Principles of Finance with Excel (Основы финансов c Excel)
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PFE, Chapter 6, Weighted average cost of capital page 7
• E, the market value of a firm’s equity. We will usually take E to equal the number of
shares of the firm times the market price per share.
•
r
D
, the cost of debt. r
D
is the cost of borrowing for the firm. In most cases we will take
r
D
to be the firm’s marginal interest rate—the interest rate at which the firm could borrow
additional funds from its banks or by selling bonds. A detailed example of the
calculation of r
D
for an actual firm is given in Section 6.4.
•
D, the market value of the firm’s debt. In most cases we will take D to be the total value
of the firm’s financial obligations. An actual example of a calculation for D is given
below in Section 6.4.
•
T
C
, the firm’s tax rate. Most often we calculate T
C
by computing the average tax rate of
the firm; see Section 6.4 for an example.
6.2. The Gordon dividend model: discounting anticipated dividends to derive
the firm’s cost of equity r
E
In this section we present a formula for computing the firm’s cost of equity r
E
. We will
call the formula the Gordon dividend model, in honor of Myron Gordon who first set out the
model in 1959.
5
This section has two subsections:
•
In the first subsection we derive a model for calculating the value of a firm’s shares based
on their future anticipated dividends.
5
The model is sometimes simply called the Gordon model; others call it the dividend discount model.
PFE, Chapter 6, Weighted average cost of capital page 8
• In the second subsection, we use the share valuation model of the first section to derive
the cost of equity r
E
.
Valuing the firm’s shares as the present value of the future anticipated dividends
We start by computing the fair market value of a stock that pays a growing dividend
stream. Here is an example which presents most of the logic of our model: It is March 2, 2000,
and you are thinking of purchasing a share of XYZ Corporation. Here are some facts about the
company and its stock:
•
XYZ is a steady payer of dividends; in the past it has paid dividends annually, and these
dividends have tended to grow at an annual rate of 7%.
•
The company just paid a dividend of $10 per share. This dividend was paid on March 1,
the company’s traditional dividend payment date.
You want value XYZ shares by discounting the stream of future anticipated dividends.
In predicting the future dividends of XYZ Corp., you assume that the dividends will grow at a
rate of 7% per year. Then the future anticipated dividends per share are:
() ( )
() () ( )
() () ( )
()
0
10
22
21 0
33
32 0
0
$10.00
1 $10* 1 7% $10.70
1 1 $10* 1 7% $11.45
1 1 $10* 1 7% $12.25
1
t
t
Dividend today Div
Dividend next year Div Div g
Div Div g Div g
Div Div g Div g
Div Div g
==
== += +=
=+=+= +=
=+=+= +=
=+
…
The three dots ... indicate that you think that the dividend stream is very long (when we write
down the actual model, we will assume that the dividend stream goes on forever).
PFE, Chapter 6, Weighted average cost of capital page 9
Suppose you think that the appropriate discount rate for the dividend stream is XYZ’s
cost of equity r
E
= 15%. Using r
E
to discount the future anticipated dividends, you get the fair
value of the XYZ Corp’s stock today (we will denote this by P
0
):
()
()()
()
()
()
()
()
()
()
3
12
0
23
23
00 0
23
0
.:
,
1
11
11 1
1
11
1
E
EE
E
EE
E
valuing XYZ Corp stock
Div
Div Div
fair share value today P
r
rr
Div g Div g Div g
r
rr
Div g
rg
=+ + +
+
++
++ +
=
+++
+
++
+
=
−
…
…
The last line of the above formula uses a formula for the present value of a
constant-growth annuity developed in Chapter 1 (p000): The present value of the cash
flows Div
0
(1+g), Div
0
(1+g)
2
, Div
0
(1+g)
3
, ... at the discount rate r
E
is :
()
()
()
00
0
1
11
,
1
t
E
t
t
E
E
Div g Div g
Pwhengr
rg
r
∞
=
++
== <
−
+
∑
.
6
Applying the valuation model to XYZ stock gives:
()
()
()
()
()
()
()
()
0
23
0
23
This is 1
This is
10 1.07 10 1.07 10 1.07
,
1.15
1.15 1.15
10 1.07
133.75
0.15 0.07
E
Div g
rg
fair share value today P
+
↓
↑
−
=+ + +
==
−
…
Here’s a spreadsheet implementation of the Gordon dividend model:
6
.The condition |g| < |r
E
| means that the absolute value of g less than the absolute value of r
E
. If a firm’s dividends
have positive growth, then this is the same as assuming that 0 < g < r
E
.
PFE, Chapter 6, Weighted average cost of capital page 10
1
2
3
4
5
6
7
AB C
Current dividend, D
0
10
Dividend growth rate, g 7%
Cost of equity, r
E
15%
Share value 133.75 <-- =B2*(1+B3)/(B4-B3)
The formula used in cell B5 is usually called the Gordon
dividend model, in honor of M. J. Gordon,who first stated its
application to share valuation (1959).
