ńņš. 12 |

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 simpliļ¬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 ļ¬‚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 simpliļ¬es

to:

1 g

x10

1

r, ā’ 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 formulaā“sellersā™ 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 sellersā™ transaction costs, be-

PART 3 Adjusting for Control and Marketability

286

cause the buyer does not care about the sellerā™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 inļ¬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 ļ¬‚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 ļ¬rst relevant transaction cost takes place at time zero instead of

t j, which is appropriate for quantifying the discount component due

to buyersā™ transaction costs. With this assumption, we would modify the

above analysis by inserting a (1 z) in front of the ļ¬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 buyersā™ equivalent formula of equation (A7-8) is:

1 r x10

1

NPVTC (1 z)

z)x10

r g 1 (1

NPV with buyersā™ transaction costs removed (A7-8a)

Obviously, equation (A7-8a) is lower than equation (A7-8), because

the ļ¬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, formulaā”buyersā™ 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 buyersā™ and sellersā™ transactions costs are appropriate for business

valuations. However, for calculating that component of DLOM for limited

life entities such as limited partnerships whose document speciļ¬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 signiļ¬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 ļ¬nal equations, (A7-23) and (A7-24). Consider this section as

reference material.

Letā™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. Letā™s deļ¬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 ļ¬nal term ā˜ā˜should haveā™ā™ a subtraction of (1 g) /

0.5

(1 r) , but that equals zero for g r. Therefore, we leave that term

out. Again, the ļ¬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 ļ¬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 ļ¬nite geometric sequence.

The ļ¬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) simpliļ¬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 simpliļ¬es to:

xj z)sx n j

1 z(1

D 1 (A7-23)

z)x j

1 (1

discount componentā”sellersā™ costsā”finite life

Note that as the life of the entity (or the interest in the entity) that

we are valuing goes to inļ¬nity, n ā’ , so xn j ā’ 0 and (A7-23) reduces

to equation (A7-11).

The equivalent expression for buyersā™ costs is:

xj z)sxn j]

(1 z)[1 z(1

D 1

z)x j

1 (1

discount componentā”buyersā™ costsā”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 DISCOUNTā”

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 simpliļ¬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 deļ¬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 ļ¬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 conļ¬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). AVGā™s

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

date of the appraisal.

Note: all names are ļ¬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. (ā˜ā˜ENCO,ā™ā™ or ā˜ā˜the Companyā™ā™) 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 deļ¬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 Stormā™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. Ā§ 1.170A-1(c)(2) (income tax, charitable contributions of property); see Reg.

Ā§Ā§ 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 ļ¬les, together with ļ¬eld data from

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

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

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

tion:

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

ā— 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.

ā— 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.

ā— Our compensation is not contingent on an action or event

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

this report.

ā— 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.

ā— No one has provided signiļ¬cant professional assistance to the

person signing this report.

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

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

American Society of Appraisers, with certiļ¬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

Blackā“Scholes Put Option Formula

Chaffeā™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

APPRAISERā™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. (ā˜ā˜ENCO,ā™ā™ or ā˜ā˜the Companyā™ā™) 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

ENCOā™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 stockā”a reasonable assumptionā”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 deļ¬ned as ā˜ā˜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.ā™ā™3

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

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

Ā§Ā§ 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 certiļ¬cate issued to Robert Smith including

copies of the ā™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 ļ¬rst is based on our own statistical analysis using multiple

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

involves using a Blackā“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 ļ¬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 1980ā“

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 ļ¬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 ļ¬rst one and one-quarter pages contain

data on 53 sales of restricted stock between 1980ā“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 signiļ¬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 soldā”$: 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 nonafļ¬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 insigniļ¬cant. We re-

strict our commentary to the seven independent variables that were sta-

tistically signiļ¬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

signiļ¬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% conļ¬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 coefļ¬cients for the independent variables. E77 to E84 contains

the t-statistics. Only the y-intercept itself is not signiļ¬cant at the 95%

conļ¬dence level. The earnings stability and market capitalization varia-

bles are signiļ¬cant at the 98% level,6 and all the other variables are sig-

niļ¬cant at the 99 % conļ¬dence level.

