. 10
( 18)


CHAPTER 7 Adjusting for Levels of Control and Marketability 233
were several other inconsistencies in the results of the two
3. The log“log form of regression that Phillips and Freeman used
can have the effect of making large variations look small. The
standard errors of their regressions were very high. The
standard error of the Mergerstat regression was 0.925. Two
standard errors is 1.85. Exponentiating, the 95% con¬dence
interval is approximately equal to multiplying the (value/sales)
estimate by two standard errors on either side of the regression
estimate. The high side of the 95% con¬dence interval is e1.85
6.36 times the regression estimate, and the low side is e 1.85
0.157 times the regression estimate. Let™s put some speci¬c
numbers into their equation to see what the con¬dence intervals
look like. Let™s assume we are forecasting the value of the
common stock as a percentage of sales for a ¬rm over $100
million in value that is neither a bank, a private placement, nor
a subsidiary. Their regression equation is ln(Value/Sales)
3.242 0.56 ln net margin 0.45 ln (1/PE of the S&P 500).
Let™s assume a 5% after-tax margin and an average PE for the
S&P 500 of 15, so 1/PE 0.067. Then, ln(Value/Sales) 3.242
(0.56 ln 0.05) (0.45 ln 0.067) 3.242 1.678 1.219
0.345. Thus, the regression estimate of (Value/Sales)
1.413, or value is approximately 1.4 times sales, which seems
high. If sales are $100, then net income after taxes is $5, which
when multiplied by a PE ratio of 15 leads to a value of $75,
which implies value should be 0.75 Sales, not 1.4. The
reliability of the forecast is low. The 95% con¬dence interval is
approximately: 0.22 Sales Value 8.99 Sales.
4. There were fairly few transactions with a private seller. In the
Mergerstat database, private targets were 18 out of 416
transactions, and in the SDC database, private targets were 33
out of 445 targets. In total, private targets were approximately
6% of the combined databases.
The small number of transactions with privately held sellers is not
necessarily worrisome in itself, but combined with the limitations of the
results in 1, the inconsistent results in 2, and the very wide con¬dence
intervals in 3, the results of this study are insuf¬cient to reject DLOM for
control interests of privately held ¬rms.

Kasper™s BAS Model
Larry Kasper (Kasper 1997, p. 106) uses an econometric equation devel-
oped by Amihud and Mendelson (Amihud and Mendelson 1991) to cal-
culate the bid-ask spread (BAS). Their equation is: r 0.006477 0.01012
0.002144 ln BAS, where r is the excess monthly returns on a stock
portfolio over the 90-day Treasury Bill rate and the BAS is multiplied by
100, i.e., a BAS of 25% is denominated as 25, not 0.25.
Kasper says that most business brokers would not list a business that
had to be discounted more than 25%. Substituting 25 into the above equa-

PART 3 Adjusting for Control and Marketability
tion, the excess return required for a BAS of 25% is 0.0069 per month, or
approximately 8.28% per year. One would then seek out business brokers
(or through IBA, Pratt™s Stats, BIZCOMPS, etc.) for actual BASs. Anyone
interested in using Kasper™s model must read his outstanding book, as
this summary is inadequate for understanding his work.
A number of differences in the environment of NASDAQ and pri-
vately held business can weaken the applicability of this regression equa-
tion from the former to the latter:
1. The BAS in NASDAQ compensates the dealer for actually taking
possession of the stock. The dealer actually stands to gain or
lose money, whereas business brokers do not.
2. It takes much longer to sell a private business than stock on
3. The market for privately held ¬rms is much thinner than it is
with Nasdaq.
4. Transactions costs are far higher in privately held business than
in Nasdaq.
Note that items 2 through 4 are the components of the economic
components approach, which we will cover shortly in my model. Also,
the reservation in 1 also applied in the Menyah and Paudyal results ear-
lier in the chapter, where the BAS depends on the number of market
makers. Again, business brokers are not market makers in the same sense
that dealers are. Additionally, as Kasper points out, the regression coef-
¬cients will change over time. Kasper also presents a different model, the
discounted time to market model (Kasper 1997, pp. 103;“04) that is worth
reading. Neither of his models considers transactions costs or the effects
of thin markets.40

Restricted Stock Discounts
We will now discuss DLOM for restricted stocks as a preparation for our
general model for DLOM. We use two valuation methodologies in cal-
culating the restricted stock discount. The ¬rst is based on my own mul-
tiple regression analysis of data collected by Management Planning, Inc.
(MPI),41 an independent valuation ¬rm in Princeton, New Jersey. The sec-
ond method involves using a Black“Scholes put option as a proxy for the

Regression of MPI Data
Ten studies of sales of restricted stocks have been published.42 The ¬rst
nine appear in Pratt, Reilly, Schweihs (1996, chap. 15) and Mercer (1997);

40. That is not to say that I downgrade his book. It is brilliant and a must read for anyone in the
41. Published in Chapter 12 of Mercer (1997). I wish to thank MPI for being gracious and helpful
in providing us with its data and consulting with us. In particular, Roy H. Meyers, Vice
President, was extremely helpful. MPI provided us with four additional data points and
some data corrections.
42. See Mercer (1997, p. 69) for a summary of the results of the ¬rst nine studies.

CHAPTER 7 Adjusting for Levels of Control and Marketability 235
in those studies, the authors did not publish the underlying data and
merely presented their analysis and summary of the data. Additionally,
only the Hall/Polacek study contains data beyond 1988 (through 1992).
The Management Planning study, which Mercer justi¬ably accords a sep-
arate chapter and extensive commentary in his book, contains data on
trades from 1980“1996 and thus is superior to the others in two ways:
the detail of the data exists and the data are more current.
Table 7-5 is two pages long. The ¬rst page contains data on 53 sales
of restricted stock between 1980“1996. Column A is numbered 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 independent variables, which I have
labeled X1, X2, . . ., X7. Below is a description of the independent variables:

# Independent Variable

1 Revenues squared.
2 Shares Sold”$: the discounted 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 R 2 of the regression of net income as a function of time, with time
measured as years 1, 2, 3, etc.
Revenue stability: the R2 of the regression of revenue as a function of time, with time
measured as years 1, 2, 3, etc.
6 Average years to sell: the weighted average years to sell by a nonaf¬liate based on SEC
Rule 144. I calculated the holding period for the last four issues (DPAC, UMED, NEDI,
and ARCCA) based on changes in Rule 144, even though it was not effective yet,
because the change was out for review at that time and was highly likely to be
accepted.43 These transactions occurred near the beginning of March 1996, well after
the SEC issued the exposure draft on June 27, 1995. This was approximately 14
months before the rule change went into effect at the end of April 1997. The average
time to resale for the shares in these four transactions was determined based on the
rule change, resulting in a minimum and maximum average holding period of 14
months and 2 years, respectively.44
7 Price stability: This ratio is calculated by dividing the standard deviation of the stock
price by the mean of the stock price”which is the coef¬cient of variation of price”
then multiplying by 100. The end-of-month stock prices for the 12 months prior to the
valuation date are used.

