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69 63 20.5938 1.13E-05 0.000232 0.4087 0.0000947
70 64 21.6235 9.38E-06 0.000203 0.4087 0.0000829
71 65 22.7047 7.81E-06 0.000177 0.4087 0.0000725
72 66 23.8399 6.51E-06 0.000155 0.4087 0.0000634
73 67 25.0319 5.43E-06 0.000136 0.4087 0.0000555
74 68 26.2835 4.52E-06 0.000119 0.4087 0.0000486
75 69 27.5977 3.77E-06 0.000104 0.4087 0.0000425
76 70 28.9775 3.14E-06 0.000091 0.4087 0.0000372
77 71 30.4264 2.62E-06 0.000080 0.3596 0.0000286
78 72 31.9477 2.18E-06 0.000070 0.3596 0.0000251
79 73 33.5451 1.82E-06 0.000061 0.3596 0.0000219
80 74 35.2224 1.51E-06 0.000053 0.3596 0.0000192
81 75 36.9835 1.26E-06 0.000047 0.3596 0.0000168
82 76 38.8327 1.05E-06 0.000041 0.3596 0.0000147
83 77 40.7743 8.76E-07 0.000036 0.3596 0.0000128
84 78 42.8130 7.3E-07 0.000031 0.3596 0.0000112
85 79 44.9537 6.09E-07 0.000027 0.3596 0.0000098
86 80 47.2014 5.07E-07 0.000024 0.3596 0.0000086
87 81 49.5614 4.23E-07 0.000021 0.3165 0.0000066
88 82 52.0395 3.52E-07 0.000018 0.3165 0.0000058
89 83 54.6415 2.93E-07 0.000016 0.3165 0.0000051
90 84 57.3736 2.45E-07 0.000014 0.3165 0.0000044
91 85 60.2422 2.04E-07 0.000012 0.3165 0.0000039
92 86 63.2544 1.7E-07 0.000011 0.3165 0.0000034
93 87 66.4171 1.42E-07 0.000009 0.3165 0.0000030
94 88 69.7379 1.18E-07 0.000008 0.3165 0.0000026
95 89 73.2248 9.83E-07 0.000007 0.3165 0.0000023
96 90 76.8861 8.19E-08 0.000006 0.3165 0.0000020
97 91 80.7304 6.82E-08 0.000006 0.2785 0.0000015
98 92 84.7669 5.69E-08 0.000005 0.2785 0.0000013
99 93 89.0052 4.74E-08 0.000004 0.2785 0.0000012
100 94 93.4555 3.95E-08 0.000004 0.2785 0.0000010
101 95 98.1283 3.29E-08 0.000003 0.2785 0.0000009
102 96 103.0347 2.74E-08 0.000003 0.2785 0.0000008
103 97 108.1864 2.29E-08 0.000002 0.2785 0.0000007
104 98 113.5957 1.9E-08 0.000002 0.2785 0.0000006
105 99 119.2755 1.59E-08 0.000002 0.2785 0.0000005
106 100 125.2393 1.32E-08 0.000002 0.2785 0.0000005

107 Totals $7.3030 $6.1626

108 Discount 1 (F107/D107) 15.6%
109 Discount-By Formula [1] 15.6%

111 Parameters Sensitivity Analysis

112 r 20% Avg Yrs Between Sales

113 g 5% 8 10 12
114 z 12% 18% 18.3% 16.5% 15.3%
115 x (1 g)/ 87.50% 20% 17.2% 15.6% 14.6%
(1 r)
116 j yrs to sale 10 22% 16.3% 14.9% 14.0%


[1] Formula For Discount: 1 ((1 x j)/((1 (1 z)*x j)))




CHAPTER 7 Adjusting for Levels of Control and Marketability 271
T A B L E 7-14

Sample Calculation of DLOM


A B C D E F G

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 13.4% 13.4% 86.6% Delay To sale-1 yr (Table 7-10, D12)
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 2.7% 3.6% 96.4% Transactions costs-buyers
12 3B 7.4% 2.4% 97.6% Transactions costs-sellers
13 Percent remaining 76.9% Total % remaining components 1 2 3A 3B
14 Final discount 23.1% Discount 1 Total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $5,000,000
19 Discount rate r [3] 23.0%
20 Constant growth rate g 7.0%
21 Intermediate calculation: x (1 g)/(1 r) 0.8699
22 Avg # years between sales j 10

