will pay me less because of those costs, and therefore I must pay my

seller less because of my costs as a seller in Year 10. Additionally, the

process goes on forever, because in Year 20, my buyer becomes a seller

and faces the same problem.™™ Thus, we need to quantify the present value

of a periodic perpetuity of buyer™s transactions costs beginning with the

immediate sale and sellers™ transactions costs that begin with the second

sale of the business.64 In the next section we will develop the mathematics

necessary to do this.

Developing Formulas to Calculate DLOM Component #3. This

section contains some dif¬cult mathematics, but ultimately we will arrive

at some very usable formulas that are not that dif¬cult. It is not necessary

to follow all of the mathematics that gets us there, but it is worthwhile

to skim through the math to get a feel for what it means. In the Mathe-

62. We could run another regression to forecast investment banking fees. This was sight estimated.

One could also use a formula such as the Lehman Brothers formula to forecast investment

banking fees.

63. I thank R. K. Hiatt for the brilliant insight that the ¬rst two components of DLOM do not have

this characteristic and thus do not require this additional present value calculation.

64. One might think that the buyers™ transactions costs are not relevant the ¬rst time, because the

buyer has to put in due diligence time whether or not a transaction results. In individual

instances that is true, but in the aggregate, if buyers would not receive compensation for

their due diligence time, they would cease to buy private ¬rms until the prices declined

enough to compensate them.

PART 3 Adjusting for Control and Marketability

262

matical Appendix we develop the formulas below step by step. In order

to avoid presenting volumes of burdensome math in the body of the

chapter, we present only occasional snapshots of the math”just enough

to present the conclusions and convey some of the logic behind it.

For simplicity, suppose that, on average, business owners hold the

business for 10 years and then sell. Every time an owner sells, he or she

incurs a transactions cost of z. The net present value (NPV) of the cash

¬‚ows to the business owner is:65

NPV NPV1 (1 z)NPV11 (7-1)

10

Equation (7-1) states that the NPV of cash ¬‚ows at Year 0 to the

owner is the sum of the NPV of the ¬rst 10 years™ cash ¬‚ows and (1

z) times the NPV of all cash ¬‚ows from Year 11 to in¬nity. If transactions

costs are 10% every time a business sells, then z 10% and 1 z

66

90%. The ¬rst owner would have 10 years of cash ¬‚ows undiminished

by transactions costs and then pay transactions costs of 10% of the NPV

at Year 10 of all future cash ¬‚ows.

The second owner operates the business for 10 years and then sells

at Year 20. He or she pays transactions costs of z at Year 20. The NPV of

cash ¬‚ows to the second owner is:

NPV11 NPV11 (1 z)NPV21 (7-2)

20

Substituting (7-2) into equation (7-1), the NPV of cash ¬‚ows to the

¬rst owner is:

NPV NPV1 (1 z)[NPV11 (1 z)NPV21 ] (7-3)

10 20

This expression simpli¬es to:

z)2 NPV21

NPV NPV1 (1 z)NPV11 (1 (7-4)

10 20

We can continue on in this fashion ad in¬nitum. The ¬nal expression

for NPV is:

z)i 1

NPV (1 NPV[10(i (7-5)

1) 1] 10i

i1

The NPV is a geometric sequence. Using a Gordon model, i.e., as-

suming constant, perpetual growth, in the Mathematical Appendix, we

show that equation (7-5) solves to:

10

1 g

1

1 r

1 r

NPVTC (7-6)

10

r g 1 g

1 (1 z)

1 r

where NPVTC is the NPV of the cash ¬‚ows with the NPV of the trans-

actions costs that occur every 10 years removed, g is the constant growth

65. Read the hyphen in the following equation™s subscript text as the word ˜˜to,™™ i.e., the NPV from

one time period to another.

66. z is actually an incremental transaction cost, as we will explain later in the chapter.

CHAPTER 7 Adjusting for Levels of Control and Marketability 263

rate of cash ¬‚ows, r is the discount rate, and cash ¬‚ows are midyear.67

The end-of-year formula is the same, replacing the 1 r in the nu-

merator with the number 1.

