The bottom of Table 9-7 shows the overall regression results. R2 and ad-

justed R 2 are 49.0% (C37) and 41.7% (C38), respectively.22 This means that

the regression model explains 41.7% of the variation in the discounts.

The standard error of the y-estimate is 10.07% (B10). We can form an

approximate 95% con¬dence interval around the regression estimate by

adding and subtracting two standard errors, or approximately 20.1%.

The dummy variable for whether the transaction was pre-1990 or not

is the only variable in the ¬nal regression:

The regression equation is:

Average Discount 0.4679 0.1846 pre-1990

The y-intercept and the x-coef¬cient appear in cells B43 to B44. The

y-intercept of 0.4679 means that when all the independent variables have

a zero value, or in this case when the transaction occurs after the end of

the 1980s, then the average discount from net asset value is 46.79%, or

46.8% rounded.

One should not place great weight on the regression equation above.

It was derived from a small data set, and the explanatory variable for

yield is missing. It is nevertheless relevant evidence of the fractional in-

terest discount in real-world transactions.

Commentary to Table 9-8: Final Calculation of Fractional

Interest Discounts

2.80% Member Interest

To calculate the ¬nal discount, we weight the ¬rst two valuation ap-

proaches equally at 45% each, as both approaches appear equally impor-

tant, valid, and reliable. The third approach we weight only 10% because

of the small and incomplete data set from which its regression model was

derived. The 49.2% (C8) discount calculated using the economic compo-

nents approach comes from Table 9-5, B12, the 47.5% (C9) discount using

the Partnership Pro¬les Regression comes from Table 9-6, D37, and the

46.8% (C10) discount comes from Table 9-6, C43.23 The weighted average

of the three discounts is 48.2% (E11), which we round to 48% in E12.

Final Calculation of FMV of Fractional Interests

Gifts Transferred on December 25, 1999. To calculate the FMV of

the 2.80% fractional interests gifted on December 25, 1999, we calculate

the dollar value of a 100% fractional interest discount, $1,389,185 48%

$666,809 (B18 B19 B20). Note that B18 is the FMV of the equity

before discounts, which comes from Table 9-2, C22.

Next we subtract this discount from the FMV of equity to determine

the FMV of a 100% Fractional Interest, $1,389,185 $666,809 $722,376

22. The adjusted R2 is a downward adjustment to remove the effects of irrelevant variables

randomly increasing R2.

23. As this is not a pre-1990 valuation, the regression estimate for the discount is equal to the

intercept coef¬cient.

CHAPTER 9 Sample Appraisal Report 345

346

T A B L E 9-7

Private Fractional Interest Sales [1]

A B C D E F G H I J K

4 Pro-Rata

5 Date Size FMV-100% Value Price Discount Pre 1990 GP

6 1 Linda Vista Rd., S Diego [2] [3] GP 1/1/1984 6.7% NA NA NA 20.0% 1 1

7 2 Eighth Ave., S. Diego TIC 9/1/1985 66.7% 60,000 40,000 27,000 32.5% 1 0

8 3 Fifth Ave., S. Diego [2] [4] GP 4/1/1988 33.3% 3,000,000 1,000,000 675,000 32.5% 1 1

9 4 West 61st St., Los Angeles TIC 10/1/1996 33.3% 90,000 30,000 10,000 66.7% 0 0

10 5 Garden Grove Ave., Reseda TIC 10/1/1996 25.0% 145,000 36,250 22,000 39.3% 0 0

11 6 K St., S. Diego TIC 2/1/1998 20.0% 325,000 65,000 36,000 44.6% 0 0

12 7 So. Calif. [5] LP 7/1/1998 2.5% 5,460,000 136,500 75,000 45.1% 0 0

13 8 So. Calif. [5] LP 7/1/1998 0.5% 14,800,000 74,000 37,000 50.0% 0 0

14 9 Brant St., S. Diego TIC 6/1/1999 50.0% 1,680,000 840,000 545,000 35.1% 0 0

[1] Source of data: Jones, Roach & Caringella, Real Estate Appraisers, S. Diego.

[2] These are interests in General Partnerships, not GP interests in Limited Partnerships.

[3] The seller reportedly would not hypothecate her interest for a required construction loan and offered to sell her interest to another partner who would. This indicates that the seller may have had unusual leverage, which may have

reduced the discount.

[4] We have reduced the nominal selling price of $750,000 by 10% to account for the ˜˜very bene¬cial ¬nancing™™ provided to the buyer by the seller. Exact details of the ¬nancing agreement are unknown.

[5] The seller had tried to sell his interest on the open market and had offers at a 75% discount from pro rata value. He then sold his interest to the GP, who had tried to discourage the sale. It is quite possible that the GP did not extract

the full market discount and that the full discount was actually 75%. This is because the GP is in the business of forming partnerships, not taking advantage of limited partners.

REGRESSION RESULTS [6]

35 Regression Statistics

36 Multiple R 0.7000

37 R square 0.4900

38 Adjusted R square 0.4171

39 Standard error 0.1007

40 Observations 9

42 Coeff Std Error t Stat P-value Lower 95% Upper

95%

43 Intercept 0.4679 0.0411 11.3857 0.0000 0.3708 0.5651

44 Pre 1990 0.1846 0.0712 2.5933 0.0358 0.3529 0.0163

[6] We found that with the constraints of the available data, the best explanatory variable for discounts was the time of the transaction, where the time is categorized pre-1990 or post-1990.

This might best be explained by investors™ increased perception of risk due to the real estate crash in 1990. Had it been available, we would expect that using yield as a second independent

variable would have signi¬cantly increased the explanatory power of the regression.

