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

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

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

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

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