The NPV of the cash ¬‚ows without removing the NPV of transactions

costs every 10 years is simply the Gordon model multiple of ( 1 r)/

(r g), which is identical with the ¬rst term on the right-hand side of

equation (7-6). The discount for lack of marketability for transactions costs

is equal to:

NPVTC

DLOM 1 (7-7)

NPV

The fraction in equation (7-7) is simply the term in the large braces

in equation (7-6). Thus, DLOM simpli¬es to:

10

1 g

1

1 r x 10

1

D 1 1 (7-8)

10

z)x10

1 (1

1 g

1 (1 z)

1 r

r, ’ 0 1.68

where x (1 g)/(1 r), D is the discount, and g x

Equation (7-8) is the formula for the discount assuming a sale every

10 years. Instead of assuming a business sale every 10 years, now we let

the average years between sale be a random variable, j, which leads to

the generalized equation in (7-9) for sellers™ transactions costs:69

j

1 g

1

1 r xj

1

D3B 1 1

j

z)x j

1 (1

1 g

1 (1 z)

1 r

DLOM formula”sellers™ costs (7-9)

Using an end-of-year Gordon Model assumption instead of midyear

cash ¬‚ows leads to the identical equation, i.e., equation (7-9) holds for

both.

Analysis of partial derivatives in the Mathematical Appendix shows

that the discount, i.e., DLOM, is always increasing with increases in

growth (g) and transactions costs (z) and is always decreasing with in-

creases in the discount rate (r) and the average number of years between

sales ( j). The converse is true as well. Decreases in the independent var-

iables have opposite effects on DLOM as increases do.

67. This appears as equation (A7-7) in the Mathematical Appendix.

68. This is identical with equation (A7-10) in the Mathematical Appendix.

69. This is identical with equation (A7-11) in the Mathematical Appendix. Note that we use the

plural possessive here because we are speaking about an in¬nite continuum of sellers (and

buyers).

PART 3 Adjusting for Control and Marketability

264

Equation (7-9) is the appropriate formula to use for quantifying the

sellers™ transactions costs, because it ignores the ¬rst sale, as discussed

above.70 The appropriate formula for quantifying the buyers™ transactions

costs incorporates an initial transaction cost at time zero instead of at

t j. With this assumption, we would modify the above analysis by

changing the (1 z)i 1 to (1 z)i in equation (7-5). The immediate trans-

action equivalent formula of equation (7-9) for buyers™ transactions costs

is:71

x j)

(1 z)(1

D3A 1

z)x j

1 (1

generalized DLOM formula”buyers™ transactions costs 7-9a

Obviously, equation (7-9a), which assumes an immediate sale, results

in much larger discounts than equation (7-9), where the ¬rst sale occurs

j years later. Equation (7-9) constitutes the discount appropriate for sell-

ers™ transactions costs, while equation (7-9a) constitutes the discount ap-

propriate for buyers™ transactions costs. Thus, component #3 splits into

#3A and #3B because we must use different formulas to value them.72,73

A Simpli¬ed Example of Sellers™ Transactions Costs. Because ap-

praisers are used to automatically assuming that all sellers™ costs merely

reduce the net proceeds to the seller but have no impact on the fair market

value, the concept of periodic sellers™ costs that do affect FMV is poten-

tially very confusing. Let™s look at a very simpli¬ed example to make the

concept clear.

Consider a business that will sell once at t 0 for $1,000 and once

at t 10 years for $1,500, after which the owner will run the company

and eventually liquidate it. For simplicity, we will ignore buyers™ trans-

actions costs. We can model the thinking of the ¬rst buyer, i.e., at t 0,

as follows: ˜˜When I eventually sell in Year 10, I™ll have to pay a business

broker $150. If I were selling publicly traded stock, I would have paid a

broker™s fee of 2% on the $1,500, or $30, so the difference is $130. Assum-

ing a 25% discount rate, the present value factor is 0.1074, and $130

0.1074 $13.96 today. On a price of $1,000, the excess transactions costs

from my eventual sale are 1.396%, or approximately 1.4%. Formulas (7-

9) and (7-9a) extend this logic to cover the in¬nite continuum of trans-

actions every 10 years (or every j years, allowing the average selling pe-

riod to be a variable).

70. Note that we have shifted from speaking in the singular about the ¬rst seller to the plural in

speaking about the entire continuum of sellers throughout in¬nite time. We will make the

same shift in language with the buyers as well.

