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