The P/E ratio rises because of the increase in the forecast growth rate

across the columns. If all cells in Row 45 were equal to 0.05, then the PE

ratios in Row 52 would actually decline going to the right across the

columns. The reason for this is that the x-coef¬cient for ln NI is 0.95708

(page 1, B37) 1. This is contrary to our expectations. If B38 were greater

than 1, then P/E ratios would rise with ¬rm size, holding forecast growth

constant. Does this disprove the log size model? No. While all the rest of

the data are real, these growth rates are not actual. They are made up.

Also, one small sample of one industry at one point in time does not

generalize to all ¬rms at all times.

In the absence of the made-up growth rates, the actual regression

yielded an adjusted R 2 of 93.3% and a standard error of 0.2896 (not

shown).

95% Con¬dence Intervals

We multiply the standard error in B26 by 2 0.16544 (B55). To convert

the standard error of ln FMV to the standard error of FMV, we have to

exponentiate the two standard errors. In B56 we raise e, Euler™s constant,

to the power of B55. Thus, e0.16544 1.1799, which means the high side of

our 95% con¬dence interval is 18% higher than our estimate.23 To calcu-

late the low side of our 95% con¬dence interval, we raise e to the power

of two standard errors below the regression estimate. Thus B57 e 0.16544

0.8475, which is approximately 15% below the regression estimate.

Thus our 95% con¬dence interval is the regression estimate 18% and

15%. Using only the actual data that were available at the time, the

same regression without 1/g yielded con¬dence intervals of the regres-

21. The Excel formula for cell B42, for example, is ln(B41). The Lotus 123 formula would be

@ln(B41).

22. In Lotus 123 the formula would be @exp(B50)

23. The Excel formula for cell B56 is EXP(B55) and the Lotus 123 formula is @EXP(B55).

Similarly, the Excel formula for B57 is EXP( B55), and the Lotus 123 formula is

@EXP( B55).

CHAPTER 2 Using Regression Analysis 51

sion estimate 78% and 56%. Obviously, growth can make a huge dif-

ference. Also, without growth, the x-coef¬cient for ln NI was slightly

above one, indicating increasing P/E multiples with size.

SUMMARY

Regression analysis is a powerful tool for use in forecasting future costs,

expenses, and sales and estimating fair market value. We should take care

in evaluating and selecting the input data, however, to arrive at mean-

ingful answer. Similarly, we should carefully scrutinize the regression out-

put to determine the signi¬cance of the variables and the amount of error

in the Y-estimate to determine if the overall relationship is meaningful.

BIBLIOGRAPHY

Bhattacharyya, Gouri K., and Richard A. Johnson. 1977. Statistical Concepts and Methods.

New York: John Wiley & Sons.

Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Business: The

Analysis and Appraisal of Closely Held Companies; 3d ed. New York: McGraw-Hill.

Wonnacott, Thomas H., and Ronald J. Wonnacott. 1981. Regression: A Second Course in

Statistics. New York: John Wiley & Sons.

PART 1 Forecasting Cash Flows

52

APPENDIX

The ANOVA table (Rows 28“32)

We have already discussed the importance of variance in regression anal-

ysis. The center section of Table A2-1, which is an extension of Table

2-1B, contains an analysis of variance (ANOVA) automatically generated

by the spreadsheet. We calculate the components of ANOVA in the top

portion of the table to ˜˜open up the black box™™ and show the reader

where the numbers come from.

First, we calculate the regression estimate of adjusted costs in Col-

umn D using the regression equation:

Costs $56,770 (0.80 Sales) (B35, B36)

Next, we subtract the average actual adjusted cost of $852,420 (C18) from

the calculated costs in Column D to arrive at the deviation from the mean

in Column E. Note that the sum of the deviations is zero in cell E17, as

expected.

In Column F we square each deviation term in Column E and total

them in F17. The total, 831,414,202,481, is known as the sum of squares

and measures the amount of variation explained by the regression. In the

absence of a regression, our best estimate of costs for any year during the

1988“1997 period would be Y, the mean costs. Therefore, the difference

between the historical mean and the regression estimate (Column E) is

the absolute deviation explained by the regression. The square of that

(Column F) is the variance explained by the regression. This term appears

in the ANOVA table in C30 under SS (sum of squares).

The next term to the right in the ANOVA table is the mean squared

error (MS), which measures the variance explained by the regression. In

our case, the number is identical to the SS term (D30 C30). This occurs

because we have only one independent variable, sales, and thus one de-

gree of freedom (B30) in the regression.

In Column G we calculate the difference between the each actual cost

and the calculated cost (the regression estimate) by subtracting the values

in Column D from Column C. Again, the sum of the deviations is zero.

