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publicly held companies on average led to increased premiums of 12.2%
compared to takeovers without competitive bidding. Based on the re-
gression in Table 4 of his article, we assumed a typical deal con¬guration
that would apply to a privately held ¬rm.59 The premium without an
auction was 21.5%. Adding 12.2%, the premium with an auction was
33.7%. To calculate the discount for lack of competition, we go in the
other direction, i.e., 12.2% divided by one plus 33.7% 0.122/1.337
9.1%, or approximately 9%. This is a useful benchmark for D 2.
However, it is quite possible that D 2 for any subject interest should
be larger or smaller than 9%. It all depends on the facts and circumstances
of the situation. Using Schwert™s measure of the effect of multiple versus
single bidders as our estimate of D 3 may possibly have a downward bias
in that the markets for the underlying minority interests in the same ¬rms
is very deep. So it is only the market for control of publicly held ¬rms
that is thin. The market for privately held ¬rms is thin for whole ¬rms
and razor thin for minority interests.


59. We assume a successful purchase, a tender offer, and a cash deal.


CHAPTER 7 Adjusting for Levels of Control and Marketability 257
Component #3: Transactions Costs
Transactions costs in selling a privately held business are substantially
more than they are for selling stock in publicly traded ¬rms. Most stock
in publicly traded ¬rms can be sold with a broker™s fee of 1“2%”or less.

Table 7-11: Quantifying Transactions Costs for Buyer and
Seller. Table 7-11 shows estimates of transactions costs for both the
buyer and the seller for the following categories: legal, accounting, and
appraisal fees (the latter split into posttransaction, tax-based appraisal for
allocation of purchase price and/or valuation of in-process R&D and the
pretransaction ˜˜deal appraisal™™ to help buyer and/or seller establish the
right price), the opportunity cost of internal management spending its
time on the sale rather than on other company business, and investment
banking (or, for small sales, business broker) fees. The ¬rst ¬ve of the
categories appear in columns B through F, which we subtotal in column
G, and the investment banking fees appear in column H. The reason for
segregating between the investment banking fees and all the others is
that the others are constantly increasing as the deal size (FMV) decreases,
while investment banking fees reach a maximum of 10% and stop in-
creasing as the deal size decreases.
Rows 6“9 are transactions costs estimates for the buyer, while rows
13“16 are for the seller. Note that the buyer does not pay the investment
banking fees”only the seller pays. Rows 20“23 are total fees for both
sides.
Note that the subtotal transactions costs (column G) are inversely
related to the size of the transaction. For the buyer, they are as low as
0.23% (I6) for a $1 billion transaction and as high as 5.7% (I9) for a $1
million transaction. We summarize the total in Rows 27“30 and include
the base 10 logarithm of the sales price as a variable for regression.60 The
purpose of the regression is to allow the reader to calculate an estimated
transactions costs for any size transaction.
The buyer regression equation is:
Buyer Subtotal Transaction Cost
0.1531 (0.0173 log10 Price)
Price
The regression coef¬cients are in cells B48 and B49. The adjusted R2
is 83% (B37), which is a good result. The standard error of the y-estimate
is 0.9% (B38), so the 95% con¬dence interval around the estimate is ap-
proximately two standard errors, or 1.8%”a very good result.
The seller regression equation is:
Seller Subtotal Transaction Cost
0.1414 (0.01599 log10 Price)
Price
The regression coef¬cients are in cells B67 and B68. The adjusted R2 is
82% (B56), which is a good result. The standard error of the y-estimate is


60. Normally we use the natural logarithm for regression. Here we chose base 10 because the logs
are whole numbers and are easy to understand. Ultimately, it makes no difference which
one we use in the regression. The results are identical either way.




PART 3 Adjusting for Control and Marketability
258
T A B L E 7-11

Estimates of Transaction Costs [1]


A B C D E F G H I

4 Buyer
Tax Deal
5 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

6 $1 billion 0.10% 0.02% 0.02% 0.00% 0.09% 0.23% 0.00% 0.23%
7 $100 million 1.00% 0.10% 0.06% 0.00% 0.16% 1.32% 0.00% 1.32%
8 $10 million 1.50% 0.23% 0.20% 0.00% 0.25% 2.18% 0.00% 2.18%
9 $1 million 4.00% 0.30% 0.70% 0.00% 0.70% 5.70% 0.00% 5.70%

11 Seller
Tax Deal
12 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

13 $1 billion 0.10% 0.01% 0.00% 0.02% 0.05% 0.18% 0.75% 0.93%
14 $100 million 1.00% 0.05% 0.00% 0.05% 0.10% 1.20% 1.10% 2.30%
15 $10 million 1.50% 0.08% 0.00% 0.20% 0.15% 1.93% 2.75% 4.68%
16 $1 million 4.00% 0.10% 0.00% 0.75% 0.42% 5.27% 10.00% 15.27%

18 Total
Tax Deal
19 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

20 $1 billion 0.20% 0.03% 0.02% 0.02% 0.14% 0.41% 0.75% 1.16%
21 $100 million 2.00% 0.15% 0.06% 0.05% 0.26% 2.52% 1.10% 3.62%
22 $10 million 3.00% 0.30% 0.20% 0.20% 0.40% 4.10% 2.75% 6.85%
23 $1 million 8.00% 0.40% 0.70% 0.75% 1.12% 10.97% 10.00% 20.97%

25 Summary For Regression Analysis-Buyer Summary For Regression Analysis-Seller

26 Sales Price Log10 Price Subtotal Sales Price Log10 Price Subtotal

27 $1,000,000,000 9.0 0.23% $1,000,000,000 9.0 0.18%
28 $100,000,000 8.0 1.32% $100,000,000 8.0 1.20%
29 $10,000,000 7.0 2.18% $10,000,000 7.0 1.93%
30 $1,000,000 6.0 5.70% $1,000,000 6.0 5.27%
259
T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]


