Stock Valuation

Common Stock Valuation is Difficult

• Uncertain cash flows– Equity is the residual claim on the firm’s

cash flows

• Life of the firm is forever

• Rate of return (the appropriate discount rate) is not easily observed

Differential Growth Dividend Model

• Forecasted Dividends grow at a constant rate, g1 for a certain number of years and then grow at a second growth rate, g2.

• Example: The dividend of a company was $1 yesterday. During the next 18 years the dividend will grow at 14% per year. After that the dividend will grow at 10% per year. What is the price of the stock if the required return is 15%?

The second dividend regime is a growing perpetuity

The first dividend regime is a growing annuity

PD

r g

D g g

r g

P

1819 0 1

182

18

18

1 1

1 114 11

015 010232 65

( ) ( )

( . ) ( . )

. ..

58.16

15.01

14.011

14.015.0

14.1

1

11

0

18

0

10

P

P

r

g

gr

DP

t

Now, we need to sum the two dividend regime values.

P PP

r

P

0 018

18 18

0

11658

232 65

115

3538

( )

..

.

.

EPS and Dividends• Dividends (share repurchase) are a function

of…– Ability to pay: Cash flow uncertainty– Decision to pay: Managerial uncertainty

• Why does a manager retain earnings?– Has better investment opportunities than the

shareholder– Makes a sub-optimal decision for the shareholder

• What is a “better investment opportunity”?– Investment has a NPV>0

Value a firm that retains earnings?

• Fundamental valuation equation: Sum of the discounted cash flows

• First component: PV(no-growth earnings stream)– Remember EPS=Net income/Shareholders equity

• Second component: PV of growth opportunities– Look for pricing shortcuts: perpetuity, annuity, etc.

• Rule: As long as PV(GO) > 0, price increases

)(GOPVr

EPSP

One Time Investment Opportunity

• Firm expects $1 million in earnings in perpetuity without new investments. Firm has 100,000 shares outstanding. Firm has investment opportunity at t=1 to invest $1 million in a project expected to increase future earnings by $210,000 per year. The firm’s discount rate is 10%. What is the share price with and without the project?

Constant Growth, Constant Investing• Firm Q has EPS of $10 at the end of the first year

and a dividend pay-out ratio of 40%, rE = 16% and a return on investment of 20%. The firm takes advantage of its growth opportunities each year by investing retained earnings.

• PV(GO) model– 1st investment = 0.6 × $10 = $6, which generates 0.2 ×

$6 = $1.20

– Per share PVGO1 = -6 + (1.20/0.16) = $1.50 (at t=1)

– 2nd investment = 0.6 × $11.20 = $6.72, generating 0.2 × $6.72 = $1.344

– Per share PVGO2 = -6.72 + (1.344/0.16) = $1.68 (at t=2)

Constant Growth, Constant Investing (cont)

• Relationship between PV(GO)’s? – 1.68 = (1+g) × 1.5 g=0.12

• Is there an easier way to estimate g for this case?– G=ROI x Investment Rate=0.2 x (1-0.4)=0.12

• PVGO0 = $1.50 / (0.16 - 0.12) =$37.50

• No-growth dividend value: $10/0.16 = $62.50• P = $62.50 + $37.50 = $100

Constant Growth, Constant Investing (cont)

• Can we price this firm a different way? – Since the investment grows at a constant rate

we can immediately estimate g– Investment rate x ROI = 0.6 × 20% = 12%

• Then estimate PV(GO) as a growing perpetuity based on dividends rather than cash flow– D1 / (rE - g) = $4 / (0.16 - 0.12) = $100

• So the entire firm is worth $100

Another ExampleFirm X currently has expected earnings of $100,000

per year in perpetuity. Firm X is switching its policy and wants to invest 20% of its earnings in projects with a 10% return. The discount rate is 18%.

• No-growth price: P=$100,000/0.18 = $555,555• PV(GO) is a constant growth perpetuity

– What’s g? g=Investment rate x ROI = 0.2 × 10% = 2%– What is the first year’s investment cash flow? Invest

$20,000 and receive $2,000 forever– -20,000+(2,000/0.18)=-8888.89– PV(GO) = (-8,888.89)/(0.18-0.02) = - 55,555

• New Policy: P=$555,555 - 55,555 = $500,000