ńņš. 10 |

CHAPTER 7 Adjusting for Levels of Control and Marketability 233

were several other inconsistencies in the results of the two

regressions.

3. The logā“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% conļ¬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% conļ¬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. Letā™s put some speciļ¬c

numbers into their equation to see what the conļ¬dence intervals

look like. Letā™s assume we are forecasting the value of the

common stock as a percentage of sales for a ļ¬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).

Letā™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

e0.345

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% conļ¬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 conļ¬dence

intervals in 3, the results of this study are insufļ¬cient to reject DLOM for

control interests of privately held ļ¬rms.

Kasperā™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, Prattā™s Stats, BIZCOMPS, etc.) for actual BASs. Anyone

interested in using Kasperā™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 ļ¬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-

ļ¬cients will change over time. Kasper also presents a different model, the

discounted time to market model (Kasper 1997, pp. 103;ā“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 ļ¬rst is based on my own mul-

tiple regression analysis of data collected by Management Planning, Inc.

(MPI),41 an independent valuation ļ¬rm in Princeton, New Jersey. The sec-

ond method involves using a Blackā“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 ļ¬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 ļ¬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 justiļ¬ably accords a sep-

arate chapter and extensive commentary in his book, contains data on

trades from 1980ā“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 ļ¬rst page contains data on 53 sales

of restricted stock between 1980ā“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 signiļ¬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 Soldā”$: 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 nonafļ¬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 priceā”which is the coefļ¬cient of variation of priceā”

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

33 27 Noble Romanā™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 Ryanā™s Family Steak Houses, Inc. 8.7% 1.02E+15 $5,250,000 159,390,000 0.90 0.87 2.13 13.6

46 40 Ryanā™s Family Steak Houses, Inc. 5.2% 1.02E+15 $7,250,000 110,160,000 0.90 0.87 2.58 14.4

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 Signiļ¬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 Coefļ¬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 Signiļ¬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 Coefļ¬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 signiļ¬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% conļ¬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 coefļ¬cients for the independent var-

iables. The t-statistics are in D20 through D27. Only the y-intercept itself

is not signiļ¬cant at the 95% conļ¬dence level. The market capitalization

and earnings stability variables are signiļ¬cant at the 98% level,45 and all

the other variables are signiļ¬cant at the 99 % conļ¬dence level.

Note that several of the variables are similar to Grabowski and Kingā™s

results (Grabowski and King 1999), discussed in Chapter 5. They found

that the coefļ¬cient of variations (in log form) of operating margin and

return on equity are statistically signiļ¬cant in explaining stock market

returns. Here we ļ¬nd that the stability of revenues and earnings (as well

as the coefļ¬cient of variation of stock market prices) explain restricted

stock discounts. Thus, these variables are signiļ¬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 ļ¬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 ā˜ā˜buyā™ā™ 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 Blackā“

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 sufļ¬cient daily price history on 13 of the stocks in Table

45. The statistical signiļ¬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 Chaffeā™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.ā”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 deviationā”CHTL 0.16900 0.20175

35 Annualized 0.84901 1.03298

36 Average of standard deviations 0.94099

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

PART 3 Adjusting for Control and Marketability

242

calculation instead of daily intervals is that the bidā“ask spread on the

stock may create apparent volatility that is not really present. This is

because the quoted closing prices are from the last trade. In Nasdaq trad-

ing, one sells to a dealer at the bid price and buys at the ask price. If on

successive days the last price of the day is switching randomly from a

bid to an ask price and back, this can cause us to measure a considerable

amount of apparent volatility that is not really there. By using 10-day

intervals, we minimize this measurement error caused by the spread.

We start with the 1/31/95 closing price in column C and the 2/7/

95 closing price in column D. For example, the 10-trading-day return from

1/31/95 (A7) to 2/14/95 (A9) is calculated as follows: return Ln(B9/

B7) Ln(2.5660/2.1650) 0.169928 (C9).

