<<

. 11
( 18)



>>

the brokerage cost of selling publicly traded stock, in 10 years my buyer
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

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