I regressed 30 other independent variables included in or derived
from the Management Planning study, and all were statistically insignif-
icant. I restrict our commentary to the seven independent variables that
were statistically signi¬cant at the 95% level.
The third page of Table 7-5 contains the regression statistics. In re-
gression #1 the adjusted R 2 is 59.47% (B9), a reasonable though not stun-
ning result for such an analysis. This means that the regression model
accounts for 59.47% of the variation in the restricted stock discounts. The

43. According to John Watson, Jr., Esq., of Latham & Watkins in Washington, D.C., the securities
community knew the rule change would take place. In a telephone conversation with Mr.
Watson, he said it was only a question of timing.
44. In other words, I assumed perfect foreknowledge of when the rule change would become

PART 3 Adjusting for Control and Marketability
T A B L E 7-5

Abrams Regression of Management Planning Study Data


4 Y X1 X2 X3 X4 X5 X6 X7
6 Discount Shares Sold-$ Mkt Cap Earn Stab Rev Stab AvgYrs2Sell Price Stab

7 1 Air Express Int™l 0.0% 8.58E+16 $4,998,000 25,760,000 0.08 0.22 2.84 12.0
8 2 AirTran Corp 19.4% 1.55E+16 $9,998,000 63,477,000 0.90 0.94 2.64 12.0
9 3 Anaren Microwave, Inc. 34.2% 6.90E+13 $1,250,000 13,517,000 0.24 0.78 2.64 28.6
10 4 Angeles Corp 19.6% 7.99E+14 $1,800,000 16,242,000 0.08 0.82 2.13 8.4
11 5 AW Computer Systems, Inc. 57.3% 1.82E+13 $1,843,000 11,698,000 0.00 0.00 2.91 22.6
12 6 Besicorp Group, Inc. 57.6% 1.57E+13 $1,500,000 63,145,000 0.03 0.75 2.13 98.6
13 7 Bioplasty, Inc, 31.1% 6.20E+13 $11,550,000 43,478,000 0.38 0.62 2.85 44.9
14 8 Blyth Holdings, Inc. 31.4% 8.62E+13 $4,452,000 98,053,000 0.04 0.64 2.13 58.6
15 9 Byers Communications Systems, Inc. 22.5% 4.49E+14 $5,007,000 14,027,000 0.90 0.79 2.92 6.6
16 10 Centennial Technologies, Inc. 2.8% 6.75E+13 $656,000 27,045,000 0.94 0.87 2.13 35.0
17 11 Chantal Pharm. Corp. 44.8% 5.21E+13 $4,900,000 149,286,000 0.70 0.23 2.13 51.0
18 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
19 13 Crystal Oil Co. 24.1% 7.47E+16 $24,990,000 686,475,000 0.42 0.57 2.50 28.5
20 14 Cucos, Inc. 18.8% 4.63E+13 $2,003,000 12,579,000 0.77 0.87 2.84 20.4
21 15 Davox Corp. 46.3% 1.14E+15 $999,000 18,942,000 0.01 0.65 2.72 24.6
22 16 Del Electronics Corp. 41.0% 4.21E+13 $394,000 3,406,000 0.08 0.10 2.84 4.0
23 17 Edmark Corp 16.0% 3.56E+13 $2,000,000 12,275,000 0.57 0.92 2.84 10.5
24 18 Electro Nucleonics 24.8% 1.22E+15 $1,055,000 38,435,000 0.68 0.97 2.13 21.4
25 19 Esmor Correctional Svces, Inc. 32.6% 5.89E+14 $3,852,000 50,692,000 0.95 0.90 2.64 34.0
26 20 Gendex Corp 16.7% 2.97E+15 $5,000,000 55,005,000 0.99 0.71 2.69 11.5
27 21 Harken Oil & Gas, Inc. 30.4% 7.55E+13 $1,999,000 27,223,000 0.13 0.88 2.75 19.0
28 22 ICN Paramaceuticals, Inc. 10.5% 1.50E+15 $9,400,000 78,834,000 0.11 0.87 2.25 23.9
29 23 Ion Laser Technology, Inc. 41.1% 1.02E+13 $975,000 10,046,000 0.71 0.92 2.82 22.0
30 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

T A B L E 7-5 (continued)

Abrams Regression of Management Planning Study Data


4 Y X1 X2 X3 X4 X5 X6 X7
6 Discount Shares Sold-$ Mkt Cap Earn Stab Rev Stab AvgYrs2Sell Price Stab
31 25 Medco Containment Svces, Inc. 15.5% 5.42E+15 $99,994,000 561,890,000 0.84 0.89 2.85 12.8
32 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
33 27 Noble Roman™s Inc. 17.2% 8.29E+13 $1,251,000 11,422,000 0.06 0.47 2.79 17.0
34 28 No. American Holding Corp. 30.4% 1.35E+15 $3,000,000 79,730,000 0.63 0.84 2.50 22.1
35 29 No. Hills Electronics, Inc. 36.6% 1.15E+13 $3,675,000 21,812,000 0.81 0.79 2.83 52.7
36 30 Photographic Sciences Corp 49.5% 2.70E+14 $5,000,000 44,113,000 0.06 0.76 2.86 27.2
37 31 Presidential Life Corp 15.9% 4.37E+16 $38,063,000 246,787,000 0.00 0.00 2.83 17.0
38 32 Pride Petroleum Svces, Inc. 24.5% 4.34E+15 $21,500,000 74,028,000 0.31 0.26 2.83 18.0
39 33 Quadrex Corp. 39.4% 1.10E+15 $5,000,000 71,016,000 0.41 0.66 2.50 44.2
40 34 Quality Care, Inc. 34.4% 7.97E+14 $3,150,000 19,689,000 0.68 0.74 2.88 7.0
41 35 Ragen Precision Industries, Inc. 15.3% 8.85E+14 $2,000,000 22,653,000 0.61 0.75 2.25 26.0
42 36 REN Corp-USA 17.9% 2.85E+15 $53,625,000 151,074,000 0.02 0.88 2.92 19.8
43 37 REN Corp-USA 29.3% 2.85E+15 $12,003,000 163,749,000 0.02 0.88 2.72 36.1
44 38 Rentrak Corp. 32.5% 1.15E+15 $20,650,000 61,482,000 0.60 0.70 2.92 30.0
45 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
46 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
47 41 Sahlen & Assoc., Inc. 27.5% 3.02E+15 $6,057,000 42,955,000 0.54 0.81 2.72 26.1
48 42 Starrett Housing Corp. 44.8% 1.11E+16 $3,000,000 95,291,000 0.02 0.01 2.50 12.4
49 43 Sudbury Holdings, Inc. 46.5% 1.39E+16 $22,325,000 33,431,000 0.65 0.17 2.96 26.6
50 44 Superior Care, Inc. 41.9% 1.32E+15 $5,660,000 50,403,000 0.21 0.93 2.77 42.2
51 45 Sym-Tek Systems, Inc. 31.6% 4.03E+14 $995,000 20,550,000 0.34 0.92 2.58 13.4
52 46 Telepictures Corp. 11.6% 5.50E+15 $15,250,000 106,849,000 0.81 0.86 2.72 6.6
53 47 Velo-Bind, Inc. 19.5% 5.51E+14 $2,325,000 18,509,000 0.65 0.85 2.81 14.5
54 48 Western Digital Corp. 47.3% 4.24E+14 $7,825,000 50,417,000 0.00 0.32 2.64 22.7
55 49 50-Off Stores, Inc. 12.5% 6.10E+15 $5,670,000 43,024,000 0.80 0.87 2.38 23.7
56 50 ARC Capital 18.8% 3.76E+14 $2,275,000 18,846,000 0.03 0.74 1.63 35.0
57 51 Dense Pac Microsystems, Inc. 23.1% 3.24E+14 $4,500,000 108,862,000 0.08 0.70 1.17 42.4
58 52 Nobel Education Dynamics, Inc. 19.3% 1.95E+15 $12,000,000 60,913,000 0.34 0.76 1.74 32.1
59 53 Unimed Pharmaceuticals 15.8% 5.49E+13 $8,400,000 44,681,000 0.09 0.74 1.90 21.0
60 Mean 27.1% 5.65E+15 $9,223,226 $78,621,472 0.42 0.69 2.54 25.4
4 Regression #1

6 Regression Statistics
7 Multiple R 0.8058
8 R square 0.6493
9 Adjusted R square 0.5947
10 Standard error 0.0873
11 Observations 53


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)

T A B L E 7-5 (continued)

Abrams Regression of Management Planning Study Data


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


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
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)
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
d1 [ln(S/E) (Rf 0.5 variance) t]/[std dev
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
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


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
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:
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)

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


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
0.777 standard deviations below the mean. In cell B14, N( d2)
N( 0.594)) N(0.594) 0.724, which means there is a 72.4% probability
of being less than or equal to 0.594 standard deviations above the mean.
For perspective, it is useful to note that since the normal distribution is
symmetric, 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 B15, we calculate the value of the put option, which is $3.73 (B15),
or 42.0% (B16) of the stock price of $8.875 (B5). Thus, our calculation of
the restricted stock discount for the Chantal block using the Black“Scholes
model is 42.0% (B16).

