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CHAPTER 1 Cash Flow: A Mathematical Derivation 11
T A B L E 1-4




Feathers R Us
STATEMENT OF STOCKHOLDERS™ EQUITY
For Calendar Year 2000

Additional Total
Capital Paid in Retained Treasury Shareholder
Symbols Stock Capital Earnings Stock Equity

Balance, 1999 100,000 200,000 1,425,000 0 1,725,000
NI Net income 90,000 90,000
Other equity
transactions
DIV Dividends 50,000 50,000
SALSTK Sales of stock 50,000 300,000 350,000
TRSTK Purchase of stock 50,000 50,000
Subtotal OET 50,000 300,000 50,000 50,000 250,000
Balance, 2000 150,000 500,000 1,465,000 50,000 2,065,000




C NI GAIN DEPR OCA CL
CAPEXP SALESFA
LTD SALSTK TRSTK DIV AET (1-17)
375,000 90,000 30,000 30,000 (85,000) 35,000
175,000 115,000
(25,000) 350,000 50,000 50,000 0
Equation (1-17) can be simpli¬ed to the more familiar form:
C Cash flows from operating activities
Cash flows from investing activities
Cash flows from financing activities (1-18)
375,000 210,000
(60,000)
225,000
Equations (1-17) and (1-18) describe the conventional Statement of
Cash Flows shown in Table 1-5.
For the moment we will de¬ne the required change in working capital
as the change in current assets other than cash, less the change in current
liabilities, as shown in equation (1-19).12
RWC OCA CL (1-19)
(120,000) (85,000) 35,000
This illustration is somewhat unusual. Here working capital is being
reduced. This reduction is a source of the cash from operating activities.


12. The de¬nition in equation (1-19) will be modi¬ed later in the chapter.




PART 1 Forecasting Cash Flows
12
T A B L E 1-5




Feathers R Us
ABBREVIATED
STATEMENT OF
CASH FLOWS
For Calendar
Symbols Year 2000

Cash ¬‚ows from operating activities
NI Net income 90,000
Adjustments to reconcile net income to net
cash provided by operating activities:
GAIN Gain on sale of property, plant, & equipment (30,000)
DEPR Depreciation expense 30,000
OCA Decrease in current assets 85,000
CL Increase in current liabilities 35,000 120,000
Net cash provided by operating activities 210,000
Cash ¬‚ows from investing activities
CAPEXP Purchase of property, plant, & equipment (175,000)
SALESFA Sale of property, plant, & equipment 115,000
Net cash used by investing activities (60,000)
Cash ¬‚ows from ¬nancing activities
LTD Increase in long term debt (25,000)
SALSTK Sale of stock 350,000
TRSTK Purchase of treasury stock (50,000)
DIV Payment of dividends (50,000)
Net cash provided by ¬nancing activities 225,000
Net increase in cash 375,000
Cash, January 1, 2000 1,125,000
Cash, December 31, 2000 1,500,000




(In the typical case working capital is being increased. This is usually true
when sales are growing. In these cases, the increase in working capital
represents a use of cash.)
Substituting equation (1-19) into equation (1-17) shows that 13

C NI GAIN DEPR RWC
CAPEXP SALESFA
LTD SALSTK TRSTK DIV AET (1-20)
375,000 90,000 30,000 30,000 (120,000)
175,000 115,000
(25,000) 350,000 50,000 50,000 0

The ¬rst line of equation (1-20) can be rephrased in the following
way:


13. Equation (1-17): C NI GAIN + DEPR OCA + CL CAPEXP + SALESFA + LTD +
SALSTK TRSTK DIV + AET
Equation (1-19): RWC OCA CL




CHAPTER 1 Cash Flow: A Mathematical Derivation 13
Activity Symbol Description

Operating NI Net income
GAIN Gains ( losses) on the sale of property, plant, and equipment
DEPR Depreciation and other noncash charges
RWC Increases ( decreases) in required working capital




When deriving the cash ¬‚ows from operating activities, we subtract
the gain (or add the loss) on the sale of property, plant, and equipment
for several reasons. First, these gains and losses simply are not the result
of ˜˜operating™™ activities. They are the result of ˜˜investing™™ activities.
These gains and losses arise when property, plant, and equipment are
sold for more or less than their net book value. Furthermore, the full
amount received for such sales (SALESFA) is included as part of the cash
¬‚ows from investing activities. To show these gains or losses again as part
of cash ¬‚ows from operating activities would erroneously double count
their impact.
Depreciation and other noncash expenses do reduce net income, but
they do not involve any payments during the current period. Therefore,
when the indirect method is used and net income is the starting point for
arriving at a ¬rm™s net cash ¬‚ow, these noncash expenses must be added
back.
The rationale for subtracting required increases (or adding decreases)
in working capital will be discussed at some length in the next section
after introducing the components of the other current assets ( OCA) and
the current liabilities ( CL).
To complete the summary of equations (1-17), (1-18), and (1-20), the
second and third lines consist of 14


Activity Symbol Description

Investing CAPEXP Capital expenditures
SALESFA Selling price of property, plant, and equipment disposed of or retired
Financing LTD Increases ( decreases) in long-term debt
SALSTK Proceeds received from the sale of stock
TRSTK Payments for treasury stock
DIV Dividends
AET Additional equity transactions




Considering the Components of Required Working Capital
Before discussing required working capital further, it will be helpful to
break down changes in ( OCA) other current assets and ( CL) current
liabilities into some typical component parts. Table 1-6 is a restatement
of Table 1-1 with this additional detail provided in the boxed sections.



14. The second line of both equations (1-17) and (1-20) is: CAPEXP + SALESFA
The third line of both equations (1-17) and (1-20) is: LTD + SALSTK TRSTK DIV +
AET


PART 1 Forecasting Cash Flows
14
T A B L E 1-6




Feathers R Us
BALANCE SHEETS
For Calendar Years

Increase
Symbols ASSETS: 1999 2000 (Decrease)

C Cash 1,125,000 1,500,000 375,000
Accounts receivable 100,000 150,000 50,000
Inventory 750,000 600,000 (150,000)
Additional current assets 25,000 40,000 15,000
Total current assets 2,000,000 2,290,000 290,000
GPPE Gross property, plant, & equipment 830,000 900,000 70,000
AD Accumulated depreciation 30,000 40,000 10,000
NPPE Net property, plant, & equipment 800,000 860,000 60,000
A Total assets 2,800,000 3,150,000 350,000

LIABILITIES

Accounts payable 200,000 225,000 25,000
Short-term notes payable 50,000 35,000 (15,000)
Accrued expenses 75,000 100,000 25,000
CL Current liabilities 325,000 360,000 35,000
LTD Long-term debt 750,000 725,000 (25,000)
L Total liabilities 1,075,000 1,085,000 10,000

STOCKHOLDERS™ EQUITY

Capital stock 100,000 150,000 50,000
Additional paid in capital 200,000 500,000 300,000
Retained earnings 1,425,000 1,465,000 40,000
Treasury stock 0 50,000 50,000
CAP Total stockholders™ equity 1,725,000 2,065,000 340,000
Total liabilities & equity 2,800,000 3,150,000 350,000




Here, other current assets consist of accounts receivable, inventory,
and additional current assets. Current liabilities include accounts payable,
short-term notes payable, and accrued expenses.
Accounts receivable, inventory, and additional current assets should
all be treated in the same way that other current assets was treated. When
using the indirect method, increases (decreases) in these component ac-
counts should be subtracted from (added to) net income to arrive at net
cash provided by operating activities.
Likewise, accounts payable, short-term notes payable, and accrued
expenses should all be treated in the same way that current liabilities was
treated. When using the indirect method, increases (decreases) in these
component accounts should be added to (subtracted from) net income to
arrive at net cash provided by operating activities.
Applying the procedures outlined in the two preceding paragraphs
results in the Statement of Cash Flows shown in Table 1-7 which is simply
a restatement of Table 1-5 with the boxed detail added.