71. This is identical with equation (A7-11A) in the Mathematical Appendix.

72. An alternative approach is to use equation (7-9a) for both and subtract the ¬rst round seller™s

costs.

73. It is not that buyers and sellers sit around and develop equations like (7-9) and (7-9a) and run

them on their spreadsheets before making deals. One might think this complexity is silly,

because real-life buyers and sellers don™t do this. However, we are merely attempting to

model economically their combination of ideal rationality and intuition.

CHAPTER 7 Adjusting for Levels of Control and Marketability 265

Tables 7-12 and 7-13: Proving Formulas (7-9) and (7-9a). Tables

7-12 and 7-13 prove equations (7-9) and (7-9a), respectively. The two ta-

bles have identical structure and logic, so we will cover both of them by

explaining Table 7-12.

Column A shows 100 years of cash ¬‚ow. While the formulas presume

perpetuities, the present value effect is so small that there is no relevant

present value after Year 100.

The assumptions of the model are: the discount rate is 20% (cell

B112), the perpetual growth rate is 5% (B113), sellers™ transactions costs

z 12% (B114),

1 g 1.05

x 0.875 (B115)

1 r 1.2

and j, the average years between sales of the business, equals 10 years

(B116).

In B7 we begin with $1.00 of forecast cash ¬‚ow in Year 1. The cash

¬‚ow grows at a rate of g 5%. Thus, every cash ¬‚ow in column B from

rows 8“106 equals 1.05 times the number above it. Column C is the pres-

ent value factor assuming midyear cash ¬‚ows at a discount rate of 20%.

Column D, the present value of cash ¬‚ows, equals column B column

C.

Column E is the factor that tells us how much of the cash ¬‚ows from

each year remains with the original owner after removing the seller™s

transactions costs. The buyer does not care about the seller™s transactions

costs, so only future sellers™ transactions costs count in this calculation.

In other words, the buyer cares about the transactions costs that he or

she will face in 10 years when he or she sells the business. In turn, he or

she knows that his or her own buyer eventually becomes a seller. There-

fore, each 10 years, or more generally, each j years, the cash ¬‚ows that

remains with the original owner declines by a multiple of (1 z). Its

Int(Yr 1)

formula is (1 z) .

Thus, the ¬rst 10 years, 100% 1.0000 (E7“E16) of the cash ¬‚ows

with respect to sellers™ transactions costs remain with the original owner.

The next 10 years, Years 11“20, the original owner™s cash ¬‚ows are re-

duced to (1 z) 88% (E17“E26) of the entire cash ¬‚ow, with the 12%

being lost as sellers™ transactions costs to the second buyer. For Years 21“

30, the original owner loses another 12% to transactions costs for the third

buyer, so the value that remains is (1 z)2 (1 0.12)2 0.882 0.7744

(E27“E36). This continues in the same pattern ad in¬nitum.

Column F is the posttransactions costs present value of cash ¬‚ows,

which is column D column E. Thus, D17 E17 0.240154 0.8800

0.2113356 (F17). We sum the ¬rst 100 years™ cash ¬‚ows in F107, which

equals $7.0030. In other words, the present value of posttransactions costs

cash ¬‚ows to the present owner of the business is $7.003. However, the

present value of the cash ¬‚ows without removing transactions costs is

$7.3030 (D107). In F108 we calculate the discount as 1 (F107/D108)

1 ($7.0030/$7.3030) 4.1%.

In F109 we present the calculations according to equation (7-9), and

it, too, equals 4.1%. Thus we have demonstrated that equation (7-9) is

accurate.

PART 3 Adjusting for Control and Marketability

266

T A B L E 7-12

Proof of Equation (7-9)