Table 7-8: Black“Scholes Put Model Results. The stock symbols
in Table 7-8, column A, relate to restricted stock sale numbers 8, 11, 15,
17, 23, 31, 32, 38, and 49“53 in Table 7-5, column A. Cells B6 through B18
show the discounts calculated using the Black“Scholes put model for the

47. 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 7-8

Put Model Results


5 Company Put Calculation Actual Error Absolute Error

6 BLYH 32.3% 31.4% 0.9% 0.0% 0.9%
7 CHTL 42.0% 44.8% 2.8% 0.1% 2.8%
8 DAVX 47.5% 46.3% 1.2% 0.0% 1.2%
9 EDMK 11.9% 16.0% 4.1% 0.2% 4.1%
10 ILT 38.3% 41.1% 2.8% 0.1% 2.8%
11 PLFE 23.7% 15.9% 7.8% 0.6% 7.8%
12 PRDE 13.3% 24.5% 11.2% 1.2% 11.2%
13 RENT 41.5% 32.5% 9.0% 0.8% 9.0%
14 FOFF 27.2% 12.5% 14.7% 2.2% 14.7%
15 ARCCA 36.1% 18.8% 17.3% 3.0% 17.3%
16 DPAC 18.3% 23.1% 4.8% 0.2% 4.8%
17 NEDI 24.6% 19.3% 5.3% 0.3% 5.3%
18 UMED 12.9% 15.8% 2.9% 0.1% 2.9%
19 Mean 28.4% 26.3% 2.1% 0.67% 6.5%

22 Comparison with the Mean as the Discount

24 Company Mean Discount Actual Error Absolute Error

25 BLYH 27.1% 31.4% 4.3% 0.2% 4.3%
26 CHTL 27.1% 44.8% 17.7% 3.1% 17.7%
27 DAVX 27.1% 46.3% 19.2% 3.7% 19.2%
28 EDMK 27.1% 16.0% 11.1% 1.2% 11.1%
29 ILT 27.1% 41.1% 14.0% 2.0% 14.0%
30 PLFE 27.1% 15.9% 11.2% 1.3% 11.2%
31 PRDE 27.1% 24.5% 2.6% 0.1% 2.6%
32 RENT 27.1% 32.5% 5.4% 0.3% 5.4%
33 FOFF 27.1% 12.5% 14.6% 2.1% 14.6%
34 ARCCA 27.1% 18.8% 8.3% 0.7% 8.3%
35 DPAC 27.1% 23.1% 4.0% 0.2% 4.0%
36 NEDI 27.1% 19.3% 7.8% 0.6% 7.8%
37 UMED 27.1% 15.8% 11.3% 1.3% 11.3%
38 Mean 27.1% 26.3% 0.8% 1.28% 10.1%

13 stocks. The actual discounts are in column C, and the error in the put
model estimate is in column D.48 Columns E and F are the squared error
and the absolute error. Row 19 is the mean of each column. The bottom
half of the table is identical to the top half, except that we use the mean
discount of 27.1% as the estimated discount instead of the Black“Scholes
put model.
A comparison of the top and bottom of Table 7-8 reveals that the put
option model performs much better than the mean discount of 27.1% for
the 13 stocks. The put model™s mean absolute error of 6.5% (F19) and
mean squared error of 0.67% (E19) are much smaller than the mean ab-
solute error of 10.1% (F38) and mean squared error of 1.28% (E38) using

48. The error is equal to the estimated discount minus the actual discount, or column B minus
column C.

CHAPTER 7 Adjusting for Levels of Control and Marketability 245
the MPI data mean discount as the forecast. The mean errors in cells D19
and D38 are not indicative of relative predictive power, since low values
could be obtained even though the individual errors are high due to neg-
ative and positive errors canceling out.

Comparison of the Put Model and the Regression Model
In order to compare the put model discount results with the regression
model, we will analyze Table 7-9, which shows the calculation of dis-
counts, using regression #1 in Table 7-5, on the 13 stocks for which price
data was available.
The intercept of the regression is in cell B6, and the coef¬cients for
the independent variables are in cells B7 through B13. The variables for
each stock are in columns C through O, Rows 7 through 13. Multiplying
the variables for each stock by their respective coef¬cients and then add-
ing them together with the y-intercept results in the regression estimated
discounts in C14 through O14.
The errors in row 16 equal the actual discounts in row 15 minus the
estimated discounts in Row 14. We then calculate the error squared and
absolute error in Rows 17 and 18.
The mean squared error of 0.57% (C20) and the mean absolute error
of 6.33% (C21) are comparable but slightly better than the put model
results of 0.67% and 6.5% in Table 7-8, E19 and F19, respectively. Having
only been able to test the put model on 13 stocks and not the entire
database of 53 reduces our ability to distinguish which model is better.
At this point it is probably best to use an average of the results of both
models when determining a discount in a restricted stock valuation.

Empirical versus Theoretical Black“Scholes. It is important to un-
derstand that in using the BSOPM put for calculating restricted stock
discounts, we are using it as an empirical model, not as a theoretical
model. That is because buying a put on a publicly traded stock does not
˜˜buy marketability™™ for the restricted stock.49 Rather, it locks in a mini-
mum price for the restricted shares once they become marketable, while
allowing for theoretically unlimited price appreciation. Therefore, issuing
a hypothetical put on the freely tradable stock does not accomplish the
same task as providing marketability for the restricted stock, but it does
compensate for the downside risk on the restricted stock during its hold-
ing period.
BSOPM has some attributes that make it a successful predictor of
restricted stock discounts, i.e., it is a better forecaster than the mean dis-
count and did almost as well as the regression of the MPI data.
The reason for BSOPM™s success is that its mathematics is compatible
with the underlying variable”primarily volatility”that would tend to
drive restricted stock discounts. It is logical that the more volatile the
restricted stock, the larger the discount, and that volatility is the single
most important determinant of BSOPM results. Therefore, BSOPM is a
good candidate for empirically explaining restricted stock discounts, even

49. I thank R. K. Hiatt for this observation

PART 3 Adjusting for Control and Marketability
T A B L E 7-9

Calculation of Restricted Stock Discounts for 13 Stocks Using Regression from Table 7-5