T A B L E 9-8

Final Calculation of Fractional Interest Discount

A B C D E

4 2.80% Member Interest

6 Indication Wtd Avg

7 Table of Discount Weight Discount

8 Economic components approach 9-5, B12 49.2% 45.0% 22.1%

9 Regression of partnership pro¬les database 9-6, D37 47.5% 45.0% 21.4%

10 Regression of private fractional interest data [1] V-3, C43 46.8% 10.0% 4.7%

11 Total 100.0% 48.2%

12 Round to 48%

14 Final Calculation of FMV of Fractional Interests

16 Date of gift 12/25/99 1/3/00

17 Fractional interest 2.80% 2.25%

18 FMV of equity (Table 9-2, C22) $1,389,185 $1,389,185

19 Fractional interest discount-% (E11) 48% 48%

20 Fractional interest account-$ $ 666,809 $ 666,809

21 100% fractional interest FMV of equity $ 722,376 $ 722,376

22 FMV of 2.80% and 2.25% member interests $ 20,227 $ 16,253

23 Rounded FMV of 2.80% and 2.25% member interests $ 20,000 $ 16,250

Note: Mrs. Smith intends to make four gifts of $16,250 on approximately 1/3/2000. We have ignored second-order effects of gifting a smaller interest (2.25% vs. 2.80%) in Table 9-5A,

B15, as it has no impact on the ¬nal calculation. Thus, in our opinion, a 2.25% interest gifted on 1/3/2000 has a FMV of $16,250.

[1] As this is not a pre-1990 valuation, the regression estimate for the discount is equal to the intercept coef¬cient. We do not weight this valuation method heavily, due to the relatively

little data that were available for the regression.

(B18 B20 B21). Finally, we multiply the FMV of a 100% fractional

interest by the 2.80% interest to calculate the FMV of the 2.80% member

interest, $722,376 2.80% $20,227 (B21 B17 B22), which we round

to $20,000 in B23.

Gifts Transferred on January 3, 2000. Mrs. Smith also made four

gifts of 2.25% LP interests on January 3, 2000, which we value in column

C. We multiply the $722,376 (C21 B21) 100% fractional interest fair

market value of the equity by a 2.25% member interest in C17 to arrive

at a fair market value of $16,253 (C22), which we round to $16,250 (C23).

The calculations in Table 9-5A depend on the size of the interests in

cell B15. However, the change from a 2.80% member interest to a 2.25%

member interest is so small that it actually has no impact. Thus, we use

the same fractional interest discount of 48% for the January 3, 2000 gifts.

Conclusion

In our opinion, subject to this report and the Statement of Limiting Con-

ditions, the appropriate fractional interest discount for a 2.80% and 2.25%

member interests in the LLC as of December 25, 1999 through January 3,

2000 is 48%, which amounts to fair market values of $20,000 and $16,250,

respectively.

CHAPTER 9 Sample Appraisal Report 347

STATEMENT OF LIMITING CONDITIONS

In accordance with recognized professional ethics, the fee for this service

is not contingent upon our conclusion of fractional interest discount, and

neither Abrams Valuation Group nor any of its employees has a present

or intended interest in the subject interest.

We have relied upon ¬nancial information provided by Bradley Jones

and David Sofer, CPA, and have accepted it as correct without further

veri¬cation. We assume there are no material transactions between De-

cember 14, 1999, the date of the LLC™s ¬nancial statements, and December

25, 1999, and January 3, 2000, the dates of the gifts. For use of this report

in the year 2000, we assume there are no material changes in property

values and that there are no material changes in the equity of the LLC or

its member interests.

All other information used in this report is from sources we deem

reliable. We have accurately re¬‚ected such information in this report;

however, we make no representation as to our sources™ accuracy or com-

pleteness and have accepted their information without further veri¬ca-

tion.

We have not made a physical visit to the properties. We assume that

the present owners would continue to maintain the character and integ-

rity of the property through any sale, reorganization, or diminution of

the owners™ participation or equity interest. We also assume there are no

present or future ˜˜skeletons in the closet,™™ e.g., environmental problems

with the property, litigation, and so on.

Our opinion of the fractional interest discount in this report is valid

only for the stated purpose and only for the effective dates of the ap-

praisal. It is our understanding that this opinion will be used for gift tax

purposes. The fractional interest discount shall not be used for other pur-

poses and cannot even be used for the same purposes and time frame for

different size member interests, as they could be misleading and danger-

ous. Though some similarities exist between the fractional interest dis-

count for this purpose and others, it would be incorrect to use the dis-

count as determined in our report for any other purposes. Speci¬c timing,

performance, and marketability issues that arise in evaluating the fair

market value of the properties and related ownership interests could

change the results. Accordingly, any such use of the fractional interest

discount as determined in this report for other purposes or effective dates

may be inaccurate and misleading, and no such use shall be made with-

out our written consent.

Our determination of the fractional interest discount does not rep-

resent investment advice of any kind to any person and does not consti-

tute a recommendation as to the purchase or sale of shares of the property

or related interests or regarding any other course of action.

Future services regarding the subject matter of this report, including,

but not limited to, testimony or attendance in court shall not be required

of Abrams Valuation Group unless previous arrangements have been

made in writing.

No part or all of the contents of this report shall be conveyed to the

public through advertising, public relations, news, sales, mail, direct

PART 3 Adjusting for Control and Marketability

348

transmittal, or other media without the prior written consent and ap-

proval of Abrams Valuation Group. This report may only be distributed

in its entirety to those directly involved with the purpose of this study.

All other users are to be considered unintended users.

This report may not be distributed in part, as only a thorough read-

ing of this report can accurately convey the logic contained within. Ex-

cerpts taken out of context can be dangerously misleading and are there-

fore forbidden without the written consent of Abrams Valuation Group.

APPRAISER™S QUALIFICATIONS

Jay B. Abrams, ASA, CPA, MBA, author and inventor, is a nationally

recognized valuation economist.

Mr. Abrams lectured at the June 1996 Toronto International Confer-

ence of the American Society of Appraisers, the organization from which

he holds the professional designation of Accredited Senior Appraiser

(ASA) in Business Valuation. He has lectured for the National Association

of Certi¬ed Valuation Analysts and the Anthony Robbins™ Financial Mas-

tery Seminar.

Mr. Abrams has provided services to clients representing a variety

of organizations from small entrepreneurs to Columbia Pictures, Dr. Pep-

per, Purex Corporation, and other Fortune 1000 ¬rms in the area of in-

tangibles, including goodwill, customer lists, licensing agreements, con-

tracts, and business enterprise and capital stock appraisals for numerous

purposes, including the following:

— Employee stock ownership plans (ESOPs).