A B C D E F G

4 (1 z) Int(Yr 1) Post Tx

5 Cash PV Cash Post-Trans PV Cash

6 Year Flow PVF Flow Costs Flow

7 1 1.0000 0.912871 0.912871 1.0000 0.9128709

8 2 1.0500 0.760726 0.798762 1.0000 0.7987621

9 3 1.1025 0.633938 0.698917 1.0000 0.6989168

10 4 1.1576 0.528282 0.611552 1.0000 0.6115522

11 5 1.2155 0.440235 0.535108 1.0000 0.5351082

12 6 1.2763 0.366862 0.468220 1.0000 0.4682197

13 7 1.3401 0.305719 0.409692 1.0000 0.4096922

14 8 1.4071 0.254766 0.358481 1.0000 0.3584807

15 9 1.4775 0.212305 0.313671 1.0000 0.3136706

16 10 1.5513 0.176921 0.274462 1.0000 0.2744618

17 11 1.6289 0.147434 0.240154 0.8800 0.2113356

18 12 1.7103 0.122861 0.210135 0.8800 0.1849186

19 13 1.7959 0.102385 0.183868 0.8800 0.1618038

20 14 1.8856 0.0852 0.160884 0.8800 0.1415783

15 15 1.9799 0.0711 0.140774 0.8800 0.1238810

22 16 2.0789 0.05925 0.123177 0.8800 0.1083959

23 17 2.1829 0.049375 0.107780 0.8800 0.0948464

24 18 2.2920 0.041146 0.094308 0.8800 0.0829906

25 19 2.4066 0.034288 0.082519 0.8800 0.0726168

26 20 2.5270 0.028574 0.072204 0.8800 0.0635397

27 21 2.6533 0.023811 0.063179 0.7744 0.0489256

28 22 2.7860 0.019843 0.055281 0.7744 0.0428099

29 23 2.9253 0.016536 0.048371 0.7744 0.0374586

30 24 3.0715 0.0138 0.042325 0.7744 0.0327763

31 25 3.2251 0.011483 0.037034 0.7744 0.0286793

32 26 3.3864 0.009569 0.032405 0.7744 0.0250944

33 27 3.5557 0.007974 0.028354 0.7744 0.0219576

34 28 3.7335 0.006645 0.024810 0.7744 0.0192129

35 29 3.9201 0.005538 0.021709 0.7744 0.0168113

36 30 4.1161 0.004615 0.018995 0.7744 0.0147099

37 31 4.3219 0.003846 0.016621 0.6815 0.0113266

38 32 4.5380 0.003205 0.014543 0.6815 0.0099108

39 33 4.7649 0.002671 0.012725 0.6815 0.0086719

40 34 5.0032 0.002226 0.011135 0.6815 0.0075879

41 35 5.2533 0.001855 0.009743 0.6815 0.0066394

42 36 5.5160 0.001545 0.008525 0.6815 0.0058095

43 37 5.7918 0.001288 0.007459 0.6815 0.0050833

44 38 6.0814 0.001073 0.006527 0.6815 0.0044479

45 39 6.3855 0.000894 0.005711 0.6815 0.0038919

46 40 6.7048 0.000745 0.004997 0.6815 0.0034054

47 41 7.0400 0.000621 0.004373 0.5997 0.0026222

48 42 7.3920 0.000518 0.003826 0.5997 0.0022944

49 43 7.7616 0.000431 0.003348 0.5997 0.0020076

50 44 8.1497 0.000359 0.002929 0.5997 0.0017567

51 45 8.5572 0.0003 0.002563 0.5997 0.0015371

52 46 8.9850 0.00025 0.002243 0.5997 0.0013449

53 47 9.4343 0.000208 0.001962 0.5997 0.0011768

54 48 9.9060 0.000173 0.001717 0.5997 0.0010297

55 49 10.4013 0.000144 0.001502 0.5997 0.0009010

56 50 10.9213 0.00012 0.001315 0.5997 0.0007884

57 51 11.4674 0.0001 0.001150 0.5277 0.0006071

58 52 12.0408 8.36E-05 0.001007 0.5277 0.0005312

59 53 12.6428 6.97E-05 0.000881 0.5277 0.0004648

59 54 13.2749 5.81E-05 0.000771 0.5277 0.0004067

61 55 13.9387 4.84E-05 0.000674 0.5277 0.0003558

62 56 14.6356 4.03E-05 0.000590 0.5277 0.0003114

CHAPTER 7 Adjusting for Levels of Control and Marketability 267

T A B L E 7-12 (continued)

Proof of Equation (7-9)

