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A Value Driver Approach to Valuation
Using a Declining Growth Rate Model
by
Larry C. Holland, PhD CFA
University of Arkansas at Little Rock
Little Rock, AR 72204-1099
Email: lcholland@ualr.edu
Telephone: (501) 569-3042
January 16, 2018
2
A Value Driver Approach to Valuation
Using a Declining Growth Rate Model
Abstract
Identifying the value drivers in a valuation analysis provides a robust approach to
valuation, although existing valuation models do not offer much flexibility to include them.
Maintaining consistent accounting relationships is also important. A new declining growth rate
valuation model is used to illustrate the impact of selected value drivers, including consistent
changes in each value driver over time. Two equivalent approaches demonstrate a complete
valuation analysis, using dividends and residual income. Overall, a comprehensive example
illustrates a robust valuation procedure that incorporates changing value drivers, maintains
consistent accounting, and uses a flexible declining growth valuation model.
Key Words: Equity valuation, valuation model, declining growth model, value drivers, and
consistent accounting relationships.
3
A Value Driver Approach to Valuation Using a Declining Growth Rate Model
Analysts use stock valuation models in fundamental analysis to determine a fair
value from the present value of cash flow streams such as dividends, free cash flow to the firm
(FCFF), free cash flow to equity (FCFE), or residual income (RI). However, the cash flows used
for valuation are determined by other factors, which can be identified as the key value drivers in
a firm. For example, the growth in sales, the EBIT profit margin, the invested capital to sales
ratio, and the debt to invested capital ratio are key value drivers from the financial statements of
a firm. Changes in these variables over time will directly affect the intermediate values of the
return on invested capital and return on equity, which in turn determine the level of cash flows
that ultimately lead to a complete valuation analysis. At the same time, it is important to
maintain consistent accounting relationships among the various forecasted cash flows on pro
forma financial statements by choosing appropriate retention and reinvestment rates for the level
of growth assumed. This often will result in different growth rates for sales, EBIT, interest
expense, net income, dividends, invested capital, and debt. In the following sections, a valuation
model is developed that utilizes key value drivers to determine the valuation cash flows for a
firm and also allows for declining growth rates and accounting consistency.
Literature Review
There is a large literature on equity valuation; however, there is much less devoted to
incorporating directly into the valuation analysis the effect of declining growth rates in the
forecasted cash flows, the impact of key value drivers, and maintaining accounting consistency
in the forecast of future cash flows. A survey of the state of the art for this portion of the
4
literature can be summarized from several practitioner oriented books devoted exclusively to
equity valuation: Pinto, Robinson, Stowe (2010), Viebig, Poddig, Varmaz (2008), Damodaran
(2010 and 2012), Koller, Goedhart, Wessels (2010), and Lundholm, Sloan (2004 and 2017).
Pinto, Henry, Robinson, and Stowe (2010) identify dividends, Free Cash Flow to Equity
(FCFE) and Free Cash Flow to the Firm (FCFF) as cash flows that can be used to value a firm
with a discounted cash flow approach. They also illustrate the use of the Gordon constant
dividend growth formula, multi-stage valuation with constant growth segments, and the H-Model
which is a rough estimate of a declining growth model. There is little mention of the value
drivers that influence the valuation or accounting consistency in the forecast of cash flows.
Viebig, Poddig, and Vamaz (2008) highlight several different models and approaches
utilized by practitioners in valuation analysis. They place much emphasis on identifying key
financial value drivers, and note that these value drivers change over time. These changes are
generally incorporated into the analysis through a manual forecast five to ten years into the
future, followed by a terminal value assuming constant mature values of selected value drivers
thereafter. Although several different models are described, those models do not generally
incorporate continuous declining growth rates over time – in some cases, some constant growth
segments are considered in order to account for changing growth rates over time. And finally, in
one case (the Morgan Stanley’s ModelWare approach) an attempt is made to maintain
accounting relationships over time in a forecast of valuation cash flows.
Damodaran (2010 and 2012) shows that companies grow at declining growth rates
according to the business life cycle theory. He accommodates the declining growth rates with a
valuation model that uses multiple constant growth segments that step down the growth rates
followed by a terminal value at the mature stage based on the Gordon constant growth formula.
5
In terms of accounting consistency, the mature growth rate is matched with a mature
reinvestment rate using a formula similar to the sustainable growth rate popularized by Higgins
(1977), which assumes a constant capital structure. He also points out that the choice of which
cash flows to use in a present value approach is a matter of convenience because using
dividends, FCFE, FCFF, or residual income should theoretically yield the same valuation.
Keller, Goedhart, and Wessels (2010) focus on the value drivers of a valuation analysis,
and identify sales growth and Return On Invested Capital (ROIC) as key value drivers. They
define ROIC as NOPLAT divided by Invested Capital. The ROIC can be further divided into an
after-tax profitability margin (NOPLAT/Sales) and a productivity ratio (Sales/IC). They define
NOPLAT as Net Operating Profit Less Adjusted Taxes, which is closely related to EBIT (1-t),
further adjusted for changes in deferred taxes. For a valuation model, they utilize a two-stage
model with an initial constant growth segment (a fixed term annuity) followed by a constant
growth terminal value. The adjustment for changes in growth rates is approximated through
constant growth segments, and there is little attempt to maintain accounting consistency in
estimating future cash flows.
