Simon Benninga. Financial Modelling 3-rd edition

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48 Chapter 2

2.4 “Supernormal Growth” and the Gordon Model

A basic condition of the Gordon formula r

Div

()1

is the

condition |g| < r

In ﬁ nance examples, violations of |g| < r

usually occur

for very fast-growing ﬁ rms, in which—at least for short periods of time—

we anticipate very high growth rates, so that g > r

. If such “supernormal”

growth were the case in the long run, the original dividend discount

formula shows that P

would have an inﬁ nite value, since, when g > r

, the

expression

Div

∗+

()

=∞

∞

∑

. Thus a period of very high dividend

growth rates (where g > r

) must be followed by a period in which the

long-term growth rate of dividends is less than the cost of equity, g < r

Suppose that the ﬁ rm is anticipated to pay high-growth dividends

during periods 1, . . . , m, and that for subsequent periods the growth rate

of dividends will be lower. A little algebra yields the following formula:

Share value today

Present value of dividends

Div

∗+

()

↑

∑

∗

PV of m years

of high-growth g

dividends

Div



(11

−

∞

↑

∑

)

()

PV of remaining

normal-growth g

divide

nnds

Div Div



∗+

∑

()

() ()

))

()

∗

⎛

⎝

⎜

⎞

⎠

⎟

∗−

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

∞

∑

Div

⎥⎥

−

()

∗+

∗

∞

∑

Div

()

EEE

⎛

⎝

⎜

⎞

⎠

⎟

∗−

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

−

()

(Div

ggg

1)( )+

−

5. In this section we interpret Div

in the Gordon formula to denote either dividends per

share or total equity payouts.

49 Calculating the Cost of Capital

The following spreadsheet uses this formula. It gives a numerical illus-

tration of initially high growth rates of dividends followed by lower, more

normal, rates. We consider a ﬁ rm whose current dividend is $8 per share.

The ﬁ rm’s dividend is expected to grow at 35 percent for the next ﬁ ve

years, after which the growth rate will slow down to 8 percent per year.

The cost of equity, the discount rate for all the dividends, is 18 percent:

CBA

Current dividend, Div

8.00

Growth rate g

, years 1-m ("supernormal")

35%

Growth rate g

, years 6 -

∞

Number of supernormal growth years, m 5

Cost of equity 18%

Dividend valuation

PV of supernormal growth years

Using Excel NPV 60.99 <-- =NPV(B6,B16:B20)

Check: Using book formula 60.99 <-- =B2*(1+B3)/(1+B6)*(1-((1+B3)/(1+B6))^B5)/(1-(1+B3)/(1+B6))

PV of normal growth years, after year 5 169.35 <-- =B20*(1+B4)/(B6-B4)/(1+B6)^B5

Share value 230.33 <-- =SUM(B11:B12)

Year

Anticipated

dividend

1 10.80 <-- =B2*(1+$B$3)

2 14.58 <-- =B16*(1+$B$3)

3 19.68 <-- =B17*(1+$B$3)

4 26.57 <-- =B18*(1+$B$3)

5 35.87 <-- =B19*(1+$B$3)

6 38.74 <-- =B20*(1+$B$4)

7 41.84 <-- =B21*(1+$B$4)

8 45.19

9 48.80

10 52.71

11 56.92

12 61.48

etc. etc.

THE GORDON MODEL WITH TWO GROWTH RATES

The value of the ﬁ rm’s share is $230.33, which is the sum of

•

Five years of supernormal growth, computed either by Excel’s

NPV function (cell B10) or by the formula

Div

∗

⎛

⎝

⎜

⎞

⎠

⎟

∗

⎧

⎨

⎩

−

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

⎫

⎬

⎭

−

⎛

⎝

⎜

⎞

⎠

⎟

derived previously (cell B11).

•

The remaining dividends at a normal growth rate (cell B12), com-

puted by the formula derived previously

()

()()

−r

Div

50 Chapter 2

Based on the historical growth rate of the equity payout of 28.28

percent (cell E13), the Gordon model cost of equity is r

= 37.52 percent.

This astoundingly large number is based on the assumption that the his-

torical equity payout growth rate will continue forever.