Using this methodology, we get two measures of standard deviation:

0.16900 (C34) and 0.20175 (D34). To convert to the annualized standard

deviation, we must multiply each interval standard deviation by the

square root of the number of intervals that would occur in a year. The

equation is as follows:

SQRT

annualized interval returns

# of interval returns in sample period

365 days per year

days in sample period

For example, the sample period in column C is the time period from

the close of trading on January 31, 1995, to the close of trading on August

7, 1995, or 188 days, and there are 13 calculated returns. Therefore the

annualized standard deviation of returns is:

0.1690 SQRT(13 365/188)

annualized

0.1690 SQRT(25.2394) 0.84901 (cell C35)

The 13 trading periods that span 188 days would become 25.2394 trading

periods in one year (25.2394 13 365/188). The square root of the

25.2394 trading periods is 5.0239. We multiply the sample standard de-

viation of 0.1690 by 5.0239 0.84901 to annualize the standard deviation.

Similarly, the annualized standard deviation of returns in column D is

1.03298 (D35), and the average of the two is 0.94099 (D36).

Calculation of the Discount. Table 7-7 is the Blackā“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 ļ¬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

Blackā“Scholes Put Optionā”CHTL

A B

5 S Stk price on valuation date $8.875

6 E Exercise price $8.875

7 t time to expiration in yrs (Table 7-5, I17) 2.125

8 r risk-free rate [1] 5.90%

9 stdev standard deviation (Table 7-6, D36) 0.941

10 var variance 0.885

11 d1 1st Black-Scholes parameter [2] 0.777

12 d2 2nd Black-Scholes parameter 3] (0.594)

13 N( d1) cum normal density function 0.219

14 N( d2) cum normal density function 0.724

[E*N( d2)*e rt ] S*N( d1)

15 P $3.73

16 P/S 42.0%

Note: Values are for European options. The put option formula can be found in Options Futures and Other Derivatives, 3rd Ed. by

John C. Hull, Prentice Hall, 1997, pp. 241 and 242.

[1] 2 Year Treasury rate on transaction date, 8/8/95 (Source: Federal Reserve)

.5 * var) * t]/[stdev *t0.5], where variance and standard deviation are expressed in annual terms.

[2] d1 [ln (S/E) (r

[std dev * t0.5]

[3] d2 d1

for CHTL of 0.941, transferred from Table 7-6, cell D36, while B10 is var-

iance, merely the square of B9.

Cells B11 and B12 are the calculation of the two Blackā“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 Blackā“Scholes

model is 42.0% (B16).

Table 7-8: Blackā“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 49ā“53 in Table 7-5, column A. Cells B6 through B18

show the discounts calculated using the Blackā“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

244

T A B L E 7-8

Put Model Results

A B C D E F

4

Black-Scholes

Error2

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

Error2

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 Blackā“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 modelā™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 coefļ¬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 coefļ¬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 Blackā“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

ā˜ā˜buy marketabilityā™ā™ 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 BSOPMā™s success is that its mathematics is compatible

with the underlying variableā”primarily volatilityā”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

246

T A B L E 7-9

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

A B C D E F G H I J K L M N O

5 Coefļ¬cients BLYH CHTL DAVX EDMK ITL PLFE PRDE RENT FOFF ARCCA DPAC NEDI UMED

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

sold-$

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

sell

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

stability

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%

247

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 speciļ¬c risk premium for restricted stocks should be some-

where in the range of 1.5ā“5% or less.50 This is because restricted stocks

have short and well-deļ¬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 ļ¬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 signiļ¬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:

1

1.152.5

Min Discount 1 (FV PVF) 1 3.2%

1.1652.5

where FV future value of the investment and PVF the present value

factor. With a 5% investment risk premium, we have:

1

1.152.5

Max Discount 1 (FV PVF) 1 10.1%

1.202.5

The QMDM forecast of restricted stock discounts thus range from 3ā“10%,

with the lower end of the range appearing most appropriate, considering

the examples in Mercerā™s Chapter 10.51 These calculated discounts are

50. Actually, the lower end of the rangeā”1.5%ā”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

1

1.202.5)

(1

1.2152.5

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

discountā”and, of course, the holding period.

PART 3 Adjusting for Control and Marketability

248

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.