CHAPTER 1 Cash Flow: A Mathematical Derivation 15
T A B L E 1-7




Feathers R Us
STATEMENT OF CASH
FLOWS
Symbols For Calendar Year 2000

Cash ¬‚ows from operating activities
NI Net Income 90,000
Adjustments to reconcile net income to net
cash provided by operating activities:
GAIN Gain on sale of property, plant, & (30,000)
equipment
DEPR Depreciation expense 30,000
Increase in accounts receivable (50,000)
Decrease in inventory 150,000
Increase in additional current assets (15,000)

Increase in accounts payable 25,000
Decrease in short-term notes payable (15,000)
Increase in accrued expenses 25,000 120,000
Net cash provided by operating activities 210,000
Cash ¬‚ows from investing activities
CAPEXP Purchase of property, plant, & equipment (175,000)
SALESFA Sale of property, plant, & equipment 115,000
Net cash used by investing activities (60,000)
Cash ¬‚ows from ¬nancing activities
LTD Decrease in long term debt (25,000)
SALSTK Sale of stock 350,000
TRSTK Purchase of treasury stock (50,000)
DIV Payment of dividends (50,000)
Net cash provided by ¬nancing activities 225,000
Net increase in cash 375,000
Cash, January 1, 2000 1,125,000
Cash, December 31, 2000 1,500,000




In many cases it is quite apparent why increases in current assets
should be subtracted from net income to arrive at net cash provided by
operating activities. Increases in inventories and other current assets (such
as supplies) do require the use of cash.
However, accounts receivable can be troublesome to think through.
Why should an increase in accounts receivable be subtracted from net
income to arrive at net cash provided by operating activities? Before an-
swering this question, it is helpful to consider why accounts receivable
increase in the ¬rst place. They increase because the company has failed
to collect cash. Its collections have been less than its reported revenues.
When applying the indirect method, the ¬rst source of cash from
operating activities is net income. This implies that each of the components
of net income represents a cash ¬‚ow. The full amount of reported sales, for
example, is implicitly being treated as a cash in¬‚ow. When net accounts
receivable have increased over the period, collections must have been less
than reported revenues. Therefore, it is necessary to subtract the increase
in accounts receivable from net income to arrive at the true ¬gure for cash
provided from operations.

PART 1 Forecasting Cash Flows
16
Also, it is usually apparent why increases in current liabilities should
be added to net income to arrive at net cash provided by operating ac-
tivities. Increases (decreases) in short-term notes payable do provide (use)
cash.
To understand the treatment of accounts payable, again it is helpful
to begin by considering why accounts payable increase. They increase
because the company has not paid these bills yet. Its disbursements have
been less than its reported expenses.
Again, under the indirect method, the full amount of a reported ex-
pense is implicitly being treated as a cash out¬‚ow. When accounts pay-
able has increased over the period, payments must have been less than
that reported expense. Therefore, it is necessary to add the increase in
accounts payable back to net income when trying to arrive at the true
¬gure for cash provided from operations.
Likewise, when accrued expenses increase, it means the company has
disbursed less cash than indicated by one of its reported expenses. Again
it is necessary to add the increase in accrued expenses back to net income
when trying to arrive at the true ¬gure for cash provided from operations.
This discussion of the treatment of the components of working cap-
ital calls to mind a major difference between the income statement and
the statement of cash ¬‚ows. Both do serve as a reconciling link between
the beginning and ending balance sheets. However, the income statement
in an accrual-based partial reconciliation between the beginning and end-
ing balances in retained earnings. (The complete reconciliation requires
consideration of dividends, and occasionally certain other items.) The
statement of cash ¬‚ows is a cash-based reconciliation between the begin-
ning and ending cash balances. Much of the immediate discussion has
simply been a recital of the differences between accrual and cash account-
ing.
Recall that cash ¬‚ows from operating activities are the cash equiva-
lent of the accrual-based income statement. Again, to complete the rec-
onciliation between the beginning and ending cash balances, the state-
ment of cash ¬‚ows (as illustrated above) must also include cash from
investing or ¬nancing activities.

Adjusting for Required Cash
For valuation purposes, it is important to recognize that all ¬rms require
a certain amount of cash be kept on hand; otherwise checks would con-
stantly bounce. Therefore, the amount of required cash (CReq) will not be
available for dividend payments.
In equation (1-19), the required change in working capital was de-
¬ned simply as the change in current assets other than cash, less the
change in current liabilities. We will now modify that de¬nition, as shown
in equation (1-21) below, to include the changes in the cash balance the
¬rm will be required to keep on hand ($20,000 in this illustration).15
RWC OCA CL CReq
(1-21)
(100,000) (85,000) 35,000 20,000


15. Typically appraisers forecast required cash as a percentage of sales. Required cash increases
(decreases) by that percentage multiplied by the increase (decrease) in sales.


CHAPTER 1 Cash Flow: A Mathematical Derivation 17
Previously (in equation [1-19]), the $85,000 decrease in other current
assets and the $35,000 increase in current liabilities gave rise to a reduc-
tion in required working capital of $120,000. After taking into consider-
ation the $20,000 additional cash which will be required, the reduction in
required working capital falls to $100,000, i.e., the net addition to cash
¬‚ow from the reduction in required net working capital is $20,000 less.
Using this modi¬ed de¬nition for RWC lowers the resulting cash
¬‚ow to $355,000 (from the $375,000 originally shown in equation [1-20]).16
NI GAIN DEPR
C* RWC
CAPEXP SALESFA
LTD SALSTK TRSTK DIV AET (1-20a)
90,000 30,000 30,000
355,000 (100,000)
175,000 115,000
(25,000) 350,000 50,000 50,000 0
This $355,000 amount represents the net cash ¬‚ow available for dividend
payments in excess of the dividends already considered ($50,000).
Alternatively, DIV could be added to both sides of equation (1-20a)
to show the total amount of net cash ¬‚ow available for distribution to stock-
holders. That amount is $405,000, as shown in equation (1-20b).
C* NI GAIN DEPR RWC
DIV
CAPEXP SALESFA
LTD SALSTK TRSTK AET (1-20b)
90,000 30,000 30,000 (100,000)
405,000
175,000 115,000
(25,000) 350,000 50,000 0

COMPARISON TO OTHER CASH FLOW DEFINITIONS
The de¬nition of net cash ¬‚ow available for distribution to stockholders
in equation (1-20b) can be summarized in the following way:

Activity Symbol Description

Operating NI Net income
GAIN Gains ( losses) on the sale of property, plant, and equipment
DEPR Depreciation and other noncash charges
RWC Increases ( decreases) in required working capital*
Investing CAPEXP Capital expenditures
SALESFA Selling price of property, plant, and equipment disposed of or retired
Financing LTD Increases ( decreases) in long-term debt
SALSTK Proceeds received from the sale of stock
TRSTK Payments for treasury stock
AET Additional equity transactions

*After adjusting for required cash.