— Estate planning, estate and gift taxes.

— Income taxes and charitable contributions.

— Mergers and acquisitions and sales.

— Divestitures.

— Warrants and stock options.

— Shareholder buy/sell agreements.

— Blocks of publicly traded securities.

— Private placements and public offerings.

— Restricted securities.

— Recapitalization and reorganizations.

— Debt and equity ¬nancing.

— Company dissolutions.

— Litigation settlement.

Additionally, Mr. Abrams has prepared and given expert testimony

in the capital stock and business enterprise valuation areas in various

courts of law.

Mr. Abrams™ valuation experience encompasses a wide array of in-

dustries and assignments, for mergers/acquisitions, sales and leaseback,

litigation support, leveraged buyouts, and stockholder agreements. Mr.

Abrams was Vice-President of Paci¬c Corporate Valuation, Inc. in charge

of the valuation practice, and he was a Project Manager at Arthur D. Little

CHAPTER 9 Sample Appraisal Report 349

Valuation, Inc. He was a cofounder and president of Raycom, a radio

communications ¬rm, and prior to this was an auditor with Arthur An-

dersen & Company. Mr. Abrams received his MBA from the University

of Chicago in ¬nance and marketing, where he also pursued graduate

studies in economics.

Mr. Abrams invented and published the Abrams Table of Equity Pre-

mia and has published an article quantifying the discount for lack of

marketability. He invented several formulas for valuing leveraged ESOPs,

as well as the Abrams Table of Accounting Transposition Errors, used for

troubleshooting such errors. He also wrote software to automatically gen-

erate a table of potential sources of error.

Mr. Abrams™ writings include:

— Quantitative Business Valuation: A Mathematical Approach for Today™s

Professionals, McGraw-Hill, November 2000.

— ˜˜ESOPs: Measuring and Apportioning the Dilution,™™ Valuation,

June 1997.

— ˜˜Discount Rates as a Function of Log Size and Valuation Error

Measurement,™™ The Valuation Examiner, February/March, 1997.

— ˜˜An Iterative Valuation Approach,™™ Business Valuation Review,

March 1995.

— ˜˜A Breakthrough in Calculating Reliable Discount Rates,™™

Valuation, August, 1994.

— ˜˜Discount for Lack of Marketability: A Theoretical Model,™™

Business Valuation Review, September, 1994.

— ˜˜Cash Flow: A Mathematical Derivation,™™ Valuation, March 1994.

— ˜˜An Iterative Procedure To Value Leveraged ESOPs,™™ Valuation,

January 1993.

— ˜˜How to Quickly Find and Fix Accounting Transposition Errors,™™

The Practical Accountant, June 1992.

— Coauthor of ˜˜Valuation of Companies for ESOP Purposes,™™

Chapter 8 in Employee Stock Ownership Plans by Robert W. Smiley,

Jr. and Ronald J. Gilbert, Prentice Hall/Rosenfeld Launer

Publications, New York, 1989.

— ˜˜The Annuity Discount Factor: Generalization, Analysis of

Special Cases, and Relationship to the Gordon Model and Fixed-

Rate Loan Amortization,™™ unpublished.

BIBLIOGRAPHY

Mercer, Z. Christopher. 1997. Quantifying Marketability Discounts. Memphis, Tenn.: Pea-

body.

Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Business, 3d

ed. Burr Ridge, Ill.: McGraw-Hill.

PART 3 Adjusting for Control and Marketability

350

APPENDIX

Tax Court™s Opinion for Discount for Lack of

Marketability24

INTRODUCTION

The U.S. Tax Court outlined a list of 10 nonexclusive factors that in the

Court™s opinion affect discount for lack of marketability (DLOM). We ¬rst

present its list and then we comment on each item as to how we consid-

ered it in our analysis.

THE COURT™S TEN FACTORS

The Court™s 10 factors are:

1. The value of the subject corporation™s privately traded

securities vis-a-vis its publicly traded securities (or, if the

`

subject corporation does not have stock that is traded both

publicly and privately, the cost of a similar corporation™s public

and private stock). These are known as ˜˜Letter Stock™™ or

restricted securities, the restrictions arising from Section 144 of

the Securities Exchange Commission Rules.

2. An analysis of the subject corporation™s ¬nancial statements.

3. The corporation™s dividend-paying capacity, its history of

paying dividends, and the amount of its prior dividends.

4. The nature of the corporation, its history, its position in the

industry, and its economic outlook.

5. The corporation™s management.

6. The degree of control transferred with the block of stock to be

valued.

7. Any restriction on the transferability of the corporation™s stock.

8. The period of time for which an investor must hold the subject

stock to realize a suf¬cient pro¬t.

9. The corporation™s redemption policy.

10. The cost of effectuating a public offering of the stock to be

valued, e.g., legal, accounting, and underwriting fees25

The Court in general had the right idea. It created a list of criteria with

which to judge the difference in marketability between the source of the

valuation data and the asset to which we are applying the data.

24. Bernard Mandelbaum, et al. v. Commissioner, TCM, CCH Dec. 50, 687(M), 1995-254.

25. See Estate of Gilford v. Commissioner [Dec. 43,622], 88 T.C. 38, 60 (1987); Northern Trust Co. v.

Commissioner [Dec. 43,261], 87 T.C. 349, 383-389 (1986); see also Rev. Rul. 77287, 1977-2 C.B.

319 (valuation of restricted securities).

CHAPTER 9 Sample Appraisal Report 351

APPLICATION OF THE COURT™S 10 FACTORS TO

THE VALUATION

In this section we will address the factors in the Tax Court™s opinion and

demonstrate how we have incorporated those factors into our analysis of

the discount for lack of marketability (DLOM), which, combined with the

discount for lack of control (DLOC), forms the fractional interest discount.

The following analysis applies regardless of the form of the subject entity,

whether it is a common stock interest in a corporation, Limited or General

Partnership interest, LLC interest, etc. We use the terms the entity and the

subject interest to maintain generality.

1. We estimate the letter stock discount in the delay-to-sale

component of the economic components approach. We do this

either by a regression analysis or a Black“Scholes put option

calculation, depending on the availability of data.