Lundholm and Sloan (2004 and 2017) identify Residual Income (RI), FCFE, and FCFF
as cash flows. As will be mentioned in the next section, they also identify key value drivers, and
incorporate these value drivers into a spreadsheet to forecast 23 years of cash flows, with a
constant growth model as a terminal value at year 23. A major innovation in their approach is
that accounting consistency is maintained throughout their valuation forecasts, which is a feature
not generally included in other valuation approaches (with the exception of the Morgan Stanley
ModelWare). However, they do not provide a closed form solution to their valuation analysis –
the valuation is mostly controlled in a hard-coded manner within the spreadsheet model itself.
6
To summarize, the current state of the art in valuation analysis does not provide a flexible
means to incorporate a continuous declining growth rate in valuation cash flows, which is a
feature of the life cycle theory of a firm. At the same time, consistency with accounting theory is
generally incorporated through assumed step function changes in the reinvestment rate. And
finally, the key value drivers in a valuation analysis are often summarized by an assumption of
an on-going Return on Equity (ROE) or Return on Invested Capital (ROIC) rather than the
components that make up those ratios.
The purpose of this paper is to provide a detailed example of applying a declining growth
valuation model while maintaining consistent accounting relationships among the key forecasted
cash flows. At the same time, key value drivers that change over time are explicitly included in
the valuation analysis, that result in reasonable values of ROE, ROIC, and a reinvestment rate.
These key value drivers are identified in the next section.
Identifying the Key Value Drivers in A Valuation Analysis
The first step in a complete valuation analysis is to identify the factors that are the key
value drivers for a firm. The following paragraphs identify the factors that will be used in an
example valuation analysis in the next three sections of this paper, with verification from the
valuation literature.
A primary factor in driving a valuation analysis is the growth in top line sales. Viebig,
Poddig, and Varmaz (2008) state that “Top line growth is arguably one of the most important
value drivers of a firm. Modeling future revenues starts with carefully analyzing historical
revenues recorded on a company’s income statements.” (p. 59). They continue this thought with
7
the idea that sales growth would also be expected to decline over time with their statement,
“Initial high revenue growth rates usually slow down when the revenue base increases (“base
effect”) and when companies enter into a more mature stage of their life cycle.” (p. 62).
Therefore, initial sales growth and a decline in this growth rate over time would be an important
key value driver to begin a valuation analysis.
Lundholm and Sloan (2004 and 2017) verify the importance of the growth in sales with
their statement, “the growth in sales is the key long-term driver of growth in all other metrics.”
(2004, p. 78). They also point out that the growth rates in key cash flows will be different from
the growth in sales because of several key variables, with bracketed comments added for
emphasis: “Asset growth will differ from sales growth when a firm’s level of assets that is
required to generate a given level of sales [Assets/Sales] is also changing. Common Equity
growth will differ from sales growth because of changes in both the amount of assets
[Assets/Sales] and the amount of debt that is used to finance the assets that generate the sales
[Debt/Assets]. Earnings growth will differ from sales growth because of changes in the firm’s
profit margin [EBIT/Sales].” Thus, they have generally identified three additional value drivers,
which they expect to change over time: Assets/Sales, Debt/Assets, and EBIT/Sales.
Piotroski (2000) identifies several key factors in the construction of his F-SCORE.
Among others, this includes the change in operating margin (EBIT/Sales), turnover
(Sales/Assets), and the relative level of debt (Debt/Assets). Also, Soliman (2008) relates key
value factors directly to the simple DuPont relationship, extending the work of Fairfield and
Yohn (2001), Nissim and Penman (2001 and 2002), and Penman and Zhang (2002). The simple
DuPont formula is presented in numerous textbooks (e.g. Ross, Westerfield, Jordan (2006), Berk,
DeMarzo, and Harford (2015), Graham, Smart, and Megginson (2010), Parrino and Kidwell
8
(2017), Block and Hirt (2015), Cornett and Nofsinger (2006), and others), and decomposes the
return on equity (ROE) as follows:
𝑅𝑂𝐸 = (𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒
𝑆𝑎𝑙𝑒𝑠) (
𝑆𝑎𝑙𝑒𝑠
𝐴𝑠𝑠𝑒𝑡𝑠) (
𝐴𝑠𝑠𝑒𝑡𝑠
𝐸𝑞𝑢𝑖𝑡𝑦) (1)
Soliman (2008) specifically focuses on return on net operating assets (NOA) rather than total
assets in the simple DuPont formula. The finance literature more often utilizes a concept similar
to NOA, which is invested capital (IC), defined as total assets less non-operating current
liabilities. Through the DuPont relationship, Soliman identifies profitability (defined as
operating income or EBIT divided by NOA) and total asset turnover (Sales divided by Assets) as
key factors, along with financial leverage.
Lundholm and Sloan (2017) and others also define an extended DuPont formula that
decomposes ROE more cleanly into the return on invested capital (ROIC) based on operating
earnings and the financial leverage. One form of the extended DuPont formula is
𝑅𝑂𝐸 = (𝐸𝐵𝐼𝑇 (1 − 𝑡)
𝑆𝑎𝑙𝑒𝑠) (
𝑆𝑎𝑙𝑒𝑠
𝐼𝐶) (1 −
𝐼𝑁𝑇
𝐸𝐵𝐼𝑇) (
𝐼𝐶
𝐸𝑞𝑢𝑖𝑡𝑦) (2)
The first two terms are equal to the ROIC, and the last two terms incorporate the reduction in
leverage from additional interest expense and the increase in leverage from additional debt.
Overall, the extended DuPont relationship identifies several value drivers in common with the
references cited earlier.
Taking into account all the cited references and the DuPont relationships, typical value
drivers in a valuation analysis would include the growth in top line sales, the operating
9
profitability, the production efficiency, and the financial leverage. In addition, some analysts
specify that the ROE over time should approach the required return on equity during maturity.