We square the deviations and sum them to arrive at a value of

2,051,637,107 (H17). This second sum of squares, which appears in the

ANOVA table in cell C31, is the unexplained variation. We calculate the

corresponding mean square error term in Column I by dividing the val-

ues in Column H by 8, the number of degrees of freedom (B30). The sum

is 256,454,638 (I17), which appears in the ANOVA table in D31. This num-

ber represents the unexplained variance. Finally, we calculate the F-

statistic of 3,241 (E30) by dividing the explained variance (D30) by the

unexplained variance (D31).

The explained variation plus the unexplained equals the total vari-

ation. The correlation coef¬cient is

Explained Variation of Y

R2

Total Variation of Y

In our case, the explained variation (C30) divided by the total variation

(C32) is equal to 99.75%, as seen in B23.

CHAPTER 2 Using Regression Analysis 53

54

T A B L E A2-1

Regression Analysis 1988“1997

A B C D E F G H I

4 Actual Calculated Deviation of Sum of Squares Deviation of Deviation from Mean

Calc. [5] Actual Actual Square [7]

ˆ

5 Costs Y [3] from Mean from Calc. Squared [5]

[4] [6]

ˆ ˆ ˆ ˆ

Y)2 Y)2 Y)2/8

6 Year Sales X [1] Adj. Costs Y [2] Y Y (Y Y Y (Y (Y

7 1988 $250,000 $242,015 $257,889 $594,532 353,467,822,773.69 $15,874 251,983,658 31,497,957

8 1989 $500,000 $458,916 $459,007 -$393,413 154,773,949,895.09 $92 8,399 1,050

9 1990 $750,000 $696,461 $660,126 $192,295 36,977,294,181.16 $36,336 1,320,285,654 165,035,707

10 1991 $1,000,000 $863,159 $861,244 $8,824 77,855,631.91 $1,915 3,668,783 458,598

11 1992 $1,060,000 $891,517 $909,512 $57,092 3,259,496,294.19 -$17,995 323,821,415 40,477,677

12 1993 $1,123,600 $965,043 $960,677 $108,257 11,719,473,702.15 $4,366 19,064,659 2,383,082

13 1994 $1,191,016 $1,012,745 $1,014,911 $162,491 26,403,295,435.15 $2,166 4,691,209 586,401

14 1995 $1,262,477 $1,072,633 $1,072,400 $219,979 48,390,920,118.80 $233 54,240 6,780

15 1996 $1,338,226 $1,122,714 $1,133,338 $280,917 78,914,430,752.87 $10,623 112,853,095 14,106,637

16 1997 $1,415,000 $1,199,000 $1,195,101 $342,680 117,429,663,696.32 $3,899 15,205,993 1,900,749

17 Total $0 831,414,202,481 $0 2,051,637,107 256,454,638

18 $852,420 Average Actual Adjusted Costs (Y)

19 SUMMARY OUTPUT

21 Regression Statistics

22 Multiple R 0.998768455

23 R square 0.997538427

24 Adjusted R square 0.99723073

25 Standard error 16014.20115

26 Observations 10

28 ANOVA

29 df SS MS F Signi¬cance F

30 Regression 1 8.31414E 11 8.31414E 11 3241.954241 1.00493E-11

31 Residual 8 2051637107 256454638.3

32 Total 9 8.33466E 11

34 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

35 Intercept 56770.40117 14863.25124 3.819514334 0.005093239 22495.66018 91045.14216 22495.66018 91045.14216

36 Sales X [1] 0.804473578 0.0141289 56.93816156 1.00493E-11 0.771892255 0.8370549 0.771892255 0.8370549

[a] This sheet is an extension of Table 2-1B.

[1] from Table 2-1A, Row 7

[2] from Table 2-1A, Row 27

[3] Calculated costs using Costs 0.80 Sales $56,806 with sales ¬gures in Column B

ˆ

[4] Deviation of calculated costs from average actual costs (Column D C17) Y Y

[5] Deviations squared

[6] Deviation of actual costs from calculated costs (Column C Column D)

[7] Deviations squared / 8 (degrees of freedom)

[8] Regression estimate of ¬xed costs

[9] Regression estimate of variable costs

55

CHAPTER 3

Annuity Discount Factors and

the Gordon Model

INTRODUCTION

De¬nitions

Denoting Time

ADF WITH END-OF-YEAR CASH FLOWS

Behavior of the ADF with Growth

Special Case of ADF when g 0: The Ordinary Annuity

Special Case when n ’ and r g: The Gordon Model

Intuitively Understanding Equations (3-6) and (3-6a)

