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