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. 12
( 18)



>>

the brackets into a single term by adding the numerators.
10
1 g
1
1 r
1 r
NPVTC (A7-7)
10
r g 1 g
1 (1 z)
1 r

Letting x (1 g)/(1 r), this simpli¬es to:
1 r x10
1
NPVTC (A7-8)
z)x10
r g 1 (1


The Discount Formula
D, the component of the discount for lack of marketability that measures
the periodic transaction costs, is one minus the ratio of the NPV of the
cash ¬‚ows net of transaction costs (NPVTC) to the NPV without removing
transaction costs (NPV). Using a midyear Gordon model formula of
(1 r)/(r g) as the NPV, we come to:
1 r x10
1
z)x10
r g 1 (1
NPVTC
D 1 1 (A7-9)
NPV 1 r
r g
The term ( 1 r)/(r g) cancels out, and the expression simpli¬es
to:
1 g
x10
1
r, ’ 0
D 1 , where x and g x 1
z)x10
1 (1 1 r
(A7-10)
Equation (A7-10) is the formula for the discount assuming a sale
every 10 years. Instead of assuming a business sale every 10 years, now
we let the average years between sale be a random variable, j, which
leads to the generalized equation in equation (A7-11):
xj
1
D 1
z)x j
1 (1
generalized discount formula“sellers™ transaction costs (A7-11)
In determining fair market value, we ask how much would a rational
buyer pay for (and for how much would a rational seller sell) a business
interest. That presumes a hypothetical sale at time zero. Equation (A7-11)
is the formula appropriate for quantifying sellers™ transaction costs, be-

PART 3 Adjusting for Control and Marketability
286
cause the buyer does not care about the seller™s costs, which means he or
she will not raise the price in order to cover the seller. However, the buyer
does care that 10 years down the road, he or she will be a seller, not a
buyer, and the new buyer will reduce the price to cover his or her trans-
action costs, and so on ad in¬nitum. Thus, we want to quantify the dis-
counts due to transaction costs for the continuum of sellers beginning
with the second sale, i.e., in year j. Equation (A7-11) accomplishes that.
Using an end-of-year Gordon model assumption instead of midyear
cash ¬‚ows leads to the identical equation, i.e., (A7-11) holds for both.


Buyer Discounts Begin with the First Transaction
An important variation of equation (A7-11) is to consider what happens
if the ¬rst relevant transaction cost takes place at time zero instead of
t j, which is appropriate for quantifying the discount component due
to buyers™ transaction costs. With this assumption, we would modify the
above analysis by inserting a (1 z) in front of the ¬rst series of bracketed
terms in equation (A7-1) and increasing the exponent of all the other (1
z) terms by one. All the other equations are identical, with the (1 z)
term added. Thus, the buyers™ equivalent formula of equation (A7-8) is:
1 r x10
1
NPVTC (1 z)
z)x10
r g 1 (1
NPV with buyers™ transaction costs removed (A7-8a)
Obviously, equation (A7-8a) is lower than equation (A7-8), because
the ¬rst relevant cost occurs 10 years earlier. The generalized discount
formula equivalent of equation (A7-11) for the buyer scenario is:
x j)
(1 z)(1
D 1
z)x j
1 (1
generalized discount, formula”buyers™ transactions costs
(A7-11a)
We demonstrate the accuracy of equations (A7-11) and (A7-11a),
which are excerpted from here and renumbered in the chapter as equa-
tions (7-9) and (7-9a), in Tables 7-12 and 7-13 in the body of the chapter.


NPV of Cash Flows with Finite Transactions
Costs Removed78
The previous formulas for calculating the present value of the discount
for buyers™ and sellers™ transactions costs are appropriate for business
valuations. However, for calculating that component of DLOM for limited
life entities such as limited partnerships whose document speci¬es a ter-
mination date, the formulas are inexact, although they are often good
approximations. In this section we develop the formulas for components
#3A and #3B of DLOM for limited life entities.79 This section is very math-
ematical and will have practical signi¬cance for most readers only when


78. This section is written by R. K. Hiatt.
79. Even in limited partnerships, it is necessary to question whether the LP is likely to renew, i.e.,
extend its life. If so, then the perpetuity formulas (A7-11) and (A7-11a) may be appropriate.


CHAPTER 7 Adjusting for Levels of Control and Marketability 287
the life of the entity is short (under 30 years) and the growth rate is close
to the discount rate. Some readers may want to skip this section, perhaps
noting the ¬nal equations, (A7-23) and (A7-24). Consider this section as
reference material.
Let™s assume a fractional interest in an entity, such as a limited part-
nership, with a life of 25 years that sells for every j 10 years. Thus,
2 sales80 of the frac-
after the initial hypothetical sale, there will be s
tional interest before dissolution of the entity. Let™s de¬ne n as the number
of years to the last sale before dissolution. We begin by repeating equa-
tions (A7-1) and (A7-2) as (A7-12) and (A7-13), with the difference that
the last incremental transaction cost occurs at n 20 years instead of
going on perpetually.
g)9
(1 g) (1
1
NPVTC
r)0.5 r)1.5 r)9.5
(1 (1 (1
g)10 g)19
(1 (1
(1 z)
r)10.5 r)19.5
(1 (1
g)20
(1
2
(1 z) (A7-12)
r)20.5
(1
g)10
1 g 1 g (1
NPVTC
r)1.5 r)10.5
1 r (1 (1
g)11 g)20
(1 (1
(1 z)
r)11.5 r)20.5
(1 (1
g)21
(1
2
(1 z) (A7-13)
r)21.5
(1
Subtracting equation (A7-13) from equation (A7-12), we get:
g)10
1 g (1
1
1 NPVTC
r)0.5 r)10.5
1 r (1 (1
g)10 g)20
(1 (1
(1 z)
r)10.5 r)20.5
(1 (1
g)20
(1
2
(1 z) (A7-14)
r)20.5
(1
Note that the ¬nal term ˜˜should have™™ a subtraction of (1 g) /
0.5
(1 r) , but that equals zero for g r. Therefore, we leave that term
out. Again, the ¬rst term of the equation reduces to (r g)/(1 r). We
then multiply both sides by its inverse:
g)10
(1
1 r 1
NPVTC
r)0.5 r)10.5
r g (1 (1
g)10 g)20
(1 (1
(1 z)
r)10.5 r)20.5
(1 (1
g)20
(1
2
(1 z) (A7-15)
r)20.5
(1


