The perpetual growth model and the cost of
computational efficiency: Rounding errors

or wild distortions?

Morris G. Danielsona, Jean L. Hecka,*
aDepartment of Finance, Saint Joseph’s University, 5600 City Avenue, Philadelphia, PA 19131, USA

Abstract

The constant growth model (Gordon, 1962) plays an important role in the stock selection process
for individual investors, in part, because of its computational simplicity. However, value estimates
from the model can be highly dependent on cash flows to be received in the distant future. If future
events might constrain a firm’s growth or lead to its demise, the unadjusted Gordon model can
substantially overstate value. Because the model is less likely to misstate value for low-growth,
high-payout firms, the ironic implication is that the model is most useful when its ability to value
growth is needed least. © 2014 Academy of Financial Services. All rights reserved.

Jel classification: G30

Keywords: Constant growth model; Perpetual growth; Stock valuation

1. Introduction

“Nothing lasts forever,” or so the old saying goes. When valuing assets, however,
individual investors often ignore this adage and assume cash flows will grow at a constant
rate in perpetuity. The Gordon (1962) growth model uses this assumption and is a first-cut
tool for estimating stock prices and calculating terminal values in two-stage growth models,
or discounted cash flow analyses. Bradley and Jarrell (2008) note that this model “… is
taught in all top-tier business schools and used widely throughout the financial community.
It is found in virtually all graduate-level corporate finance textbooks and valuation manuals.”

* Corresponding author. Tel.: �1-610-660-3148; fax: �1-610-660-1986.
E-mail address: jean.heck@sju.edu (J. Heck)

Financial Services Review 23 (2014) 189–206

1057-0810/14/$ – see front matter © 2014 Academy of Financial Services. All rights reserved.



Recent surveys (e.g., Block, 1999; Demirakos, Strong, and Walker, 2004; Dukes, Peng, and
English, 2006; Imam, Barker, and Clubb, 2008) suggest that value estimates from dividend
discount models (such as the Gordon model) are used to inform the recommendations of
many analysts.

The constant-growth assumption greatly simplifies the calculation of the present value of
an infinite dividend stream. Indeed, one of the main reasons why the Gordon model plays
such a prominent role in valuation theory and practice is because it is so easy to use.
However, value estimates from the perpetual-growth-model can rely heavily on cash flows
to be received in the distant future. Shaffer (2006) notes that when the required return is 4%
and the growth rate is 3%, 62% of the value estimate is accounted for by cash flows to be
received more than 50 years in the future. Danielson, Heck, and Shaffer (2008) observe that
almost 70% of an asset’s value stems from cash flows to be received in years 11 to infinity
if the discount rate is 8% and the growth rate is 4%.

From a mathematical perspective, a present value (today) of $20 is worth $20, regardless
of whether the cash flow supporting this present value will be received in five years or 500
years. As long as cash flow estimates five and 500 years in the future are equally reliable, the
timing of the future cash flows is not a cause for concern. From a practical standpoint though,
it is decidedly easier to forecast future events (and cash flows) five years out, than to predict
what will happen in 500 years. In the short term, consumer preferences are likely to evolve
in subtle ways, current patents will remain in force, and technological innovations may
already be in the works (e.g., new medications in the clinical trial process), making their
implications at least somewhat identifiable. Over longer periods of time, market leaders can
be replaced (e.g., Sears and K-Mart by Wal-Mart) and entire product markets can become
obsolete (e.g., buggy whips and VCR machines).

In response to this concern, numerous valuation models allow the assumed growth rate to
decrease (or change) over time. Examples include Miller and Modigliani (1961), Holt,
(1962), Mao (1966), Fielitz and Muller (1985), Gordon and Gordon (1997), Danielson
(1998), and O’Brien (2003). More recently, Shaffer (2006) extends the Gordon model to
include a constant, annual probability of permanent failure. Both of these approaches
(allowing for decreasing growth or permanent failure) effectively reduce the portion of an
estimated present value accounted for by cash flows to be received far in the future.
However, the Gordon model maintains computational advantages over even the simplest of
these alternative models, accounting for its ongoing popularity.

Nevertheless, if future events might constrain a firm’s growth rate, or lead to its eventual
demise, the use of the unadjusted constant growth model can overstate an asset’s value. The
goal of this article is to quantify the size of these potential valuation errors. Are these errors
modest (i.e., rounding differences), in which case the benefits of the model’s computational
simplicity would outweigh the potential costs created by its imprecise value estimates? Or,
are the valuation errors large enough to materially distort investment decisions?

The analytical results in this article suggest that price estimates from the constant growth
model can overstate a stock’s intrinsic value by a sizeable amount—in some cases the
valuation errors can be two or three times the underlying intrinsic value! The potential
overstatement increases as the firm’s dividend yield decreases, shifting a greater portion of
the expected cash flow into later years. These results do not imply that the constant growth

190 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



model should be abandoned as a valuation tool. Instead, the goal of this article is to promote
a better understanding of the limitations of the model, and to place guardrails on its use. In
particular, the constant growth model is most useful when the firm faces a low default
probability, when only a small portion of the growth will be the result of positive net present
value investments, and when the firm’s dividend yield is sufficiently large. In all other cases,
value estimates obtained from the constant growth model have the potential to significantly
overstate value. The ironic implication is that the perpetual growth model is most useful
when it is needed least. That is, when the model is used to value low-growth, high-payout
firms.

2. The constant-growth model: an overview

In the constant-growth model, a firm’s stock price (or an asset’s value) is a function of the
firm’s future dividends (i.e., cash dividends or stock repurchases); the key assumption is that
the dividends will increase at a constant annual rate, forever. To begin, we define each
variable in real terms: D1 is the dividend expected to be paid next year, stated in current
dollars; r is the required (real) rate of return; and g is the (real) perpetual-growth rate. Using
these definitions, the model can be written as Eq. (1).

P0 �
D1

r � g
(1)

In future years, a firm’s cash flows can grow as the result of new investments or inflation.
Section 2.1 rewrites Eq. (1) to focus on growth from new investments when all inputs are
stated in real terms. Section 2.2 extends the model further, to allow the firm’s cash flow
stream to increase with inflation.