2. Our analysis of the entity™s ¬nancial statements is incorporated

into the calculation of DLOM in the calculation of transactions

costs, expected growth rates, the discount rate, and the delay-

to-sale component of the economic components approach.

3. We incorporate the dividends (distributions) into the analysis

in the Partnership Pro¬les (PP) approach. This is the single

most important factor in the regression model. Dividends or

distributions are also incorporated indirectly into the

calculation of DLOM through their effect on the growth rate

and, therefore, the transactions costs.

4. The nature of the entity and its history, industry position,

composition of assets, and economic outlook are factors that

are very signi¬cant in the valuation of the underlying assets.

The comments to item 2 apply here as well.

5. Management can be signi¬cant in determining the fractional

interest discount because of two factors: its dividend policy,

which is already considered in 3, and more importantly, its

potential for making decisions that favor one group of owners

over another or withholding bad news from any ownership

group. Owners of private interests generally have more

in¬‚uence with their management than the LPs in the

Partnership Pro¬les database would with theirs. We consider

this factor in the adjustment for increased in¬‚uence in

Partnership Pro¬les approach.

6. The degree of control of the block of stock is, strangely enough,

signi¬cant in calculating DLOM. The reason why that appears

strange is that the degree of control has its own discount”the

DLOC”for lack of control. Why then does the degree of

control in¬‚uence the DLOM? The reason is found in the above-

mentioned book, Chapter 7 in Quantitative Business Valuation: A

Mathematical Approach for Today™s Professionals. We calculate

control premiums and discount for lack of control by looking

at control premiums paid for marketable minority interests in

the stock market. But control matters less in publicly held ¬rms

PART 3 Adjusting for Control and Marketability

352

than in privately held ¬rms because the former generally have

both current cash ¬‚ow in the form of dividends and instant

marketability in the ability to cash out in three days.

Furthermore, management of publicly held ¬rms are generally

managing to maximize the per-share values of the minority

shareholders. With privately held ¬rms, control shareholders

often divert wealth from minority shareholders (similarly

general partners in LPs and managing members in LLCs can

divert wealth from limited partners and other members). It is

rare to see dividends in closely held corporations, and there is

generally no ability to cash out. Therefore, there is an

interactive effect in being both a minority shareholder (or LP/

minority interest member) and owning an interest in a private

¬rm. The whole is worse than the sum of the two parts. As in

5 above, we have considered this in the adjustment for

increased in¬‚uence in the Partnership Pro¬les approach.

The marketability of private interests is limited, since there is

7.

no formal market for such interests. We consider this limitation

in the economic components approach in the following ways:

(a) in the calculation of the delay-to-sale component; and (b) in

accounting for the buyer™s monopsony power (component 2).

In the Partnership Pro¬les approach, we implicitly considered

the lack of marketability in selecting the discount for lack of

public registration.

We incorporate time horizons into the delay-to-sale component

8.

in the economic components approach and the adjustment for

lack of public registration in the Partnership Pro¬les approach.

We also incorporate time horizons in another fashion in the

selection of the variable j”the average years between sales”in

the economic components approach.

The entity™s redemption policy is relevant in determining one™s

9.

ability to cash out of an investment. The subject entity does not

provide a redemption option.

The cost of undergoing an initial public offering is about 15“

10.

18% for a small ¬rm. There is a possibility that an IPO might

lower the discount by making the subject interest more

marketable. However, the cost of the IPO and the subsequent

regulatory administrative costs would be prohibitive in the

case, and we don™t need to account for the IPO possibility here.

CHAPTER 9 Sample Appraisal Report 353

PART FOUR

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

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

$25,000, which is the mean selling price for ¬rms in the $0 to $50,000

category. At the high end, H6 $750,000, which is the mean price in the

$500,000 to $1 million sales price category.

Row 7 is the mean P/E multiple reported in the article. Note that

the P/E multiple constantly rises as the mean selling price rises. Figure

10-1 shows this relationship clearly. Row 8 is owner™s discretionary in-

come, which is row 6 divided by row 7, i.e., P P/E E, where P is

price and E is earnings.

The IBA™s de¬nition of owner™s discretionary income is net income

before income taxes and owner™s salary. It does not conform to the arm™s-

length income that appraisers use in valuing businesses. Therefore, we

subtract our estimate of an arm™s-length salary for owners, which we do

in row 9. This is an educated guess, but Raymond Miles felt my estimates

were reasonable.

In row 10, we add back personal expenses charged to the business.

Unfortunately, no one has any data on this. I have asked many account-

ants for their estimates, and their answers vary wildly. Ultimately, I de-

cided to estimate this at 10% (cell B33) of owner™s discretionary income

(row 8).

4. For public ¬rms, this is market capitalization, i.e., price per share number of shares.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 361