6 Intercept 0.0673
7 Rev2 4.629E 18 8.62E 13 5.21E 13 1.14E 15 3.56E 13 1.02E 13 4.37E 16 4.34E 15 1.15E 15 6.10E 15 3.76E 14 3.24E 14 1.95E 15 5.49E 13
8 Shares 3.619E 09 4,452,000 $4,900,000 $999,000 $2,000,000 $975,000 $38,063,000 $21,500,000 $20,650,000 $5,670,000 $2,275,000 $4,500,000 $12,000,000 $8,400,000
9 Mkt cap 4.789E 10 98,053,000 149,286,000 18,942,000 12,275,000 10,046,000 246,787,000 74,028,000 61,482,000 43,024,000 18,846,000 108,862,000 60,913,000 44,681,000
10 Earn stab 0.1038 0.04 0.70 0.01 0.57 0.71 0.00 0.31 0.60 0.80 0.03 0.08 0.34 0.09
11 Rev stabil 0.1824 0.64 0.23 0.65 0.92 0.92 0.00 0.26 0.70 0.87 0.74 0.70 0.76 0.74
12 Avg yrs to 0.1722 2.125 2.125 2.750 2.868 2.844 2.861 2.833 2.950 2.375 1.633 1.167 1.738 1.898
13 Price 0.0037 58.6 51.0 24.6 10.5 22.0 17.0 18.0 30.0 23.7 35.0 42.4 32.1 21.0
14 Calculated discount 42.22% 42.37% 37.67% 23.65% 26.25% 26.57% 34.43% 30.97% 15.83% 20.27% 18.68% 15.20% 18.27%
15 Actual discount 31.40% 44.80% 46.30% 16.00% 41.10% 15.90% 24.50% 32.50% 12.50% 18.80% 23.10% 19.30% 15.80%
16 Error (actual calculated) 10.82% 2.43% 8.63% 7.65% 14.85% 10.67% 9.93% 1.53% 3.33% 1.47% 4.42% 4.10% 2.47%
17 Error squared 1.17% 0.06% 0.75% 0.59% 2.21% 1.14% 0.99% 0.02% 0.11% 0.02% 0.20% 0.17% 0.06%
18 Absolute error 10.82% 2.43% 8.63% 7.65% 14.85% 10.67% 9.93% 1.53% 3.33% 1.47% 4.42% 4.10% 2.47%
19 Mean error 0.80%
20 Mean squared error 0.57%
21 Mean absolute error 6.33%
though that is not the original intended use of the model, nor is this
scenario part of the assumptions of the model.

Comparison to the Quantitative Marketability Discount
Model (QMDM)
Mercer shows various examples of investment risk premium calculations
Mercer 1997, chapter 10). When he adds this premium to the required
return on a marketable minority basis, he gets the required holding period
return for a nonmarketable minority interest. Judging from his example
calculations of the risk premium for other types of illiquid interests, the
investment speci¬c risk premium for restricted stocks should be some-
where in the range of 1.5“5% or less.50 This is because restricted stocks
have short and well-de¬ned holding periods. Also, the payoff at the end
of the holding period is almost sure to be at the marketable minority level.
To test the applicability of QMDM to restricted stocks, we ¬rst esti-
mate a typical marketable minority level required return. The MPI data-
base average market capitalization is approximately $78 million. This puts
the MPI stocks in the mid-cap to small-cap category, given the dates of
the transactions in the database. A reasonable expected rate of return for
stocks of this size is 15% or so on a marketable minority basis.
We will assume that the stocks, given their size, were probably not
paying any signi¬cant dividends. Therefore, the expected growth rate
equals the expected rate of return at the marketable minority level of 15%.
Given the average years to liquidity of approximately 2.5 years in the
data set, we can calculate a typical restricted stock discount using QMDM.
Assuming a 1.5% investment risk premium, and therefore a required
holding period return of 16.5%, QMDM would predict the following re-
stricted stock discount:
Min Discount 1 (FV PVF) 1 3.2%
where FV future value of the investment and PVF the present value
factor. With a 5% investment risk premium, we have:
Max Discount 1 (FV PVF) 1 10.1%
The QMDM forecast of restricted stock discounts thus range from 3“10%,
with the lower end of the range appearing most appropriate, considering
the examples in Mercer™s Chapter 10.51 These calculated discounts are

50. Actually, the lower end of the range”1.5%”appears most appropriate.
51. The QMDM restricted stock discount is insensitive to the absolute level of the discount rate. It
is only sensitive to the premium above the discount rate. For example, changing the
minimum discount formula to


has little impact on the QMDM result. It is the 1.5% premium that is the difference between
the 20% growth and the 21.5% required return that constitutes the bulk of the QMDM
discount”and, of course, the holding period.

PART 3 Adjusting for Control and Marketability
nowhere near the average discount of 27.1% in the MPI database. This
sheds doubt on the applicability of QMDM for restricted stocks and the
applicability of the model in general. At least it shows that the model
does not work well for small holding periods.
I invited Chris Mercer to write a rebuttal to my analysis of the
QMDM results. His rebuttal is at the end of this chapter, just before the
conclusion, after which I provide my comments, as I disagree with some
of his methodology.

Abrams™ Economic Components Model
The remainder of this chapter will be spent on Abrams™ economic com-
ponents model (ECM). The origins of this model appear in Abrams
(1994a) (the ˜˜original article™™). While the basic structure of the model is
the same, this chapter contains major revisions of that article. One of the
revisions is that for greater clarity and ease of exposition, components #2
and #3 have switched places. In the original article, transactions costs was
component #2 and monopsony power to the buyer due to thin markets
was component #3, but in this chapter they are reversed.
We will be assuming that we are applying DLOM to a valuation
determined either directly or indirectly by comparison to publicly traded
¬rms. This could be a guideline company method or a discounted cash
¬‚ow method, with discount rates determined by data on publicly traded
¬rms. The ECM is not meant to be used as described on data coming
from sales of privately held businesses.

Component #1: The Delay to Sale
The ¬rst component of DLOM is the economic disadvantage of the con-
siderable time that it takes to sell a privately held business in excess of
the near instantaneous ability to sell the publicly held stocks from which
we calculate our discount rates.

Psychology. Investors don™t like illiquidity. Medical and other emer-
gencies arise in life, causing people to have to sell their assets, possibly
including their businesses. Even without the pressure of a ¬re sale, it
usually takes three to six months to sell a small business and one year or
more to sell a business worth $1 million or more.
The selling process may entail dressing up the business, i.e., tidying
up the accounting records, halting the standard operating procedures of
charging personal expenses to the business, and getting an appraisal. Ei-
ther during or after the dress-up stage, the seller needs to identify poten-
tial buyers or engage a business broker or investment banker to do so.
This is also dif¬cult, as the most likely buyers are often competitors. If
the match doesn™t work, the seller is worse off, having divulged con¬-
dential information to his competitors. The potential buyers need to go
through their due diligence process, which is time consuming and ex-
During this long process, the seller is exposed to the market. He or
she would like to sell immediately, and having to wait when one wants
to sell right away tries one™s patience. The business environment may be

CHAPTER 7 Adjusting for Levels of Control and Marketability 249
better or worse when the transaction is close to consummation. It is well
established in behavioral science”and it is the major principle on which
the sale of insurance is based”that the fear of loss is stronger than the
desire for gain (Tversky and Kahneman 1987). This creates pressure for
the seller to accept a lower price in order to get on with life.
Another important ¬nding in behavioral science that is relevant in
explaining DLOM and DLOC is ambiguity aversion (Einhorn and Ho-
garth 1986). The authors cite a paradox proposed by the psychologist
Daniel Ellsberg (Ellsberg 1961) (of Pentagon Papers fame), known as the
Ellsberg paradox.
Ellsberg asked subjects which of two gambles they prefer. In gamble
A the subject draws from an urn with 100 balls in it. They are red or
black only, but we don™t know how many of each. It could be 100 black
and 0 red, 0 black and 100 red, or anything in between. The subject calls
˜˜red™™ or ˜˜black™™ before the draw and, if he or she calls it right, wins $100;
otherwise, he or she gets nothing. In gamble B, the subject draws one ball
from an urn that has 50 red balls and 50 black balls. Again, if the subject
forecasts the correct draw, he or she wins $100 and otherwise wins noth-
Most people are indifferent between choosing red or black in both
gambles. When asked which gamble they prefer, the majority of people
had an interesting response (before we proceed, ask yourself which gam-
ble you would prefer and why). Most people prefer to draw from urn #2.
This is contrary to risk-neutral logic. The ¬nding of Ellsberg and Einhorn
and Hogarth is that people dislike ambiguity and will pay to avoid it.
Ambiguity is a second-order uncertainty. It is ˜˜uncertainty about un-
certainties,™™ and it exists pervasively in our lives. Gamble B has uncer-
tainty, but it does not have ambiguity. The return-generating process is
well understood. It is a clear 50“50 gamble. Gamble A, on the other hand,
is fuzzier. The return-generating process is not well understood. People
feel uncomfortable with that and will pay to avoid it.
It is my opinion that ambiguity aversion probably explains much of
shareholder level discounts. As mentioned earlier in the chapter, Jan-
kowske mentions wealth transfer opportunities and the protection of in-
vestment as economic bene¬ts of control. Many minority investors are
exposed to the harsh reality of having their wealth transferred away.
Many of those who do not experience that still have to worry about it
occurring in the future. The minority investor is always in a more am-
biguous position than a control shareholder.
In our regressions of the partnership pro¬les database that tracks the
results of trading in the secondary limited partnership markets (see Chap-
ter 9), we ¬nd that regular cash distributions are the primary determinant
of discounts from net asset value. Why would this be so? After all, there
have already been appraisals of the underlying properties, and those ap-
praisals certainly included a discounted cash ¬‚ow approach to valua-
tion.52 If the appraisal of the properties already considered cash ¬‚ow, then