2.1. Growth and new investments

Eq. (1) is often expanded to calculate a stock price as a function of the amount invested
in new projects and the return on these investments (see, e.g., Brealey and Myers, 2003). To
do this, the variable E1 is defined as the perpetual earnings stream expected to be generated
by the current operations (stated in current dollars), the plowback rate, b, is the portion of the
earnings reinvested each year, and RN is the economic (not accounting) real return on new
investments. Using these additional definitions, Eq. (1) becomes Eq. (2).

P0 �
D1

r � g
�

�1 � b� E1

r � bRN
(2)

When RN exceeds r, each new project is expected to have a positive net present value and
P0 will increase with the reinvestment rate b.1 If RN equals r, Eq. (2) simplifies to P0 � E1/r
and value does not depend on the reinvestment rate b. In this case, the firm’s dividend still
increases each year at the rate g � bRN. However, this dividend growth simply compensates

191M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



investors for deferring dividends through the reinvestment process; the growth will not
increase the estimated value P0.

2.2. Growth and inflation

To convert Eq. (2) into nominal terms, the following definitions are used:

h � The annual inflation rate
R*N � The nominal return on new investments � (1 � RN)(1 � h) – 1
r* � The discount rate in nominal terms � (1 � r)(1 � h) – 1
E*1 � Next year’s nominal expected earnings � (1 � h)E1

g* � The nominal growth rate � (1 – b)h � b R*N

The nominal growth rate, g*, has two terms on the right-hand side to allow the firm’s entire
cash flow stream (including amounts currently paid as dividends) to increase with inflation. The
first term captures the growth (because of inflation) of the portion of the cash flow stream
currently paid as a dividend. The second term is the growth created through the reinvestment
process; the inflation rate h is embedded in the nominal investment return R*N. Bradley and Jarrell
(2008) also define the nominal growth rate in this way (but with different notation).

When these definitions are substituted into the constant growth model, the inflation terms
ultimately cancel out and the model simplifies to Eq. (2), with r, RN, and g defined as real
rates.2 Eq. (3) outlines this process.

P0 �
�1 � b� E*1

r* � g*

�
�1 � b��1 � h� E1

��1 � r��1 � h� � 1� � ��1 � b�h � b��1 � RN��1 � h� � 1��

�
�1 � b� E1

r � bRN
(3)

Bradley and Jarrell (2008) show that when the constant growth model is not adjusted
properly to incorporate nominal discount and growth rates, the model will produce incorrect
value estimates. In particular, if the discount rate is stated in nominal terms (i.e., the discount
rate is r*), but the growth rate is defined simply as the product of the reinvestment rate b and
R*N, rather than as g* (as defined above), the constant growth model can understate an asset’s
value.3 However, this result does not mean that the constant growth model produces
“conservative” value estimates. Instead, the analysis in Bradley and Jarrell (2008) simply
means that when the model is applied incorrectly (and inputs are not defined consistently in
either nominal or real terms) the model’s value estimates will be wrong (i.e., garbage in,
garbage out).

Because the constant growth model produces identical value estimates using properly
defined real or nominal inputs, and because the model’s notation is simpler using real interest
rates, the analysis in the remainder of this article defines all valuation inputs in real terms.

192 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



3. Value creation and the timing of future cash flows

As noted by Shaffer (2006), Danielson, Heck, and Shaffer (2008), and others, a substantial
portion of the value estimates obtained from the constant growth model can be attributed to
cash flows that may not be received for decades. To calculate the percentage of a perpetual-
growth value estimate attributed to cash flows after year t, first note that the present value of
a finite (t-period), growing dividend stream (i.e., the dividends will grow at a constant rate
for t periods before dropping to $0) can be written as follows (Welch, 2009):

P0 �
D1

r � g �1 � �1 � g

1 � r�
t� (4)

Then, subtract Eq. (4) from Eq. (1) and divide the result by Eq. (1). This yields Eq. (5),
which quantifies the portion of an asset’s value created by cash flows to be received after
year t.

PV�t � 1 to ��/PV�0 to �� � �1 � g

1 � r�
t

(5)

Table 1 uses Eq. (5) to calculate the percentage of an asset’s estimated value—using Eq.
(1)—that is expected to be received after year t, for three discount rates (7, 10, and 13%) and
a range of growth rates. For each combination of r and g, the table lists the portion of the
value estimate created by cash flows received after years 10, 20, 30, 40, 50, and 100. In
addition, the table lists the Macaulay (1938) duration for each combination of r and g.

Although r and g can be (and typically are) estimated individually, Payne and Finch
(1999) point out that the key determinant of P0 in Eq. (1) is the difference between r and g.
For example, if D1 is estimated to be $1 and the difference between r and g is 5%, the
estimated stock price is $20 regardless of whether r � 7% and g � 2%, or r � 13% and g �
8%. In each case, the implied dividend yield (D/P) is 5%.4

Table 1 reveals that the portion of a value estimate attributable to cash flows after a
specified year is also closely related to the difference r – g (� D/P), but this relation is not
exact. For example, when r � 7% and g � 2%, 38.4% (9.1%) of the asset’s value is created
after year 20 (year 50); when r � 13% and g � 8%, 40.4% (10.4%) of the value is expected
to be realized after year 20 (year 50).

As the difference between r and g becomes smaller, a progressively larger portion of the
value estimate will depend on cash flows from the out years. For example, when r – g is four
percentage points (D/P � 4%), over 20% of the estimated value will be realized after year
40. When r – g is two percentage points (D/P � 2%), more than twice as much (i.e., nearly
50%) of the asset’s value will be received after year 40.

As r and g converge (and D/P decreases toward zero), value estimates from the Gordon
model become highly dependent on cash flows to be received in the distant future. When r
– g � 1%, almost 70% of the value will be realized after year 40, and �40% will be received
after year 100. The Macaulay duration in this case exceeds 100. When r – g � 0.5%, over
60% of the estimated value will be received after year 100 and the Macaulay duration
exceeds 200!