362

T A B L E 10-2

Reconciliation to IBA Database

A B C D E F G H I

4 Part 1: IBA P/CF Multiples

6 Mean selling price: Illiquid 100% Int 25,000 75,000 125,000 175,000 225,000 375,000 750,000 Avg

7 Mean P/E ratio 1.66 2.11 2.44 2.74 3.06 3.44 4.26

8 Owner™s discretionary inc [6]/[7] 15,060 35,545 51,230 63,869 73,529 109,012 176,056

9 Arm™s length salary 22,500 25,000 30,000 35,000 40,000 50,000 75,000

10 Personal exp charged to bus”assume B33* [8] 1,506 3,555 5,123 6,387 7,353 10,901 17,606

11 Adjusted net income [8] [9] [10] (5,934) 14,100 26,352 35,255 40,882 69,913 118,662

12 Effective corp. inc tax rate 0% 0% 0% 0% 0% 0% 0%

13 Adjusted inc taxes 0 0 0 0 0 0 0

14 Adj net inc after tax (5,934) 14,100 26,352 35,255 40,882 69,913 118,662

15 Cash ¬‚ow/net income (assumed) 95% 95% 95% 95% 95% 95% 95%

16 Adj cash ¬‚ow after tax [14] * [15] (5,637) 13,395 25,035 33,493 38,838 66,417 112,729

17 Avg disc to cash equiv value (Table 10-3) 6.7% 6.7% 6.7% 6.7% 6.7% 6.7% 6.7%

18 Adj sell price (illiq 100% int) {1 [17]}*[6] 23,317 69,951 116,585 163,220 209,854 349,756 699,512

19 Adjusted price/cash ¬‚ow multiple [18]/[16] NM 5.2 4.7 4.9 5.4 5.3 6.2

21 Part 2: Log Size P/CF Multiples

22 Control prem-% (1982“1991 Avg) [note 1] 25% 25% 25% 25% 25% 25% 25%

23 DLOM-% (Tables 10-6, 10-6A, 10-6B, etc.) 9.9% 10.1% 10.2% 10.2% 10.5% 12.4% 18.6%

24 Adj sell price (mkt min) [18]/{(1 [22])*(1 [23])} 20,704 62,221 103,838 145,440 187,511 319,458 687,614

25 Discount rate r .5352 .0186 ln (FMVMkt Min) 35.0% 33.0% 32.0% 31.4% 30.9% 29.9% 28.5%

26 Growth rate g (assumed) 2.0% 2.5% 3.0% 4.0% 4.5% 5.0% 6.0%

27 Theoretical P/CF (1 g)*SQRT(1 r)/(r g) 3.6 3.9 4.1 4.4 4.5 4.8 5.3

28 P/CF-Illiquid control [27]*(1 [22])*(1 [23]) 4.0 4.4 4.6 4.9 5.1 5.3 5.4

29 Error {1 [28]/[19]} NM 16.5% 1.7% 0.2% 6.3% 0.2% 12.5% 4.1%

30 Absolute error [note 2] NM 16.5% 1.7% 0.2% 6.3% 0.2% 12.5% 4.2%

31 Squared error [note 2] 2.7% 0.0% 0.0% 0.4% 0.0% 1.6% 0.4%

33 Personal exp % of Owner™s discretionary inc 10%

35 Sensitivity Analysis: How the error varies with Cell B33 Error

personal exp

37 2% 17.3%

38 4% 14.0%

39 6% 10.7%

40 8% 7.4%

41 10% 4.1%

[1] Approximate midpoint of the 21% to 28% control premium estimated in Chapter 7

[2] The averages are for the last 5 columns only, as the sales under $100,000 are mostly likely asset-based, not income based.

F I G U R E 10-1

P/E Ratio as a Function of Size (From the IBA Database)

4.5

4

3.5

3

P/E Multiple

2.5

2

1.5

1

0.5

0

25,000 75,000 125,000 175,000 225,000 375,000 750,000

Average Selling Price

Row 11 is adjusted net income, which is row 8 row 9 row 10.

Row 12 is an estimate of the effective corporate income tax rate. This is

a judgment call. An accountant convinced me that even for the $1 million

sales, the owner™s discretionary income is low enough that it would not

be taxed at all. Any excess remaining over salary would be taken out of

taxable income as a bonus. I acceded to his opinion, though this point is

arguable”especially for the higher dollar sales. It is true that what counts

here is not who the seller is, but who the buyer is. A large corporation

buying a small ¬rm would still impute corporate taxes at the maximum

rate; however, only the last category is at all likely to be bought by a large

¬rm, and even then, most buyers of $0.5 to $1 million ¬rms are probably

single individuals. Therefore, it makes sense to go with no corporate

taxes, with a possible reservation in our minds about the last column.

With this zero income taxes assumption, row 13 equals zero and row

14, adjusted income after taxes, equals row 11.

Next we need to convert from net income to cash ¬‚ow. Again, the

information does not exist, so we need to make reasonable assumptions.

For most businesses, cash ¬‚ow lags behind net income. Most of these are

small businesses that sold for fairly small dollar amounts, which means

that expected growth”another important missing piece of information”

must be low, on average. The lower the growth, the less strain on cash

¬‚ow. We assume cash ¬‚ow is 95% of adjusted net income. It would be

reasonable to assume this ratio is smaller for the higher value businesses,

which presumably have higher growth. We do not vary our cash ¬‚ow

ratio, as none of these are likely to be very high-growth businesses. Thus,

all cells in row 15 equal 95%. In row 16 we multiply row 14 by row 15

to calculate adjusted after-tax cash ¬‚ow.

The next step in adjusting the IBA multiples is to reduce the nominal

selling price to a cash-equivalent selling price, which we calculate in Table

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 363

10-3. Exhibit 33-3 in Pratt (Pratt 1993) shows a summary of sale data from

Bizcomps. Businesses selling for less than $100,000 have a 60% average

cash down, and businesses selling for more than $500,000 have an aver-

age 58% cash down. Using a 60% cash down, we assume the seller ¬-

nances the 40% (Table 10-3, B11) balance for 7 years, which is 84 months

(B8, C8) at 8% (B5) with a market rate of 14% (C5).

The annuity discount factor (ADF), the formula for which is

r)n]

1 [1/(1

ADF

r

is 53.3618 (C9) at the market rate of interest and 64.15926 (B9) at the

nominal rate. One minus the ratio of two equals the discount to cash

equivalent value if the loan is 100% ¬nanced, or

53.3618

1 16.8%

64.15926

(B10). We multiply this by the 40% ¬nanced (B11) to calculate the average

discount to cash equivalent value of 6.7% (B12), which we transfer back

to Table 10-2, row 17.

Multiplying the mean selling price in row 6 by one minus the dis-

count to cash equivalent value in row 17 leads to an adjusted mean selling

price in row 18. For example, $25,000 (1 6.7%) $23,317 [B6

(1 B17) B18].