52. In the regression we included a dummy variable to determine whether the discount from net
asset value depended on whether the properties were appraised by the general partner or
by independent appraiser. The dummy variable was statistically insigni¬cant, meaning that
the market trusts the appraisals of the general partners as much as the independent

PART 3 Adjusting for Control and Marketability
why would we consider cash ¬‚ow again in determining discounts? I
would speculate the following reasons:
1. If the general partner (GP) takes greater than arm™s-length fees
for managing the property, that would not be included in the
appraisal of the whole properties and would reduce the value of
the limited partner (LP) interest. It is a transfer of wealth from
the LP to the GP.
2. Even if the GP takes an arm™s-length management fee, he or she
still determines the magnitude and the timing of the
distributions, which may or may not be convenient for the
individual LPs.
3. LPs may fear potential actions of the GP, even if he or she never
takes those actions. The LP only knows that information about
the investment that the GP discloses and may fear what the GP
does not divulge”which, of course, he or she won™t know. The
LPs may hear rumors of good or bad news and not know what
to do with it or about it.
The bottom line is that investors don™t like ignorance, and they will
pay less for investments that are ambiguous than for ones that are not”
or that are, at least, less ambiguous”even if both have the same expected
Our paradigm for valuation is the two-parameter normal distribu-
tion, where everything depends only on expected return and expected
risk. Appraisers are used to thinking of risk only as either systematic risk,
measured by , or total risk in the form of , the historical standard
deviation of returns. The research on ambiguity avoidance adds another
dimension to our concept of risk, which makes our task more dif¬cult
but affords the possibility of being more realistic.
It is also noteworthy that the magnitude of special distributions, i.e.,
those coming from a sale or re¬nancing or property, was statistically in-
signi¬cant. Investors care only about what they feel they can count on,
the regular distributions.

Black“Scholes Options Pricing Model. One method of modeling
the economic disadvantage of the period of illiquidity is to use the Black“
Scholes options pricing model (BSOPM) to calculate the value of a put
on the stock for the period of illiquidity. A European put, the simplest
type, is the right to sell the stock at a speci¬c price on a speci¬c day. An
American put is the right to sell the stock on or before the speci¬c day.
We will be using the European put.
The origins of using this method go back to David Chaffe (Chaffe
1993), who ¬rst proposed using the BSOPM for calculating restricted
stock discounts for SEC Rule 144 restricted stock. The restricted stock
discounts are for minority interests of publicly held ¬rms. There is no
admixture of minority interest discount in this number, as the restricted
stock studies in Pratt™s Chapter 15 (Pratt, Reilly, and Schweihs 1996) are
minority interests both pre- and posttransaction.
Then Abrams (1994a) suggested that owning a privately held busi-
ness is similar to owning restricted stock in that it is very dif¬cult to sell

CHAPTER 7 Adjusting for Levels of Control and Marketability 251
a private ¬rm in less than the normal due diligence time discussed above.
The BSOPM is a reasonable model with which to calculate Component
#1 of DLOM, the delay to sale discount.
There is disagreement in the profession about using BSOPM for this
purpose. Chapter 14 of Mercer™s book (Mercer 1997) is entitled, ˜˜Why
Not the Black“Scholes Options Pricing Model Rather Than the QMDM?™™
Mercer™s key objections to the BSOPM are:53
1. It requires the standard deviation of returns as an input to the
model. This input is not observable in privately held companies.
2. It is too abstract and complex to meaningfully represent the
thinking of the hypothetical willing investor.
Argument 2 does not matter, as the success of the model is an em-
pirical question. Argument 1, however, turned out to be more true than
I would have imagined. It is true that we cannot see or measure return
volatility in privately held ¬rms. However, there are two ways that we
indirectly measured it. We combined the regression equations from re-
gressions #1 and #2 in Table 4-1 to develop an expression for return vol-
atility as a function of log size, and we performed a regression of the
same data to directly develop an expression for the same. We tried using
both indirect estimates of volatility as inputs to the BSOPM to forecast
the restricted stock discounts in the Management Planning, Inc. data, and
both approaches performed worse than using the average discount. Thus,
argument 1 was an assertion that turned out to be correct.
When volatility can be directly calculated, the BSOPM is superior to
using the mean and the QMDM. So, BSOPM is a competent model for
forecasting when we have ¬rm-speci¬c volatility data, which we will not
have for privately-held ¬rms.

Other Models of Component #1. The regression equation developed
from the Management Planning, Inc. data is superior to both the non-
¬rm-speci¬c BSOPM and the QMDM. Thus, it is, so far, the best model
to measure component #1, the delay to sale component, as long as the
expected delay to sale is one to ¬ve (or possibly as high as six) years.
The QMDM is pure present value analysis. It has no ability to quan-
tify volatility”other than the analyst guessing at the premium to add to
the discount rate. It also suffers from being highly subjective. None of the
components of the risk premium at the shareholder level can be empiri-
cally measured in any way.
Is the QMDM useless? No. It may be the best model in some sce-
narios. As mentioned before, one of the limitations of my restricted stock
discount regression is that because the restricted stocks had so little range
in time to marketability, the regression equation performs poorly when
the time to marketability is substantially outside that range”above ¬ve
to six years. Not all models work in all situations. The QMDM has its
place in the toolbox of the valuation professional. It is important to un-