The two-stage Gordon model may be the answer to this conundrum:

Perhaps the large historical equity payout growth rate is unsustainable

in the long run and will slow down to “normal” growth rates after a

number of years. If this is the case, we have to solve the following equa-

tion for r

FEDCBA

Common

stock issued

Total

dividend

payments

Common

stock

repurchased

Total equity

cash flow

1991 16,462,000 150,730,000 1,215,000 135,483,000 <-- =D3+C3-B3

1992 29,717,000 170,756,000 31,197,000 172,236,000 <-- =D4+C4-B4

1993 24,961,000 191,488,000 98,804,000 265,331,000

1994 25,339,000 210,503,000 52,908,000 238,072,000

1995 24,115,000 235,495,000 65,032,000 276,412,000

1996 25,826,000 254,458,000 376,716,000 605,348,000

1997 59,281,000 327,303,000 532,682,000 800,704,000

1998 80,375,000 381,798,000 531,122,000 832,545,000

1999 59,478,000 418,447,000 634,623,000 993,592,000

Compound growth rate

17.42% 13.61% 118.65% 28.28% <-- =(E11/E3)^(1/8)-1

End 1999 stock price 68.00

Number of shares

outstanding, end 1999

202,795,000

Future dividend growth, g? 28.28% <-- =E13

End 1999 equity value, P

13,790,060,000 <-- =B16*B17

Projected next total

equity cash flow, D

1,274,598,774 <-- =E11*(1+B18)

Gordon model cost

of equity, r

37.52% <-- =B20/B19+B18

WACHOVIA BANK--DIVIDENDS, STOCK ISSUED, AND STOCK REPURCHASED

Does historical growth overstate the cost of equity?

Cost of capital using the Gordon model

Using total equity payout and total equity value

2.4.1 Computing the Cost of Equity for a Firm with Supernormal Growth Rates

The supernormal growth model can be used to compute the cost of

equity r

for companies whose historical equity payout data overstate

any anticipation of future growth rates. Consider, for example, the annual

data for Wachovia Bank:

51 Calculating the Cost of Capital

∑

Current equity value

Div Div

0High 0H

()

(

iigh Normal

)( )

()

mtm

−

∞

∑

(Note that we now use P

to stand for total equity value at time 0 and

Div

for total equity payouts at time 0.) As shown previously, we can

simplify this expression as follows:

⎛

⎝

⎜

⎞

⎠

⎟

∗

−

⎛

⎝

⎜

⎞

⎠

⎟

−

Div

High

⎛

⎝

⎜

⎞

⎠

⎟

+∗

−

Div

High

Normal

()

Given the dividend growth rates g

High

and g

Normal

, the current dividend

, and the number of years m of supernormal growth, the cost of equity

is the internal rate of return of the preceding expression.

Here’s an implementation for Wachovia that assumes three years of

High

= 28.28 percent, after which the growth rate will be g

Normal

= 4

percent. This gives the cost of equity for Wachovia of r

= 17.251

percent:

CBA

End-1999 equity value, P

13,790,060,000

End 1999 total equity payout 993,592,000

High growth rate, g

High

28.28%

Number of high-growth years, m 3

Normal growth rate, g

Normal

Cost of equity, r

using the function TwoStageGordon

17.251% <-- =TwoStageGordon(B2,B3,B5,B6,B7)

WACHOVIA, TWO-STAGE GORDON MODEL

The function TwoStageGordon (on the ﬁ le with this chapter, fm3_

chapter02.xls) computes the cost of equity r

for a two-stage Gordon

model. The function computes the discount rate r

that equates the

current share price to the present value of future equity cash ﬂ ows:

52 Chapter 2

2.5 Using the Capital Asset Pricing Model to Determine the Cost

of Equity r

Sections 2.2 and 2.3 explored the Gordon dividend model as a means of

computing the ﬁ rm’s cost of equity r

. The capital asset pricing model

(CAPM) is a viable alternative to the Gordon model for calculating the

cost of capital.

It is also the most widely used cost-of-equity model,

Function TwoStageGordon(P0, Div0, Highgrowth,

Highgrowthyrs, Normalgrowth)

High = 4

Low = 0

Do While (High − Low) > 0.0000001

Estimate = (High + Low) / 2

factor = (1 + Highgrowth) / _

(1 + Estimate)

Term1 = Div0 * factor * _

(1 − factor ^ Highgrowthyrs) / _

(1 − factor)

Term2 = Div0 * factor ^ Highgrowthyrs * _

(1 + Normalgrowth) / _

(Estimate − Normalgrowth)

If (Term1 + Term2) > P0 Then

Low = (High + Low) / 2

Else: High = (High + Low) / 2

End If

Loop

TwoStageGordon = Estimate

End Function

6. This chapter does not discuss the derivation of the CAPM. For many more details, see

Chapters 8–12.

53 Calculating the Cost of Capital

because of both its theoretical elegance and its implementational sim-

plicity. The CAPM derives the ﬁ rm’s cost of capital from its covariance

with the market return.

