Summary of Mathematical Analysis in Remainder

of Appendix

The remainder of the appendix is devoted to calculating partial deriva-

tives necessary to evaluate the behavior of the discount formula (A7-11).

The partial derivatives of D with respect to its underlying independent

variables, g, r, z, and j, give us the slope of the discount as a function of

each variable. The purpose in doing so is to see how D behaves as the

independent variables change.

It turns out that D is a monotonic function with respect to each of

its independent variables. That is analytically convenient, as it means that

an increase in any one of independent variables always affects D in the

same direction. For example, if D is monotonically increasing in g, that

means that an increase in g will always lead to an increase in D, and a

decrease in g leads to a decrease in D. If D is monotonically increasing,

there is no value of g such that an increase in g leads either to no change

in D or a decrease in D.

The results that we develop in the remainder of the appendix are

that the discount, D, is monotonically increasing with g with z and de-

creasing with r and j. The practical reader will probably want to stop

here.

MATHEMATICAL ANALYSIS OF THE DISCOUNT”

CALCULATING PARTIAL DERIVATIVES

We can compute an alternative form of equation (A7-11) by multiplying

the numerator by 1 and changing the minus sign before the fraction to

a plus sign. This will ease the computations of the partial derivatives of

the expression.

PART 3 Adjusting for Control and Marketability

290

xj 1

D 1 (A7-25)

z)x j

1 (1

z)x j]jx j 1} {(x j z)jx j 1]}

{[1 (1 1)[ (1

D

(A7-26)

z)x j]2

x [1 (1

Factoring out jx j 1, we get:

jx j 1{[1 z)x j] (x j

(1 1)(1 z)}

D

(A7-27)

z)x j]2

x [1 (1

jx j 1[1 z)x j z)x j

(1 (1 (1 z)]

D

(A7-28)

z)x j]2

x [1 (1

Note that (1 z)x j and (1 z)x j cancel out in the numerator. Also,

the 1 (1 z) z. This simpli¬es to:

jx j 1z

D

0 (A7-29)

z)x j]2

x [1 (1

Since j, x, and z are all positive, the numerator is positive. Since the

denominator is squared, it is also positive. Therefore, the entire expression

is positive. The means that the discount is monotonically increasing in x.

We begin equation (A7-30) with a repetition of the de¬nition of x in

order to compute its partial derivatives.

1 g

x (A7-30)

1 r

Differentiating equation (A7-30) with respect to g, we get:

x (1 r)(1) 1

0 (A7-31)

r)2

g (1 1 r

Differentiating equation (A7-30) with respect to r, we get:

(1 g)(1) (1 g)

x

0 (A7-32)

r)2 r)2

r (1 (1

Using the chain rule, the partial derivative of D with respect to g is

the partial derivative of D with respect to x multiplied by the partial

derivative of x with respect to g, or:

D Dx

0 (A7-33)

g xg

The ¬rst term on the right-hand side of the equation is positive by

equation (A7-29), and the second term is positive by equation (A7-31).

Therefore, the entire expression is positive and thus the discount is mon-

otonically increasing in g. Using the chain rule again with respect to r,

we get:

D Dx

0 (A7-34)

r xr

Thus, the discount is monotonically decreasing in r. Now we make

an algebraic substitution to simplify the expression for D in order to fa-

cilitate calculating other partial derivatives.

CHAPTER 7 Adjusting for Levels of Control and Marketability 291

Let y 1 z (A7-35)

dy

1 (A7-36)

dz

Substituting equation (A7-35) into equation (A7-25), we get:

xj 1

D 1 (A7-37)

yx j

1

x j)( x j) x j(x j

D (1 1)

(A7-38)

yx j)2 yx j)2

y (1 (1

D dy x j(x j

D 1)( 1)

0 (A7-39)

z)x j]2

z y dz [1 (1

The denominator of (A7-39), being squared, is positive. The numer-

ator is also positive, as x j is positive and less than one, which means that

xj 1 is negative, which when multiplied by 1 results in a positive

number. Thus, the entire partial derivative is positive, which means that

D is monotonically increasing in z, the transaction costs. This result is

intuitive, as it makes sense that the greater the transaction costs, the

greater the discount.

Differentiating equation (A7-37) with respect to j, the average num-

ber of years between sales, we get:

yx j)x j ln x (x j 1)( y)x j ln x

(1

D

(A7-40)

yx j)2

j (1

Factoring out x j ln x, we get:

x j ln x(1 yx j yx j x j ln x(1

y) y)

D

(A7-41)

yx j)2 yx j)2

j (1 (1

x j z ln x

D

0 (A7-42)

z)x j]2

j [1 (1

The denominator is positive. The numerator is negative; since x 1,

ln x 0. Thus, the discount is monotonically decreasing in j, the average

years between sale. That is intuitive, as the less frequently business sell,

the smaller the discount should be.