80. It is important not to include the initial hypothetical sale in the computation of s.


PART 3 Adjusting for Control and Marketability
288
As before, we divide the ¬rst term on the right-hand side of the equation
by 1 r and multiply all terms inside the brackets by the same. This
has the same effect as reducing the exponents in the denominators by 0.5
years.
10
1 r 1 g
NPVTC 1
r g 1 r
10 20
1 g
1 g
(1 z)
1 r 1 r
20
1 g
2
(A7-16)
(1 z)
1 r
Letting y 1 z and x (1 g)/(1 r), equation (A7-16) becomes:
1 r
x10) y(x10 x20) y2x20]
NPVTC [(1 (A7-17)
r g
1 r
yx10 y2x20) (x10 yx20)]
NPVTC [(1 (A7-18)
r g
Within the square brackets in equation (A7-18), there are two sets of
terms set off in parentheses. Each of them is a ¬nite geometric sequence.
The ¬rst sequence solves to
y3x30
1
yx10
1
and the second sequence solves to
x10 y2x30
yx10
1
They both have the same denominator, so we can combine them. Thus,
equation (A7-18) simpli¬es to:
x10 y2x30 y3x30
1 r 1
NPVTC (A7-19)
yx10
r g 1
Note that if we eliminate the two right-hand terms in the square brackets
in the numerator, equation (A7-10) reduces to equation (A7-8). We can
now factor the two right-hand terms and simplify to:
x10 y2x30(1
1 r 1 y)
NPVTC
yx10
r g 1
x10 zy2x30
1 r 1
yx10
r g 1
1 r x10 z)2x30
1 z(1
(A7-20)
z)x10
r g 1 (1
Since j 10, s 2, n 20, and n j 30, we can now generalize this
equation to:
1 r xj z)sxn j
1 z(1
NPVTC (A7-21)
z)x j
r g 1 (1

CHAPTER 7 Adjusting for Levels of Control and Marketability 289
As before, the discount component is D 1 NPVTC/NPV. This comes
to:
1 r xj z)sxn j
1 z(1
z)x j
r g 1 (1
D 1 (A7-22)
1 r
r g
Canceling terms, this simpli¬es to:
xj z)sx n j
1 z(1
D 1 (A7-23)
z)x j
1 (1
discount component”sellers™ costs”finite life
Note that as the life of the entity (or the interest in the entity) that
we are valuing goes to in¬nity, n ’ , so xn j ’ 0 and (A7-23) reduces
to equation (A7-11).
The equivalent expression for buyers™ costs is:
xj z)sxn j]
(1 z)[1 z(1
D 1
z)x j
1 (1
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
to a one-year holding period for limited sales of stock. We should expect
the shortening of the period of restriction to decrease the discount. The
latest Management Planning data contain four observations with ex-


4. Published in Mercer (1997), chap. 12. Also, MPI provided us with four additional data points
and some data corrections.
5. See Mercer, p. 69 for a summary of the results of the ¬rst nine studies.


PART 3 Adjusting for Control and Marketability
298
pected holding periods of less than two years, which will enable us to
statistically infer the effect of the change in Rule 144 on DLOM.

The Data
Table 8-1 is two pages long. The ¬rst one and one-quarter pages contain
data on 53 sales of restricted stock between 1980“1996. Column A is num-
bered 1 through 53 to indicate the sale number. Column C, our dependent
(Y) variable, is the restricted stock discount for each transaction.
Columns D through J are our seven statistically signi¬cant indepen-
dent variables, which we have labeled X1, X2, . . . X7. Below is a descrip-
tion of the independent variables:


# Independent Variable

1 Revenues squared.
2 Shares sold”$: the post-discount dollar value of the traded restricted shares.
3 Market capitalization price per share times shares outstanding summed for all
classes of stock.
Earnings stability: the unadjusted R2 of the regression of net income as a function
4
of time, with time measured as years 1, 2, 3, . . . We calculate this in Table 8-1A,
regression #1.
Revenue stability: the unadjusted R2 of the regression of revenue as a function of
5
time, with time measured as years 1, 2, 3, . . . We calculate this in Table 8-1A,
regression #2.
6 Average years to sell: the weighted average years to sell by a nonaf¬liate, based on
SEC Rule 144.
7 Price stability: This ratio is calculated by dividing the standard deviation of the stock
price by the mean of the stock price. Management Planning used the end-of-
month stock prices for the 12 months prior to the valuation date.