Finally, we divide row 18 by row 16 to calculate the adjusted price

to cash ¬‚ow (P/CF) multiple for the IBA database. In general, the P/CF

multiple rises as price rises, although not always. There is no meaningful

P/CF multiple in B19, because adjusted cash ¬‚ow in B16 is negative. The

P/CF multiples begin in C19 at 5.2 for a mean selling price of $75,000,

then decline to 4.7 (D19) for a mean selling price of $125,000, and rise

steadily to 6.2 (H19) for a mean selling price of $750,000. The only excep-

tion is that the P/CF is greater at 5.4 for the $225,000 selling price than

at 5.3 for the $375,000 selling price. The ¬rst anomaly is probably not

signi¬cant, because many, if not most, ¬rms selling under $100,000 are

T A B L E 10-3

Proof of Discount Calculation

A B C

4 Nominal Market

5 r 8% 14%

6 i r/12 0.6667% 1.1667%

7 Yrs 7 7

8 n Yrs *12 84 84

9 ADF @ 14%, 84 mos. 64.15926114 53.36176

10 Discount on total prin 16.8%

11 % ¬nanced 40%

12 Discount on % ¬nanced 6.7%

PART 4 Putting It All Together

364

priced based on their assets rather than their earnings capacity. The sec-

ond anomaly, from P/CF of 5.4 to 5.3, is a very small reversal of the

general pattern of rising P/CF multiples in the IBA database.

Part 2: Log Size P/CF Multiples

In this section of Table 10-2 we will calculate ˜˜theoretical™™ P/CF multiples

based on the log size model and the DLOM calculations in Chapter 7.

The term theoretical is somewhat of a misnomer, as the calculation of both

the log size equation and DLOM is empirically based. Nevertheless, we

use the term for convenience.

Before we can apply the log size equation from Table 10-1, we need

a marketable minority interest FMV, while the adjusted selling price

(FMV) in row 18 is a illiquid control value. Therefore, we need to divide

row 18 by one plus the control premium times one minus DLOM, which

we do in row 24. We assume a control premium of 25% (row 22), which

is the approximate midpoint of the 21“28% range of control premiums

discussed in Chapter 7.

The calculation of DLOM is unique for each size category and ap-

pears in Tables 10-6 and 10-6A“10-6F. We will cover those tables later. In

the meantime, DLOM rises steadily from 9.9% (B23) for the $25,000 mean

selling price to 18.6% (H23) for the $750,000 mean selling price category.

Row 24, the marketable minority FMV, is row 18 [(1 row 22)

(1 row 23)]. The marketable minority values are all lower than the

illiquid control values, as the control premium is much greater in mag-

nitude than DLOM.

We calculate the log size discount rate in row 25 using the regression

equation from Table 10-1. It ranges from a high of 35.2% (B25) for the

smallest category to a low of 28.7% (H25) for the largest category.

Next we estimate the constant growth rates that the buyers and sell-

ers collectively implicitly forecast when they agreed on prices. It is un-

fortunate that none of the transactional databases that are publicly avail-

able contain even historical growth rates, let alone forecast growth rates.

Therefore, we must make another estimate. We estimate growth rates to

rise from 2% (B26) to 6% (H26), growing at 0.5% for each category, except

the last one going from 5% to 6%. It is logical that buyers will pay more

for faster growing ¬rms.

In row 27 we calculate a midyear Gordon model:

1 r

(1 g)

r g

with r and g coming from rows 25 and 26, respectively.5 This is a mar-

ketable minority interest P/CF multiple when cash ¬‚ow is expressed as

the trailing year™s cash ¬‚ow. In row 28 we convert this to an illiquid

control P/CF by doing the reverse of the procedure we performed in row

5. The purpose of the (1 g) term is correct for the fact that we are applying it to each dollar of

prior year™s cash ¬‚ow and not to the customary next year™s cash ¬‚ow.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 365

24”we multiply by one plus the control premium and one minus DLOM,

i.e., P/CFIlliq Control P/CFMM (1 CP) (1 DLOM) row 27

(1 row 22) (1 row 23).

In row 29 we calculate the error, which is one minus the ratio of row

28 divided by row 19, or one minus the ratio of the forecast log size-

based P/CF to the IBA™s adjusted P/CF. Row 30 is the absolute value of

the errors in row 29. The absolute values of the errors are most extreme

for the low and high values of the mean selling price, with a 16.5% (C30)

absolute error for the $75,000 mean selling price and a 12.5% (H30) ab-

solute error for the $750,000 selling price, with small absolute errors in

between ranging around 0.2“6.3%. The mean error is 4.1% (I29).6

Conclusion

The mean absolute error is 4.2% (I30). Rounding this to 4%, that is a very

respectable result. It is evidence supporting the log size model in Chapter

4 and control premium and economic components model of DLOM in

Chapter 7.

Nevertheless, as mentioned before, there are too much missing data

and resulting guesswork to come to solid conclusions. The estimates are

all reasonable, but one could make different reasonable estimates and

come to very different results. Thus, this analysis is worthwhile evidence,

but it proves nothing.

In the remainder of the chapter we will describe the DLOM calcu-

lations in Tables 10-4, 10-6, and their variations as 10-4A, 10-6A, etc.

CALCULATION OF DLOM

As discussed in Chapter 7, there are three components in the economic

components model to the calculation of DLOM. Components #1 and #3,

the delay to sale and transactions costs components, require unique anal-

ysis for each IBA size category. Therefore, we have one spreadsheet for

each of the two components for each IBA size category. Tables 10-4 and

10-6 are the calculations of components #1 and #3, respectively, for the

$25,000 mean selling price ¬rm. Additionally, Table 10-6 contains the

DLOM calculations. We will describe these tables in detail. Tables 10-4A

and 10-6A are identical to Tables 10-4 and 10-6, the only difference being

that these are calculations for the $75,000 mean selling price ¬rms. This

series continues all the way through Tables 10-4F and 10-6F for the

$750,000 mean selling price IBA category. Table 10-5 contains the calcu-

lations of the buyer and seller transactions costs for all size categories.

Table 10-4: Computation of the Delay-to-Sale

Component”$25,000 Firm

Table 10-4 is identical to Table 7-10, except that we are customizing the

calculation for this IBA category of ¬rm. We begin by inserting the selling

6. This excludes the $75,000 mean selling price errors, as that is likely due to the sale being priced

on an asset rather than an income basis. We also exclude this category in the other measures

of mean error.

