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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

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