53. Actually, Chapter 14 is co-authored by J. Michael Julius and Matthew R. Crow, employees at
Mercer Capital.

PART 3 Adjusting for Control and Marketability
derstand its limitations in addition to its strengths, which are ¬‚exibility
and simplicity.
The BSOPM is based on present value analysis, but contains far more
heavy-duty mathematics to quantify the probable effects of volatility on
investor™s potential gains or losses. While the general BSOPM did not
perform well when volatility was measured indirectly, we can see by
looking at the regression results that Black“Scholes has the essence of the
right idea. Two of the variables in the regression analysis are earnings
stability and revenue stability. They are the R2 from regressions of earn-
ings and revenues as dependent variables against time as the independent
variable. In other words, the more stabile the growth of revenues and
earnings throughout time, the higher the earnings and revenue stability.
These are measures of volatility of earnings and revenues, which are the
volatilities underlying the volatility of returns. Price stability is another
of the independent variables, and that is the standard deviation of stock
price divided by the mean of returns (which is the coef¬cient of variation
of price) and then multiplied by 100.
Thus, the regression results demonstrate that using volatility to mea-
sure restricted stock discounts is empirically sound. The failure of the
non-¬rm-speci¬c BSOPM to quantify restricted stock discounts is a mea-
surement problem, not a theoretical problem.54
An important observation regarding the MPI data is that MPI ex-
cluded startup and developmental ¬rms from its study. There were no
¬rms that had negative net income in the latest ¬scal year. That may
possibly explain the difference in results between the average 35% dis-
counts in most of the other studies cited in Pratt™s Chapter 15 (Pratt,
Reilly, and Schweihs 1996) and MPI™s results. When using my regression
of the MPI data to calculate component #1 for a ¬rm without positive
earnings, I would make a subjective adjustment to increase the discount.
As to magnitude, we have to make an assumption. If we assume that the
other studies did contain restricted stock sales of ¬rms with negative
earnings in the latest ¬scal year, then it would seem that those ¬rms
should have a higher discount than the average of that study. With the
average of all of them being around 33“35%, let™s say for the moment
that the ¬rms with losses may have averaged 38“40% discounts, all other
things being equal (see the paragraph below for the rationale). Then 38“
40% minus 27% in the MPI study would lead to an upward adjustment
to component #1 of 11% to 13%. That all rests on an assumption that this
is the only cause of the difference in the results of the two studies. Further
research is needed on this topic.
We can see the reason that ¬rms with losses would have averaged
higher discounts than those who did not in the x-coef¬cient for earnings
stability in Table 7-10, cell B9, which is 0.1381. This regression tells us
the market does not like volatility in earnings, which implies that the

54. There is a signi¬cant difference between forecasting volatility and forecasting returns. Returns
do not exhibit statistically signi¬cant trends over time, while volatility does (see Chapter 4).
Therefore, it is not surprising that using long-term averages to forecast volatility fail in the
BSOPM. The market is obviously more concerned about recent than historical volatility in
pricing restricted stock. That is not true about returns.

CHAPTER 7 Adjusting for Levels of Control and Marketability 253
market likes stability in earnings. Logically, the market would not like
earnings to be stable and negative, so investors obviously prefer stable,
positive earnings. Thus, we can infer from the regression in Table 7-10
that, all other things being equal, the discount for ¬rms with negative
earnings in the prior year must be higher than for ¬rms with positive
earnings. Ideally, we will eventually have restricted stock data on ¬rms
that have negative earnings, and we can control for that by including
earnings as a regression variable.
It is also worth noting that the regression analysis results are based
on the database of transactions from which we developed the regression,
while the BSOPM did not have that advantage. Thus, the regression had
an inherent advantage in this data set over all other models.

Abrams™ Regression of the Management Planning, Inc. Data. As
mentioned earlier in the chapter, there are two regression equations in
our analysis of the MPI data. The ¬rst one includes price stability as an
independent variable. This is ¬ne for doing restricted stock studies. How-
ever, it does not work for calculating Component #1 in a DLOM calcu-
lation for the valuation of a privately held ¬rm, whether a business or a
family limited partnership with real estate. In both cases there is no ob-
jective market stock price with which to calculate the price stability.
Therefore, in those types of assignments, we use the less accurate second
regression equation that excludes price stability.
Table 7-10 is an example of using regression #2 to calculate compo-
nent #1, the delay to sale of DLOM, for a privately held ¬rm. Note that
˜˜Value of Block”Post Discount™™ (Table 7-10, A7) is analogous to ˜˜Shares
Sold”$™™ (Table 7-5, A50), and ˜˜FMV“100% Marketable Minority Inter-
est™™ (Table 7-10, B8) is analogous to ˜˜Market Capitalization™™ (Table 7-5,
A51). The regression coef¬cients are in B5“B11. We insert the subject com-

T A B L E 7-10

Calculation of Component #1”Delay To Sale [1]


4 Coef¬cients Subject Co. Data Discount

5 Intercept 0.1292 NA 12.9%
Revenues2 [2]
6 5.39E 18 3.600E 13 0.0%
7 Value of block-post-discount [3] 4.39E 09 $4,331,435 1.9%
8 FMV-100% marketable minority interest 6.10E 10 $5,000,000 0.3%
9 Earnings stability 0.1381 0.4500 6.2%
10 Revenue stability 0.1800 0.3000 5.4%
11 Average years to sell 0.1368 1.0000 13.7%

12 Total Discount 13.4%
14 Value of block”pre-discount [4] $5,000,000

[1] Based on Abrams™ Regression #2 of Management Planning, Inc. data
Revenues2 $6,000,0002 (6 106)2 1013
[2] 3.6
[3] Equal to (value of block pre-discount) * (1 discount).
[4] Marketable minority interest FMV

PART 3 Adjusting for Control and Marketability
pany data in C6“C11, except for row 7, which we will discuss below. Our
subject company has $5 million in revenues (which, squared, equals 3.6
1013, per (C6), 100% marketable minority interest FMV of $5 million
(C8, analogous to market capitalization for the public companies in the
Management Planning, Inc. data), and earnings and revenue stability of
0.45 (C9) and 0.30 (C10), respectively.55 We estimate it will take one year
to sell the interest (C11).
Since we are valuing 100% of the capital stock of the ¬rm, the value
of the block of stock also has an FMV of $5 million (B14) before DLOM.56
The regression calls for the postdiscount FMV, which means we must
subtract the discount. The formula in cell C7 is: B14*(1 D12), i.e., the
postdiscount FMV equals the prediscount FMV (1 Discount). How-
ever, this is a simultaneous equation since the discount and the shares
sold in dollars each depend on the other. In order to be able to calculate
this, your spreadsheet should be set to allow recalculation with multiple
iterations. Otherwise you will get an error message with a circular ref-
erence.57 Column D is equal to column B column C, except for the y-
intercept in D5, which transfers directly from B5. Adding each of the
components in column D, we obtain a forecast discount of 13.4% (D12).

Limitations of the Regression. There may be combinations of subject
company data that can lead to strange results. This is especially true be-
1. The subject company data are near the end or outside of the
ranges of data in the regression of the MPI data.
2. There is very little variation in the range of the ˜˜average time to
sale™™ variable in our set. Most all of the restricted stock could be
sold between two and three years from the transaction date,
which is very little variation. Only 4 of the 53 sales were
expected to take less than two years (see below).
3. The R 2 is low.
4. The standard error of the y-estimate is fairly high”10%.
Regarding number 1, 47 of the 53 restricted stock sales in the MPI
database took place before the SEC circulated its Exposure Draft on June
27, 1995,58 to amend Rule 144(d) and (k) to shorten the waiting period

55. We do not explicitly show the detail of the calculations of earnings and revenue stability. Our
sample Restricted Stock Discount Study in Chapter 8, Table 8-1, shows these calculations.
56. Had we been valuing a 10% block of stock, B14 would have been $500,000.
57. If you create your own spreadsheet and make changes to the data, the simultaneous equation
is fragile, and it can easily happen that you may get error messages. When that happens,
you must put in a simple number in C7, e.g., $200,000, allow the spreadsheet to
˜˜recalibrate™™ and come back to equilibrium, then put in the correct formula. We do not have
this iterative problem with the other components of DLOM.
58. Revision of Holding Period Requirements in Rule 144; Section 16(a) Reporting of Equity Swaps
and Other Derivative Securities. File No. S7-17-95, SEC Release Nos. 33-7187; 34-35896; 17
CFR Parts 230 and 241; RIN 3235-AG53. The author expresses his gratitude to John Watson,
Jr., Esq., of Latham & Watkins in Washington, D.C., for providing him with a copy of the
exposure draft.

