. 70
( 100 .)


The standard valuation industry calculation of the minority interest
discount begins with measuring control premiums in acquisitions of pub-
licly held ¬rms. Such acquisitions generally take place at substantial pre-
miums. There is a value to control, and buyers pay for it.
On the contrary, there is negative value to a lack of control, and
buyers will discount value because of it. If we assume a 40% premium,
that means a company trading at $100 per share before being acquired
will be acquired at $140 per share, or a $40 per share or 40 per cent
premium. The other perspective is to say that there is a $40 discount for
minority interest from the control price of $140, i.e., the discount for lack
of control (DLOC). DLOC is then $40/140, or 28.6%. A more general for-
mula to calculate the minority interest discount is DLOC P/(1 P),
where P is the control premium in percentage.
The average control premium paid in 1998 was 40.7%9 (E17), which
implies a discount for lack of control of 28.9% (E18).
A 2.80% member interest has more in¬‚uence over policy than a typ-
ical minority interest in the stock market. Because of the 2.80% member
interest™s greater control, we reduce the discount for lack of control by
10%, leaving 90% (E19) of the minority interest discount. Multiplying
28.9% 90% 26.0% (E20), the discount for lack of control, which we
transfer to A10.

Commentary to Table 9-5A: Delay-to-Sale
Table 9-5A displays our calculation of the ¬rst of four components of
DLOM, the delay-to-sale. The chapter discusses how stock in privately
held ¬rms is illiquid. Most ¬rms of substance require a year or more to
sell. We begin the calculation by making a comparison of owning a pri-
vate ¬rm to holding restricted securities of a publicly traded ¬rm.
There have been many studies that consistently ¬nd that the sellers
of restricted securities, who can choose to wait for two years10 and sell

9. Houlihan Lokey Howard & Zukin, Mergerstat Review”1999, p. 23. There is new research in
Chapter 7 of Abrams™ book Quantitative Business Valuation: A Mathematical Approach for
Today™s Professionals which suggests that control premiums for private ¬rms probably should
be on the order of 21“28% above the marketable minority level. This would imply a lower
discount for lack of control. However, in private ¬rms the possibility of wealth transfer from
minority interests to control interests could very well increase DLOC. In Chapter 7, Abrams
also cites international voting rights premia (VRP) as high as 82% and an American outlier
VRP 42% that might indicate the value of control to be higher than 28%. Taking these data
into consideration, we use the Mergerstat acquisition premium to arrive at our DLOC.
10. The SEC changed Rule 144 on April 29, 1997 to require only a one year instead of a two year
waiting period to sell restricted securities for nonaf¬liate owners. The studies we refer to
were conducted prior to April 29, 1997, and therefore measure the discount taken at the
time of sale instead of waiting two years.

CHAPTER 9 Sample Appraisal Report 329
T A B L E 9-5A

Calculation of Component #1: Delay to Sale [1]


5 Coef¬cients Subject Co. Data Discount

6 Intercept 0.1292 NA 12.9%
Revenues2 (Table 9-3, B7)2
7 5.39E 18 6.76E 09 0.0%
8 Value of block-post-discount [2] 4.39E 09 $ 30,351 0.0%
9 FMV-100% interest in property (Table 9-2, C22) [3] 6.10E 10 $1,389,185 0.1%
10 Earnings stability (Table 9-5B, B40) 0.1381 0.1124 1.6%
11 Revenue stability (Table 9-5B, B21) 0.1800 0.1749 3.1%
12 Average years to sell [4] 0.1368 1.0000 13.7%

13 Total discount (transfer to Table 9-5C, B9) 22.0%
15 Block size in percent 2.80%

[1] This table is identical to Table 7-5, Regression #2 from Abrams™ book, with only subject™s data changed.
[2] Equal to fractional interest of FMV * (1-discount for delay to sale).
[3] In the restricted stock study, this was a marketable minority interest value. Due to the limitations of the data available, we must use the FMV of the whole property, which is a control
[4] We normally assume it takes one year to sell such illiquid, fractional interests. A 3-month right of ¬rst refusal would tend to make this interest somewhat more dif¬cult than most to
sell. However, we take a conservative approach and assume it has no further impact. Thus, we remain with a one-year delay to sale.

all or part of their securities according to Rule 144 at the prevailing mar-
ket price, sell privately at an average discount of 35% (Pratt et al. 1996,
chap. 15). However, if a business takes one year on average to sell, what
is the discount? Furthermore, should every business be discounted
equally for an equal delay-to-sale, or do other business characteristics
in¬‚uence the delay-to-sale discount?
To answer these questions, Jay Abrams developed an original equa-
tion for the delay-to-sale discount. The equation was derived by perform-
ing regression analysis on the data from the Management Planning Study.
The Management Planning Study, presented as an entire chapter in Mer-
cer (1997), contains data on 49 restricted stock trades from 1980-1995. An
additional four restricted stock sales in 1996, obtained from Management
Planning, were added to the analysis.11 Abrams tested 37 independent
variables included in or derived from the Management Planning study.
Only the following 7 independent variables were statistically signi¬cant
at the 95% level.

# Independent Variable

1 Revenues squared.
2 Shares sold $: This is the post-discount dollar value of the transaction.
3 Market capitalization price per share times shares outstanding summed for all classes of
Earnings stability: the unadjusted R2 of the regression of net income as a function of time,
with time measured as years 1, 2, 3, . . . This is calculated in Quantitative Business
Valuation: A Mathematical Approach for Today™s Professionals in Table 7-5, regression
Revenue stability: the unadjusted R2 of the regression of revenue as a function of time,
with time measured as years 1, 2, 3, . . . This is calculated in Quantitative Business
Valuation: A Mathematical Approach for Today™s Professionals in Table 7-5, regression

11. In addition, Management Planning provided a few small corrections to the original data.

PART 3 Adjusting for Control and Marketability
# Independent Variable

6 Average years to sell: This is the weighted average years to sell by a nonaf¬liate, based on
SEC Rule 144.
7 Price stability: this ratio is calculated by dividing the standard deviation of the stock price
by the mean of the stock price. The end-of-month stock prices for the 12 months prior
to the valuation date are used.

