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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
[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

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
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)

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
T A B L E 8-2A

Standard Deviation of Continuously Compounded Returns


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


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

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
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.

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
— ˜˜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
This page intentionally left blank.
January 4, 2000
Mr. Bradley J. Jones
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
not under any compulsion to buy and the latter is not under any com-
pulsion to sell, and when both parties have reasonable knowledge of
relevant facts.™™1
The scope of our engagement did not include a physical visit to the prop-
erties or your of¬ces, nor a separate valuation of the former.
For the fair market value of the properties, we relied on the appraisals
by ABC Real Estate Appraisals as of December 1998 and the estimate of
fair market value for the Dutch Flat property by Bradley Jones. All in-
formation regarding the LLC was provided by Bradley Jones and the
LLC™s attorney, David Hollander, Esq., and its accountant, David Sofer,

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).

PART 3 Adjusting for Control and Marketability
CPA. This information has been accepted, without additional veri¬cation,
as correctly re¬‚ecting the ¬nancial statements and value and nature of
the underlying assets, and is your responsibility.
In our opinion, based upon our investigation and analysis and subject to
the attached report and Statement of Limiting Conditions, the appropriate
fractional interest discount for the subject 2.80% and 2.25% member in-
terests is 48%. The fair market value of each 2.80% interest is $20,000,
and the fair market value of each 2.25% interest is $16,250.
We retain a copy of this letter in our ¬les, together with the ¬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 and Principals of Appraisal Practice) Certi-
I certify that to the best of my knowledge and belief that
— 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

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