193M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



Only when the dividend yield is large, can the majority of the estimated value be linked
to cash flows within a foreseeable time horizon. For example, assume that g � 0 and D/P �
r. If r � 13% (r – g � D/P � 0.13), over 90% of the value will be received over the next
20 years; if r � 10% (r – g � D/P � 0.10), over 90% of the value will be received during
the next 30 years; and when r � 7% (r – g � D/P � 0.07), over 90% of the value will be
received in the next 40 years.

4. By how much can the perpetual growth model overstate value?

Table 1 demonstrates that perpetual-growth-model value estimates can be highly depen-
dent on the present value of cash flows that will not be received for decades (i.e., 20, 50, 100,

Table 1 Percent of value created after year t
Panel A (r � 7%)

g � 0 0.02 0.03 0.04 0.05 0.06 0.065

t � 0 100% 100% 100% 100% 100% 100% 100%
10 50.8% 62.0% 68.3% 75.2% 82.8% 91.0% 95.4%
20 25.8% 38.4% 46.7% 56.6% 68.6% 82.9% 91.1%
30 13.1% 23.8% 31.9% 42.6% 56.8% 75.5% 86.9%
40 6.7% 14.7% 21.8% 32.1% 47.0% 68.7% 82.9%
50 3.4% 9.1% 14.9% 24.1% 38.9% 62.5% 79.1%
100 0.1% 0.8% 2.2% 5.8% 15.2% 39.1% 62.6%
Duration 15.3 21.4 26.8 35.7 53.5 107.0 214.0

Panel B (r � 10%)

g � 0 0.05 0.06 0.07 0.08 0.09 0.095

t � 0 100% 100% 100% 100% 100% 100% 100%
10 38.6% 62.8% 69.0% 75.8% 83.2% 91.3% 95.5%
20 14.9% 39.4% 47.7% 57.5% 69.3% 83.3% 91.3%
30 5.7% 24.8% 32.9% 43.6% 57.7% 76.0% 87.2%
40 2.2% 15.6% 22.7% 33.1% 48.0% 69.4% 83.3%
50 0.9% 9.8% 15.7% 25.1% 40.0% 63.3% 79.6%
100 0.0% 1.0% 2.5% 6.3% 16.0% 40.1% 63.4%
Duration 11.0 22.0 27.5 36.7 55.0 110.0 220.0

Panel C (r � 13%)

g � 0 0.03 0.08 0.09 0.11 0.12 0.125

t � 0 100% 100% 100% 100% 100% 100% 100%
10 29.5% 39.6% 63.8% 69.7% 83.6% 91.5% 95.7%
20 8.7% 15.7% 40.4% 48.6% 70.0% 83.7% 91.5%
30 2.6% 6.2% 25.7% 33.9% 58.5% 76.6% 87.5%
40 0.8% 2.5% 16.4% 23.7% 49.0% 70.1% 83.7%
50 0.2% 1.0% 10.4% 16.5% 40.9% 64.1% 80.1%
100 0.0% 0.0% 1.1% 2.7% 16.8% 41.1% 64.2%
Duration 8.7 11.3 22.6 28.3 56.5 113.0 226.0

Notes. This table lists the percentage of a perpetual-growth model value estimate attributed to cash flows
received after year t (where t � 0, 10, 20, and so forth) calculated using Eq. (5). The calculations use required
returns of 7% (Panel A), 10% (Panel B) and 13% (Panel C), and a range of assumed growth rates, g. For each
combination or r and g, the table also lists the Macaulay duration of the value estimate.

194 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



or more years in the future). This fact would not be a cause for concern if the future cash
flows were certain to be paid. However, economic conditions can change dramatically over
the course of any 20, 50, or 100 year period; the survival of no firm is guaranteed.

Indeed, the experience of the past 100 years highlights the challenges individual firms face
to remain in business (let alone to maintain a positive growth rate) over an extended period.
During the past century, the economy has been transformed by two world wars, a global
depression, and innovations in the transportation, communication, and information systems
industries. Orville and Wilbur Wright’s first flight was in 1903, the Ford Model T debuted
in 1908, AT&T completed the first transcontinental telephone line in 1915, the first electrical,
binary, programmable computer was invented during the 1930s, and the construction of the
interstate highway system commenced in 1956. For an individual firm to have maintained a
constant (or even positive) growth rate across the past century, the firm would have been
required to reinvent itself multiple times to adjust to these (and other) events.

Simply maintaining a competitive advantage (and a constant, positive growth rate) over a
25 year period can be a challenge. As noted by Sheth (2009), many of the 62 firms
highlighted in the 1982 book “In Search of Excellence” (including Sears, Xerox, Kodak,
Dana, and Digital Computer Corp.) experienced financial hardships in the ensuing 25 years.
Thus, investors should be wary of purchasing any asset at a price that can only be justified
by cash flows to be received in the unforeseeable, distant future.

Previous studies identify two alternative ways to minimize the importance of far distant
cash flows in the valuation process. First, Shaffer (2006) extends the Gordon model to
include a constant, annual probability of permanent failure. Second, numerous valuation
models allow the assumed growth rate to decrease over time (e.g., Miller and Modigliani,
1961; Holt, 1962; Mao, 1966; Fielitz and Muller, 1985; Gordon and Gordon, 1997; Dan-
ielson, 1998; O’Brien, 2003). This section reports on potential valuation errors embedded in
perpetual growth model price estimates if future cash flows will be constrained by default,
finite growth, or both.

4.1. Valuation models: default and finite growth

Shaffer (2006) extends the perpetual growth model to allow for the possibility that the firm
will irreversibly fail (and the cash flow stream will end) in the future. Assuming that the
annual failure probability (p) is constant over time, Eq. (1) can be rewritten as Eq. (6), and
Eq. (2) can be rewritten as Eq. (7).

