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13 ANOVA

14 df SS MS F Signi¬cance F
15 Regression 7 0.6354 0.0908 11.9009 1.810E-08
16 Residual 45 0.3432 0.0076
17 Total 52 0.9786
19 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%
20 Intercept 0.0673 0.1082 0.6221 0.5370 0.2854 0.1507
21 Rev2 4.629E-18 9.913E-19 4.6698 0.0000 6.626E-18 2.633E-18
22 Shares sold-$ 3.619E-09 1.199E-09 3.0169 0.0042 6.035E-09 1.203E-09
23 Mkt cap 4.789E-10 1.790E-10 2.6754 0.0104 1.184E-10 8.394E-10
24 Earn stab 0.1038 0.0402 2.5831 0.0131 0.1848 0.0229
25 Rev stab 0.1824 0.0531 3.4315 0.0013 0.2894 0.0753
26 AvgYrs2Sell 0.1722 0.0362 4.7569 0.0000 0.0993 0.2451
27 Price stab 0.0037 8.316E-04 4.3909 0.0001 0.0020 0.0053

Source: Management Planning, Inc. Princeton NJ (except for AvgYrs2Sell and Rev 2 , which we derived from their data)
239
240




T A B L E 7-5 (continued)

Abrams Regression of Management Planning Study Data


A B C D E F G

32 Regression #2 (Without Price Stability)
34 Regression Statistics
35 Multiple R 0.7064
36 R square 0.4990
37 Adjusted R square 0.4337
38 Standard error 0.1032
39 Observations 53

41 ANOVA

42 df SS MS F Signi¬cance F
43 Regression 6 0.4883 0.0814 7.6365 0.0000
44 Residual 46 0.4903 0.0107
45 Total 52 0.9786
47 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%
48 Intercept 0.1292 0.1165 1.1089 0.2732 0.1053 0.3637
Rev2
49 5.39E-18 1.15E-18 4.6740 0.0000 7.71E-18 3.07E-18
50 Shares sold-$ 4.39E-09 1.40E-09 3.1287 0.0030 7.21E-09 1.57E-09
51 Mkt cap 6.10E-10 2.09E-10 2.9249 0.0053 1.90E-10 1.03E-09
52 Earn stab 0.1381 0.0466 2.9626 0.0048 0.2319 0.0443
53 Rev stab 0.1800 0.0628 2.8653 0.0063 0.3065 0.0536
54 AvgYrs2Sell 0.1368 0.0417 3.2790 0.0020 0.0528 0.2208
other 40.53% of variation in the discounts that remains unexplained is
due to two possible sources: other signi¬cant independent variables of
which I (and Management Planning, Inc.) do not know, and random var-
iation. The standard error of the y-estimate is 8.7% (B10 rounded). We
can form approximate 95% con¬dence intervals around the y-estimate by
adding and subtracting two standard errors, or 17.4%.
Cell B20 contains the regression estimate of the y-intercept, and B21
through B27 contain the regression coef¬cients for the independent var-
iables. The t-statistics are in D20 through D27. Only the y-intercept itself
is not signi¬cant at the 95% con¬dence level. The market capitalization
and earnings stability variables are signi¬cant at the 98% level,45 and all
the other variables are signi¬cant at the 99 % con¬dence level.
Note that several of the variables are similar to Grabowski and King™s
results (Grabowski and King 1999), discussed in Chapter 5. They found
that the coef¬cient of variations (in log form) of operating margin and
return on equity are statistically signi¬cant in explaining stock market
returns. Here we ¬nd that the stability of revenues and earnings (as well
as the coef¬cient of variation of stock market prices) explain restricted
stock discounts. Thus, these variables are signi¬cant in determining the
value of the underlying companies, assuming they are marketable, and
in determining restricted stock discounts when restrictions exist.
I obtained regression #2 in Table 7-5 by regressing all the indepen-
dent variables in the ¬rst regression except for price stability. The adjusted
R 2 has dropped to 43.37% (B37), indicating that regression #1 is superior
when price data are available, which generally it is for restricted stock
studies and is not for calculating DLOM for privately held businesses.
The second regression is not recommended for the calculation of re-
stricted stock discounts, but it will be useful in other contexts.

Using the Put Option Model to Calculate DLOM
of Restricted Stock
Chaffe (1993) wrote a brilliant article in which he reasoned that buying a
hypothetical put option on Section 144 restricted stock would ˜˜buy™™ mar-
ketability and that the cost of that put option is an excellent measure of
the discount for lack of marketability of the stock. For puts, the Black“
Scholes option pricing model has the following formula:
Rf t
P E N( d2)e S N( 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
t0.5]
d1 [ln(S/E) (Rf 0.5 variance) t]/[std dev
t0.5]
d2 d1 [std dev
We have suf¬cient daily price history on 13 of the stocks in Table


45. The statistical signi¬cance is one minus the P-value, which is in E20 through E27.


CHAPTER 7 Adjusting for Levels of Control and Marketability 241
7-5 to derive the proper annualized standard deviation (std dev) of con-
tinuously compounded returns to test Chaffe™s approach.

