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Putting It All Together




Part 4 of this book consists of Chapters 10 and 11. Chapter 10 empirically
tests the log size and economic components models by reconciling price
to cash ¬‚ow (P/CF) multiples calculated using these models with P/CF
multiples for groups of ¬rms of different sizes in the Institute of Business
Appraisers™ (IBA) database. The results provide weak support for the two
models, but missing data make it impossible to provide strong support.
There is simply too much data we need that does not exist in the IBA
database or any other one of which I am aware.
In Chapter 11 we look at two issues. In the ¬rst half of the chapter
we calculate 95% con¬dence intervals around our valuation estimate us-
ing the log size model (both for all 72 years of New York Stock Exchange
data and for the past 60 years), assuming we forecast cash ¬‚ows and
adjust for control and marketability perfectly. The importance of this is
to understand how much statistical uncertainty there is in our valuation
estimates.
The second half of Chapter 11 is concerned with measuring the val-
uation errors that arise from errors in forecasting cash ¬‚ow and growth
rates and calculating discount rates. We look at the effects of both relative
and absolute errors and show how the majority of these errors affect the
valuation of large ¬rms more than small ¬rms.
Whereas Part 3 of this book consists of practical, hands-on, ˜˜how-
to™™ chapters, Part 4 does not. It can be skipped by the time-pressed reader.
Nevertheless, for one who wants to be well educated and familiar with
important theoretical and empirical issues in valuation, these chapters are
important.




355




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CHAPTER 10


Empirical Testing of Abrams™
Valuation Theory1




INTRODUCTION
Steps in the Valuation Process
Applying a Valuation Model to the Steps
TABLE 10-1: LOG SIZE FOR 1938“1986
TABLE 10-2: RECONCILIATION TO THE IBA DATABASE
Part 1: IBA P/CF Multiples
Part 2: Log Size P/CF Multiples
Conclusion
CALCULATION OF DLOM
Table 10-4: Computation of the Delay-to-Sale Component“$25,000
Firm
Table 10-5: Calculation of Transactions Costs
Table 10-6: Calculation of DLOM
Table 10-6A“10-6F: Calculations of DLOM for Larger Firms
Calculation of DLOM for Large Firms
INTERPRETATION OF THE ERROR
CONCLUSION




1. I offer my profound thanks to Mr. Raymond Miles for his considerable help. Without his vitally
important research, this article would be impossible. Also, Professor Haim Mendelson of
Stanford University provided extremely helpful comments.




357




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INTRODUCTION
Many appraisers have long believed that when small businesses sell, they
are priced very differently than large businesses and that the rules gov-
erning their valuation are totally different. I, too, held this opinion at one
time, but this chapter is evidence”though not proof”that it is not true.
A skeptic could level the charge that the log size discount rate equa-
tion is based on a mathematical relationship that exists between returns
and size of New York Stock Exchange (NYSE) ¬rms, but it may not apply
to the universe of small and medium privately held ¬rms. Additionally,
the calculations of the transactions costs component of the discount for
lack of marketability (DLOM) is based on interviews, then quanti¬ed in
an equation and extrapolated downwards for small ¬rms. Thus, it™s nice
in theory, but does it really work in practice?
The purpose of this chapter is to subject the log size and economic
components models to empirical testing to see whether they do a good
job of explaining real world transactions of smaller businesses. Our pri-
mary data comes from an article published by Raymond Miles (Miles
1992) (˜˜the article™™) about the relationship of size to price earnings (PE)
multiples in the Institute of Business Appraisers™ (IBA) database.


Steps in the Valuation Process
Using a simple discounted cash ¬‚ow model as the valuation paradigm,
valuation consists of four steps:
Forecast cash ¬‚ows.
1.
Discount to net present value.
2.
Adjust for marketability or lack thereof.
3.
Adjust for degree of control.
4.


