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Required Expected Concluded
Holding Holding Period Growth in Value (R G) Dividend Marketability
Example Period Return (R) Assumed (G) Difference Yield Discount

1 5“8 years 20.0% 10.0% 10.0% 0.0% 45.0%
2 5“9 years 20.5% 4.0% 16.5% 8.8% 25.0%
3 7“15 years 18.5% 7.0% 11.5% 8.0% 15.0%
4 1.5“5 years 19.5% 7.5% 12.0% 0.0% 20.0%
5 5“10 years 20.5% 9.8% 10.7% 3.2% 40.0%
6 5“10 years 18.5% 10.0% 8.5% 2.1% 25.0%
7 5“15 years 19.5% 6.0% 13.5% 0.0% 60.0%
8 10“15 years 19.5% 5.0% 14.5% 10.0% 25.0%
9 10 years 26.4% 5.0% 21.4% 0.6% 80.0%
10 3“5 years 22.5% 6.0% 16.5% 0.0% 35.0%
Averages 20.5% 7.0% 13.5% 3.3% 37.0%
Medians 19.8% 6.5% 12.8% 1.4% 30.0%

Source: Quantifying Marketability Discounts, Chapter 10

CHAPTER 7 Adjusting for Levels of Control and Marketability 277
The unpublished [and Mr. Abrams™] criticisms of the QMDM out-
lined above are, I believe, not correct. They do not recognize the critical
distinctions that appraisers must draw between their analyses in valuing
companies and valuing minority interests in those companies. And they
do not consider the implications of the market evidence of required re-
turns provided by the familiar restricted stock studies.
Marketable minority (and controlling interest) appraisals are devel-
oped based on the capitalized expected cash ¬‚ows of businesses, or en-
terprises. Minority interests in those businesses must be valued based
on consideration of the cash ¬‚ows expected to be available to minority
investors. The QMDM allows the business appraiser to bridge the gap
between these two cash ¬‚ow concepts, enterprise and shareholder, to
develop reasoned and reasonable valuation conclusions at the non-
marketable minority interest level.

My Counterpoints
In responding to Mr. Mercer™s rebuttal, it is clear that we will need a
speci¬c numerical example to make my criticism clear of the QMDM™s
inability to forecast restricted stock discounts.
Table 7-19, columns H and I, which we take from Mercer™s Chapter
10, Example 1, show his calculation of the required holding period return
of a minority stake for a private, closely held C corporation. The corpo-
ration is expected to grow in value by 10% each year mainly through an
increase in earnings. It is not expected to pay dividends, and the majority
owner is expected to retire and sell the business in ¬ve to eight years.
In columns K and L we show our own calculation of a restricted
stock™s required holding period return using Mercer™s Example 1 as a
guide. Our purpose is to show that the QMDM cannot even come close
to forecasting ex ante the ex post discount rates of 27“50% from Table
7-16 that are necessary to explain restricted stock discounts using the
We assume a non-dividend-paying stock with an equivalent base eq-
uity discount rate as the stock in Mercer™s example of 16.7% (row 14). It
is in the investment speci¬c risk premiums where the restricted stock
differs from the private minority shares. The restricted stock should be
much easier to sell than a minority stake in a private closely held C
corporation, since the ability to sell at the then-market rate in 2.5 years
is guaranteed and public minority shareholder rights are generally better
protected they are in private ¬rms. We therefore reduce this premium for
illiquidity from the premium in Mercer™s example of between 1 and 2%
(H18 and I18) to 0% (K18, L18) for the restricted stock. While it is possible
that the restricted stocks should have a positive premium for this factor,
they are nevertheless far more liquid than all of the private ¬rms in Mer-
cer™s examples. If we should increase K18 and L18 to, say, 1%, then we
should increase H18 and I18 to at least 2“3%, respectively, or probably
higher yet.
Relative to the private C corporation shares, the expected holding
period for the restricted stock is short and certain. We therefore reduce
the premium for holding period uncertainty from between 0 and 1% (H19
and I19) for Example 1 to 0 (K19, L19) for the restricted shares. As both

PART 3 Adjusting for Control and Marketability
T A B L E 7-19

QMDM Comparison of Restricted Stock Discount Rate versus Mercer Example 1


5 Mercer Example Restricted Stock

6 Range of Range of
Returns Returns

7 Components of the Required Holding Period Return Lower Higher Lower Higher

8 Base equity discount rate (adjusted capital asset pricing model)
9 Current yield-to-maturity composite long term treasuries 6.7% 6.7% 6.7% 6.7%
10 Adjusted Ibbotson large stock premium 6.5%
11 applicable beta statistic 1
12 Beta adjusted large stock premium 6.5% 6.5% 6.5% 6.5%
13 Adjusted Ibbotson small stock premium 3.5% 3.5% 3.5% 3.5%
14 Base equity discount rate 16.7% 16.7% 16.7% 16.7%
17 Investment Speci¬c Risk Premiums