24 Section 3: Sensitivity Analysis

26 j Average Years Between Sales

27 j 5 10 15 20
28 Discount 26.6% 23.1% 22.0% 21.6%

[1] Pure discounts: for component #1, Table 7-10, cell D12; for component #2, 9% per Schwert article. For component #3A and #3B, Table 7-11, cells I73 and I74 1% for public
brokerage costs.
[2] PV of perpetual discount formula: 1 (1 x j)/((1 (1 z)*x j)), per equation (7-9), used for component #3B. PV of perpetual discount formula: 1 (1 z)*(1 x j)/((1 (1
z)*x j)), per equation (7-9a), used for component #3A. Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.4172 (.01204 ln FMV), based on Table 4-1




components, as mentioned earlier, do not repeat through time, so their
perpetual discount is equal to their pure discount. Thus, C9 B9 and
C10 B10.
Column D is the remaining value after subtracting the perpetual dis-
count column from one, i.e., Column D 1 Column C. We multiply
D9 D10 D11 D12 D13 76.9%. The Final Discount is 1
Remaining Value 1 76.9% (D13) 23.1% (D14).
The sensitivity analysis in section 3, row 28 of the table shows how
the ¬nal discount varies with different assumptions of j the average
number of years between sales. At j 10 years, it appears that DLOM is
more sensitive to reducing j than increasing it. At j 5, the discount
increased from 23.1% (at j 10) to 26.6%, whereas it only dropped
slightly for j 15 and 20“22.0% and 21.6%, respectively.

Evidence from the Institute of Business Appraisers
In Chapter 10, we examine data published by Raymond Miles, founder
of the Institute of Business Appraisers (IBA), and apply log size discount
rates and the DLOM calculations in this chapter to determine how well
the they explain price/earnings multiples of real world sales of small
businesses. The evidence in Chapter 10 is that within an order of mag-

PART 3 Adjusting for Control and Marketability
272
rate of return (discount rate) implied in the valuation of an enterprise and
the expected returns attributable to minority investors of that enterprise.
There can be many sources of these differentials, several of which were
noted above [in the text of the article leading to this point].
In most cases in which the QMDM is applied, there is a differential
between the expected growth rate in value assumed and the required
holding period return (discount rate) applied. This differential is the pri-
mary source of discounting using the QMDM. Several of my colleagues
have pointed to this aspect of the QMDM. Their comments range from:
(1) Mercer™s Bermuda Triangle of disappearing value; to (2) there should
be no difference at all; to (3) using the range of speci¬c illiquidity dis-
counts used in Chapter 10 of Quantifying Marketability Discounts (roughly
1.5“5.0% or so), when applied to the base equity discount rate (as a proxy
for the expected growth rate), should yield much smaller marketability
discounts than implied by the QMDM. Note that the essence of this third
criticism [which is Mr. Abrams™ criticism] is that the differential between
the expected growth rate in value and the discount rate used would be
only 1.5“5.0% or so in this case.
The criticisms seem to re¬‚ect a lack of understanding of the concep-
tual workings of the QMDM and a lack of familiarity with its consistency
with existing empirical research. We can rely on market evidence from
the various restricted stock studies to support the need for a differential
in the expected growth rate and the required holding period return (dis-
count) rate. The implications of two recent restricted stock studies are
illustrated next, followed by a similar analysis of actual appraisals using
the QMDM.
The Management Planning Study, ˜˜Analysis of Restricted Stocks of
Public Companies (1980“1995), was published, with permission of Man-
agement Planning, Inc. (˜˜MPI™™), as Chapter 12 of Quantifying Marketability
Discounts. The median and average restricted stock discounts in the MPI
study were 27.7% and 28.9%, respectively. For this analysis we will round
the average to 30%.74 We can further assume that the typical expected
holding period before the restrictions of Rule 144 were lifted was on the
order of 2.5 years, or 2 years plus a reasonable period to sell the shares
into the market.
A recently published study by Bruce A. Johnson, ASA (Johnson 1999)
focusing on transactions in the 1991“1995 timeframe yields a smaller av-
erage restricted stock discount of 20%. We will consider the implications
of the Johnson study using a shorter two-year holding period (versus the
MPI average of a 30% average discount and a 2.5-year holding period).
Tables 7-15 and 7-16 use the MPI study and Table 7-17 uses the Johnson
study to illustrate the differential between the expected growth of public
companies and the discount rate embedded in their average restricted
stock pricing.



74. The average of the averages of the 10 restricted stock studies discussed in Chapters 2 and 12 of
Quantifying Marketability Discounts is 31%.




PART 3 Adjusting for Control and Marketability
274
T A B L E 7-15a




Assume market price of public entity $1.00
Average management planning discount (rounded) 30.0% ($0.30)
Assumed purchase price of restricted shares $0.70
Holding period until restricted shares are freely tradable (years) 2.5

a
Using the MPI study 30% average discount.




Now we can examine a variety of assumptions about the ˜˜average™™
restricted stock transaction in the Management Planning study.75 The av-
erage public price has been indexed to $1.00 per share. As a result, the
average restricted stock transaction price, as indexed, is $0.70 per share.
We can estimate the implied returns that were required by investors
in restricted stocks based on a variety of assumptions about the expected
growth rates in value (or the expected returns of the publicly traded
stocks). For purposes of this analysis we have assumed that the consensus
expectations for the public stock returns were somewhere in the range of
0% (no expected appreciation) to 30% compounded. The most relevant
portion of this range likely begins at about 10% since stocks expected to
appreciate less than that were probably not attractive for investments in
their restricted shares. See Table 7-16.
Note that the implied holding period returns for the restricted stock
transactions, on average, ranged from about 27% per year compounded
(with value growing at 10%) to 50% per year compounded (with expected
growth of 30%). As noted in Chapter 8 of Quantifying Marketability Dis-

T A B L E 7-16a




Annualized
Assumed Expected Implied Incremental Return
Expected Future Return for Attributable to
Growth in Value in Holding Restricted Stock
Value (G) 2.5 Years Period (R) Discount (R G)

0% $1.00 15.3% 15.3%
5% $1.13 21.1% 16.1%
10% $1.27 26.9% 16.9%
15% $1.42 32.7% 17.7%
20% $1.58 38.5% 18.5%
25% $1.75 44.3% 19.3%
30% $1.93 50.0% 20.0%

a
Using the MPI study 30% average discount and a 2.5 year holding period.