P0 �
D1�1 � p�

r � p � g�1 � p�
(6)

P0 �
E1�1 � b��1 � p�

r � p � bRN�1 � p�
(7)

Shaffer (2006) separately modifies the perpetual growth model to allow the growth rate to
permanently decrease—from a supernormal level (i.e., RN � r) down to the required
return—in any future year. Again, Shaffer assumes that this event occurs with a constant

195M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



annual probability. However, Shaffer does not develop a form of the perpetual growth model
that allows for both default (where the cash flow stream ends) and finite growth (where the
firm’s growth rate permanently decreases, but the cash flow stream continues).

To develop such a model, we simplify the assumptions, and stipulate that the firm’s return
on new investments, RN, be equal to the required return, r, starting in the specific future year
t � 1. As derived in Appendix 1, the adjusted perpetual growth model, allowing for default
and finite growth, can be written as Eq. (8).

P0 �
E1�1 � b��1 � p�

r � p � bRN�1 � p� �1 � � (1 � p)(1 � bRN)

(1 � r) � t�
� � (1 � p)(1 � bRN)

(1 � r) � t� E1(1 � b)(1 � p)

r � p � br(1 � p)� (8)

If the default probability is zero (p � 0), Eq. (8) can be written as Eq. (9). This is the
traditional finite growth model from Miller and Modigliani (1961) and Gordon and Gordon
(1997).

P0 �
E1�1 � b�

r � g �1 � �1 � g

1 � r�
t� �

E1(1 � g)t

r(1 � r)t (9)

4.2. Estimation errors with default (but perpetual growth)

In this section, we compare value estimates from Eq. (7) to those obtained using Eq. (2).
If the intrinsic value of the firm is defined by Eq. (7) (because the firm might fail in the
future) by how much will the perpetual growth model overstate firm value? The percentage
overstatement embedded in a perpetual growth value estimate in this case (positive default
probability; perpetual growth) is Eq. (10). Appendix 2 derives this equation.

% Overstatement �p � 0; perpetual growth� �
p�1 � r�

�r � g��1 � p�
(10)

Table 2 reports on the potential valuation errors—calculated using Eq. (10)—that can
arise when the perpetual growth model is used to value a firm that might fail in the future.
Panel A lists the assumptions used in this exercise, and calculates firm value using the
unadjusted perpetual growth model. In particular, the firm is expected to have earnings per
share of $1 next year, the required return (stated as a real interest rate) is 7%, and the return
on new investments (again stated as a real interest rate) is 7%. The unadjusted perpetual
growth model, Eq. (2), estimates the current stock price as $14.29 regardless of whether the
reinvestment rate is 0%, 50%, or 99%. The results in Table 2, Panel A illustrate this for eight
different reinvestment rates, b. Although each reinvestment rate produces the same value
estimate when plugged into the perpetual growth model, they do not create identical
challenges for the firm’s managers, or impose equal risks on the firm’s shareholders.

Mathematically, value is preserved if retained earnings are reinvested in projects earning
a return equal to the cost of capital, and if the default probability is zero. However,

196 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



identifying projects that can earn a return equal to the cost of capital is not a trivial task in
a competitive economy. For example, if the economy is growing at a real rate of 2% per year,
but the firm must earn 7% (i.e., the cost of capital) on new investments to preserve value, the
process of deferring dividends into the future might require the firm to grow faster than the
economy as a whole, simply to maintain its value.5 It is possible that the aggressive policies
required to achieve this growth could ultimately cause the firm to “grow broke,” contributing
to (or magnifying) the firm’s default probability.

Table 2 Estimation errors–default
Panel A: Assumptions and unadjusted value estimates (default probability � 0)

Scenario 1 2 3 4 5 6 7 8

r � 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07
R � 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07
b � 0.0000 0.1429 0.2857 0.4286 0.5714 0.7143 0.8571 0.9286
g � bR � 0 0.01 0.02 0.03 0.04 0.05 0.06 0.065
r � g � 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.005
PPGM � 14.286 14.286 14.286 14.286 14.286 14.286 14.286 14.286

Panel B: Default probability � 0.0025

Scenario 1 2 3 4 5 6 7 8

Padj � 13.759 13.675 13.559 13.388 13.113 12.597 11.265 9.299
%OS � 3.8% 4.5% 5.4% 6.7% 8.9% 13.4% 26.8% 53.6%

Panel C: Default probability � 0.005

Scenario 1 2 3 4 5 6 7 8

Padj � 13.267 13.111 12.899 12.593 12.114 11.259 9.290 6.883
%OS � 7.7% 9.0% 10.8% 13.4% 17.9% 26.9% 53.8% 107.5%

Panel D: Default probability � 0.0075

Scenario 1 2 3 4 5 6 7 8

Padj � 12.806 12.589 12.297 11.884 11.253 10.173 7.899 5.459
%OS � 11.6% 13.5% 16.2% 20.2% 27.0% 40.4% 80.9% 161.7%

Panel E: Default probability � 0.01

Scenario 1 2 3 4 5 6 7 8

Padj � 12.375 12.105 11.747 11.247 10.502 9.274 6.865 4.518
%OS � 15.4% 18.0% 21.6% 27.0% 36.0% 54.0% 108.1% 216.2%

Notes. This table compares price estimates from the unadjusted perpetual growth model, Eq. (2), to price
estimates from the perpetual growth model with default, Eq. (7), and reports the percentage overstatement of
value embedded in the unadjusted perpetual growth model price estimate. The firm’s estimated earnings next
year, E1, is assumed to be $1, the required return, r, is 7%, and the return on new investments, R, is 7%. Panel
A lists combinations of reinvestment rate b and growth rate g that produce a stock price of $14.286 using Eq. (2).
Panels B through E list the adjusted price estimates from Eq. (7), Padj, for each scenario using default probabilities
of 0.0025 (Panel B), 0.005 (Panel C), 0.0075 (Panel D), and 0.01 (Panel E), and report the % difference between
the unadjusted and adjusted value estimates: %OS � (PPGM � Padj)/Padj. The values reported in the %OS rows
can also be calculated using Eq. (10).