6. The statistical signiļ¬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 Intā™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 & Ermaā™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. Intā™l, Inc. 37.8% 1.10E 14 $5,950,000 101,259,000 0.00 0.87 2.00 30.2

32 27 Noble Romanā™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 Ryanā™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 Ryanā™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 Signiļ¬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 Coefļ¬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 Coefļ¬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 ā˜ā˜Avg Yrs To Sellā™ā™ and ā˜ā˜Rev2ā™ā™ 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 ENCOā™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 ENCOā™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 inļ¬‚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 ļ¬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.ā™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 Blackā“Scholes Options pricing

model (Blackā“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 Blackā“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 speciļ¬c number of

shares of a company at a speciļ¬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

speciļ¬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 Blackā“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 speciļ¬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. Blackā“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 Blackā“Scholes calculated value of a put option is the area under the

left half of the bell-shaped curve multiplied by the proļ¬t at each price,

7. We have not researched IBMā™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 proļ¬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% conļ¬dence interval, and two standard de-

viations on either side includes 95% of the entire population.

Letā™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 21ā„2% probability of being below $98 and a

21ā„2% probability of being above $102. The chances of hitting a jackpot on

this stock are very low.

Now letā™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

wentā”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 shareā”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.

Blackā“Scholes Put Option Formula

The Blackā“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

Chaffeā™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 ā˜ā˜buyā™ā™ 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: Blackā“Scholes Calculation of DLOM

for ENCO, Inc.

Table 8-2 is the Blackā“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

ENCOā™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 Blackā“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

Blackā“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 Blackā“

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 ļ¬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 Blackā“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 Smithā™s informa-

tion as to shares outstanding and other relevant information. We have

been accepted this information without veriļ¬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 ownersā™ participation or equity in-

terest. We know of no signiļ¬cant pending legal action against the Com-

pany of which the market is unaware;13 nor do we know of any other

ā˜ā˜skeleton in the closet,ā™ā™ and we assume none is or will be occurring. If

this did happen, then might change the value of the Company and Robert

Smithā™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 speciļ¬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.

APPRAISERā™S QUALIFICATIONS

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

recognized consultant within the valuation ļ¬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 efļ¬cient markets hypothesis, if the market knows about a lawsuit or even a potential

lawsuit, the stock price will reļ¬‚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 Certiļ¬ed Valuation Analysis and the Anthony Robbinsā™ 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 ļ¬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:

ā— Employee stock ownership plans (ESOPs).

ā— Estate planning, estate and gift taxes.

ā— Income taxes and charitable contributions.

ā— Mergers and acquisitions and sales.

ā— Divestitures.

ā— Warrants and stock options.

ā— Shareholder buy/sell agreements.

ā— Blocks of publicly traded securities.

ā— Private placements and public offerings.

ā— Restricted securities.

ā— Recapitalization and reorganizations.

ā— Debt and equity ļ¬nancing.

ā— Company dissolutions.

ā— 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. Abramsā™ 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 Paciļ¬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 ļ¬rm, and prior to this was an auditor with Arthur An-

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

of Chicago in ļ¬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. Abramsā™ writings include:

ā— Quantitative Business Valuation, McGraw-Hill, November 2000.

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

June 1997.

PART 3 Adjusting for Control and Marketability

312

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

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

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

March 1995.

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

Valuation, August, 1994.

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

Business Valuation Review, September 1994.

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

ā— ā˜ā˜An Iterative Procedure to Value Leveraged ESOPs,ā™ā™ Valuation,

January 1993.

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

The Practical Accountant, June 1992.

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

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

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

Publications, New York, 1989.

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

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

Rate Loan Amortization,ā™ā™ 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 (ā˜ā˜ABC,ā™ā™ or ā˜ā˜the LLCā™ā™), 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, ā˜ā˜the Frac-

tional Interest Discountā™ā™) 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 deļ¬ned as follows: ā˜ā˜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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