For example, Viebig and Poddig (2008) make a point about the effect of competition on the
return on capital when they state, “under perfect competition return on capital fades to cost of
capital over time”, p. 71. The return on capital relates to the ROE with the addition of financial
leverage, i.e. increasing the level of debt increases the ROE. As a result, the approach used in
this paper will demonstrate a method for choosing a long-term financial leverage (Debt/IC) that
will result in the long-term ROE approaching the required return on equity.
In the following sections, an example problem will illustrate the application of key value
drivers to valuation analysis using a declining growth valuation model. In some cases, a ratio is
inverted in comparison to references in the literature to simplify the application of the model.
The following variables are specified as value drivers, all of which will be allowed to decline
over time to a mature level: (1) Growth in sales, (2) Profitability (EBIT/Sales), (3) Production
efficiency (IC/Sales), and (4) Relative level of debt (Debt/IC) which can be specified such that
the ROE will decline asymptotically to the Required Rate of Return. These four value drivers
will be utilized to develop future cash flows that maintain consistent accounting relationships
and lead to an estimate of future potential dividends (i.e., Free Cash Flow to Equity)1 as a plug
factor in a pro forma analysis. And finally, two equivalent approaches will be used to illustrate a
complete and comprehensive valuation analysis – the present value of future dividends and the
present value of residual income.
1 Free Cash Flow to Equity (FCFE) represents the level of dividends that potentially could be
paid with a balance in the level of internal cash flows of the firm. A firm could choose to pay
dividends less that the FCFE and accumulate excess cash, or pay dividends higher than the FCFE
supported by additional funding such as borrowing more debt.
10
A Declining Growth Valuation Model
The model used to illustrate valuation with a value driver approach is a model developed
by Holland (2017), which incorporates a declining growth function. This model is based on the
subtraction of two cash flow streams that grow at different constant rates. The simplest
formulation of the model assumes a constant growth for a larger cash flow stream at a long-term
mature growth rate and a zero growth for a second smaller cash flow stream, as follows:
𝐶𝑡 = 𝐶0[(𝐻𝐶)(1 + 𝑔𝐿)𝑡 − (𝐻𝐶 − 1)] (3)
or
𝐶𝑡 = 𝐶0 [(𝑔𝑆
𝑔𝐿) (1 + 𝑔𝐿)𝑡 − (
𝑔𝑆
𝑔𝐿− 1)] (4)
where Ct = a cash flow at time t which is a component of the cash flows to be valued,
HC = the declining growth factor related to cash flow Ct, defined as gS/gL,
gS = the initial growth rate of Ct from time 0 to time 1, and
gL = the long-term mature growth rate.
The cash flow stream to be valued in this model (Ct) will have an initial growth rate of gS from
time 0 to time 1 and the growth rate will decline over time, asymptotically approaching a long-
term mature growth rate of gL. The present value of the two component cash flow streams can
be found by using the constant growth valuation model on the first term and the present value of
a perpetuity for the second term. Thus, the valuation model with a declining growth rate
function is as follows:
𝑉0 = 𝐶0 ((𝐻𝐶)(1 + 𝑔𝐿)
𝑅 − 𝑔𝐿 −
(𝐻𝐶 − 1)
𝑅) (5)
11
or
𝑉0 = 𝐶0 ( (
𝑔𝑆
𝑔𝐿) (1 + 𝑔𝐿)
𝑅 − 𝑔𝐿 −
(𝑔𝑆
𝑔𝐿− 1)
𝑅) (6)
where R = the required rate of return for the cash flow stream, Ct.
A nice feature of this model is that one cash flow series in a valuation analysis can easily be
related to a second multiplicative or additive cash flow series while specifying a function of
changing growth rates. This allows a significant level of flexibility in forecasting pro forma cash
flows that incorporate accounting consistency as well as changes in the value drivers. For
example, the components of the financial statements of a firm (that are additive factors) can
easily be modeled, which allows consistent accounting relationships to be maintained over time
in a forecast of cash flows. Therefore, relationships such as a clean accounting surplus can be
maintained as well as a reinvestment or payout choice for a firm. At the same time, the key
value drivers (which are multiplicative factors) can be specified as a function that changes over
time. This allows for a robust valuation with changing growth rates for multiple intermediate
cash flow streams leading to an ultimate overall valuation analysis.
Modeling the Value Drivers
An example problem with complete financial statements is utilized in this section to
illustrate how the four value drivers identified earlier can be directly incorporated into a
valuation analysis. The income statement and balance sheet for this example are shown in the
12
first column of Table 1. Let Sales follow a growth path with an initial growth rate of gS = 24%
from time t=0 to time t=1, and which declines over time to a long-term growth rate of gL = 2%.
Following Equation 3, a declining growth rate series can be established as an estimate of future
Sales, as follows:
𝐻𝑆𝑎𝑙𝑒𝑠 = 𝑔𝑆
𝑔𝐿 =
24%
2% = 12 (7)
𝑆𝑎𝑙𝑒𝑠𝑡 = 𝑆𝑎𝑙𝑒𝑠0[𝐻𝑆𝑎𝑙𝑒𝑠(1 + 𝑔𝐿)𝑡 − (𝐻𝑆𝑎𝑙𝑒𝑠 − 1)] (8)
𝑆𝑎𝑙𝑒𝑠𝑡 = 10 [ 12 (1.02)𝑡 − 11] = 120 (1.02)𝑡 − 110 (9)
This Sales forecast has a growth path over the next 30 as shown in Figure 1. Table 1 shows the
first four years of the forecast while holding the other value drivers constant.