18 General illiquidity of the investment [1] 1.0% 2.0% 0.0% 0.0%
19 Uncertainties related to length of expected holding period [2] 0.0% 1.0% 0.0% 0.0%
20 Lack of expected interim cash ¬‚ows [3] 0.5% 1.0% 0.5% 1.0%
21 Small shareholder base [4] 0.0% 1.0% 0.0% 0.0%
22 Range of speci¬c risk premiums for the investment 1.5% 5.0% 0.5% 1.0%
24 Initial range of required returns 18.2% 21.7% 17.2% 17.7%
26 Concluded range of required holding period returns (rounded) 18.0% 22.0% 17.0% 18.0%

[1] The restricted stock should be much easier to sell than a minority stake in a private closely held C corporation, since public minority shareholder rights are generally better protected.
While it is possible that the restricted stocks should have a positive premium for this factor, they are nevertheless far more liquid than all of the private ¬rms in Mercer™s examples. If we
should increase K18 and L18 to 1%, then we should increase H18 and I18 to at least 2% to 3% or probably higher yet.
[2] Relative to the private shares, the expected holding period for the restricted stock is short and certain.
[3] We assume a non dividend paying restricted stock. The example also concerned a non dividend paying C corporation. We therefore assign the same risk premium for this factor.
[4] The restricted stock shares are shares of public corporations, which in general have large shareholder bases.

investments are expected to pay no dividends, there is no difference in
the premium for lack of expected interim cash ¬‚ows (Row 20), although
the latter experiences that lack of dividends for a far shorter and much
more certain time period, which could well justify a lower premium than
the former.
At this point I can digress to pose my objections to the ¬rst two
factors. General illiquidity of the investment is a very fuzzy term. It can mean
almost anything. There is no empirical measure of it. Therefore, it can be
almost anything that one wants it to be”which I admit has its advan-
tages in practical application, but it™s not good science. It is also unclear
where general illiquidity stops and uncertainties in the holding period
begin. Do they overlap? How does one prevent him- or herself from
double-counting them? That is a problem with loosely-de¬ned terms.
Returning to the main train of thought, the private, closely held C
corporation would have a much smaller shareholder base than the re-
stricted stock corporations. We therefore reduce the premium for a small
shareholder base from between 0 and 1% (H21 and I21) for Example 1 to
0 (K21, L21) for the restricted stock. The total speci¬c risk premium for

CHAPTER 7 Adjusting for Levels of Control and Marketability 279
the restricted stock comes to 0.5% (K22) to 1.0% (L22) versus the 1.5%
(H22) to 5% (I22) for the private shares. After adding the base equity
discount rates and rounding, we arrive at a concluded range of required
holding period returns of 18“22% and 17“18% (Row 26) for Mercer™s
Example 1 and the restricted stock, respectively.
Next we need to determine the expected growth rate in value of the
unrestricted marketable minority shares. Since there are no dividends, the
expected growth rate must be equal to the discount rate”by de¬nition.76
In this example the equity discount rate of the unrestricted marketable
shares or the ˜˜base equity discount rate™™ is 16.7%.
Let™s now calculate the QMDM discount on the restricted stock with
the following assumptions:

1. A midrange (of K26 and L26) required holding period return of
2. The 2.5-year average holding period.
3. The growth rate in value of 16.7%.

The calculation is as follows:

DLOM 1 (FV PVF) 1 1.7%

Assuming the correct discount is 30%, the QMDM is almost 95% too low!

Mercer™s Response
After reviewing Mr. Abrams™ response to my rebuttal of his criticism of
the QMDM, it is apparent that he and I continue to disagree over how
the QMDM is applied in practice. The average marketability discounts in
the 10 examples cited in my rebuttal of his criticism was 37%, and the
median discount was 30%, not 1.7%. Mr. Abrams™ mistake is in assuming
that the discount rate embedded in the pricing of a publicly traded stock
is the required return of restricted stock investors. The fact that the av-
erage restricted stock discount is 30% or so indicates that investors have
extracted a signi¬cant premium in return relative to the expected returns
of the counterpart publicly traded securities.
What may be true ˜˜by de¬nition™™ in a perpetuity calculation may
well not be true for shorter holding periods. The QMDM deals, not with
perpetuity calculations, but with investor assessments of expected cash
¬‚ows over ¬nite time horizons. And it makes explicit the assumptions
made about the relationship between the expected growth in value of
investments and the required returns of investors in those investments. I
maintain that the model does indeed provide an excellent tool for esti-
mating marketability discounts (from an estimated freely traded value)
for minority interests in closely held companies.