We regressed 30 other independent variables included in the Man-
agement Planning study, and all were statistically insigni¬cant. We re-
strict our commentary to the seven independent variables that were sta-
tistically signi¬cant at the 95% level.
Table 8-1, page 2 contains the regression statistics. Adjusted R2 is
59.47% (C66), a reasonable though not stunning result for such an anal-
ysis. That means the regression model accounts for 59.47% of the varia-
tion in the restricted stock discounts. The other 40.53% of variation in the
discounts that remains unexplained are due to two possible sources: other
signi¬cant independent variables of which we (and Management Plan-
ning) do not know and random variation.
The standard error of the y-estimate is 8.7% (C67 rounded). We can
form approximate 95% con¬dence intervals around the y-estimate by add-
ing and subtracting two standard errors, or 17.4%.
Cell C77 contains the y-intercept, and C78 through C84 contain the
regression coef¬cients for the independent variables. E77 to E84 contains
the t-statistics. Only the y-intercept itself is not signi¬cant at the 95%
con¬dence level. The earnings stability and market capitalization varia-
bles are signi¬cant at the 98% level,6 and all the other variables are sig-
ni¬cant at the 99 % con¬dence level.


6. The statistical signi¬cance is one minus the P-value, which is in F79 through F86.


CHAPTER 8 Sample Restricted Stock Discount Study 299
300
T A B L E 8-1

Abrams Valuation Group Regression of Management Planning, Inc. Data [1]


A B C D E F G H I J

4 Y X1 X2 X3 X4 X5 X6 X7
Rev2
5 Discount Shares Sold-$ Mkt Cap Earn Stab Rev Stabil Avg Yrs To Sell Price Stability [2]

6 1 Air Express Int™l 0.0% 8.58E 16 $4,998,000 25,760,000 0.08 0.22 2.84 12.0
7 2 AirTran Corp 19.4% 1.55E 16 $9,998,000 63,477,000 0.90 0.94 2.64 12.0
8 3 Anaren Microwave, Inc. 34.2% 6.90E 13 $1,250,000 13,517,000 0.24 0.78 2.64 28.6
9 4 Angeles Corp 19.6% 7.99E 14 $1,800,000 16,242,000 0.08 0.82 2.13 8.4
10 5 AW Computer Systems, Inc. 57.3% 1.82E 13 $1,843,000 11,698,000 0.00 0.00 2.91 22.6
11 6 Besicorp Group, Inc. 57.6% 1.57E 13 $1,500,000 63,145,000 0.03 0.75 2.13 98.6
12 7 Bioplasty, Inc, 31.1% 6.20E 13 $11,550,000 43,478,000 0.38 0.62 2.85 44.9
13 8 Blyth Holdings, Inc. 31.4% 8.62E 13 $4,452,000 98,053,000 0.04 0.64 2.13 58.6
14 9 Byers Communications Systems, Inc. 22.5% 4.49E 14 $5,007,000 14,027,000 0.90 0.79 2.92 6.6
15 10 Centennial Technologies, Inc. 2.8% 6.75E 13 $656,000 27,045,000 0.94 0.87 2.13 35.0
16 11 Chantal Pharm. Corp. 44.8% 5.21E 13 $4,900,000 149,286,000 0.70 0.23 2.13 51.0
17 12 Choice Drug Delivery Systems, Inc. 28.8% 6.19E 14 $3,375,000 21,233,000 0.29 0.89 2.86 23.6
18 13 Crystal Oil Co. 24.1% 7.47E 16 $24,990,000 686,475,000 0.42 0.57 2.50 28.5
19 14 Cucos, Inc. 18.8% 4.63E 13 $2,003,000 12,579,000 0.77 0.87 2.84 20.4
20 15 Davox Corp. 46.3% 1.14E 15 $999,000 18,942,000 0.01 0.65 2.72 24.6
21 16 Del Electronics Corp. 41.0% 4.21E 13 $394,000 3,406,000 0.08 0.10 2.84 4.0
22 17 Edmark Corp 16.0% 3.56E 13 $2,000,000 12,275,000 0.57 0.92 2.84 10.5
23 18 Electro Nucleonics 24.8% 1.22E 15 $1,055,000 38,435,000 0.68 0.97 2.13 21.4
24 19 Esmor Correctional Svces, Inc. 32.6% 5.89E 14 $3,852,000 50,692,000 0.95 0.90 2.64 34.0
25 20 Gendex Corp 16.7% 2.97E 15 $5,000,000 55,005,000 0.99 0.71 2.69 11.5
26 21 Harken Oil & Gas, Inc. 30.4% 7.55E 13 $1,999,000 27,223,000 0.13 0.88 2.75 19.0
27 22 ICN Paramaceuticals, Inc. 10.5% 1.50E 15 $9,400,000 78,834,000 0.11 0.87 2.25 23.9
28 23 Ion Laser Technology, Inc. 41.1% 1.02E 13 $975,000 10,046,000 0.71 0.92 2.82 22.0
29 24 Max & Erma™s Restaurants, Inc. 12.7% 1.87E 15 $1,192,000 31,080,000 0.87 0.87 2.25 18.8
30 25 Medco Containment Svces, Inc. 15.5% 5.42E 15 $99,994,000 561,890,000 0.84 0.89 2.85 12.8
31 26 Newport Pharm. Int™l, Inc. 37.8% 1.10E 14 $5,950,000 101,259,000 0.00 0.87 2.00 30.2
32 27 Noble Roman™s Inc. 17.2% 8.29E 13 $1,251,000 11,422,000 0.06 0.47 2.79 17.0
33 28 No. American Holding Corp. 30.4% 1.35E 15 $3,000,000 79,730,000 0.63 0.84 2.50 22.1
34 29 No. Hills Electronics, Inc. 36.6% 1.15E 13 $3,675,000 21,812,000 0.81 0.79 2.83 52.7
35 30 Photographic Sciences Corp 49.5% 2.70E 14 $5,000,000 44,113,000 0.06 0.76 2.86 27.2
36 31 Presidential Life Corp 15.9% 4.37E 16 $38,063,000 246,787,000 0.00 0.00 2.83 17.0
37 32 Pride Petroleum Svces, Inc. 24.5% 4.34E 15 $21,500,000 74,028,000 0.31 0.26 2.83 18.0
38 33 Quadrex Corp. 39.4% 1.10E 15 $5,000,000 71,016,000 0.41 0.66 2.50 44.2
39 34 Quality Care, Inc. 34.4% 7.97E 14 $3,150,000 19,689,000 0.68 0.74 2.88 7.0
40 35 Ragen Precision Industries, Inc. 15.3% 8.85E 14 $2,000,000 22,653,000 0.61 0.75 2.25 26.0
41 36 REN Corp-USA 17.9% 2.85E 15 $53,625,000 151,074,000 0.02 0.88 2.92 19.8
42 37 REN Corp-USA 29.3% 2.85E 15 $12,003,000 163,749,000 0.02 0.88 2.72 36.1
43 38 Rentrak Corp. 32.5% 1.15E 15 $20,650,000 61,482,000 0.60 0.70 2.92 30.0
44 39 Ryan™s Family Steak Houses, Inc. 8.7% 1.02E 15 $5,250,000 159,390,000 0.90 0.87 2.13 13.6
45 40 Ryan™s Family Steak Houses, Inc. 5.2% 1.02E 15 $7,250,000 110,160,000 0.90 0.87 2.58 14.4
46 41
Sahlen & Assoc., Inc. 27.5% 3.02E 15 $6,057,000 42,955,000 0.54 0.81 2.72 26.1
47 42
Starrett Housing Corp. 44.8% 1.11E 16 $3,000,000 95,291,000 0.02 0.01 2.50 12.4
48 43
Sudbury Holdings, Inc. 46.5% 1.39E 16 $22,325,000 33,431,000 0.65 0.17 2.96 26.6
49 44
Superior Care, Inc. 41.9% 1.32E 15 $5,660,000 50,403,000 0.21 0.93 2.77 42.2
50 45
Sym-Tek Systems, Inc. 31.6% 4.03E 14 20,550,000 0.34 0.92 2.58 13.4
51 46
Telepictures Corp. 11.6% 5.50E 15 $15,250,000 106,849,000 0.81 0.86 2.72 6.6
52 47
Velo-Bind, Inc. 19.5% 5.51E 14 $2,325,000 18,509,000 0.65 0.85 2.81 14.5
53 48
Western Digital Corp. 47.3% 4.24E 14 $7,825,000 50,417,000 0.00 0.32 2.64 22.7
54 49
50-Off Stores, Inc. 12.5% 6.10E 15 $5,670,000 43,024,000 0.80 0.87 2.38 23.7
55 50
ARC Capital 18.8% 3.76E 14 $2,275,000 18,846,000 0.03 0.74 1.63 35.0
56 51
Dense Pac Microsystems, 23.1% 3.24E 14 $4,500,000 108,862,000 0.08 0.70 1.17 42.4
Inc.
57 52 Nobel Education 19.3% 1.95E 15 $12,000,000 60,913,000 0.34 0.76 1.74 32.1
Dynamics, Inc.
58 53 Unimed Pharmaceuticals 15.8% 5.49E 13 $8,400,000 44,681,000 0.09 0.74 1.90 21.0