197M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



Table 2, Panels B to E, report potential overstatements embedded in perpetual-growth
value estimates when the default probability is positive. The table reports results for failure
probabilities ranging from 0.25% to 1%. We use this range of failure probabilities because
Shaffer (2006) reports that the average annual failure rate for U.S. businesses during the 1955
to 1995 period was 0.6%. The Table 2 results reveal that the potential overstatement is
directly related to the timing of the implied future cash flows. If r – g is 0.5 percentage
points—that is, D/P is 0.5% (Scenario 8) and the bulk of the estimated value is created by
cash flows from distant years—the perpetual growth model can overstate value by almost
54% (over 216%) if the firm faces a constant, annual, failure probability of 0.25% (1%). In
contrast, if r – g is seven percentage points (Scenario 1), the potential overstatement implied
by the perpetual growth model is only 3.8% (15.4%) when the annual failure probability is
0.25% (1%).

If a firm faces a positive default probability, the process of deferring dividends creates
additional risk (i.e., default risk) for shareholders. Thus, for a firm to maintain value after
deferring dividends, the return on investment earned by the new projects must exceed the
cost of capital in those outcomes in which the firm does not fail. That is, the project must
produce an “ex-post” positive NPV to compensate investors for the firm’s default risk. Eq.
(1), which is derived in Appendix 3, calculates the return on new investments required to
preserve value for a firm that retains and reinvests earnings, if the firm faces an annual default
probability of p.

RN�breakeven� �
r � p

1 � p
(11)

Thus, if a firm’s annual default probability is 1% and its required return is 7%, new
projects must earn an 8.08% return in those outcomes in which the firm is successful for the
investments to preserve value.

4.3. Estimation errors with finite growth (no default)

Because the Gordon model assumes growth will continue forever, the model can overstate
the value of an asset if RN � r, and if competition will limit the period of time positive net
present value projects will be available (Stigler, 1963). In this section, we assume that the
default probability is zero and focus directly on the value overstatement created by the use
of a growth phase that is too long (i.e., perpetual vs. finite growth period).

If Eq. (9) defines an asset’s intrinsic value, the use of the perpetual growth model, Eq. (2),
will produce a value estimate that is too high. To calculate this valuation error as a percentage
of the intrinsic value, subtract Eq. (9) from Eq. (2), and divide the difference by Eq. (9). After
much algebra, this yields Eq. (12).

%Overstatement �p � 0; finite growth� �
1

1 � �1 � g

1 � r�
t� b

1 � b��RN

r
� 1� � 1

(12)

198 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



Table 3 reports potential valuation errors using Eq. (12). The inputs include four combi-
nations of r, g, b, and RN (with p � 0) and competitive advantage periods ranging from 10
to 100 years. Each combination of r, g, b, and RN produces a P/E ratio of 30 using the
unadjusted perpetual growth model.6

If competition will limit a firm’s growth phase to 10 years (but the default probability is
zero) Table 3 shows that the unadjusted perpetual growth model will overstate firm value by
over 65% when the dividend yield is 3% (Scenario 1), and by almost 100% when D/P is 0.5%
(Scenario 4).7 If the firm’s growth phase can be maintained for 20 years, Eq. (2) will still
overstate firm value by over 42% when D/P is 3% and by over 91% when D/P is 0.5%.8 Even
if the true growth period is 100 years, the unadjusted perpetual growth model will overstate
value by almost 49% when D/P is 0.5%. In contrast, the unadjusted perpetual growth model
only overstates value by 3% when the growth period equals 100 years and the dividend yield
is 3%. These results suggest that the perpetual growth model is an appropriate first-cut
valuation tool for a fairly limited set of firms: high-dividend firms with very strong com-
petitive advantages (i.e., 50 or more years).

Table 3 Estimation errors–finite growth

Scenario 1 2 3 4

r � 0.07 0.07 0.07 0.07
R � 0.4 0.125 0.0857 0.0765
b � 0.1 0.4 0.7 0.85
g � bR � 0.04 0.05 0.06 0.065
r � g � 0.03 0.02 0.01 0.005
PPGM � 30 30 30 30
t � 10:
Padj � 18.175 16.988 15.694 15.005
%OS � 65.1% 76.6% 91.2% 99.9%
t � 20:
Padj � 21.102 19.225 16.976 15.691
%OS � 42.2% 56.0% 76.7% 91.2%
t � 50:
Padj � 26.209 23.883 20.173 17.567
%OS � 14.5% 25.6% 48.7% 70.8%
t � 100:
Padj � 29.085 27.619 23.855 20.163
%OS � 3.1% 8.6% 25.8% 48.8%

Notes. This table compares value estimates from the unadjusted perpetual growth model, Eq. (2), to value
estimates from the extended model allowing finite growth (but no default), Eq. (9), and reports the percentage
overstatement embedded in the unadjusted perpetual growth model value estimate. The firm’s estimated earnings
next year, E1, is $1, the required return, r, is 7%, and the firm’s P/E ratio is 30. The table lists four combinations
of reinvestment rate b, return on new investments R, and growth rate g that produce a stock price estimate of $30
(P/E � 30) using Eq. (2). For each combination, the table reports the adjusted value estimate from Eq. (9), Padj,
and the % difference between the unadjusted and adjusted value estimates: %OS � (PPGM � Padj)/Padj. The
values reported in the %OS rows can also be calculated using Eq. (12). This information is reported assuming the
period in which the firm can invest in projects where R � r will last 10, 20, 50, and 100 years.

199M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



4.4. Estimation errors with default and finite growth

Table 4 compares value estimates from the perpetual growth model, Eq. (2), to value
estimates from Eq. (8), which adjusts the perpetual growth model for both finite growth and
potential default. The examples in Table 4 calculate potential valuation errors, for compet-
itive advantage periods ranging from 10 to 100 years, using the same four combinations of
r, g, b, and RN as in Table 3. Panel A lists the assumptions used in each scenario. Panels B
through E list potential value overstatements—embedded in value estimates from Eq.
(2)—when firms face positive default probabilities ranging from 0.25% to 1%.