The classic CAPM formula for the ﬁ rm’s cost

of equity is

rr Er r

Ef M f

=+ −

[]

()

where r

is the market risk-free rate of interest, E(r

) is the expected

return on the market portfolio, and b is a ﬁ rm-speciﬁ c risk measure

Cov

Var

(,)

()

stock M

In the remainder of this section we focus on measuring the ﬁ rm’s b;

the next section shows how to apply the CAPM to ﬁ nd the ﬁ rm’s cost of

equity r

2.5.1 Beta Is the Regression Coefﬁ cient of the Firm’s Stock Returns on the

Market Returns

In the following spreadsheet, we show the ﬁ ve-year monthly prices and

returns for Intel Corporation and the S&P 500 returns, which we take as

proxy for the stock market as a whole. In cells B2 : B6 we have regressed

the Intel returns on those of the S&P 500:

tSPt

SP t

Intel, Intel Intel ,

=− + =

αβ

0 0029 2 2516 0 5304

.. .

7. The CAPM is discussed in detail in Chapters 7–11. At this point we outline the applica-

tion of the model to ﬁ nding the cost of capital without entering into the theory.

54 Chapter 2

BCDEF G

lpha -0.0029 <-- =INTERCEPT(E13:E72,F13:F72)

Beta

Using Excel's Slope function 2.2516 <-- =SLOPE(E13:E72,F13:F72)

Using Cov/Var 2.2516 <-- =COVAR(E13:E72,F13:F72)/VARP(F13:F72)

R-squared 0.5304 <-- =RSQ(E13:E72,F13:F72)

t-statistic for alpha -0.2438 <-- =tintercept(E13:E72,F13:F72)

t-statistic for beta 8.0942 <-- =tslope(E13:E72,F13:F72)

Date Intel SP500 Intel SP500

9-Jan-01 35.38 1366.01

1-Feb-01 27.32 1239.94 -25.85% -9.68% <-- =LN(C13/C12)

1-Mar-01 25.17 1160.33 -8.20% -6.64% <-- =LN(C14/C13)

2-Apr-01 29.57 1249.46 16.11% 7.40% <-- =LN(C15/C14)

1-May-01 25.86 1255.82 -13.41% 0.51%

1-Jun-01 28 1224.38 7.95% -2.54%

2-Jul-01 28.54 1211.23 1.91% -1.08%

1-Aug-01 26.78 1133.58 -6.37% -6.63%

4-Sep-01 19.58 1040.94 -31.31% -8.53%

1-Oct-01 23.39 1059.78 17.78% 1.79%

1-Nov-01 31.31 1139.45 29.16% 7.25%

3-Dec-01 30.15 1148.08 -3.78% 0.75%

2-Jan-02 33.59 1130.2 10.80% -1.57%

1-Feb-02 27.38 1106.73 -20.44% -2.10%

1-Mar-02 29.17 1147.39 6.33% 3.61%

1-Apr-02 27.44 1076.92 -6.11% -6.34%

1-May-02 26.51 1067.14 -3.45% -0.91%

3-Jun-02 17.54 989.82 -41.30% -7.52%

1-Jul-02 18.04 911.62 2.81% -8.23%

1-Aug-02 16.02 916.07 -11.88% 0.49%

3-Sep-02 13.35 815.28 -18.23% -11.66%

1-Oct-02 16.62 885.76 21.91% 8.29%

1-Nov-02 20.09 936.31 18.96% 5.55%

2-Dec-02 14.98 879.82 -29.35% -6.22%

2-Jan-03 15.06 855.7 0.53% -2.78%

3-Feb-03 16.62 841.15 9.86% -1.71%

3-Mar-03 15.68 848.18 -5.82% 0.83%

1-Apr-03 17.72 916.92 12.23% 7.79%

1-May-03 20.08 963.59 12.50% 4.96%

2-Jun-03 20.07 974.5 -0.05% 1.13%

1-Jul-03 24 990.31 17.88% 1.61%

1-Aug-03 27.59 1008.01 13.94% 1.77%

2-Sep-03 26.56 995.97 -3.80% -1.20%

1-Nov-05 26.27 1249.48 13.03% 3.46%

1-Dec-05 24.57 1248.29 -6.69% -0.10%

3-Jan-06 25.9 1285.45 5.27% 2.93%

ReturnsPrices

COMPUTING THE BETA FOR INTEL

monthly returns for Intel and S&P 500, 2001-2006

Intel Returns vs SP500, 2001-2006

y = 2.2516x - 0.0029

= 0.5304

-45%

-35%

-25%

-15%

-5%

15%

25%

-13%

-11% -9% -7% -5% -3% -1% 1% 3% 5% 7%

SP500 returns

Intel returns

Here’s what we can learn from this regression:

•

Intel’s beta, b

Intel

, shows the sensitivity of its stock return to the market

return. It is calculated by the following formula.