Summary of Comparative Statics

Summarizing, the discount for periodic transaction costs is related in the

following ways to its independent variables:

Variable Varies with Discount Monotonically

r Negatively Decreasing

g Positively Increasing

z Positively Increasing

j Negatively Decreasing

PART 3 Adjusting for Control and Marketability

292

CHAPTER 8

Sample Restricted Stock

Discount Study

ENCO, INC.

As of AUGUST 11, 1997

The information contained in this report is con¬dential. Neither all nor

any part of the contents shall be conveyed to the public without the prior

written consent and approval of Abrams Valuation Group (AVG). AVG™s

opinion of value in this report is valid only for the stated purpose and

date of the appraisal.

Note: all names are ¬ctional

Note: Because this sample report is in a book, there are slight changes in

the table numbering and appearance of the report to accommodate the

book format.

293

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

Letter of Opinion

November 18, 1998

Mr. Robert Smith

2633 Elm Way

La Jolla, CA 92037

Dear Mr. Smith:

In accordance with your instructions, we have made a determination of

the Discount for Lack of Marketability (DLOM) necessary to calculate the

fair market value (FMV) of the common stock that you received in ENCO,

Inc. (˜˜ENCO,™™ or ˜˜the Company™™) as of August 11, 1997, the date that

you sold your company, Smith Metals, to ENCO. The stock is restricted

according to SEC Rule 144, and it becomes marketable one year after the

date of your sale. ENCO trades on Nasdaq, and the closing price of its

freely trading shares on August 11, 1997 was 2 3/8, or $2.375.

It is our understanding that this appraisal will be used for income tax

purposes. The DLOM and related FMV, as determined within our report,

shall not be used for other purposes or dates without our written consent,

as they can be misleading and dangerous.

The de¬nition of fair market value is:

The price at which property [in this case, the capital stock of the Company]

would change hands between a willing seller and a willing buyer, when neither

is under compulsion to buy and when both have reasonable knowledge of the

relevant facts.1

The scope of our engagement included discussions with you and Len

Storm, Esq., Vice President and Legal Secretary of ENCO, as to the se-

curities laws that apply, as he understands them. Per your instructions,

we assume Len Storm™s understanding of the timing of your ability to

sell your ENCO stock to be correct. If his information were incorrect, that

would cause a change in the related DLOM.

Based upon our investigation and analysis and subject to the attached

report and Statement of Limiting Conditions, it is our opinion that the

restricted stock discount (the DLOM) is 20.5%. The closing price of

ENCO, Inc. common stock on August 11, 1997, was $2.375 per share.2 The

1. American Society of Appraisers Business Valuation Standards. Also, the wording is virtually

identical in Reg. § 1.170A-1(c)(2) (income tax, charitable contributions of property); see Reg.

§§ 20.2031-1(b) (second sentence) (estate tax), 25.2512-1 (second sentence) (gift tax).

2. Source: American Online, Prophet Line.

PART 3 Adjusting for Control and Marketability

294

20.5% discount is $0.486 per share, leaving the fair market value of the

restricted stock on that date at $1.889 per share (see Table 8-3 of the report

for those calculations).

We retain a copy of this letter in our ¬les, together with ¬eld data from

which it was prepared. We consider these records con¬dential, and we

do not permit access to them by anyone without your authorization.

USPAP (Uniform Standards of Professional Appraisal Practice) Certi¬ca-

tion:

I certify that to the best of my knowledge and belief:

— The statements of fact contained in this report are true and

correct, the reported analyses, opinions and conclusions are

limited only by the reported conditions, and they are our

personal, unbiased professional analyses, opinions, and

conclusions.

— We have no present or prospective interest in the property that is

the subject of this report, and we have no personal interest or

bias with respect to the parties involved.

— Our compensation is not contingent on an action or event

resulting from the analyses, opinions, conclusions in or use of

this report.

— Our analyses, opinions, and conclusions were developed and this

report has been prepared in conformity with the Uniform

Standards of Professional Appraisal Practice and the Business

Valuation Standards of the American Society of Appraisers.

— No one has provided signi¬cant professional assistance to the

person signing this report.

— I have passed the USPAP examination and am certi¬ed through

the year 2001. I am an Accredited Senior Appraiser with the

American Society of Appraisers, with certi¬cation current to the

year 2000.

Sincerely yours,

Jay B. Abrams, ASA, CPA, MBA

CHAPTER 8 Sample Restricted Stock Discount Study 295

INTRODUCTION

Background

Stock Ownership

Purpose of the Appraisal

No Economic Outlook Section

Sources of Data

VALUATION

Commentary to Table 8-1: Regression Analysis of Management

Planning Data

Previous Restricted Stock Studies

Change in SEC Rule 144

The Data

Commentary to Table 8-1A: Revenue and Earnings Stability

Commentary to Table 8-1B: Price Stability

Valuation Using Options Pricing Theory

Options Theory

Black“Scholes Put Option Formula

Chaffe™s Article: Put Options to Calculate DLOM of Restricted

Stock

Commentary to Table 8-2: Black-Scholes Calculation of DLOM for

ENCO, Inc.