PART 4 Putting It All Together

366

T A B L E 10-4

Calculation of Component #1”Delay to Sale”$25,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2

6 5.33E 18 5.625E 09 0.0%

7 Value of block-post-discount [2] 4.26E 09 $ 25,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $ 25,000 0.0%

9 Earnings stability [3] 0.1376 0.4200 5.8%

10 Revenue stability [3] 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.2500 3.3%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $ 25,000

16 Selling price $ 25,000

17 Adjusted net income $ (5,934)

18 Assumed pre-tax margin NA

19 Sales $ 75,000

Sales2

20 5.625E 09

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

price in B16 and adjusted net income in B17. For the larger IBA categories,

net income (owner™s discretionary income) is positive, and we divide that

by an assumed pretax margin of 5% in B18 to estimate sales in B19. We

cannot do that for the $25,000 sales category only, because of net losses.

We estimate sales at three times the selling price, or $75,000 (B19). The

square of sales is then $5.625 109, which is calculated in B20 and trans-

ferred to C6.7

We insert the $25,000 mean selling price in C8, C14, and C16. Here

we are calculating the value of 100% of the stock, so the block value and

the value of the entire ¬rm will be identical, which is not true in the

restricted stock calculations in Table 7-10.8

Cell C7 is the post-discount value of the block. However, both C7

and C14 equal $25,000. This is because the discount calculation came to

zero (D12). Normally, C7 would be lower than C14.

A correlation analysis of the Management Planning data, not shown

in the book, revealed that ¬rm size and earnings and revenue stability

are uncorrelated. Thus, we use the averages from Table 7-5, G60 and H60

of 0.42 (C9) and 0.69 (C10), respectively.

7. The calculations in B16 to B20 did not appear in Table 7-10, as they were unnecessary there.

8. Technically, we should be using the marketable minority FMV rather than the illiquid control

FMV in Table 10-4 (and its variants 10-4A, etc.), cell C14 (which also affects C7 and C8).

However, we do not yet know the marketable minority FMV, as that is the point of the

exercise. To even attempt to calculate it would require multiple iterations, which would

greatly complicate the analysis and add nothing, as the regression coef¬cients in B7 and B8

are so small that the difference is immaterial. Therefore, we use the illiquid control values.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 367

Finally we assume that a $25,000 ¬rm takes only three months, or

0.25 (C11) years to sell. Summing D5 through D11 actually results in a

slightly negative discount, which does not make sense. Therefore, we use

a spreadsheet formula to calculate D12 as the maximum of the sum of

D5:D11 and zero. The delay to sale component is zero for all size cate-

gories except $375,000 and $750,000. The calculations of component #1 of

DLOM for those two categories appear in Tables 10-4E and 10-4F. The

main reason for this is that we assume is takes either 0.25 years or 0.33

years to sell ¬rms under the $375,000 category, while we assume that it

takes 0.5 years and 1.0 years to sell in the $375,000 and $750,000 catego-

ries, respectively (Tables 10-4E and 10-F, C11). The resulting discounts are

still small in magnitude. In Table 10-4E, D12, we calculate component #1

as 1.9%, and in Table 10-4F, D12, we calculate component #1 as 8.4%.

Though we did not elect to do so here, it would be a reasonable

approach to rely on our ¬ndings in Chapter 7 that the regression analysis

does not work well for delays to sale of much less than a year. That being

the case, it would make sense to use a different model”even something

so simple as a present value”to calculate the delay to sale component

for under one year. For example, if we assume a 25% discount rate, a

three-month delay to sale implies a 5% discount as component #1, and a

four-month delay to sale implies a 7% discount as component #1. It is

important to recognize that not all models work well across all ranges of

data, and sometimes circumstances force us to use different models. For

simplicity in this analysis, we did not elect to use another model.

Table 10-5: Calculation of Transactions Costs

Table 10-5 contains our calculations of transactions costs for both buyer

and seller for all of the IBA size categories. Column A denotes whether

the transactions costs are for buyer or sellers. Column B is the mean

selling price of the IBA study. Column C is the base 10 logarithm of

column B.

Columns D and F contain, respectively, the x-coef¬cient and the con-

stant from the regression in Table 7-11. In column E we multiply column

C by column D. We add columns E and F together to obtain column G,

which is the regression forecast of all transactions costs except for the

business broker (or investment banker). Column H contains the business

broker fees, which we assume at 10% for sellers and zero for buyers.

Finally, column I is the grand total forecast of transactions costs for buyers

and sellers by size category. Note that both buyer and seller transactions

costs decline as ¬rm size grows.

While the $10 million ¬rm in rows 20 and 21 are outside of the scope

of the IBA study, we use them later on in our own analysis to extrapolate

the results that we derive from our analysis of the IBA study.

