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-