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

References

Berk, Jonathan, Peter DeMarzo, and Jarrad Harford (2015), Fundamentals of Corporate Finance

3rd Ed., Pearson Education, One Lake Street, Upper Saddle River, NJ 07458.

Block, Stanley and Geoffrey Hirt (2017), Foundations of Financial Management 16th Ed.,

McGraw-Hill Irwin, 1221 Avenue of the Americas, New York, NY 10020.

Cornett, M., Adair, T., and Nofsinger, J. (2012), Finance: Applications & Theory, 2ed., McGraw

Hill, New York, N.Y.

Damodaran, Aswath (2010), The Dark Side of Valuation 2nd Ed., Pearson Education, Inc. Upper

Saddle River, New Jersey 07458.

Damodaran, Aswath (2012), Investment Valuation 3rd University Ed., John Wiley and Sons, 111

River Street, Hoboken, NJ 07030.

Fairfield, P. an T. Yohn (2001), “Using Asset Turnover and Profit Margin to Forecast Changes

in Profitability”, Review of Accounting Studies (December), p. 371-385.

Graham, John, Scott Smart, and William Megginson (2010), Corporate Finance: Linking Theory

to What Companies Do, South-Western Cengage Learning, 5191 Natorp Boulevard,

Mason, OH 45040.

Higgins, Robert (1977), “How much growth can a firm afford”, Financial Management 6 (3),

p. 7-16

Holland, Larry (2018), “A Flexible Valuation Model Incorporating Declining Growth Rates”,

Accounting and Finance Research 7 (1), p. 116-129.

22

Koller, Tim, Marc Goedhart, and David Wessels (2010), Valuation: Measuring and Managing

the Value of Companies, 5th University Ed., John Wiley and Sons, 111 River Street,

Hoboken, NJ 07030.

Lundholm, Russell and Richard Sloan (2004), Equity Valuation and Analysis 1st Ed., McGraw-

Hill/Irwin, 1221 Avenue of the Americas, New York, NY 10020.

Lundholm, Russell and Richard Sloan (2017), Equity Valuation and Analysis 4th Ed., Copyright

by Russell Lundholm and Richard Sloan.

Nissim, Doron and Stephen Penman (2002), “Financial Statement Analysis of Leverage and

How it Informs About Profitability and Price to Book Ratios”, Working Paper, Columbia

University.

Nissim, Doron and Stephen Penman (2001), “Ratio Analysis and Equity Valuation: From

Research to Practice”, Review of Accounting Studies (March), p. 109-154.

Ohlson, J. and B. Juettner-Nauroth (2005). “Expected EPS and EPS growth as determinants of

value”, Review of Accounting Studies 10, 349-365.

Parrino, Robert and David Kidwell, Fundamentals of Corporate Finance 3rd Ed., John Wiley &

Sons, Inc., 111 River Street, Hoboken, NJ 07030.

Penman, Stephen (2015), "Valuation Models: An Issue of Accounting Theory," In Routledge

Companion to Financial Accounting Theory. Abingdon: Routledge.

Penman, Stephen (1998), “A synthesis of equity valuation techniques and terminal value

calculation for the dividend discount model”, Review of Accounting Studies 2, 303-323.

23

Penman, Stephen and X. Zhang (2004), “Modeling Sustainable Earnings and P/E Ratios Using

Financial Statement Information”, Working Paper, Columbia University.

Pinto J., E. Henry, T. Robinson, and J. Stowe (2010), Equity Asset Valuation, 2nd Ed., John

Wiley and Sons, Hoboken, N.J.

Piotroski, Joseph (2001), “Value Investing: The Use of Historical Financial Statement

Information to Separate Winners from Losers”, Journal of Accounting Research Vol. 38,

Supplement 2000, p. 1-41.

Ross, S., R. Westerfield, and B. Jordan (2006), Fundamentals of Corporate Finance, McGraw

Hill, New York, N.Y.

Soliman, Mark (2008), “The Use of DuPont Analysis by Market Participants”, The Accounting

Review Vol 83, No. 3, p. 823-853.

Viebig, Jan, Thorsten Poddig, and Armin Varmaz (2008), Equity Valuation: Models from

Leading Investment Banks, John Wiley and Sons, Hoboken, N.J.

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