This is easily compared to other de¬nitions that have been provided
in the authoritative literature. For example, one group of authors (Pratt,


16. C* C CReq


PART 1 Forecasting Cash Flows
18
Reilly, and Schweihs 1996) have proposed the following de¬nition of net
cash ¬‚ow available for distribution to stockholders in their Formula 9-3
(at 156“157):


Description

Net income
Depreciation and other non-cash charges
Increases ( decreases) in required working capital
Capital expenditures
Selling price of property, plant, and equipment disposed of or retired
Increases ( decreases) in long term debt



Implicitly, this de¬nition assumes that gains and losses on the sale
of property, plant, and equipment and the selling price of property, plant,
and equipment disposed of or retired are immaterial. Likewise, this def-
inition assumes that the proceeds from the sale of stock, payments made
for treasury stock, and additional equity transactions are also immaterial.
These assumptions are quite reasonable and can safely be made in a
large number of cases.17 However, it is important for the analyst to realize
that these assumptions are being made.
It is well known that when calculating value by capitalizing a single
initial cash ¬‚ow, the consequences of making adjustments to the initial
cash ¬‚ow are magni¬ed considerably. It is important for the analyst to
understand how these hidden assumptions might in¬‚uence the amount
of initial cash ¬‚ow being capitalized. Perhaps it is even more important
for the analyst to take into account how these assumptions might impact
the future cash ¬‚ows available for distribution to stockholders.
For example, if a company were to routinely to sell its equipment
for signi¬cant sums, the analyst would be remiss if he or she overlooked
the cash ¬‚ows from these sales.

CONCLUSION
Careful consideration of mathematics in this chapter should enhance the
analyst™s understanding of important accounting relationships and the
˜˜whys™™ of the Statement of Cash Flows. It should also make the analyst
aware of the simplifying assumptions embedded in abbreviated de¬ni-
tions of cash ¬‚ow available for distribution to stockholders. Hopefully,
this awareness will result in superior valuations in those instances where
the making of these simplifying assumptions is unwarranted.

BIBLIOGRAPHY
Abrams, Jay B. 1997. ˜˜Cash Flow: A Mathematical Derivation.™™ Valuation (March 1994):
64“71.
Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Business: The
Analysis and Appraisal of Closely Held Companies, 3rd ed. New York: McGraw-Hill.


17. With respect to the proceeds from the sale of stock, it is unlikely that a ¬rm would sell its
stock in order to obtain cash for distribution to its stockholders. However, sometimes large
sales of stock do occur.


CHAPTER 1 Cash Flow: A Mathematical Derivation 19
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CHAPTER 2


Using Regression Analysis




INTRODUCTION
FORECASTING COSTS AND EXPENSES
Adjustments to Expenses
Table 2-1A: Calculating Adjusted Costs and Expenses
PERFORMING REGRESSION ANALYSIS
USE OF REGRESSION STATISTICS TO TEST THE ROBUSTNESS OF
THE RELATIONSHIP
Standard Error of the y Estimate
The Mean of a and b
The Variance of a and b
Precise Con¬dence Intervals
Selecting the Data Set and Regression Equation
PROBLEMS WITH USING REGRESSION ANALYSIS FOR
FORECASTING COSTS
Insuf¬cient Data
Substantial Changes in Competition or Product/Service
USING REGRESSION ANALYSIS TO FORECAST SALES
Spreadsheet Procedures to Perform Regression
Examining the Regression Statistics
Adding Industry-Speci¬c Independent Variables
Try All Combinations of Potential Independent Variables
APPLICATION OF REGRESSION ANALYSIS TO THE GUIDELINE
COMPANY METHOD
Table 2-6: Regression Analysis of Guideline Companies
95% Con¬dence Intervals
SUMMARY
APPENDIX: The ANOVA table




21




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
INTRODUCTION
Regression analysis is a statistical technique that estimates the mathe-
matical relationship between causal variables, known as independent var-
iables, and a dependent variable. The most common uses of regression
analysis in business valuation are:
1. Forecasting sales in a discounted cash ¬‚ow analysis
2. Forecasting costs and expenses in a discounted cash ¬‚ow
analysis
3. Measuring the relationship between market capitalization (fair
market value) as the dependent variable and several possible
independent variables for a publicly traded guideline company
valuation approach. Typical independent variables that are
candidates to affect the fair market value are net income
(including nonlinear transformations such as its square, square
root, and logarithm), book value, the debt-to-equity ratio, and so
on.
This chapter is written to provide the appraiser with some statistical
theory, but it is primarily focused on how to apply regression analysis to
real-life appraisal assignments using standard spreadsheet regression
tools. We have not attempted to provide a rigorous, exhaustive treatment
on statistics and have put as much of the technical background discussion
as possible in the appendix to keep the body of the chapter as simple as
possible. Those who want a comprehensive refresher should consult a
statistics text, such as Bhattacharyya and Johnson (1977) and Wonnacott
and Wonnacott (1981). We present only bits and pieces of statistics that
are necessary to facilitate our discussion of the important practical issues.
Even though you may not be familiar with using regression analysis
at all, let alone with nonlinear transformations of the data, the material
in this chapter is not that dif¬cult and can be very useful in your day-to-
day valuation practice. We will explain all the basics you need to use this
very important tool on a daily basis and will lead you step-by-step
through an example, so you can use this chapter as a guide to get ˜˜hands-
on™™ experience.
For those who are unfamiliar with the mechanical procedures to per-
form regression analysis using spreadsheets, we explain that step-by-step
in the section on using regression to forecast sales.


FORECASTING COSTS AND EXPENSES
In performing a discounted cash ¬‚ow analysis, an analyst should forecast
sales, expenses, and changes in balance sheet accounts that affect cash
¬‚ows. Frequently analysts base their forecasts of future costs on historical
averages of, or trends in, the ratio of costs as a percentage of sales.
One signi¬cant weakness of this methodology is that it ignores ¬xed
costs, leading to undervaluation in good times and possible overvaluation
in bad times. If the analyst treats all costs as variable, in good times when
he or she forecasts rapid sales growth, the ¬xed costs should stay constant
(or possibly increase with in¬‚ation, depending on the nature of the costs),
but the analyst will forecast those ¬xed costs to rise in proportion to sales.

