Valuation of Stocks
The Present Value
of Common Stocks
 Dividends

versus Capital Gains
 Valuation of Different Types of Stocks




Zero ...
Dividends versus Capital Gains

Div1
P
1
P0 =
+
1+ r 1+ r
Div2
P2
P=
+
1
1+ r 1+ r
Dividends versus Capital Gains (contd.)
1 
 Div2 + P2 
P0 =
Div1 + 


1+ r 
 1 + r 
Div1
Div2
P2
=
+
+
2
2
1 +...
Case 1: Zero Growth


Assume that dividends will remain at the same
level forever

Div1 = Div2 = Div3 = 
Since future ca...
Case 2: Constant Growth
Assume that dividends will grow at a constant
rate, g, forever. i.e.
Div1 = Div0 (1 + g )
= Div1 (...
Case 3: Differential Growth


Assume that dividends will grow at different rates
in the future





One constant growt...
Case 3: Differential Growth
Assume that dividends will grow at rate g1
for N years and grow at rate g2 thereafter
Div1 = D...
Case 3: Differential Growth
Dividends will grow at rate g1 for N years
and grow at rate g2 thereafter
+ g1 ) Div0 (1 + g1 ...
Case 3: Differential Growth
We can value this as the sum of:
an N-year annuity growing at rate g1

 (1 + g1 )N 
Div1
PA ...
Case 3: Differential Growth
To value a Differential Growth Stock, we can use

 DivN +1

 (1 + g1 )N   r − g2
Div1

+...
A Differential Growth Example
A common stock just paid a dividend of $2. The
dividend is expected to grow at 8% for 3 year...
With the Formula
 DivN +1

 (1 + g1 )N   r − g2
Div1
=
−
+
P
1
N 
− g1  (1 + r )  (1 + r )N
r






 $2(1.0...
A Differential Growth Example
(continued)
$2(1.08)
0

1

$2.16
0

1

2

$2(1.08)
2

$2.33
2

3

3

$2(1.08) $2(1.08) (1.04...
Estimates of Parameters in the
Dividend-Discount Model
 The

value of a firm depends upon its growth
rate, g, and its dis...
Estimating Growth Rate
Earnings
next
year

=

Earnings
this
year

+

Retained
earnings
this year

×

Return on
retained
ea...
Estimating Growth Rate (contd.)
1 + g = 1 + Retention ratio × Return on retained earnings

g = Retention ratio × Return on...
Estimating Discount Rate
 The

estimation of discount rate can be
broken into two parts.


Dividend yield (Div/P0)



G...
Estimating Discount Rate (contd.)
Div
P0 =
r−g
Div
r=
+g
P0
Value of Stock when Firm Acts
as a Cash Cow
Value of a firm that pays out 100% of its
earnings as dividends

EPS Div
Value...
Price Earnings Ratio
 Many

analysts frequently relate earnings per
share to price
 The price earnings ratio or multiple...
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Лекц 5-6 Valuation of stocks

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Лекц 5-6 Valuation of stocks

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  • This type of problem separates the “A” students from the rest of the class.
  • Лекц 5-6 Valuation of stocks