Changes in the other three value drivers can also be accommodated within the declining
growth model. For example, another value driver identified earlier is the profitability, or the
EBIT margin (m = EBIT/Sales). Assume that the EBIT margin is forecast to decline over time
from the current short-term margin of m0 = EBIT0/Sales0 to a long-term margin of mL =
EBITL/SalesL. Note that Sales multiplied times the margin will yield an estimate of EBIT.
Appendix 1 shows that this multiplicative relationship can be modeled from the Sales estimate
while at the same time changing the EBIT margin. For the example problem, assume that the
EBIT margin declines from the current m0 = 15% to a long-term margin of mL = 10%. From
Appendix 1, future EBIT can be modeled from Sales while the margin declines over time as
𝐻𝐸𝐵𝐼𝑇 = 𝑔𝑆
𝑔𝐿
𝑚𝐿
𝑚0 = 𝐻𝑆𝑎𝑙𝑒𝑠
𝑚𝐿
𝑚0 = 12 (
10%
15%) = 8 (10)
13
𝐸𝐵𝐼𝑇𝑡 = 𝐸𝐵𝐼𝑇0 [𝐻𝐸𝐵𝐼𝑇(1 + 𝑔𝐿)𝑡 − (𝐻𝐸𝐵𝐼𝑇 − 1)] (11)
𝐸𝐵𝐼𝑇𝑡 = 1.25 [ 8 (1.02)𝑡 − 7] = 12 (1.02)𝑡 − 10.50 (12)
Note that Equation 10 is equivalent to a declining series for EBIT with an initial growth of 16%
declining to a long-term growth of 2%. The initial growth of EBIT is simply scaled down from
the initial growth in Sales by a ratio of the decline in EBIT margin (i.e., 24% x10%/15% = 16%).
The growth path over the next 30 years for EBIT is also shown in Figure 1.
Other value drivers can be estimated in a similar manner. For example, another value
driver identified earlier is the production efficiency (or capital intensity). Assume that the
production efficiency (p = IC/Sales) changes from p0 = IC0/Sales0 to pL = ICL/SalesL gradually
over time while at the same time Sales is following a declining growth pattern. For example, in
the same manner as the estimate of future EBIT, it can be shown that an increase in the future
capital intensity from p0 = 0.80 to pL = 1.20 can be estimated as
𝐻𝐼𝐶 = 𝑔𝑆
𝑔𝐿
𝑝𝐿
𝑝0 = 𝐻𝑆𝑎𝑙𝑒𝑠
𝑝𝐿
𝑝0 = 12 (
1.20
0.80) = 18 (13)
𝐼𝐶𝑡 = 𝐼𝐶0 [𝐻𝐼𝐶(1 + 𝑔𝐿)𝑡 − (𝐻𝐼𝐶 − 1)] (14)
𝐼𝐶𝑡 = 8 [18 (1.02)𝑡 − 17] = 144 (1.02)𝑡 − 136 (15)
Note that HIC = 18 in Equation 13 is equivalent to a cash flow stream with an initial growth of
18 (2%) = 36% per year declining asymptotically to a growth of 2% per year.
14
Following the same procedure as before, the level of Debt over time can be estimated as
an increase in the Debt to Invested Capital ratio or financial leverage (f = Debt/IC). Assume the
financial leverage changes from f0 = Debt0/IC0 to f0 = Debt0/IC0. Any reasonable level of debt
can be specified at this point. However, suppose an analyst prefers to assume the ROE will
decline asymptotically to the required rate of return over the long run. Appendix 2 shows that
the long-term level of debt that will result in the ROE declining over time to the required rate of
return (while the EBIT margin and IC/Sales changes as shown earlier) can be determined as
𝐷𝑒𝑏𝑡𝐿
𝐼𝐶𝐿 =
[𝑅𝑂𝐸𝐿 − 𝑚𝐿 (1 − 𝑡)
𝑝𝐿]
𝑅𝑂𝐸𝐿 − 𝑖 (1 − 𝑡)
(16)
Using the numbers in our example, this establishes a long-term Debt/IC ratio of
𝐷𝑒𝑏𝑡𝐿
𝐼𝐶𝐿 =
[0.10 − 0.10 (1 − 0.20))
1.2 ]
0.10 − 0.0625 (1 − 0.20) =
0.0333
0.05 = 0.6667
(17)
An increase in the financial leverage (Debt/IC) from f0 = 0.50 to fL = 0.6667 while the
Invested Capital and Sales are also changing can be estimated as
𝐻𝐷𝑒𝑏𝑡 = 𝑔𝑆
𝑔𝐿
𝑝𝐿
𝑝0 𝑓𝐿
𝑓0 = 𝐻𝐼𝐶
𝑓𝐿
𝑓0 = 18 (
0.6667
0.50) = 24 (18)
𝐷𝑒𝑏𝑡𝑡 = 𝐷𝑒𝑏𝑡0 [𝐻𝐷𝑒𝑏𝑡(1 + 𝑔𝐿)𝑡 − (𝐻𝐷𝑒𝑏𝑡 − 1)] (19)
𝐷𝑒𝑏𝑡𝑡 = 4 [24 (1.02)𝑡 − 23] = 96 (1.02)𝑡 − 92 (20)
Again note that HDebt = 24 in Equation 18 is equivalent to the Debt growing at an initial rate of
48% per year and declining asymptotically to 2% per year.