PART 1 Forecasting Cash Flows
22
That leads to forecasting expenses too high and income too low in good
times, which ultimately causes an undervaluation of the ¬rm. In bad
times, if sales are forecasted ¬‚at, then costs will be accidentally forecasted
correctly. If sales are expected to decline, then treating all costs as variable
will lead to forecasting expenses too low and net income too high, leading
to overvaluation.
Ordinary least squares (OLS) regression analysis is an excellent tool
to forecast adjusted costs and expenses (which for simplicity we will call
˜˜adjusted costs™™ or ˜˜costs™™) based on their historical relationship to sales.
OLS produces a statistical estimate of both ¬xed and variable costs, which
is useful in planning as well as in forecasting. Furthermore, the regression
statistics produce feedback used to judge the robustness of the relation-
ship between sales and costs.


Adjustments to Expenses
Prior to performing regression analysis, we should analyze historical in-
come statements to ascertain if various expenses have maintained a con-
sistent pattern or if there has been a shift in the structure of a particular
expense. When past data is not likely to be representative of future ex-
pectations, we make pro forma adjustments to historical results to model
how the Company would have looked if its operations in the past had
conformed to the way we expect them to behave in the future. The pur-
pose of these adjustments is to examine longstanding ¬nancial trends
without the interference of obsolete information from the past. For ex-
ample, if the cost of advertising was 10% of sales for the ¬rst two years
of our historical analysis, decreased to 5% for the next ¬ve years, and is
expected to remain at 5% in the future, we may add back the excess 5%
to net income in the ¬rst two years to re¬‚ect our future expectations. We
may make similar adjustments to other expenses that have changed dur-
ing the historical period or that we expect to change in the future to arrive
at adjusted net income.


Table 2-1A: Calculating Adjusted Costs and Expenses
Table 2-1A shows summary income statements for the years 1988 to 1997.
Adjustments to pretax net income appear in Rows 15“20. The ¬rst ad-
justment, which appears in Rows 15“18, converts actual salary paid”
along with bonuses and pension payments”to an arm™s length salary.
This type of adjustment is standard in all valuations of privately held
companies.
The second type of adjustment is for a one-time event that is unlikely
to repeat in the future. In our example, the Company wrote off a discon-
tinued operation in 1994. As such, we add back the write-off to income
(H19) because it is not expected to recur in the future.
The third type of adjustment is for a periodic expense. We use a
company move as an example, since we expect a move to occur about
every 10 years.1 In our example, the company moved in 1993, 4 years


1. Losses from litigation are another type of expense that often has a periodic pattern.


CHAPTER 2 Using Regression Analysis 23
24


T A B L E 2-1A

Adjustments to Historical Costs and Expenses


A B C D E F G H I J K

4 Summary Income Statements

6 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997

7 Sales $250,000 $500,000 $750,000 $1,000,000 $1,060,000 $1,123,600 $1,191,016 $1,262,477 $1,338,226 $1,415,000
8 Cost of sales 100,000 250,000 375,000 500,000 490,000 505,000 520,000 535,000 550,000 600,000
9 S, G & A expenses 100,000 150,000 250,000 335,000 335,000 360,000 370,000 405,000 435,000 450,000
10 Operating expenses 58,000 68,000 78,000 88,000 83,000 110,000 112,000 117,000 122,000 132,000
11 Other expense 5,000 15,000 20,000 25,000 20,000 43,000 100,000 50,000 50,000 50,000
12 Pretax income $13,000 $17,000 $27,000 $52,000 $132,000 $105,600 $89,016 $155,477 $181,226 $183,000
13 Pre-tax pro¬t margin 5.20% 3.40% 3.60% 5.20% 12.45% 9.40% 7.47% 12.32% 13.54% 12.93%
14 Adjustments:
15 Actual salary 75,000 80,000 85,000 130,000 100,000 100,000 105,000 107,000 109,000 111,000
16 Bonus 3,000 4,000 4,000 20,000 5,000 5,000 5,000 7,000 9,000 10,000
17 Pension 1,000 1,000 1,500 2,000 2,000 2,000 2,000 2,000 2,000 2,000
18 Arms length salary [1] (58,015) (60,916) (63,961) (67,159) (70,517) (74,043) (77,745) (81,633) (85,714) (90,000)
19 Discontinued operations [2] 55,000
20 Moving expense [3] 20,000
21 Adjusted pretax income $7,985 $41,084 $53,539 $136,841 $168,483 $158,557 $178,271 $189,844 $215,511 $216,000
22 Adjusted pretax pro¬t margin 3.19% 8.22% 7.14% 13.68% 15.89% 14.11% 14.97% 15.04% 16.10% 15.27%
Calculation of adjusted costs
24 and expenses
25 Sales $250,000 $500,000 $750,000 $1,000,000 $1,060,000 $1,123,600 $1,191,016 $1,262,477 $1,338,226 $1,415,000
26 Adjusted pretax net income $7,985 $41,084 $53,539 $136,841 $168,483 $158,557 $178,271 $189,844 $215,511 $216,000
27 Adjusted costs and $242,015 $458,916 $696,461 $863,159 $891,517 $965,043 $1,012,745 $1,072,633 $1,122,714 $1,199,000
expenses

[1] Arms length salary includes bonus and pension
[2] A write-off for discontinued operations was an unusual a one-time expense already included in other expense. We reverse it our here.
[3] Moving expense is a periodic expense which occurs approximately every 10 years. For the 1993 move, we add back the $20,000 cost to pre-tax income, and use a Periodic Perpetuity Factor to calculate an adjustment to FMV, which we
apply later in the valuation process (see Chapter 3).
ago. We add back the $20,000 cost of the move in the adjustment section
(G20) and treat the cost separately as a periodic perpetuity.
In Chapter 3, we develop two periodic perpetuity factors (PPFs)2 for
periodic cash ¬‚ows occurring every j years, growing at a constant rate of
g, discounted to present value at the rate r, where the last cash ¬‚ow
occurred b years ago. Those formulas are:
r)b
(1
PPF PPF”end-of-year (3-18a)
r) j g) j
(1 (1
r)b
1 r (1
PPF PPF”midyear (3-19a)
r) j g) j
(1 (1
We assume the move occurs at the end of the year and use equation
(3-18a), the end-of-year PPF. We also assume a discount rate of r 20%,
moves occur every j 10 years, the last move occurred b 4 years ago,
and the cost of moving grows at g 5% per year. The cost of the next
1.210 $20,000 1.62889
move, which is forecast in Year 6, is $20,000
$32,577.89. We multiply this by the PPF, which is:
1.24
PPF 0.45445
1.210 1.0510
(see Table 3-9, cell A20), which results in a present value of $14,805.14.
Assuming a 40% tax rate, the after-tax present value of moving costs
is $14,805.14 (1 40%) $8,883. Since this is an expense, we must
remember to subtract it from”not add it to”the FMV of the ¬rm before
moving expenses. For example, if we calculate a marketable minority in-
terest FMV of $1,008,883 before moving expenses, then the marketable
minority FMV would be $1 million after moving expenses.
The other possible treatment for the periodic expense, which is
slightly less accurate but avoids the complex PPF, is to allocate the peri-
odic expense over the applicable years”10 in this example. The appraiser
who chooses this method must allocate expenses from the prior move to
the years before 1993. This approach causes the regression R 2 to be arti-
¬cially high, as the appraiser has created what appears to be a perfect
¬xed cost. For example, suppose we allocated $2,000 per year moving
costs to the years 1993“1998. If we run a regression on those years only,
R 2 will be overstated, as the perfect ¬xed cost of $2,000 per year is merely
an allocation, not the real cash ¬‚ow. Other regression measures will also
be exaggerated. If the numbers being allocated are small, however, the
overstatement is also likely to be small.
Adjusted pretax income appears in Row 21. Note that as a result of
these adjustments, the adjusted pretax pro¬t margin in Row 22 is sub-
stantially higher than the unadjusted pretax margin in Row 13.