A B C D E F G

4 (1 z) Int(Yr 1) Post Tx

5 Cash PV Cash Post-Trans PV Cash

6 Year Flow PVF Flow Costs Flow

63 57 15.3674 3.36E-05 0.000516 0.5277 0.0002724

64 58 16.1358 2.8E-05 0.000452 0.5277 0.0002384

65 59 16.9426 2.33E-05 0.000395 0.5277 0.0002086

66 60 17.7897 1.94E-05 0.000346 0.5277 0.0001825

67 61 18.6792 1.62E-05 0.000303 0.4644 0.0001405

68 62 19.6131 1.35E-05 0.000265 0.4644 0.0001230

69 63 20.5938 1.13E-05 0.000232 0.4644 0.0001076

70 64 21.6235 9.38E-06 0.000203 0.4644 0.0000941

71 65 22.7047 7.81E-06 0.000177 0.4644 0.0000824

72 66 23.8399 6.51E-06 0.000155 0.4644 0.0000721

73 67 25.0319 5.43E-06 0.000136 0.4644 0.0000631

74 68 26.2835 4.52E-06 0.000119 0.4644 0.0000552

75 69 27.5977 3.77E-06 0.000104 0.4644 0.0000483

76 70 28.9775 3.14E-06 0.000091 0.4644 0.0000423

77 71 30.4264 2.62E-06 0.000080 0.4087 0.0000325

78 72 31.9477 2.18E-06 0.000070 0.4087 0.0000285

79 73 33.5451 1.82E-06 0.000061 0.4087 0.0000249

80 74 35.2224 1.51E-06 0.000053 0.4087 0.0000218

81 75 36.9835 1.26E-06 0.000047 0.4087 0.0000191

82 76 38.8327 1.05E-06 0.000041 0.4087 0.0000167

83 77 40.7743 8.76E-07 0.000036 0.4087 0.0000146

84 78 42.8130 7.3E-07 0.000031 0.4087 0.0000128

85 79 44.9537 6.09E-07 0.000027 0.4087 0.0000112

86 80 47.2014 5.07E-07 0.000024 0.4087 0.0000098

87 81 49.5614 4.23E-07 0.000021 0.3596 0.0000075

88 82 52.0395 3.52E-07 0.000018 0.3596 0.0000066

89 83 54.6415 2.93E-07 0.000016 0.3596 0.0000058

90 84 57.3736 2.45E-07 0.000014 0.3596 0.0000050

91 85 60.2422 2.04E-07 0.000012 0.3596 0.0000044

92 86 63.2544 1.7E-07 0.000011 0.3596 0.0000039

93 87 66.4171 1.42E-07 0.000009 0.3596 0.0000034

94 88 69.7379 1.18E-07 0.000008 0.3596 0.0000030

95 89 73.2248 9.83E-07 0.000007 0.3596 0.0000026

96 90 76.8861 8.19E-08 0.000006 0.3596 0.0000023

97 91 80.7304 6.82E-08 0.000006 0.3165 0.0000017

98 92 84.7669 5.69E-08 0.000005 0.3165 0.0000015

99 93 89.0052 4.74E-08 0.000004 0.3165 0.0000013

100 94 93.4555 3.95E-08 0.000004 0.3165 0.0000012

101 95 98.1283 3.29E-08 0.000003 0.3165 0.0000010

102 96 103.0347 2.74E-08 0.000003 0.3165 0.0000009

103 97 108.1864 2.29E-08 0.000002 0.3165 0.0000008

104 98 113.5957 1.9E-08 0.000002 0.3165 0.0000007

105 99 119.2755 1.59E-08 0.000002 0.3165 0.0000006

106 100 125.2393 1.32E-08 0.000002 0.3165 0.0000005

107 Totals $7.3030 $7.0030

108 Discount 1 (F107/D107) 4.1%

109 Discount-By Formula [1] 4.1%

111 Parameters Sensitivity Analysis

112 r 20% Avg Yrs Between Sales

113 g 5% 8 10 12

114 z 12% 18% 7.2% 5.1% 3.8%

115 x (1 g)/ 87.50% 20% 5.9% 4.1% 2.9%

(1 r)

116 j yrs to sale 10 22% 4.9% 3.3% 2.3%

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

PART 3 Adjusting for Control and Marketability

268

Table 7-13 is identical to Table 7-12, except that it demonstrates the

accuracy of equation (7-9a), which is the formula appropriate for buyers™

transactions costs. Buyers care about their own transactions costs from

the outset. Therefore, the continuum of buyers™ transactions costs begins

immediately. Thus, E7 to E16 equal 0.88 in Table 7-13, while they were

equal to 1.00 in Table 7-12.