15
Given the above forecasts for Sales, EBIT, Invested Capital (IC), and Debt, the four value
drivers have been defined and allowed to change over time. The trend over the next 20 years in
the profitability margin (EBIT/Sales), the capital productivity (IC/Sales), and the financial
leverage (Debt/IC) is shown in Figures 2, 3, and 4. Figure 5 also shows that the Return on
Equity is forecast to decline asymptotically from 25% to the required return on equity of 10%.
Maintaining Accounting Consistency in the Analysis
The forecast of Debt in Equation 20 makes possible a forecast of Interest Expense. If the
interest rate on debt is assumed to be constant, interest expense will remain a constant proportion
of Debt (0.25/4.0 = 6.25%). This means that Interest Expense can be estimated as
𝐼𝑁𝑇𝑡 = 𝐼𝑁𝑇0 [𝐻𝐷𝑒𝑏𝑡(1 + 𝑔𝐿)𝑡 − (𝐻𝐷𝑒𝑏𝑡 − 1)] (21)
𝐼𝑁𝑇𝑡 = 0.25 [24 (1.02)𝑡 − 23] = 6 (1.02)𝑡 − 5.75 (22)
Given the above individual components of EBIT and Interest Expense, Net Income (NI) can be
estimated by subtracting Equation 22 (INT) from Equation 12 (EBIT) and multiplied by one
minus the tax rate. Given a tax rate of 20% yields
𝑁𝐼𝑡 = (𝐸𝐵𝐼𝑇𝑡 − 𝐼𝑁𝑇𝑡)(1 − 𝑡) (23)
𝑁𝐼𝑡 = ([12 (1.02)𝑡 − 10.50] − [6 (1.02)𝑡 − 5.75])(1 − 0.20) (24)
𝑁𝐼𝑡 = 4.8 (1.02)𝑡 − 3.8 (25)
16
Note that the forecast for Net Income is equivalent to a cash flow stream with an initial growth
rate of 4.8 (2%) = 9.6% per year declining asymptotically to 2% per year. The growth path for
Net Income over the next 30 years is again shown in Figure 1.
The last step in preparing a pro forma income statement is to subtract the Additions to
Retained Earnings (ARE) from NI to arrive at the potential Dividends (or FCFE) paid out to
investors. If a clean surplus relationship is assumed and no new equity is issued, then the
increase in Equity will be the amount necessary to meet the Debt/IC assumption. This means
that the change in Equity will equal the ARE subtracted from NI, and potential Dividends (or
FCFE) will be determined as a plug factor.
Equity over time (EQt) is defined as ICt minus Debtt. Substituting ICt from Equation 14
and Debtt from Equation 19 yields
𝐸𝑄𝑡 = 𝐼𝐶𝑡 − 𝐷𝑒𝑏𝑡𝑡
𝐸𝑄𝑡 = 𝐼𝐶0 [𝐻𝐼𝐶 (1 + 𝑔𝐿)𝑡 − (𝐻𝐼𝐶 − 1)] − 𝐷𝑒𝑏𝑡0[𝐻𝐷𝑒𝑏𝑡 (1 + 𝑔𝐿)𝑡 − (𝐻𝐷𝑒𝑏𝑡 − 1)]
(26)
The change in Equity (∆EQ) is the difference between EQt+1 and EQt.
𝐸𝑄𝑡+1 = 𝐼𝐶0[𝐻𝐼𝐶(1 + 𝑔𝐿)𝑡+1 − (𝐻𝐼𝐶 − 1)] − 𝐷𝑒𝑏𝑡0[𝐻𝐷𝑒𝑏𝑡(1 + 𝑔𝐿)𝑡+1 − (𝐻𝐷𝑒𝑏𝑡 − 1)] (27)
𝐸𝑄𝑡 = 𝐼𝐶0 [𝐻𝐼𝐶 (1 + 𝑔𝐿)𝑡 − (𝐻𝐼𝐶 − 1)] − 𝐷𝑒𝑏𝑡0 [𝐻𝐷𝑒𝑏𝑡 (1 + 𝑔𝐿)𝑡 − (𝐻𝐷𝑒𝑏𝑡 − 1)] (28)
Subtracting EQt from EQt+1 and simplifying yields
∆𝐸𝑄𝑡 = 𝐸𝑄𝑡 − 𝐸𝑄𝑡−1 = (𝐼𝐶0 𝐻𝐼𝐶 − 𝐷𝑒𝑏𝑡0 𝐻𝐷𝑒𝑏𝑡) (1 + 𝑔𝐿)𝑡 𝑔𝐿
(1 + 𝑔𝐿) (29)
Applying the numbers in the example yields
∆𝐸𝑄𝑡 = (8 (18) − 4 (24)) (1.02)𝑡 . 02
(1.02) = 0.941176 (1.02)𝑡 (30)
17
The Dividend (DIVt) is equal to Net Income (NIt) minus the Additions to Retained Earnings
(AREt). Again assuming a clean surplus relationship and no new equity, then the ∆EQt = AREt.