Abramsā™ Economic Components Model

The remainder of this chapter will be spent on Abramsā™ economic com-

ponents model (ECM). The origins of this model appear in Abrams

(1994a) (the ā˜ā˜original articleā™ā™). 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

ļ¬rms. This could be a guideline company method or a discounted cash

ļ¬‚ow method, with discount rates determined by data on publicly traded

ļ¬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 ļ¬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 donā™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 ļ¬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 difļ¬cult, as the most likely buyers are often competitors. If

the match doesnā™t work, the seller is worse off, having divulged conļ¬-

dential information to his competitors. The potential buyers need to go

through their due diligence process, which is time consuming and ex-

pensive.

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 oneā™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 scienceā”and it is the major principle on which

the sale of insurance is basedā”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 ļ¬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 donā™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

ā˜ā˜redā™ā™ or ā˜ā˜blackā™ā™ 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-

ing.

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 ļ¬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 ā˜ā˜uncertainty about un-

certainties,ā™ā™ 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 50ā“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 beneļ¬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 proļ¬les database that tracks the

results of trading in the secondary limited partnership markets (see Chap-

ter 9), we ļ¬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 ļ¬‚ow approach to valua-

tion.52 If the appraisal of the properties already considered cash ļ¬‚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 insigniļ¬cant, meaning that

the market trusts the appraisals of the general partners as much as the independent

appraisers.

PART 3 Adjusting for Control and Marketability

250

why would we consider cash ļ¬‚ow again in determining discounts? I

would speculate the following reasons:

1. If the general partner (GP) takes greater than armā™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 armā™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 divulgeā”which, of course, he or she wonā™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 donā™t like ignorance, and they will

pay less for investments that are ambiguous than for ones that are notā”

or that are, at least, less ambiguousā”even if both have the same expected

value.

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 difļ¬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 reļ¬nancing or property, was statistically in-

signiļ¬cant. Investors care only about what they feel they can count on,

the regular distributions.

Blackā“Scholes Options Pricing Model. One method of modeling

the economic disadvantage of the period of illiquidity is to use the Blackā“

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 speciļ¬c price on a speciļ¬c day. An

American put is the right to sell the stock on or before the speciļ¬c day.

We will be using the European put.

The origins of using this method go back to David Chaffe (Chaffe

1993), who ļ¬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 ļ¬rms. There is no

admixture of minority interest discount in this number, as the restricted

stock studies in Prattā™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 difļ¬cult to sell

CHAPTER 7 Adjusting for Levels of Control and Marketability 251

a private ļ¬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 Mercerā™s book (Mercer 1997) is entitled, ā˜ā˜Why

Not the Blackā“Scholes Options Pricing Model Rather Than the QMDM?ā™ā™

Mercerā™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 ļ¬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 ļ¬rm-speciļ¬c volatility data, which we will not

have for privately-held ļ¬rms.

Other Models of Component #1. The regression equation developed

from the Management Planning, Inc. data is superior to both the non-

ļ¬rm-speciļ¬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 ļ¬ve (or possibly as high as six) years.

The QMDM is pure present value analysis. It has no ability to quan-

tify volatilityā”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 rangeā”above ļ¬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

252

derstand its limitations in addition to its strengths, which are ļ¬‚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

investorā™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 Blackā“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 coefļ¬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-ļ¬rm-speciļ¬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 ļ¬rms from its study. There were no

ļ¬rms that had negative net income in the latest ļ¬scal year. That may

possibly explain the difference in results between the average 35% dis-

counts in most of the other studies cited in Prattā™s Chapter 15 (Pratt,

Reilly, and Schweihs 1996) and MPIā™s results. When using my regression

of the MPI data to calculate component #1 for a ļ¬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 ļ¬rms with negative

earnings in the latest ļ¬scal year, then it would seem that those ļ¬rms

should have a higher discount than the average of that study. With the

average of all of them being around 33ā“35%, letā™s say for the moment

that the ļ¬rms with losses may have averaged 38ā“40% discounts, all other

things being equal (see the paragraph below for the rationale). Then 38ā“

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 ļ¬rms with losses would have averaged

higher discounts than those who did not in the x-coefļ¬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 signiļ¬cant difference between forecasting volatility and forecasting returns. Returns

do not exhibit statistically signiļ¬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 ļ¬rms with negative

earnings in the prior year must be higher than for ļ¬rms with positive

earnings. Ideally, we will eventually have restricted stock data on ļ¬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.