55 Calculating the Cost of Capital

Intel

Covariance(SP500 returns, Intel returns)

Variance(SP500

rreturns)

We can compute b either by using this formula directly (cell B5) or

by using the Excel Slope function (cell B4). Over the period covered,

a 1 percent increase or decrease in the monthly returns of the

S&P 500 was accompanied by a 2.25 percent increase or decrease in

Intel’s returns. The statistic TSlope (cell B8) shows that the b

Intel

highly signiﬁ cant (see section 2.5.2 for how this function was

constructed).

•

Intel’s alpha, a

Intel

, shows that irrespective of changes in the S&P

500, the monthly return on Intel over the period was a

Intel

= −0.29

percent. On an annual basis, this is 12

−0.29 = −3.5 percent; this seems

to indicate that, in the jargon of ﬁ nancial markets, Intel had negative

performance over the period. Note, however, the TIntercept (cell B7):

This function (its construction in Excel is discussed in section 2.5.2)

shows that the negative intercept is not signiﬁ cantly different from

zero.

•

The R

of the regression shows that 53 percent of the variation in Intel’s

returns is accounted for by variability in the S&P 500. An R

of 53 percent

may seem low, but in the CAPM literature this is actually quite a respect-

able number. It says that roughly 53 percent of the variation in Intel’s

returns is explicable by the variation in the S&P 500 return. The rest of

the variability in the Intel returns can be diversiﬁ ed away by including

Intel’s shares in a diversiﬁ ed portfolio of shares. The average R

for

stocks is approximately 30–40 percent, meaning that market factors

account for approximately this percentage of a stock’s variability, with

factors idiosyncratic to the stock accounting for the rest.

The spreadsheet shows three ways of doing the regression: One way

is to use the functions Intercept, Slope, Rsq. A second method involves

8. For the precise meaning of a t-statistic, you should refer to a good statistics text. For

our purposes, a t-statistic over 1.96 indicates that with 95 percent probability the vari-

able under discussion (the intercept when using TIntercept or the slope when using

TSlope) is signiﬁ cantly different from zero. Thus the t-statistic for the intercept of

−0.2438 indicates that the intercept is not signiﬁ cantly different from zero, whereas the

t-statistic for the slope of 8.0942 indicates that the slope is signiﬁ cantly different from

zero.

56 Chapter 2

Figure 2.1

The sequence of commands for producing regression results from the XY Scatter Plot in

Excel. Having marked the points, we right-click to select Add Trendline (left panel). We

choose the linear regression (middle panel) and then click on the Options tab to indicate

that the regression equation and the R

should be indicated on the graph.

using the Excel functions Covar and VarP. A third way involves Excel’s

Trendline function. Having graphed the returns of Intel and the S&P 500

on an XY Scatter plot, we then follow the procedure described in Figure

2.1.

2.5.2 The Homemade Functions TIntercept and TSlope

The preceding spreadsheet uses two functions to compute the t-statistics

for the intercept and slope. These functions are built on the Linest func-

tion discussed in Chapter 33. Applying Linest to the return data, we get

the following:

LKJI

Slope Intercept

Slope --> 2.2516 -0.0029 <-- Intercept

Standard error of slope --> 0.2782 0.0120 <-- Standard error of intercept

R-squared --> 0.5304 0.0927 <-- Standard error of y values

F statistic --> 65.5155 58.0000 <-- Degrees of freedom

-->

0.5627 0.4982 <-- SSE = Residual sum of squares

Cells J14:K18created with the

formula

=LINEST(E12:E71,F12:F71,,1)}

57 Calculating the Cost of Capital

By using the Excel function Index we deﬁ ne a function TIntercept that

divides the value of the intercept term produced by Linest (ﬁ rst row,

second column of Linest output) by the standard error of the intercept

(second row, second column):

Function tintercept(yarray, xarray)

tintercept = Application.

Index(Application.

LinEst(yarray, xarray, , 1), 1, 2) / _

Application.Index(Application.

LinEst(yarray,

xarray, , 1), 2, 2)

End Function

Similarly we can deﬁ ne a function TSlope that gives the t-statistic for the

slope:

Function tslope(yarray, xarray)

tslope = Application.Index(Application.

LinEst(yarray, xarray, , 1), 1, 1) / _

Application.Index(Application.

LinEst(yarray,

xarray, , 1), 2, 1)

End Function

2.5.3 Using Excel’s Data Analysis Add-in

There’s a fourth way to produce the regression output: By clicking on

Tools|Data Analysis|Regression, we can use a sophisticated Excel routine

that computes more statistics, including the t-statistics. The output for

this routine is illustrated as follows: