D1 = E1(1 – b).

g = ROE b. ( ROE and b g)

The impact of ROE on the P/E ratio:

ROE P/E ratio.

(High-ROE projects give the firm good opportunities for growth).

gkD

P

10

bROEkbE

P

)1(10 bROEk

bEP

1

1

0

39

The impact of the plowback rate (b) on the P/E ratio:

(i) If ROE > k, then b P/E ratio.

When ROE exceeds k, the firm offers attractive investment opportunities, so the market will reward it with a higher P/E multiple if it exploits those opportunities more aggressively by plowing back more earnings into those opportunities.

210

][

)/(

bROEk

kROEb

EP

40

(ii) If ROE < k, then b P/E ratio.

When ROE < k, investors prefer that the firm pay out earnings as dividends rather than reinvest earnings in the firm at an inadequate rate of return.

That is, for ROE < k, P/E falls as plowback increases.

(iii) If ROE = k, then b P/E ratio does not change.

41

The impact of stock risk on the P/E ratio:

Riskier stocks

have higher required rates of return (k)

® have lower P/E multiples (P/E).

Note:

This is true even outside the context of the constant-growth model.

For any expected earnings and dividend stream, the present value of those cash flows will be lower when the stream is perceived to be riskier.

Hence the stock price and P/E ratio will be lower.

42

Combining P/E analysis and the DDM

Recall the example:

D2013 = $0.78 D2014 = $0.85 D2015 = $0.92 D2016 = $1.00 Dividends enter their constant-growth phase at the end of 2016.

][ fMf rrErk

%6.921095.02

43

The forecasted data for year 2016:

P/E ratio: 14.

EPS: $4.

the estimate of share price for year 2016:

14 $4 = $56.

62.41$)096.1(

5600.1

)096.1(

92.0

)096.1(

85.0

096.1

78.04322012

V

420162016

32015

220142013

2012)1()1()1()1( k

PD

k

D

k

D

k

DV

42016

32 )096.1(

00.1

)096.1(

92.0

)096.1(

85.0

096.1

78.0 P

44

Other comparative valuation ratios

The P/E ratio is an example of a comparative valuation ratio. Such ratios are used to assess the valuation of one firm versus another based on a fundamental indicator such as earnings.

Other such comparative ratios are commonly used:

Price-to-book ratio:

The ratio of price per share divided by book value per share.

Book value is the net worth of a company as shown on the balance sheet.

45

Price-to-cash-flow ratio:

Earnings as reported on the income statement can be affected by the company’s choice of accounting practices, and thus are commonly viewed as subject to some imprecision and even manipulation.

In contrast, cash flow—which tracks cash actually flowing into or out of the firm—is less affected by accounting decisions.

Some analysts use operating cash flow when calculating this ratio.

Others prefer “free cash flow” (= operating cash flow - new investment).

46

Price-to-sales ratio:

Many start-up firms have no earnings. Thus, the P/E ratio for these firms is meaningless.

The price-to-sales ratio (the ratio of stock price to the annual sales per share) has recently become a popular valuation benchmark for these firms.

Of course, price-to-sales ratios can vary markedly across industries, since profit margins vary widely. Nevertheless, use of this ratio has increased substantially in recent years.

47

Market valuation statistics:

While the levels of these ratios differ considerably, for the most part, they track each other fairly closely, with upturns and downturns at the same times.

48

Free Cash Flow Valuation Approaches

Free cash flow: cash flow available to the firm or its equityholders net of capital expenditures.

This approach is particularly useful for firms that pay no dividends, for which the dividend discount model would be difficult to implement.

But free cash flow models may be applied to any firm and can provide useful insights about firm value beyond the DDM.

49

The first approach:

Discount the free cash flow for the firm (FCFF) at the weighted average cost of capital (WACC) to obtain the value of the firm, and subtract the then-existing value of debt to find the value of equity.

The free cash flow to the firm:

FCFF = EBIT(1 - tc) + Depreciation – Capital expenditures

- Increase in NWC

where EBIT = earnings before interest and taxes

tc = corporate tax rate

NWC = net working capital

50

WACC is the weighted average of the after-tax cost of debt and the cost of equity.

To compute the cost of equity, we will use the CAPM, but accounting for the fact that equity beta increases with the firm’s financial leverage:

where L: leveraged beta

U: unleveraged beta

D/E: debt-equity ratio

(D: market value of debt; E: market value of equity)

)]/)(1(1[ EDtcuL

51

The free cash flow to the firm approach discounts year-by-year cash flows plus some estimate of terminal value, VT .

As in the dividend discount model, free cash flow models use a terminal value to avoid adding the present values of an infinite sum of cash flows.

We use the constant-growth model to estimate terminal value and discount at the WACC.

52

(g: the steady growth rate)

Equity value = value of the firm market value of debt

T

tT

Tt

t

WACC

V

WACC

FCFFvalueFirm

1 )1()1(

gWACC

FCFFVwhere T

T 1

53

Example:

2013 2014 2015 2016

(1) EBIT(1 - tc) 4,945.9 5,567.5 6,189.2 6,810.8

(2) Depreciation 2,625.0 2,880.0 3,135.0 3,390.0

(3) Capital spending 2,800.0 2,783.3 2,766.7 2,750.0

(4) Change in NWC 1,411.0 663.3 663.3 663.3

(5) FCFF【 = (1)+(2)-(3)-

(4)】

3,359.9 5,000.9 5,894.2 6,787.5

54

Current year 2012:

Current market value of equity: $66,383

Current market value of debt: $3,392

Current beta (leveraged beta): 1.4

Corporate tax rate: 35%

)]/)(1(1/[: EDtbetaunleveredcurrent cLu

355.1)]383,66/392,3)(35.01(1/[4.1 u

55

)]/)(1(1[: EDtbetalevered cuL

392.1)]265,73/090,3)(35.01(1[355.1:2013 L

2013 2014 2015 2016

(1) P/E 15.075 16.050 17.025 18.000

(2) Profits (after tax) 4,860 5,490 6,120 6,750

(3) Market value of equity 【 =(1) × (2)】

73,265 88,115 104,193 121,500

(4) Market value of debt 3,090 2,790 2,490 2,190

(5) Levered beta 1.392 1.383 1.376 1.371

(6) Cost of equity 0.140 0.140 0.139 0.139

(7) WACC 0.136 0.137 0.137 0.137

56

])([ fMf rrErequityofcost

Risk-free rate: 5%

Market risk premium: 6.5%

140.0)065.0(392.105.0:2013 equityofcost

57

WACC

)]/()[1)(( EDDtdebtofcosttaxbefore c

)]/()[( EDEequityofcost

WACC:2013

Before-tax cost of debt: 5.7%

)]265,73090,3/(265,73)[140.0(

)]265,73090,3/(090,3)[35.01)(057.0(

136.0

58

gWACC

gFCFF

gWACC

FCFFV TT

T

)1(1

The steady growth rate: 5%

05.0137.0

)05.01(5.787,6

05.0137.0

)05.01(20162016

FCFF

V

= 81,918.1

59

The intrinsic value of the firm:

T

tT

Tt

t

WACC

V

WACC

FCFF

1 )1()1(

)137.01)(136.01(

9.000,5

)136.01(

9.359,3

332 )137.01)(136.01(

1.918,81

)137.01)(136.01(

5.787,6

)137.01)(136.01(

2.894,5

967,63$

60

The intrinsic value of equity

= firm value – debt value

= 63,967 –3,392 = $60,575

The number of shares outstanding: 2,850

The intrinsic value of equity per share

= ($60,575/2,850)

= $21.25

61

The second approach:

Start on the free cash flow to equity holders (FCFE), discounting those directly at the cost of equity to obtain the value of equity.

Cash flow available to equityholders:

This differs from free cash flow to the firm by after- tax interest expenditures, as well as by cash flow associated with net issuance or repurchase of debt (i.e., principal repayments minus proceeds from issuance of new debt).

FCFE = FCFF – Interest expense (1 – tc) + Increases in net debt

62

T

tT

E

Tt

E

t

k

V

k

FCFEvalueEquity

1 )1()1(

Discount free cash flows to equity (FCFE) at the cost of equity (kE):

gk

FCFEVwhere

E

TT

1

(g: the steady growth rate)

63

No-inflation case:

Consider a firm X that pays out all earnings as dividends.

Earnings and dividends per share are $1, and there is no growth.

Consider an equilibrium real capitalization rate (k*) of 10% per year.

The price per share of X stock:

Inflation and Equity Valuation

10$%101$

0 P

64

Inflation-case:

Inflation (i) is 6% per year, but the values of the other economic variables adjust so as to leave their real values unchanged.

The nominal capitalization rate (k):

k = (1 + k*)(l + i) - 1 = 1.10 1.06 - 1 = 16.6%.

The expected nominal growth rate of dividends (g) is now 6%, which is necessary to maintain a constant level of real dividends.

Thus, the nominal dividend expected at the end of this year is $1.06 per share.

65

Thus, as long as real values are unaffected, the stock’s current price is unaffected by inflation.

Notes:

The expected nominal dividend yield:

The expected nominal capital gains rate = g = 6%.

Almost the entire 6.6% increase in nominal return comes in the form of expected capital gains.

%6.100

1 PD

10$060.0166.0

06.1$10

gkD

P

66

The effect of inflation on earnings, plowback ratio, and P/E ratio:

Firm X produces a product that requires purchase of inventory at the beginning of each year, and sells the finished product at the end of the year.

Last year there was no inflation. The inventory cost $10 million. Labor, rent, and other processing costs (paid at year-end) were $1 million, and revenue was $12 million.

Assuming no taxes: 　 Revenue $12 million

– Labor and rent 1 million

– Cost of goods sold 10 million

= Earnings 1 million

67

All earnings are distributed as dividends to the 1 million shareholders.

Because the only invested capital is the $10 million in inventory, the ROE is 10% (= $1 million/$10 million).

This year, inflation of 6% is expected, and all prices are expected to rise at that rate.

Because inventory is paid for at the beginning of the year, it will still cost $10 million.

However, revenue will be $12.72 (= $12 1.06) million, and other costs will be $1.06 (= $1 1.06) million.

68

　 Nominal Earnings

　 Revenue $12.72 million

– Labor and rent 1.06 million

– Cost of goods sold 10.00 million

Earnings $1.66 million

　 ROE 16.6%

69

Note:

The amount required to replace inventory at year’s end is $10.6 million (rather than the beginning cost of $10 million), so the amount of cash available to distribute as dividends is $1.06 million (not the reported earnings of $1.66 million).

A dividend of $ 1.06 million would be just enough to keep the real value of dividends unchanged and at the same time allow for maintenance of the same real value of inventory.

That is, the reported earnings of $1.66 million overstate true economic earnings.

70

Thus, we have the following set of relationships:

　 No Inflation 6% Inflation

Dividends $1 million $1.06 million

Reported earnings $1 million $1.66 million

ROE 10% 16.6%

Plowback ratio 0 0.36145

Price of a share $10 $10

P/E ratio 10 6.0241

71

Notes:• The plowback ratio rises from 0 to 0.36145.

Inventory must rise from a nominal level of $10 million to a level of $10.6 million to maintain its real value.

This inventory investment requires reinvested earnings of $0.6 million.

• The P/E ratio drops from 10 in the no-inflation scenario to 6.0241 in the 6% inflation scenario.

This is entirely a result of the fact that the reported earnings figure gets distorted by inflation and overstates true economic earnings.

72

• We have seen that as long as real values are unaffected, the stock’s price is unaffected by inflation.

However, inflation may affect stock price because:

An increase in inflation may be associated with a decrease in real expected dividend ( ), an increase in real capitalization rate (k*), a decrease in real growth rate (g*), or some combination of all three.

Economic “shocks” such as oil price hikes can cause a simultaneous increase in the inflation rate and decline of expected real earnings (and dividends).

*1D

73

Higher inflation is associated with greater uncertainty about the economy, which tends to induce a higher required rate of return on equity and hence a lower level of stock prices.

Many investors in the stock market suffer from a form of “money illusion.”

Investors mistake the rise in nominal rate of interest for a rise in the real rate.

As a result, they undervalue stocks in a period of higher inflation.