A B C D E F G H

32 SUMMARY OUTPUT: Buyer Subtotal Fees as a Function of Log10 FMV
34 Regression Statistics

35 Multiple R 0.9417624
36 R square 0.88691642
37 Adjusted R square 0.83037464
38 Standard error 0.00975177
39 Observations 4

41 ANOVA

42 df SS MS F Signi¬cance F
43 Regression 1 0.001491696 0.0014917 15.68603437 0.058237596
44 Residual 2 0.000190194 9.5097E 05
45 Total 3 0.00168189

47 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

48 Intercept 0.1531 0.033069874 4.62959125 0.043626277 0.010811717 0.295388283
49 Log10 price 0.0172725 0.004361126 3.96055986 0.058237596 0.036036923 0.001491923

51 SUMMARY OUTPUT: Seller Subtotal Fees as a Function of Log10 FMV
53 Regression Statistics

54 Multiple R 0.93697224
55 R square 0.87791699
56 Adjusted R square 0.81687548
57 Standard error 0.00943065
58 Observations 4

60 ANOVA

61 df SS MS F Signi¬cance F

62 Regression 1 0.00127912 0.00127912 14.38229564 0.063027755
63 Residual 2 0.000177874 8.8937E 05
64 Total 3 0.001456994

66 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

67 Intercept 0.14139 0.031980886 4.42107833 0.04754262 0.00378726 0.27899274
68 Log10 price 0.0159945 0.004217514 3.79239972 0.063027755 0.034141012 0.002152012




also 0.9% (B57), which gives us the same con¬dence intervals around the
y-estimate of 1.8%.
Rows 73 and 74 show a sample calculation of transactions costs for
the buyer and seller, respectively. We estimate FMV before discounts for
our subject company of $5 million (B73, B74). The base 10 logarithm of 5
million is 6.69897 (C73, C74).61 In D73 and D74, we insert the x-coef¬cient
from the regression, which is 0.0172725 (from B49) for the buyer and
0.0159945 (from B68) for the seller. We multiply column C column


61. In other words, 106.69897 5 million.




PART 3 Adjusting for Control and Marketability
260
T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]


A B C D E F G H I J

70 Sample Forecast of Transactions Costs For $5 Million Subject Company:

72 FMV log10 FMV X-Coeff. log FMV Coef Regr. Constant Forecast Subtotal Inv Bank [5] Forecast Total

73 Buyer $5,000,000 6.698970004 0.0172725 0.115707959 0.1531 3.7% 0.0% 3.7%
74 Seller $5,000,000 6.698970004 0.0159945 0.107146676 0.14139 3.4% 5.0% 8.4%

Notes:
[1] Based on interviews with investment banker Gordon Gregory, attorney David Boatwright, Esq; and Douglas Obenshain, CPA. Costs include buy and sell side. These are estimates of average costs. Actual costs vary with the complexity
of the transaction.
[2] Legal fees will vary with the complexity of the transaction. An extremely complex $1 billion sale could have legal fees of as much as $5 million each for the buyer and the seller, though this is rare. Complexity increases with: stock deals
(or asset deals with a very large number of assets), seller carries paper , contingent payments, escrow, tax-free (which is treated as a pooling-of-interests), etc.
[3] We are assuming the seller pays for the deal appraisal. Individual sales may vary. Sometimes both sides hire a single appraiser and split the fees, and sometimes each side has its own appraiser.
[4] Internal management costs are the most speculative of all. We estimate 6,000 hours (3 people fulltime for 1 year) at an average $150/hr. internal cost for the $1 billion sale, 2,000 hours @ $80 for the $100 million sale, 500 hours at $50
for the $10 million sale, and 200 hours @$35 for the $1 million sale for the buyer, and 60% of that for the seller. Actual results may vary considerably from these estimates.
[5] Ideally calculated by another regression, but this is sight-estimated. Can often use the Lehman Bros. Formula”5% for 1st $1 million, 4%, for 2nd, etc., leveling off at 1% for each $1 million.
261
D column E. F73 and F74 are repetitions of the regression constants
from B48 and B67, respectively. We then add column E to column F to
obtain the forecast subtotal transactions costs in G73 and G74. Finally, we
add in investment banking fees of 5%62 for the seller (the buyer doesn™t
pay for the investment banker or business broker) to arrive at totals of
3.7% (I73) and 8.4% (I74) for the buyer and seller, respectively.

Component #3 Is Different than #1 and #2. Component #3, trans-
actions costs, is different than the ¬rst two components of DLOM. For
component #3, we need to calculate explicitly the present value of the
occurrence of transactions costs every time the company sells. The reason
is that, unlike the ¬rst two components, transactions costs are actually
out-of-pocket costs that leave the system.63 They are paid to attorneys,
accountants, appraisers, and investment bankers or business brokers. Ad-
ditionally, internal management of both the buyer and the seller spend
signi¬cant time on the sale to make it happen, and they often have to
spend time on failed acquisitions before being successful.
We also need to distinguish between the buyer™s transactions costs
and the seller™s costs. The reason for this is that the buyer™s transactions
costs are always relevant, whereas the seller™s transactions costs for the
immediate transaction reduce the net proceeds to the seller but do not
reduce FMV. However, before the buyer is willing to buy, he or she should
be saying, ˜˜It™s true, I don™t care about the seller™s costs. That™s his or her
problem. However, 10 years or so down the road when it™s my turn to
be the seller, I do care about that. To the extent that seller™s costs exceed
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

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