76. This is the discount rate applicable to marketable minority shares, not the higher discount rate
applicable to illiquid shares, i.e., the required holding period return.

PART 3 Adjusting for Control and Marketability
We have reviewed the professional and some of the academic literature
dealing with control premiums and DLOM. My opinion is that with our
current information set, we should use control premiums in the 21“28%
range. We developed this as being three to four times the value of the
voting rights premium adjusted to U.S. laws and for liquidity differences
between voting and nonvoting stock. This measure is consistent with the
median going private premium of 24.1% (Table 7-1, E21), although it is
preferable to make a clean separation of expected performance improve-
ments, which increase the ˜˜top line,™™ i.e., cash ¬‚ows, versus the pure
value of control, which is represented by a reduction in the discount rate.
We reviewed three quantitative models of DLOM: Mercer™s, Kas-
per™s, and Abrams™. The QMDM was unable to provide any meaningful
restricted stock discounts for the Management Planning, Inc. data, as dis-
counting modest risk premiums for two to three years provides little var-
iation in discount. Abrams™ non-company-speci¬c Black-Scholes options
pricing model performed worse at explaining restricted stock discounts
than the mean, while using BSOPM with ¬rm-speci¬c calculations of stan-
dard deviations was superior to the mean. While that makes Black“
Scholes a viable candidate for restricted stock studies, it is not a possible
model for valuing the delay-to-sale component of DLOM, and we must
use the regression of the MPI data.
We quanti¬ed component #2, monopsony power to the buyer, as 9%,
according to Schwert™s ¬ndings of a 12.2% greater premium in takeovers
when there are multiple buyers than when there is only one buyer.
Finally, we quanti¬ed transactions costs separately for the buyer and
the seller. The premise of fair market value is such that we ask, ˜˜What
would a hypothetical buyer be willing to pay for this interest,™™ which
means that we are presuming a ¬rst sale immediately. Buyers care about
their own transactions costs, but they do not care about sellers™ transac-
tions cost on the immediate transaction. However, buyers do care that in
10 years or so they become the sellers. They therefore care about all sub-
sequent sellers™ (and buyers™) transactions costs. We presented two dis-
count formulas”equations (7-9) and (7-9a), which are appropriate for
seller and buyer, respectively, to translate the pure discount that applies
to each transaction into a discount based on the present value of the
in¬nite continuum of periodic transactions.
In Table 7-14 we applied our DLOM model to a control interest in a
hypothetical private company. The result was a DLOM of 23.1%, which
is a reasonable result.
Of course, the economic components model is merely a model. It is
certainly imperfect, and it must be used with common sense. It is possible
to obtain strange or nonsensical results, and if the appraiser is asleep at
the wheel, he or she may not realize it. There is plenty of room for ad-
ditional research to improve our modeling and results. Nevertheless, in
my opinion this is the most realistic and comprehensive model to date
for calculating DLOM.

Abrams, Jay B. 1994a. ˜˜Discount for Lack of Marketability: A Theoretical Model.™™ Business
Valuation Review (September): 132“39.

CHAPTER 7 Adjusting for Levels of Control and Marketability 281
” ”. 1994b. ˜˜A Breakthrough in Calculating Reliable Discount Rates.™™ Valuation (Au-

gust): 8“24.
Amihud, Y., and H. Mendelson. 1991. ˜˜Liquidity, Asset Prices, and Financial Policy.™™ Fi-
nancial Analysts Journal (November“December): 56“66.
Barca, F. 1995. ˜˜On Corporate Governance in Italy: Issues, Facts, and Agency.™™ Mimeo,
Bank of Italy.
Bergstrom, C., and K. Rydqvist. 1990. ˜˜Ownership of Equity in Dual-Class Firms.™™ Journal
of Banking and Finance 14:237“53.
Berkovitch, E., and M. P. Narayanan. 1993. ˜˜Motives for Takeovers: An Empirical Inves-
tigation. Journal of Financial and Quantitative Analysis 28:347“62.
Bolotsky, Michael J. 1991. ˜˜Adjustments for Differences in Ownership Rights, Liquidity,
Information Access, and Information Reliability: An Assessment of ˜Prevailing Wis-
dom™ versus the ˜Nath Hypothesis™.™™™ Business Valuation Review (September): 94“110.
” ”. 1995. ˜˜Is the ˜Levels of Value™ Concept Still Viable? Bolotsky™s Response.™™ Business


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