Moving Cash Flows: Review
Formulas
FV
PV 
t
(1  r )
C
1 
PV(Annuity ) = 1 r  (1 + r) t 
C
PV ( Perpetuity) 
rg
t
C  1 g 
PV ( growannuit y ) 

1  
r  g   1  r 





C
t
FV(Annuity ) = (1 + r) - 1
r
n
quoted rate of interest 

ER = 1 +
-1

m


Growing Annuity
• Annuities are a constant cash flow over
time
• Growing annuities are a constant growth
cash flow over time
C
PV 
rg
 1 g 

1  
  1  r 
t



What are you worth today?
• You will make $100,000 the first year.
• You expect to work for 40 years, get 9%
raises every year and 20% per year on
investments.
t

C
1 g  
PV 

1  
r  g   1  r  
40

100,000
 1  0.09  
PV 
 
1  
0.2  0.09   1  0.2  
PV  889,663.39
Cash Flow Timing
• When does the first cash flow occur
relative to the present value of the
_______
– Perpetuity? Growing perpetuity?
– Annuity? Growing annuity?
• One period later!
Review: Bond Features
• Coupon Payments: Regular interest
payments
– Semi annual for most US corporate bonds
– Types of Coupon payments
• Fixed Rate: 8% per year
• Floating Rate: 6-mo. Treasury bill rate + 100
basis points.
• Face or Par Value: $1,000/bond
• Maturity: no. of years from issue date
until principal is paid
• Coupon Rate
Bond Valuation
C
Bond Value =
YTM


1
Par
1 - (1 + YTM) t  + (1 + YTM) t


Annuity Formula
What is the price of a $1000 bond maturing in ten
years with a 12% coupon that is paid semiannually
if the YTM is 10%

60 
1
1,000
BondValue 
1


20 
0.05  (1  0.05)  (1  0.05) 20
BondValue  1,124.62
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 first dividend regime is a growing annuity
t

D1
1  g  
P0 
 
1  
r  g   1  r  
18

1.14
 1  0.14  
P0 
 
1  
0.15  0.14   1  0.15  
P0  16.58
The second dividend regime is a growing perpetuity
D19
D0 (1  g1 )18 (1  g2 )
P18 

rg
rg
1(114
. )18 (11
.)
P18 
 232.65
015
.  010
.
Now, we need to sum the two dividend regime
values.
P18
232.65
P0  P0 
18  16.58 
(1  r )
115
. 18
P0  35.38
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
EPS
P 
 PV (GO)
r
• 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
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 Example
Firm 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