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