CHAPTER 7 Adjusting for Levels of Control and Marketability 255
for selling restricted stock to one year from two years and for nonaf¬l-
iated shareholders to sell shares without restriction after two years in-
stead of three.
Two sales took place in 1995 (Esmor Correctional Services, Inc. and
Chantal Pharmaceuticals Corp.) after the SEC Exposure Draft, and four
sales took place in 1996 (ARC Capital, Dense Pac Microsystems, Inc., No-
bel Education Dynamics, Inc., and Unimed Pharmaceuticals). That means
the market knew there was some probability that this would become law
and might shorten the waiting period to sell the restricted stock it was
issuing, and the later the sale, the more likely it was at the time that the
Exposure Draft would become law and provide relief to the buyer of the
restricted stock.
Thus, we should expect that those sales would carry lower discounts
than earlier sales”and that is correct. The discounts on the 1996 sales
were signi¬cantly lower than discounts on the earlier sales, all other
things being equal. The discounts ranged from 16“23% on the 1996 sales.
However, the two post-Exposure Draft 1995 sales had higher-than-
average discounts, which is somewhat counterintuitive. It is true that the
1996 sales would be more affected because the relief from restrictions for
the 1995 sales were more likely to have lapsed from the passage of time
than the 1996 sales, if it would take a long time for the Exposure Draft
to become law. Nevertheless, the two 1995 sales remain anomalies.
The average years needed to sell the stock ranged from a low of 1.2
years for Dense Pac Microsystems to 2.96 years for Sudbury Holdings,
Inc., with the vast majority being between 2 and 3 years. Extrapolating
this model to forecast a restricted stock discount for a sale with a restric-
tion of 10 years, for example, leads to ridiculous results, and even more
than 4 years is very questionable.
The coef¬cient for average years to sell is 0.1368 (B11), which means
that for each year more (less) than the forecast we made for this subject
company of 1 year, the discount increases (decreases) by 13.68%, holding
all else constant. Thus, if we were to forecast for a 10-year restriction, we
would get a discount of 136.8%”a nonsense result.
Thus, the appraiser must exercise good judgment and common sense
in using these results. Mechanically using these regression formulas to all
situations can be dangerous. It may be necessary to run other regressions
with the same data, i.e., using different independent variables or different
transformations of the data, to accommodate valuation assignments with
facts that vary considerably with those underlying these data. Another
possible solution is to assume, for example, that when a particular subject
company™s R 2 is beyond the maximum in the MPI database, that it is
equal to the maximum in the MPI database. It may be necessary to use
the other models, i.e., BSOPM with inferred rather than explicit standard
deviations or the QMDM, for more extreme situations where the regres-
sion equation is strained by extreme data. Hopefully we will soon have
much more data, as there will be increasingly more transactions subject
to the relaxed Rule 144 restrictions.

Component #2: Buyer Monopsony Power
The control stockholder of a privately held ¬rm has no guarantee at all
that he or she can sell his or her ¬rm. The market for privately held

PART 3 Adjusting for Control and Marketability
businesses is very thin. Most small and medium-size ¬rms are unlikely
to attract more than a small handful of buyers”and even then probably
not more than one or two every several months”while the seller of pub-
licly traded stock has millions of potential buyers. Just as a monopolist
is a single seller who can drive up price by withholding production, a
single buyer”a monopsonist”can drive price down by withholding pur-
The presence of 100 or even 10 interested buyers is likely to drive
the selling price of a business to its theoretical maximum, i.e., ˜˜the right
price.™™ The absence of enough buyers may confer monopsony power on
the few who are interested. Therefore, a small, unexciting business will
have an additional component of the discount for lack of marketability
for the additional bargaining power accruing to the buyers in thin mar-
It is easy to think that component #2 may already be included in
component #1, i.e., they both derive from the long time to sell an illiquid
asset. To demonstrate that they are indeed distinct components and that
we are not double counting, it is helpful to consider the hypothetical case
of a very exciting privately held ¬rm that has just discovered the cure for
cancer. Such a ¬rm would have no lack of interested buyers, yet it still is
very unlikely to be sold in less than one year. In that year other things
could happen. Congress could pass legislation regulating the medical
breakthrough, and the value could decrease signi¬cantly. Therefore, it
would still be necessary to have a signi¬cant discount for component #1,
while component #2 would be zero. It may not take longer to sell the
corner dry-cleaning store, but while the ¬rst ¬rm is virtually guaranteed
to be able to sell at the highest price after its required marketing time,
the dry-cleaning store will have the additional uncertainty of sale, and its
few buyers would have more negotiating power than the buyers of the
¬rm with the cure for cancer.
The results from Schwert, described earlier in the chapter, are rele-
vant here. He found that the presence of multiple bidders for control of
publicly held companies on average led to increased premiums of 12.2%
compared to takeovers without competitive bidding. Based on the re-
gression in Table 4 of his article, we assumed a typical deal con¬guration
that would apply to a privately held ¬rm.59 The premium without an
auction was 21.5%. Adding 12.2%, the premium with an auction was
33.7%. To calculate the discount for lack of competition, we go in the
other direction, i.e., 12.2% divided by one plus 33.7% 0.122/1.337
9.1%, or approximately 9%. This is a useful benchmark for D 2.
However, it is quite possible that D 2 for any subject interest should
be larger or smaller than 9%. It all depends on the facts and circumstances
of the situation. Using Schwert™s measure of the effect of multiple versus
single bidders as our estimate of D 3 may possibly have a downward bias
in that the markets for the underlying minority interests in the same ¬rms
is very deep. So it is only the market for control of publicly held ¬rms
that is thin. The market for privately held ¬rms is thin for whole ¬rms
and razor thin for minority interests.

59. We assume a successful purchase, a tender offer, and a cash deal.

CHAPTER 7 Adjusting for Levels of Control and Marketability 257
Component #3: Transactions Costs
Transactions costs in selling a privately held business are substantially
more than they are for selling stock in publicly traded ¬rms. Most stock
in publicly traded ¬rms can be sold with a broker™s fee of 1“2%”or less.

Table 7-11: Quantifying Transactions Costs for Buyer and
Seller. Table 7-11 shows estimates of transactions costs for both the
buyer and the seller for the following categories: legal, accounting, and
appraisal fees (the latter split into posttransaction, tax-based appraisal for
allocation of purchase price and/or valuation of in-process R&D and the
pretransaction ˜˜deal appraisal™™ to help buyer and/or seller establish the
right price), the opportunity cost of internal management spending its
time on the sale rather than on other company business, and investment
banking (or, for small sales, business broker) fees. The ¬rst ¬ve of the
categories appear in columns B through F, which we subtotal in column
G, and the investment banking fees appear in column H. The reason for
segregating between the investment banking fees and all the others is
that the others are constantly increasing as the deal size (FMV) decreases,
while investment banking fees reach a maximum of 10% and stop in-
creasing as the deal size decreases.
Rows 6“9 are transactions costs estimates for the buyer, while rows
13“16 are for the seller. Note that the buyer does not pay the investment
banking fees”only the seller pays. Rows 20“23 are total fees for both
Note that the subtotal transactions costs (column G) are inversely
related to the size of the transaction. For the buyer, they are as low as
0.23% (I6) for a $1 billion transaction and as high as 5.7% (I9) for a $1
million transaction. We summarize the total in Rows 27“30 and include
the base 10 logarithm of the sales price as a variable for regression.60 The
purpose of the regression is to allow the reader to calculate an estimated
transactions costs for any size transaction.
The buyer regression equation is:
Buyer Subtotal Transaction Cost
0.1531 (0.0173 log10 Price)
The regression coef¬cients are in cells B48 and B49. The adjusted R2
is 83% (B37), which is a good result. The standard error of the y-estimate
is 0.9% (B38), so the 95% con¬dence interval around the estimate is ap-
proximately two standard errors, or 1.8%”a very good result.
The seller regression equation is:
Seller Subtotal Transaction Cost
0.1414 (0.01599 log10 Price)
The regression coef¬cients are in cells B67 and B68. The adjusted R2 is
82% (B56), which is a good result. The standard error of the y-estimate is

60. Normally we use the natural logarithm for regression. Here we chose base 10 because the logs
are whole numbers and are easy to understand. Ultimately, it makes no difference which
one we use in the regression. The results are identical either way.