75. This analysis is for purposes of illustration only. Chapters 2 and 3 of Quantifying Marketability
Discounts raise signi¬cant questions about reliance on averages of widely varying
transactions indications for both the restricted stock and the pre-IPO studies.




CHAPTER 7 Adjusting for Levels of Control and Marketability 275
T A B L E 7-17a




Annualized
Assumed Expected Implied Incremental Return
Expected Future Return for Attributable to
Growth in Value in Holding Restricted Stock
Value (G) 2.0 Years Period (R) Discount (R G)

0% $1.00 11.8% 11.8%
5% $1.10 17.4% 12.4%
10% $1.21 23.0% 13.0%
15% $1.32 28.6% 13.6%
20% $1.44 34.2% 14.2%
25% $1.56 39.8% 14.8%
30% $1.69 45.3% 15.3%

a
Using the Johnson study 20% average discount and a 2 year holding period.




counts, the implied returns are in the range of expected venture capital
returns for initial investments (not average venture capital returns, which
include unsuccessful investments). Interestingly, the differential between
the implied holding period returns above and the expected growth rate
in values used are quite high, ranging from 15.3“20.0%.
This analysis is ex post. We do not know how the actual investment
decisions were made in the transactions included in the Management
Planning study or any of the restricted stock studies. But, ex post, it is
clear that the investors in the ˜˜average™™ restricted stock transactions were,
ex ante, either: (1) placing very high discount rates on their restricted
stock transactions (ranging from 15“20% in excess of the expected returns
of the public companies they were investing in; (2) questioning the con-
sensus expectations for returns; or (3) some combination of 1 and 2.
The Johnson study cited above focused on transactions in the 1991“
1995 timeframe when the Rule 144 restriction period was still two years
in length. If we assume an index price of $0.80 per share ($1.00 per share
freely tradable price less the 20% average discount) and a holding period
of two years (and instant liquidity thereafter) and replicate our analysis
of Table 7-16 we obtain the following result in Table 7-17.
Even with a shortened assumed holding period and a smaller aver-
age restricted stock discount, the implied required returns for the Johnson
study are in the range of 23“45% for companies assumed to be growing
at 10“30% per year. And the average differential between this calculated
discount rate and the expected growth rate of the investment companies
is in the range of 13.0“15.3%.
We can make several observations about the seemingly high differ-
entials between the restricted stock investors™ required returns and the
expected value growth of the typical entity:
— The average discounts appear to be indicative of defensive
pricing.
— The discounts would likely ensure at least a market return if the
expected growth is not realized.
— Very high implied returns are seen as expected growth increases,
suggesting that high growth is viewed with skepticism.

PART 3 Adjusting for Control and Marketability
276
— The implied incremental returns of R over expected G are
substantial at any level, suggesting that the base ˜˜cost™™ of 2.0 or
2.5 years of illiquidity is quite expensive.
Given varying assumptions about holding periods longer than 2.5
years and allowing for entities that pay regular dividends, we would
expect some variation from the premium range found in appraisals of
private company interests.
By way of comparison, we have made the same calculations for the
example applications of the QMDM from Chapter 10 of Quantifying Mar-
ketability Discounts.
As noted in Table 7-18, the range of differences between the average
required returns and the expected growth rates in value assumed in the
10 appraisals was from 8.5“21.4%, with an average of about 13%. The
table also indicates the range of other assumptions that yielded the con-
cluded marketability discounts in the illustrations. I believe that these
results, which came from actual appraisals, are generally consistent with
the market evidence gleaned from the restricted stock studies above. In-
deed, the premium returns required by the restricted stock investors, on
average, exceed those applied in the above examples, suggesting the con-
clusions yielded conservative (i.e., relatively low) marketability discounts
on average. [section omitted]

Conclusion
The QMDM, which is used primarily in valuing (nonmarketable) minor-
ity interests of private companies, develops concrete estimates of expected
growth in value of the enterprise and reasonable estimates of additional
risk premia to account for risks faced by investors in nonmarketable mi-
nority interests of companies. In its fully developed form, it incorporates
expectations regarding distributions to assist appraisers in reaching log-
ical, supportable, and reasonable conclusions regarding the appropriate
level of marketability discounts for speci¬c valuations.

T A B L E 7-18

Summary of Results of Applying the QMDM in 10 Example Appraisals


Average
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
QMDM.
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

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