The discount in Table 7-13 is considerably larger”15.6%, which we

calculate in F108 using the ˜˜brute force™™ method and in F109 using equa-

tion (7-9a). The spreadsheet formula appears in note [1] as it also does in

Table 7-12. Table 7-13 thus demonstrates the accuracy of equation

(7-9a).

Value Remaining Formula and the Total Discount. The fraction in

(7-9) is the percentage of value that remains after removing the perpetuity

of transactions costs. Equation (7-10) shows the equation for the value

remaining, denoted as VR:

xj

1

VR valuing remaining formula (7-10)

z)x j

1 (1

We can multiply all three value remaining ¬gures for each of the

three components, and the result is the value remaining for the ¬rm over-

all. The ¬nal discount is then one minus the value remaining for the ¬rm

overall.

Next we will demonstrate the ¬nal calculation of DLOM.

Table 7-14: Sample Calculation of DLOM

Table 7-14 is an example of calculating DLOM for a privately held

¬rm with a $5 million FMV on a marketable minority basis. Column B is

the pure discount of each component as calculated according to the meth-

odology in the previous tables. Component #1, the discount due to the

delay to sale, is equal to 13.4% (B9), which comes from Table 7-10, cell

D12. Component #2, monopsony power to the buyer, equals 9% (B10),

per our discussion of Schwert™s article earlier in this chapter. Component

#3A, buyers™ transactions costs, equals 3.7% (Table 7-11, I73) for private

buyers, minus the approximately 1% brokerage fee to buy a $5 million

interest in publicly traded stocks 2.7% (B11). Component #3B, sellers™

transactions costs, equals 8.4% (Table 7-11, I74) for private buyers minus

the approximate 1% brokerage fee to buy publicly traded stocks 7.4%

(B12). The reason that we subtract stock market transactions costs from

the private market transactions costs is that we are using public market

values as our basis of comparison, i.e., our point of reference.

Column C is the present value of the perpetual discount, which

means that for Components #3A and #3B, we quantify the in¬nite peri-

odic transactions costs. Using equations (7-9a) for the buyers and (7-9)

for the sellers, the 2.7% (B11) pure discount for buyers results in a net

present value of buyers™ transactions costs of 3.6% (C11), and the 7.4%

(B12) pure discount for sellers results in a net present value of sellers™

transactions costs of 2.4% (C12). Again, that excludes the seller™s costs

on the assumed sale to the hypothetical buyer at t 0. The ¬rst two

CHAPTER 7 Adjusting for Levels of Control and Marketability 269

T A B L E 7-13

Proof of Equation (7-9a)

A B C D E F G

4 (1 z) Int(Yr 1) Post Tx

5 Cash PV Cash Post-Trans PV Cash

6 Year Flow PVF Flow Costs Flow

7 1 1.0000 0.912871 0.912871 0.8800 0.8033264

8 2 1.0500 0.760726 0.798762 0.8800 0.7029106

9 3 1.1025 0.633938 0.698917 0.8800 0.6150468

10 4 1.1576 0.528282 0.611552 0.8800 0.5381659

11 5 1.2155 0.440235 0.535108 0.8800 0.4708952

12 6 1.2763 0.366862 0.468220 0.8800 0.4120333

13 7 1.3401 0.305719 0.409692 0.8800 0.3605291

14 8 1.4071 0.254766 0.358481 0.8800 0.3154630

15 9 1.4775 0.212305 0.313671 0.8800 0.2760301

16 10 1.5513 0.176921 0.274462 0.8800 0.2415264

17 11 1.6289 0.147434 0.240154 0.7744 0.1859753

18 12 1.7103 0.122861 0.210135 0.7744 0.1627284

19 13 1.7959 0.102385 0.183868 0.7744 0.1423873

20 14 1.8856 0.08532 0.160884 0.7744 0.1245889

21 15 1.9799 0.0711 0.140774 0.7744 0.1090153

22 16 2.0789 0.05925 0.123177 0.7744 0.0953884

23 17 2.1829 0.049375 0.107780 0.7744 0.0834648

24 18 2.2920 0.041146 0.094308 0.7744 0.0730317

25 19 2.4066 0.034288 0.082519 0.7744 0.0639028

26 20 2.5270 0.028574 0.072204 0.7744 0.0559149

27 21 2.6533 0.023811 0.063179 0.6815 0.0430545

28 22 2.7860 0.019843 0.055281 0.6815 0.0376727

29 23 2.9253 0.016536 0.048371 0.6815 0.0329636

30 24 3.0715 0.0138 0.042325 0.6815 0.0288431

31 25 3.2251 0.011483 0.037034 0.6815 0.0252378

32 26 3.3864 0.009569 0.032405 0.6815 0.0220830

33 27 3.5557 0.007974 0.028354 0.6815 0.0193227

34 28 3.7335 0.006645 0.024810 0.6815 0.0169073

35 29 3.9201 0.005538 0.021709 0.6815 0.0147939

36 30 4.1161 0.004615 0.018995 0.6815 0.0129447

37 31 4.3219 0.003846 0.016621 0.5997 0.0099674

38 32 4.5380 0.003205 0.014543 0.5997 0.0087215

39 33 4.7649 0.002671 0.012725 0.5997 0.0076313

40 34 5.0032 0.002226 0.011135 0.5997 0.0066774

41 35 5.2533 0.001855 0.009743 0.5997 0.0058427

42 36 5.5160 0.001545 0.008525 0.5997 0.0051124

43 37 5.7918 0.001288 0.007459 0.5997 0.0044733

44 38 6.0814 0.001073 0.006527 0.5997 0.0039142

45 39 6.3855 0.000894 0.005711 0.5997 0.0034249

46 40 6.7048 0.000745 0.004997 0.5997 0.0029968

47 41 7.0400 0.000621 0.004373 0.5277 0.0023075

48 42 7.3920 0.000518 0.003826 0.5277 0.0020191

49 43 7.7616 0.000431 0.003348 0.5277 0.0017667

50 44 8.1497 0.000359 0.002929 0.5277 0.0015459

51 45 8.5572 0.0003 0.002563 0.5277 0.0013526

52 46 8.9850 0.00025 0.002243 0.5277 0.0011835

53 47 9.4343 0.000208 0.001962 0.5277 0.0010356

54 48 9.9060 0.000173 0.001717 0.5277 0.0009062

55 49 10.4013 0.000144 0.001502 0.5277 0.0007929

56 50 10.9213 0.00012 0.001315 0.5277 0.0006938

57 51 11.4674 0.0001 0.001150 0.4644 0.0005342

58 52 12.0408 8.36E-05 0.001007 0.4644 0.0004674

59 53 12.6428 6.97E-05 0.000881 0.4644 0.0004090

60 54 13.2749 5.81E-05 0.000771 0.4644 0.0003579

61 55 13.9387 4.84E-05 0.000674 0.4644 0.0003131

62 56 14.6356 4.03E-05 0.000590 0.4644 0.0002740

63 57 15.3674 3.36E-05 0.000516 0.4644 0.0002397

PART 3 Adjusting for Control and Marketability

270

T A B L E 7-13 (continued)

Proof of Equation (7-9a)

A B C D E F G

4 (1 z) Int(Yr 1) Post Tx

5 Cash PV Cash Post-Trans PV Cash

6 Year Flow PVF Flow Costs Flow

64 58 16.1358 2.8E-05 0.000452 0.4644 0.0002098

65 59 16.9426 2.33E-05 0.000395 0.4644 0.0001836

66 60 17.7897 1.94E-05 0.000346 0.4644 0.0001606

67 61 18.6792 1.62E-05 0.000303 0.4087 0.0001237

68 62 19.6131 1.35E-05 0.000265 0.4087 0.0001082

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

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

278

T A B L E 7-19

QMDM Comparison of Restricted Stock Discount Rate versus Mercer Example 1

A B C D E F G H I J K L

5 Mercer Example Restricted Stock

1

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

17.5%.

2. The 2.5-year average holding period.

3. The growth rate in value of 16.7%.

The calculation is as follows:

1

1.1672.5

DLOM 1 (FV PVF) 1 1.7%

1.1752.5

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

280

Conclusion

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.

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MATHEMATICAL APPENDIX

DEVELOPING THE DISCOUNT FORMULAS

Initially we assume the current business owner will operate the business

for 10 years, sell it, and pay transaction costs of z.77 The next owner will

run the business another 10 years, sell it, and pay transaction costs. We

assume this pattern occurs ad in¬nitum. Of course, there will be variations

from the sale every 10 years”some will sell after 1 year, others after 30

years. In the meantime, in the absence of prior knowledge, we assume

every 10 years to be a reasonable estimate of the average of what will

occur.

NPV of Cash Flows with Periodic Transaction

Costs Removed

The net present value (NPV) of cash ¬‚ows to the existing business owner

with periodic transaction costs removed is the full amount of the ¬rst 10

years™ cash ¬‚ows, plus (1 z) times the next 10 years™ cash ¬‚ows, where

z)2 times the next 10 years™

z is the periodic transaction cost, plus (1

cash ¬‚ows, etc. We will denote the NPV net of transaction costs, i.e., with

transaction costs removed from the stream of cash ¬‚ows, as NPVTC .

g)9

(1 g) (1

1

NPVTC

r)0.5 r)1.5 r)9.5

(1 (1 (1

g)10 g)19

(1 (1

(1 z)

r)10.5 r)19.5

(1 (1

g)20 g)29

(1 (1

2

(1 z) (A7-1)

r)20.5 r)29.5

(1 (1

Multiplying each term in equation (A7-1) by (1 g)/(1 r), we get:

77. As explained in the body of the chapter, z is an incremental transaction cost. For example,

when we value a small fractional ownership in a privately owned business, often our

preliminary value is on a marketable minority basis. In this case z would be the difference

in transaction cost (expressed as a percentage) between selling a private business interest

and selling publicly traded stock through a stockbroker.

PART 3 Adjusting for Control and Marketability

284

g)10

1 g 1 g (1

NPVTC

r)1.5 r)10.5

1 r (1 (1

g)11 g)20

(1 (1

(1 z)

r)11.5 r)20.5

(1 (1

g)21 g)20

(1 (1

2

(1 z)

r)21.5 r)30.5

(1 (1

(A7-2)

Subtracting equation (A7-2) from equation (A7-1), we get:

g)10

1 g (1

1

1 NPV

r)0.5 r)10.5

1 r (1 (1

g)10 g)20

(1 (1

(1 z)

r)10.5 r)20.5

(1 (1

g)20 g)30

(1 (1

2

(1 z) (A7-3)

r)20.5 r)30.5

(1 (1

Note that all terms in each sequence drop out except for the ¬rst

terms in equation (A7-1) and the last terms in equation (A7-2). In equation

(A7-4), we collect the positive terms from equation (A7-3) in the ¬rst set

of square brackets and the negative terms from equation (A7-3) in the

second one. Additionally, the left-hand side of equation (A7-3) reduces to

(r g)/(1 r)NPVTC . Multiplying through by (1 r)/(r g), we get:

1 r

NPVTC

r g

g)10 g)20

(1 (1

1 2

(1 z) (1 z)

r)0.5 r)10.5 r)20.5

(1 (1 (1

g)10 g)20 g)30

(1 (1 (1

2

(A7-4)

(1 z) (1 z)

r)10.5 r)20.5 r)30.5

(1 (1 (1

Next we will manipulate the right-hand side of the equation only.

We divide the term (1 r)/(r g) by 1 r, which leaves that term as

(1 r)/(r g) and we multiply all terms inside the brackets by

1 r. The latter action has the effect of reducing the exponents in the

denominators by 0.5 years. Thus, we get:

1 r

NPVTC

r g

10 20

1 g 1 g

2

1 (1 z) (1 z)

1 r 1 r

10 20 30

1 g 1 g 1 g

2

(A7-5)

(1 z) (1 z)

1 r 1 r 1 r

Recognizing that each term in brackets is an in¬nite geometric se-

quence, this solves to:

CHAPTER 7 Adjusting for Levels of Control and Marketability 285

1 r

NPVTC

r g

10

1 g

1 r

1

(A7-6)

g)10

(1 z)(1 g)10

(1 z)(1

1 1

r)10 r)10

(1 (1

Since the denominators are identical, we can combine both terms in