Subtracting Equation 30 from Equation 25 yields
𝐷𝐼𝑉𝑡 = 𝑁𝐼𝑡 − 𝐴𝑅𝐸𝑡 = 𝑁𝐼𝑡 − ∆𝐸𝑄𝑡
(31)
𝐷𝐼𝑉𝑡 = [4.8 (1.02)𝑡 − 3.8] − [0.941176 (1.02)𝑡] (32)
𝐷𝐼𝑉𝑡 = 3.858824 (1.02)𝑡 − 3.8 (33)
Equation 33 indicates that the dividend cash flow stream of this firm can be simulated with two
cash flow streams: One cash flow stream with an initial value of 3.5858824 that grows at a
constant rate of 2% per year, and a second cash flow stream that remains constant at -3.8 each
year into perpetuity. These two cash flow streams can be valued easily with the constant growth
model and a perpetuity. Therefore, from Equation 33, a fair value for this stock given Sales
declining in growth from 24% to 2%, the EBIT margin declining from 15% to 10%, the asset
intensity increasing from 0.80 to 1.2, the Debt ratio increasing from 50% to 66.67%, and a
required rate of return on equity of 10% evaluates at
𝑉0 = 3.5858824 (1.02)
. 10 − .02 −
3.8
. 10 = 49.2 − 38 = 11.2 (34)
Residual Income Approach to Valuation
Another approach equivalent to a valuation based on the present value of future dividends
is the residual income approach. In this case, the total valuation of equity is the current book
value of equity added to the present value of future residual income, as follows:
18
𝑉0 = 𝐸𝑄0 + ∑𝑅𝐼𝑡
(1 + 𝑅𝐸)𝑡
∞
𝑡=1
(35)
𝑅𝐼𝑡 = 𝑁𝐼𝑡 − 𝑅𝐸 𝐸𝑄𝑡−1 (36)
where RIt = Residual Income at time t, and
RE = the required rate of return on equity.
The equity over time was defined earlier in Equation 26. Applying numbers from the example
problem (including HIC=18 from Equation 13 and HDebt=24 from Equation 18) yields
𝐸𝑄𝑡 = 𝐼𝐶𝑡 − 𝐷𝑒𝑏𝑡𝑡
𝐸𝑄𝑡 = 𝐼𝐶0 [𝐻𝐼𝐶 (1 + 𝑔𝐿)𝑡 − (𝐻𝐼𝐶 − 1)] − 𝐷𝑒𝑏𝑡0[𝐻𝐷𝑒𝑏𝑡 (1 + 𝑔𝐿)𝑡 − (𝐻𝐷𝑒𝑏𝑡 − 1)]
(37)
𝐸𝑄𝑡 = 8 [18 (1.02)𝑡 − (18 − 1)] − 4[24 (1.02)𝑡 − (24 − 1)] (38)
𝐸𝑄𝑡 = 48 (1.02)𝑡 − 44 (39)
From the definition of residual income in Equation 36 and net income from Equation 25,
𝑅𝐼𝑡 = 𝑁𝐼𝑡 − 𝑅𝐸 𝐸𝑄𝑡−1 (40)
𝑅𝐸 𝐸𝑄𝑡−1 = (0.10)[48 (1.02)𝑡−1 − 6] = 4.8 (1.02)𝑡−1 − 4.4 (41)
𝑅𝐼𝑡 = [4.8 (1.02)𝑡 − 3.8] − [4.8 (1.02)𝑡−1 − 4.4] (42)
𝑅𝐼𝑡 = 0.941176 (1.02)𝑡 + 0.6 (43)
The present value of the residual income cash flow stream can be determined with a constant
growth model for the first term in Equation 43 and a perpetuity for the second term. For a
19
valuation of the equity of this firm, adding the beginning book value of equity to the present
value of the residual income according to Equation 35 yields
𝑉0 = 4.0 + 0.941176 (1.02)
. 10 − .02 +
0.6
. 10 = 4 + 1.2 + 6 = 11.2 (44)
Note that this yields the same result as the present value of future dividends. As a point of
comparison, Figure 6 shows the trend of the present values in the two approaches to valuation.
Note that the present value of dividends increases at first, as the growth rate exceeds the required
return on equity. However, a peak is reached in year 11 when the growth in the dividend equals
the required rate of return, and declines thereafter as the growth rate continues to decline. Also
note in Figure 7 that the retention rate declines from a high of 0.9412 to 0.1964 at maturity. This
means that the firm retains more dividends while the growth rate is high, but gradually decreases
the retention as the growth rate declines into maturity. These charts indicate that the cash flows,
key value drivers, and resultant ROE and reinvestment rates are all reasonable as the growth
rates decline and accounting consistency is maintained.
Summary
This paper adds several contributions to the literature on equity valuation. The first
contribution is to demonstrate a procedure for incorporating the effect of key value drivers
directly into a valuation analysis. An example valuation problem was used to incorporate a
declining growth in sales and three other factors identified from the valuation literature and the
DuPont relationships. As an added feature, a procedure was included for choosing an
appropriate long-term capital structure (i.e., Debt/IC) such that the ROE declines asymptotically
20
to the required rate of return on equity over time. A second contribution is to demonstrate the
flexibility of a new declining growth valuation model. The growth in sales was modeled to
decrease from the current level to a mature, long-term growth rate comparable to the growth in
the overall economy. In addition, the other value drivers in the valuation analysis were allowed
to change over time to a level appropriate for a mature firm. A third contribution is to
demonstrate a procedure for maintaining consistency in accounting relationships over time for
the estimates of future cash flows. This means that the estimates for future cash flows are
consistent with accounting theory in the preparation of future pro forma income statements and
balance sheets. And finally, reasonable values for ROE, ROIC, and the reinvestment rate are a
result of the assumed changes in value drivers.
In summary, this paper demonstrates the process of completing a robust valuation
analysis using a declining growth valuation model while maintaining consistent accounting and
incorporating key value drivers.
