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vided by Row 41.
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

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
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.

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.

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
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
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
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

T A B L E A2-1

Regression Analysis 1988“1997


4 Actual Calculated Deviation of Sum of Squares Deviation of Deviation from Mean
Calc. [5] Actual Actual Square [7]
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)
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
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
[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

Annuity Discount Factors and
the Gordon Model

Denoting Time
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
Table 3-2: Example of Equations (3-10) through (3-10b)
Special Cases for Midyear Cash Flows: No Growth, g 0
Gordon Model
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
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


Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Calculating Loan Payments
Present Value of a Loan
Table 3-10: Example of Equation (3-23)
Mathematical Derivation

PART 1 Forecasting Cash Flows
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
— 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
— 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.

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
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
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.

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
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
PV (3-1)
r)1 r)2 r)n
(1 (1 (1
Factoring out the $1:
g)n 1
(1 g) (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
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

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