59 Mean 27.1% 5.65E 15 $9,223,226 $78,621,472 0.42 0.69 2.54 25.4
60 Standard deviation 13.7% 0.35 0.27 0.39 16.1
61 Management Planning Study: Summary Output of Regression

63 Regression Statistics

64 Multiple R 0.8058
65 R square 0.6493
66 Adjusted R square 0.5947
67 Standard error 0.0873
68 Observations 53
301
302

T A B L E 8-1 (continued)

Abrams Valuation Group Regression of Management Planning, Inc. Data [1]


A B C D E F G H I J

70 ANOVA

71 df SS MS F Signi¬cance F

72 Regression 7 0.6354 0.0908 11.9009 0.0000
73 Residual 45 0.3432 0.0076
74 Total 52 0.9786

76 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

77 Intercept 0.0673 0.1082 0.6221 0.5370 0.2854 0.1507
78 Rev2 4.629E 18 9.913E 19 4.6698 0.0000 6.626E 18 2.633E 18
79 Shares sold-$ 3.619E 09 1.199E 09 3.0169 0.0042 6.035E 09 1.203E 09
80 Mkt cap 4.789E 10 1.790E 10 2.6754 0.0104 1.184E 10 8.394E 10
81 Earn stab 0.1038 0.0402 2.5831 0.0131 0.1848 0.0229
82 Rev stabil 0.1824 0.0531 3.4315 0.0013 0.2894 0.0753
83 Avg yrs to sell 0.1722 0.0362 4.7569 0.0000 0.0993 0.2451
84 Price stability [2] 0.0037 0.0008 4.3909 0.0001 0.0020 0.0053

86 Management Planning Study: Applying Regression Results to Company Data
88 Y X1 X2 X3 X4 X5 X6 X7
Rev2
89 Discount Shares Sold-$ Mkt Cap Earn Stab Rev Stabil Avg Yrs To Sell Price Stability [1]