    1. 1. Valuation of Stocks
    2. 2. The Present Value of Common Stocks  Dividends versus Capital Gains  Valuation of Different Types of Stocks    Zero Growth Constant Growth Differential Growth
    3. 3. Dividends versus Capital Gains Div1 P 1 P0 = + 1+ r 1+ r Div2 P2 P= + 1 1+ r 1+ r
    4. 4. Dividends versus Capital Gains (contd.) 1   Div2 + P2  P0 = Div1 +    1+ r   1 + r  Div1 Div2 P2 = + + 2 2 1 + r (1 + r ) (1 + r ) ∞ Div3 Divt Div1 Div2 P0 = + + + ⋅⋅⋅ = ∑ 2 3 t 1 + r (1 + r ) (1 + r ) t =1 (1 + r )
    5. 5. Case 1: Zero Growth  Assume that dividends will remain at the same level forever Div1 = Div2 = Div3 =  Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity: Div1 Div 2 Div 3 + + + P0 = 1 2 3 (1 + r ) (1 + r ) (1 + r ) Div P0 = r
    6. 6. Case 2: Constant Growth Assume that dividends will grow at a constant rate, g, forever. i.e. Div1 = Div0 (1 + g ) = Div1 (1 + g ) = Div0 (1 + g )2 Div2 = Div2 (1 + g ) = Div0 (1 + g )3 Div3 . . . Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity: Div1 P = 0 r −g
    7. 7. Case 3: Differential Growth  Assume that dividends will grow at different rates in the future    One constant growth rate in the foreseeable future A different constant growth rate thereafter To value a Differential Growth Stock, we need to:    Estimate future dividends in the foreseeable future. Estimate the future stock price when the stock becomes a Constant Growth Stock thereafter (case 2). Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.
    8. 8. Case 3: Differential Growth Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter Div1 = Div0 (1 + g1 ) = Div1 (1 + g1 ) = Div0 (1 + g1 )2 Div2 . . . = DivN −1 (1 + g1 ) = Div0 (1 + g1 )N DivN N DivN +1 = DivN (1 + g2 ) = Div0 (1 + g1 ) (1 + g2 ) . . .
    9. 9. Case 3: Differential Growth Dividends will grow at rate g1 for N years and grow at rate g2 thereafter + g1 ) Div0 (1 + g1 )2 Div0 (1 … 0 1 Div0 (1 + g1 ) 2 N DivN (1 + g2 ) = Div0 (1 + g1 )N (1 + g2 ) … … N N+1
    10. 10. Case 3: Differential Growth We can value this as the sum of: an N-year annuity growing at rate g1  (1 + g1 )N  Div1 PA = 1 − N  r − g1  (1 + r )  plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1  DivN +1   −  r g2 P = B + r )N (1    
    11. 11. Case 3: Differential Growth To value a Differential Growth Stock, we can use  DivN +1   (1 + g1 )N   r − g2 Div1  + P= 1 − N  − g1  (1 + r )  (1 + r )N r    
    12. 12. A Differential Growth Example A common stock just paid a dividend of $2. The dividend is expected to grow at 8% for 3 years, then it will grow at 4% in perpetuity. What is the stock worth? The discount rate is 12%.
    13. 13. With the Formula  DivN +1   (1 + g1 )N   r − g2 Div1 = − + P 1 N  − g1  (1 + r )  (1 + r )N r      $2(1.08)3 (1.04)     .12 −.04   (1.08)3   $2 ×(1.08) −  = + P 1 3  3 −.08  (1.12)  .12 (1.12) ($32.75 ) P = $54 ×[ −.8966]+ 1 3 (1.12) P = $5.58 + $23.31 P = $28.89
    14. 14. A Differential Growth Example (continued) $2(1.08) 0 1 $2.16 0 1 2 $2(1.08) 2 $2.33 2 3 3 $2(1.08) $2(1.08) (1.04) … 3 4 The constant $2.62 $2.52 + .08 3 growth phase beginning in year 4 can be valued as a growing perpetuity at time 3. $2.62 P3 = = $32.75 .08 $2.16 $2.33 $2.52 + $32.75 + + = $28.89 P = 0 2 3 1.12 (1.12) (1.12)
    15. 15. Estimates of Parameters in the Dividend-Discount Model  The value of a firm depends upon its growth rate, g, and its discount rate, r.   Where does g come from? Where does r come from?
    16. 16. Estimating Growth Rate Earnings next year = Earnings this year + Retained earnings this year × Return on retained earnings Increase in earnings Earnings next year Earnings this year  Retained earnings this year   = +   Earnings this year Earnings this year  Earnings this year  × Return on retained earnings
    17. 17. Estimating Growth Rate (contd.) 1 + g = 1 + Retention ratio × Return on retained earnings g = Retention ratio × Return on retained earnings
    18. 18. Estimating Discount Rate  The estimation of discount rate can be broken into two parts.  Dividend yield (Div/P0)  Growth rate of dividends (g)  In practice, there is a great deal of estimation error involved in estimating the discount rate, r
    19. 19. Estimating Discount Rate (contd.) Div P0 = r−g Div r= +g P0
    20. 20. Value of Stock when Firm Acts as a Cash Cow Value of a firm that pays out 100% of its earnings as dividends EPS Div Value = = r r
    21. 21. Price Earnings Ratio  Many analysts frequently relate earnings per share to price  The price earnings ratio or multiple  Calculated as current stock price divided by annual EPS Price per share P/E ratio = EPS

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