PART 3 Adjusting for Control and Marketability
T A B L E 7-11

Estimates of Transaction Costs [1]


4 Buyer
Tax Deal
5 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

6 $1 billion 0.10% 0.02% 0.02% 0.00% 0.09% 0.23% 0.00% 0.23%
7 $100 million 1.00% 0.10% 0.06% 0.00% 0.16% 1.32% 0.00% 1.32%
8 $10 million 1.50% 0.23% 0.20% 0.00% 0.25% 2.18% 0.00% 2.18%
9 $1 million 4.00% 0.30% 0.70% 0.00% 0.70% 5.70% 0.00% 5.70%

11 Seller
Tax Deal
12 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

13 $1 billion 0.10% 0.01% 0.00% 0.02% 0.05% 0.18% 0.75% 0.93%
14 $100 million 1.00% 0.05% 0.00% 0.05% 0.10% 1.20% 1.10% 2.30%
15 $10 million 1.50% 0.08% 0.00% 0.20% 0.15% 1.93% 2.75% 4.68%
16 $1 million 4.00% 0.10% 0.00% 0.75% 0.42% 5.27% 10.00% 15.27%

18 Total
Tax Deal
19 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

20 $1 billion 0.20% 0.03% 0.02% 0.02% 0.14% 0.41% 0.75% 1.16%
21 $100 million 2.00% 0.15% 0.06% 0.05% 0.26% 2.52% 1.10% 3.62%
22 $10 million 3.00% 0.30% 0.20% 0.20% 0.40% 4.10% 2.75% 6.85%
23 $1 million 8.00% 0.40% 0.70% 0.75% 1.12% 10.97% 10.00% 20.97%

25 Summary For Regression Analysis-Buyer Summary For Regression Analysis-Seller

26 Sales Price Log10 Price Subtotal Sales Price Log10 Price Subtotal

27 $1,000,000,000 9.0 0.23% $1,000,000,000 9.0 0.18%
28 $100,000,000 8.0 1.32% $100,000,000 8.0 1.20%
29 $10,000,000 7.0 2.18% $10,000,000 7.0 1.93%
30 $1,000,000 6.0 5.70% $1,000,000 6.0 5.27%
T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]


32 SUMMARY OUTPUT: Buyer Subtotal Fees as a Function of Log10 FMV
34 Regression Statistics

35 Multiple R 0.9417624
36 R square 0.88691642
37 Adjusted R square 0.83037464
38 Standard error 0.00975177
39 Observations 4


42 df SS MS F Signi¬cance F
43 Regression 1 0.001491696 0.0014917 15.68603437 0.058237596
44 Residual 2 0.000190194 9.5097E 05
45 Total 3 0.00168189

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

48 Intercept 0.1531 0.033069874 4.62959125 0.043626277 0.010811717 0.295388283
49 Log10 price 0.0172725 0.004361126 3.96055986 0.058237596 0.036036923 0.001491923

51 SUMMARY OUTPUT: Seller Subtotal Fees as a Function of Log10 FMV
53 Regression Statistics

54 Multiple R 0.93697224
55 R square 0.87791699
56 Adjusted R square 0.81687548
57 Standard error 0.00943065
58 Observations 4


61 df SS MS F Signi¬cance F

62 Regression 1 0.00127912 0.00127912 14.38229564 0.063027755
63 Residual 2 0.000177874 8.8937E 05
64 Total 3 0.001456994

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

67 Intercept 0.14139 0.031980886 4.42107833 0.04754262 0.00378726 0.27899274
68 Log10 price 0.0159945 0.004217514 3.79239972 0.063027755 0.034141012 0.002152012

also 0.9% (B57), which gives us the same con¬dence intervals around the
y-estimate of 1.8%.
Rows 73 and 74 show a sample calculation of transactions costs for
the buyer and seller, respectively. We estimate FMV before discounts for
our subject company of $5 million (B73, B74). The base 10 logarithm of 5
million is 6.69897 (C73, C74).61 In D73 and D74, we insert the x-coef¬cient
from the regression, which is 0.0172725 (from B49) for the buyer and
0.0159945 (from B68) for the seller. We multiply column C column

61. In other words, 106.69897 5 million.

PART 3 Adjusting for Control and Marketability
T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]


70 Sample Forecast of Transactions Costs For $5 Million Subject Company:

72 FMV log10 FMV X-Coeff. log FMV Coef Regr. Constant Forecast Subtotal Inv Bank [5] Forecast Total

73 Buyer $5,000,000 6.698970004 0.0172725 0.115707959 0.1531 3.7% 0.0% 3.7%
74 Seller $5,000,000 6.698970004 0.0159945 0.107146676 0.14139 3.4% 5.0% 8.4%

[1] Based on interviews with investment banker Gordon Gregory, attorney David Boatwright, Esq; and Douglas Obenshain, CPA. Costs include buy and sell side. These are estimates of average costs. Actual costs vary with the complexity
of the transaction.
[2] Legal fees will vary with the complexity of the transaction. An extremely complex $1 billion sale could have legal fees of as much as $5 million each for the buyer and the seller, though this is rare. Complexity increases with: stock deals
(or asset deals with a very large number of assets), seller carries paper , contingent payments, escrow, tax-free (which is treated as a pooling-of-interests), etc.
[3] We are assuming the seller pays for the deal appraisal. Individual sales may vary. Sometimes both sides hire a single appraiser and split the fees, and sometimes each side has its own appraiser.
[4] Internal management costs are the most speculative of all. We estimate 6,000 hours (3 people fulltime for 1 year) at an average $150/hr. internal cost for the $1 billion sale, 2,000 hours @ $80 for the $100 million sale, 500 hours at $50
for the $10 million sale, and 200 hours @$35 for the $1 million sale for the buyer, and 60% of that for the seller. Actual results may vary considerably from these estimates.
[5] Ideally calculated by another regression, but this is sight-estimated. Can often use the Lehman Bros. Formula”5% for 1st $1 million, 4%, for 2nd, etc., leveling off at 1% for each $1 million.
D column E. F73 and F74 are repetitions of the regression constants
from B48 and B67, respectively. We then add column E to column F to
obtain the forecast subtotal transactions costs in G73 and G74. Finally, we
add in investment banking fees of 5%62 for the seller (the buyer doesn™t
pay for the investment banker or business broker) to arrive at totals of
3.7% (I73) and 8.4% (I74) for the buyer and seller, respectively.

Component #3 Is Different than #1 and #2. Component #3, trans-
actions costs, is different than the ¬rst two components of DLOM. For
component #3, we need to calculate explicitly the present value of the
occurrence of transactions costs every time the company sells. The reason
is that, unlike the ¬rst two components, transactions costs are actually
out-of-pocket costs that leave the system.63 They are paid to attorneys,
accountants, appraisers, and investment bankers or business brokers. Ad-
ditionally, internal management of both the buyer and the seller spend
signi¬cant time on the sale to make it happen, and they often have to
spend time on failed acquisitions before being successful.
We also need to distinguish between the buyer™s transactions costs
and the seller™s costs. The reason for this is that the buyer™s transactions
costs are always relevant, whereas the seller™s transactions costs for the
immediate transaction reduce the net proceeds to the seller but do not
reduce FMV. However, before the buyer is willing to buy, he or she should
be saying, ˜˜It™s true, I don™t care about the seller™s costs. That™s his or her
problem. However, 10 years or so down the road when it™s my turn to
be the seller, I do care about that. To the extent that seller™s costs exceed


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