Relationship between the ADF and the Gordon Model

Table 3-1: Proof of ADF Equations (3-6) through (3-6b)

A Brief Summary

MIDYEAR CASH FLOWS

Table 3-2: Example of Equations (3-10) through (3-10b)

Special Cases for Midyear Cash Flows: No Growth, g 0

Gordon Model

STARTING PERIODS OTHER THAN YEAR 1

End-of-Year Formulas

Valuation Date 0

Table 3-3: Example of Equation (3-11)

Tables 3-4 through 3-6: Variations of Table 3-3 with S 0, Negative

Growth, and r g

Special Case: No Growth, g 0

Generalized Gordon Model

Midyear Formula

PERIODIC PERPETUITY FACTORS (PPFs): PERPETUITIES FOR

PERIODIC CASH FLOWS

The Mathematical Formulas

Tables 3-7 and 3-8: Examples of Equations (3-18) and (3-19)

Other Starting Years

New versus Used Equipment Decisions

57

Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

ADFs IN LOAN MATHEMATICS

Calculating Loan Payments

Present Value of a Loan

Table 3-10: Example of Equation (3-23)

RELATIONSHIP OF THE GORDON MODEL TO THE

PRICE/EARNINGS RATIO

De¬nitions

Mathematical Derivation

CONCLUSIONS

PART 1 Forecasting Cash Flows

58

INTRODUCTION

This chapter describes the derivation of annuity discount factors (ADFs)

and the Gordon model (Gordon and Shapiro 1956).1 The ADF is the pres-

ent value of a ¬nite stream of cash ¬‚ows (CF) with constant or zero

growth, assuming the ¬rst cash ¬‚ow $1.00. Thus, the actual ¬rst year™s

cash ¬‚ow times the ADF is the present value as of time zero of the stream

of cash ¬‚ows from years 1 to n. Growth rates in cash ¬‚ows may be pos-

itive, zero, or negative, the latter being a decline in cash ¬‚ows.

The Gordon model is identical to the ADF, except that it produces

the present value of a perpetuity for each $1.00 of initial cash ¬‚ow. The

resulting present value is known as the Gordon model multiple. When

using the Gordon model multiple, the discount rate must be larger than

the constant growth rate, which is not true of the ADF.

There are several varieties of ADFs, depending on whether the cash

¬‚ows:

— Are constant or grow/decline.

— Occur midyear or at the end of the year.

— Begin in the ¬rst year or at some other time.

— Occur every year or at regular, skipped intervals.

— Finish on a whole year or a fractional year.

This chapter begins with the derivation of the ADF and later shows

that the Gordon model, which is the present value of a perpetual annuity

with constant growth, is simply a special case of the ADF. We will dem-

onstrate that an ADF is actually the difference of two perpetuities.

There are several uses of ADFs, including:

— Calculating the present value of annuities. This application has

become far more important since the quantitative marketability

discount model (Mercer 1997) requires an ADF with growth (see

Chapter 8). While Mercer™s book has an approximation of the

ADF (at 276) that appears to be fairly accurate, this chapter

contains the exact formulas.

— Valuing periodic cash ¬‚ows such as moving expenses, losses

from lawsuits, etc. This requires a specialized ADF called a

periodic perpetuity factor (PPF), which we develop later in the

chapter. Additionally, PPFs are useful for decisions in buying

new versus used income-producing equipment (such as CAT

scans, ships, or taxicabs) and for calculating the value of used

equipment.

— Calculating loan payments.

— Calculating loan principal amortization.

— Calculating the present value of a loan. This is important in

calculating the correct selling price of a business, as seller

¬nancing typically takes place at less-than-market rates. The

present value of a loan is also important in ESOP valuation.

1. Gordon and Shapiro were preceded by Williams (1938). See also Gordon (1962).

CHAPTER 3 Annuity Discount Factors and the Gordon Model 59

At ¬rst glance this chapter appears mathematically very intensive

and daunting in its use of geometric sequences. However, because the

primary concepts appear in equations (3-1) through (3-9), once you un-

derstand those equations, the remainder are merely special cases or slight

variations on the original theme and can easily be comprehended. While

the formulas look complex, we decompose them into units that behave

as modular building blocks, each of which has an intuitive explanation.

You will bene¬t from understanding the math in the body of the chapter,

as this material is useful in several areas of business valuation. Addition-

ally, you will also gain a much better understanding of the Gordon model,

which appraisers often use in discounted future net income or discounted

cash ¬‚ow valuation.