21
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23
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24
Table 1: Four-Year Forecast for Example Financial Statements: Sales Growth Only
Year 1 Year 2 Year 3 Year 4
Actual g1 = 24% g2 = 19.74% g3 = 16.82% g4 = 14.68%
Income Statement
Sales 10.00 12.40 14.85 17.34 19.89
CGS 7.50
SG&A 0.50
Depreciation 0.50
EBIT 1.50 1.86 2.23 2.60 2.98
Interest Expense 0.25 0.31 0.37 0.37 0.43
EBT 1.25 1.55 1.86 2.17 2.49
Taxes @ 40% 0.25 0.31 0.37 0.43 0.50
Net Income 1.00 1.24 1.48 1.73 1.99
Dividends 0.06 0.28 0.51 0.74 0.97
Additions to RE 0.94 0.96 0.98 1.00 1.02
FCFF
Retention Rate (b) 0.9412 0.7742 0.6595 0.5758 0.5121 Balance Sheet
Cash 0.25 0.31 0.38 0.39 0.40
Accounts Receivable 1.00 1.24 1.54 1.57 1.60
Inventory 1.25 1.55 1.92 1.96 2.00
Current Assets 2.50 3.10 3.71 4.34 4.97
Net Fixed Assets 6.50 8.06 9.65 11.27 12.93
Total Assets 9.00 11.16 13.36 15.61 17.90 Current Liabilities 1.00 1.24 1.48 1.73 1.99
LT Debt 4.00 4.96 5.94 6.94 7.96
Common Stock 0.50 0.50 0.50 0.50 0.50
Retained Earnings 3.50 4.46 5.44 6.44 7.46
Total L&OE 9.00 11.16 13.36 15.61 17.90 Invested Capital
LT Debt 4.00 4.96 5.94 6.94 7.96
Common Stock 0.50 0.50 0.50 0.50 0.50
Retained Earnings 3.50 4.46 5.44 6.44 7.46
Total Equity 4.00 4.96 5.94 6.94 7.96
Invested Capital 8.00 9.92 11.88 13.88 15.91
ROIC 15.00% 15.00% 15.00% 15.00% 15.00%
Return on Equity 25.00% 25.00% 25.00% 25.00% 25.00%
Debt/IC Ratio 50.00% 50.00% 50.00% 50.00% 50.00%
EBIT/Sales 15.00% 15.00% 15.00% 15.00% 15.00%
IC/Sales 0.8000 0.8000 0.8000 0.8000 0.8000
25
Table 2: Four-Year Forecast for Example Financial Statements: Value Drivers Change
Year 1 Year 2 Year 3 Year 4
Actual g1 = 24% g2 = 19.74% g3 = 16.82% g4 = 14.68%
Income Statement
Sales 10.00 12.40 14.85 17.34 19.89
CGS 7.50
SG&A 0.50
Depreciation 0.50
EBIT 1.50 1.74 1.98 2.23 2.49
Interest Expense 0.25 0.37 0.49 0.62 0.74
EBT 1.25 1.37 1.49 1.62 1.74
Taxes @ 40% 0.25 0.27 0.30 0.30 0.32
Net Income 1.00 1.10 1.19 1.29 1.40
Dividends 0.06 0.14 0.21 0.29 0.38
Additions to RE 0.94 0.96 0.98 1.00 1.02
FCFF
Retention Rate (b) 0.9412 0.8759 0.8202 0.7720 0.7299 Balance Sheet
Cash 0.25 0.31 0.38 0.39 0.40
Accounts Receivable 1.00 1.24 1.54 1.57 1.60
Inventory 1.25 1.55 1.92 1.96 2.00
Current Assets 2.50 3.50 3.52 4.52 5.56
Net Fixed Assets 6.50 8.84 11.23 13.66 16.14
Total Assets 9.00 12.24 15.54 18.92 22.35 Current Liabilities 1.00 1.36 1.73 2.10 2.48
LT Debt 4.00 5.92 7.88 9.88 11.91
Common Stock 0.50 0.50 0.50 0.50 0.50
Retained Earnings 3.50 4.46 5.44 6.44 7.46
Total L&OE 9.00 12.24 15.54 18.92 22.35 Invested Capital
LT Debt 4.00 5.92 7.88 9.88 11.91
Common Stock 0.50 0.50 0.50 0.50 0.50
Retained Earnings 3.50 4.46 5.44 6.44 7.46
Total Equity 4.00 4.96 5.94 6.94 7.96
Invested Capital 8.00 10.88 13.82 16.81 19.87
ROIC 15.00% 12.79% 11.49% 10.63% 10.02%
Return on Equity 25.00% 22.93% 21.34% 20.07% 19.04%
EBIT/Sales 15.00% 14.03% 13.37% 12.88% 12.51%
