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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

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( 100 .)



>>