General DCF valuation formula

1
LECTURE 2 (CHAPTER 2, 5, 8)Valuation Basics

• General DCF valuation formula
• PVs of single CF, annuity, perpetuity and uneven
  CFs
• Bond valuation
• Stock valuation

2

• Time lines show timing of cash flows.

0
1
2
3
i
CF0
CF1
CF2
CF3
Tick marks at ends of periods, so Time 0 is
today Time 1 is the end of Period 1 or the
beginning of Period 2.
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Factors that Affect Value

• Amount of cash flows expected
• Risk of the cash flows
• Timing of the cash flow stream

4
General Formula
Finding PVs is discounting. The discount factor i
is determined by the cost of capital invested
(or, the required rate of return by the investor).
5
Three Ways to Find PV

• Use a formula.
• Use a financial calculator.
• Use a spreadsheet.

6
Single Cash Flow
Whats the PV of 100 due in 3 years if i 10?
0
1
2
3
10
100
PV ?
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Formula Solution


8
Financial Calculator Solution
INPUTS
3 10 0 100 N I/YR PV
PMT FV -75.13
OUTPUT
Either PV or FV must be negative. Here PV
-75.13. Put in 75.13 today, take out 100
after 3 years.
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Spreadsheet Solution
A B C D 1 0 1 2 3 2 0 0 100 3 75.13
Excel Formula in cell A3 NPV(10,B2D2)PV(10,
3,0,-100) 75.13
10
PV(10,3,0,-100) 10 discount rate 3
number of periods 0 periodic cash
flows -100 additional cash flow in the last
period (future value).
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Annuity

• A series of equal payments made at fixed
  intervals for a specified number of periods.
• Ordinary (deferred) annuity the payments occur
  at the end of each period.
• mortgages, car loans, student loans
• Annuity Due the payments are made at the
  beginning of each period.
• apartment rental, life insurance premiums

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Ordinary Annuity vs. Annuity Due
Ordinary Annuity
0
1
2
3
i
PMT
PMT
PMT
Annuity Due
0
1
2
3
i
PMT
PMT
PMT
PV
FV
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Whats the PV of a 3-year ordinary annuity of
100 at 10?
0
1
2
3
10
100
100
100
90.91
82.64
75.13
248.69 PV
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INPUTS
3 10 100 0
N
I/YR
PV
PMT
FV
OUTPUT
-248.69
Have payments but no lump sum FV, so enter 0 for
future value.
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A B C D 1 0 1 2 3 2 100 100 100 3 248.69
Excel Formula in cell A3 NPV(10,B2D2)PV(10,
3,-100,0) 248.69
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Find the PV if theannuity were an annuity due.
0
1
2
3
10
100
100
100
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Switch from End to Begin. Then enter
variables to find PVA3 273.55.
INPUTS
3 10 100 0
-273.55
N
I/YR
PV
PMT
FV
OUTPUT
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Excel Function for Annuities Due
Change the formula to PV(10,3,-100,0,1)273.55
The fourth term, 0, tells the function there
are no other cash flows. The fifth term tells
the function that it is an annuity due.
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Perpetuity
A series of equal payments or payments with
constant growth rate made at fixed intervals for
unlimited number of periods.
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Whats the PV of a perpetuity that starts with a
payment 100 at the end of year 1, and the
payments growing at an annual 5 and invested
every year at a rate of 10?
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Uneven Cash Flow Stream
0
1
2
3
4
10
100
300
300
-50
90.91
247.93
225.39
-34.15
530.08 PV
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• Input in CFLO register
• CF0 0
• CF1 100
• CF2 300
• CF3 300
• CF4 -50
• Enter I 10, then press NPV button to get NPV
  530.09. (Here NPV PV.)