Table 4 Estimation errors–default and finite growth
Panel A: Assumptions and perpetual growth model value estimate

Scenario 1 2 3 4

r � 0.07 0.07 0.07 0.07
R � 0.4 0.125 0.0857 0.0765
b � 0.1 0.4 0.7 0.85
g � bR � 0.04 0.05 0.06 0.065
r � g � 0.03 0.02 0.01 0.005
PPGM � 30 30 30 30

Panel B: Default probability � 0.0025

Scenario 1 2 3 4

t � 10:
Padj � 17.384 15.935 13.900 11.944
%OS � 72.6% 88.3% 115.8% 151.2%
t � 20:
Padj � 20.087 17.959 14.994 12.470
%OS � 49.4% 67.1% 100.1% 140.6%
t � 50:
Padj � 24.593 21.9791 17.593 13.839
%OS � 22.0% 36.5% 70.5% 116.8%
t � 100:
Padj p � 26.911 24.916 20.311 15.556
%OS � 11.5% 20.4% 47.7% 92.9%

Panel C: Default Probability � 0.005

Scenario 1 2 3 4

t � 10:
Padj � 16.653 14.997 12.465 9.910
%OS � 80.1% 100.0% 140.7% 202.7%
t � 20:
Padj � 19.151 16.834 13.410 10.330
%OS � 56.6% 78.2% 123.7% 190.4%
t � 50:
Padj � 23.135 20.317 15.550 11.372
%OS � 29.7% 47.7% 92.9% 163.8%
t � 100:
Padj � 25.007 22.636 17.583 12.556
%OS � 20.0% 32.5% 70.6% 138.9%

(continued on next page)

200 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



The results in Table 4 show that the introduction of a positive default probability can
magnify the potential overstatements reported in Table 3 (where the default probability is 0).
If the firm’s dividend yield is 3% (Scenario 1), Table 4, Panel B reveals that the percentage
overstatement in a perpetual growth model value estimate is 72.6% when p � 0.25% and t �
10 years (vs. 65.1% in Table 3, p � 0), and is 11.5% when p � 0.25% and t � 100 (vs. 3.1%
in Table 3, p � 0). If the firm’s dividend yield is only 0.5% (Scenario 4), the differences
between the percentage overstatements in Table 4, Panel B and Table 3 become more

Table 4 (continued)
Panel D: Default probability � 0.0075

Scenario 1 2 3 4

t � 10:
Padj � 15.975 14.157 11.291 8.461
%OS � 87.8% 111.9% 165.7% 254.6%
t � 20:
Padj � 18.288 15.828 12.115 8.806
%OS � 64.0% 89.5% 147.6% 240.7%
t � 50:
Padj � 21.815 18.856 13.895 9.621
%OS � 37.5% 59.1% 115.9% 211.8%
t � 100:
Padj � 23.330 20.693 15.432 10.463
%OS � 28.6% 45.0% 94.4% 186.7%

Panel E: Default probability � 0.01

Scenario 1 2 3 4

t � 10:
Padj � 15.344 13.399 10.312 7.375
%OS � 95.5% 123.9% 190.9% 306.8%
t � 20:
Padj � 17.488 14.925 11.038 7.665
%OS � 71.5% 101.0% 171.8% 291.4%
t � 50:
Padj � 20.615 17.564 12.531 8.316
%OS � 45.5% 70.8% 139.4% 260.7%
t � 100:
Padj � 21.844 19.025 13.704 8.928
%OS � 37.3% 57.7% 118.9% 236.0%

Notes. This table compares value estimates from the unadjusted perpetual growth model, Eq. (2), to value
estimates from the extended model allowing both default and finite growth, Eq. (8), and reports the percentage
overstatement embedded in the unadjusted perpetual growth model value estimate. The firm’s estimated earnings
next year, E1, is $1, the required return, r, is 7%, and the firm’s P/E ratio is 30. Panel A lists combinations of
reinvestment rate b, return on new investments R, and growth rate g that produce a stock price estimate of $30
(P/E � 30) using Eq. (2). Panels B through E list the adjusted value estimates from Eq. (8), Padj, for each scenario
using default probabilities of 0.0025 (Panel B), 0.005 (Panel C), 0.0075 (Panel D), and 0.01 (Panel E), and report
the % difference between the unadjusted and adjusted value estimates: %OS � (PPGM � Padj)/Padj. Within each
panel, this information is reported assuming the period in which the firm can invest in projects where R � r will
last 10, 20, 50, and 100 years.

201M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



pronounced. The percentage overstatement in a perpetual growth model value estimate is
151.2% when p � 0.25% and t � 10 years (vs. 99.9% in Table 3, p � 0), and is 92.9% when
p � 0.25% and t � 100 (vs. 48.8% in Table 3, p � 0).

As the estimated default rate increases from Panel B to Panel E, the potential overstate-
ment embedded in perpetual-growth-model value estimates continues to grow. If the firm’s
dividend yield is 3% (Scenario 1), Table 4, Panel E reveals that the percentage overstatement
in a perpetual-growth-model value estimate is 95.5% when p � 1% and t � 10 years (vs.
65.1% in Table 3, p � 0), and is 37.3% when p � 1% and t � 100 (vs. 3.1% in Table 3,
p � 0). If the firm’s dividend yield is only 0.5% (Scenario 4), the percentage overstatement
is 306.8% when p � 1% and t � 10 years (vs. 99.9% in Table 3, p � 0), and is 236.0%
when p � 1% and t � 100 (vs. 48.8% in Table 3, p � 0).

The results in Table 4 further limit the population of firms for which the perpetual growth
model is an appropriate valuation tool. In addition to having a high dividend yield and a
strong competitive advantage, the firm should also be financially strong, with a very low
probability of default.

5. Conclusions

The perpetual-growth model provides investors with a simple way to calculate the present
value of a perpetual dividend stream. The model is a very useful teaching tool, as it clearly
illustrates the links between reinvestment policies, reinvestment returns, and stock prices. In
addition, the economic interpretation of valuation ratios (such as the price-to-earnings ratio)
is often explained within the framework of the constant growth model (e.g., Leibowitz and
Kogelman, 1990). Thus, it is not surprising that the model “… is taught in all top-tier
business schools …” as noted by Bradley and Jarrell (2008).