90 ENCO parameters Constant-NA 5.90E 14 933,311 267,187,500 0.12 0.54 1.0000 27.01
91 Coef¬cients C77 to C84 0.0673 4.629E 18 3.619E 09 4.789E 10 0.1038 0.1824 0.1722 0.0037
92 row 90* row 91 0.0673 0.0027 0.0034 0.1280 0.0125 0.0988 0.1722 0.0986
93 Restricted stock discount 21.41%
(sum of row 94)

[1] Source: Management Planning, Inc. Princeton NJ (except for ˜˜Avg Yrs To Sell™™ and ˜˜Rev2™™ which we derived from their data).
[2] See Table 8-1B for the calculation of Price Stability.
We transpose the results in C77 though C84 into row 91. Row 90
contains the ENCO parameters for each variable. The shares sold $
variable actually depends on the restricted stock discount, the dependent
variable, and the latter also depends on the former. Therefore, we must
derive ENCO™s input for this independent variable through an iterative
process. With the aid of a spreadsheet program, the task is simple. We
input the numbers of shares sold times the Share price times one minus
the restricted stock discount, or 500,000*$2.375*(1-C93) for ENCO™s shares
sold $ value and activate the iterative capability of the spreadsheet
program. For columns D through J, we multiply row 90 row 91 row
92, which is the regression determined in¬‚uence of each independent
variable on the discount. C91 is the y-intercept, which equals C92 and
does not get multiplied like the independent variables do.
The sum of all the values in Row 92 is 21.41% and appears in C93.
This is the ¬nal answer according to this valuation approach.


Commentary to Table 8-1A: Revenue and
Earnings Stability
Table 8-1A contains two regression analyses. Regression #1, starting at
row 19, is net income as a function of time (measured in years). Regres-
sion #2, starting at row 40, is revenue as a function of time, also measured
in years. The R2 is 0.12 (B23) and 0.54 (B44) for regressions #1 and #2,
respectively. We transfer these amounts to Table 8-1, cells G90 and H90,
respectively.


Commentary to Table 8-1B: Price Stability
Table 8-1B contains the calculation of price stability. Cells B5 through B16
show the month-end stock prices for ENCO from August 30, 1996,
through July 31, 1997. The standard deviation of these prices is 0.84 (B17),
and the arithmetic mean of the stock prices is 3.11 (B18). Dividing the
standard deviation by the mean and multiplying by 100 produces Man-
agement Planning, Inc.™s measure of price stability, which is 27 (B19).


Valuation Using Options Pricing Theory
Options Theory
The economic theory on which we rely is options pricing theory. The
paradigm options pricing model is the Black“Scholes Options pricing
model (Black“Scholes, or BSOPM), developed by University of Chicago
Professors Fisher Black and Myron Scholes, the latter of whom received
the Nobel Prize in Economics for developing the model (Black had died
in the meantime).
The Black“Scholes model is based on a heat exchange equation in
physics. (It is truly a wonder that an equation developed in the physical
world would be the one to explain the value of stock options.)
A call option is a contract enabling one to buy a speci¬c number of
shares of a company at a speci¬c price and time. For example, one might
buy an option to purchase 100 shares of IBM at $100 per share on a



CHAPTER 8 Sample Restricted Stock Discount Study 303
T A B L E 8-1B

Calculation of Price Stability


A B

4 Date Closing Price

5 8/30/96 4.3750
6 9/30/96 3.7500
7 10/31/96 3.7500
8 11/29/96 3.1250
9 12/31/96 2.8750
10 1 / 31 / 97 4.0625
11 2 / 28 / 97 3.8750
12 3 / 31 / 97 2.8750
13 4 / 30 / 97 2.1250
14 5 / 30 / 97 2.1875
15 6 / 30 / 97 2.3750
16 7 / 31 / 97 1.9375

17 Std dev 0.84
18 Mean price 3.11
19 Price stability 27.01




speci¬c date. A European option is such that one can buy only on that
date, while an American option allows one to buy anytime up to and
including that date. The original Black“Scholes model works on the as-
sumption of a European option. A put option is the opposite of a call. It
enables one to sell the stock at a speci¬c price and time. Let us examine
a put option.
Suppose IBM were selling today at $100 per share.7 What would be
the value of the ability to sell 100 shares of IBM on the last day of this
year at $100 per share? If the stock price in a year were greater than $100,
the value would be zero. If the price were less than or equal to $100, it
would be $100 minus the actual stock price, multiplied by the number of
shares.8 There are two ways to cash out on the put option: you can buy
the stock at its new lower market value and then sell it for $100 to the
writer of the option, or you can sell the option itself.
The problem is that we do not know what the price of the stock will
be. Black“Scholes assumes a normal probability distribution (the bell-
shaped curve) of prices on the expiration date of the option. The bell-
shaped curve is symmetrical and peaks in the center, which is the statis-
tical mean, median, and mode, these being three different types of
averages, which are not identical for asymmetric distributions.9
If we assume the center of the distribution is the exercise price, then
the Black“Scholes calculated value of a put option is the area under the
left half of the bell-shaped curve multiplied by the pro¬t at each price,


7. We have not researched IBM™s actual price. We use $100 per share for ease of illustration.
8. We are ignoring transactions costs and, for the moment only, the time value of money.
9. Technically it is the natural logarithm of prices that is normally distributed, but for a more
intuitive explanation, we speak in terms of prices rather than log prices.