IC/Sales 0.8000 0.8774 0.9306 .9694 .9989
Debt/IC 50.00% 54.41% 57.02% 58.74% 59.96%
26
Appendix 1
Derive a Multiplicative Declining Growth Factor, HEBIT
Establish a multiplicative relationship between Sales and EBIT, where
𝐸𝐵𝐼𝑇𝑡 = 𝑚𝑡 𝑆𝑎𝑙𝑒𝑠𝑡 (45)
From Equation 8,
𝑆𝑎𝑙𝑒𝑠𝑡 = 𝑆𝑎𝑙𝑒𝑠0[𝐻𝑆𝑎𝑙𝑒𝑠[(1 + 𝑔𝐿)𝑡 − 1] + 1] (46)
Let EBIT also follow a declining growth pattern,
𝐸𝐵𝐼𝑇𝑡 = 𝐸𝐵𝐼𝑇0[𝐻𝐸𝐵𝐼𝑇[(1 + 𝑔𝐿)𝑡 − 1] + 1] (47)
Substituting Equation 45 into Equation 47 yields
𝑚𝑡 𝑆𝑎𝑙𝑒𝑠𝑡 = 𝑚0 𝑆𝑎𝑙𝑒𝑠0[𝐻𝐸𝐵𝐼𝑇[(1 + 𝑔𝐿)𝑡 − 1] + 1] (48)
Dividing Equation 48 by Equation 46 yields
𝑚𝑡 𝑆𝑎𝑙𝑒𝑠𝑡
𝑆𝑎𝑙𝑒𝑠𝑡 = 𝑚𝑡 = 𝑚0
𝑆𝑎𝑙𝑒𝑠0[𝐻𝐸𝐵𝐼𝑇[(1 + 𝑔𝐿)𝑡 − 1] + 1]
𝑆𝑎𝑙𝑒𝑠0[𝐻𝑆𝑎𝑙𝑒𝑠[(1 + 𝑔𝐿)𝑡 − 1] + 1] (49)
As t gets very large, the effect of adding 1 to the numerator and denominator diminishes to zero,
and mt approaches mL. Therefore, taking the limit as t approaches infinity yields
lim𝑡→∞
𝑚𝑡 = 𝑚𝐿 = 𝑚0
𝐻𝐸𝐵𝐼𝑇
𝐻𝑆𝑎𝑙𝑒𝑠 (50)
Solving for HEBIT yields
𝐻𝐸𝐵𝐼𝑇 =
𝑔𝑆
𝑔𝐿 𝑚𝐿
𝑚0 (51)
27
Appendix 2
Derive the Debt/IC Which Causes ROE to Approach RE
The Return on Equity (ROE) is defined as Net Income divided by Equity. Substituting for the
definitions of Net Income and Equity yields
𝑅𝑂𝐸𝑡 =
𝑁𝐼𝑡
𝐸𝑄𝑡 =
(𝐸𝐵𝐼𝑇𝑡 − 𝐼𝑁𝑇𝑡) (1 − 𝑡)
𝐼𝐶𝑡 − 𝐷𝑒𝑏𝑡𝑡 (52)
Define Interest as a constant yield (i) times the level of Debt, or
𝑅𝑂𝐸𝑡 =
(𝐸𝐵𝐼𝑇𝑡 − 𝑖 𝐷𝑒𝑏𝑡𝑡) (1 − 𝑡)
𝐼𝐶𝑡 − 𝐷𝑒𝑏𝑡𝑡 (53)
Multiplying both sides by the denominator of ICt – Debtt yields
𝑅𝑂𝐸𝑡 (𝐼𝐶𝑡 − 𝐷𝑒𝑏𝑡𝑡) = (𝐸𝐵𝐼𝑇𝑡 − 𝑖 𝐷𝑒𝑏𝑡𝑡) (1 − 𝑡) (54)
Dividing both sides by ICt yields
𝑅𝑂𝐸𝑡 (1 −
𝐷𝑒𝑏𝑡𝑡
𝐼𝐶𝑡) = (
𝐸𝐵𝐼𝑇𝑡
𝐼𝐶𝑡− 𝑖
𝐷𝑒𝑏𝑡𝑡
𝐼𝐶𝑡) (1 − 𝑡) (55)
Solving for (Debtt/ICt) yields
𝐷𝑒𝑏𝑡𝑡
𝐼𝐶𝑡 =
[𝑅𝑂𝐸𝑡 − 𝐸𝐵𝐼𝑇𝐿 (1 − 𝑡)
𝐼𝐶𝐿]
𝑅𝑂𝐸𝑡 − 𝑖 (1 − 𝑡)
(56)
Dividing EBIT and IC by Sales in the numerator of the term on the right side yields
𝐷𝑒𝑏𝑡𝑡
𝐼𝐶𝑡 =
[𝑅𝑂𝐸𝑡 −
𝐸𝐵𝐼𝑇𝑡
𝑆𝑎𝑙𝑒𝑠𝑡 (1 − 𝑡)
𝐼𝐶𝑡
𝑆𝑎𝑙𝑒𝑠𝑡
]
𝑅𝑂𝐸𝑡 − 𝑖 (1 − 𝑡)
(57)
Substituting the definition of m=(EBIT/Sales) and p=(IC/Sales), and evaluating at time L yields
𝐷𝑒𝑏𝑡𝐿
𝐼𝐶𝐿 =
[𝑅𝑂𝐸𝐿 − 𝑚𝐿 (1 − 𝑡)
𝑝𝐿]
𝑅𝑂𝐸𝐿 − 𝑖 (1 − 𝑡)
(58)
28
Figure 1
Growth Rate of Sales, EBIT, and Net Income
Declining Towards a 2% Long-Term Growth Rate
Figure 2
Trend in EBIT/Sales
Declining Asymptotically from 15% To 10%
0%
5%
10%
15%
20%
25%
0 5 10 15 20 25 30
Sales
EBIT
NI
10%
11%
12%
13%
14%
15%
0 5 10 15 20
29
Figure 3
Trend in IC/Sales
Increasing Asymptotically from 0.8 To 1.2
Figure 4
Trend in Debt/IC
Increasing Asymptotically from 0.50 To 0.6667
0.8
0.9
0.9
1.0
1.0
1.1
1.1
1.2
1.2
0 5 10 15 20
0.50
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0 5 10 15 20
30
Figure 5
Trend in ROE and ROIC
ROE Declining from 25% To 10%
ROIC Declining from 15% to 6.667%
Figure 6
Present Value of Dividends and Residual Income
(Required Return = 10%)
0%
5%
10%
15%
20%
25%
0 5 10 15 20 25 30 35 40
ROE
ROIC
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 10 20 30 40 50 60 70 80
RI
DIV
31
Figure 7
Trend in the Retention Rate (b)
Decreasing from 0.9412 to 0.1964
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 20 40 60 80 100
Recommended