Annualized Standard Deviation of Continuously Compounded
Returns. Table 7-6 is a sample calculation of the annualized standard
deviation of continuously compounded returns for Chantal Pharmaceu-
tical, Inc. (CHTL), which is one of the 13 stocks. The purpose of this table
is to demonstrate how to calculate the standard deviation.
Column A shows the date, column B shows the closing price, and
columns C and D show the continuously compounded returns. The sam-
ple period is just over 6 months and ends the day prior to the transaction
date.
We calculate continuously compounded returns over 10-trading-day
intervals for CHTL stock.46 The reason for using 10-day intervals in our

T A B L E 7-6

Calculation of Continuously Compounded Standard Deviation
Chantal Pharmaceutical, Inc.”CHTL


A B C D

6 Date Close Interval Returns

7 1/31/95 $2.1650
8 2/7/95 $2.2500
9 2/14/95 $2.5660 0.169928
10 2/22/95 $2.8440 0.234281
11 3/1/95 $2.6250 0.022733
12 3/8/95 $2.9410 0.033538
13 3/15/95 $2.4480 0.069810
14 3/22/95 $2.5000 0.162459
15 3/29/95 $2.2500 0.084341
16 4/5/95 $2.0360 0.205304
17 4/12/95 $2.2220 0.012523
18 4/20/95 $2.1910 0.073371
19 4/27/95 $2.6950 0.192991
20 5/4/95 $2.6600 0.193968
21 5/11/95 $2.5660 0.049050
22 5/18/95 $2.5620 0.037538
23 5/25/95 $2.9740 0.147560
24 6/2/95 $3.3120 0.256764
25 6/9/95 $5.1250 0.544223
26 6/16/95 $6.0000 0.594207
27 6/23/95 $5.8135 0.126052
28 6/30/95 $6.4440 0.071390
29 7/10/95 $6.5680 0.122027
30 7/17/95 $6.6250 0.027701
31 7/24/95 $8.0000 0.197232
32 7/31/95 $7.1250 0.072759
33 8/7/95 $7.8120 0.023781 0.092051
34 Interval standard deviation”CHTL 0.16900 0.20175
35 Annualized 0.84901 1.03298
36 Average of standard deviations 0.94099




46. The only exception is the return from 7/31/95 to 8/7/95, which is in cell D33.




PART 3 Adjusting for Control and Marketability
242
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 trad-
ing, one sells to a dealer at the bid price and buys at the ask price. If on
successive days the last price of the day is switching randomly from a
bid to an ask price and back, this can cause us to measure a considerable
amount of apparent volatility that is not really there. By using 10-day
intervals, we minimize this measurement error caused by the spread.
We start with the 1/31/95 closing price in column C and the 2/7/
95 closing price in column D. For example, the 10-trading-day return from
1/31/95 (A7) to 2/14/95 (A9) is calculated as follows: return Ln(B9/
B7) Ln(2.5660/2.1650) 0.169928 (C9).
Using this methodology, we get two measures of standard deviation:
0.16900 (C34) and 0.20175 (D34). To convert to the annualized standard
deviation, we must multiply each interval standard deviation by the
square root of the number of intervals that would occur in a year. The
equation is as follows:
SQRT
annualized interval returns



# of interval returns in sample period
365 days per year
days in sample period
For example, the sample period in column C is the time period from
the close of trading on January 31, 1995, to the close of trading on August
7, 1995, or 188 days, and there are 13 calculated returns. Therefore the
annualized standard deviation of returns is:
0.1690 SQRT(13 365/188)
annualized

0.1690 SQRT(25.2394) 0.84901 (cell C35)
The 13 trading periods that span 188 days would become 25.2394 trading
periods in one year (25.2394 13 365/188). The square root of the
25.2394 trading periods is 5.0239. We multiply the sample standard de-
viation of 0.1690 by 5.0239 0.84901 to annualize the standard deviation.
Similarly, the annualized standard deviation of returns in column D is
1.03298 (D35), and the average of the two is 0.94099 (D36).

Calculation of the Discount. Table 7-7 is the Black“Scholes put op-
tion calculation of the restricted stock discount. We begin in cell B5 with
S, the stock price on the valuation date of August 8, 1995, of $8.875. We
then assume that E, the exercise price, is identical (B6).
B7 is the time in years from the valuation date to marketability. Ac-
cording to SEC Rule 144, the shares have a two-year period of restriction
before the ¬rst portion of the block can be sold. At 2.25 years the rest can
be sold. The weighted average time to sell is 2.125 years (B7, transferred
from Table 7-5, I17) for this particular block of Chantal.
B8 shows the two-year Treasury rate, which was 5.90% as of the
transaction date. B9 contains the annualized standard deviation of returns



CHAPTER 7 Adjusting for Levels of Control and Marketability 243
T A B L E 7-7

Black“Scholes Put Option”CHTL


A B

5 S Stk price on valuation date $8.875
6 E Exercise price $8.875
7 t time to expiration in yrs (Table 7-5, I17) 2.125
8 r risk-free rate [1] 5.90%
9 stdev standard deviation (Table 7-6, D36) 0.941
10 var variance 0.885
11 d1 1st Black-Scholes parameter [2] 0.777
12 d2 2nd Black-Scholes parameter 3] (0.594)
13 N( d1) cum normal density function 0.219
14 N( d2) cum normal density function 0.724
[E*N( d2)*e rt ] S*N( d1)
15 P $3.73
16 P/S 42.0%

Note: Values are for European options. 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.
[1] 2 Year Treasury rate on transaction date, 8/8/95 (Source: Federal Reserve)
.5 * var) * t]/[stdev *t0.5], where variance and standard deviation are expressed in annual terms.
[2] d1 [ln (S/E) (r
[std dev * t0.5]
[3] d2 d1




for CHTL of 0.941, transferred from Table 7-6, cell D36, while B10 is var-
iance, merely the square of B9.
Cells B11 and B12 are the calculation of the two Black“Scholes par-
ameters, d1 and d 2. B13 and B14 are the cumulative normal density func-
tions for d1 and d 2. For example, look at cell B13, which is N( 0.777)
0.219. This requires some explanation. The cumulative normal table
from which the 0.219 came assumes the normal distribution has been
standardized to a mean of zero and standard deviation of 1.47 This means
that there is a 21.9% probability that our variable is less than or equal to

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