CHAPTER 8 Sample Restricted Stock Discount Study 305
with some present value adjustments. In other words, it is the statistical
probability of each point on the curve times the pro¬t at each point.
All normal distributions are measured by two and only two para-
meters: the mean and the standard deviation. The mean is the average,
and the standard deviation is a statistical measure of the width of the
curve. In a normal distribution, one standard deviation on either side of
the mean creates includes 68% con¬dence interval, and two standard de-
viations on either side includes 95% of the entire population.
Let™s assume the mean expected stock price at the expiration of the
option is $100 per share. If the standard deviation is $1 per share, then
there is a 68% probability that the stock value will be between $99 and
$101 and a 95% probability that the stock value at expiration will be
between $98 and $102. That would be a tight distribution and would look
like a tall, thin bell-shaped curve. There would only be a 5% probability
that the price would be below $98 or above $102. Since the distribution
is symmetric, that means a 21„2% probability of being below $98 and a
21„2% probability of being above $102. The chances of hitting a jackpot on
this stock are very low.
Now let™s assume the standard deviation is $20 per share, or 20% of
the price. Now there is a 95% probability the price will be within $40 per
share (two standard deviations) of $100, or between $60 and $140. The
probability of hitting the jackpot is much higher.
We now have the background to understand how the stock volatility
is the main determinant of the value of the option. The more volatile the
stock, the shorter and fatter is the normal curve and the greater is the
probability of making a lot of money on the investment. If your stock
ends up on the right side of the curve, it does not matter how far up it
went”you will choose to not exercise the option and you lose only the
price of the option itself. In contrast to owning the stock itself, as an
option holder it matters not at all whether the stock ends up at $100 per
share or $140 per share”your loss is the same. Only the left side matters.
Therefore, a put option on a volatile stock is much more valuable than
one on a stable stock.


Black“Scholes Put Option Formula
The Black“Scholes options pricing model has the following forbidding
formula:
Rft
P EN( d2)e SN( d1)

where:

S stock price
N( ) cumulative normal density function
E exercise price
Rf risk-free rate, i.e., treasury rate of the same term as the option
t time remaining to expiration of the option
t 0.5]
d1 [ln(S/E) (Rf 0.5 variance) t]/[std dev
t0.5]
d2 d1 [std dev



PART 3 Adjusting for Control and Marketability
306
Chaffe™s Article: Put Options to Calculate DLOM of Restricted
Stock
David Chaffe (Chaffe 1973, p. 182) wrote a brilliant article in which he
reasoned that buying a hypothetical put option on Section 144 restricted
stock would ˜˜buy™™ marketability, and the cost of that put option is an
excellent measure of the discount for lack of marketability of restricted
stock.

Commentary to Table 8-2: Black“Scholes Calculation of DLOM
for ENCO, Inc.
Table 8-2 is the Black“Scholes put option calculation of the restricted stock
discount. We begin in row 5 with S, the stock price on the valuation date
of August 11, 1997, of $2.375. We then assume that E, the exercise price,
is identical (row 6).
Row 7 is the time in years from the valuation date to marketability.
According to SEC Rule 144, Robert Smith has a one-year period of re-
striction before he can sell all of his ENCO shares.
Row 8 shows the one year treasury bill rate as of August 11, 1997,
which was 5.32% (see note 1, Table 8-2 for the data source). Row 9 is the
square of row 10. Row 10 contains the annualized standard deviation of
ENCO™s continuously compounded returns, which we calculate in Table
8-2A to be 0.57.
Rows 11 and 12 are the calculation of the two Black“Scholes para-
meters, d1 and d2, the formulas of which appear in notes [2] and [3] of
Table 8-2. Rows 13 and 14 are the cumulative normal density functions
for d1 and d2.10 For example, look at cell B13, which is N( 0.380)

T A B L E 8-2

Black“Scholes Call and Put Options


A B

5 S stk price on valuation date $2.375
6 E exercise price $2.375
7 t time To expiration (yrs) 1.0
8 Rf risk-free rate [1] 5.32%
9 var variance 0.33
10 std dev standard deviation (Table 8-2A, C35) 0.57
11 d1 1st Black-Scholes Parameter [2] 0.380
12 d2 2nd Black-Scholes Parameter [3] (0.194)
13 N(-d1) cum normal density function 0.3521
14 N(-d2) cum normal density function 0.5771
[E * N(-d2)*e-Rft S * N(-d1) [4]
15 P $0.46
16 P/S 19.51%

[1] Source: 1 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
[2] d1 [ ln (S/E) (Rf .5 * variance) * t ] / [ std dev * SQRT(t) ], where variance is expressed as an annual term.
[3] d2 d1 [ std dev * SQRT(t) ] , where std dev is expressed in annual terms.
[4] The put option formula can be found in Options Futures and Other Derivatives, 3rd Ed. by John C. Hull, Prentice Hall, 1997, pp.
241 and 242. The formula is for a European style put option.




10. We use d1 and d2 to calculate call option values and their negatives to calculate put option
values.




CHAPTER 8 Sample Restricted Stock Discount Study 307
0.3521. This requires some explanation. The cumulative normal table from
which the 0.3521 came assumes the normal distribution has been stan-
dardized to a mean of zero and standard deviation of 1.11 This means
that there is a 35.21% probability that our variable is less than or equal
to 0.380 standard deviations below the mean. In cell B14, N( d2)
N(0.194) 0.5771, which means there is a 57.71% probability of being
less than or equal to 0.194 standard deviations above the mean. For per-
spective, it is useful to note that since the normal distribution is sym-
metric, N(0) 0.5000, i.e., there is a 50% probability of being less than or
equal to the mean, which implies there is a 50% probability of being above
the mean.
In row 15 we calculate the value of the put option, which is $0.46
(B15) for the one-year option. In row 16 we calculate the ratio of the fair
market values of the put option to the stock price on the valuation date.
That ratio is our calculation of the restricted stock discount using Black“
Scholes. Thus, our calculation of the restricted stock discount is 19.51%
(B16) for the one-year period of restriction.