ADFs are an area that many practitioners ¬nd dif¬cult, leading to

many mistakes. Timing errors in ADFs frequently result from the fact that

the guideline company method uses the most recent historical earnings

for calculating P/E multiples, whereas the Gordon model uses the ¬rst

future period (forecast) cash ¬‚ow as its earnings base. Many practitioners

confuse the two and use historical rather than forecast earnings as their

base in a discounted cash ¬‚ow or discounted future net income approach.

Another common error is the use of end-of-year multiples when midyear

Gordon model multiples are appropriate.

The ADF formulas given within the chapter apply only to cash ¬‚ow

streams that have a whole number of years associated with them. If the

cash ¬‚ow stream ends in a fractional year, you should use the formulas

in the appendix for ADFs with stub periods.

Unless otherwise speci¬ed, all ADF formulas are for cash ¬‚ows with

constant growth. At speci¬c points in the chapter, we make the simpli-

fying assumption that growth is zero and clearly state when that is the

case. Otherwise the reader may assume growth is constant and non-zero.

De¬nitions

Let us initially consider an ADF with constant growth in cash ¬‚ows,

where the last cash ¬‚ow occurs in period n. We will use the following

de¬nitions:

r discount rate

g annual growth rate in cash ¬‚ows

ADF annuity discount factor

PV present value

CF cash ¬‚ow

LHS left-hand side of the equation

RHS right-hand side of the equation

n terminal year of the cash ¬‚ows

t time (which can refer to a point in time or a year)

Denoting Time

Timing is frequently a source of confusion. Time t denotes the time period

under discussion. It generally refers to a speci¬c year.2 Time t refers to

2. In the context of loan amortization, periods are usually months.

PART 1 Forecasting Cash Flows

60

the entire year, except for two contexts that we discuss in the paragraph

below. Thus, time t is a span of time, not a point in time.

There are two contexts in which time t means a point in time. The

¬rst occurs with the statement t 0, which means the beginning of the

period t 1, i.e., usually the beginning of the ¬rst year of cash ¬‚ows.

For example, if t 1 represents the calendar year 2000, then t 0 means

January 1, 2000, the ¬rst day of t 1. Usually, but not always, t 0 is

the valuation date. The other context in which t means a point in time is

when we specify either the beginning, midpoint, or end of t.

In business valuation, we generally assume that cash ¬‚ows occur

approximately evenly throughout time t. In present value terms, that is

equivalent to assuming they occur at the midpoint of time t. Occasionally

it is appropriate to assume that cash ¬‚ows occur at the end of the year,

which can be the case with annuities, royalties, etc. The former is com-

monly known as the midyear assumption, while the latter is known as the

end-of-year (or end year) assumption.

Another important concept related to time that can be confusing is

the valuation date, the point in time to which we discount the cash ¬‚ows.

The valuation date is rarely the same as the ¬rst cash ¬‚ow. The most

common valuation date in this chapter is as of time zero, i.e., t 0. The

cash ¬‚ows usually, but not always, either begin during Year 1 or occur at

the end of Year 1.

ADF WITH END-OF-YEAR CASH FLOWS

The ADF is the present value of a series of cash ¬‚ows over n years with

constant growth, beginning with $1 of cash ¬‚ow in Year 1. We multiply

by the ¬rst year™s forecast cash ¬‚ow by the ADF to arrive at the PV of

the cash ¬‚ow stream. For example, if the ADF is 9.367 and the ¬rst year™s

cash ¬‚ow is $10,000, then the PV of the annuity is 9.367 $10,000

$93,670.

We begin the calculation of the ADF by de¬ning the cash ¬‚ows and

discounting them to their present value. Initially, for simplicity, we as-

sume end-of-year cash ¬‚ows. The PV of an annuity of $1, paid at the end

of the year for each of n years, is:

g)n 1

$1 (1 g) $1 (1

$1

PV (3-1)

r)1 r)2 r)n

(1 (1 (1

Factoring out the $1:

g)n 1

(1 g) (1

1

PV $1 (3-1a)

r)1 r)2 r)n

(1 (1 (1

The ADF is the PV of the constant growth cash ¬‚ows per $1 of starting

year cash ¬‚ow. Dividing both sides of equation (3-1a) by $1, the left-hand

side becomes PV/$1, which equals the ADF. Thus, equation (3-1a) sim-

pli¬es to:

g)n 1

(1 g) (1

1

ADF (3-1b)

r)1 r)2 r)n

(1 (1 (1

The numerators in equation (3-1b) are the forecast cash ¬‚ows them-

CHAPTER 3 Annuity Discount Factors and the Gordon Model 61

selves, and the denominators are the present value factors for each cash