Table 10-6: Calculation of DLOM

Table 10-6 is exactly the same format and logic as Table 7-14, which we

already described in Chapter 7. B9 through B12 contain the pure dis-

counts for the four economic components. B9, the pure discount for com-

PART 4 Putting It All Together

368

T A B L E 10-5

Calculation of Transaction Costs for Firms of All Sizes in the IBA Study

A B C D E F G H I

5 FMV log10 FMV X-Coeff. log FMV Coeff. Regr. Constant Forecast Subtotal Bus. Broker Forecast Total

6 Buyer $ 25,000 4.39794 0.01727 0.07596 0.15310 7.7% 0.0% 7.7%

7 Seller $ 25,000 4.39794 0.01599 0.07034 0.14139 7.1% 10.0% 17.1%

8 Buyer $ 75,000 4.87506 0.01727 0.08420 0.15310 6.9% 0.0% 6.9%

9 Seller $ 75,000 4.87506 0.01599 0.07797 0.14139 6.3% 10.0% 16.3%

10 Buyer $ 125,000 5.09691 0.01727 0.08804 0.15310 6.5% 0.0% 6.5%

11 Seller $ 125,000 5.09691 0.01599 0.08152 0.14139 6.0% 10.0% 16.0%

12 Buyer $ 175,000 5.24304 0.01727 0.09056 0.15310 6.3% 0.0% 6.3%

13 Seller $ 175,000 5.24304 0.01599 0.08386 0.14139 5.8% 10.0% 15.8%

14 Buyer $ 225,000 5.35218 0.01727 0.09245 0.15310 6.1% 0.0% 6.1%

15 Seller $ 225,000 5.35218 0.01599 0.08561 0.14139 5.6% 10.0% 15.6%

16 Buyer $ 375,000 5.57403 0.01727 0.09628 0.15310 5.7% 0.0% 5.7%

17 Seller $ 375,000 5.57403 0.01599 0.08915 0.14139 5.2% 10.0% 15.2%

18 Buyer $ 750,000 5.87506 0.01727 0.10148 0.15310 5.2% 0.0% 5.2%

19 Seller $ 750,000 5.87506 0.01599 0.09397 0.14139 4.7% 10.0% 14.7%

20 Buyer $10,000,000 7.00000 0.01727 0.12091 0.15310 3.2% 0.0% 3.2%

21 Seller $10,000,000 7.00000 0.01599 0.11196 0.14139 2.9% 2.0% 4.9%

Note: Regression constants and x-coef¬cients come from Table 7-11. The $10 million ¬rm, using a Lehman Bros. Formula, has a 2% investment banker fee instead of a 10% business broker™s fee.

369

T A B L E 10-6

Calculation of DLOM

A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability

6 1 Col. [C]

7 Pure Discount PV of Perpetual Remaining

8 Component z [1] Discount [2] Value

9 1 0.0% 0.0% 100.0% Delay to sale

10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets

11 3A 5.7% 6.1% 93.9% Transactions costs”buyers

12 3B 15.1% 1.0% 99.0% Transactions costs”sellers

13 Percent remaining 90.1% Total % remaining components 1 2 3A 3B

14 Final discount 9.9% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $ 25,000

19 Discount rate r [3] 34.7%

20 Constant growth rate g 2.0%

21 Intermediate calculation: x (1 g)/(1 r) 0.7574

22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I6 and I7 2% for public

brokerage costs.

[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.

PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.

Components #1 and #2 simply transfer the pure discount.

[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.

ponent #1, equals zero, and that comes from our calculation in Table

10-4, D12. B10, the pure discount for component #2, equals 9%. That is

the same as it was in Table 7-14, and it comes from the Schwert article.

Components 3A and 3B come from Table 10-5, cells I6 and I7, respectively,

less a 2% brokerage cost for publicly traded stock. These two components

are equal to 5.7% (B11) and 15.1% (B12), respectively.

As in Table 7-12, the ¬rst two components transfer from B9 and B10

to C9 and C10 directly. However, as discussed in the commentary to Table

7-12, transactions costs ˜˜leave the system™™ with every sale. Thus, we must

present value a perpetuity of transactions costs that occur every j 10

years. We do so using the formulas in note [2] to the spreadsheet, which

are equations (7“9) and (7“9a) from Chapter 7. The present value of all

buyers™ transactions costs is 6.1% (C11), and the present value of all sell-

ers™ transactions costs is 1.0% (C12). The ¬nal calculation of DLOM is

11.9% (D14)

Tables 10-6A“10-6F: Calculations of DLOM for

Larger Firms

Tables 10-6A“10-6F are structured and calculated identically to Table

10-6. There are ¬ve differences in the parameters, the ¬rst four of which

tend to increase DLOM as ¬rm size increases, and the last to decrease

DLOM as ¬rm size increases.

PART 4 Putting It All Together

370

T A B L E 10-4A

Calculation of Component #1”Delay to Sale”$75,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2

6 5.33E 18 7.952E 10 0.0%

7 Value of block-post-discount [2] 4.26E 09 $ 75,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $ 75,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.2500 3.3%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $ 75,000

16 Selling price $ 75,000

17 Adjusted net income $ 14,100

18 Assumed pre-tax margin 5%

19 Sales $281,991

Sales2

20 7.95E 10

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

T A B L E 10-4B

Calculation of Component #1”Delay to Sale”$125,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 2.778E 11 0.0%

7 Value of block-post-discount [2] 4.26E 09 $125,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $125,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.3330 4.5%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $125,000

16 Selling price $125,000

17 Adjusted net income $ 26,352

18 Assumed pre-tax margin 5%

19 Sales $527,049

Sales2

20 2.78E 11

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 371

T A B L E 10-4C

Calculation of Component #1”Delay to Sale”$175,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 4.972E 11 0.0%

7 Value of block-post-discount [2] 4.26E 09 $175,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $175,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.3330 4.5%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $175,000

16 Selling price $175,000

17 Adjusted net income $ 35,255

18 Assumed pre-tax margin 5%

19 Sales $705,109

Sales2

20 4.97E 11

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

T A B L E 10-4D

Calculation of Component #1”Delay to Sale”$225,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 6.685E 11 0.0%

7 Value of block-post-discount [2] 4.26E 09 $225,000 0.1%

8 FMV-marketable minority 100% interest 5.97E 10 $225,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.3330 4.5%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $225,000

16 Selling price $225,000

17 Adjusted net income $ 40,882

18 Assumed pre-tax margin 5%

19 Sales $817,647

Sales2

20 6.69E 11

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

PART 4 Putting It All Together

372

T A B L E 10-4E

Calculation of Component #1”Delay to Sale”$375,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 1.955E 12 0.0%

7 Value of block-post-discount [2] 4.26E 09 $368,041 0.2%

8 FMV-marketable minority 100% interest 5.97E 10 $375,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.5000 6.7%

12 Total Discount [4] 1.9%

14 Value of block-pre-discount [5] $375,000

16 Selling price $375,000

17 Adjusted net income $ 69,913

18 Assumed pre-tax margin 5%

19 Sales $1,398,256

Sales2

20 1.96E 12

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

T A B L E 10-4F

Calculation of Component #1”Delay to Sale”$750,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 1.955E 12 0.0%

7 Value of block-post-discount [2] 4.26E 09 $686,724 0.3%

8 FMV-marketable minority 100% interest 5.97E 10 $750,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 1.0000 13.7%

12 Total Discount [4] 8.4%

14 Value of block-pre-discount [5] $ 750,000

16 Selling price $ 750,000

17 Adjusted net income $ 69,913

18 Assumed pre-tax margin 5%

19 Sales $1,398,256

Sales2

20 1.96E 12

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.