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

234

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

Nasdaq.

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

discount.

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

profession.

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

4

measured as years 1, 2, 3, etc.

Revenue stability: the R2 of the regression of revenue as a function of time, with time

5

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

effective.

PART 3 Adjusting for Control and Marketability

236

T A B L E 7-5

Abrams Regression of Management Planning Study Data

A B C D E F G H I J

4 Y X1 X2 X3 X4 X5 X6 X7

Rev2

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

237

238

T A B L E 7-5 (continued)

Abrams Regression of Management Planning Study Data

A B C D E F G H I J

4 Y X1 X2 X3 X4 X5 X6 X7

Rev2

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

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.