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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

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