Applying a Valuation Model to the Steps
The sales described in the article are all $1 million or less. It is a reason-
able assumption that the vast majority of the small ¬rms in the IBA trans-
actional database are mature. The number of high-growth startup ¬rms
in that database is likely to be small. Therefore, it is reasonable to assume
a constant growth rate to perpetuity. Using a Gordon model to apply to
the next year™s forecast cash ¬‚ows should give us a fairly accurate FMV
on a marketable minority level. Using a midyear assumption, the formula
is:

1 r
FMV CFt 1
r g

where r is the discount rate, which we will estimate using the log size
model, and g is the constant growth rate, which we will estimate. That
takes care of the ¬rst two valuation steps.




PART 4 Putting It All Together
358
We will use the economic components model from Chapter 7 for our
calculations of DLOM. We assume a control premium of 25%, which is
the approximate midpoint of the 21“28% range estimated in Chapter 7.
There are only two major principals in steps 2 and 3 of business
valuation: risk and marketability, which are both functions of size. Thus,
size is the overriding principle in steps 2 and 3 of the valuation process,
and step 1 determines size. If value depends only on the forecast cash
¬‚ows, risk, and marketability, and the latter two are in turn dependent
on size, then in essence value depends only on size (and possibly control).
That statement sounds like a tautology, but it is not.
This chapter is an attempt to identify the fewest, most basic princi-
ples underlying the inexact science of valuation. The remainder of this
chapter covers the calculations that test the log size model and DLOM
calculations.


TABLE 10-1: LOG SIZE FOR 1938“1986
In Table 10-1 we develop the log size equation for the years 1938“1986.
We use 1938 as the starting year to eliminate the highly volatile Roaring
Twenties and Depression years 1926“1937. The reason we stop at 1986
has to do with the IBA database. The article is based on sales from 1982“
1991.2 We take 1986 as the midpoint of that range and calculate our log
size equation from 1938“1986.
Cells B7“B16 and C7“C16 contain the mean and standard deviation
of returns for the 10 deciles for the period 1938“1986. We need to be able
to regress the returns against 1986 average market capitalization for each
decile. Unfortunately, those values are unavailable and we must estimate
them.
D7“D16 contain the market capitalization for the average ¬rm in
each decile for 1994, the earliest year for which decile breakdowns are
available. E7“E16 are the 1986 year-end index values in Ibbotson™s Table
7-4. F7“F16 are the 1994 year-end index values, with our estimate of in-
come returns removed.3
Column G is our estimate of 1986 average market capitalization per
¬rm for each decile. We calculate it as Column D Column E Column
F. Thus, the average ¬rm size in decile #1 for 1986 is $7.3 billion (G7),
and for decile #10 it is $32.49 million (G16).
Rows 18“35 contain our regression analysis of arithmetic mean re-
turns as a function of the logarithm of the market capitalization”exactly


2. A footnote in the article states that in relation to Figure 1 (and I con¬rmed this with the author,
Raymond Miles), those dates apply to the rest of the article.
3. SBBI, Table 7-4, approximate income returns have been removed from the 1994 values. The
adjustment was derived by comparing the large company stock total return indices with the
capital appreciation indices for 1994 and 1986 per SBBI Tables B-1 and B-2. It was found
that 77.4% of the total return was due to capital appreciation. There were no capital
appreciation indices for small company stocks. We removed 1 77.4% 22.6% of the gain
in the decile index values for deciles #1 through #5, 22.6%/2 11.3% for deciles #6 through
#8, and made no adjustment for #9 and #10. Larger stocks tend to pay larger dividends.




CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 359
360
T A B L E 10-1

Log Size Equation for 1938“1986 NYSE Data by Decile and Statistical Analysis: 1938“1986


A B C D E F G H I

5 Year-End Index Values [1] [D] [E]/[F] Ln [G]

6 Decile Mean Std Dev 94 Mkt Cap 1986 1994 1986 Mkt Cap Ln(Mkt Cap)
7 1 11.8% 15.8% 14,847,774,614 198.868 404.436 7,300,897,357 22.7113
2 14.0% 18.3% 3,860,097,544 434.686 920.740 1,822,371,137 21.3234
9 3 15.0% 19.7% 2,025,154,234 550.313 1,248.528 892,625,877 20.6097
10 4 15.8% 22.0% 1,211,090,551 637.197 1,352.924 570,396,575 20.1618
11 5 16.7% 23.0% 820,667,228 856.893 1,979.698 355,217,881 19.6882
12 6 17.1% 23.8% 510,553,019 809.891 1,809.071 228,566,124 19.2473
13 7 17.6% 26.4% 339,831,804 786.298 1,688.878 158,216,901 18.8795
14 8 19.0% 28.5% 208,098,608 1,122.906 2,010.048 116,253,534 18.5713
15 9 19.7% 29.9% 99,534,481 1,586.521 2,455.980 64,297,569 17.9790
16 10 22.7% 38.0% 33,746,259 6,407.216 6,654.508 32,492,195 17.2965

18 SUMMARY OUTPUT

20 Regression Statistics
21 Multiple R 0.9806
22 R square 0.9617
23 Adjusted R square 0.9569
24 Standard error 0.0064
25 Observations 10

27 ANOVA

28 df SS MS F Signi¬cance F
29 Regression 1 0.0082 0.0082 200.6663 0.0000
30 Residual 8 0.0003 0.0000
31 Total 9 0.0085
33 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%
34 Intercept 0.5352 0.0259 20.6710 0.0000 0.4755 0.5949
35 Ln(Mkt Cap) (0.0186) 0.0013 (14.1657) 0.0000 (0.0216) (0.0156)

[1] SBBI, Table 7-3*, approximate income returns have been removed from the 1994 values. The adjustment was derived by comparing the large company stock total return indices with the capital appreciation indices for 1994 and 1986 per
SBBI Tables B-1 and B-2. It was found that 77.4% of the total return was due to capital appreciation. There were no capital appreciation indices for small company stocks. We removed (1-77.4%) of the gain in the decile index values for
deciles 1 through 5, [(1-77.4%)/2] for deciles 6 through 8, and made no adjustment for 9 and 10. Larger stocks tend to pay larger dividends.
*Used with permission. 1998 Ibbotson Associates, Inc. All rights reserved. [Certain portions of this work were derived from copyrighted works of Roger G. Ibbotson and Rex Sinque¬eld.] Source: CRSP University of Chicago, Used with
permission. All rights reserved.
the same as Table 4-1, regression #2. The regression equation is: r 0.5352
“ 0.0186 ln FMV.4 We use this regression equation in Table 10-2.


TABLE 10-2: RECONCILIATION TO THE IBA DATABASE
Table 10-2 is the main table in this chapter. All other tables provide details
that ¬‚ow into this table.
The purpose of the table is to perform two series of calculations,
which make up part 1 and part 2 of the table, respectively. The ¬rst series
calculates adjusted price to cash ¬‚ow (P/CF) multiples for each size cat-
egory of IBA database results described in the article. The second series
is to calculate theoretical P/CF multiples using the log size equation and
the DLOM methodology in Chapter 7. Ultimately we compare them, and
they match reasonably well.
Unfortunately, there are much data that we do not have, which will
force us to make estimates. There are so many estimates in the following
analysis, that we will not be able to make strong conclusions. It would
be easy to manipulate the results in Table 10-2 to support different points
of view. Nevertheless, it is important to proceed with the table, as we
will still gain valuable insights. Additionally, it points out the de¬ciencies
in the information set available. This is not a criticism of the IBA database.
All of the other transactional databases of which I am aware suffer from
the same problems. This analysis highlights the type of information that
would be ideal to have in order to come to stronger conclusions.


Part 1: IBA P/CF Multiples
We begin in row 6. The mean selling prices in row 6 are the means of the
corresponding range of selling prices reported in the article. Thus, B6

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