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Keystrokes
Explanation CE/C
Clears display CF 2nd CLR
Clears cash flow variables
0 ENTER ? Stores
the initial cash flow CF00 100 ENTER ?
Stores the first years cash flow
CF1100 ENTER ?
Stores the number of years CF1 is repeated 300
ENTER ? Stores CF2300 2
ENTER ? Stores the number
of years CF2 is repeated 50 /- ENTER ?
Stores CF3-50 ENTER
Stores the number of years CF3 is
repeated 2nd QUIT
Stores storage of individual cash flows NPV 10
ENTER ? Stores interest
rate CPT
Calculate the NPV
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A B C D E 1 0 1 2 3 4 2 100 300 300 -50 3 53
0.09
Excel Formula in cell A3 NPV(10,B2E2)
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Bond Valuation
Key Features of a Bond 1. Par value
Face amount paid at maturity. Assume
1,000. 2. Coupon interest rate Stated
interest rate. Multiply by par value to get
dollars of interest. Generally fixed.
(More)
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3. Maturity Years until bond must be repaid.
Declines. 4. Issue date Date when bond was
issued. 5. Default risk Risk that issuer will
not make interest or principal payments.
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Whats the value of a 10-year, 10 coupon bond if
kd 10?
10
...
100 1,000
100
100
V ?
100
1
,
000
100
V
?
.
.
.




B
1
10
10
?
?
?
?
?
?
k
k
1
1
1
k



d
d
d
90.91 . . . 38.55 385.54
1,000.
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The bond consists of a 10-year, 10 annuity of
100/year plus a 1,000 lump sum at t 10
INPUTS
10 10 100 1000 N I/YR PV
PMT FV -1,000
OUTPUT
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What would happen if expected inflation rose by
3, causing k 13?
INPUTS
10 13 100 1000 N I/YR PV
PMT FV -837.21
OUTPUT
When kd rises, above the coupon rate, the bonds
value falls below par, so it sells at a discount.
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What would happen if inflation fell, and kd
declined to 7?
INPUTS
10 7 100 1000 N I/YR PV
PMT FV -1,210.71
OUTPUT
If coupon rate gt kd, price rises above par, and
bond sells at a premium.
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Semiannual Bonds
1. Multiply years by 2 to get periods
2n. 2. Divide nominal rate by 2 to get periodic
rate kd/2. 3. Divide annual INT by
2 to get PMT INT/2.
INPUTS
2n kd/2 OK INT/2 OK N I/YR
PV PMT FV
OUTPUT
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Find the value of 10-year, 10 coupon, semiannual
bond if kd 13.
2(10) 13/2 100/2 20 6.5
50 1000 N I/YR PV
PMT FV -834.72
INPUTS
OUTPUT
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Common Stock Valuation

• Different Approaches
• Dividend growth model
• Using the multiples of comparable firms
• Free cash flow method
• CAPM (for public firms only)

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Stock Value PV of Dividends
We apply our present value formula to dividends,
with P-hat0 as todays price of a share of stock,
and the Ds equal to the dividend per share, to
obtain the following equation
The dividends can grow either at a constant rate
(from negative to zero to positive), or a
non-constant rate. When they grow at a constant
rate, the above equation becomes
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What happens if g gt ks?

• If kslt g, get negative stock price, which is
  nonsense.
• We cant use model unless (1) g ? ks and (2) g is
  expected to be constant forever. Because g must
  be a long-term growth rate, it cannot be ? ks.

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Whats the stocks market value? D0 2.00, ks
13, g 6.
Constant growth model
2.12
2.12
30.29.
0.13 - 0.06
0.07
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What is the stocks market value one year from
now, P1?

• D1 will have been paid, so expected dividends are
  D2, D3, D4 and so on. Thus,

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What would P0 be if g 0?
The dividend stream would be a perpetuity.
0
1
2
3
ks13
2.00
2.00
2.00
PMT
2.00

P0 15.38.
k
0.13
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What if the growth is negative,with g -5?

• The dividend stream would decrease each year.
  But the firm still has value.

• This negative growth version is less than 1/3 the
  value of the growth version.

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The value of Gordons Model

• The role of risk (ks), growth, and cash-flows is
  obvious.
• When ks ?, g ?, or CFs (dividends) ?, then the
  stock price is greater.
• When ks ?, g ? , or CFs (dividends) ?, then the
  stock price is lower.
• We can see that stock value is due largely to
  long-term growth, not short-term results.

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Super-Normal Growth Model

• Use this approach when short-term growth is
  greater than ks.
• Consider an example for a firm which has
• D0 2.40, and k 12,
• dividend growth at 25 for 4 years, and
• dividend growth at 5 per year forever
  after the initial 4 years.

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Super-Normal Growth Model
Note that D1 3.00 and k 12 Dividends grow
at 25 for 4 years, and then at 5 per year
forever.
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Super-Normal Growth (Contd).
First, find the value of the stock at the end of
the super-normal growth period (t 4)
Note The discount timing for the continuing
value of 87.89 is t 4.