Moving from the classroom to the investing world, however, the usefulness of the model
can be called into question. To exploit the computational elegance of the model, one must
assume that cash flows will grow at a constant rate forever. This assumption is of dubious
validity in a competitive economy, where a firm’s growth can be derailed by internal
mistakes, innovations by other firms, or macroeconomic shocks.

The analytical results in this article suggest that the perpetual-growth model should be
applied in a very limited set of circumstances: when the firm faces a low default probability,
when only a small portion of the growth will be the result of positive net present value
investments, and when the firm’s dividend yield is sufficiently large. Along these lines,
Foerster and Sapp (2005) show that the constant-growth model produced reasonable value
estimates for the Bank of Montreal (e.g., value estimates that approximated historical stock
prices), a large, mature, dividend-paying company. For firms that do not fit this profile,
individual investors, financial analysts, and financial planners should use the perpetual-
growth model with caution.

202 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



Notes

1 We acknowledge that this statement is true (for the perpetual growth model) only over
the range of reinvestment rates, b, in which the implied growth rate g (� bRN) is less
than the required return r.

2 Danielson and Scott (2000) and Weston, Chung, and Siu (1998) use a similar approach
to reconcile the constant growth model in nominal terms to the constant growth model
in real terms.

3 For example, assume that b � 0.4, E1 � $1, r � 8%, RN � 12%, and h � 2%. Thus,
next year’s nominal earnings are $1.02, the nominal required return is 10.16%, and the
nominal growth rate is 6.896% (� (1 – 0.4)(2%) � 0.4(14.24%)). Using real inputs,
the stock price is $18.75 (� [$1(1 – 0.4)]/[0.08 – 0.4(0.12)]. The stock price using Eq.
(3) and nominal inputs is also $18.75 (� [$1.02(1 – 0.4)]/[0.1016 – 0.06896]. In
contrast, if the nominal growth rate is defined simply as the retention rate times the
nominal return on investments, 5.696% � 0.4(14.24%), the estimated stock price
using this (incomplete) nominal growth rate, the nominal discount rate, and the
nominal earnings is $13.71 (� [$1.02(1 – 0.4)]/[0.1016 –0.05696]. Thus, the incorrect
application of the constant growth model will understate value by almost 27%.

4 According to the Gordon model, if the expected dividend next year is D1, the required
return is r, and the perpetual growth rate is g, then the estimated stock price today is
P0 � D1/(r – g). This expression can be rearranged to write the dividend yield as
D1/P0 � (r – g).

5 The firm’s growth rate will depend on the reinvestment rate and the return on new
investments. For example, if the retention rate b is 90% and the return on new
investments is 7%, the firm’s growth rate is 6.3%. For an extreme illustration of the
challenge facing firms, assume that a firm currently has a market share of 2%. If the
firm’s product market is growing at an annual rate of 2%, and the firm grows at an
annual rate of 6.3%, the firm’s revenues will become larger than those of its industry
in 95 years. However, the examples in Table 1 show that a sizeable portion of firm
value must be realized after year 100 when the difference between r and g is small.

6 The P/E ratio can be derived from the constant growth model by dividing each side of
Eq. (2) by E1. This yields P/E � (1 – b)/(r – bRN).

7 Empirical evidence in Fuller, Huberts, and Levinson (1993) and Lakonishok, Shleifer
and Vishney (1994) reveals that high P/E firms (e.g., P/E � 30) experience higher
future growth rates than low-P/E firms. However, the superior growth of these firms
typically lasts for less than 10 years.

8 The traditional finite growth model, and Eqs. (8) and (9), make the unrealistic
simplifying assumption that a firm’s competitive advantage period will end abruptly at
a specified future date, and that the return on new investments will at that point fall
immediately to the cost of capital. It is perhaps more likely that the return on new
investments will decrease gradually over some future period. For example, if the
return on new investments remains equal to RN for the next 10 years, and then
gradually decreases to r over a 10 year period, the true valuation errors will fall
between the overstatements reported in the t � 10 and t � 20 rows of Table 3.

203M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



Appendix 1

This appendix derives the perpetual growth model adjusted to allow for both permanent
failure and finite growth. To begin, note that the perpetual growth model, allowing for
permanent failure, is defined by Eq. (7). This expression is repeated here as Eq. (A1).

P0 �
E1�1 � b��1 � p�

r � p � bRN�1 � p�
(A1)

If the return on new investments remains RN forever, the stock price at the end of year t
can be written as Eq. (A2). If the return on new investments changes to R* starting in year
t � 1, whereas the plowback rate (b) and the failure probability (p) remain constant, the stock
price at the end of year t (P*t ) can be written as Eq. (A3).

Pt �
Et�1�1 � b��1 � p�

r � p � bRN�1 � p�
(A2)

P*t �
Et�1�1 � b��1 � p�

r � p � bR*�1 � p�
(A3)

Given that the annual failure probability is p, there is a (1 – p)t chance that the firm will
survive past year t. In addition, only the discounted values of Eqs. (A2) and (A3) impact the
stock price today; the future stock prices must be divided by (1 � r)t to calculate their present
values. Thus, the stock price today, allowing for the shift in the investment return, can be
written as Eq. (A4).

P0 �
E1�1 � b��1 � p�

r � p � bRN�1 � p�
� � (1 � p)

(1 � r)�
t

�P*t � Pt� (A4)

In each of years 1 through t, the firm’s earnings will grow at the annual rate bRN. Thus,
the firm’s earnings in year t � 1 can be written as Eq. (A5).

Et�1 � E1�1 � bRN�t (A5)

If the firm cannot invest in positive net present value projects after year t (that is the typical
assumption in finite growth models), R* equals r. Using this assumption, and substituting Eq.
(A5) into (A4), produces Eq. (A6), which is Eq. (8) in the text.

