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discount component”buyers™ costs”finite life (A7-24)

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

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