Commentary to Table 8-2A: Annualized Standard Deviation of

Continuously Compounded Returns

Commentary to Table 8-3: Final Calculation of Discount

Conclusion of Discount for Lack of Marketability

ASSUMPTIONS AND LIMITING CONDITIONS

APPRAISER™S QUALIFICATIONS

PART 3 Adjusting for Control and Marketability

296

INTRODUCTION

Background

On August 11, 1997, Robert Smith sold his company, Smith Metals, LLC,

to ENCO, Inc. (˜˜ENCO,™™ or ˜˜the Company™™) and received 500,000 shares

of ENCO Common Stock that is subject to the greater of two sets of

restrictions in transfer:

1. The sales contract with ENCO: According to Section 2.12(b)(v),

Robert Smith must wait one year to sell his ENCO stock.

2. In accordance with SEC Rule 144, Robert Smith must wait one

year to begin selling his stock, at which point he can make

quarterly sales equal to the greater of:

(a) Rule 144 (e)(1)(i): 1% of the outstanding shares. With 112.5 million

shares outstanding at the valuation date (the date of sale), 1% is

1.125 million shares.

(b) Rule 144 (e)(1)(ii): The average weekly trading volume for the four

weeks preceding the date of sale. The average weekly trading

volume for the month preceding the sale was 900,000 shares. Thus,

(2)(a) predominates, and 1.125 million is the maximum sale per

quarter according to Rule 144 after the one-year waiting period.

The sale did not qualify as a tax-free reorganization, and you need the

fair market value of the ENCO stock to compute your capital gains tax.

Stock Ownership

ENCO is publicly traded on Nasdaq with a ticker symbol of ENCO.

Through the sale of Smith Metals, Robert Smith acquired less than 1% of

ENCO™s stock.

Purpose of the Appraisal

The purpose of this appraisal is to calculate the discount for lack of mar-

ketability (DLOM) needed to ascertain the fair market value for income

tax purposes of the 500,000 shares of ENCO stock owned by Robert Smith.

Your instructions are that we are to assume the market price is the fair

market value of the unrestricted stock”a reasonable assumption”and

that the only calculation necessary to produce the fair market value of

the restricted stock is the DLOM.

The term fair market value is de¬ned as ˜˜the amount at which prop-

erty [in this case, the capital stock of the Company] would change hands

between a willing buyer and a willing seller, when the former is not under

any compulsion to buy and the latter is not under any compulsion to sell,

and when both parties have reasonable knowledge of relevant facts.™™3

3. American Society of Appraisers Business Valuation Standards. Also, the wording is virtually

identical in Reg. § 1.170A-1(c)(2) (income tax, charitable contributions of property); see Reg.

§§ 20.2031-1(b) (second sentence) (estate tax), 25.2512-1 (second sentence) (gift tax).

CHAPTER 8 Sample Restricted Stock Discount Study 297

No Economic Outlook Section

The Economic Outlook, a standard section in business valuations, is ir-

relevant in this study. This section would be relevant in valuing ENCO

stock, but that is not our assignment. The same is true of a History of the

Company section. Thus, we proceed to the Valuation section.

Sources of Data

1. Financial statements sent by Len Storm, Esq., Vice President and

Legal Secretary of ENCO.

2. Copy of ENCO stock certi¬cate issued to Robert Smith including

copies of the ™33 Act legend and contractual legend.

3. One-year secondary market Treasury Bill rate as of 8/11/97

from the Federal Reserve Bank of St. Louis, internet web site

http://www.stls.frb.org.

4. America Online, Prophet Line stock quotes.

5. Restricted stock transaction data from Management Planning,

Inc., Princeton, New Jersey.

VALUATION

We use two valuation methodologies in calculating the restricted stock

discount. The ¬rst is based on our own statistical analysis using multiple

regression of data collected by Management Planning, Inc.4 The second

involves using a Black“Scholes put option as a proxy for the discount.

Commentary to Table 8-1: Regression Analysis of

Management Planning Data

Previous Restricted Stock Studies

There have been 10 studies of sales of restricted stocks.5 In the ¬rst nine

studies the authors did not publish the underlying data and merely pre-

sented their analysis and summary of the data. Additionally, only the

Hall/Polacek study contains data beyond 1988, theirs going through 1992.

The Management Planning Study contains data on trades from 1980“

1996. Thus, it is superior to the other studies in two ways: the detail of

the data exists, and the data are more current. Therefore, we use the

Management Planning study exclusively.

Change in SEC Rule 144

On April 29, 1997, SEC Rule 144 changed from a two-year holding period