The regression has an adjusted R 2 of 59%. This means that 59% of
the variation in restricted stock discounts is explained by the regression
model. The subject of this report does not have the data necessary to
calculate the Price stability variable. Therefore, we need to use a modi¬ed
version of the regression which excludes price stability. We also rename
variables #2 and #3 ˜˜value of block”post-discount™™ and ˜˜FMV-100% in-
terest in the LLC™™ to better suit the context of this application. The ad-
justed R 2 of this alternate regression is 43%. The coef¬cients of the re-
gression equation appear in column B.
In order to employ Abrams™ equation, we must determine the para-
meters for the LLC. Column C contains the LLC™s parameters. Cell C7
contains the square of the LLC™s 1999 revenue, $6.76 billion, shown as
6.76E 09.
The value of block”post-discount variable actually depends on the
¬nal delay-to-sale discount, the dependent variable. Therefore, we must
derive the LLC™s input for this independent variable through an iterative
process. With the aid of a spreadsheet program, the task is simple. We
input the FMV of equity, $1,389,185, which comes from Table 9-2, C22,
times the percentage interest times one minus the delay-to-sale discount,
or $1,389,185 2.80% (1 D13) $30,351 (C8) for the LLC™s value of
block”post-discount and activate the iterative capability of the spread-
sheet program.
For the FMV-100% interest in the LLC (C9), we simply input the FMV
of the LLC™s equity, $1,389,185 from Table 9-2, C22.
To determine the LLC™s earnings and revenue stability, we perform
a regression analysis of the LLC™s earnings as a function of time and its
revenue as a function of time. The results of the regressions are in Table
9-5B. The R 2 of the earnings regression (Table 9-5B, B40) is the earnings
stability of 0.1124 in C10. The R 2 of the revenue regression (Table 9-5B,
B21) is the revenue stability of 0.1749 in C11.
Due to the circumstances of the subject member interests, one who
desires to sell such a member interest could easily search for several years
to ¬nd a buyer. We assume a one-year incremental delay to sale, which
is a conservative estimate (C12).
To calculate the actual discount for delay to sale, we multiply the
coef¬cients in column B by the LLC™s parameters in Column C. Then, we
add together the y-intercept value and the products of the coef¬cients
and the parameters, which yields a delay to sale discount of 22.0% (D13).
This ¬gure is inserted in Table 9-5C, cell B9.

Commentary to Table 9-5C: Calculation of DLOM
Table 9-5C is our calculation of DLOM. Component 1 was discussed in
our commentary to Table 9-5A. Therefore we begin with a discussion of

CHAPTER 9 Sample Appraisal Report 331

T A B L E 9-5B

Earnings and Revenue Stability


4 Year Year Revenue Income

5 1 1989 $89,044 $1,165
6 2 1990 $79,646 $8,033
7 3 1991 $89,894 $(34,588)
8 4 1992 $90,645 $(25,486)
9 5 1993 $73,825 $(24,984)
10 6 1994 $70,739 $19,203
11 7 1995 $61,853 $(18,186)
12 8 1996 $70,476 $6,916
13 9 1997 $82,054 $25,025
14 10 1998 $75,147 $15,400
15 11 1999 $82,220 $(9,704)


19 Regression Statistics

20 Multiple R 0.418245668
21 R square 0.174929439
22 Adjusted R square 0.083254932
23 Standard error 8831.270953
24 Observations 11


27 df SS MS F Signi¬cance F

28 Regression 1 148819808.3 148819808.3 1.908157952 0.200494368
29 Residual 9 701922119.9 77991346.65
30 Total 10 850741928.2

32 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

33 Intercept 85664.6 5710.916139 15.00015022 1.128E 07 72745.6003 98583.5997 72745.6 98583.6
34 Year 1163.145455 842.0286469 1.381360906 0.200494368 3067.948041 741.6571321 3067.95 741.6571
38 Regression Statistics

39 Multiple R 0.335265099
40 R square 0.112402687
41 Adjusted R square 0.013780763
42 Standard error 20145.2116
43 Observations 11


46 df SS MS F Signi¬cance F

47 Regression 1 462537437.2 462537437.2 1.139733262 0.313506838
48 Residual 9 3652465953 405829550.4
49 Total 10 4115003391

51 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

52 Intercept 15685.85455 13027.29977 1.204075658 0.259270779 45155.67649 13783.9674 45155.7 13783.97
53 Year 2050.581818 1920.770561 1.067582906 0.313506838 2294.506377 6395.670013 2294.51 6395.67
T A B L E 9-5C

Calculation of DLOM: 2.80% Member Interest


4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 22.0% 22.0% 78.0% Delay to sale-1 yr (Table 9-5A, D13)
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power-thin markets
11 3A 2.0% 3.2% 96.8% Transactions costs-buyers [3]
12 3B 0.0% 0.0% 100.0% Transactions costs-sellers [4]
13 Percent remaining 68.7% Total % remaining components 1 2 3A 3B
14 Final discount 31.3% Discount 1 Total % Remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 Discount rate r [5] 13.38%
19 Constant growth rate g [6] 3.18%
20 Intermediate calculation: x (1 g)/(1 r) 0.9101
21 Avg # years between sales j 10


. 70
( 100 .)