Commentary to Table 8-2A: Annualized Standard Deviation of
Continuously Compounded Returns
In Table 8-2A we calculate the annualized standard deviation of contin-
uously compounded returns for use in Table 8-2. Column A shows the
date, Column B shows the closing price, and Columns C and D show the
continuously compounded returns.
We calculated continuously compounded returns over 10 trading
days intervals for ENCO stock. In column C we start with the 1/23/97
closing price and in column D we start with the 1/30/97 closing price.
For example, the 10-trading-day return from 1/23/97 (A5) to 2/6/97 (A7)
is calculated as follows:
return Ln(B7/B5) Ln(3.75/4.25) 0.12516 (cell C7)
In cells C33 and D33 we get two measures of standard deviation of
0.09414 and 0.13500 respectively. To get the annualized standard deviation
we must multiply each interval standard deviation by the square root of
the number of intervals which would occur in a year. The equation is as
follows:
SQRT (# of interval returns in sample period
annualized interval returns

365 days/days in sample period)
For example, the sample period in column C is the time period from the
close of trading on 1/23/97 to the close of trading on 7/31/97 or 189
days, and the number of calculated returns is 13. Therefore the annualized
standard deviation of returns is:
0.09414 SQRT(13 365/189) 0.47169 (cell C34)
annualized

Similarly, the annualized standard deviation of returns in column D is


11. One standardizes a normal distribution by subtracting the mean from each value and dividing
by the standard deviation.


PART 3 Adjusting for Control and Marketability
308
T A B L E 8-2A

Standard Deviation of Continuously Compounded Returns


A B C D

4 Date Closing Price Interval Returns
5 1/23/97 4.2500
6 1/30/97 4.1250
7 2/6/97 3.7500 0.12516
8 2/13/97 3.6250 0.12921
9 2/21/97 3.2500 0.14310
10 2/28/97 3.8750 0.06669
11 3/7/97 3.7500 0.14310
12 3/14/97 3.3750 0.13815
13 3/21/97 3.2500 0.14310
14 3/31/97 2.8750 0.16034
15 4/7/97 2.7500 0.16705
16 4/14/97 2.7500 0.04445
17 4/21/97 2.7500 0.00000
18 4/28/97 2.1875 0.22884
19 5/5/97 2.7500 0.00000
20 5/12/97 2.6250 0.18232
21 5/19/97 2.3125 0.17327
22 5/27/97 2.0625 0.24116
23 6/3/97 2.0625 0.11441
24 6/10/97 2.2500 0.08701
25 6/17/97 2.1250 0.02985
26 6/24/97 2.3750 0.05407
27 7/2/97 2.0625 0.02985
28 7/10/97 2.1875 0.08224
29 7/17/97 1.9375 0.06252
30 7/24/97 2.1250 0.02899
31 7/31/97 1.9375 0.00000
32 8/7/97 2.3750 0.11123
33 Interval std deviation 0.09414 0.13500
34 Annualized std deviation 0.47169 0.67644
35 Average of 2 std deviations 0.57406




0.67644 (D34), while the average of the two is 0.57406 (C35), which trans-
fers to Table 8-2 cell B10.
The reason that we use 10-day intervals in our calculation instead of
daily intervals is that the bid ask spread on the stock may create apparent
volatility that is not really present. This is because the quoted closing
prices are from the last trade. In Nasdaq trading, when one sells to a
dealer it is at the bid price, but when one buys it is at the ask price. If
the last price of the day is switching randomly from a bid to an ask price
and vice versa, this can cause us to measure an apparent volatility that
is not really there. By using 10-day intervals, we reduce any measurement
effect caused by the spread.

Commentary to Table 8-3: Final Calculation of Discount
Table 8-3 is our ¬nal calculation of the restricted stock discount. We use
a weighted average of the two valuation approaches discussed earlier in
the report.
According to the multiple regression analysis in Table 8-1, cell C93,
the discount should be 21.41%. We show that in Table 8-3 in cell C6. In

CHAPTER 8 Sample Restricted Stock Discount Study 309
T A B L E 8-3

Final Calculation of Discount


A B C D E

4 Weighted
5 Method Source Table Discount Weight Discount
6 Multiple regression analysis 8-1, C93 21.41% 50% 10.7%
7 Black-Scholes put option 8-2, B16 19.51% 50% 9.8%
8 Total 100% 20.5%
10 Freely trading closing price, 8/11/97 [1] $ 2.375
11 Less discount for lack of marketability-20.5% $ (0.486)
12 Fair market value of restricted stock $ 1.889
13 Number of shares 500,000
14 FMV of restricted shares (rounded) $945,000

Source: America Online, Prophet Line.




C7 we show the Black“Scholes calculation of 19.51%, which we calculated
in Table 8-2, B16. We weight the two approaches equally, which results
in a discount of 20.5% (E8). The closing price of ENCO, Inc. common
stock on August 11, 1997, was $2.375 (E10) per share.12 The 20.5% discount
is $0.486 (E11) per share, leaving the fair market value of the restricted
stock on that date at $1.889 per share (E12). Multiplying that by 500,000
shares (E13), the fair market value of the ENCO stock received by Robert
Smith is $945,000 (E14).