P0 �
E1�1 � b��1 � p�

r � p � bRN�1 � p� �1 � � (1 � p)(1 � bRN)

(1 � r) � t�
� � (1 � p)(1 � bRN)

(1 � r) � t� E1(1 � b)(1 � p)

r � p � br(1 � p)� (A6)

204 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



Appendix 2

This appendix derives an equation that quantifies the percentage overstatement embedded
in perpetual growth model value estimates for firms that may fail in the future, with an annual
failure probability of p, but will grow at the perpetual annual rate g if they do not fail. The
valuation error is stated as a percentage of the true stock price, calculated by Eq. (7). To do
this, subtract Eq. (7) from Eq. (2), and divide the difference by Eq. (7). This step yields Eq.
(A7), which simplifies to Eq. (A8).

% Overstatement �p � 0; perpetual growth� �

D1

r � g
�

D1�1 � p�

r � p � g�1 � p�

D1�1 � p�

r � p � g�1 � p�

(A7)

% Overstatement �p � 0; perpetual growth� �
r � p � g�1 � p�

�r � g��1 � p�
� 1 (A8)

After rewriting the integer 1 on the right-hand side of Eq. (A8) as 1 � [(r – g)(1 – p)]/
[(r – g)(1 – p)], the equation further simplifies to Eq. (A9), which is Eq. (10) in the text.

% Overstatement �p � 0; perpetual growth� �
p�1 � r�

�r � g��1 � p�
(A9)

Appendix 3

This appendix derives the breakeven return on new investments for a firm with a positive
probability of failure. If a firm’s default probability is zero, the firm’s breakeven reinvest-
ment return will equal the required return. If a firm faces a positive default probability, the
process of deferring dividends through the reinvestment process creates additional default
risk for shareholders. For a firm to maintain value after increasing its plowback ratio, the
expected return on new investments must exceed the cost of capital. The breakeven return on
new investments is the return a firm must earn, in outcomes in which the firm does not
default, to maintain value if the firm reduces its payout ratio.

To illustrate this, assume that Eq. (A1) defines firm value before it changes its plowback
rate b. If the firm increases its plowback rate to b � �, firm value can be written as Eq. (A10).

P*0 �
E1�1 � b � ���1 � p�

r � p � �b � �� RN�1 � p�
(A10)

To solve for the breakeven return on new investments, set Eq. (A1) equal to (A10), and
solve algebraically for RN. This expression simplifies to Eq. (A11), which is Eq. (11) in the
text.

RN�breakeven� �
r � p

1 � p
(A11)

205M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206



References

Block, S. (1999). A study of financial analysts: practice and theory. Financial Analysts Journal, 55, 86–95.
Bradley, M., & Jarrell, G. (2008). Expected inflation and the constant-growth valuation model. Journal of Applied

Corporate Finance, 20, 66–78.
Brealey, R. A., & Myers, S. C. (2003). Principles of Corporate Finance. New York: McGraw-Hill/Irwin.
Danielson, M. (1998). A simple valuation model and growth expectations. Financial Analysts Journal, 54, 50–57.
Danielson, M., Heck, J., & Shaffer, D. (2008). Shareholder theory: How opponents and proponents both get it

wrong. Journal of Applied Finance, 18, 62–66.
Danielson, M., & Scott, J. (2000). Expanding the scope of valuation analysis. Financial Practice and Education,

10, 52–61.
Demirakos, E. G., Strong, N. C., & Walker, M. (2004). What valuation models do analysts use? Accounting

Horizons, 18, 221–240.
Dukes, W., Peng, Z., & English, P. (2006). How do practitioners value common stock? Journal of Investing, 15,

90–104.
Fielitz, B., & Muller, F. (1985). A simplified approach to common stock valuation. Financial Analysts Journal,

41, 35–41.
Foerster S., & Sapp, S. (2005). The dividend discount model in the long-run: A clinical study. Journal of Applied

Finance, 15, 55–75.
Fuller, R., Huberts, L., & Levinson, M. (1993). Returns to E/P strategies, higgledy-piggledy growth, analysts’

forecast errors, and omitted risk factors. Journal of Portfolio Management, 19, 13–24.
Gordon, M. (1962). The Investment, Financing, and Valuation of the Corporation. Homewood, IL: Irwin.
Gordon, J., & Gordon, M. (1997). The finite horizon expected return model. Financial Analysts Journal, 53,

52–61.
Holt, C. (1962). The influence of growth duration on share prices. Journal of Finance, 17, 465–475.
Imam, S., Barker, R., & Clubb, C. (2008). The use of valuation models by UK investment analysts. European

Accounting Review, 17, 503–535.
Lakonishok, J., Shleifer, A., & Vishny, R. (1994). Contrarian investment, extrapolation, and risk. Journal of

Finance, 49, 1541–1578.
Leibowitz, M., & Kogelman, S. (1990). Inside the P/E ratio: The franchise factor. Financial Analysts Journal, 46,

17–35.
Macaulay, F. (1938). Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and

Stock Prices in the United States Since 1856. New York: Columbia University Press.
Mao, J. (1966). The valuation of growth stocks: The investment opportunities approach. Journal of Finance, 21,

95–102.
Miller, M., & Modigliani, F. (1961). Dividend policy, growth, and the valuation of shares. Journal of Business,

34, 411–433.
O’Brien, T. (2003). A simple and flexible DCF valuation model. Journal of Applied Finance, 13, 54–62.
Payne, T., & Finch, J. H. (1999). Effective teaching and use of the constant growth dividend discount model.

Financial Services Review, 8, 283–291.
Shaffer, S. (2006). Corporate failure and equity valuation. Financial Analysts Journal, 62, 71–80.
Sheth, J. (2009). The Self-Destructive Habits of Good Companies. Philadelphia: Wharton School Publishing.
Stigler, G. J. (1963). Capital and Rates of Return in Manufacturing Industry. Princeton, NJ: University Press.
Weston, J. F., Chang, S. C., & Siu, J. (1998). Takeovers, Restructuring, and Corporate Governance. Upper

Saddle River, NJ: Prentice Hall.
Welch, I. (2009). Corporate Finance. New York: Prentice Hall.

206 M.G. Danielson, J.L. Heck / Financial Services Review 23 (2014) 189–206