Abramsā™ 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 ļ¬rst one includes price stability as an

independent variable. This is ļ¬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 ļ¬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 ļ¬rm. Note that

ā˜ā˜Value of Blockā”Post Discountā™ā™ (Table 7-10, A7) is analogous to ā˜ā˜Shares

Soldā”$ā™ā™ (Table 7-5, A50), and ā˜ā˜FMVā“100% Marketable Minority Inter-

estā™ā™ (Table 7-10, B8) is analogous to ā˜ā˜Market Capitalizationā™ā™ (Table 7-5,

A51). The regression coefļ¬cients are in B5ā“B11. We insert the subject com-

T A B L E 7-10

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

A B C D

4 Coefļ¬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 blockā”pre-discount [4] $5,000,000

[1] Based on Abramsā™ 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

254

pany data in C6ā“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 ļ¬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-

cause:

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 ā˜ā˜average time to

saleā™ā™ 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 highā”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

ā˜ā˜recalibrateā™ā™ 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 nonafļ¬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 salesā”and that is correct. The discounts on the 1996 sales

were signiļ¬cantly lower than discounts on the earlier sales, all other

things being equal. The discounts ranged from 16ā“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 coefļ¬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%ā”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

companyā™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 ļ¬rm has no guarantee at all

that he or she can sell his or her ļ¬rm. The market for privately held

PART 3 Adjusting for Control and Marketability

256

businesses is very thin. Most small and medium-size ļ¬rms are unlikely

to attract more than a small handful of buyersā”and even then probably

not more than one or two every several monthsā”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 buyerā”a monopsonistā”can drive price down by withholding pur-

chase.

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., ā˜ā˜the right

price.ā™ā™ 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-

kets.

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 ļ¬rm that has just discovered the cure for

cancer. Such a ļ¬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 signiļ¬cantly. Therefore, it

would still be necessary to have a signiļ¬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 ļ¬rst ļ¬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

ļ¬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 conļ¬guration

that would apply to a privately held ļ¬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 Schwertā™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 ļ¬rms

is very deep. So it is only the market for control of publicly held ļ¬rms

that is thin. The market for privately held ļ¬rms is thin for whole ļ¬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 ļ¬rms. Most stock

in publicly traded ļ¬rms can be sold with a brokerā™s fee of 1ā“2%ā”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 ā˜ā˜deal appraisalā™ā™ 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 ļ¬rst ļ¬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 6ā“9 are transactions costs estimates for the buyer, while rows

13ā“16 are for the seller. Note that the buyer does not pay the investment

banking feesā”only the seller pays. Rows 20ā“23 are total fees for both

sides.

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 27ā“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)

Price

The regression coefļ¬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% conļ¬dence interval around the estimate is ap-

proximately two standard errors, or 1.8%ā”a very good result.

The seller regression equation is:

Seller Subtotal Transaction Cost

0.1414 (0.01599 log10 Price)

Price

The regression coefļ¬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

258

T A B L E 7-11

Estimates of Transaction Costs [1]

A B C D E F G H I

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%

259

T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]

A B C D E F G H

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

41 ANOVA

42 df SS MS F Signiļ¬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 Coefļ¬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

60 ANOVA

61 df SS MS F Signiļ¬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 Coefļ¬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 conļ¬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-coefļ¬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

260

T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]

A B C D E F G H I J

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%

Notes:

[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. Formulaā”5% for 1st $1 million, 4%, for 2nd, etc., leveling off at 1% for each $1 million.

261

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 doesnā™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 ļ¬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 ļ¬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

signiļ¬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 buyerā™s transactions costs

and the sellerā™s costs. The reason for this is that the buyerā™s transactions

costs are always relevant, whereas the sellerā™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, ā˜ā˜Itā™s true, I donā™t care about the sellerā™s costs. Thatā™s his or her

problem. However, 10 years or so down the road when itā™s my turn to

be the seller, I do care about that. To the extent that sellerā™s costs exceed

ńņš. 10 |