Conclusion of Discount for Lack of Marketability
It is our opinion, subject to this report and the statement of limiting con-
ditions, that the proper discount to fair market value of the restricted
shares from the traded price of ENCO, Inc. stock on August 11, 1997, is
20.5%. Assuming the closing price of ENCO stock on that date of $2.375
per share is the fair market value of the freely trading shares, the discount
of 20.5% is $0.486 per share, leaving a fair market value of the 500,000
shares of restricted stock of $1.889 per share, or $945,000 (E14) for Robert
Smith.


ASSUMPTIONS AND LIMITING CONDITIONS
In accordance with recognized professional ethics, the fee for this service
is not contingent upon our conclusion of value, and neither Abrams Val-
uation Group nor any of its employees has a present or intended interest
in the Company.
Per your instructions, we have relied upon Robert Smith™s informa-
tion as to shares outstanding and other relevant information. We have
been accepted this information without veri¬cation as being correct. The
same is true as to the dates of marketability, though our information came
from Len Storm, Vice President and Legal Secretary of ENCO, Inc.


12. Source: America Online, Prophet Line.


PART 3 Adjusting for Control and Marketability
310
The conclusions are based on our analysis and discussions with Rob-
ert Smith. We did not make any site visit, as we deemed that unnecessary.
We further assume that present ENCO Management would continue to
maintain the character and integrity of the enterprise through any sale,
reorganization, or diminution of the owners™ participation or equity in-
terest. We know of no signi¬cant pending legal action against the Com-
pany of which the market is unaware;13 nor do we know of any other
˜˜skeleton in the closet,™™ and we assume none is or will be occurring. If
this did happen, then might change the value of the Company and Robert
Smith™s underlying stock.
Our opinion of the discount for lack of marketability in this report
is valid only for the stated purpose and only at the date of the appraisal.
It is our understanding that this opinion will be used for income tax
purposes. The fair market value, as determined within our report, shall
not be used for other purposes or dates.
Though some similarities exist between value as set forth for this
purpose and others, it would be incorrect to use the price per share as
determined within our report for any other purposes due to speci¬c tim-
ing, performance, and marketability issues that arise in evaluating the
fair market value of a company. Accordingly, any such use of the value
as determined within this report for other purposes would be inaccurate
and possibly misleading and no such use shall be made without written
permission from Abrams Valuation Group.
Our determination of fair market value as reported herein does not
represent investment advice of any kind to any person and does not con-
stitute a recommendation as to the purchase or sale of shares of the Com-
pany or as to any our 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.
This report may only be presented to persons whose use is relevant
to its purpose, and only the entire report can be so conveyed. Giving part
of this report for someone to read can lead to dangerous misunderstand-
ing and is prohibited.
Neither all nor any part of the contents of this report shall be con-
veyed to the public through advertising, public relations, news, sales,
mail, direct transmittal, or other media without the prior written consent
and approval of Abrams Valuation Group.


APPRAISER™S QUALIFICATIONS
Jay B. Abrams, ASA, CPA, MBA, author and inventor, is a nationally
recognized consultant within the valuation ¬eld.
Mr. Abrams lectured at the June 1996 Toronto International Confer-
ence of the American Society of Appraisers, the organization from which


13. By the ef¬cient markets hypothesis, if the market knows about a lawsuit or even a potential
lawsuit, the stock price will re¬‚ect that. Here we are saying we know of no insider relevant
information that would change the market price if the public knew about that.


CHAPTER 8 Sample Restricted Stock Discount Study 311
he holds the professional designation of Accredited Senior Appraiser
(ASA) in Business Valuation. He has lectured for the National Association
of Certi¬ed Valuation Analysis 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
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, McGraw-Hill, November 2000.
— ˜˜ESOPs: Measuring and Apportioning the Dilution,™™ Valuation,
June 1997.

PART 3 Adjusting for Control and Marketability
312
— ˜˜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.




CHAPTER 8 Sample Restricted Stock Discount Study 313
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January 4, 2000
Mr. Bradley J. Jones
Manager
ABC Company, LLC
PO Box 99214
San Diego, CA 92169
Dear Mr. Jones:
On January 6, 1999, ABC Company, LLC (˜˜ABC,™™ or ˜˜the LLC™™), a Cal-
ifornia Limited Liability Company, was established for purposes of in-
vesting in real estate and other assets. On December 25, 1999, Tina M.
Smith made four gifts of member interests of 2.80% in the LLC to the
other existing members, who are her children. On January 3, 2000, Mrs.
Smith made four gifts of 2.25% member interests.
In accordance with your instructions, we have performed a Complete
Appraisal, documented in a Self-Contained Report, to calculate the dis-
counts for lack of control and lack of marketability (collectively, ˜˜the Frac-
tional Interest Discount™™) for the four 2.80% and 2.25% member interest
gifts for gift tax purposes.
Our opinion of the Fractional Interest Discount will be effective from
December 25, 1999, through January 3, 2000, for gift tax purposes. The
fractional interest discounts, as determined within our report, shall not
be used for other purposes or dates without our written consent, as they
may be misleading and dangerous.
The term fair market value is de¬ned as follows: ˜˜the amount at which
property [in this case, the member interests in the LLC] would change
hands between a willing buyer and a willing seller, when the former is

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