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

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

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

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

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

Black“Scholes Put Option”CHTL


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

Note: Values are for European options. The put option formula can be found in Options Futures and Other Derivatives, 3rd Ed. by
John C. Hull, Prentice Hall, 1997, pp. 241 and 242.
[1] 2 Year Treasury rate on transaction date, 8/8/95 (Source: Federal Reserve)
.5 * var) * t]/[stdev *t0.5], where variance and standard deviation are expressed in annual terms.
[2] d1 [ln (S/E) (r
[std dev * t0.5]
[3] d2 d1

for CHTL of 0.941, transferred from Table 7-6, cell D36, while B10 is var-
iance, merely the square of B9.
Cells B11 and B12 are the calculation of the two Black“Scholes par-
ameters, d1 and d 2. B13 and B14 are the cumulative normal density func-
tions for d1 and d 2. For example, look at cell B13, which is N( 0.777)
0.219. This requires some explanation. The cumulative normal table
from which the 0.219 came assumes the normal distribution has been
standardized to a mean of zero and standard deviation of 1.47 This means
that there is a 21.9% probability that our variable is less than or equal to
0.777 standard deviations below the mean. In cell B14, N( d2)
N( 0.594)) N(0.594) 0.724, which means there is a 72.4% probability
of being less than or equal to 0.594 standard deviations above the mean.
For perspective, it is useful to note that since the normal distribution is
symmetric, N(0) 0.5000, i.e., there is a 50% probability of being less
than or equal to the mean, which implies there is a 50% probability of
being above the mean.
In B15, we calculate the value of the put option, which is $3.73 (B15),
or 42.0% (B16) of the stock price of $8.875 (B5). Thus, our calculation of
the restricted stock discount for the Chantal block using the Black“Scholes
model is 42.0% (B16).

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

47. One standardizes a normal distribution by subtracting the mean from each value and dividing
by the standard deviation.

PART 3 Adjusting for Control and Marketability
T A B L E 7-8

Put Model Results


5 Company Put Calculation Actual Error Absolute Error

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

22 Comparison with the Mean as the Discount

24 Company Mean Discount Actual Error Absolute Error

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

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

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

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

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

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

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

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

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



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

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

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


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

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

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

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

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

CHAPTER 7 Adjusting for Levels of Control and Marketability 249
better or worse when the transaction is close to consummation. It is well
established in behavioral science”and it is the major principle on which
the sale of insurance is based”that the fear of loss is stronger than the
desire for gain (Tversky and Kahneman 1987). This creates pressure for
the seller to accept a lower price in order to get on with life.
Another important ¬nding in behavioral science that is relevant in
explaining DLOM and DLOC is ambiguity aversion (Einhorn and Ho-
garth 1986). The authors cite a paradox proposed by the psychologist
Daniel Ellsberg (Ellsberg 1961) (of Pentagon Papers fame), known as the
Ellsberg paradox.
Ellsberg asked subjects which of two gambles they prefer. In gamble
A the subject draws from an urn with 100 balls in it. They are red or
black only, but we don™t know how many of each. It could be 100 black

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