2. This is a term to describe the present value of a periodic cash ¬‚ow that runs in perpetuity. To
my knowledge, these formulas are my own invention and PPF is my own name for it. As
mentioned in Chapter 3, where we develop this, it is in essence the same as a Gordon
model, but for a periodic, noncontiguous cash ¬‚ow. As noted in Chapter 3, when sales
occur every year, j 1 and formulas (3-18a) and (3-19a) simplify to the familiar Gordon
model multiples.




CHAPTER 2 Using Regression Analysis 25
We repeat sales (Row 7) in Row 25 and adjusted pretax income (Row
21) in Row 26. Subtracting Row 26 from Row 25, we arrive at adjusted
costs and expenses in Row 27. These adjusted costs and expenses are
what is used in forecasting future costs and expenses regression analysis.


PERFORMING REGRESSION ANALYSIS
Ordinary least squares regression analysis measures the linear relation-
ship between a dependent variable and an independent variable. Its
mathematical form is y x, where:
y the dependent variable (in this case, adjusted costs).
x the independent variable (in this case, sales).
the true (and unobservable) y-intercept value, i.e., ¬xed costs.
the true (and unobservable) slope of the line, i.e., variable
costs.
Both and , the true ¬xed and variable costs of the Company, are
unobservable. In performing the regression, we are estimating and
from our historical analysis, and we will call our estimates:
a the estimated y-intercept value (estimated ¬xed costs).
the estimated slope of the line (estimated variable costs).3
b
OLS estimates ¬xed and variable costs (the y-intercept and slope) by
calculating the best ¬t line through the data points.4 In our case, the de-
pendent variable (y) is adjusted costs and the independent variable (x) is
sales. Sales, which is in Table 2-1A, Row 7, appears in Table 2-1B as B6
to B15. Adjusted costs and expenses, Table 2-1A, Row 27, appears in Table
2-1B as C6 to C15. Table 2-1B shows the regression analysis of these var-
iables using all 10 years of data. The resulting regression yields an inter-
cept value of $56,770 (B33) and a (rounded) slope coef¬cient of $0.80
(B34). Using these results, the equation of the line becomes:
Adjusted Costs and Expenses $56,770 ($0.80 Sales)
The y-intercept, $56,770, represents the ¬xed costs of operation, or
the cost of operating the business at a zero sales volume. The slope co-
ef¬cient, $0.80, is the variable cost per dollar of sales. This means that for
every dollar of sales, there are directly related costs and expenses of $0.80.
We show this relationship graphically at the bottom of the table. The
diamonds are actual data points, and the line passing through them is
the regression estimate. Note how close all of the data points are to the
regression line, which indicates there is a strong relationship between
sales and costs.5


3. The regression parameters a and b are often shown in statistical literature as and with a
circum¬‚ex (ˆ) over each letter.
4. The interested reader should consult a statistics text for the multivariate calculus involved in
calculating a and b. Mathematically, OLS calculates the line that minimizes the sum of the
squared deviations between the actual data points and the regression estimate.
5. We will discuss the second page of Table 2-1B later in the chapter.




PART 1 Forecasting Cash Flows
26
T A B L E 2-1B

Regression Analysis 1988“1997


A B C D E F G

4 Actual

5 Year Sales X [1] Adj. Costs Y [2]

6 1988 $250,000 $242,015
7 1989 $500,000 $458,916
8 1990 $750,000 $696,461
9 1991 $1,000,000 $863,159
10 1992 $1,060,000 $891,517
11 1993 $1,123,600 $965,043
12 1994 $1,191,016 $1,012,745
13 1995 $1,262,477 $1,072,633
14 1996 $1,338,226 $1,122,714
15 1997 $1,415,000 $1,199,000

17 SUMMARY OUTPUT

19 Regression Statistics

20 Multiple R 99.88%
21 R square 99.75%
22 Adjusted R square 99.72%
23 Standard error 16,014
24 Observations 10

26 ANOVA

27 df SS MS F Signi¬cance F

28 Regression 1 8.31E 11 8.31E 11 3.24E 03 1.00E 11
29 Residual 8 2.05E 09 2.56E 08
30 Total 9 8.33E 11

32 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

33 Intercept [3] 56,770 14,863 3.82 5.09E-03 22,496 91,045
34 Sales [4] 0.80 0.01 56.94 1.00E-11 0.77 0.84

Regression Plot
[1] From Table 2-1A, Row 7 $1,400,000
[2] From Table 2-1A, Row 27
[3] Regression estimate of ¬xed costs
[4] Regression estimate of variable costs
$1,200,000




$1,000,000
y = 0.8045x + 56770
R2 = 0.9975



$800,000
Adj. Costs




$600,000




$400,000




$200,000




$0
$0 $200,000 $400,000 $600,000 $800,000 $1,000,000 $1,200,000 $1,400,000 $1,600,000
Sales




CHAPTER 2 Using Regression Analysis 27
T A B L E 2-1B (continued)