VALUING XYZ CORP. SHARES
Using the Gordon dividend model to calculate the cost of equity r
E
In the previous subsection we derived the value of a share P
0
based on the current
dividend per share Div
0
, the anticipated growth rate of dividends g, and the cost of equity r
E
. In
this section we turn this formula around: We derive the cost of equity r
E
based on the current
value of a share P
0
, the current dividend per share Div
0
, the anticipated growth rate of dividends
g.
According to the Gordon dividend model of the previous subsection, the stock price is
given by
()
0
0
1
E
Div g
P
rg
+
=
−
. Turning this formula around to solve for the cost of equity r
E
gives:
()
0
0
1
E
Div g
rg
P
+
=+ .
This is the Gordon dividend model cost of equity formula. In the Gordon dividend model
the cost of equity r
E
–the discount rate to be applied to equity cash flows—is the sum of two
terms:
PFE, Chapter 6, Weighted average cost of capital page 11
•
()
0
0
1Div g
P
+
. This is the anticipated dividend yield of the stock. Suppose you buy the
stock today, paying P
0
. Then you anticipate getting a next-period dividend of Div
0
(1+g),
where g is the anticipated growth rate of dividends. The term
()
0
0
1Div g
P
+
is the
anticipated next period dividend return.
•
g . This the growth rate of all future dividends paid on the stock.
Applying the Gordon dividend model cost of equity formula—a simple example
Consider a firm for which the current share price is P
0
= $25.00 and which has just paid a
per-share dividend of Div
0
= $3.00. Shareholders of the firm believe that dividends will grow at
a rate g =8% per year. In this case the Gordon model cost of equity is r
E
= 20.96%:
1
2
3
4
5
ABC
Current dividend, Div
0
3.00
Current share price, P
0
25.00
Anticipated dividend growth rate, g 8%
Gordon model cost of equity, r
E
20.96% <-- =B2*(1+B4)/B3+B4
USING THE GORDON MODEL TO
COMPUTE THE COST OF EQUITY r
E
6.3. Applying the Gordon cost of equity formula—Courier Corporation
Courier Corporation (stock symbol CRRC) is a book manufacturer that has experienced
rapid growth of sales and profits. Courier’s financial year ends September 30. We use the
Gordon dividend model to calculate Courier’s cost of equity at the end of September 2000.
Here’s a spreadsheet which gives the relevant data and the calculations:
PFE, Chapter 6, Weighted average cost of capital page 12
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
ABC
Year ended
30 Sept
Dividend
per share
1995 0.27
1996 0.32
1997 0.32
1998 0.35
1999 0.40
2000 0.48
g
, growth rate of dividends
12.47% <-- =(B8/B3)^(1/5)-1
Div
0
, current dividend
0.48 <-- =B8
Div
0
*(1+g)
, dividend anticipated in 2001
0.54 <-- =B11*(1+B10)
P
0
, stock price, 30 Sept. 2000
21.66
r
E
, Gordon dividend model cost of equity
14.97% <-- =B12/B13+B10
COURIER CORPORATION (CRRC)
Calculation of Cost of Equity using Gordon Model
In order to use the Gordon model to calculate the cost of equity in cell B15, we need the
following assumptions:
•
The price of the share, P
0
, is known. In this case the P
0
is the stock price on the date of
the calculation (30 September 2000). On this date P
0
= $21.66.
•
The current dividend per share, Div
0
, is known. By “current dividend” we mean the
last dividend paid by the firm, which in this case is the year 2000 Courier dividend Div
0
= $0.48 per share.
•
The average growth rate of the dividends, g, can be derived. We derive this below
from the dividend series in cells B3:B8. Our assumption is that
()
()
()
1995
1996 1995
2
1997 1995
5
2000 1995
0.27
1
1
...
10.48
Div
Div Div g
Div Div g
Div Div g
=
=+
=+
=+=
PFE, Chapter 6, Weighted average cost of capital page 13
This means that
2000
5
5
1995
0.48
1 1 12.47%
0.27
Div
g
Div
=−=−=
.
Given these assumptions, the cost of equity, r
E
, for Courier is given by:
()
(
)
0
0
1 0.48* 1 12.47%
12.47% 14.97%
21.66
E
Div g
rg
P
++
=+= +=
This is the calculation performed in cell B15.
Alternative calculations of the growth rate
The implementation of the Gordon model illustrated above uses the geometric growth
rate
2000
5
1995
1
D
g
D
=−
to calculate the g in the Gordon model. Here are two alternative ways to
compute the growth rate g:
• Alternative 1: Use a different time period. In the example above, we’ve assumed that the
future expected growth rate of dividends is predicted by the dividends between 1995 and
2000. However, we could—after some thought—decide that the dividends are better
predicted by the period 1994 – 2000.
