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

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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.