Calculation of 95% Con¬dence Intervals for Forecast 1998 Costs


A B C D E F

4 Actual

x2 x21998 / Sum x2
5 Year Sales X [1] Adj. Costs Y [2] x

6 1988 $250,000 $242,015 739,032 5.5E 11
7 1989 $500,000 $458,916 489,032 2.4E 11
8 1990 $750,000 $696,461 239,032 5.7E 10
9 1991 $1,000,000 $863,159 10,968 1.2E 08
10 1992 $1,060,000 $891,517 70,968 5.0E 09
11 1993 $1,123,600 $965,043 134,568 1.8E 10
12 1994 $1,191,016 $1,012,745 201,984 4.1E 10
13 1995 $1,262,477 $1,072,633 273,445 7.5E 10
14 1996 $1,338,226 $1,122,714 349,194 1.2E 11
15 1997 $1,415,000 $1,199,000 425,968 1.8E 11
16 Average/Total $989,032 $ 0 1.28E 12
17 Forecast 1998 $1,600,000 $1,343,928 610,968 3.7E 11 0.2905650

x2 x2
1 1
o o
21 Con¬dence Interval t0.025s t0.025s 1
x2 x2
n n
i i

24 Con¬dence Intervals For: Mean Speci¬c Year

25 t0.025 [t-statistic for 8 degrees of freedom] 2.306 2.306
26 s [From Table 2-1B, B23] $16,014 $16,014
27 1/n 0.1 0.1
x02 / Sum (Xi2)
28 [F17] 0.2905650 0.2905650
29 Add 0 for mean, 1 for speci¬c year™s exp. 0.0000000 1.0000000
30 Add rows 27 To 29 0.3905650 1.3905650
31 Square root of row 30 0.6249520 1.1792222
32 Con¬d interval row 25 * row 26 * row 31 $23,078 $43,547
33 Con¬d interval / forecast 1998 costs row 32 / C17 1.7% 3.2%
35 Regression Coef¬cients Coef¬cients
36 Intercept [From Table 2-1B, B33] 56,770
37 Sales [From Table 2-1B, B34] 0.80




We can use this regression equation to calculate future costs once we
generate a future sales forecast. Of course, to be useful, the regression
equation should make common sense. For example, a negative y-intercept
in this context would imply negative ¬xed costs, which makes no sense
whatsoever (although in regressions involving other variables it may well
make sense). Normally one should not use a result like that, despite oth-
erwise impressive regression statistics.
If the regression forecasts variable costs above $1.00, one should be
suspicious. If true, either the Company must anticipate a signi¬cant de-
crease in its cost structure in the near future”which would invalidate
applicability of the regression analysis to the future”or the Company
will be out of business soon. The analyst should also consider the pos-
sibility that the regression failed, perhaps because of either insuf¬cient or
incorrect data, and it may be unwise to use the results in the valuation.




PART 1 Forecasting Cash Flows
28
USE OF REGRESSION STATISTICS TO TEST THE
ROBUSTNESS OF THE RELATIONSHIP
Having determined the equation of the line, we use regression statistics
to determine the strength of the relationship between the dependent and
independent variable(s). We give only a brief verbal description of re-
gression statistics below. For a more in-depth explanation, the reader
should refer to a book on statistics.
In an OLS regression, the ˜˜goodness of ¬t™™ of the line is measured
by the degree of correlation between the dependent and independent
variable, referred to as the r value. An r value of 1 indicates a perfect
direct relationship, where the independent variable explains all of the
variation of the dependent variable. A value of 1 indicates a perfect
inverse relationship. Most r values fall between 1 and 1, but the closer
to 1 (or 1), the better the relationship. An r value of zero indicates no
relationship between the variables.
In a multivariable regression equation, the multiple R measures how
well the dependent variable is correlated to all of the independent vari-
ables in the regression equation. Multiple R measures the total amount
of variation in the dependent variable that is explained by the indepen-
dent variables. In our case, the value of 99.88% (B20) is very close to 1,
indicating that almost all of the variation in adjusted costs is explained
by sales.6
The square of the single or multiple R value, referred to as R-square
(or R 2), measures the percentage of the variation in the dependent vari-
able explained by the independent variable. It is the main measure of the
goodness of ¬t. We obtain an R 2 of 99.75% (B21), which means that sales
explains 99.75% of the variation in adjusted costs.
Adding more independent variables to the regression equation usu-
ally adds to R 2, even when there is no true causality. In statistics, this is
called ˜˜spurious correlation.™™ The adjusted R2, which is 99.72% in our
example (B22), removes the expected spurious correlation in the ˜˜gross™™
R2.
k n 1
Adj R 2 R2
n 1 n k 1
where n is the number of observations and k is the number of indepen-
dent variables (also known as regressors).
Although the data in Table 2-1A are ¬ctitious, in practice I have
found that regressions of adjusted costs versus sales usually give rise to
R 2 values of 98% or better.7

Standard Error of the y-Estimate
The standard error of the y-estimate is another important regression sta-
tistic that gives us information about the reliability of the regression es-


6. Although the spreadsheet labels this statistic Multiple R, because our example is an OLS
regression, it is simply R.
7. This obviously does not apply to start-ups.




CHAPTER 2 Using Regression Analysis 29
timate. We can multiply the standard error of $16,014 (B23) by two to
calculate an approximate 95% con¬dence interval for the regression es-
timate. Thus, we are 95% sure that the true adjusted costs are within
$32,028 of the regression estimate of total adjusted costs.8 Dividing
$64,000 by the mean of adjusted costs (approximately $1 million) leads
to a 95% con¬dence interval that varies by about 3%, or 6% total. Later
in the chapter we will calculate precise con¬dence intervals.


The Mean of a and b
Because a and b are speci¬c numbers that we calculate in a regression
analysis, it is easy to lose sight of the fact that they are not simply num-
bers, but rather random variables. Remember that we are trying to esti-
mate and , the true ¬xed and variable cost, which we will never know.
If we had 20 years of ¬nancial history for our Subject Company, we could
take any number of combinations of years for our regression analysis.
Suppose we had data for 1978“1997. We could use only the last ¬ve years,
1993“1997, or choose 1992“1995 and 1997, still keeping ¬ve years of data,
but excluding 1996”although there is no good reason to do so. We could
use 5, 6, 7, or more years of data. There are a large number of different
samples we can draw out of 20 years of data. Each different sample would
lead to a different calculation of a and b in our attempt to estimate and
, which is why a and b are random variables. Of course, we will never
be exactly correct in our estimate, and even if we were, there would be
no way to know it!
Equations (2-1) and (2-2) state that a and b are unbiased estimators
of and , which means that their expected values equal and . The
capital E is the expected value operator.