7
In this case the dividend growth rate is
2000
6
6
1994
0.48
1 1 23.80%
0.13
Div
g
Div
=−=−=
7
If you look at the dividend series, you’ll see that there was a very large increase in the Courier dividend between
1994 and 1995. In choosing 1995 as the base year, we’ve decided that the one-time increase in dividends of 100%
between 1994 and 1995 isn’t likely to be repeated. If—as in the current example—we use 1994 as the base year
instead of 1995, we indicate that over a longer period the large dividend increase between 1994 and 1995 is likely to
recur. As you can see, this assumption means that the cost of equity increases considerably.
PFE, Chapter 6, Weighted average cost of capital page 14
This changes the cost of equity considerably—since anticipated dividend growth g is higher in
this alternative than before, the cost of equity
(
)
0
0
1
E
Div g
rg
P
+
=
+ will be higher as shown
below:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
ABC
Year ended
30 Sept
Dividend
per share
1994 0.13
1995 0.27
1996 0.32
1997 0.32
1998 0.35
1999 0.40
2000 0.48
g
, growth rate of dividends
23.80% <-- =(B9/B3)^(1/6)-1
Div
0
, current dividend
0.48 <-- =B9
Div
0
*(1+g)
, dividend anticipated in 2001
0.59 <-- =B12*(1+B11)
P
0
, stock price, 30 Sept. 2000
21.66
r
E
, Gordon dividend model cost of equity
26.54% <-- =B13/B14+B11
COURIER CORPORATION (CRRC)
Alternative 1: Using a different base year
•
Alternative 2: Ignore historical dividends altogether. You might decide that the past
history of Courier dividends is not indicative of its future dividend payouts. In this case,
you might want to use a different number altogether for the anticipated dividend growth
rate g.
8
In the example below you’ve decided that the growth rate for Courier’s future
dividends is 15%. This gives a cost of equity of 17.55%.
8
In Chapters 8 - 9 we’ll show how to build a model for the firm. You might want to use this model to project the
dividend growth rate.
PFE, Chapter 6, Weighted average cost of capital page 15
1
2
3
4
5
6
7
ABCD
g, growth rate of dividends
15.00%
Div
0
, current dividend
0.48
Div
0
*(1+g), dividend anticipated in 2001
0.55 <-- =B3*(1+B2)
P
0
, stock price, 30 Sept. 2000
21.66
r
E
, Gordon dividend model cost of equity
17.55% <-- =B4/B5+B2
COURIER CORPORATION (CRRC)
Alternative 2: Making up a future growth rate of dividends
A final alternative to computing the cost of equity r
E
: Using the total equity payout
instead of per-share data
In some of the years 1995 – 2000, Courier purchased stock from its shareholders in open-
market repurchase transactions. Look at the data below:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
ABCDEF
Year ended
30 Sept
Dividend
per share
Total
dividends
Share
repurchases
Total equity
payout:
Dividends
+
repurchases
1995 0.27 793,000 0 793,000
1996 0.32 970,000 0 970,000
1997 0.32 969,000 882,000 1,851,000
1998 0.35 1,205,000 0 1,205,000
1999 0.40 1,354,000 455,000 1,809,000
2000 0.48 1,572,000 114,000 1,686,000
Growth rate 12.47% 14.67% 16.28% <-- =(E8/E3)^(1/5)-1
Stock price, 30 Sept. 2000
21.66
Number of shares, 30 Sept. 2000
2,938,000
Market value of equity, 30sep00
63,637,080 <-- =B13*B12
2000 total dividend
1,686,000 <-- =E8
Anticipated dividend growth rate 16.28% <-- =E10
Gordon model cost of equity, r
E
19.36% <-- =B16*(1+B17)/B14+B17
COURIER CORPORATION (CRRC)
Computing the Total Equity Payout
The data in column C are for the total dividend paid out by Courier (total dividend =
dividend per share*number of shares); in column D we see the amount of cash paid out to
PFE, Chapter 6, Weighted average cost of capital page 16
shareholders for the repurchase of their shares. Column D gives the total equity payout: the sum
of the dividends + repurchases. Using the figures from column D and using the total market
value of the equity on 30 September 2000, the Gordon model cost of equity for Courier is
19.36%.
9
The anticipated
growth rate
of total equity
payouts
1
E
Gordon dividend model for cost of equity
using all payouts to equity holders :
Total current
equity payout
g
total dividends
repurchases of stock
r
↑
⎛
⎜
⎡⎤
⎜
⎢⎥
=
⎜
⎢⎥
+
⎜
⎢⎥
+
⎜
⎢⎥
⎜
⎣⎦
⎝
=
()
1,686,000* 1 16.28%
16.28% 19.36%
63,637,080
g
Total equity value today
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎜⎟
⎠
+
+
=+=
Although there is some controversy attached to the use of total equity payouts to compute
the cost of equity r
E
, we think it is the correct method. In the examples for Courier Corporation
which follow, we will assume that the r
E
for Courier is 19.36%.
9
Of course our previous comments on alternative ways of computing the dividend growth rate g still apply—we
could, for example, choose a different base year for the growth calculation.