E (a) the mean of a is alpha (2-1)
E (b) the mean of b is beta (2-2)


The Variance of a and b
We want to do everything we can to minimize the variances of a and b
in order to improve their reliability as estimators of and . If their
variances are high, we cannot place much reliability on our regression
estimate of costs”something we would like to avoid.
Equations (2-3) and (2-4) below for the variance of a and b give us
important insights into deciding how many years of ¬nancial data to
gather and analyze. Common practice is that an appraisal should encom-
pass ¬ve years of data. Most appraisers consider anything older than ¬ve
years to be stale data, and anything less than ¬ve years insuf¬cient. You
will see that the common practice may be wrong.
The mathematical de¬nition for the variance of a is:



8. This is true at the sample mean of X, and the con¬dence interval widens as we move away
from that.


PART 1 Forecasting Cash Flows
30
2
Var (a) (2-3)
n
where 2 is the true and unobservable population variance around the
true regression line and n number of observations.9 Therefore, the var-
iance of our estimate of ¬xed costs decreases with n, the number of years
10, the variance of our estimate of is 1„2 of its variance
of data. If n
if we use a sample of ¬ve years of data. The standard deviation of a,
which is the square root of its variance, decreases somewhat less dra-
matically than the variance, but signi¬cantly nonetheless. Having 10 years
of data reduces the standard deviation of our estimate of ¬xed costs by
29% vis-a-vis ¬ve years of data. Thus, having more years of data may
`
increase the reliability of our statistical estimate of ¬xed costs if the data
are not ˜˜stale,™™ that is, out of date due to changes in the business, all else
being constant.
The variance of b is equal to the population variance divided by the
sum of the squared deviations from the mean of the independent variable,
or:
2
Var (b) (2-4)
n
x2
i
i1

where xi Xi X, the deviation of the independent variable of each
observation, Xi, from the mean, X, of all its observations. In this context,
it is each year™s sales minus the average of sales in the period of analysis.
Since we have no control over the numerator”indeed, we cannot even
know it”the denominator is the only portion where we can affect the
variance of b. Let™s take a further look at the denominator.
Table 2-2 is a simple example to illustrate the meaning of x versus
X. Expenses (Column C) is our Y (dependent) variable, and sales (Column


T A B L E 2-2

OLS Regression: Example of Deviation from Mean


A B C D E F

5 Variable

x2
6 Y X x

7 Deviation Squared Dev.
8 Observation Year Expenses Sales From Mean From Mean

9 1 1994 $ 80,000 $100,000 $(66,667) 4,444,444,444
10 2 1996 $115,000 $150,000 $(16,667) 277,777,778
11 3 1997 $195,000 $250,000 $ 83,333 6,9444,444,444
12 Total $500,000 $ - 11,666,666,667
13 Average $166,667




9. Technically this is true only when the y-axis is placed through the mean of x. The following
arguments are valid, however, in either case.


CHAPTER 2 Using Regression Analysis 31
D) is our X (independent) variable. The three years sales total $500,000
(cell D12), which averages to $166,667 (D13) per year, which is X. Column
E shows x, the deviation of each X observation from the sample mean,
X, of $166,667. In 1995, x1 $100,000 $166,667 $66,667. In 1996, x2
$150,000 $166,667 $16,667. Finally in 1997, x3 $250,000
$166,667 $83,333. The sum of all deviations is always zero, or
3
xi 0
i1


Finally, Column F shows x 2, the square of Column E. The sum of the
squared deviations,
3
x2 $11,666,666,667.
i
i1


This squared term appears in several OLS formulas and is particularly
important in calculating the variance of b.
When we use relatively fewer years of data, there tends to be less
variation in sales. If sales are con¬ned to a fairly narrow range, the
squared deviations in the denominator are relatively small, which makes
the variance of b large. The opposite is true when we use more years of
data. A countervailing consideration is that using more years of data may
lead to a higher sample variance, which is the regression estimate of 2.
Thus, it is dif¬cult to say in advance how many years of data are optimal.
This means that the common practice in the industry of using only
¬ve years of data so as not to corrupt our analysis with stale data may
be incorrect if there are no signi¬cant structural changes in the competi-
tive environment. The number of years of available data that gives the
best overall statistical output for the regression equation is the most de-
sirable. Ideally, the analyst should experiment with different numbers of
years of data and let the regression statistics”the adjusted R 2, t-statistics,
and standard error of the y-estimate”provide the feedback to making
the optimal choice of how many years of data to use.
Sometimes prior data can truly be stale. For example, if the number
of competitors in the Company™s geographic area doubled, this would
tend to drive down prices relative to costs, resulting in a decreased con-
tribution margin and an increase in variable costs per dollar of sales. In
this case, using the old data without adjustment would distort the re-
gression results. Nevertheless, it may be advisable in some circumstances
to use some of the old data”with adjustments”in order to have enough
data points for analysis. In the example of more competition in later years,
it is possible to reduce the sales in the years prior to the competitive
change on a pro forma basis, keeping the costs the same. The regression
on this adjusted data is often likely to be more accurate than ˜˜winging
it™™ with only two or three years of fresh data.
Of course, the company™s management has its view of the future. It
is important for the appraiser to understand that view and consider it in
his or her statistical work.




PART 1 Forecasting Cash Flows
32
Con¬dence Intervals
Constructing con¬dence intervals around the regression estimates a and
b is another important step in using regression analysis. We would like
to be able to make a statement that we are 95% sure that the true variable
(either or ) is within a speci¬c range of numbers, with our regression
estimate (a or b) at the midpoint. To calculate the range, we must use the
Student™s t-distribution, which we de¬ne in equation (2-6).
We begin with a standardized normal (Z) distribution. A standard-
ized normal distribution of b”our estimate of ”is constructed by sub-
tracting the mean of b, which is , and dividing by its standard deviation.
b
Z (2-5)
x2
/ i
i


Since we do not know , the population standard deviation, the best
we can do is estimate it with s, the sample standard deviation. The result
is the Student™s t-distribution, or simply the t-distribution. Figure 2-1
shows a z-distribution and a t-distribution. The t-distribution is very sim-
ilar to the normal (Z) distribution, with t being slightly more spread out.
The equation for the t-distribution is:
b
t (2-6)
x2
s/ i
i
where the denominator is the standard error of b, commonly denoted as
sb (the standard error of a is sa).
Since is unobservable, we have to make an assumption about it in
order to calculate a t-distribution for it. The usual procedure is to test for
the probability that, regardless of the regression™s estimate of ”which
is our b”the true is really zero. In statistics, this is known as the ˜˜null
hypothesis.™™ The magnitude of the t-statistic is indicative of our ability
to reject the null hypothesis for an individual variable in the regression
equation. When we reject the null hypothesis, we are saying that our
regression estimate of is statistically signi¬cant.
We can construct 95% con¬dence intervals around our estimate, b, of
the unknown . This means that we are 95% sure the correct value of
is in the interval described in equation (2-7).
b t0.025 sb (2-7)

Formula for 95% confidence interval for the slope
Figure 2-2 shows a graph of the con¬dence interval. The graph is a
t-distribution, with its center at b, our regression estimate of . The mark-
ings on the x-axis are the number of standard errors below or above b.
As mentioned before, we denote the standard error of b as sb. The lower
boundary of the 95% con¬dence interval is b t0.025 sb, and the upper
boundary of boundary of the 95% con¬dence interval is b t0.025 sb. The




CHAPTER 2 Using Regression Analysis 33
34




F I G U R E 2-1

Z-distribution vs t-distribution


0.4




0.35




0.3




0.25
probability density




0.2




0.15




0.1



t t
0.05

Z Z

0
-6 -4 -2 0 2 4 6
for Z, standard deviations from mean
For t, standard errors from mean
F I G U R E 2-2

t-distribution of B around the Estimate b


0.4




0.35




0.3
probability density


0.25




0.2




0.15

area = 2.5%
area =2.5%
0.1




0.05




0
-6 -4 -2 0 2 4 6
B =b+t0.025 sb
B =b
B= b“t 0.025sb B measured in standard
errors away from b
35
area under the curve for any given interval is the probability that will
be in that interval.
The t-distribution values are found in standard tables in most statis-
tics books. It is very important to use the 0.025 probability column in the
tables for a 95% con¬dence interval, not the 0.05 column. The 0.025 col-
umn tells us that for the given degrees of freedom there is a 21„2% prob-
ability that the true and unobservable is higher than the upper end of
the 95% con¬dence interval and a 21„2% probability that the true and
unobservable is lower than the lower end of the 95% con¬dence interval
(see Figure 2-2). The degrees of freedom is equal to n k 1, where n
is the number of observations and k is the number of independent vari-
ables.
Table 2-3 is an excerpt from a t-distribution table. We use the 0.025
column for a 95% con¬dence interval. To select the appropriate row in
the table, we need to know the number of degrees of freedom. Assuming
n 10 observations and k one independent variable, there are eight
degrees of freedom (10 1 1). The t-statistic in Table 2-3 is 2.306 (C7).
That means that we must go 2.306 standard errors below and above our
regression estimate to achieve a 95% con¬dence interval for . The re-
gression itself will provide us with the standard error of . As n, the
number of observations, goes to in¬nity, the t-distribution becomes a z-
distribution. When n is large”over 100”the t-distribution is very close
to a standardized normal distribution. You can see this in Table 2-3 in
that the standard errors in Row 9 are very close to those in Row 10, the
latter of which is equivalent to a standardized normal distribution.
The t-statistics for our regression in Table 2-1B are 3.82 (D33) and
56.94 (D34). The P-value, also known as the probability (or prob) value,
represents the level at which we can reject the null hypothesis. One minus
the P-value is the level of statistical signi¬cance of the y-intercept and
independent variable(s). The P-values of 0.005 (E33) and 10 11 (E34) mean
that the y-intercept and slope coef¬cients are signi¬cant at the 99.5% and
99.9% levels, respectively, which means we are 99.5% sure that the true
y-intercept is not zero and 99.9% sure that the true slope is not zero.10

T A B L E 2-3

Abbreviated Table of T-Statistics


A B C D

4 Selected t Statistics

5 d.f.\Pr. 0.050 0.025 0.010
6 3 2.353 3.182 4.541
7 8 1.860 2.306 2.896
8 12 1.782 2.179 2.681
9 120 1.658 1.980 2.358
10 In¬nity 1.645 1.960 2.326




10. For spreadsheets that do not provide P-values, another way of calculating the statistical
signi¬cance is to look up the t-statistics in a Student™s t-distribution table and ¬nd the level
of statistical signi¬cance that corresponds to the t-statistic obtained in the regression.


PART 1 Forecasting Cash Flows
36
The F test is another method of testing the null hypothesis. In mul-
tivariable regressions, the F-statistic measures whether the independent
variables as a group explain a statistically signi¬cant portion of the var-
iation in Y.
We interpret the con¬dence intervals as follows: there is a 95% prob-
ability that true ¬xed costs (the y-intercept) fall between $22,496 (F33) and
$91,045 (G33); similarly, there is a 95% probability that the true variable
cost (the slope coef¬cient) falls between $0.77 (F34) and $0.84 (G34).
The denominator of equation (2-6) is called the standard error of b,
or sb. The standard error of the Y-estimate, which is de¬ned as
n
1 ˆ
Yi)2
s (Yi
n 2 i1



is $16,014 (B23). The larger the amount of scatter of the points around
the regression line, the greater the standard error.11

Precise Con¬dence Intervals12
Earlier in the chapter, we estimated 95% con¬dence intervals by subtract-
ing and adding two standard errors of the y-estimate around the regres-
sion estimate. In this section, we demonstrate how to calculate precise
95% con¬dence intervals around the regression estimate using the equa-
tions:

x2
1 o
t0.25s (2-8)
x2
n i

95% confidence interval for the mean forecast

x2
1 o
t0.025s 1 (2-9)
x2
n i

95% confidence interval for a specific year™s forecast
In the context of forecasting adjusted costs as a function of sales,
equation (2-8) is the formula for the 95% con¬dence interval for the mean
adjusted cost, while equation (2-9) is the 95% con¬dence interval for the
costs in a particular year. We will explain what that means at the end of
this section, after we present some material that illustrates this in Table
2-1B, page 2.
Note that these con¬dence intervals are different than those in equa-
tion (2-7), which was around the forecast slope only, i.e., b. In this section,


11. This standard error of the Y-estimate applies to the mean of our estimate of costs, i.e., the
average error if we estimate adjusted costs and expenses many times. This is appropriate in
valuation, as a valuation is a forecast of net income and / or cash ¬‚ows for an in¬nite
number of years. The standard error”and hence 95% con¬dence interval”for a single
year™s costs is higher.
12. This section is optional, as the material is somewhat advanced, and it is not necessary to
understand this in order to be able to use regression analysis in business valuation.
Nevertheless, it will enhance your understanding should you choose to read it.


CHAPTER 2 Using Regression Analysis 37
we are calculating con¬dence intervals around the entire regression fore-
cast.
The ¬rst 15 rows of Table 2-1B, page 2, are identical to the ¬rst page
and require no explanation. The $989,032 in B16 is the average of the 10
years of sales in B6“B15.
Column D is the deviation of each observation from the mean, which
is the sales in Column B minus the mean sales in B16. For example, D6
( $739,032) is equal to B6 ($250,000) minus B16 ($989,032). D7
( $489,032) equals B7 ($500,000) minus B16 ($989,032). The total of all
deviations from the mean must always equal zero, which it does (D16).
Column E is the squared deviations, i.e., the square of Column D. In
statistics, the independent variable(s) is known as X, while the deviations
from the mean are known as x, which explains the column labels in B5
and D5. The sum of squared deviations,
1997
x i2
i 1988


1012 (E16).
equals 1.28
The next step is to compute the squared deviations for our sample
forecast year. We assume that forecast sales for 1997 is $1.6 million (B17).
We repeat the coef¬cients from the regression formula from the ¬rst page
of the table in B36 and B37. Applying the regression equation, we would
then forecast expenses at $1,343,928 (C17).
In order to compute a 95% con¬dence interval around the expense
forecast of $1,343,928, we apply equations (2-8) and (2-9). 1998 forecast
sales are $610,968 (D17 B17 B16) above the mean of the historical
period. That is the x0 in (3-8) and (3-9). We square the term to get 3.73
1011 (E17). Then we divide that by the sum of the squared deviations in
1012 (E16) to get 0.2905650 (F17), which we
the historical period 1.28
repeat below in Row 28.
In Row 25, we insert the t-statistic of 2.306, which one can ¬nd in a
table for a 95% con¬dence level (the 0.025 column in a two-tailed distri-
bution) and eight degrees of freedom (n 10 observations 1 indepen-

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